INTEGRO-DIFFERENTIAL OPERATORS: CONNECTIONS TO

DEGENERATE ELLIPTIC EQUATIONS AND SOME FREE

BOUNDARY PROBLEMS

By

Reshma Menon

A DISSERTATION

Submitted to

Michigan State University

in partial fulfillment of the requirements

for the degree of

Mathematics — Doctor of Philosophy

2020

ABSTRACT

INTEGRO-DIFFERENTIAL OPERATORS: CONNECTIONS TO DEGENERATE

ELLIPTIC EQUATIONS AND SOME FREE BOUNDARY PROBLEMS

By

Reshma Menon

In this dissertation, we study aspects of integro-differential operators, and how they

relate to different types of equations. In each case, we use information and results about

the operators in a lower dimension to analyse an equation in a higher dimension, and vice-

versa. We begin in chapter 1 with an introduction to the operators and equations we will

be considering.

In Chapters 2 and 3, we discuss certain integro-differential operators of functions in a
relatively smooth space like C1,α(Rn). However, to understand more about the structure

of these operators, particularly about the measure associated with them, we study certain

equations in a higher dimension such as degenerate elliptic equations in the upper half space.

We analyse the solution of such an equation and its gradient, followed by estimates on its

Green’s function and Poisson kernel. These estimates then help reveal some properties of

the measure associated with the integro-differential operator in the lower dimension. The

structure of the degenerate elliptic equations is similar to that of uniformly elliptic equations,

but with an additional complexity of a term which involves distance to the boundary. This

degeneracy complicates the analysis; as such, the classical techniques of finding pointwise es-

timates as mentioned above do not work so well anymore. So we provide some revised results

for the same. Thus understanding an equation in a higher dimension gives us information

about an integro-differential operator in a lower dimension.

In Chapters 4 and 5, we prove some results about the solutions of free boundary problems

in Rn+1 × [0, T ], where the free boundary for a fixed time t can be seen as the graph of a

function over a sphere. This time, we connect the solution of the free boundary problem to the

solution of a parabolic equation on the sphere – that is, in a lower dimension. This parabolic

equation involves an integro-differential operator, which has a min-max representation that
is consistent with all the results about viscosity solutions of parabolic equations in Rn. We

modify these results for parabolic equations on the sphere, which then gives us existence and

uniqueness results about the free boundary problem in a higher dimension.

For Amma, the most resilient person I know.

iv

ACKNOWLEDGMENTS

First and foremost, my sincerest gratitude to my dissertation advisor, Russell Schwab

– for guiding me through this journey, answering my stupid questions with more patience

than I deserve, knowing when to overlook or reprimand my procrastination and balancing

these reactions perfectly, encouraging my non-research career choice in teaching for my own

happiness, and above all, always standing up for me as a true mentor and friend. Thank you

to my dissertation committee – Jun Kitagawa for being instrumental to my understanding

of my comprehensive exam syllabus with his teaching and class notes and for being the most

chill professor to get a drink with; Keith Promislow for never failing to brighten my day with

a big smile and a genuine compliment whenever we crossed paths; and Jeffrey Schenker for

always giving my grievances and suggestions a fair hearing as a co-member on committees

we have served on together, and the graduate director of the mathematics department.

Graduate school to me was initially a stepping stone to what I really wanted – a teaching

career where I could have more freedom to plan and teach my own classes, design courses

and policies. I could not have imagined having a multitude of these opportunities in the

past 6 years itself, leading me to secure a job that I still cannot believe I landed. The people

I owe this to are my ultimate teaching mentors, Tsvetanka Sendova and Andrew Krause.

Tsveta, thank you for leaving your office door open for me to waltz into anytime to unload

my good and bad thoughts on you, for celebrating all my successes and offering a quick

pick-me-up from my setbacks, and for giving me the best no-nonsense advice which always

worked, regarding teaching, graduate school life, or just human interaction. Andy, than

k you for the million conversations about classroom teaching and assessment, for all the

last-minute meetings about how-tos for a class, and being editor-in-chief for all my pieces of

v

writing about my teaching. I owe you both so much, for encouraging and believing in me and

getting me to where I am today. To those who have unknowingly been my teaching mentors

over the years, I have learnt so much from all of you – Sharon Griffin, Ryan Maccombs,

Jeanne Wald, Jane Zimmerman, and Sue Allen. Sue, thank you for all the wine and cheese

dinners and the laughs we shared, and for making it so effortless to be your lead TA. I will

always cherish our unique friendship.

There are many teachers who have inspired my journey as a mathematician and a teacher.

Some were mentors, some my own teachers, and some were colleagues.

I will be forever

grateful towards Casim Abbas for giving me this opportunity to start graduate school at

Michigan State University, despite my mixed-levels-of-impressive portfolio. My first year of

graduate school at MSU saw my continued love for mathematics thanks to Teena Gerhardt

and Ignacio Uriarte-Tuero, who are both responsible in their own ways with their enthusiasm

and kindness for making me fall in love with the fields of Algebraic Topology and Analysis

respectively. Before I came to even dream of a PhD in mathematics as an option for me, I

am thankful to both Sanjay Shinde Sir and Felix Almeida Sir from St. Xavier’s College for

inspiring me to fall in love with mathematics and believing that I could get a degree in the

subject even though I was registered in the Arts stream. Thanks to Ms. Chethna Amin from

Udayachal High School who taught me everything I know about public speaking and life as

a young adult, for being the first cheerleader to all my dreams, and for the beautiful bond

we continue to share across generations and oceans. A million thanks to all my colleagues at

Don Bosco Institute of Technology for giving me some of the best years of my professional life

and fortifying my belief that teaching is my life’s calling – especially Sheetal Dighe, Minirani

Nair, Anice Matthew, Rohini Chandramouli, Revathy Sunderraman, and Sameer Hadkar.

In the last year of my PhD program, I decided to take a peek into the world of Mathe-

vi

matics Education. I was fortunate to work with people who welcomed me into their research

group like old friends, and were open and respectful towards my untrained opinions. Thank

you Jack, Mari, Shiv, Bob, Jihye, Sarah, Sofia, Valentin, and Yao – I have enjoyed being a

part of your work in the past year, and it has made me so proud to be able to contribute

in whatever way I can. Special thanks to Jack Smith for nudging me about my dissertation

every chance he got and hence keeping me on my toes. For opening the door to these op-

portunities for me, I am ever grateful to Shiv Smith Karunakaran, who became my de facto

mentor in this field. Thank you, Shiv, for taking a chance by including me in your research

group, for always being generous with your time and advice, especially with navigating the

job market. Thanks to Monica Smith Karunakaran for all the helpful job market advice,

especially navigating the two-body problem as a female mathematician, and for teaching me

everything I’ve learnt about how to teach a brilliantly coordinated course.

I want to make a special mention of the staff in the mathematics department, especially

those who serve as a liaison between graduate students and faculty. Leslie, for patiently

responding to all the phone calls and emails from a very nervous international student; Ami,

for our long walks and gossip sessions; Estrella, for her regular kind check-ins and always

adding the extra touch of care; Carolyn, for being patient with my endless questions about

reimbursements; and Debie, for her warm smile and strong presence in C212.

This has been a long and arduous journey, which would have been impossible without

the presence of many, many friends in my life who inspire and cheer for me alike. I have

been blessed with some beautiful life-long friendships, especially the ones I had back home

in India and have traveled halfway across the world with me. Elisha, we are truly lucky to

have come to Michigan together, and if it wasn’t for you, I would have never survived the

homesickness of those initial years. Maya, Suraj, and Ankita, with whom there is always

vii

more laughter than I can handle. Ankita, for finding the silver lining and reminding me

about things to be thankful for, and Maya, for being the calm voice of reason, I am so happy

that my two oldest friends are in a time zone closer to me.

I am even more blessed to have friends scattered across the globe, long distance friend-

ships which have stood the test of time and distance. Sreya, for being the 4 AM friend

irrespective of time zones; Pari, for long weekend conversations about book recommenda-

tions and reminiscing about the 90s; Garima and Cheryl, for keeping all the Sundays free for

long video calls across 3 continents; Shruti, for honestly calling me out when I am slacking

yet telling me what I should be proud of; and Nikita, for salvaging my waning belief in

romantic comedies. Thanks to Ajay and Kasturi Aunty for all the help when I was planning

my first move to the US, and to Abhishek, for being a fierce friend and pushing me to apply

for a PhD in the first place.

Uprooting your very well-loved life with an abundance of friends, family, and colleagues

you can count on and moving to the other side of the planet is an incredibly difficult choice,

made harder once you realise going back home will cost you 3 paychecks and is not an

option you have presently available. Thankfully, my army of friends in East Lansing made

this easier over time. First, my PhD soulmate, Hitesh – for pushing me through qualifying

exams and panic attacks, fighting with me like only family can, inspiring me to make better

rajma and broadening my definition of Broadway, thank you for walking every step with

me as a true partner in this program. My first roommate and co-parent to my cats, Sarvie,

thank you and your family for giving me a home away from home. Thank you to Rani,

for all the long conversations and being my most insightful friend no matter what I need

advice on; Charlotte, for your supportive outrage when unfair things happened and inspiring

my fitness goals; Sarah Klanderman, for being my quals-study partner and introducing me

viii

to so many of my American firsts; Christos, for always keeping it real; Eylem, for always

fighting for what’s right; Allison, for the best surprises and presents; and Sarah Vedolich,

for being such a wonderful caretaker to my cats. Thank you to my friends Salil, Charuta,

Amit, and Manasi for all the wonderful evenings filled with nostalgia for home. Thank you

to Ben and Lauren for bringing music and so much positivity into my life; Guillermo for

further inspiring my love for real analysis; Sebastian and Michael for welcoming me into

their office space and regaling me with wonderful tales about their little ones; and Rajinder

for introducing me to the Pomodoro method for productivity. Thanks to Danika for always

reminding me of my awesomeness; Chloe for being the most wonderfully kind neighbour and

friend; and Will, Teja, and Grace for our weekly dinners and giving me the community that

makes East Lansing home. I am also grateful for the company and advice from good friends

like Livy, Stevie, Rami, Sami, Mollee, Darshan, Farhan, Emily, Leo, and Rodrigo.

Tyler, thank you for being my go-to person with my analysis questions, for being the

kindest and most righteous person I know – inspiring me to be better at my job and in life,

for reminding and showing me every single day that we’re on the same team. I am grateful to

have you as my life-partner and best friend, and for your ever-loving and supportive parents

and grandparents who are now my family. To my sister, Rasya, thank you for believing I

am a rockstar and keeping me humble by not saying it out loud. To my parents, Suresh

and Jayasree Menon, thank you for raising us with the belief that education always comes

first, for pushing my reluctant self to complete my Master’s degree, for unflinchingly letting

me go halfway across the world to follow my dreams, and for never failing to encourage our

unconventional career choices as women. Finally, I am also grateful for my two cats, Don

Quixote and Leopold, for all their warmth and cuddles during the long hours of isolated

dissertation writing.

ix

TABLE OF CONTENTS

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii

KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

Chapter 1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 Degenerate elliptic equations and integro-differential operators . . . . . . . .
1.2 Parabolic equations and some free boundary problems . . . . . . . . . . . . .

2.1 Uniformly elliptic equations

Chapter 2 Background for degenerate elliptic equations . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1.1 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
2.1.2 Dirichlet-to-Neumann maps
2.2 Degenerate Elliptic Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Dirichlet-to-Neumann maps

Chapter 3 Connection to Dirichlet-to-Neumann (D-to-N) maps . . . . . .
3.1
Introduction to our equation and coefficients . . . . . . . . . . . . . . . . . .
3.2 Estimates for the Green’s function . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1
Interior estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 Boundary estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.3 Relationship with harmonic measure . . . . . . . . . . . . . . . . . .
3.3 D-to-N maps of the weighted equation (main result and proof) . . . . . . . .

Chapter 4 Background on free boundary problems . . . . . . . . . . . . . .
4.1
Introduction and background . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Definitions and examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 General Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . .
4.2.2
4.3 Results to be modified . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Some examples of free boundary problems

1
1
5

9
10
12
18
20
22
25

27
27
30
30
31
44
45

51
51
56
56
61
66

5.1

Chapter 5 Parabolic equations and free boundary problems . . . . . . . .
Introduction to the problem and results . . . . . . . . . . . . . . . . . . . . .
5.1.1 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.2 Details of two phase free boundary problem on the sphere
. . . . . .
5.2
. . . . . . . . . . . .
Integro-differential representations of certain operators
5.3 Comparison theorem and existence for parabolic viscosity solutions on the
sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
5.3.1 Using the assumptions of section 5.2 . . . . . . . . . . . . . . . . . .
88
5.3.2 Comparison results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.3.3 Existence (Perron’s method) . . . . . . . . . . . . . . . . . . . . . . . 108

73
73
74
77
79

x

5.4 Revisiting the main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
. . . . . . . . . . . . . . . . . 114
. . . . . . . . . . . . . . . . . . . . 117

5.4.1 Different notions of viscosity solutions
5.4.2 Vertical shifts in the intersurface
5.4.3 Correspondence between viscosity solutions of the free boundary evo-

lution and viscosity solutions of the parabolic equation . . . . . . . . 118
5.4.4 Propogation of the modulus of continuity . . . . . . . . . . . . . . . . 120

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

xi

LIST OF FIGURES

Figure 3.1: Boundary conditions for lemma 3.2.2 . . . . . . . . . . . . . . . . . . . .

33

Figure 3.2: Different locations for Bd(X0) . . . . . . . . . . . . . . . . . . . . . . . .

34

Figure 4.1: One phase Hele-Shaw on the half-space . . . . . . . . . . . . . . . . . . .

62

Figure 4.2: Two phase Hele-Shaw on an infinite strip . . . . . . . . . . . . . . . . . .

63

Figure 5.1: Two phase free boundary problem on the sphere . . . . . . . . . . . . . .

78

xii

KEY TO SYMBOLS

• Rn+1

+ , the half space in dimension n + 1, i.e. Rn+1
0}. We sometimes use, N = n + 1 for brevity.

+ = {X = (x, y) : x ∈ Rn, y ∈ R, y >

+ of side-length R.

R = QR ∩ R+
N .

+ = Rn, the boundary of the half space.

, the partial derivative with respect to xi. For a function u, we will also

• ∂RN
• ∂i = ∂
∂xi
sometimes denote this as ui or uxi. Further, ∇u = (ux1, ux2,··· , uxn).
• QR, a cube centred at 0 in RN
• Q+
• Sn, the n-dimensional sphere.
• ∂ν, the normal derivative. The notation is also used later for the co-normal derivative.
• A, a uniformly elliptic matrix, and ˜A, the matrix with weights. In this work, ˜A = yaA,

where a ∈ (0, 1) or a ∈ (−1, 1), as specified.

• p.v., Principal Value integral, i.e. if x0 is a point of singularity for an integral(cid:82)

Ω f dx,

then

p.v.

• σ, the Lebesgue measure.
• ||f||p, the Lp-norm, i.e.

Ω

|f (x)|pdx

||f||p :=
• Lp(Ω) = {f|f : Ω → R, ||f||p < ∞}.
• Lp(Ω, w), the weighted Lp-space.
• Lip(Ω) = {f : Ω → R, f Lipschitz}.
• H1,p(Ω), the Sobolev space of Lp functions whose weak first derivatives are also in Lp.

• D(u, v), the Dirichlet form, i.e. (cid:82)

Ω aij(x)u(x)v(x)dx.

• cap(E), the capacity of a set E.
• [f ]Cγ , the γ-th H¨older seminorm of f , i.e. [f ]Cγ = sup
x,y∈Ω
x(cid:54)=y

|f (x) − f (y)|

|x − y|γ

.

(cid:90)

(cid:90)

f dx = lim
ε→0

Ω

Ω\Bε

f dx

(cid:19)1/p

(cid:18)(cid:90)

xiii

• C1,γ(Ω) = {f : Ω → R : ||f||L∞ + ||∇f||L∞ + [∇f ]Cγ < ∞}.

(equivalently)

C1,γ(Ω) =

f ∈ L∞(Ω) : sup

z∈Ω

r−1−γ

sup
r>0

inf

P (x)=c+p·x
c∈R, p∈Rn

||f − P||L∞(Br(z)) < ∞

 .

• For γ ∈ (0, 1), δ > 0, m > 0, the convex set K(γ, δ, m) is defined as
K(γ, δ, m) := {f ∈ C1,γ(Ω) : f (x) ≥ δ ∀x ∈ Ω and ||f||C1,γ (Ω) ≤ m}.
• K(γ, δ) =

K(γ, δ, m).

(cid:91)
(cid:91)

m>0

(cid:91)

γ∈(0,1)

m>0

• K(δ) =

K(γ, δ, m).

• For f : Rn → R, Df = {(x, y) ∈ Rn+1, 0 < y < f (x)}
• Γf = {(x, y) ∈ Rn+1, y = f (x)}
• Similarly, when δ ≤ f ≤ L,

f = {(x, y) ∈ Rn+1, 0 < y < f (x)} and D−
D+

f = {(x, y) ∈ Rn+1, f (x) < y < L}

• ν is the inward facing normal to the boundary of the set D+
•

f , and

∂+
ν u = lim
t→0+
∂−
ν u = − lim
t→0+

u(X0 + tν(X0)) − u(X0)

,
u(X0 − tν(X0)) − u(X0)

t

.

t

xiv

Chapter 1

Introduction

1.1. Degenerate elliptic equations and integro-differential

operators

The first area of my work is in degenerate elliptic equations on a half-space and their Dirichlet

to Neumann (D-to-N) maps expressed as integro-differential operators. The D-to-N map

appears in wide range of contexts in analysis, probability, Calder´on or inverse problems,

and mathematical physics. For example, the D-to-N map arises naturally in the study of

operators that describe boundary processes of diffusions in a bounded domain. The map is

also central to studying free boundary problems, acting as the natural quantity that drives

the free boundary. It is a fundamental object in problems relating voltage to current, where

the D-to-N map takes the voltage on the boundary and gives the resulting current density on

the boundary. When D-to-N maps are expressed as integro-differential operators, I study the

properties of the measure associated with this representation; in particular its relationship

with the Lebesgue measure and the density in question. To analyse these properties I

proved some estimates of the Green’s function associated with these equations, as well as

the connection between the Green’s function and the harmonic measure for the equation.

There is a connection between certain partial differential equations on some “nice” domain

1

(like the half space) of the form

Lu = −∂j(aij∂iu) = 0 in Ω

u = f

on ∂Ω

(1.1.1)

and operators of the kind

(cid:90)

∂Ω

I(f, x) =

f (h) − f (x)µ(x, dh)

(1.1.2)

for aij and µ that satisfy certain conditions. For example, one possible connection between
the two is that the operator I in (1.1.2) will often arise as the D-to N map for (1.1.1).

These operators have generated considerable interest lately and it is important to understand

when the two situations overlap. We can ask the following two questions – for what µ does
the representation in (1.1.2) hold? And what can we say about the order of I, i.e.
is
I(u(r·), x) = rαI(u, rx)? for some α? Classically, it is known that when L in (1.1.1) is
the Laplacian, then the corresponding D-to-N map I will have the structure in (1.1.2),

and it is the 1/2-Laplacian of the boundary data f . Subsequently, it is equally natural
to consider the α/2−Laplacian for some α instead of the 1/2−Laplacian. As for µ, it is
typically reflected in a weight such as µ(x, dh) ≈ |h|−(n+α)dh in this case, as opposed to
µ(x, dh) ≈ |h|−(n+1)dh in case of the 1/2-Laplacian. However, to make the connection
between the D-to-N map I and the equation in (1.1.2), one must study a weighted equation,
which consists of L = div(ya∇u). This has been studied by many authors, for example,

Caffarelli and Silvestre [4] and in the context of probability theory, by Song and Vondraˇcek

in [36], and also in the book, ‘Bernstein functions. Theory and applications’ [31] by Schilling,

Song, and Vondraˇcek.

2

The main goal of the following chapters (2 and 3) is to generalize these results for opera-

tors that are not translation invariant, and thus we present work on the weighted extensions
Lu = div(yaA∇u), where a ∈ (−1, 1), and the matrix A(X) = (aij(X)) satisfies the follow-

ing conditions:

• Uniform ellipticity i.e. ∃λ, Λ > 0 for all X, ξ ∈ Rn+1

+ , λ|ξ|2 ≤ aij(X)ξiξj ≤ Λ|ξ|2.

• A modified version of the Dini-continuity given in [22, Section 3]; we have for X = (x, y)

and Z = (z, s)

|yaA(X) − taA(Z)| ≤ ω(|X − Z|),

where ω is a type of Dini-modulus of continuity, particular to our degenerate equation.

ω satisfies the properties in [22, Section 3], and additionally,

(cid:90)

y−a ω(|X − Z|)
|X − Z|N

dX ≤ C.

Ω

Thus, the equation we will study in this work is:

Lu = −∂j(yaaij∂iu) = 0 in Rn+1
on ∂Rn+1

u = f

+

+ = Rn

(1.1.3)

The uniformly elliptic equation (a = 0) has been studied by Guillen, Kitagawa, and

Schwab [23]. In this case solutions behave linearly at the boundary, yet the analysis is quite

delicate. However, in the weighted setting, we lose uniform ellipticity and the equation

degenerates as we approach the boundary. As such, the techniques of [23] no longer apply,
the normal derivative can blow up, solutions behave like y1−a instead of being linear, and

the analysis becomes more complicated as the proofs to many foundational results need to

3

be adapted to these new challenges. So far we have been able to overcome these technical

difficulties to the following result.

Theorem 1.1.1. In equation (1.1.2) above, µ satisfies the following conditions

(a) µ is absolutely continuous with respect to the surface measure, σ, i.e.

for all X ∈

∂Ω, µ(X,·) has a density, µ(X, dh) = K(X, h)σ(dh).

(b) There exist universal constants C1, C2 > 0 so that ∀X, h ∈ ∂Rn+1

+ , X (cid:54)= h

K(X, h) ≤ C2|X − h|−n+a−1.

This extends the uniformly elliptic case treated in [23, Theorem 1.1] to the degenerate
elliptic case, for which I have established part (a) and the bound in part (b) for a ≥ 0.

The proof of the result in [23] relies on two key steps, namely the relationship between the

harmonic measure and the Green’s function found in [3], as well as boundary estimates for

the Green’s function found in [22]. When a < 0, we see that writing analogous proofs to [22,

Lemmas 3.1,3.2] seem to fail as the estimates blow up the closer you get to the boundary.

To prove the first condition on µ, we prove two important lemmas. The first of these

gives a relationship between the harmonic measure and the Green’s function – not unlike the

well-known result for the uniformly elliptic equation (a = 0) which comes from [3, Lemma
2.2] – but with a change reflecting the term ya. Namely, for x ∈ Rn, y ∈ Rn+1

+ \ Bsr for

some s > 1, we have constants c1, c2 such that

C1r(n+1)+a−2G(y, x + rν(x)) ≤ ωy(∂Ω ∩ Br(x)) ≤ C2r(n+1)+a−2G(y, x + rν(x))

(1.1.4)

where ν(x) represents the inward normal vector at x ∈ ∂Rn+1

+ . The above result for certain

4

classes of degenerate elliptic equations has been established in [20].

The second result is to obtain lower and upper estimates for the Green’s function associ-

ated with the operator L, especially as we get close to the boundary. This will in turn give

us the bounds in condition (b) of theorem 1. Thus, there are constants C1, C2 that depend

only on λ, Λ, n, ω such that

C1

δp(X)δp(Y )
|X − Y |(n+1)−a

≤ G(X, Y ) ≤ C2

δp(X)δp(Y )
|X − Y |(n+1)−a

(1.1.5)

for a good choice of p, where δ(X) = dist(X, ∂Rn+1
+ ). For this, I have obtained the upper
bounds when a ≥ 0 with p = 1 − a by first proving that for the solutions of the degenerate
elliptic equation in a cube with specific boundary conditions, we have δa(X)|∇u(X)| ≤ C,

where C = C(λ, Λ, n, ω). The proof here follows in a similar fashion to the one given by

Gr¨uter and Widman in [22, Lemma 3.2] for uniformly elliptic equations. However, the proof
only works when a ≥ 0, as when a < 0 the estimate we get on δa(X)|∇u(X)| seems to blow

up as you get close to the boundary.

1.2. Parabolic equations and some free boundary problems

We analyse a function u : Rn+1 × [0, T ] → R, which is harmonic on the sets {u > 0},{u <
0} ⊂ Rn+1 × [0, T ]. These sets have boundary ∂{u > 0} which, for a fixed time t, is a
hypersurface in Rn+1. There is an additional constraint on u which arises naturally from

either physical principles or energy minimization, and it concerns the balance of the normal

derivatives ∂+

n u+ and ∂−

n u− in the inward direction along the free boundary, which is given

by the boundary velocity. More formally, for our two-phase problem, we seek a function u

5

that solves

under the velocity condition

∆u = 0 in {u > 0} ∪ {u < 0}

(1.2.1)

∂{u > 0} moves with normal velocity V = |∇u+| − |∇u−|.

(1.2.2)

This model includes a special case of the stationary two phase problem under the condition

that u does not depend upon time and the balance condition becomes

|∇u+| − |∇u−| = 1.

(1.2.3)

It also includes the one phase version that usually carries the name Hele-Shaw, and that
corresponds to ignoring ∂{u < 0} and setting the velocity condition to be V = |∇u+|.

I work with these canonical examples, and the goal is to find a solution u that satisfies

(1.2.1) above with either of the conditions in (1.2.2) or (1.2.3). Viscosity solutions give a way

of handling this when other techniques fail, and there is substantial literature on the variants

of these equations in [1, 5, 27, 28]. Recently, the work of Chang-Lara, Guillen, and Schwab

[9] gives a new technique which reduces the variants in the family of problems with conditions

(1.2.2) and (1.2.3) to an equivalent problem of the integro-differential type by considering
the hypersurface ∂{u(·, t) > 0} as the graph of a function f : Rn+1 × [0, T ]. Then, for a

sufficiently smooth f , the equation (1.2.1) with the condition in (1.2.2) is equivalent to

∂tf = H(f, x)

on Rn × [0, T ]

f (·, 0) = f0

on Rn

(1.2.4)

6

where H : C1,γ → Cγ, and H(f ) = (|∇U +

f |)(cid:112)1 + |∇f|2. The main technique

f | − |∇U−

used to show the equivalence of solving (1.2.1) under condition (1.2.2) with solving (1.2.4)

was given in [9]; the key idea is to use that the operator H has a special structure of the

min-max form

H(f, x) = min

i

{aij + cijf (x) + bij ·∇f (x) + p.v.

max

j

(cid:90)

Rn

f (x + h)− f (x)µij(x, dh)} (1.2.5)

for an appropriate family of aij, bij, cij and µij. Equations that admit a similar min-max form

are frequently amenable to a large collection of tools from the viscosity solutions context.
The assumption that ∂{u > 0} = graphf is not ideal; we see that it does not appear

as a requirement in [1, 5, 11, 12, 27, 28]. However, it gives a natural reduction to explore
new techniques, especially with an assumption that {u > 0} is a star-shaped domain with

respect to X = 0. Thus it makes sense to expand the ideas in [9] to the case of the functions
u, f defined on the sphere Sn instead of Rn+1.

My work in this area is about expanding this new technique; in particular, when the
functions u, f are defined over Sn × [0, T ]. So if time t is fixed, then f (·, t) : Sn → [δ,∞).

As such, any previous assumption about translation invariance of the domain or operators

now becomes a matter of rotational invariance.

The advantage of using the approach of (1.2.4) is that it now allows us to generalise these

techniques to many more variations in the free boundary problems described above. One

important variant of the Hele-Shaw type problem is to have ∆u = ρ(u) with monotone ρ in
{u > 0} and the condition V = k(x, t)|∇u+| in (1.2.3); this was used as a semilinear model

for tumor growth [30]. Another equation to study is when ∆u = g(x) in (1.2.1) and the

7

Batchelor-Prandtl equation is a special case of this with the condition (1.2.2) [9, 16, 17].

The second area of my work is in free boundary problems, which have an additional layer

of complexity as their domain and boundary are not fixed; but rather, the domain of the

function is an unknown in the equation, it is in fact the set where the solution is positive.

Furthermore in the type of problem we study, it may evolve in time. When expressed as

a function over the d-dimensional sphere, this boundary is a hypersurface that may not

be regular. One physical example of this is the motion of a pressurized fluid through a

medium with friction, which is modeled by the Hele-Shaw type equations. In fact, we study

a generalized version of Hele-Shaw that allows for two phases of a fluid.

8

Chapter 2

Background for degenerate elliptic

equations

Overall , we are interested in learning more about the connection between partial differential

equations on some “nice” domain of the form

and operators of the kind

Lu = −∂j(aij∂iu) = 0 in Ω

u = f

on ∂Ω

(cid:90)

∂Ω

I(f, x) =

f (h) − f (x)µ(x, dh)

for aij and µ that satisfy certain conditions. One of the possible connections we know of is
that the operator I will often arise as the D-to N map for the equation above. Once we know

we can express the D-to-N map as an integro-differential operator, we are further interested

in studying the properties of the measure µ associated with this representation. We also
want to investigate the order of I, i.e. is I(u(r·), x) = rαI(u, rx) for some α?

These questions have already been answered for the case of an equation where the coeffi-

cients aij are uniformly elliptic, and hence the operator L is translation invariant. Here the

9

order is 1, and the operator is the half-Laplacian, −∆1/2. In this work, we want to show
we can create non translation invariant operators of order α, where we have α = 2s = 1 − a
for a ∈ (−1, 1). This is to be done by the process of a weighted D-to-N map for degen-

erate elliptic equations in the half space with variable coefficients, which is the main goal
for chapters 2 and 3. In particular, we will be able to express I in the degenerate case as

an integro-differential operator like in (1.1.2), and establish some properties of this measure

µ. In the second part of this background chapter, we will provide some background about

equations with weighted coefficients with results mainly from [4], [19], and [20].

But first, in the first part of this chapter, we will provide some background about uni-

formly elliptic equations, listing the known results about the integro-differential operators

of their D-to-N maps. The proofs of these results rely heavily of some results of the Green’s

function for uniformly elliptic equations, so we start by first providing a list of results from

[22] and [3], and then go on to describe what is known about the structure of µ for the

uniformly elliptic operators from [23].

2.1. Uniformly elliptic equations

In this section we will provide some background for the uniformly elliptic equation and the

corresponding Green’s function. Most of the following results are standard and appear in

many places, but as [3, 22] was a main reference in this dissertation, we will list the relevant

definitions and provide references to results only from [3, 22]. We first define the notions of

uniformly elliptic equations and operators.

Definition 2.1.1. A : Rn → Rn is called uniformly elliptic if ∃ positiveand integro- differ-

10

ential operators constants λ, Λ such that

λ|ξ|2 ≤ (cid:104)A(x)ξ, ξ(cid:105) ≤ Λ|ξ|2 ∀x, ξ ∈ Rn

(2.1.1)

A partial differential equation of the type

Luφ = − div(A∇uφ) = 0 in Ω ⊂ Rn

uφ = φ on ∂Ω

(2.1.2)

where A is uniformly elliptic; is a uniformly elliptic partial differential equation. One of

the classic and simplest examples of a uniformly elliptic equation is when A = Id, making
L = −∆, the Laplacian operator.

The set-up for Dirichlet-to-Neumann maps (henceforth D-to-N maps) is as follows.

Definition 2.1.2. Given the equation (2.1.2) as above with φ ∈ C1,α(∂Ω), we define a map

andintegro − dif f erentialoperatorsI : C1,α(∂Ω) → Cα(∂Ω) as φ (cid:55)→ ∂νuφ,

(2.1.3)

where ν(x) is the inward normal vector to ∂Ω at x.

The simplest possible case of the map I is when A = Id and the domain is Ω = Rn+1,
the upper half space. In this case, the inward normal to the boundary is (0, 0,··· , 1), we

will have ∂νu = uy. Here u is the harmonic extension of φ, and it is well known that
I = (−∆)1/2.

This D-to-N map I is the one that has a certain inteand integro- differential operatorsgro-

differential representation by the results in [14]. To prove our main goal which is to describe
properties of the special measure associated with I, we will need another important measure

11

which is used in the context of a partial differential equation, i.e. the harmonic measure.

Another key factor in the proofs we will present are estimates on the Green’s function of the

equation. We define tand integro- differential operatorshese concepts below.

Definition 2.1.3. Given the operator L as in (2.1.2) and φ ∈ C(∂Ω), there exists a unique
uφ that solves (2.1.2). Hence, for a fixed x ∈ Ω, the mapping x (cid:55)→ uφ(x) is well defined.

By the comparison principle, this is a non-negative linear functional on C(∂Ω) (i.e.
the
mapping φ (cid:55)→ uφ(x)). The unique Borel measure that represents this functional is called the

L-harmonic measure, and we denote it by ωx. In other words, ωx is uniquely characterized

by

∀φ ∈ C(∂Ω), uφ(x) =

2.1.1. Green’s function

(cid:90)

φ(z)ωx(dz)

∂Ω

and integro- differential operators The primary tool of all of our analyses is to understand

the boundary behaviour of the Green’s function. So in what follows, I will give the reader

the known results of the Green’s function as well as my new results.

Definition 2.1.4. Given an operator L and a function f (in some appropriate function

space), suppose u is the unique solution of the equation

Lu = f

u = 0

in Ω

on ∂Ω

Then the Green’s function is the unique function such that u can be represented as

u(x) =

G(x, z)f (z)dz

(cid:90)

Ω

12

The equation (2.1.2) above can be rewritten by using u(cid:48) = u − φ. Then

Lu(cid:48) = L(u − φ) = Lu − Lφ = Lφ
u(cid:48) = u − φ = 0

in Ω

on ∂Ω

(cid:90)

(cid:90)

G(x, z)Lφ(z)dz

Thus the Green’s function here is the unique function such that u(cid:48) can be represented as

u(cid:48)(x) =

G(x, z)Lφ(z)dz =⇒ u(x) = φ(x) +

Ω

Ω

One of first results we are interested in and will later modify and use in this work is the

relationship between the Green’s function and the harmonic measure.

Proposition 2.1.5. [3, Lemma 2.2] Let {ωx}x∈Ω be the L−harmonic measure where L is

as given in the divergence equation in (2.1.2). Then there are universal constants r0, C1, C2,
and s > 1 such that for any r ∈ (0, r0), z ∈ ∂Ω, and x ∈ Ω \ Bsr(z) ⊂ Rn, the following

holds:

C1rn−2G(x, z + rν(z)) ≤ ωx(∂Ω ∩ Br(z)) ≤ C2rn−2G(x, z + rν(z))

and integro- differential operators

Further, we will be interested in studying the behaviour of the Green’s function in the

interior of the domain, as well as if an how it can be approximated when you get closer to

the boundary of the domain. Of course, the latter will depend on how regular the boundary

of the domain is. Therefore in the results that follow, we will discuss the estimates on the

Green’s function starting with some crude domains that fulfil some minimum requirements

(e.g. the cone condition in definition 2.1.6). At first, we will see that we find some very

basic H¨older regularity up to the boundary. However, as we improve the conditions on the

13

boundary (e.g. the exterior ball condition (see definition 2.1.7)) we get an improved estimate

near the boundary on both the Green’s function and it’s gradient. Naturally, these results

are not special to a Green’s function, but to any solution with zero boundary data.

Definition 2.1.6. [22, Assumption 1.6] A set Ω is said to satisfy the exterior cone condition
if ∃h > 0, 0 < θ < π/2 such that the following is true:
for each z ∈ ∂Ω, ∃ a cone
C = C(z, h, θ) such that Ω◦ ∩ C = ∅, Ω ∩ C = {z}. Here, C(z, h, θ) denotes a cone with

cusp z, height h, and opening angle θ.

Definition 2.1.7. A set Ω is said to satisfy the exterior sphere condition if ∃r > 0 such that
the following is true: for each z ∈ ∂Ω, there exists a sphere of radius r B = B(z, r) such
that Ω◦ ∩ B = ∅, Ω ∩ B = {z}.

For a domain that satisfies the exterior cone condition, we can get the following estimates

on the solution and the Green’s functions as we approach the boundary.

Lemma 2.1.8. [22, Lemma 1.7] Let r > 0 and Dr = Br(0) \ C(0, r/2, θ). Consider the

weak solution ur of the following equation with the given boundary conditions

Lur = 0

ur = 0

ur = 1

in Dr

in ∂C(0, r/2, θ)

in ∂Br(0)

Then, ∃ K(n, λ, Λ, θ) > 0, α(n, λ, Λ, θ) ∈ (0, 1) such that ∀x ∈ Dr

ur(x) ≤ K

|x|α
rα

14

The above lemma zooms into a portion of the domain close to the boundary, by con-

structing cone at a point x0 on the boundary and taking a little ball around the point which

of course overlaps with a little part of the domain close to the boundary. For convenience,

this point is taken to be x0 = 0 and this works for any operator L with bounded, measurable

coefficients. We look at Br(0) because the ellipticity class is invariant by rescaling, i.e. if we

have ˜aij(x) = aij(rx) or ˆaij(x) = aij(x/r), then the ellipticity constants of ˜aij, ˆaij are also

λ, Λ. The lemma redefines the boundary values in this little domain Dr which comprises a

small neighbourhood around this boundary point. As mentioned earlier, this is a very basic

H¨older estimate. This lemma then leads us to a a result about the boundary regularity of

the Green’s function, as below.

Theorem 2.1.9. [22, Theorem 1.8] There are constants K(n, λ, Λ, θ, diam Ω, ∂Ω) > 0,
α(n, λ, Λ, θ) ∈ (0, 1) such that ∀x, z ∈ Ω

G(x, z) ≤ Kδα(z)|x − z|2−n−α

where δ(z) = dist(z, ∂Ω).

If we improve the boundary of our domain to one that satisfies the exterior sphere con-

dition, we will get better estimates for the solutions and the Green’s function. But first, we

also require some special assumptions on the coefficients aij [22, Section 3].

Definition 2.1.10. The coefficient matrix A is called Dini-continuous if it satisfies,

|aij(x) − aij(z)| ≤ ω(|x − z|)

(2.1.4)

15

where ω : R+ → R+ is non-decreasing and it satisfies the doubling condition,

ω(2t) ≤ Kω(t) for some K > 0 and all t > 0

and

0

t

(cid:90) 1

ω(t)

dt < ∞.

(2.1.5)

(2.1.6)

With coefficients as given above, we have the following two lemmas from [22] regarding

the size of the gradient of the solution. These estimates are then used to study the behaviour

of the Green’s function close to the boundary.

Lemma 2.1.11. [22, Lemma 3.1] Suppose u is a bounded solution of Lu = 0 in Ω. with
L as in (2.1.2) and Dini-continuous coefficients aij as described above. Then ∃ K =
(n, λ, Λ, ω, Ω) > 0 such that for any x ∈ Ω

|∇u(x)| ≤ Kδ−1(x) sup

|u|,

Ω

where δ−1(x) := dist(x, ∂Ω).

In the above result we assume no special conditions about the domain Ω. As such, we

note that when you get closer to ∂Ω, it is quite possible for the gradient to blow up. The

following lemma describes a bound of the gradient of the solution on a much nicer and

integro- differential operatorsdomain, i.e. an annulus.

Lemma 2.1.12. [22, Lemma 3.2] Let u be the solution of the Dirichlet problem (with Dini

16

continuous coefficients aij)

Lu = 0

u = 0

u = 1

in Dr = B2r \ Br

in ∂Br

in ∂B2r

Then ∃ K = (n, λ, Λ, ω) > 0 such that for any x ∈ Dr, r ∈ (0, 1],

|∇u(x)| ≤ K
r

.

The above lemmas are used powerfully in proving the following theorem about the es-

timates of the Green’s function.

If we have a domain that satisfies the exterior sphere

condition, we can construct an annulus of a suitable radius at every point on the boundary

of the domain, with the inner sphere touching the boundary on the outside. We then em-

ploy the same techniques used in the proof of 2.1.9 to prove the boundary regularity of the

Green’s function.

Theorem 2.1.13. [22, Theorem 3.3] Let Ω be a domain that satisfies the exterior sphere

condition and let L satisfy (2.1.1)–(2.1.6). Let G be the corresponding Green’s function.
Then the following inequalities are true for any x, z ∈ Ω

(a) G(x, z) ≤ K|x − z|2−n, K = K(n, λ, Λ)

(b) G(x, z)δ(x) ≤ K|x − z|1−n, K = K(n, λ, Λ, ω, Ω)

(c) G(x, z) ≤ Kδ(x)δ(z)|x − z|−n,

|∇xG(x, z)| ≤ K|x − z|1−n,

(d)

17

|∇xG(x, z)| ≤ Kδ(z)|x − z|−n,

|∇x∇zG(x, z)| ≤ K|x − z|−n,

(e)

(f )

2.1.2. Dirichlet-to-Neumann maps

We seek to study the relationship between the equation (1.1.1) and the operator in (1.1.2)
in a degenerate elliptic setting. However, we already have a lot of information about this I,

which arises as the Dirichlet-to-Neumann map in the case of the uniformly elliptic equation
(see definition 2.1.2). In this section, we provide some background and results about I as a

D-to-N map, its integro-differential representation, and properties of the associated measure.
To begin with, this map I is not only well defined from C1,α(∂Ω) to Cα(∂Ω), but it also

satisfies the global comparison property as defined below.

Definition 2.1.14. The global comparison property (GCP) for I : C1,α(X) → C0(X) re-
quires that for all f, g ∈ C1,α(X) such that f (x) ≤ g(x) for all x ∈ X and f (x0) = g(x0) for
some x0 ∈ X, then the operator I satisfies I(f, x0) ≤ I(g, x0). That is to say that I preserves

the ordering on the functions f, g on X at any points where the graphs of f, g touch.

In the case of the divergence equations, L, and hence also I are linear operators. It was

proved in the 1960s, by Courr`ege [14] and Bony-Courr`ege-Priouret [2], through linearity and
the global comparison property, that I must be an integro-differential operator of the form

(cid:90)

I(φ, x) = b(x) · ∇φ(x) − p.v.

φ(x + h) − φ(x)µ(x, dh).

∂Ω

Following is the main theorem from [23]. We will prove the analogous result for degenerate

elliptic equations in the next chapter.

18

Theorem 2.1.15. [23, Theorem 1.1] Suppose L is the divergence operator as in (2.1.2) with
Ω = Rn+1
a family of measures parametrised by x, µ(x, dh) such that ∀φ ∈ C1,α(Rn),

+ and I is as defined via (2.1.2) and (2.1.3), then there exists a vector field, b, and

(cid:90)

Rn

I(φ, x) = (cid:104)b(x),∇φ(x)(cid:105) −

Further, µ satisfies

φ(x + h) − φ(x) − 1B1(x)(h)(cid:104)∇φ(x), h − x(cid:105) µ(x, dh).

(i) For all x ∈ Rn, µ(x,·) has density µ(x, dh) = K(x, h)σ(dh),

(ii) There exists universal c1 > 0, c2 ≥ c1 such that for all x, h ∈ Rn, x (cid:54)= h,

c1|x − h|−n−1 ≤ K(x, h) ≤ c2|x − h|−n−1

In this dissertation, we explore the answers to these questions in a setting where the

coefficients are not uniformly elliptic; in fact they come attached with variable coefficients

and weights making them degenerate as we get closer to the boundary of the domain. As a

result, we are dealing with operators that are no longer translation invariant. This means

many of the classically known results, particularly about the Green’s function may no longer

be valid or have modified versions to suit the new conditions. In the next section, we discuss

some degenerate elliptic equations with weights similar to the ones in our work, and well

known results about the Green’s function of these equations.

19

2.2. Degenerate Elliptic Equations

Consider the following type of degenerate elliptic equation in divergence form

Lu = −∂i(˜aijuj) = 0 in Rn+1
+
on Rn

u = f

(2.2.1)

The coefficients ˜aij will be real-valued, measurable, symmetric, and satisfy

λ|Z|2w(X) ≤ ˜aij(X)ZiZj ≤ Λ|Z|2w(X)

(2.2.2)

for all X = (x, y), Z = (z, s) ∈ Rn+1

+ . The weight w(X) will be a non-negative, measurable

function satisfying Muckenhoupt’s condition, or the A2 condition, which is

1

dX

≤ C.

(2.2.3)

(cid:90)

(cid:90)

(cid:18) 1

(cid:19)(cid:18) 1

(cid:19)
Here the supremum is taken over all Euclidean balls B and(cid:82)
a set E, we denote w(E) =(cid:82)

w(X)

B

w(X)dX

sup
B

|B|

B

|B|

B 1dX = |B|. Further, for
E w(X)dX. Another way of writing the equation above would

be to denote ˜A(X) = w(X)A(X), where A is uniformly elliptic.

Now if w(X) satisfies the A2 condition (2.2.3), then following are two well known facts

about the measure w(X)dX:

(a) w(X)dX and σ(dX) are mutually absolutely continuous.

(b) w(B(X, 2r)) ≤ cw(B(X, r)) (doubling condition.

The above description of the degenerate elliptic equation is taken from [19].

In this

section, we will list many of the results in [19], many of which we will adapt later in chapter

20

3 to our particular version of the degenerate elliptic equation given in (2.2.1). First, we start

by listing some commonly known function spaces with their definitions which will show up

in multiple results in this section and the next chapter.

Definition 2.2.1. (Some function spaces):

(a) Lp(Ω) is the set of all functions f such that their Lp-norm, ((cid:82)
(b) Lp(Ω, w) is the Lebesgue class with the norm ((cid:82)

Ω |f (x)|pdx)1/p, is finite.

Ω |f (x)|pw(x)dx)1/p, or the weighted

Lp-space.

(c) Lip(Ω) is the set of all functions f on Ω that satisfy the Lipschitz condition, i.e.

|f (x) − f (y)| ≤ M|x − y| for some M .

(d)

If we consider the inclusion from Lip(Ω) → [Lp(Ω, w)]n+1 given by the mapping f (cid:55)→
(f,∇f ) = (f, fx1, fx2,··· , fxn), then H1,p(Ω) denotes the closure of the image of
Lip(Ω) in [Lp(Ω, w)]n+1. Essentially, H1,p(Ω) is the space of Lp functions on Ω whose

weak first derivatives are also in Lp(Ω).

(e) H1,p

0 (Ω) denotes the closure of compactly supported functions in the image of Lip(Ω)
in [Lp(Ω, w)]n+1. i.e. f ∈ H1,p
0 (Ω) if and only if there exist functions fm ∈ Lip(Ω)
such that fm are all compactly supported in Ω and fm → f in H1,p(Ω).

We further describe a notion of convergence in these Sobolev spaces, and the capacity of

a set, which later appears in the interior estimates of the Green’s function.

Definition 2.2.2. Let K ⊂ Ω. We say that u ≥ c on K in the H1,2(Ω) sense if ∃ ϕj ∈
Lip(Ω) such that ϕj ≥ c ∀x ∈ K and ϕj −→ u in H1,2(Ω).

21

Definition 2.2.3. The Dirichlet form D : H1,2(Ω) × H1,2(Ω) → R is defined by

D(u, v) :=

˜aiju(x)v(x)dx.

Definition 2.2.4. Let K ⊂ Ω be compact. The capacity of K in Ω is given by

Ω

cap(K) := inf{D(u, u) : u ∈ H1,2

0 (Ω), u ≥ 1 on K in the H1,2(Ω) sense.}

For an open set Θ in Ω, cap(Θ) := sup{cap(K) : K compact, K ⊂ Θ}.

2.2.1. Green’s function

Definition 2.2.5. The Green’s function for the degenerate equation is defined in a similar

manner as the elliptic case. Given the equation Lu = f in Ω with L as in (2.2.1), the Green’s

function is the unique function such that u can be represented as

(cid:90)

(cid:90)

Ω

(cid:90)

u(x) =

G(x, z)f (z)dz

As before, we can rewrite this equation u(cid:48) = u − φ and get

u(x) = φ(x) +

G(x, z)Lφ(z)dz

Ω

Remark 2.2.6. Another way of looking at the Green’s function is that is we fix z ∈ Ω, then

the Green’s function denoted by G(x, z) is the weak solution of Lxu(x) = δz(x) as a function

22

of x, where δz is the unit mass at z. Thus, we claim that LxG(x, z) = δz(x).This is because

Ω

Ω

(cid:90)
(cid:90)
(cid:90)
(cid:90)
(cid:90)

δz(x)u(x)dx.

(by the definition of δz)

G(x, z)f (x)dx

(by the definition of G(x, z))

G(x, z)Lxu(x)dx

(since Lxu(x) = f (x))

LxG(x, z)u(x)dx

(since L is self-adjoint)

LxG(x, z)u(x)dx

first, u(z) =

Next, u(z) =

=

=

δz(x)u(x)dx =

(cid:90)

Ω

=⇒

Ω

Ω

Ω

Thus we have, as claimed, LxG(x, z) = δz(x).

Now we list results from [19, Section 3] about the size of the Green’s function for the

degenerate elliptic equation described in (2.2.1), which will be useful for finding the estimates

of the Green’s function to the case of the specific degenerate elliptic equation we will look

at in the next chapter.

Lemma 2.2.7. [19, Lemma 3.1] If B(x, 2r) ∈ Ω and z ∈ ∂B(x, r), then

G(x, z) ∼= 1/ cap(B(x, r)).

Lemma 2.2.8. [19, Lemma 3.2] If x ∈ Ω and 3

2 r ≤ dist(x, ∂Ω) ≤ 8r, then cap(B(x, r)) (cid:39)

w(B(x, r))/r2.

Using these lemmas, we get an estimate for the Green’s function in the interior of the

domain.

23

Theorem 2.2.9. [19, Theorem 3.3] Let BR ⊂ Ω and x, z ∈ BR/4. Denote r = |x− z|. Then

(cid:90) R

G(x, z) (cid:39)

s2

w(B(x, s))

r

ds
s

.

(2.2.4)

The above result gives interior estimates for the Green’s function, and we will still need

results about the behaviour of the Green’s function as we approach the boundary, which will

take up the entire first section in chapter 3.

Additionally, we also need to describe some relationship between the Green’s function

and the harmonic measure (same as defined in definition 2.1.3) for the equation (2.2.1).

There is such a result in [20] which concerns nontangentially accessible (NTA) domains.
One of the conditions in the definition of an NTA domain Ω is that ∃A > 1, r0 > 0 such
that ∀r, 0 < r < r0, ∀z ∈ ∂Ω, ∃z(cid:48) ∈ Ω such that |z − z(cid:48)| < Ar and B(z(cid:48), r/A) ⊂ Ω. In a

sense, this is an interior ball condition analogous to the exterior sphere condition described

in definition 2.1.7.

Lemma 2.2.10. [20, Lemma 3] Let z ∈ ∂Ω, z(cid:48) ∈ Ω be as above in the definition of an NTA
domain, and let {ωx}x∈Ω be the L-harmonic measure where L is as given in (2.2.1).
x ∈ Ω \ B(z, 4Ar) then

If

G(z(cid:48), x) (cid:39) ωx(∂Ω ∩ B(z, r))

r2

w(B(x, r))

This is of course analogous to proposition 2.1.5, by considering z(cid:48) = z + rν(z).

24

2.2.2. Dirichlet-to-Neumann maps

Now that we are equipped with sufficient background about degenerate elliptic equations,

we will look at the D-to-N maps of these equations. A canonical result of this is the frac-

tional Laplacian which we mentioned in Chapter 1. Let us consider a fairly simple type of
degenerate elliptic equation in the half-space in Rn+1. For X ∈ Rn+1
where x ∈ Rn, and y > 0. Also note that ∂Rn+1 = Rn.

+ , we denote X = (x, y),

Lu = − div(ya∇u) = 0

u = f

in Rn+1

+

in Rn

(2.2.5)

This is an equation that looks like (2.2.1), and the weights as given in (2.2.2) are ya. where
a ∈ (−1, 1). Thus, ˜aij(X) = aij(X)w(X) = yaId. Thus w(X) = w(x, y) = ya and

A(x) = Id, which is uniformly elliptic. Among the many authors to study this equation in

the context of the fractional Laplacian are Caffarelli and Silvestre [4].

Definition 2.2.11. The fractional Laplacian of a function f : Rn → R is expressed by

(cid:90)

(−∆)sf (x) = Cn,s

f (x) − f (z)
|x − z|n+2s dz

Rn

where the parameter s ∈ (0, 1) and Cn,s is some normalization constant.

One of the main results of [4] is the connection between the fractional Laplacian and the

D-to-N map for the equation (2.2.5). The D-to-N map is given by the co-normal derivative

of the solution of the equation.

Definition 2.2.12. The co-normal derivative for an equation is defined with the help of

Green’s identity. In particular, it is the term that appears in the integral on the boundary.

25

For the above equation (2.1.2), using, integration by parts, we have

− div(ya∇u) · ϕdX =

ya∇u · ∇ϕdX −

ya∇u · ϕ · νdS

(cid:90)

Rn+1

+

(cid:90)

Rn+1

+

(cid:90)

Rn

Since the inward facing normal to the upper half space is (0, 0, . . . , y), the last term above is

(cid:82)Rn ϕ · (yauy)dS. The co-normal derivative is the term multiplied to the test function, i.e.

−yauy.

Theorem 2.2.13. [4, Section 3] For u, the solution of (2.2.4), we have up to a constant
factor, that the co-normal derivative is the fractional Laplacian of order s = (1 − a)/2, i.e.

yauy = (−∆)sf (x)

lim
y→0+

Note that s ∈ (0, 1) =⇒ a ∈ (−1, 1).

yauy = (−∆)

1−a
2 f (x) = Cn,s

lim
y→0+

(cid:90)

Rn

f (x) − f (z)
|x − z|n+1−a dz

This also fits with our earlier mention of the D-to-N map in the case of the Laplacian,

i.e. when a = 0, we have I = (−∆)1/2.

Now for this equation, say we would like to develop a Poisson formula P to explicitly

solve (2.2.5) i.e. we want P that satisfies for X = (x, y) ∈ Rn+1

+ , Z ∈ ∂Rn+1

+ = Rn

(cid:90)

Rn

u(X) =

P (X, Z)f (Z)dZ

We will also denote this Poisson Kernel P as P (X, Z) = PX (Z). Another notable comment

about the Poisson kernel P is that it is the co-normal derivative of the Green’s function.

26

Chapter 3

Connection to Dirichlet-to-Neumann

(D-to-N) maps

3.1. Introduction to our equation and coefficients

In this work, we will be proving many of the above results for a special case of the degenerate
elliptic equation set in the upper half-space. For some f ∈ C1,α(Rn+1

+ ), let u solve the

following equation:

Lu = − div(yaaijui) = 0

u = f

in Rn+1

+

in Rn

(3.1.1)

where [aij] = A is measurable, symmetric, and uniformly elliptic. At times, we will use the

notation ˜aij to denote yaaij, or ˜A to denote yaA. Thus, ˜aij are like the weighted coefficients

in (2.2.2) with weights w(X) = ya. Notice that when a = 0, we get the uniformly elliptic
case that was discussed in chapter 2. However, if a (cid:54)= 0, then this equation degenerates as
you get closer to the boundary, i.e. Rn, where y = 0.

Now, we will consider a special version of Dini-continuity for our coefficients ˜aij, namely

for any bounded subset Ω ⊂ Rn+1

+ we have

(cid:90)

Ω

y−a|˜aij(X) − ˜aij(Z)|

|X − Z|N

dX ≤ C

(3.1.2)

27

where X, Z ∈ Rn+1

+ , X = (x, y), Z = (z, s), a ∈ (0, 1). The reason for imposing this

extra condition on the coefficients will become evident in the proofs of the results about the

Green’s function to this equation, particularly lemma 3.2.2.

Before we look at the D-to-N map for this equation, we will need a definition for the

co-normal derivative in this setting.

Definition 3.1.1. The co-normal derivative for an equation is defined with the help of

Green’s identity, just as in the definition 2.2.12. In particular, it is the term that appears in

the integral on the boundary. For the above equation (3.1.1), using, integration by parts, we

have

(cid:90)

Rn+1

+

− div(yaaijui) · ϕdX =

(cid:90)

Rn+1

+

yaaijui · ϕjdX −

(cid:90)

Rn

yaaijui · ϕ · νdS

Since the inward facing normal to the upper half space is (0, 0, . . . , en+1), the last term above

j an+1,juj)dS. The co-normal derivative is the term multiplied to the test

is (cid:82)Rn ϕ · (ya(cid:80)
function, i.e. −ya(cid:80)

j an+1,juj.

Remark 3.1.2. There are, in essence, two choices for the “normal” derivative. This is

because for the divergence equation, there are two possible normal vectors we can consider.

One of them is the natural inward normal vector we see geometrically, which in the case
of the half-space is just (0,··· , 0, 1). We can see that this is the normal vector taken into

consideration for the definition 2.2.12 of the co-normal derivative for the equation 2.2.5.
The other choice of normal vector comes from the equation, i.e. (an+1,1,··· , an+1,n+1) as

in the definition 3.1.1 above. This is still an inward normal vector, but it is weighted by the

coefficients yaaij. Since aij are uniformly elliptic, the distance between this normal vector

and the conventional geometric normal vector are comparable up to a constant.

28

The Dirichlet-to-Neumann map for this equation (3.1.1) is given by this weighted normal
derivative, i.e. I(f, x) = −ya∂νu = −yauy. This map I is not only well defined from

C1,α(∂Ω) to Cα(∂Ω), but it also satisfies the global comparison property as defined in
definition 2.2.1. Again, here in (3.1.1), L is a linear operator and hence so is I. By a

result of Bony-Courr`ege-Priouret [2] in the 1960s using linearity and the global comparison
property, it is known that I can be expressed as an integro-differential operator of the form

(cid:90)

I(φ, x) = b(x) · ∇φ(x) − p.v.

φ(x + h) − φ(x)µ(x, dh).

∂Ω

The main result in this chapter is a modification of the theorem 2.1.15 from [23] for this

particular elliptic equation which degenerates as you get close to the boundary.

Theorem 3.1.3. Suppose L is the divergence operator as in (3.1.1) with Ω = Rn+1
and
a ∈ (0, 1). If I is as defined via (2.1.2) and (2.1.3), then we know there exists a vector field,
b, and a family of measures parametrised by x, µ(x, dh) such that ∀φ ∈ C1,α(Rn),

+

φ(x + h) − φ(x) − 1B1(x)(h)(cid:104)∇φ(x), h − x(cid:105) µ(x, dh).

(cid:90)

Rn

I(φ, x) = (cid:104)b(x),∇φ(x)(cid:105) −

Further, µ satisfies

(i) µ is absolutely continuous with respect to the surface measure, σ, i.e.

for all x ∈

Rn, µ(x,·) has density µ(x, dh) = K(x, h)(dh),

(ii) There exists universal C > 0 such that for all x, h ∈ Rn, x (cid:54)= h,

K(x, h) ≤ C|x − h|−n+a−1.

29

The proof of this theorem relies on many properties and results about the Green’s function

associated with (3.1.1), which we will outline in the next section.

3.2. Estimates for the Green’s function

We start by finding estimates for the Green’s function of (3.1.1) as defined in definition 2.1.4,

first in the interior of our domain, and then closer to the boundary.

3.2.1. Interior estimates

Since we have the the weighted coefficients, i.e. yaaij, we know that the A2 weights for

(3.1.1) are w(X) = ya. The following result for the interior estimate for the Green’s function

are a direct result of applying the results in [19, Section 3]. Since our domain is a half-space,

we will consider cubes that are contained in the domain instead of the spheres/balls that

appear in lemmas 2.2.7, 2.2.8 for the proofs. Following is the modified version of theorem

2.2.9, which is [19, Theorem 3.3].

Theorem 3.2.1. Let X, Z ∈ Q

Then

1/4

R = {(m, n+3/4) : M = (m, n) ∈ QR} and let r = |X−Z|.

(cid:90) R

G(X, Z) (cid:39)

w(B(X, t))

r

t2

dt
t

The proof of this theorem relies on the two aforementioned lemmas 2.2.7, 2.2.8. Using

this theorem and w(X) = w(x, y) = ya we can approximate the size of the Green’s function
for our degenerate equation if we let P = (p, q) ∈ B(X, t), then

w(B(X, t)) =

w(P )dP =

qadP (cid:39) Ctn+1+a

(cid:90)

(cid:90)

B(X,t)

B(X,t)

30

(cid:90) R

t2

w(B(X, t))

r

dt
t

=

Then

(cid:90) R

t2

|X−Z|

Ctn+1+a

(cid:90) R
|X−Z| Ct−n−adt (cid:39) C|X − Z|−n−a+1

dt
t

=

But 1 − n − a = 2 − (n + 1) − a = 2 − N − a and thus ∃ C = C(R, n)

G(X, Z) (cid:39) C|X − Z|2−N−a

One can also write an alternate lengthier proof using the techniques in [29].

3.2.2. Boundary estimates

In this section, we will modify some of the results in [22] to suit the degenerate elliptic

equation in (3.1.1). Recall from section 2.1.1 that first, in order to find estimates on the

Green’s function, it is helpful to get some estimates on the gradient of the solution. We saw

what with a domain that satisfies the exterior cone condition defined in definition 2.1.6, we

get some crude H¨older-type estimates on the solution of the equation, and subsequently the

Green’s function, as seem in lemma 2.1.8 and 2.1.9. After modifying the coefficients, when

we have no assumptions on the boundary, we see in lemma 2.1.11 that the gradient of the

solution can possibly blow up as you get closer to the boundary. Finally, we considered a

domain with the exterior sphere condition; by first proving some estimates for the uniformly

elliptic equation with Dini-continuous coefficients on an annulus (lemma 2.1.12), we are then

able to get estimates for the Green’s function and it’s gradient up to the boundary in theorem

2.1.13.

Since the equation we are dealing with is set in the half-space Rn+1

+ , the boundary of our
domain i.e. Rn, is incredibly nice. It satisfies both the interior and exterior cone and sphere

conditions. However, the roadblock in our work is due to the degeneracy of the equation as

31

you get closer to the boundary. To be able to uniformly study points close to the boundary,
we make use of cubes in Rn+1. Particularly, we look at the intersection of cubes centred at

the origin with the half-space. Of course, the operator in (3.1.1) is not translation invariant

in general, but for any fixed distance from the boundary, we do have translation invariance.

This is why it is acceptable to look at cubes centred around 0 alone, as we can “slide” the
cubes “horizontally” along the boundary Rn without changing the equation.

Our goal is to be able to prove the estimates for the Green’s function as given in theorem

3.2.7. For this, we will need some specific estimates on the solution of (3.1.1) and its gradient.

In the following results, we make some discoveries regarding how the gradient of the solution

and the solution itself behaves close to the boundary. In lemma 3.2.2 and corollary 3.2.6, we

see that when we restrict the values of a to (0, 1), we can get the estimates we want which

lead us to the estimates for the Green’s function in the following theorem 3.2.7.

Lemma 3.2.2. Suppose u solves the following equation in Q+

4R = Q4R ∩ Rn+1

+

given in (3.1.1))

Lu = 0 in Q+
4R

u = 0 on Q+

u = 1 on ∂Q+
0 ≤u ≤ 1 on (Q+

+

2R ∩ ∂Rn+1
4R \ ∂Rn+1
4R \ Q+

+

2R) ∩ ∂Rn+1

+

Then ∃K = K(n, λ, Λ, ω) such that for any X = (x, y) ∈ Q+
R,

|∇u(X)| ≤ K

y−a
R1−a

(with L as

(3.2.1)

(3.2.2)

Here, Lu = − div(yaA∇u), with A uniformly elliptic, and a ∈ (0, 1). (We will sometimes

denote ˜A = yaA.)

32

u = 1

Q+
4R

Lu = 0

t

(x, y)

Q+
R

Q+
2R

0 ≤ u ≤ 1

u = 0

0 ≤ u ≤ 1

Figure 3.1: Boundary conditions for lemma 3.2.2

Remark 3.2.3. We shed light on some things we may take for granted in this proof, which

we will elaborate upon later.

(a) We will first consider the case where R = 1, and see later that we can re-scale to get

the result (see Remark 3.2.4).

(b) We do not yet know if Schauder theory applies to (3.2.1) with the given coefficients,

but we can fix this with a limiting argument (see Remark 3.2.5). For now, we shall
assume that u ∈ C1

4 ) and proceed with the proof.

loc(Q+

Proof.

ya|∇u(X)| =: M < ∞

Let

sup
X∈Q+
1

Let X0 = (x0, t) ∈ Q+

1 be such that ya|∇u(X0)| ≥ 1

2 M , i.e. X0 is a point where this value

is particularly large.

Consider a ball of radius d around X0, where d ≤ 1/2 will be determined later. We
c (Bd(X0)) such that η ≡ 1 on Bd/2(X0), |∇η| ≤ k1d−1 and

choose a cut-off function η ∈ C∞

33

# q X0
"!

Q+
1

u = 0

t = 1

# q X0
"!

t = 1/2

t = 0

Figure 3.2: Different locations for Bd(X0)

|∇2η| ≤ k2d−2. (k1, k2 are universal constants.)

Note that we also have uη ≡ 0 on ∂Q+

4 . Indeed, if for X0 = (x0, t) we have t > 1/2 then

Bd(X0) is safely contained inside Q+

4 and since η = 0 outside Bd(X0), uη = 0 on ∂Q+

4 . On

the other hand, if t < 1/2 then part of Bd(X0) may be outside Q+

4 , but the only part of

the boundary it intersects is where Q+
different places that Bd(X0) could be with X0 ∈ Q+

2 ∩ ∂Rn+1

1 and given d < 1/2.

+ , where u = 0. The figure above shows the

Now suppose F is the Green’s function in Q+

to the operator with the constant coefficients ˜aij
L0 = − div(taaij
Then for Z = (z, s) ∈ Q+

0 ∇), and L0F = 0 in Q+
(cid:90)

4 we use η(·)F (·, Z) as a test function to get

4 for the equation in (3.2.1) corresponding
0 (X) = ˜aij(X0) = taaij(X0).

i.e we have

4 , F ≡ 0 on ∂Q+

4 . Hence, ηF is a valid test function.

0 =

Q+
4

˜aij(X)uxi(X)[η(X)F (X, Z)]xj dX =

˜aijui(ηF )j

(cid:90)

Q+
4

34

We will use the second notation for brevity.

˜aijui(ηF )j

˜aij
0 ui(ηF )j +

(cid:90)

Q+
4

˜aij
0 ui(ηjF + ηFj) +

(˜aij − ˜aij

0 )ui(ηF )j

(˜aij − ˜aij

0 )ui(ηF )j

(cid:90)

Q+
4

(cid:90)
(cid:90)
(cid:90)

Q+
4

Q+
4

Q+
4

0 =

=

=

(cid:90)

Q+
4

We will use integration by parts and note that uη ≡ 0 on ∂Q+

4 , as is F .

0 = −

0 uηjFi−
˜aij

0 uηjiF −
˜aij

0 uηiFj−
˜aij

˜aij
0 uηFji +

(˜aij−˜aij

0 )ui(ηF )j

(cid:90)

Q+
4

(cid:90)

Q+
4

(cid:90)

Q+
4

(cid:90)

Q+
4

Since −aij

0 Fji(·, Z) = δZ , we have

(cid:90)

(cid:90)

Q+
4

˜aij
0 uηiFj

(3.2.3)

u(Z)η(Z) =

+

˜aij
0 uηjiF +

˜aij
0 uηjFi +

Q+
4
0 − ˜aij)ui(ηF )j
(˜aij

(cid:90)
(cid:90)

Q+
4

Q+
4

Now we differentiate (take gradient) (3.2.3) with respect to Z and set Z = X0. Note that
η(X0) = 1 and ∇η(X0) = 0 since η ≡ 1 on Bd/2(X0).
(cid:90)

∇u(X0) =

taaij

0 uηj∇Z Fi(·, X0) +

taaij

0 uηji∇Z F (·, X0)

Q+
4

taaij

0 uηi∇Z Fj(·, X0)

(taaij

0 − yaaij)ui[ηj∇Z F (·, X0) + η∇Z Fj(·, X0)]

(3.2.4)

(cid:90)
(cid:90)
(cid:90)

Q+
4

Q+
4

Q+
4

+

+

= I1 + I2 + I3 + I4

35

Since aij

0 is just a constant coefficient and ta is a fixed constant multiplied to it, the

estimates we have on the Green’s function F from [37] are as follows:

|∇Z F (X, X0)| ≤ t−aC1|X − X0|1−N , and ∇Z Fxi(X, X0) ≤ t−aC2|X − X0|−N

For I1 and I3, since ηj = ηi = 0 on Bd/2(X0) as well as outside Bd(X0), we can integrate
4 . Also using the fact that u ≤ 1, in this case we have the

over the annulus Bd \ Bd/2 ∩ Q+

following estimates

(cid:90)

|I1| ≤

taΛk1d−1C2t−a|X − X0|−N dX
(cid:90) d

r−N rN−1dr = Λk1C2d−1 ln(2) = Cd−1

Bd\Bd/2
= Λk1C2d−1

d/2

Similarly, we also have |I3| ≤ Cd−1

For I2, we will just integrate over Bd by using the fact that η ≡ 0 outside Bd and hence

ηij ≡ 0 too.

(cid:90)

Bd

|I2| ≤

taΛk2d−2C1t−a|X − X0|1−N dX

= Λk2C1d−2

r1−N rN−1dr = Cd−1

(cid:90) d

0

As for I4, we know that ui(X) ≤ M y−a by the assumption and recall we have the

condition

(cid:90)

+ ∩Bd
Rn+1

y−a|˜aij(X) − ˜aij(Z)|

|X − Z|N

dX ≤ C

36

Therefore,

(cid:90)
(cid:90)

( ˜A(X0) − ˜A(X))ui[ηj∇Z F (·, X0) + η∇Z Fj(·, X0)]

k1d−1C1t−a|X0 − X|1−N + t−aC2|X0 − X|)−N(cid:105)

dX

|I4| ≤

≤

Bd

Bd

( ˜A(X0) − ˜A(X))M y−a(cid:104)
(cid:90)

y−a

˜A(X0) − ˜A(X)
|X0 − X|N

Bd

≤ C(cid:48)M t−a

dX

(since d ≥ |X0 − X|)

Putting all the above estimates together, we have

|∇u(X0)| ≤ Cd−1 + C(cid:48)M t−a
(cid:90)

+ C(cid:48)M

ta
d

Bd

=⇒ 1
2

M ≤ ta|∇u(X0)| ≤ C
(cid:40)
d : C(cid:48)(cid:90)

y−a

Bd

˜A(X0) − ˜A(X)
|X0 − X|N

dX ≤ 1/4

Let d0 = sup

Then,

(cid:90)

Bd
y−a

dX

˜A(X0) − ˜A(X)
|X0 − X|N

y−a
˜A(X0) − ˜A(X)
|X0 − X|N
(cid:41)
, choose d = min{d0, 1/2}.

dX

1
2

M ≤ C
M ≤ C

ta
d
ta
d

1
4

M

+
≤ C sup{2, 1/d0} = K

(since X0 ∈ Q+

1 , ta < C)

Note that in order to be controlled by this universal constant, it is important that a ≥ 0,
hence our assumption that a ∈ (0, 1). Finally, this gives us

ya|∇u(X)| ≤ K

M = sup
X∈Q1

=⇒ |∇u(X)| ≤ Ky−a.

(for any X = (x, y) ∈ Q+
1 )

37

make a change of variables as follows:

values as in (3.1.1). Now if instead we look at Q+

Remark 3.2.4. Scaling
u solves the equation Lu(X) = − div(yaA(X)∇u(X)) = 0 in Q+
4 , then X ∈ Q+
(cid:19)
(cid:18) X
2 ∩ ∂Rn+1
4 \ ∂Rn+1
4 \ Q+

ˆu = 1 on ∂Q+
0 ≤ˆu ≤ 1 on (Q+

Let u(RZ) = ˆu(Z) =⇒ u(X) = ˆu

Then, we will have ˆu = 0 on Q+

2 ) ∩ ∂Rn+1

+

4R with all the boundary
4R =⇒ X

R = Z ∈ Q+

4 . We

R

+

+

Since we have the same boundary conditions, we must ask - what equation does ˆu solve in

R∇Z ˆu, and since X = (x, y) = RZ =
Q+
4 if u solves (3.1.1) in Q+
(Rz, Rs), we have ya = Rasa. If we also denote aij(X) = aij(RZ) = ˆaij(Z), and we know

4R? Note that ∇X u = 1

that ˆA will satisfy the same conditions as A, then ˆu solves the following equation in Q+
4

R1−a (saˆaij(Z)∇Z ˆu(Z)) = 0
− 1

Repeat all the calculations in the proof to see that the factor

1

R1−a remains throughout, thus

giving us the result in lemma 3.2.2.

Remark 3.2.5. Limit argument
Now we don’t really know whether u ∈ C1
coefficients ˜Am(X) = ˜Am(x, y) = max{ 1

4 ). However, we can consider ∀m ∈ N the
loc(Q+
m , ya} · A(x, y), where A is uniformly elliptic. Note

38

that ˜Am → ˜A in the L∞-norm. Also, for every m,

˜Am(X) ≥ 1
m

A(X) =⇒ ˜aij

m(X)ξiξj ≥ 1
m

λ|ξ|2 ∀X, ξ ∈ Rn+1

+

and as such the equation Lmum = −(˜aij

m(um)i)j = 0 with the same boundary values

given in (3.1.1) is non-degenerate. If fact, each ˜Am is uniformly elliptic, and hence we know
that each um ∈ C1(Q+

4 ) by Schauder. Also, um ∈ H1(Q+

4 ) as it solves the elliptic PDE.

On the other hand, we also know that u which solves (3.1.1) is in the weighted Sobolev space

H1(Q+

H1(Q+

4 , w), where w(X) = ya is the weight. We could also say that um ∈ H1(Q+
4 , w) ⊂ H1(Q+
4 ).

4 , w) as

If we proceed as in the proof above, then we see that we have |∇um| ≤ Ky−a, which is
a uniform bound independent of m, and we also know that for all m, 0 ≤ um ≤ 1. Thus
the sequence um is uniformly bounded in H1(Q+
a u(cid:63) ∈ H1(Q+

4 , w) such that a subsequence umk converges weakly to u(cid:63) in H1(Q+

4 , w), a Hilbert space, which means there is

4 , w) (cid:98) L2(Q+

4 , w). On
4 , w). Thus, not only does umk → u(cid:63)
4 ) such that umk → v strongly in L2(Q+
4 ).
4 , w). Hence the uniform

the other hand, we also know that H1(Q+

4 ), there is also a v ∈ L2(Q+

weakly in L2(Q+
Therefore, we must have u(cid:63) = v, and umk → u(cid:63) strongly in H1(Q+
bound on |∇um| also applies to its limit, |∇u(cid:63)|.

Finally, what guarantees that this u(cid:63) which is the limit of um is the same function u that

solves the equation (3.1.1)? Using ϕ as a test function

39

(cid:90)

Q+
4

(cid:90)
(cid:90)

= lim
m→∞

Q+
4

lim
m→∞

m(um)iϕj − ˜aiju(cid:63)
[˜aij

i ϕj]

m(um)iϕj − ˜aij(um)iϕj + ˜aij(um)iϕj − ˜aiju(cid:63)
˜aij

i ϕj

(cid:90)

Q+
4

˜aij((um)i − u(cid:63)

i )ϕj

= lim
m→∞
≤ lim
m→∞|| ˜Am − ˜A||L∞

m − ˜aij)(um)iϕj + lim
(˜aij
m→∞
|∇um||∇ϕ|

(cid:90)

Q+
4

Q+
4

(cid:90)

yaΛ|∇um − ∇u(cid:63)||∇ϕ|

Q+
4

+ lim
m→∞
≤ lim
m→∞|| ˜Am − ˜A||L∞||um||
m→∞||um − u(cid:63)||

H1(Q+

+ lim

||∇ϕ||L∞|Q+
4 |

4 ,w)

H1(Q+
CN Λ||∇ϕ||L∞

= 0

Thus, we have

0 = lim
m→∞

(cid:90)

Q+
4

4 ,w)

(cid:90)

Q+
4

˜aij
m(um)iϕj =

aiju(cid:63)

i ϕj

Thus, u(cid:63) solves the equation (3.1.1) weakly. But u solves (3.1.1), and the solution is

unique, which means u = u(cid:63).

As for the argument with freezing the coefficients, we have

˜A0,m = ˜Am(X0) = max

(cid:26) 1

(cid:27)

, ta

m

· A(x0, t)

which means we either have ˜aij

0,m(X) = taA(x0, t) or ˜aij

0,m(Z) =

1
m

A(x0, t).

In the proof above, we only considered the former case, however, it is easy to see that if we

consider the latter, we would have the Laplacian multiplied by a constant which gives us the

40

same estimates for the Green’s function, up to a universal constant.

Corollary 3.2.6. Let u solve the same equation above (3.1.1). Then ∃K = K(n, λ, Λ, ω)
such that ∀X = (x, y) ∈ Q+
R,

u(X) ≤ K

y1−a
R1−a

Proof. For X ∈ Q+

R, let X(cid:63) ∈ ∂Rn+1

+

be such that |X − X(cid:63)| = δ(X) = y. If [X, X(cid:63)] is the

line joining X and X(cid:63),

|u(X) − u(X(cid:63))|

|X − X(cid:63)|

≤ sup

Z∈[X,X(cid:63)]

|∇u(Z)| ≤ K

y−a
R1−a

(using the lemma above)

Since we know that X(cid:63) ∈ Q+

2R ∩ ∂Rn+1

+ , we have u(X(cid:63)) = 0, which gives

u(X) ≤ K

y−a
R1−a|X − X(cid:63)| = K

y1−a
R1−a .

Theorem 3.2.7. Let G be the Green’s function for (3.1.1) with a ∈ (0, 1). Then ∃K1, K2
that depend only on n, µ, λ, ω such that ∀ X, Z ∈ RN

+

(i) G(X, Z) ≤ K1δ(X)1−a|X − Z|1−N

(ii) G(X, Z) ≤ K2δ(X)1−aδ(Z)1−a|X − Z|−N +a

Proof.

(i) Let X be fixed and look at G as a function of Z alone, i.e. G(·) = G(X,·). Let

R = |X − Z|/5. Consider the following 2 cases:

(a) δ(Z) ≥ R. This means X, Z are safely inside the domain. So we can apply the

41

interior estimates to get

G(X, Z) = K|X − Z|2−N−a R1−a
R1−a
(cid:16) X−Z
(cid:17)1−a
≤ Kδ(Z)1−a|X − Z|2−N−a
≤ K1δ(Z)1−a|X − Z|1−N

5

(b) δ(Z) ≤ R. In this case, choose Z(cid:63) ∈ ∂RN

+ such that |Z − Z(cid:63)| = δ(Z). With the

point Z(cid:63) as the center, consider Q+
4R(Z(cid:63)). Let uR be as in lemma 3.1.
+ , we have G(P ) ≤ K|X − P|2−N−a since
Now, for any point P ∈ ∂Q+
4R \ ∂RN
we have interior estimates. Also, since uR = 1 here, we have G(P ) ≤ K|X −
P|2−N−auR(P ). We also know that

|X − Z| ≤ |X − P| + |P − Z| ≤ |X − P| + 4R =⇒ |X − Z| ≤ 5|X − P|
=⇒ |X − P|2−N−a ≤ |X − Z|2−N−a

Thus, we have G(·) ≤ K|X − Z|2−N−auR(·) for any point on ∂Q+
4R ∩ ∂RN
the other hand, on the bottom boundary, i.e., on Q+
and uR ≥ 0, and thus we have G(·) ≤ K|X − Z|2−N−auR(·).

4R \ ∂RN

+ . On
+ , we have G ≡ 0,

Thus, not only are G(·) and K|X − Z|2−N−auR(·) both solutions of Lu = 0 in
4R(Z(cid:63)), we also have G(·) ≤ K|X − Z|2−N−auR(·) on ∂Q+
Q+
4R(Z(cid:63)). By compari-

42

son principle, we get

G(P ) ≤ K|X − Z|2−N−auR(P ) ∀P ∈ Q+

4R(Z(cid:63))

In particular, this statement is true for Z ∈ Q+

4R(Z(cid:63)).

From lemma 3.1, we get

G(X, Z) ≤ K|X − Z|2−N−auR(Z)

≤ K|X − Z|2−N−a|uR(Z) − uR(Z(cid:63)|
≤ K|X − Z|2−N−a|∇(Z)|δ(Z)
≤ K|X − Z|2−N−a δ(Z)1−a

R1−a = K1δ(Z)1−a|X − Z|1−N

(ii) The proof of this is obtained from (i) the same way in which we prove (i) from the

interior estimates.

Remark 3.2.8. Recall that the Poisson kernel P is one that helps explicitly solve the equa-
tion, which is this case is (3.1.1) i.e. we want P that satisfies for X = (x, y) ∈ Rn+1
(z, 0) ∈ Rn

+ , Z =

u(X) =

P (X, Z)f (Z)dZ

(cid:90)

Rn

We will use the notation PX (Z) = P (X, Z) at times. This Poisson Kernel can be rec-

ognized as the co-normal derivative of the Green’s function. Since from the above theorem,
we have an estimate for the Green’s function as G(X, Z) ≤ Cy1−as1−a|X − Z|−N +a. Using

43

this estimate we will directly compute the co-normal derivative of the Green’s function from

the definition 3.1.1. We will use the notation an+1,j = (cid:126)an+1, X = (x, y), Z = (z, s)

PX (Z) = ya(cid:88)

j

an+1,jGxj (X, Z)
G(X + y(cid:126)an+1, Z) − G(X, Z)

= ya lim
y→0
ya−1[Cy1−as1−a|X − Z|−N +a]

y

≤ lim
y→0

≤ Cs1−a|X − Z|−N +a

(3.2.5)

3.2.3. Relationship with harmonic measure

As seen in the case of the uniformly elliptic equation, one of the other results about the

Green’s function which is crucial to the proof of our main result is the relationship between

the Green’s function and harmonic measure. We recall from the previous chapter (stated as

lemma 2.2.10) that we know what this relationship is directly from [20, lemma 3].

However, since we do not need the assumption of an NTA domain with our domain

being very smooth and regular, we will adapt this result to the half-space to establish the

connection between the Green’s function and harmonic measure that we will call upon later.

Recall that in section 3.2, we worked out that w(B(X, r)) = Crn+1+a = rN +a. Thus

applying [20, lemma 3] to (3.1.1), we get

Lemma 3.2.9. [20, lemma 3] Let Z = (z, s) ∈ Rn = ∂Rn+1

+ , r > 0, and {ωx}x∈Ω be the L-

harmonic measure where L is as given in (3.1.1). Also let ν(Z) be the inward normal vector
at Z (i.e. ν(Z) = (0,··· , 0, 1)), and let Z(cid:48)

r = Z + rν(Z). Then, if X ∈ Rn+1

+ \ B(Z, 4r), we

44

have

G(X, Z(cid:48)

r) (cid:39) ωX (∂Ω ∩ B(Z, r))

r2

w(B(x, r))

= r2−N−aωX (∂Ω ∩ B(Z, r))

In other words, we can also say that ∃C1, C2 > 0 such that

C1r2−N−aωX (∂Ω ∩ B(Z, r)) ≤ G(X, Z + rν(Z)) ≤ C2r2−N−aωX (∂Ω ∩ B(Z, r))

which is exactly the analogous result to [3, Lemma 2.2] that we need (stated in this work as

2.1.5).

3.3. D-to-N maps of the weighted equation (main result and

proof )

The main goal of this chapter and the previous one had been to modify the theorem 2.1.15

to include the degenerate elliptic equations with weights. The work of [14] already confirms
that the D-to-N map I associated with the equation (3.1.1) can be represented as the integro-

differential operator

I(φ, x) = (cid:104)b(x),∇φ(x)(cid:105) −

(cid:90)

Rn

φ(x + h) − φ(x) − 1B1(x)(h)(cid:104)∇φ(x), h − x(cid:105) µ(x, dh). (3.3.1)

We aim to prove the properties of the measure µ in the following main result:

Theorem 3.3.1. Suppose L is the divergence operator as in (3.1.1) with Ω = Rn+1
and
I is as defined via (2.1.2) and (2.1.3), then it is known that there exists a vector field, b,
and a family of measures parametrised by x, µ(x, dh) such that ∀φ ∈ C1,α(Rn), we have the

+

representation in (3.3.1). The measure µ in this expression satisfies

45

(a) µ is absolutely continuous with respect to the Lebesgue measure, i.e. for all X = (x, 0) ∈
Rn, µ(x,·) has a density, µ(x, dh) = K(x, h)(dh). (We will use the notation (x, 0) = x
since we are talking about points on the boundary, Rn.)

(b) There exist universal constants C1, C2 > 0 so that ∀x, h ∈ Rn, x (cid:54)= h

K(x, h) ≤ C2|x − h|−n+a−1.

Proof. First, we will want to know if µ is comparable to the Lebesgue measure on Rn. To
do this, fix X = (x, 0) = x ∈ Rn, show that µ(x,·) is absolutely continuous with respect to
dh on Rn \ {x}. We will do this by showing absolute continuity on the set Rn \ Br(x) for
any arbitrary r > 0, and then we can exhaust Rn \ {x} by a union of such sets. Thus, fix
r > 0 and any set E ⊂ Rn \ Br(x) with |E| = 0. Our goal is to show that µ(E) = 0. We
fix δ > 0, and find a countable cover {B(xj, rj)}∞
(cid:80)∞
in Rn. Now let φ ∈ C2(Rn) be such that 0 ≤ φ ≤ 1∪∞

j < δ, and we will write Bj = B(xj, rj) for brevity. Note that all of these balls are

j=1 of E by open geodesic balls such that

j=1 rn

.

j=1Bj

If δ is sufficiently small, then we will have φ ≡ 0 on Br/2(x). This is because since
E ⊂ Rn \ Br(x), the worst case scenario is that the balls cover all of Rn \ Br(x), and maybe

some part of Br(x). But if δ is small, then the radii of the balls are really small, and so they

won’t cross over too much, and we can have Bj in such a way that none of them intersect
= 0 on Br/2(x), we have φ ≡ 0 there. Thus in (3.3.1),
Br/2(x). Since 0 ≤ φ ≤ 1∪∞
φ(X) = 0, ∇φ(x) = 0. Also, φ(x) = 0 when x ∈ Br/2(x) and so we have

j=1Bj

46

Rn\Br/2(x)

φ(z)µ(z, dz)

(3.3.2)

(cid:90)
(cid:90)

I(φ, x) =

=

φ(z)µ(z, dz)

Rn

(cid:90)

Let {ωX}

X ∈ Rn+1
+ ,

X∈Rn+1

+

be the L-harmonic measure for L given by (3.1.1), and recall for any

Uφ(X) =

φ(z)ωX (dz)

∂Rn

We are now going to employ lemma 3.2.9, and we choose any t > 0 as for the X = x ∈ Rn,

we have X(cid:48) = X + tν(X) ∈ Rn+1

+ always. Then we have

ωX+tν(X)(∂Ω ∩ Brj (xj)) ≤ CrN +a−2

j

G(X + tν(X), Xj + rjν(Xj))

(3.3.3)

Where G is the Green’s function and C only depends on the ellipticity of the equation.

This works because xj are far away enough from x, and so Xj + rjν(Xj) will be far away

enough from X + rν(X). Thus we have the estimate

47

Uφ(X + tν(X)) =

≤

Rn

(cid:90)
∞(cid:88)
φ(z)ωX+tν(X)(dz)
ωX+tν(X)(Rn ∩ Brj (zj))
∞(cid:88)
∞(cid:88)

rN−2+a

j=1

j=1

j

rN−2+a

j

≤ C

≤ C

j=1

≤ Crt1−aδ

(φ ≤ 1∪Bj

)

(from (3.3.3))

G(X + tν(X), Xj + rjν(Xj))

t1−a · r1−a

j

|X + tν(X) − (Xj + rjν(Xj))|N−2+a

(from theorem 3.2.7 (ii))

(the distance between Xj + rjν(Xj) and X + rν(X) is some constant that only depends on r)

From the definition, we have I is the weighted normal derivative, i.e. I(φ, X) = ∂νUφ

I(φ, X) = ta lim
t→0

Uφ(X + tν(X)) − Uφ(X)

t

Crt1−aδ − 0

t1−a

= lim
t→0

= Crδ

But Bj covers E, so we can take a sequence of functions φk ∈ C2(Rn) such that 1E ≤

φk ≤ 1∪Bj

and that they decrease pointwise to 1E, so we have

(cid:90)
(cid:90)
(cid:90)

Rn

Rn

I(φk, X) =

=

≥

Rn\Br/2(X)

φk(Y )µ(X, dY )

φk(Y )µ(X, dY )

(φk ≡ 0 on Br/2(X))

1Eµ(X, dY ) = µ(X, E)

=⇒ µ(x, E) ≤ I(φk, X) ≤ Crδ.

(from earlier)

48

This shows µ is absolutely continuous with respect to the Lebesgue measure, this estab-

lishing part (a) of the theorem.

Now, in order to prove part (b), we use the Poisson kernel. From 3.2.8, we expect

(cid:90)

Uφ(X) =

P (X, Z)φ(Z)dZ

Rn

and we also have an analogous estimate to the Poisson kernel similar to [4, Section 2.4] from

(3.2.5), which is

PX (Z) ≤ C

s1−a

|X − Z|n+1−a

In the following calculation, we denote X + tν(X) = X(cid:48) ∈ Rn+1

+

for X = (x, 0) ∈ Rn,

and thus for X(cid:48) = (x, t) and ξ = (ξ, 0) ∈ Rn we have

PX(cid:48)(ξ) ≤ Ct1−a|X(cid:48) − ξ|−N +a = Ct1−a(cid:16)|x − ξ|2 + t2(cid:17)−n−1+a

2

Thus I(φ, x) = lim
t→0

= lim
t→0

ta−1(cid:0)Uφ(X + tν(X)) − Uφ(X)(cid:1)
(cid:18)(cid:90)
(cid:19)
(cid:90)

PX(cid:48)(Z)φ(Z)dZ

ta−1

t1−a|X(cid:48) − Z|−n−1+adZ

ta−1

Rn

Rn

|X − Z|−n−1+adZ

(cid:90)

≤ lim
t→0
≤ C

Rn

(Since Uφ(X) = φ(X) = 0 and using the definition of the Poisson kernel)

(Since X(cid:48) = X + ν(X), |X − Z| and |X(cid:48) − Z| are comparable)

Also, since X = (x, 0), Z = (z, 0) ∈ Rn, we can say |X − Z| = |x − z|. Now from (3.3.2),

(cid:90)

Rn

I(φ, x) =

(cid:90)

Rn

φ(z)µ(x, dz) ≤ C

49

|x − z|−n−1+aφ(z)dz

The above statement is true for every φ on Rn such that φ ≡ 0 on Br/2(x). Thus, for
E ⊂ Rn \ Br/2(x), we have

(cid:90)

(cid:90)

(cid:90)

E

µ(E) ≤

1Eµ(x, z) =

Rn

Rn

1E|x − z|−n−1+adz =

|x − z|−n−1+adz

Since this is true for every set E ⊂ Rn \ Br/2(x), and we can exhaust Rn by the union over
r of such sets Rn \ Br/2(x), we have proved part (b) of the theorem.

Remark 3.3.2. The above “proof” is comprised of two different proofs of the same result.

We note here that using the estimates on the Poisson Kernel in the second half of the proof

would be enough to shove both parts (a) and (b) of the theorem. However, we also include
an alternate proof of why µ (cid:28) σ in the first half of the above proof.

Remark 3.3.3. In the proof of the above theorem, we have used in the definition of the

weighted normal derivative the inward normal vector which is the natural geometric candi-

date as explained in 3.1.2. However, we reiterate here that this theorem and it’s proof also

works identically using the co-normal derivative defined with the help of the equation in def-

inition 3.1.1. Since the operators we use fulfil the global comparison property, the canonical

derivative to investigate is always the one that comes from the natural geometric normal.

50

Chapter 4

Background on free boundary

problems

4.1. Introduction and background

The second project in this dissertation studies some two-phase free boundary problems that

are similar to Hele-Shaw equations. To do this, we will be generalising some recent work that

used methods from integro-differential parabolic equations. We shall now briefly describe

what these equations are.

The simplest type of such an equation is as follows. Consider a function U : Rn+1 ×
[0, T ] → R, which is harmonic (i.e ∆U = 0) on the sets {U > 0},{U < 0} ⊂ Rn+1 × [0, T ].
These sets have boundary ∂{U > 0} which, for a fixed time t, is a hypersurface in Rn+1, and

may not be regular. Thus these equations have an additional layer of complexity as their

domain and boundary are not fixed; but rather, the domain of the function is an unknown

in the equation and changes with time. There is an additional constraint on u which arises

naturally from either physical principles or energy minimization, and it concerns the balance

of the normal derivatives ∂+

n U + and ∂−

n U− in the inward direction along the free boundary,

which is given by the boundary velocity. More formally, for our two-phase problem, we seek

a function u that solves

51

∆U = 0 in {U > 0} ∪ {U < 0}

(4.1.1)

under the velocity condition

∂{U > 0} moves with normal velocity V = |∇U +| − |∇U−|.

(4.1.2)

This model includes a special case of the stationary two phase problem under the condition

that U does not depend upon time and the balance condition becomes

|∇U +| − |∇U−| = 1.

(4.1.3)

It also includes the one phase version that usually carries the name Hele-Shaw, and that
corresponds to ignoring ∂{U < 0} and setting the velocity condition to be V = |∇U +|.

I work with these canonical examples, and the goal is to find a solution u that satisfies

(4.1.1) above with the condition in (4.1.2). The work presented in this dissertation applies

to more general equations in which the stationary equation for U us given by

F1(D2U ) = 0

F2(D2U ) = 0

V = G(∂+

n U, ∂−

n U )

in {U (·, t) > 0}
in {U (·, t) < 0}
in ∂{U > 0}

(4.1.4)

where F1, F2 are uniformly elliptic rotationally invariant nonlinear operators (see the

section 4.2 for precise definitions). The example stated before in equations (4.1.1)-(4.1.3) is

a special case of the same, where both F1(D2U ) = F2(D2U ) = ∆U .

Numerous works have studied these types of equations and their solutions in the case

52

that U : Rn+1 → R and some of them include [9, 11, 18, 27]. Viscosity solutions give a way

of handling these equations when other techniques fail, and there is substantial literature on

the variants of these equations in [1, 5, 27, 28].

Recently, the work of Chang-Lara, Guillen, and Schwab [9] gives a new technique which

reduces the variants in the family of problems with the condition (4.1.2) to an equivalent
problem of the integro-differential type by considering the hypersurface ∂{U (·, t) > 0} as
the graph of a function f : Rd × [0, T ] → R. Then, for a sufficiently smooth f , the equation

(4.1.1) with the condition in (4.1.2) is equivalent to

∂tf = H(f, x)

on Rn × [0, T ]

f (·, 0) = f0

on Rn

(4.1.5)

where H : C1,γ → Cγ. In our example of (4.1.2), we will have

(cid:113)
f | − |∇U−
f |)

H = (|∇U +

1 + |∇f|2.

This particular H is a form of the Hamilton-Jacobi equation, however, before the methods

of viscosity solutions are applicable to our case, we will need more information about the
structure of |∇U +

f | as operators on f .

f |,|∇U−

In chapters 4 and 5 of this dissertation, we will be looking at the case where the free
boundary set ∂{U (·, t) > 0} is given again by the graph of a function, but that the function
is now a function on Sn × [0, T ] instead of f : Rn × [0, T ], which was the assumption for

the graph in the previous works. This graph assumption is a technical restriction for the

methods herein. In this regard, the point of the work in this part is to extend the work of

[9] to the case of above by expanding this new technique, with the free boundary assumed

53

to be the graph of a function over the sphere Sn instead of Rn. Thus we have that U, f are
functions defined over Sn × [0, T ], and when time t is fixed, then f (·, t) : Sn → [δ,∞). As

such, any previous assumption about translation invariance of the domain or operators now

becomes a matter of rotational invariance.

(4.1.5) is a parabolic equation, but now we shall describe a little more explicitly the

examples of parabolic equations that play a role for the free boundary analysis. In the earlier

chapters regarding degenerate elliptic equations, we considered the canonical operator to be

the 1/2-Laplacian, which can also be considered the canonical integro-differential operator

in this context. Recall that

−(−∆)1/2f (x) = cn

(cid:16)

(cid:90)

Rn

f (x + h) − f (x) − 1B1

(h)|h|−n−1(cid:17)

dh.

The canonical parabolic equation in our context is the 1/2-heat equation given by

∂tf + (−∆)1/2f = g

Just like in the earlier chapters, the Dirichlet to Neumann operator in Rn+1 is given by the
linear operator −(−∆)1/2. Define an operator that is suitable for (4.1.2) using the normal
derivative i.e. f (cid:55)→ I(f ) = ∂νUf . Here Uf is the unique function that solves a one-phase

problem, namely

∆Uf = 0

Uf = 0

Uf = 0

in {0 < y < f (x)}
on {y = f (x)}
on {y = 0}

(4.1.6)

The condition Uf = 1 on {y = 0} can simply be interpreted as there being some background

54

pressure in the system. (As earlier, we denote X ∈ Rn+1 as X = (x, y) where x ∈ Rn, y ∈ R.

The main technique used to show the equivalence of solving (4.1.1) under condition (4.1.2)

with solving (4.1.5) was given in [9]. It turns out that the set up above is in some sense a

non-linear version of the D-to-N mapping, and the key idea is to use that the operator H

has a special structure of the min-max form

H(f, x) = min

i

{aij + cijf (x) + bij ·∇f (x) + p.v.

max

j

(cid:90)

Rn

f (x + h)− f (x)µij(x, dh)} (4.1.7)

for an appropriate family of aij, bij, cij and µij. Equations that admit a similar min-max form

are frequently amenable to a large collection of tools from the viscosity solutions context.

Hence, in the next chapter, we will utilize the solutions of

∂tf − min

i

{aij + cijf (x) + bij · ∇f (x) + p.v.

max

j

f (x + h) − f (x)µij(x, dh)} = g

(cid:90)

Rn

to deduce existence, uniqueness, and some low regularity results for the solutions of the free

boundary problems described above.

The assumption that ∂{U > 0} = graph f is not ideal; we see that it does not appear

as a requirement in [1, 5, 11, 12, 27, 28]. However, it gives a natural reduction to explore
new techniques, especially with an assumption that {U > 0} is a star-shaped domain with

respect to X = 0. Thus it makes sense to expand the ideas in [9] to the case of the functions
u, f defined on the sphere Sn instead of Rn+1.

The advantage of using the approach of (4.1.5) is that it now allows us to generalise these

techniques to many more variations in the free boundary problems described above. One

important variant of the Hele-Shaw type problem is to have ∆U = ρ(U ) with monotone ρ

55

in {U > 0} and the condition V = k(x, t)|∇U +| in (4.1.3); this was used as a semilinear

model for tumor growth [30]. Another equation to study is when ∆U = g(x) in (4.1.1) and

the Batchelor-Prandtl equation is a special case of this with the condition (4.1.2) [9, 16, 17].

4.2. Definitions and examples

Since the main goal of this project is to extend results owing to the connections between

certain parabolic equations and free boundary problems using integro-differential operators,

we use this section to provide definitions of the different equations and operators, as well as

detailed descriptions of the existing results in the field. There is a tremendous amount of

work on integro-differential parabolic equations [6, 7, 8, 10, 32, 33, 34, 35]. For this project,

the main works that contain most of the techniques which we generalize to the context of
rotationally invariant equations on Sn is that of Silvestre [34] and [9].

4.2.1. General Definitions

In this section, we provide a general list of definitions of various concepts, operators, and

their properties which are present in this chapter and the next one. First, we start with the

definitions of some of the properties of the integro-differential operators which appear in the

parabolic equations we study.

Definition 4.2.1. As a function, Rx : Sn → Sn is a rotation such that Rx(0) = x. The
rotation operator Rx, which acts on functions on Sn (say U : Sn → R), is defined for a fixed

x as

RxU (·) := U ◦ Rx(·)

56

Definition 4.2.2. We say that an operator J : C1,γ(Sn) → C0(Sn) is rotationally invariant
if ∀f ∈ C1,γ(Sn) we have J(Rxf, z) = J(f, Rxz).

Definition 4.2.3. (GCP). We say that J : C1,γ(Sn) → C0(Sn) has the global compari-
son property (GCP) if ∀ f, g ∈ C1,γ(Sn) such that f (x) ≤ g(x) ∀x ∈ Sn, and for some
x0, f (x0) = g(x0), the operator J satisfies J(f, x0) ≤ J(g, x0). In other words, J preserves
the ordering of functions on Sn wherever their graphs touch. We also say that J has GCP

at x0 if the above property only holds for one fixed x0, instead of all possible x0.

Recall that we are generally interested in equations that involve uniformly elliptic opera-

tors, and hence we shall define what a uniformly elliptic operator means in this context, for

which we first need to define Pucci operators.

Definition 4.2.4. (Extremal Operators) For a function u that is second-differentiable at X,
the second order (λ, Λ) Pucci operators are defined as M+, M− via

M−(D2U, X) =

min

λId≤B≤ΛId

tr(BD2U (X)) = Λ

M+(D2U, X) =

max

λId≤B≤ΛId

tr(BD2U (X)) = λ

ei + λ

ei + Λ

ei

ei

(cid:88)
(cid:88)

ei≤0

ei≤0

(cid:88)
(cid:88)

ei>0

ei>0

Where D2U is the Hessian matrix of all the second partial derivatives of u, {ei}i=1,2,···d+1
are the eigenvalues of D2U (X).

Definition 4.2.5. (Uniformly Elliptic) We define uniform ellipticity for F that is either

linear of non-linear in the following ways:

• When F is linear, i.e. when F (D2U,∇U ) = tr(AD2U ) + B · ∇U , then we need

||B||L∞ ≤ Λ and λId ≤ A ≤ ΛId to say F is uniformly elliptic, and

57

• when F is non-linear, if for all U, V ∈ C2,

M−(D2U − D2V ) − Λ|∇U − ∇V | ≤ F (D2U,∇V ) − F (D2V,∇V )

≤ M+(D2U − D2V ) + Λ|∇U − ∇V |

The integro-differential form with which we can represent our operators consists of a
linear operator on C1,γ(Sn), but we can later restrict this operator to some special subsets
of C1,γ(Sn) to suit the parabolic equations we work with. We shall now list some definitions

for these special subsets.

Definition 4.2.6. (C1,γ(Ω)) The γ−th H¨older semi-norm of f : Ω → R (where Ω ⊂ Sn is

given by

|f (x) − f (y)|

|x − y|γ

[f ]γ := sup
x,y∈Ω
x(cid:54)=y

The C1,γ-norm of f is given by

||f||C1,γ (Ω) := ||f||L∞ + ||∇f||L∞ + [∇f ]Cγ

The set C1,γ(Ω) consists of all functions f : Ω → R whose C1,γ-norm is finite. Equivalently,

we can also say that

C1,γ(Ω) =

f ∈ L∞(Ω) : sup

z∈Ω

||f − P||L∞(Br(z)) < ∞

 .

r−1−γ

sup
r>0

inf

P (x)=c+p·x
c∈R, p∈Rn

58

Definition 4.2.7. For γ ∈ (0, 1), δ > 0, m > 0, the convex set K(γ, δ, m) is defined as

K(γ, δ, m) := {f ∈ C1,γ(Ω) : f (x) ≥ δ ∀x ∈ Ω and ||f||C1,γ (Ω) ≤ m}

Further, K(γ, δ) =

K(δ) =

(cid:91)
(cid:91)

m>0

(cid:91)

γ∈(0,1)

m>0

K(γ, δ, m) and

K(γ, δ, m).

Definition 4.2.8. (Upper Gradient). Suppose K ∈ C1,γ(Sn) is an open convex subset and
φ : K → R is Lipschitz. Then the upper gradient of φ at f ∈ K in the direction of g ∈
C1,γ(Sn) is defined as

φ0(f ; g) := lim sup

t(cid:38)0

t

φ(f + tg) − φ(f )

This can be viewed as a function φ0 : K × C1,γ(Sn) → R

Definition 4.2.9. (Subdifferential). Let φ be as in definition 4.2.3. The Clarke differential

(or the generalized gradient) of φ at f ∈ K is a subset of(cid:0)C1,γ(Sn)(cid:1)(cid:63)

given by

∂φ(f ) :=

C1,γ(Sn)

(cid:96) ∈(cid:16)
(cid:110)
(cid:91)

f∈K

(cid:17)(cid:63) |∀ψ ∈ C1,γ(Sn), φ0 ≥ (cid:104)(cid:96), ψ(cid:105)(cid:111)


and [∂φ]K := hull

∂φ(f )

(the convex hull)

Definition 4.2.10. (C1,γ-semi-concave) Given γ ∈ (0, 1], m > 0, a Lipschitz function
f : Sn → R is said to be C1,γ-semi-concave with constant m if there is a real-valued function,
r : Sn → R, such that

f (x) = inf
y∈Sn

{r(y) + m|x − y|1+γ}

f is said to be C1,γ-semi-convex with constant m if (−f ) is C1,γ-semi-concave with constant

59

m.

Remark 4.2.11. The original definition above is for functions from Rn → R, but we can
see that we can do an easy replacement of Rn by Sn and the definition still holds. Indeed,
since Sn is a manifold, we have charts from all open subsets U ∈ Sn to B1 ∈ Rn. Let us
call this map ΦU : U (⊂ Sn) → B1 ⊂ Rn for an arbitrary open set. We will use x, y ∈ Rn
and p, q ∈ Sn and let ΦU (p) = x, ΦU (q) = y. Then we can say that f : Sn → R is
C1,γ-semi-concave if f ◦ Φ−1
U : Rn → R is C1,γ-semi-concave by the usual definition. Thus,
∃r(cid:48) : Rn → R, a real-valued function such that

f (p) = f ◦ Φ−1

U (x) = inf
y∈Rn

= inf
q∈Sn

= inf
q∈Sn

{r(cid:48)(y) + m|x − y|1+γ}
(cid:110)
r(cid:48)(ΦU (q)) + m|ΦU (p) − ΦU (q)|1+γ(cid:111)
(cid:110)
(r(cid:48) ◦ ΦU )(q) + m · C|p − q|1+γ(cid:111)

Thus, we now have a real-valued function r = r(cid:48) ◦ ΦU : Sn → R that satisfies the condition
in definition 4.2.5, with the constant m · C > 0, where C depends on the Lipschitz constant

of ΦU .

Definition 4.2.12. (Pointwise or punctually C1,γ). Let γ ∈ (0, 1] and m > 0 be fixed.
We say that f is pointwise m − C1,γ at x0 (denoted f ∈ m − C1,γ(x0)) if ∇f (x0) exists,
|f (x0)| ≤ m, |∇f (x0)| ≤ m, and ∃r > 0 such that

∀x ∈ Br(x0),|f (x) − f (x0) − ∇f (x0) · (x − x0)| ≤ m|x − x0|1+γ

60

4.2.2. Some examples of free boundary problems

To support the motivation behind studying the two-phase free boundary problem on the

sphere, we shall give some examples of free boundary problems, their D-to-N maps, and the

parabolic equations associated with them.

(a) One phase Hele-Shaw on the half-space

As before, we will use the notation X = (x, y) for any X in the half space Rn+1
+ , with
x ∈ Rn, y > 0. We consider functions f : Rn → R which are continuous, non-negative,

bounded and bounded away from zero. To any such function, we can associate a

domain

Df = {(x, y) ∈ Rn+1

+ , 0 < y < f (x)}

We can also define the hypersurface Γf , which is the graph of f as

Γf = {(x, y) ∈ Rn+1

+ , y = f (x)}

Now let Uf : Df → R be the unique bounded solution to the Dirichlet problem

∆Uf = 0

Uf = 1

Uf = 0

in Df
on {y = 0}, i.e. Rn

on Γf

(4.2.1)

Now, if ν is the inward facing unit normal to Γf (i.e. it points towards Df ), then we

61

V = ∂+

ν U

∆U = 0

Df

U = 1

Γf

y = 0

Figure 4.1: One phase Hele-Shaw on the half-space

define the normal derivative ∂+

ν U as follows for X0 ∈ Γf ,

∂+
ν U = lim
t→0+

U (X0 + tν(X0)) − U (X0)

t

, and I(f, x) := ∂+

ν Uf (x, f (x)).

(4.2.2)

Now, we can put a time evolution into the setup to make a free boundary problem, i.e.
consider U : Rn+1 × [0, T ] → R, which solves

∆U = 0

U = 1

V = ∂+

ν U

in {u > 0}
on {y = 0}, i.e. Rn
on ∂{u > 0}

(4.2.3)

(4.2.3) is the one-phase Hele-shaw problem on the upper half space.

For a sufficiently smooth f , U = Uf is a solution of (4.2.2) if and only if f is a solution

to the parabolic equation

∂tf := I(f, x)

(cid:113)

1 + |∇f|2, on Rn × [0, T ]

(4.2.4)

62

U− = −1

∆U− = 0

∆U + = 0

U + = 1

D−

f

D+
f

y = L

Γf

y = 0

V = |∂+

ν U| − |∂−
ν U|

Figure 4.2: Two phase Hele-Shaw on an infinite strip

where I is as given in (4.2.2), and it is is translation invariant, non-linear, non-local,

and it enjoys GCP.

(b) Two phase Hele-Shaw on an infinite strip

+ . Now for every f : Rn → R such that f is
We first fix an upper boundary L > 0 in Rn+1
continuous, non-negative, and bounded away from 0 and L, i.e. 0 < δ ≤ f ≤ L−δ < L.
We then define two domains for {U > 0} and {U < 0}

f = {(x, y) ∈ Rn+1
D+
D−
f = {(x, y) ∈ Rn+1
and Γf = {(x, y) ∈ Rn+1

+ , 0 < y < f (x)},
+ , f (x) < y < L},
+ , y = f (x)}

Now we suppose U + : D+

f → R and U− : D−

f → R are respectively the unique bounded

63

solution to the Dirichlet problem

∆U± = 0
U± = ±1
U± = 0

in D±
on {y = 0} and {y = L} respectively

f

(4.2.5)

on Γf

Then a Lipschitz function G : (0,∞)2 → R which satisfies the monotonicity condition

λ0 ≤ ∂
∂a

G(a, b),

G(a, b) ≤ Λ0

∂
∂b

will give the normal velocity of the flow depending on ∂±
0}. ∂+

ν U is as given above in (4.2.2), and ∂−

ν U is given by

ν U along the boundary ∂{U >

∂−
ν U = − lim
t→0+

U (X0 − tν(X0)) − U (X0)

t

(4.2.6)

Now, we can talk about the two-phase free boundary problem along an infinite strip

as follows:

∆U = 0

U = 1
U = −1

in {U (cid:54)= 0}
on {y = 0}, i.e. Rn
on {y = L}
on ∂{U > 0}

(4.2.7)

ν U, ∂−
ν U )
ν U (x, f (x)), ∂−
If we define H(f, x) as H(f, x) = G(∂+

V = G(∂+

ν U (x, f (x)), then similarly as in

the one-phase problem, a sufficiently smooth f solving the parabolic equation

(cid:113)
∂tf := H(f, x)

1 + |∇f|2, on Rn × [0, T ]

(4.2.8)

64

is equivalent to U = Uf solving the free boundary problem (4.2.7).

(c) Prandtl-Batchelor as a non-linear integro-differential equation on the sphere.

Finally, we come to an example where the free boundary is not given by the graph of a
function over Rn but by the graph of a function over a sphere. The Prandtl-Batchelor

is a two dimensional model in fluid mechanics that models a vortex patch occupying a

convex region, the patch being surroundedby a steady flow. The interface of the vortex

patch is what plays the role of the free boundary. We denote the stream function of

the flow by u, and it solves the following equation

∆U = 0

ν U|2 − |∂−
|∂+

∆U = 1
ν U|2 = 1

in {U > 0}
in {U < 0}
on ∂{U > 0}

(4.2.9)

Now consider any function f : Sn → R, which is non-negative, bounded, and bounded
away from zero, and satisfies 0 < f (x) ≤ f (x) for a given positive and continuous

function f . Thus we can define the sets

D−
f := {x ∈ Rn + 1 : x = re, e ∈ Sn, 0 ≤ r < f (x)}
f := D−
D+

\ D−

f

f

65

Suppose U±

f : D±

f → R give the unique solution to the Dirichlet problem

∆U− = 1
U− = 0

∆U + = 0

U + = 0

U + = 1

in D−
on ∂D−

f

f

(4.2.10)

in D+
f
on ∂D−
on ∂D−

f

f

and we define an operator P : C2(Sn) → C0(Sn) as P(f, x) := |∂+
ν U (x, f (x))|2 −
|∂−
ν U (x, f (x))|2. P admits GCP, and if there is a sufficiently smooth f : Sn → R
solving P(f, x) = 1 in x ∈ Sn, then the radial graph of f is the boundary of a vortex
patch in a Batchelor-Prandtl flow, and U±

f gives the positive and negative phases of

the stream function. Since the Laplacian operator and the boundary conditions on
the domain are both rotationally invariant, the operator P will also remain invariant
under the action of rotations on C2(Sn).

4.3. Results to be modified

Here we will state some of the main results from [9], which are about the equation (4.1.4).

There is a notion of weak solutions for both the free boundary problems and the parabolic

equations and these are called viscosity solutions. We will define these notions in detail later,

in sections 5.4.1 and 5.3.1 respectively.

Theorem 4.3.1. [9, Theorem 1.1] If F1, F2 are uniformly elliptic and rotationally invariant
in the Hessian variable, G is Lipschitz and monotone, and f : Rd × [0, T ] → [δ,∞), Uf :

66

Rd+1 × [0, T ] → R are such that

∀t ∈ [0, T ], Γ(t) = ∂{Uf (·, t) > 0} = graph f (·, t),

then,

(a) Uf is a viscosity solution of the free boundary evolution, (4.1.4) if and only if f is a

viscosity solution of the fractional parabolic equation (4.1.5).

(b) If additionally, graph f (·, t) = ∂{Uf (·, t)} enjoys a modulus of continuity, i.e. |f (x, 0) −
f (y, 0)| ≤ ω(|x − y|), then ∀t ∈ [0, T ] this modulus is preserved for f (·, t) and hence for
∂{Uf (·, t).

(c) If graph f (·, t) = ∂{Uf (·, t)} enjoys a modulus of continuity, i.e.

|f (x, 0) − f (y, 0)| ≤

ω(|x − y|), then there exists a unique viscosity solution to both (4.1.4) and (4.1.5).

We refer the reader to [9] for the proof of the above theorem. The main tool used in

the analysis of the above results is what could be called a non-linear version of the D-to-N
operator I, defined via the inward normal derivative of Uf which is given by

I(f, x) := ∂νUf (x, f (x))

(4.3.1)

for the one-phase problem. For the two phase problem, we define the D-to-N operators as

I+(f, x) := ∂+

ν Uf (x, f (x)), I−(f, x) := ∂−

ν Uf (x, f (x))

One of the results that leads to the proof of theorem 4.3.1 is the property of I which we

list in the following result.

67

Theorem 4.3.2. [9, Theorem 1.4] If F is uniformly elliptic and rotationally invariant in
the Hessian variable, I is as in (4.3.1), and γ ∈ (0, 1) is fixed, then ∃γ(cid:48) with 0 < γ(cid:48) < γ so

that

I : K(γ, δ, m) → Cγ(cid:48)

(Rd)

and I is locally Lipschitz. Also, I enjoys the following representation for aij, bij, cuj, µij(dh)

which are all independent of x:

(cid:110)

aij + cijf (x) + bij · ∇f (x)

f (x + h) − f (x) − 1B1

(h)∇f (x) · h

(cid:17)

(cid:27)

µij(dh)

I(f, x) = min

(cid:90)

i

max

j

(cid:16)

Rd

+

The bounds of aij, bij, cuj, µij(dh) all depend on the bounded set

{f : f ≥ δ,||f||

C1,γ (Rd)

≤ m}.

Furthermore, ∃C > 0 such that |aij|,|bij| ≤ C, −C ≤ cij ≤ 0, and

(cid:90)

min

Rd

(cid:110)|h|1+γ, 1
(cid:111)

(cid:90)

Rd

µij(dh) ≤ C and

µij(dh) ≤ C

The proof of the above result can also be found in [9]. Inorder to prove these properties
about I and subsequently H as given in (4.1.5), one can study a general class of operators

that satisfies certain conditions, which we outline in the assumption below.

Assumption 4.3.3.

(i) 0 < γ < 1 is fixed.

68

(cid:32)(cid:83)

(cid:83)

δ>0

m>δ

(cid:33)

K(γ, δ, m)

(ii) J :

→ C0(Rd).

(iii) For each δ and m, J is a Lipschitz mapping on the sets K(γ, δ, m), whose Lipschitz

constant, C(δ, m) increases as δ decreases or m increases.

(iv) J satisfies GCP.

(v) J is translation invariant.

(vi) If f ∈ K(γ, δ, m) and c > 0 is a constant, then

∀x ∈ Sn, J(f + c, x) ≤ J(f, x).

(vii) J enjoys the operator splitting property: ∃C = C(γ, δ, m), and a modulus of continuity

ω with ω(R) → 0 as R → ∞ such that for all f, g ∈ K(γ, δ, m), for R > 1,

||J(f,·) − J(g,·)||L∞(BR) ≤ C

(cid:16)||f − g||C1,γ (B2R) + ω(R)||f − g||

L∞(Rd)

(cid:17)

+ ω(R)

This operator J will enjoy a special min-max structure, and we will state the relevant

result below. However, first, the proof for deriving this structure relies on the following the

results from [25].

Theorem 4.3.4. [25, Theorem 1.10] If I : C2

b (Rd) → C0

b (Rd) (bounded functions in C2, C0,

respectively) satisfies the conditions (iii), (iv), and (v) of Assumption 4.3.3, then there exists
a family {fab, Lab}a,b∈K(I), that depends only on I, where for all a, b, fa,b are constants, and

69

Lab are linear translation invariant operators mapping I : C2

b (Rd) → C0

b (Rd) of the form

L(u, x) =Cu(x) + B · ∇u(x) + tr(AD2u(x))

(cid:90)

u(x + h) − u(x) − 1B1(0)(h)∇u(x) · hµ(dh)

+

Rd

i.e. with constant coefficients.
Further, ∀u ∈ C2

b (Rd) and x ∈ Rd we have

I(u, x) = min

a

{fab + Lab(u, x)}.

max

b

Finally, for a universal C, and for all fab and Lab, we have

(cid:90)

|fab| + |Aab| + |Bab| + |Cab| +

min{1,|h|2}µab(dh) ≤ C||I||

Lip,C2
b

.

→C0
b

(4.3.2)

Rd

The above theorem relies on the following fact from [25], which elaborates on the char-

acterization of linear functionals (whose codomain is R) which have the GCP.

Lemma 4.3.5. [25, Lemma 3.9] Assume that β ∈ [0, 3). let (cid:96) : Cβ
linear functional which has the GCP with respect to 0. For β ≥ 2 and any u ∈ Cβ

b (Rd) → R be a bounded
b (Rd) ∩

C2(0), we have the representation

(cid:90)

(cid:104)l, u(cid:105) =C(cid:96)u(0) + (B(cid:96),∇u(0)) + tr(A(cid:96)D2u(0))

u(h) − u(0) − χB1(0) (cid:104)∇u(0), h(cid:105) µ(cid:96)(dh)

+

Rd

This representation is unique, which means that if there were ˜C, ˜B, ˜A, and ˜µ a measure in

70

Rd \ {0} all such that

(cid:104)l, u(cid:105) = ˜Cu(0) + ( ˜B,∇u(0)) + tr( ˜AD2u(0))

(cid:90)

u(h) − u(0) − χB1(0) (cid:104)∇u(0), h(cid:105) ˜µ(dh)

+

Rd

for all u, then ˜C = C(cid:96), ˜B = B(cid:96), ˜A = A(cid:96), and ˜µ = µ(cid:96). Furthermore, if β < 2 and u ∈
Cβ(Rd) ∩ C1(0), then A(cid:96) = 0, and if β < 1 then B(cid:96) = 0 and the integrand on the right can
be replaced by just u(h) − u(0).

The proof of the theorem 4.3.4 above employs a lot of information from [25, Lemma 3.6]
and its proof, particularly the part which says that for β ∈ [1, 2) (as it is in the case of C1,γ,
which is the case relevant to us) and u ∈ Cβ(Rd) ∩ C1(0),

(cid:104)l, u(cid:105) = C(cid:96)u(0) + (B(cid:96),φ,∇u(0)) +

where Pφ,η,u is given by

(cid:90)

Rd

u(h) − Pφ,η,uµ(cid:96)(dh),

(4.3.3)

Pφ,η,u,x(·) = u(x) + φ(· − x)(∇u(x),· − x)

if β ∈ [1, 2).

(4.3.4)

Finally, we state the result from [9] about the structure of the operator J that satisfies

the assumption 4.3.3.

Theorem 4.3.6. (Translation invariant min-max) If J follows all the conditions in the

assumption 4.3.3, then J admits a min-max representation as follows:

∀f ∈ K(γ, δ, m), J(f, x) = min

g∈K(γ,δ,m)

max

L∈L(K(γ,δ,m))

{J(g, x) + L(f − g, x)}

(4.3.5)

71

in which each L is also translation invariant. This means L ∈ Linv which is the class that

contains all linear operators of the form

L(f, x) =cf (x) + b · ∇f (x)

(cid:90)

+

Rd

f (x + h) − f (x) − 1B1

(h)∇f (x) · hµ(dh),

(4.3.6)

where each of c, b, µ are independent of x.

Furthermore, given a C1 > 0, ∃C2 > 0 such that for all J with a Lipschitz norm bounded

by C1, all such c, b, µ resulting from an L ∈ Linv, we have |c|,|b| < C2, and

(cid:90)

Rd

(cid:90)

Rd\BR

max{|h|1+γ, 1}µ(dh) ≤ C2, and

µ(dh) ≤ C2ω(R).

(4.3.7)

The class of operators, Linv in (4.3.6) and the min-max in (4.3.5) both depend on γ, δ, m via
K(γ, δ, m).

72

Chapter 5

Parabolic equations and free

boundary problems

5.1. Introduction to the problem and results

The goal of this chapter is to reproduce the results from [9] as mentioned above in section

4.3; particularly theorems 4.3.1 and 4.3.2, but this time for the specific equation given in

(4.1.4) with the following details:

(a) F1 = F2 = ∆,

(b) G = |∇U +

f | − |∇U−
f |,

(c) and for functions f defined over the sphere Sn instead of Rd.

Thus, consider the following free boundary problem for U : Rn+1 → R

∆U = 0
V = |∇U +| − |∇U−|

in {U (·, t) > 0} and {U (·, t) < 0}
in ∂{U > 0}

(5.1.1)

Once again, we consider the hypersurface ∂{U (·, t) > 0} as the graph of a function f , but
this time f : Sn × [0, T ]. Then, for a sufficiently smooth f , the equation (5.1.1) is equivalent

73

to

∂tf = H(f, x) = (|∇U +

(cid:113)
f |)
f | − |∇U +

f (·, 0) = f0
where H : C1,γ(Sn) → Cγ(Sn).

5.1.1. Main results

1 + |∇f|2

on Sn × [0, T ]

(5.1.2)

on Sn

Theorem 5.1.1. For the equations given above, if we have f : Sn × [0, T ] → [δ,∞), Uf :
Sn × [0, T ] → R are such that

∀t ∈ [0, T ], Γ(t) = ∂{Uf (·, t) > 0} = graph f (·, t),

then,

(a) Uf is a viscosity solution of the free boundary evolution, (5.1.1) if and only if f is a

viscosity solution of the fractional parabolic equation (5.1.2).

(b) If additionally, graph f (·, t) = ∂{Uf (·, t)} enjoys a modulus of continuity, i.e. |f (x, 0) −
f (y, 0)| ≤ ω(|x − y|), then ∀t ∈ [0, T ] this modulus is preserved for f (·, t) and hence for
∂{Uf (·, t)}.

(c) If graph f (·, t) = ∂{Uf (·, t)} enjoys a modulus of continuity, i.e.

|f (x, 0) − f (y, 0)| ≤

ω(|x − y|), then there exists a unique viscosity solution to both (5.1.1) and (5.1.2).

Once again, the precise definitions of the notions of viscosity solutions that are mentioned

in these results will be given later, in sections 5.4.1 and 5.3.1. Now we can define the D-to-N

74

operator I, via the inward normal derivative of Uf which is given by

I+(f, x) := ∂+

ν Uf (x, f (x)), I−(f, x) := ∂−

ν Uf (x, f (x))

for the two-phase problem. Here ∂+

ν Uf are as defined in (4.2.2), (4.2.6) respectively.

ν Uf , ∂−

Since the gradient of Uf at a point is orthogonal to the level curve of Uf at the same point,

we have that the normal to the curve at point is given by the gradient. i.e. we have

ν = ± ∇Uf
|∇Uf|,

and since ∂νUf = ∇Uf · ν, we have ∂νUf = |∇Uf|.

Similarly, ∂+

ν Uf = |∇U +

f |, and ∂−

ν Uf = |∇U−
f |.

This of course, establishes the connection between I and H, and hence one of the results
that leads to the proof of theorem 5.1.1 is the property of I which we list in the following

result.

Theorem 5.1.2. If I is as given above, and γ ∈ (0, 1) is fixed, then ∃γ(cid:48) with 0 < γ(cid:48) < γ so

that

I : K(γ, δ, m) → Cγ(cid:48)

(Sn)

and I is locally Lipschitz. Also, I enjoys the following representation for aij, bij, cuj, µij(dh)

which are all independent of x:

75

(cid:110)

(cid:90)

i

j

(cid:16)

+

Sn

I(f, x) = min

max

aij + cijf (x) + bij · ∇f (x)

f (x + h) − f (x) − 1B1

(h)

(cid:68)∇f (x), exp−1

x (h)

(cid:69)(cid:17)

(cid:27)

µij(dh)

The bounds of aij, bij, cuj, µij(dh) all depend on the bounded set {f : f ≥ δ,||f||C1,γ (Sn) ≤

m}. Furthermore, ∃C > 0 such that |aij|,|bij| ≤ C, −C ≤ cij ≤ 0, and

(cid:90)

min

Sn

(cid:110)|h|1+γ, 1
(cid:111)

(cid:90)

Sn

µij(dh) ≤ C and

µij(dh) ≤ C

We shall give the proofs of these results in section 5.4. However, we will give a brief

outline of the steps involved in the proof.

(1) We will first study a general class of operators acting on the functions on the sphere

(with some reasonable level of smoothness). The connection of these operators to the

problem at hand is that the H described in (5.1.2) will belong to this class of operators.
Moreover, this H depends on the D-to-N map, I. We will analyse these operators in

terms of their enjoyment of the min-max structure, which is the result given by [9,

Theorem 1.4], modified here as 5.1.1.

(2) Next, we will study the existence and uniqueness properties of viscosity solutions to

parabolic equations on the sphere. These equations will involve the operators mentioned

in the previous step, as given by (5.1.2).

(3) We will go on to show how that the viscosity solutions of the two different equations

(free boundary problem, i.e. (5.1.1) and the parabolic equation, i.e. (5.1.2)) produce

76

an equivalent outcome, accounting for the correspondence between the free boundary in

(5.1.1) and the graph of the function f from (5.1.2). This is the equivalence of solutions

listed in [9, Theorem 1.1], modified here as 5.1.2.

(4) Finally, we will prove that a modulus of continuity for the initial data will be propagated

for all time. This will lead us to part (c) of the theorem 5.1.1

The work in this chapter will focus primarily on steps (1) and (2), and the detailed expla-

nations of steps (3) and (4) can be directly modified from [9], although we given a brief

explanation in section 5.4.

5.1.2. Details of two phase free boundary problem on the sphere

Here we give a few more technical details of the equations in question along with some
pictures. Consider any function f : Sn → R, which is non-negative, bounded, and bounded
away from zero. Thus, we have r0, δ, L > 0 so that 0 < r0 < δ ≤ f ≤ L − δ < L. Now

consider the following sets:

f := {X ∈ Rn+1 : X = re, e ∈ Sn, 0 ≤ r < f (x)}
D+
D−
f := {X ∈ Rn+1 : X = re, e ∈ Sn, f (x) ≤ r < L}
Γ−
f := {X ∈ Rn+1 : X = re, e ∈ Sn, r = f (x)}

Now suppose we have a function Uf : Sn × [0, T ] → R that solves the following free

77

∆Uf = 0

D+
f

∆Uf = 0
D−

f

f (e)

e

Sn

Uf = 1

Uf = −1

BL

V = |∂+

ν Uf| − |∂−

ν Uf|

Figure 5.1: Two phase free boundary problem on the sphere

boundary problem

∆Uf = 0

Uf = 0

Uf = 1
Uf = −1
V = |∂+

ν Uf| − |∂−

ν Uf|

in D±

f

on Γf

on Bδ(0)

on BL(0)

on Γf

Now consider the parabolic equation:

∂tf =(cid:0)|∂+

ν Uf| − |∂−

ν Uf|(cid:1)(cid:113)

1 + |∇f|2 on Sn × [0, T ]

(5.1.3)

(5.1.4)

The main result adapted from [9] describes the connection between the parabolic equa-

tions and the (two-phase) free boundary problem as given in the equations above.

78

Note that the one-phase problem is given by

∆Uf = 0

Uf = 1

in Df

in Br0

Uf = 0

on graph f

(5.1.5)

5.2. Integro-differential representations of certain operators

Before we prove the results for the specific two-phase free boundary problem in this work, we

will first consider a broad abstract class of operators that fulfil certain conditions. Such an

operator will appear in the parabolic equation as given in (4.1.5). In the next two sections, we

will prove a min-max representation for the operators, existence and uniqueness of solutions,

as well as the comparison principle for the equations involving these abstract operators. In

the final section 5.4, we will elaborate on why the operators in two phase equation we have

chosen definitely falls into the category of operators we will describe in this section, and as

such it will be evident why all the subsequent results will be applicable to (5.1.1), (5.1.2) as

well.

Assumption 5.2.1. We look at the broad set of operators that act on functions on Sn and

satisfy the following assumptions:

(i) 0 < γ < 1 is fixed.

(cid:32)(cid:83)

(cid:83)

(cid:33)

(ii) J :

K(γ, δ, m)

→ C0(Sn).

δ>0

m>δ

(iii) For each δ and m, J is a Lipschitz mapping on the sets K(γ, δ, m), whose Lipschitz

constant, C(δ, m) increases as δ decreases or m increases.

79

(iv) J satisfies GCP.

(v) J is rotational invariant.

(vi) If f ∈ K(γ, δ, m) and c > 0 is a constant, then

∀x ∈ Sn, J(f + c, x) ≤ J(f, x).

In this section we will first record some of the relevant results about the structure and
properties of J, but in the case when J acts on functions in Rn. All of the results below can

be found in [25].

Proposition 5.2.2. If (cid:96) : K(γ, δ, m) → R is a bounded linear function that enjoys GCP at
x0 = 0, then ∃b, c ∈ R and a measure µ so that ∀f ∈ K(γ, δ, m)

(cid:96)(f ) = cf (0) + b · ∇f (0) +

0 (h)

µ(dh),

(cid:90)

Sn

(f (h) − f (0) −(cid:68)∇f (0), exp−1

(cid:69)

where b, c, and µ satisfy the same bounds as in the above theorem, but none of them depend

on x.

This result for functions in Rn and its proof can both be found in [24, Lemma 3.6].

However, for functions defined on the sphere, the Taylor polynomial given in (4.3.4) will now

change to

Pφ,η,u,x(·) = u(x) + φ(Rx(·))(∇u(x), exp−1

x (·))

The techniques for the proof in [24, Lemma 3.6] will follow in a straightforward manner. We
also note that since φ ∈ C1,γ(Sn), the term with the gradient (∇u(x), exp−1
x (·)), will decay

80

at the rate of |x − ·|1+γ, which is integrable because we will have

(cid:90)

Sn

max{|h|1+γ, 1}µ(dh) ≤ C.

Theorem 5.2.3. (Rotationally invariant min-max) If J follows all the assumptions above,

then J admits a min-max representation as follows:

∀f ∈ K(γ, δ, m), J(f, x) = min

g∈K(γ,δ,m)

max

L∈L(K(γ,δ,m))

{J(g, x) + L(f − g, x)}

(5.2.1)

in which each L is also rotationally invariant. This means L ∈ Linv which is the class that

contains all linear operators of the form

L(f, x) =cf (x) + b · ∇f (x)

f (Rx(h) − f (x) −(cid:68)∇f (x), exp−1

x (h)

(cid:69)

µ(dh),

(5.2.2)

(cid:90)

+

Sn

where each of c, b, µ are independent of x. Furthermore, given a C1 > 0, ∃C2 > 0 such that
for all J with a Lipschitz norm bounded by C1, all such c, b, µ resulting from an L ∈ Linv,
we have |c|,|b| < C2, and

max{|h|1+γ, 1}µ(dh) ≤ C2.

(5.2.3)

(cid:90)

Sn

The class of operators, Linv in (5.2.2) and the min-max in (5.2.1) both depend on γ, δ, m via
K(γ, δ, m).

The consequences of the above theorem are as follows:

81

• Following as in [9], this identifies a natural class of extremal operators for J as

M +

inv = max
L∈Linv

L(f, x), and M−

inv = min
L∈Linv

L(f, x)

(5.2.4)

Just by the definition, we can use the min-max expression for J and write

J(f, x) − J(g, x) = min

{J(g(cid:48), x) + L(f − g(cid:48), x)} − J(g, x)

g(cid:48) max

L

≤ max

L

L(f − g, x)

(Taking g(cid:48) = g in the minimum)

and J(f, x) − J(g, x) = J(f, x) − min

{J(f(cid:48), x) + L(g − f(cid:48), x)}

L

f(cid:48) max
L(g − f, x)

≥ − max

L

(Taking f(cid:48) = f in the minimum)

L(f − g, x)

= min

L

These extremal operators are defined specifically to produce, ∀f, g ∈ K(γ, δ, m), the

following inequalities:

M−
inv(f − g, x) ≤ J(f, x) − J(g, x) ≤ M +

inv(f − g, x)

(5.2.5)

We note that because of the rotational invariance we have that

∀x ∈ Rn, M±

inv(f, x) = M±(f ◦ Rx, 0)

• Secondly, due to the rotational invariance of J, we see that all of the desired prop-
erties of Linv can be obtained from studying the (possibly non-linear) functional
j : K(γ, δ, m) → R with j(f ) := J(f, 0).

82

This functional, j, also satisfies GCP based at x0 = 0. So once J is fixed, the class L

can be further restricted to include only those L that are in the Clarke differential of

j (from definition 4.2.4), i.e. those L such that

L(f, x) = (cid:96)(f ◦ Rx) for some choice of (cid:96) ∈ [∂j]K(γ,δ,m)

Thus, we can now define a new extremal operator, depending explicitly on J and
K(γ, δ, m) as

M +

J,K(γ,δ,m)(f, x) =

and M−

J,K(γ,δ,m)(f, x) =

max

(cid:96)∈[∂j]K(γ,δ,m)

min

(cid:96)∈[∂j]K(γ,δ,m)

((cid:96)(f ◦ Rx))

((cid:96)(f ◦ Rx))

(5.2.6)

Now for the operator given above, we note that the inequalities in (5.2.4) still hold, they

are again, rotation invariant, and they also serve as extremal operators of J, which we talk

about in the result below.

Proposition 5.2.4. If J is fixed, and M±
M±
J,K(γ,δ,m) also obey the inequalities in (5.2.5), and M±
mappings from C1,γ(Sn) → C0(Sn) with a Lipchitz norm bounded by that of J.

J,K(γ,δ,m) are defined as in (5.2.6) above, then J and

J,K(γ,δ,m) are Lipschitz functions, as

Proof. Firstly, from [13, Proposition 2.1.2], we know that [∂j]K(γ,δ,m) (cid:54)= ∅. This is a result

of the Hahn-Banach Theorem, which asserts that every positively homogeneous and subad-

ditive functional on a vector space majorises a linear functional on the vector space. In this
case, the vector space being C1,γ(Sn) × C1,γ(Sn) and j0 is the positively homogeneous and
subadditive functional, so there exists some linear functional (cid:96) so that for all ψ ∈ C1,γ(Sn),

83

j0(f ) ≥ (cid:104)(cid:96), ψ(cid:105), which means [∂j]K(γ,δ,m) (cid:54)= ∅ as claimed.

Secondly, j and ∂j enjoy a mean value property due to a theorem by Lebourg [13,
Proposition 2.3.7], which says that for any points x, y ∈ X and f Lipschitz on an open set
containing the line segment [x,y], ∃ u ∈ (x, y) such that f (y) − f (x) ∈ (cid:104)∂f (u), y − x(cid:105). Thus,
for f, g ∈ K(γ, δ, m) and j Lipschitz, we can say that there is an element, (cid:96) ∈ [∂j]K(γ,δ,m)
such that j(f ) − j(g) = (cid:96)(f − g). Now we take a max over [∂j]K(γ,δ,m), and again from [13,
Proposition 2.1.2], [∂j]K(γ,δ,m) is weak-(cid:63) compact, thus giving us

∀f, g ∈ K(γ, δ, m),

j(f ) − j(g) ≤

max

(cid:96)∈[∂j]K(γ,δ,m)

(cid:96)(f − g)

A similar argument gives us the lower inequality in (5.2.5).

To see why M± are Lipschitz, consider

J,K(γ,δ,m)(f1, x) − M +
M +

J,K(γ,δ,m)(f2, x)

=

≤

max

(cid:96)∈[∂j]K(γ,δ,m)

max

(cid:96)∈[∂j]K(γ,δ,m)

max

((cid:96)(f1 ◦ Rx)) −

(cid:96)∈[∂j]K(γ,δ,m)
{((cid:96)(f1 ◦ Rx)) − ((cid:96)(f2 ◦ Rx))}

((cid:96)(f2 ◦ Rx))

= max
c,b,µ

(cid:90)
−(cid:68)

+

x ∇f1(x) − R−1
{c(f1(x) − f2(x)) + b · [R−1
(cid:68)

f1(Rx(h)) − f2(Rx(h)) − f1(x) + f2(x)
Sn
x ∇(f1(x)), exp−1
R−1

x ∇(f1(x)), exp−1
R−1

x ∇f2(x)]
(cid:69)

0 (h)

+

0 (h)

(cid:69)

µ(dh)}

Owing to the fact that f1, f2 ∈ K(γ, δ, m) and due to the bound on µ given in (5.2.3),

84

all the terms above are bounded up to a constant by the C1,γ-norm of f1 − f2.

Proof of theorem 5.2.3:

Proof. From the previous proposition, we can see that

∀f, g ∈ K(γ, δ, m), j(f ) ≤

max

(cid:96)∈[∂j]K(γ,δ,m)

(j(g) + (cid:96)(f − g))

Thus j(f ) ≤ min

g∈K(γ,δ,m)

max

(cid:96)∈[∂j]K(γ,δ,m)

(j(g) + (cid:96)(f − g))

(by taking minimum over g)

On the other hand, by letting g = f in the minimum,

min

g∈K(γ,δ,m)

max

(cid:96)∈[∂j]K(γ,δ,m)

(j(g) + (cid:96)(f − g)) ≤ j(f )

Since J is rotation invariant, we see that J(f, x) = j(Rxf ).
Now, any (cid:96) ∈ [∂j]K(γ,δ,m) enjoys the comparison principle, and thus we have that (cid:96) has a

representation as in (5.2.2), and thus the proof follows.

Corollary 5.2.5. For each γ, δ, and m fixed, we can extend the operator J (with respect to
the set K(γ, δ, m)) to a function on all of C1,γ(Sn). We define

˜JK(γ,δ,m) : C1,γ(Sn) → C0(Sn)

via ˜JK(γ,δ,m)(f, x) = min

g∈K(γ,δ,m)

max

(cid:96)∈[∂j]J,K(γ,δ,m)

{j(g) + (cid:96)((f ◦ Rx) − g)}

(5.2.7)

˜J is Lipschitz, it enjoys GCP, and ˜J = J on K(γ, δ, m). (It also satisfies all of the other

assumptions.)

85

Proof. We can see that ˜J is Lipschitz because M + is Lipschitz and from the following:

˜J(f1) − ˜J(f2) =

min

h∈K(γ,δ,m)

max

(cid:96)∈[∂j]J,K(γ,δ,m)

{j(h) + (cid:96)((f1 ◦ Rx) − h)}

− min

g∈K(γ,δ,m)

max

k∈[∂j]J,K(γ,δ,m)

{j(g) + k((f2 ◦ Rx) − g)}

=

min

h∈K(γ,δ,m)

max

(cid:96)∈[∂j]J,K(γ,δ,m)

{j(h) + (cid:96)((f1 ◦ Rx) − h)}

−

≤

−

≤

max

k∈[∂j]J,K(γ,δ,m)

{j(g(cid:63)) + k((f2 ◦ Rx) − g(cid:63))}

(Assume g = g(cid:63) is the optimizer for the second minimum)
{j(g(cid:63)) + (cid:96)((f1 ◦ Rx) − g(cid:63))}

max

(cid:96)∈[∂j]J,K(γ,δ,m)

max

k∈[∂j]J,K(γ,δ,m)

{j(g(cid:63)) + k((f2 ◦ Rx) − g(cid:63))}

(setting h = g(cid:63) in the first min)

max

(cid:96)∈[∂j]J,K(γ,δ,m)

{j(g(cid:63)) + (cid:96)((f1 ◦ Rx) − g(cid:63)) − j(g(cid:63)) − (cid:96)((f2 ◦ Rx) − g(cid:63))}

((cid:96)(f1 ◦ Rx) − (cid:96)(f2 ◦ Rx))

≤

max

(cid:96)∈[∂j]J,K(γ,δ,m)
= M +(f1 − f2)

Further, ˜J satisfies GCP because j, (cid:96) satisfy GCP at x0 = 0.

Corollary 5.2.6. If x is fixed, and γ, δ, m are given, and f ∈ m − C1,γ(x), ˜JK(γ,δ,m)(f, x)

is classically defined via (5.2.6).

Proof. Since f is m − C1,γ(x), then by definition, there are two C1,γ functions f + and f−
such that they touch f above and below respectively at x. In other words, f± ∈ C1,γ(Rn),
and ∀y, f−(y) ≤ f (y) ≤ f +(y) with f−(x) = f (x) = f +(x). By Proposition 5.2.2, we know

86

that if a bounded linear functional (cid:96) has GCP at a point then (cid:96)(Rxf ) = L(f, x). We can

work this out explicitly from

(cid:69)

µ(dh)

0 (h)

(cid:96)(f ◦ Rx) =c(f ◦ Rx)(0) + b · ∇(f ◦ Rx)(0)

b, R−1

x ∇f (x)

(f ◦ Rx)(h) − f ◦ Rx(0) −(cid:68)∇(f ◦ Rx)(0), exp−1
(cid:68)
(cid:69)
f (Rxh) − f (x) −(cid:68)
f (Rxh) − f (x) −(cid:68)∇f (x), exp−1

x ∇f (x), exp−1
R−1
(cid:69)

0 (h)

x (h)

µ(dh)

µ(dh)

(cid:69)

=cf (x) + (cid:104)Rx · b,∇f (x)(cid:105)

+

Sn

=cf (x) +

(cid:90)
(cid:90)
(cid:90)

+

Sn

+

Sn

=L(f, x),

with the corresponding bL = Rx · b and the same c. Thus the formula in (5.2.7) holds

classically, i.e. as in theorem 5.2.3 with the estimate on µ that appears in 5.2.3.

Proposition 5.2.7. If J is as in Assumption 5.2.1 and M +

J,K(γ,δ,m) is as defined in (5.2.5)

then

∀x ∈ Rn, M +

J,K(γ,δ,m)(1, x) ≤ 0

As a result, we see that ∀L ∈ Linv, we have c ≤ 0 in (5.2.2).

Proof. Note that in (5.2.2), L(1, 0) = c. Because of the rotation invariance of M +

J,K(γ,δ,m),

it is enough to show that M +

J,K(γ,δ,m)(1, 0) ≤ 0.

It can also be observed here that the

consequence of the result follows from having

c = L(1, 0) = max
(cid:96)∈[∂j]

(cid:96)(R01) = M +

J,K(γ,δ,m)(1, 0)

87

Suppose we write m0 to denote the Lipschitz functional

m0(f ) = M +

J,K(γ,δ,m)(f, 0) = max
(cid:96)∈[∂j]

(cid:96)(f ), where j = j(f ) = J(f, 0).

Assumption 5.2.1 says that J(f + c, x) ≤ J(f, x) for c > 0 and f ∈ K(γ, δ, m). Thus

j0(f ; 1) = lim sup
x→0+

By definition, ∂j(f ) := {(cid:96) ∈(cid:16)

C1,γ(Rn))

and [∂j]K := hull

j(f + s · 1) − j(f )

≤ 0.

s

(cid:17)(cid:63) |∀ψ ∈ C1,γ(Rn), j0(f ; ψ) ≥ (cid:104)(cid:96), ψ(cid:105)}
(cid:91)



∂j

f∈K

Thus, if (cid:96) ∈ [∂j]K(γ,δ,m), then (cid:96)(1) ≤ 0. So from the definition of M +
can say that M +

J,K(γ,δ,m)(1, 0) = m0(1) ≤ 0.

J,K(γ,δ,m) in (5.2.5), we

5.3. Comparison theorem and existence for parabolic viscosity

solutions on the sphere

5.3.1. Using the assumptions of section 5.2

Now, we look at operators of the type J that satisfy the above assumptions, and for such J,

we prove uniqueness for the following equation:

∂tf = J(f )

in Sn × (0, T ]

f (·, 0) = f0

on Sn

(5.3.1)

Definition 5.3.1. We say that f is a viscosity subsolution of (5.3.1) if f is upper semi-

88

continuous, inf f > 0, and if f has the following property: For all (x, t) ∈ Sn × (0, T ] for
which there exists a function φ ∈ C1,γ+β(Sn) in space and C1 in time (for some β > 0 with
γ + β ≤ 1), and δ ≤ φ ≤ L− δ such that for some t > r > 0, f − φ attains a global maximum
over Sn × (t − r, t], then φ must satisfy

∂tφ(x, t) ≤ J(φ, x)

On the other hand, we say g is a viscosity supersolution if g is lower semi-continuous,
inf g > 0, and if we replace the above properties with g − φ attains a minimum and

∂tφ(x, t) ≥ J(φ, x)

Of course, f is a viscosity solution if is satisfies both of the above.

Lemma 5.3.2. If f is a viscosity subsolution of (5.3.1), and suppose (x, t) ∈ Sn × (0, T ) is
a point such that f − φ attains a maximum at the point (x, t) for some φ that is punctually
C1,γ+β(x) in space and C1 in time, with β > 0, γ + β ≤ 1. Then there exists a choice of δ0
small enough and m0 large enough such that ˜JK(γ,δ0,m0)(φ, x) is classically defined and

∂tφ(x, t) ≤ ˜JK(γ,δ0,m0)(φ, x)

Remark 5.3.3. The analogous result holds for g that are supersolutions and for those φ we

also require them to additionally satisfy inf φ > 0. Note that this is not a problem with the
subsolution because f − φ ≤ 0 =⇒ δ ≤ f ≤ φ =⇒ inf φ > 0.

Remark 5.3.4. ˜J is defined in (5.2.6) and we need this because φ is not necessarily in the

89

set, K(γ, δ0, m0). Also, this result will hold for all δ < δ0 and m > m0.

Proof. The well defined nature of ˜JK(γ,δ0,m0)(φ, x) is because of Corollary 5.2.6, from which
we can also see that we only need φ ∈ C1,γ(x) and not γ + β. The extra regularity will come

in later.

Now, since φ is punctually C1,γ+β at x, this means there are two C1,γ+β functions

f +, f−, such that

∀y ∈ Rn, f−(y) ≤ φ(y) ≤ f +(y) and f−(x) = φ(x) = f +(x)

Now for each r > 0, we can define a function φr such that

f +(y) if y ∈ Br(x)

otherwise.

φr(y) =

φ(y)

Thus, we also have the ordering f− ≤ φ ≤ φr ≤ f + and
∇f−(x) = ∇φ(x) = ∇φr(x) = ∇f +(x) (they all touch at the point x). Finally, we can also

assume wlog that

∂tφ(x, t) = ∂tf +(x, t)

f + ∈ C1,γ+β(Rn) is a function such that f − f + attains a maximum at (x, t). Now,
f + ≥ φ and inf φ > 0, and also, f + ∈ C1,γ+β. These facts imply that there exists some δ0
and m0 such that f + ∈ K(γ, δ0, m0) = K (we call is so for simplicity). Thus by definition of

90

viscosity subsolution,

∂tφ(x, t) = ∂tf +(x, t) ≤ J(f +, x) = ˜JK(f +, x)

Now we use Corollaries 5.2.5 and 5.2.6, which tell us that ˜J has many of the properties of J

and satisfies many of the same inequalities.

˜JK(f +, x) ≤ ˜JK(φr, x) + M +
≤ ˜JK(φ, x) + M +

J,K(f + − φr, x)
J,K(φr − φ, x) + M +

(since ˜J also satisfies (5.2.4))

J,K(f + − φr, x)

All of the above operators are well defined on these functions, as they are all punctually
C1,γ(x). If we can show that the last two terms → 0 as r → 0, then we complete the proof.

This is where we will need the slightly higher regularity. Now following the definitions:

J,K(φr − φ, x) = max
M +
(cid:96)∈[∂j]K

(cid:96)((φr − φ) ◦ Rx) ≤ max
L∈Linv

L(φr − φ, x)

91

Also, from Theorem 5.2.3, we obtain the following bounds:

L((φr − φ, x) =c((((((((

(φr(x) − φ(x)) − b · ((((((((((
(∇φr(x) − ∇φ(x))
(cid:90)
(cid:90)

φr(Rx(h)) − φr(x) − 1Br(x)
φ(Rx(h)) − φ(x) − 1Br(x)

(cid:68)∇φr(x), exp−1
(cid:68)∇φ(x), exp−1

x (h)

Sn

+

−

(cid:69)

Sn

(cid:69)

x (h)

µ(dh)

µ(dh)

(at x, we have φr = φ,∇φr = ∇φ)

(cid:90)

≤

Br(x)

(Br(x) since outside of this set, ∇φr − ∇φ = 0)

Cφ|h|1+γ+βµ(dh)

(Cφ is the constant depending on φ, φr ∈ C1,γ+β)

We can make a similar statement about f + − φr. Now, for each L, there is a µ and hence
M +
J,K is a max over a family of the measures µ, arising from the set [J]K, and each of these

µ satisfy the uniform bound in (5.2.3). Suppose we say

meas(K) := {µ|∃(cid:96) ∈ [∂j]K s.t. µ corresponds to (cid:96) as in Proposition 5.2.2}

Then we can say that we have the following bounds

J,K(φr − φ, x) + M +
M +

J,K(f + − φ, x) ≤

(cid:90)

Cφ|h|1+γ+βµ(dh)
(cid:90)

|h|1+γµ(dh)

Br

≤ Cφrβ

Br

(cid:90)

Br

≤ Cφrβ

sup

µ∈meas(K)

≤ C2 · Cφ · rβ

92

min{|h|1+γ, 1}µ(dh)

Hence, as r → 0, we see that M +

J,K(φr − φ, x) + M +

J,K(f + − φ, x) → 0 and thus we have

proved the lemma.

We now propose that the inf and sup convolutions that are appropriate for the parabolic

equation on Sn × [0, T ] are the following:

φε(x, t) :=

sup

y=Rzx,d(z,0)≤c0,t(cid:48)∈[0,T ]

φ(y, s) − 1
2ε

φε(x, t) :=

y=Rzx,d(z,0)≤c0,t(cid:48)∈[0,T ]

inf

φ(y, s) +

1
2ε

(cid:16)||x − y||2 + |t − t(cid:48)|2(cid:17)
(cid:16)||x − y||2 + |t − t(cid:48)|2(cid:17)

(5.3.2)

(5.3.3)

Another way to write these (and a notations we will often use) is as follows:

φε(x, t) :=

sup

d(z,0)≤c0,s∈[−t,T−t]

φ(Rzx, t + s) − 1
2ε

φε(x, t) :=

inf

d(z,0)≤c0,s∈[−t,T−t]

φ(Rzx, t + s) +

1
2ε

(cid:16)||Rz||2 + |s|2(cid:17)
(cid:16)||Rz||2 + |s|2(cid:17)

Lemma 5.3.5. We note the following properties of the sup/inf-convolutions:

(i) The sup-convolution given in (5.3.2) is semi-convex and the inf-convolution in (5.3.3)

is semi-concave.

(ii) the sup and inf convolution are Lipschitz functions.

(iii) Both φε and φε converge to φ pointwise as ε → 0.

(iv) If u and v are a subsolution and supersolution to the equation ∂tu(x, t) = J(u, x) in Sn

respectively, then uε and vε are also a subsolution and supersolution respectively to the

same equation.

93

Proof.

(i) It is easy to see from the definition directly that the inf-convolution, φε is semi-

concave. We can write the sup-convolution as

φε(x) = −

inf

d(z,0)≤c0,s∈[0,T ]

−φ(Rzx, s) +

1
2ε

(cid:16)||x − Rzx||2 + |t − s|2(cid:17)(cid:27)

Thus, −φε is semi-concave by definition, which makes φε semi-convex. We can also see
this from knowing that φε + |X|2/2ε is convex (where X = (x, t)). Similarly, we can

see that φε is semi-concave.

(ii) Consider for X1 = (x1, t1), X2 = (x2, t2) ∈ Sn × [0, T ], and let Y ε

1 = (y1, s1) =

(Rzεx1, s1) be the point at which the supremum is realized for X1, or

φε(x1, t1) = φ(Rzεx1, s1) − 1
2ε

(cid:16)||x1 − Rzεx1||2 + |t1 − s1|2(cid:17)

.

On the other hand, we also have

(cid:16)||x2 − Rzx2||2 + |t2 − s|2(cid:17)(cid:27)

φε(x2, t2) =

sup

d(z,0)≤c0,s∈[0,T ]
≥ φ(Rzεx1, s1) − 1
2ε

φ(Rzx2, t2) − 1
2ε

(cid:16)||x2 − Rzεx1||2 + |t2 − s1|2(cid:17)

where zε

1 is such that Rzx2 = Rzε
1

x1 for some z ∈ Bc0. Thus, we can write

φε(x1, t1) − φε(x2, t2) ≤(((((((

(cid:26)

(cid:26)

(cid:16)||x1 − Rzε
x1||2 + |t1 − s1|2(cid:17)
(cid:16)||x2 − Rzεx1||2 + |t2 − s1|2(cid:17)

1

φ(Rzεx1, s1) − 1
2ε
− (((((((
φ(Rzεx1, s1) +
≤ C1|x1 − x2| + C2|t1 − t2|
≤ C||(x1, t1) − (x2, t2)||

1
2ε

94

(C = C(Sn, T ))

thereby showing that φε is Lipschitz. We can similarly show that φε is Lipschitz.

(iii) This proof is based on the one given in [15, Lemma A.5].

Assume ψ is an appropriate function that touches φε from above at (x, t) = X, and that
Dψ(x, t) = q ∈ Rn+1, D2ψ(x, t) = M ∈ Sd+1, the set of symmetric (d + 1) × (d + 1)
matrices. We will first show that this means ∃ some ψ(cid:48) that touches φ at X + qε and

φε(X) +

ε
2

|q|2 = φ(X + qε)

Indeed, suppose Y ε

(x,t) is the point such that the supremum is achieved in the definition

of φε(X). We will use the notation Y ε

(x,t) = Y ε = (yε, tε), i.e.

φε(x, t) = φ(yε, tε) − 1
2ε

(||yε − x||2 + |tε − t|2)

From similar calculations as in [15], we get Y ε = X + qε. Thus, we have

φε(X) +

ε
2

|q|2 = φ(Y ε) − 1
2ε

ε
2
|q|2

|X − Y ε|2 +

|q|2

= φ(X + qε) − ε
2

|q|2 +

ε
2

= φ(X + qε)

95

We also know that |q|ε → 0 when ε → 0 because

φ(X) ≤ φε(X)

= φ(Y ε) − 1
2ε

|X − Y ε|2

=⇒ 1
2ε

|X − Y ε|2 ≤ φ(Y ε) − φ(X) ≤ 2||φ||L∞

=⇒ |X − Y ε| ≤(cid:112)4ε||φ||L∞ → 0 as ε → 0

Finally, we see that

(cid:110)

lim sup
ε→0

φε(X) +

|q|2(cid:111)

ε
2

= lim sup
ε→0
≤ φ(X)
≤ lim sup
ε→0

φ(X + qε)

φε(X)

(φ is USC)
(since we have φε ≥ φ)

This forces lim supε→0

ε

2|q|2 = 0, and eventually, we can see that

lim sup
ε→0

φε(X) = lim sup
ε→0

φ(X + qε) − ε
2

|q|2 = φ(X)

(iv) Suppose u, v are respectively the subsolution, supersolution of the given equation, i.e.,

we have in the viscosity sense, that

∂tu(x, t) ≤ J(u, x), and ∂tv(x, t) ≥ J(v, x).

We shall use u◦Rz to denote the function u◦Rz(x, t) = u(Rzx, t), where Rz : Sn → Sn is
the rotation on the sphere that takes 0 to z. Note that we have J(u◦Rz, x) = J(u, Rzx)

96

since J is a rotation invariant operator. So as u is a subsolution, we should have

∂t(u ◦ Rz)(x, t) = ∂tu(Rzx, t) ≤ J(u, Rzx) = J(u ◦ Rz, x).

(5.3.4)

Now, suppose φ is the required function such that uε − φ has its maximum at a point
(x0, t0) = X0, i.e., we have uε ≤ φ and uε(x0, t0) = φ(x0, t0). Suppose that in the
definition of uε(x, t), zε, sε are the parameters so that the supremum is realized for

(x, t), and in particular, zε

0, sε

0 are the parameters for the supremum in the definition

of uε(x0, t0). Then we have

uε(x0, t0) = u(Rzε
0

x0, t0 + sε

0) − 1
2ε

Further, ∀(x, t) ∈ Sn × [0, T ], since uε is the sup

(cid:16)||Rzε

0

0|2(cid:17)

||2 + |sε

φ(x, t) ≥ uε(x, t) ≥ u(Rzε

0

x, t + sε

and φ(x0, t0) = uε(x0, t0) = u(Rzε
0

x0, t0 + sε

(cid:16)||Rzε
(cid:16)||Rzε

0

0

0) − 1
2ε
0) − 1
2ε

0|2(cid:17)
0|2(cid:17)

.

||2 + |sε
||2 + |sε

Let φ(cid:48) = φ + cε, where cε = 1/2ε(||Rzε

0

||2 + |sε

0|2).

From earlier, we have ∂tu ◦ Rzε

0

(x, t) ≤ J(u ◦ Rzε

0

, x).

97

Note that φ(cid:48) touches u ◦ Rzε

0

(· + (0, sε

0)) from above at x0, i.e. we have

φ(cid:48)(x, t) ≥ u(Rzε
and φ(cid:48)(x0, t0) = u ◦ Rzε

0

(x0, t0 + sε
0)

0

x, t + sε

0) = u ◦ Rzε

0

(x, t + sε
0)

Since u is the viscosity subsolution to a rotation invariant equation, we have from
(5.3.4) ∂tφ(cid:48)(x0, t0) ≤ J(φ(cid:48), x0). Also, since cε (or sε

0) is constant, the function uε solves

in the viscosity sense

∂t(uε)(x, t) = ∂t(uε + cε)(x, t) ≤ J(uε + cε, x) ≤ J(uε, x)

from the properties of J. We have already seen from the proof of part (iii) that cε → 0
as ε → 0. See previous proof.

We also need to know how the family of operators described in Assumption 5.2.1 will

operate on rescaled versions of smooth bump functions.

Lemma 5.3.6. For x0 ∈ Sn, if we define smooth functions Φ and ΦR as

Φ(x) =

|x − x0|2
1 + |x − x0|2 , and ΦR(x) = Φ

(cid:16) x

(cid:17)

R

Then, for a fixed J, given and δ > 0, m > δ, ρ > 0,∃R > 1, with R = R(J, ρ, δ, m), so that

J,K(γ,δ,m)(ΦR, x) ≤ ρ
M +

sup
x∈Sn

Proof. First, we will observe the following facts about ΦR

98

0 ≤ ΦR(x) ≤ 1 and ||∇ΦR||L∞ ≤ C
R

(cid:12)(cid:12)(cid:12)ΦR(Rhx) − ΦR(x) −(cid:68)∇ΦR(x), exp−1
(1 + |x − x0|2)2 =⇒ |∇Φ| ≤ 1 =⇒ |∇ΦR| ≤ C

2(x − x0)

x (h)

R

(cid:69)(cid:12)(cid:12)(cid:12) ≤ C|h|2

R2

Note that ∇Φ =

The first fact is owing to the fact that since |∇Φ| ≤ 1, we have |∇ΦR| ≤ C/R when we

rescale. The second fact follows from the Taylor expansion of ΦR.

Now, from the estimates in Theorem 5.2.3, we can choose R large enough so that, uni-

formly across [∂J]K(γ,δ,m)

(cid:90)

Sn

(cid:12)(cid:12)(cid:12)ΦR(Rhx) − ΦR(x) − 1B1(x)(h)

(cid:68)∇ΦR(x), exp−1

(cid:69)(cid:12)(cid:12)(cid:12) µ(dh)

x (h)
|h|2
R2 µ(dh) ≤ ρ

2

(cid:90)

≤

C

Sn

Next, given this t we can choose R large enough so that uniformly across b ∈ [∂j]K(γ,δ,m)
(actually, it is (cid:96) ∈ [∂j]K(γ,δ,m), but there is a one to one correspondence between b and (cid:96)
from Proposition 5.2.2), we have b·∇ΦR ≤ ρ/2. Finally, thanks to Proposition 5.2.7, we see
that for any of the constants, c, that appear in (5.2.2), we have cΦR(x) ≤ 0.

Now if we add everything up, we get (cid:96)(f ◦ Rx), and if we take the max then by the

definition of M +

J,K(γ,δ,m), we have the required inequality after taking the supremum.

99

Lemma 5.3.7. Given any δ, ρ, C > 0, m > δ, ∃R = R(J, ρ, δ, m) so that

∀h > 0, ∀(x, t) ∈ Sn × [0, T ], Ψ(x, t) = C + hΦR(x) + hρt

is a classical, strict supersolution of

∂tΨ > M +

J,K(γ,δ,m)(Ψ)

Proof. It is a direct calculation to show that

J,K(γ,δ,m)(C + hΦR, x) ≤ hM +
M +

JK(γ,δ,m)(ΦR)

Indeed, L(C + hΦR)

=c(C + hΦR) + hb · ∇ΦR

(cid:90)

hΦR(Rxy) − hΦR(x) − 1B1(x)(y)h

+

Sn

(cid:68)∇ΦR, exp−1

x (y)µ(dy)

(cid:69)

=c · C + hL(ΦR, x) ≤ hL(ΦR, x)

(since c ≤ 0 by Proposition 5.2.7 and C > 0 is given)

Now we invoke Lemma 5.3.5 with ρ/2 and get

M +

J,K(γ,δ,m)(Ψ) = M +

J,K(γ,δ,m)(C + hΦR) ≤ hM +

J,K(γ,δ,m)(ΦR) ≤ hρ/2 < ∂t(Ψ)

100

5.3.2. Comparison results

Lemma 5.3.8. If δ > 0, m > δ are fixed, and w : Sn × [0, T ] → R is a bounded, upper

semi-continuous function such that, in the viscosity sense,

∂tw ≤ M +
w+(·, t) ≤ sup
Sn

J,K(γ,δ,m)(w)
w+(·, 0)

then,

sup

Sn×[0,T ]

Proof. First, we will start the proof by assuming supSn w+(·, 0) = 0. Then, we need to
show that supSn×[0,T ] w+ ≤ 0. Suppose we assume, for the sake of contradiction, that
supSn×[0,T ] w+ > 0. Since for h, C, ρ > 0, Ψ as described in Lemma 5.3.6 is strictly above 0,
we can choose h and C in a way so that w+ − Ψ attains a global maximum for some t > 0.

If we use Ψ as a test function and apply the definition of a viscosity subsolution for w+, we
get a contradiction i.e. ∂tΨ ≤ M +

J,K(γ,δ,m)(Ψ), which contradicts what we proved in Lemma

5.3.7.

On the other hand, suppose we do not necessarily assume that supSn w+(·, 0) = 0. Now

we can replace w by the function

˜w = w − sup
Sn

w+(·, 0)

Thus, let c = supSn w+(·, 0) ≥ 0, we see that w = ˜w + c, and so in the viscosity sense,
∂tw = ∂t ˜w. Also, since c ≥ 0, by Proposition 5.2.7 (which tells us that c ≤ 0 in the

101

expression for L), just like in the proof of Lemma 5.3.7, we get

M +

J,K(γ,δ,m)(w) = M +

J,K(γ,δ,m)( ˜w + c) ≤ M +

J,K(γ,δ,m)( ˜w)

Hence, in the viscosity sense, we have

∂t ˜w = ∂tw ≤ M +

J,K(γ,δ,m)(w) ≤ M +

J,K(γ,δ,m)( ˜w)

and now, we can apply the first case again since we have supSn ˜w+(·, 0) = 0, which gives us

with the assumption above

˜w+(·, t) ≤ 0

sup

Sn×[0,T ]

=⇒ sup

Sn×[0,T ]

w+(·, t) ≤ sup
Sn

w+(·, 0)

Proposition 5.3.9. Suppose f, g : Sn × [0, T ] → R are bounded, and for some δ > 0, f, g ≥

δ, and they are respectively a subsolution and supersolution in the viscosity sense, of the

equation ∂t(u) = J(u), then

f ε(x, 0) ≤ gε(x, 0) =⇒ f ε(x, t) ≤ gε(x, t) ∀(x, t) ∈ Sn × [0, T ],

(5.3.5)

where f ε, gε are as their definitions given in (5.3.2), (5.3.3).

Proof. We first note from lemma 5.3.5 that f ε, gε are given in (5.3.2) and (5.3.3), then the

102

following inequalities are satisfied in the viscosity sense:

∂tf ε ≤ J(f ε), and ∂tgε ≥ J(gε)

Let ε ∈ (0, 1) be fixed, and we assume ∀x ∈ Sn, f ε(x, 0) − gε(x, 0) ≤ 0. We want to show

that

f ε(x, t) − gε(x, t) ≤ 0 ∀(x, t) ∈ Sn × [0, T ]

we will assume for the sake of contradiction that there exists a time t(cid:63) ∈ (0, T ] such that

sup

(x,t)∈Sn×[0,T ]

(f ε(x, t) − gε(x, t)) = sup
x∈Sn

(f ε(x, t(cid:63)) − gε(x, t(cid:63))) = m0 > 0

Note that this cannot happen at t = 0 because of the assumption in this case (when t = 0,
we have f ε − gε ≤ 0). Since ε is fixed, and m0 is given, we can invoke Lemma 5.3.6 with
m = cε−1 and C = cm0, i.e. have some Ψ = cm0 + hΦR + hρt (where R = R(J, δ, ρ, cε−1))

as a strict supersolution.
classically for all (x, t) ∈ Sn × [0, T ]

i.e. we know Ψ exists such that the following inequality holds

∂tΨ > M +

J,K(γ,δ/2,cε−1)

(Ψ)

Now we can translate Ψ around so it touches f ε − gε from above. Since we are assuming
that f ε − gε attains a positive supremum, and Ψ is always positive, we know they can touch
if we translate,i.e. we can choose C so that ∃(xε

0) ∈ Sn × (0, T ] such that

0, tε

f ε − gε ≤ Ψ, and (f ε − gε)(xε

0, tε

0) = Ψ(xε

0, tε
0)

103

By definition, for any ε > 0, f ε and −gε are semi-convex, which means that they both

have tangent paraboloids of opening 1/ε touching from below. Thus this is also true for
f ε − gε. Now since a smooth function Ψ touches f ε − gε from above at the point (xε
we have that both f ε and −gε must be C1,1 at the point (xε
∂tf ε, ∂tgε,∇f ε,∇gε classically at (xε
0). Thus, by Lemma (5.3.2), we see that the equations
hold classically for f ε and −gε. For f ε, we use gε +Ψ as the test function, and for gε, the test
function is f ε − Ψ. Furthermore, by Proposition 5.2.4 and Lemma 5.3.2, we get classically

0). Thus, we can evaluate

0, tε
0),

0, tε

0, tε

∂tf ε − ∂tgε ≤ ˜JK(γ,δ,cε−1)(f ε, xε

0) − ˜JK(γ,δ,cε−1)(gε, xε
0)

≤ M +

J,K(γ,δ,cε−1)

(f ε − gε, xε
0)

≤ M +

J,K(γ,δ,cε−1)

(Ψ, xε
0)

< ∂tΨ

(Since M +

J,K(γ,δ,cε−1)

enjoys GCP.)

Since we know that at (xε

0), we have ∂tf ε − ∂tgε = ∂tΨ, this gives us a contradiction. So

0, tε

we can conclude that for each ε fixed,

sup

(x,t)∈Sn×[0,T ]

(f ε(x, t) − gε(x, t)) ≤ 0

i.e. ∀(x, t) ∈ Sn × [0, T ], f ε(x, t) ≤ gε(x, t)

Corollary 5.3.10. As a consequence of the above proposition, we have that if f, g are a

subsolution and supersolution respectively of ∂tu = J(u), and f ε, gε are the sup-convolution

104

and inf-convolution respectively, then

sup

(x,t)∈Sn×[0,T ]

[f ε(x, t) − gε(x, t)] ≤ sup
x∈Sn

[f ε(x, 0) − gε(x, 0)]

(5.3.6)

Proof. We assume that supx∈Sn (f ε(x, 0) − gε(x, 0)) = cε > 0. Now let

˜wε(x, t) = f ε(x, t) − gε(x, t) − cε

Since cε is the supremum of f ε − gε at 0, we obviously have ˜wε(x, 0) ≤ 0, and now we are

back to the case in the previous proposition. Using the same argument as before leads us to

the conclusion that

sup

(x,t)∈Sn×[0,T ]

[f ε(x, t) − gε(x, t) − cε] ≤ 0

i.e.

sup

(x,t)∈Sn×[0,T ]

[f ε(x, t) − gε(x, t)] ≤ sup
x∈Sn

[f ε(x, 0) − gε(x, 0)]

Remark 5.3.11. We want to be able to make similar statements like (5.3.5), (5.3.6) about

f, g. However, though we know that f and g are LSC and USC respectively, we do not

know about the regularity of these functions yet. Thus if in (5.3.6) we take the limit as
ε → 0, the limit commutes with the supremum on the left (since it is the supremum over all
(x, t) ∈ Sn × [0, T ]) but not on the right because we are on the boundary. Thus we get

sup

(x,t)∈Sn×[0,T ]

[f (x, t) − g(x, t)] ≤ sup
x∈Sn

[f ε(x, 0) − gε(x, 0)] ,

105

when what we really want is

sup

(x,t)∈Sn×[0,T ]

[f (x, t) − g(x, t)] ≤ sup
x∈Sn

[f (x, 0) − g(x, 0)] ,

In order to take limits on the right of (5.3.6) we will need the extra assumption in the

following proposition.

Proposition 5.3.12. Let f and g be a subsolution and supersolution of (5.3.1) respectively,
and assume f (−, 0) ≤ g(−, 0) in a somewhat uniform way, i.e. ∀ε > 0,∃ δ > 0 s.t.

f (x, t) ≤ g(y, s) + ε

∀|x − y| < δ,|t|,|s| < δ.

(5.3.7)

Then

lim
ε→0

sup
x∈Sn

f ε(x, 0) − gε(x, 0) ≤ 0.

Proof. Let ρ > 0. Then we will show that

lim sup
ε→0

sup
x∈Sn

f ε(x, 0) − gε(x, 0) ≤ ρ

By the definitiosn of f ε, gε, we know that ∃ , xε, tεxε, tε such that

f ε(x, 0) = f (xε, tε) +

gε(x, 0) = g(xε, tε) −

(cid:18) 1
(cid:18) 1

2ε

2ε

(cid:19)
(cid:19)

|xε − x|2 +

|xε − x|2 +

1
2ε

1
2ε

|tε|2

|tε|2

Since f ε → f, gε → g as ε → 0, this means that entire term in parentheses above → 0. Also,
xε, xε → x and tε, tε → 0. Now for ε = ρ/3, ∃δ > 0 so that (5.3.7) holds.

106

Now we can choose ε small, and |xε − x|, |xε − x|, |tε|, |tε| are all < δ. Thus we get

f ε(x, 0) ≤ f (xε, tε) + ρ/3

≤ g(xε, tε) + ρ/3 + ρ/3
≤ gε(x, 0) + ρ/3 + ρ/3 + ρ/3.

(as f ε → f )

(because of (5.3.7))
(as gε → g)

Putting together all the above inequalities, we can take the supremum over x ∈ Sn and then
the limit supremum as ε → 0 to get

lim sup
ε→0

sup
x∈Sn

[f ε(x, 0) − gε(x, 0)] ≤ ρ.

Since ρ > 0 was arbitrary, we have

f ε(x, 0) − gε(x, 0) ≤ 0.

lim
ε→0

sup
x∈Sn

Theorem 5.3.13. Let f, g be a subsolution and supersolution respectively of (5.3.1), and
assume f (x, 0) ≤ g(x, 0) on Sn in a somewhat uniform way, i.e. suppose (5.3.7) holds. Then
f (x, t) ≤ g(x, t) ∀(x, t) ∈ Sn × [0, T ].

The proof of the theorem follows easily if you consider remark 5.3.11, thus by putting

together the results from the previous two propositions.

Remark 5.3.14. If we need to start the previous theorem with just the assumption that
f (x, 0) ≤ g(x, 0) ∀ x ∈ Sn, then we need to additionally assume that f, g are bounded and

uniformly continuous so that (5.3.7) holds.

107

5.3.3. Existence (Perron’s method)

The proof for existence will follow just like in [34] (which comes from ideas in [26], [15]),

with a minor modification just like in [9]. In the Perron method, the solution is given by

u(x) = supv∈Sϕ v(x), where Sϕ consists of all subsolutions. This u will be a subsolution in
the interior, it may not match the values at the boundary, i.e. u(y) = ϕ(y),∀y ∈ ∂Ω.

Definition 5.3.15. If ∃ y ∈ ∂Ω such that there is supersolution wy and wy(y) = 0 whereas
wy(x) > 0 ∀x (cid:54)= y, then such y are called regular points, and the function wy is called a
barrier function. At such points we also get u(x) → ϕ(y) as x → y.

We will sketch the proof for existence for (5.3.1). But first, we will list some results and

definitions from [26]. Note that we will always have f, g ∈ C(Sn × (0, T ]).

Proposition 5.3.16. [26, Proposition 2.3] If f is a subsolution and g is a supersolution of
(5.3.1), f ≤ g on Sn×(0, T ], then ∃ a viscosity solution h such that f ≤ h ≤ g on Sn×(0, T ].

Next, we will give the definitions of the relaxed supremum/infimum of a function.

Definition 5.3.17. [26, Section 2] Let Ω ⊂ Rn be open. Then for any function f : Ω → R

we can define

f (cid:63)(X) = lim sup

r↓0

{f (Z) : Z ∈ B(X, r)} for X ∈ Ω

f(cid:63)(X) = lim inf
r↓0

{f (Z) : Z ∈ B(X, r)} for X ∈ Ω

These functions are upper semicontinuous and lower semicontinuous respectively, and we
can see that f(cid:63) ≤ f ≤ f (cid:63).

108

Theorem 5.3.18. [26, Theorem 3.1] Suppose f, g are respectively viscosity subsolution and
supersolution to (5.3.1), and we have f (cid:63)(X) ≤ g(cid:63)(X) for X ∈ Sn. Then we will also have
f (cid:63) ≤ g(cid:63) on Sn × (0, T ].

Remark 5.3.19. Under the assumptions of this theorem, there is a modulus on continuity,
ω0, such that f (cid:63)(x) − f(cid:63)(z) ≤ ω0(|x − z|) for x, z ∈ Sn. We can also now find a modulus of
continuity ω so that this is true for X = (x, t), Z = (z, s) ∈ Sn × (0, T ].

Theorem 5.3.20. [26, Theorem 3.2] If f is a viscosity subsolution, and g is a viscosity
supersolution to (5.3.1), such that f ≤ g in Sn × (0, T ] and f = g on Sn, then ∃ viscosity
solution h ∈ C(Sn × (0, T ]) such that f ≤ h ≤ g in Sn × (0, T ].

We can now sketch the proof of existence using the following ideas from [34].

Proof. Let f0 be a uniformly continuous function in Sn. We will prove that there exists
a continuous function f : Sn × [0, T ] → R that solves (5.3.1) and f (−, 0) = f0. Perron’s

method gives us a continuous viscosity solution in the interior by taking a relaxed infimum

over the family of supersolutions of the equation. Now we need to show that this infimum
is continuous at t = 0 and the initial condition f (−, 0) = f0 is satisfied.

Assume f0 : Sn → R is uniformly continuous and f0 ≥ δ. Let F be the set of all
supersolutions h such that there is some modulus of continuity ω so that for every x, z ∈ Sn
and 0 < s ≤ T

h(z, s) > f0(x) − ω(|z − x| + s)

Suppose b is a smooth bump function such that b(0) = 1, b ≤ 1, and supp b = B1. For an
arbitrary x0 ∈ Sn, let bδ(x) = b((x − x0)/δ) Then depending on the modulus of continuity

109

for f0, ∀ε > 0, ∃δ > 0 so that

U0(x) := bδ(x)f0(x0) + (1 − bδ(x)) sup
Sn
f0 − ε ≤ f0(x)
L0(x) : bδ(x)f0(x0) + (1 − bδ(x)) infSn

f0 + ε ≥ f0(x)

Thus, near x0, f0 is trapped between U0 and L0 with a gap of width 2ε. Since U0 and
L0 are smooth functions (because b is smooth), |∇U0| and |∇L0| are both bounded by some

constant C. Thus we can construct a supersolution and subsolution respectively by

U (x, t) := U0(x) + Ct
L(x, t) := L0(x) − Ct

Going back to the equation (5.3.1), J in our case will only include terms like I(f ), G(I(f )),

which are Lipschitz. We also have J being Lipschitz regardless from Assumption 5.2.1.

Moreover J(U0), J(L0) have min-max representations consisting of integro-differential terms

which will be controlled by C (from the estimates for µ in Theorem 5.2.3). Thus, we have

Ut − J(U ) = C − J(U0) ≥ 0
Lt − J(L) = −C − J(L0) ≤ 0.

(we can choose C thus)

So U and h ∈ F are supersolutions and L is a subsolution to (5.3.1), and they are uniformly

continuous functions.

L(x, t) = L0(x) − Ct ≤ f0(x) − Ct ≤ h(z, s) + ω(|x − z| + s) − Ct

110

We can choose δ small enough so that |x − z| < δ, |s|,|t| < δ =⇒ ω(|x − z| + s) − Ct < ε.
The comparison principle in proposition 5.3.12 gives L ≤ h in Sn×[0, T ] for all lower barriers
L, which means F is bounded below. Moreover, F (cid:54)= ∅ because every upper barrier U ∈ F.

Indeed we have already said why U is a supersolution, but also

U (z, s) = U0(z) + Cs ≥ f0(x) + f0(z) − f0(x) + Cs = f0(x) − ω(|x − z| + s)

F is a non-empty set that is bounded below, so we can find an infimum. In fact, here we

use the relaxed infimum h(cid:63), i.e.

h(cid:63)(x, t) = lim inf
r→0

inf

|x−z|<r
|t−s|<r

inf
h∈F h(z, s)

Note that for each x0 ∈ Rn and ∀ε > 0, we can define these functions U, L, and conse-

quently h(cid:63). We have already seen that ∀h ∈ F, ∀ lower barriers L

L(x, t) ≤ h(z, s) + ω(|x − z| + s) − Ct
= h(z, s) + ω(|x − z| + |s − t|)

=⇒ L(x, t) ≤ lim inf
r→0

= lim inf
r→0

inf

|x−z|<r
|t−s|<r

inf

|x−z|<r
|t−s|<r

h∈F h(z, s) + ω(|x − z| + |s − t|)

inf

h∈F h(z, s) = h(cid:63)(x, t)
inf

111

We also have

h∈F h(z, s) ≤ U (z, s)

inf

(since U ∈ F)

= U0(z) + Cs
= U0(x) + Ct + [U0(z) − U0(x) + C(s − t)]
= U (x, t) + ω(|z − x| + |s − t|)

=⇒ h(cid:63)(x, t) ≤ lim inf
r→0

inf

|x−z|<r
|t−s|<r

U (x, t) + ω(|z − x| + |s − t|) = U (x, t)

Thus, now for each point x0 ∈ Sn and ∀ε > 0, we have trapped a solution h(cid:63) in between

a subsolution and a supersolution. Since we can make this ε as small as possible, this means

that h(cid:63) is uniformly continuous and we have h(cid:63)(x0, 0) = f0(x0). Since x0 was arbitrary, we
have h(cid:63)(−, 0) = f0.

This proves part (c) of the main result, i.e. theorem 5.1.1.

5.4. Revisiting the main results

So far, we have proved numerous properties for the broad class of operators J which satisfy

the conditions in assumption 5.2.1, and then proved uniqueness for the viscosity solution of

the equation (5.1.2), wherein such an operator J makes an appearance. As mentioned in

the introduction in chapter 4, the reason for studying these operators and equations is that

there is a strong connection between the solutions to the parabolic equations of the type in

(4.1.5) and the free boundary problem of the type in (4.1.4), and this connection is explained

in [9, Theorems 1.1, 1.4] which were mentioned earlier in section 4.3. The modified versions

of these results for the sphere were given in section 5.1.1.

112

In particular, we are interested to see if these results apply to the case of the free boundary

problem (5.1.1) and parabolic equation (5.1.2) introduced in section 5.1.1. Recall that the

parabolic equation in question is

∂tf =(cid:0)|∂+

ν U| − |∂−

ν U|(cid:1)(cid:113)

1 + |∇f|2 on Sn × [0, T ]

suggesting that the corresponding operator J in this situation for which we proved all the

results in sections 5.2 and 5.3 is given by

J(f, x) =(cid:0)|∂+

ν U (x, f (x))| − |∂−

ν U (x, f (x))|(cid:1)(cid:113)

1 + |∇f (x)|2

(5.4.1)

But of course, inorder to apply all of the results in sections 5.2 and 5.3 to this operator J

and subsequently to the equations (5.1.1) and (5.1.2), we need to first make sure that the

chosen operator J from (5.4.1) satisfies all the properties listed in assumption 5.2.1. We

check these properties below.

Indeed, if we fix 0 < γ < 1, and if f ∈

(cid:32)(cid:83)

, (cid:83)

(cid:33)

K(γ, δ, m)

, then f ∈ C1,γ(Sn) and

δ>0

m>δ

δ0 < f < M for some δ0, M > 0. U and f are at least C1,γ ( [21, Section 8.11]), therefore
their first derivatives are H¨older continuous, and thus J(f ) ∈ C0(Sn). Furthermore, in
[21, Theorem 8.33] we see that ||U||C1,γ depends in a linear way on the C1,γ-norm of the

boundary of the domain, which in our case is given by f . This implies that there is a Lipschitz

dependence from f → ∂νUf . This and due to the fact that the mapping f (cid:55)→(cid:112)1 + |∇f (x)|2

is bounded and Lipschitz on each of the sets K(γ, δ, m, L) (where L is the upper bound for
f : Sn → R as described in (5.1.4)), we can see that J is a Lipschitz operator. Further, since

all terms in J are rotationally invariant, so is J, and we can also see that the rest of the

113

assumptions work out easily as in [9, Theorem 7.10].

In the following sections, we list some results that follow analogously from the results

in [9, Sections 9,10,11], which in turn are useful for proving the main result, theorem 5.1.1.

We do not provide any full proofs for the results as they can all be easily adapted from the

proofs in [9], but we will mention some ideas behind the proofs and leave some quick remarks

where relevant.

5.4.1. Different notions of viscosity solutions

We will elaborate a little more on the topic of the equivalence between solving the parabolic

equations of the type in (5.1.2) and the free boundary problem of the type in (5.1.1), but in

order to do so, we will need a new notion of viscosity solution. In this notion, we look at
test interfaces in lieu of test functions. We shall henceforth denote the set Σ0 = Sn × [δ, L].
A test interface S in [a, b] where 0 < a ≤ b < T will be described as follows.

Definition 5.4.1. A test interface is a hypersurface S ⊂ Σ0×[a, b], such that each time slice

S(t) is a positive distance away from ∂Σ0 and it separates Σ0 into two connected components,
say S(t)+ and S(t)−, i.e. we will have Σ0 \ S(t) = S+ ∪ S−.

Recall the definitions of D±

f from (5.1.1), and we will also have in the two phase case

f ⊂ S(t)+ and S(t)− ⊂ D−
D+
f .

Now, given a test interface S in [a, b] we will define the function US : Σ0 × [a, b] → R
as the function which for each fixed time t ∈ [a.b] is the unique solution to the Dirichlet

problem

114

∆US = 0

US = 0

in S±

on S

US = 1
US = −1

on Bδ(0)

on BL(0)

(5.4.2)

Below we will give the definitions of the classical subsolution (and supersolution), as well

as the original and new notions of viscosity solutions.

Definition 5.4.2. A function U : Σ0 × [a, b] → R is said to be a classical subsolution

(respectively supersolution) of (5.1.1) if

• The set ∂{U > 0} is a differentiable submanifold of Σ0 × [a, b] with codimension 1,
and each time-slice ∂{U > 0} ∩ Σ0 × {t} (a ≤ t ≤ b) is a codimension 1 differentiable

submanifold of Σ0. Also, U is twice differentiable in space and differentiable in time
in Σ0 × [a, b] \ {U (cid:54)= 0}.

• For each fixed t, the function U (·, t) solves, in the viscosity sense

∆U ≤ 0 in {U > 0} and {U < 0} (respectively ≥)

and if V denotes the normal velocity of ∂{U > 0} (in the outer normal direction) then

V ≤ |∂+

ν U| − |∂−

ν U| (respectively ≥)

U is a classical solution of (5.1.1) if it is both a supersolution and a subsolution.

Now we move on to the definition of a viscosity solution.

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Definition 5.4.3. A function U is said to be touched from above at (X0, t0) by S if S is a
test interface in [t0 − τ, t0] for some τ > 0 such that

{U > 0} ∩ {t0 − τ ≤ t ≤ t0} ⊂ S+ and (X0, t0) ∈ ∂{U > 0} ∩ S

Definition 5.4.4. A viscosity subsolution (respectively supersolution) of the two-phase equa-

tion (5.1.1) is an upper semicontinuous function (respectively lower semicontinous function)
U : Σ0 × [a, b] → R, and it is required to have the following properties.

U ≤ 1 (respectively U ≥ 1) on Bδ(0), U ≤ −1 (respectively U ≥ −1) on BL(0),

And it satisfies the following relations in the viscosity sense

−∆U ≥ 0 in {U > 0}◦ and {U < 0}◦ (respectively ≤),

and for any test interface S touching U from above at (X0, t0) ∈ ∂{U > 0} we have

VS(X0, t0) ≤ |∂+

ν Us| − |∂−

ν Us| (respectively ≥),

(5.4.3)

where Us is as given in equation (5.4.2).

We will use the above definitions 5.4.2, 5.4.3, and 5.4.4 for classical and viscosity solutions

of (5.1.1), and in [9], it was shown that this is equivalent to the definitions given in [27]

116

5.4.2. Vertical shifts in the intersurface

It will be necessatry to understand how ∂U±

ν US varies with the vertical shifts of S, and thus

we shall introduce some notation for such shifts. Given a test interface S, we define the

intersurface Sh, which will result from shifting S in the upward direction by h as follows

Sh := {X = (x, xn+1)|(x, xn+1 − h) ∈ S},

provided h > 0; if h < 0, then Sh results from shifting S down by |h|. The resulting variation
in ∂+

ν US is recorded below.

Lemma 5.4.5. Let S be a test interface. Then there is a constant C = C(S) such that for
all sufficiently small h and any (X, t) ∈ Sh, we have

|∂+
ν USh

(X − hen+1, t) − ∂+

ν US(X, t)| ≤ C|h|,

where both US and USh

are functions given by (5.4.2).

The proof of the above lemma involves looking at a function

˜U (X, t) := USh

(X − hen+1, t), defined in S+ ∩ {(X, t)|X = (x, xn+1), xn+1 ≥ h},

and then applying the maximum principle comparing ˜U and US in S+ \ {0 ≤ xn+1 ≤ h}.

One can use the above estimate for the one-phase problem to then obtain the bound for the

two-phase problem relevant to our equation, i.e.

(cid:12)(cid:12)(cid:12)|∇U +

Sh

| − |∇U−
Sh

| − |∇U +

S | + |∇U−

S |(cid:12)(cid:12)(cid:12) ≤ CS|h|.

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This can be done using a similar approach to the one in [9, Lemma 6.1].

5.4.3. Correspondence between viscosity solutions of the free boundary evolution

and viscosity solutions of the parabolic equation

In this subsection, we state the results that ultimately help establish the equivalence between

the viscosity solutions of the free boundary evolution in (5.1.1) and viscosity solutions of the

parabolic equation given by (5.1.2). These results will show that a viscosity solution to the

parabolic equation will yield a viscosity solution to the free boundary problem (in the sense

of definition 5.4.4).

Remark 5.4.6. We also note that we can look at these equations in the context of the
level-set equations. In this case, we look at the set ∂{U > 0} as the zero level set of the

function U , or we can also consider it the domain and the boundary as a function of time,
i.r. ∂{U (·, t) > 0} = ∂Ω(t). Assume that we parametrize t by some defining function
Φ(X, t), then we have ∂Ω(t) = {Φ(·, t) = 0}. Now, if the normal velocity is given by V and

the outer normal vector is ν, then the level-set equation becomes

∇Φ · (νV ) + ∂tΦ.

Further, as Φ is the defining function of a level set, we have

ν = ± ∇Φ
|∇Φ|

this reduces the flow to ∂tΦ = ±V |∇Φ|. To be
depending on whether Φ > 0 or Φ < 0.
consistent with the parametrization, we choose Φ > 0, and also Φ(X, t) = f (x, t) − xn+1.

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With this and using the normal velocity condition given in (5.1.1), we finally obtain (5.1.2),

i.e.

∂tf =(cid:0)|∂+

ν U| − |∂−

ν U|(cid:1)(cid:113)

1 + |∇f|2 on Sn × [0, T ]

Lemma 5.4.7. [9, Lemma 9.9] If f is a globally Lipschitz viscosity subsolution (respectively
supersolution) of (5.1.2) in Sn × [a, b], then Uf is a viscosity subsolution (respectively su-
persolution) of the free boundary evolution(5.1.1) in Σ0 × [a, b] (in the sense of definition

5.4.4).

For the proof of this lemma (see [9, Lemma 9.9] for full proof), we start with the Γf being
the graph of f and also the set ∂{Uf > 0}, and a test interface S touching Uf from above

at a point, and we try to find an intermediate test interface between S and Γf which is also

the graph of some function that lies between Uf and US. This is possible because of the

Lipschitz nature of f and constructing a function using sup-convolutions. The intermediate

function, which touches the graph of f at the same point as the test interface touches the

level set of U , will satisfy the equation (5.1.2) by the definition of viscosity solutions, and

we get many more inequalities using the comparison principle, which leads us to Uf being

the viscosity subsolution (respectively supersolution) of the free boundary evolution (5.1.1).

Lemma 5.4.8. [9, Lemma 9.10] If f is a viscosity subsolution (respectively supersolution)
of (5.1.2) in Sn × [a, b], then Uf is a viscosity subsolution (respectively supersolution) of the
free boundary evolution(5.1.1) in Σ0 × [a, b].

Note that this lemma does not have the extra assumption of f being globally Lipschitz

as the previous lemma 5.4.7. To prove this lemma, we look at the sup-convolution of f ,

which is in fact a Lipschitz function, and apply lemma 5.4.7. This will tell us that the sup-

convolution of Uf is a viscosity subsolution of (5.1.1), and we can now make some limiting

119

arguments to show that Uf is a viscosity subsolution to (5.1.1).

Now the next result will tell us how the test functions for the lower dimensional, non-local

parabolic equations will yield to test functions of the free boundary problem.

Proposition 5.4.9. [9, Proposition 9.11]Let φ be an admissible test function touching f
from above at some t0 ∈ [a, b] and x0 ∈ Sn. Then Uφ touches Uf from above at (X0, t0)

where X0 = (x0, f (x0, t0))

The proof of the above proposition is a straightforward consequence of the comparison

principle and can be found in [9, Lemma 9.11].

Lemma 5.4.10. [9, Lemma 9.12] Let U be a viscosity subsolution (respectively supersolution)
of (5.1.1) in Σ0 × [a, b] whose free boundary is given as the graph of some upper semicon-

tinuous (lower semicontinuous) function f , then f is a viscosity subsolution (respectively
supersolution) of (5.1.2) in Sn × [a, b].

Using the previous proposition and the level set equations for (5.1.1) (see remark 5.4.6),

we can prove this lemma.

Remark 5.4.11. All the given results in this section 5.4.3 will give the proof for part (a) of

the main result, i.e. theorem 5.1.1.

5.4.4. Propogation of the modulus of continuity

In this final section, we will give an explanation for the remaining parts of the main result,

theorem 5.1.1, which pertain to the modulus of continuity. In particular, we can show that

the modulus of continuity of the initial data will be preserved by the fractional parabolic

equation. This result follows without difficulty once the comparison theorem for viscosity

solutions has been established, which we have done in section 5.3.

120

Lemma 5.4.12. Let J be as in assumption 5.2.1, T > 0 and f : Sn × [0, T ] → R, a

continuous viscosity solution of

∂tf = J(f )

f (x, 0) = f0(x)

in Sn × [0, T ]

in Sn

If f0 is continuous with the modulus of continuity ω(·), then the same will be true of f (·, t)
for all t ∈ [0, T ]. In particular, we have the estimate

|f (x, t) − f (z, t)| ≤ ω(|x − z|).

Proof. Let z ∈ Sn be fixed, and consider the function

w(x, t) := (Rzf )(x, t) − f (x, t)
= f (Rzx, t) − f (x, t).
Then by assumption, w(x, 0) ≤ ω(||Rz||), ∀x ∈ Sn

Now, f being a viscosity solution is a subsolution as well as a supersolution, and since J is

rotationally invariant, we can also say that Rzf is a viscosity solution. Thus by an argument

similar to the one in the proof of proposition 5.3.9, we get

∂tw ≤ M +

J (w).

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Now w satisfies all the assumptions of lemma 5.3.8, thus we have

w+ ≤ sup
Sn

w+(·, 0)

sup

Sn×[0,T ]
i.e. w(x, t) ≤ ω(||Rz||), ∀(x, t) ∈ Sn × [0, T ]

=⇒ f (Rzx, t) − f (x, t) ≤ ω(||Rz||).

Since z ∈ Sn was arbitrary, we have proved the lemma.

Note that the above lemma is the proof of part (b) of the main result, i.e. theorem 5.1.1.

122

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