GEOMETRIC EVOLUTION OF SINGLE-LAYER INTERFACES IN THE
FUNCTIONALIZED CAHN-HILLIARD EQUATION
By
Gurgen Ruben Hayrapetyan

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Applied Mathematics
2011

ABSTRACT
GEOMETRIC EVOLUTION OF SINGLE-LAYER INTERFACES IN THE
FUNCTIONALIZED CAHN-HILLIARD EQUATION
By
Gurgen Ruben Hayrapetyan
We study the Functionalized Cahn-Hilliard Energy (FCH), which is a higher-order reformulation of the Cahn-Hilliard energy, as a model for network formation in polymer-solvent
mixtures. The model affords a finite interfacial width, accommodates merging and other topological reorganization, and couples naturally to momentum balance and other macroscopic
mass transport equations.
The corresponding constrained L2 gradient flow has a rich family of approximately steadystate solutions that include not only the single-layer heteroclinic front profile seen in gradient
flows of the Cahn-Hilliard energy, but also a novel one parameter family of homoclinic bilayer solutions. In this thesis we rigorously derive the geometric evolution of the single-layer
polymer-solvent interface.
We form a manifold of quasi-equilbria by “dressing” a large family of co-dimension one interfaces immersed in Rd with heteroclinic solutions of a one-dimensional equilibrium equation
derived from the first variation of the FCH energy. We show that solutions of the gradient
flow that start sufficiently close to the manifold remain close, and moreover the flow can be
decomposed, at leading order, as a normal velocity for the underlying co-dimension one interface. Assuming the smoothness of the interface under this flow, we develop rigorous estimates
on the proximity of the true solution to the manifold, in an appropriate norm, for long time.

ACKNOWLEDGMENT

I am heartily thankful to my supervisor, Dr. Keith Promislow, for his encouragement, guidance
and continuing support, for his time, his professional knowledge, and his mathematical ideas,
which he generously shared with me throughout all of my years of graduate school.
I am also very thankful to all members of the Department of Mathematics at Michigan
State University and especially to members of my dissertation committee, Dr. Peter Bates,
Dr. Andrew Christlieb, Dr. Zhengfang Zhou, and Dr. Gabor Francsics.
It is also a pleasure to thank my fellow graduate students, especially Yang Li and Tom
Bellsky, who became my great friends.
Finally, I would like to thank my family for their love and encouragement throughout all
years of my life.

iii

TABLE OF CONTENTS

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
List of Figures

vi

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vii

1 Introduction
2 Geometry of Interfaces
2.1 Weingarten map and fundamental
2.2 Whiskered Coordinates . . . . . .
2.3 Example: Spherical Coordinates .
2.4 Admissible Interfaces . . . . . . .

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4 Evolution by Normal Velocity
4.1 Evolution Equations for the Fundamental Forms . . . . . . . . . . . . . . . . .
4.2 Evolution of Principal Curvatures . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Eigenvalues of the Laplace-Beltrami Operator
5.1 Weyl’s Asymptotic Formula for the Eigenvalues
5.2 Rayleigh Characterization of Eigenvalues . . . .
5.3 Smooth Dependence of Eigenfunctions on Time
5.4 Smooth Dependence of Eigenvalues on Time . .

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6 Initial Data Decomposition
6.1 Normal Variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 Overview of Results
3.1 Single-layer Dressed Ansatz . . . . . . . . . .
3.2 Linearization and the Functionalized Operator
3.3 Main Results . . . . . . . . . . . . . . . . . .
3.3.1 Statement of the Main Theorem . . . .
3.3.2 The time to exit the Manifold . . . . .
3.3.3 Interface Evolution . . . . . . . . . . .
3.3.4 Background Evolution . . . . . . . . .
3.4 Evolution in R2 . . . . . . . . . . . . . . . . .

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7 Analysis of the Full Linearized Operator
7.1 Functionalized Operator: Overview of Results
7.2 Additional Estimates . . . . . . . . . . . . . .
7.3 Main Results . . . . . . . . . . . . . . . . . .
7.3.1 Whiskered Operator . . . . . . . . . .
7.3.2 Low Energy Function Decomposition .
7.4 Proof of Theorem 7.11 . . . . . . . . . . . . .

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8 Analysis of the Full Linearized Operator - Coercivity of The Bilinear Form136
8.1 Orthogonality to Low Energy Eigenspace . . . . . . . . . . . . . . . . . . . . . 137
8.2 Coercivity of the Bilinear Form . . . . . . . . . . . . . . . . . . . . . . . . . . 142
8.3 Proof of Theorem 3.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
9 Interface Evolution - Bounds on Lower
9.1 Linear Operator . . . . . . . . . . . . .
9.2 Nonlinearity Control . . . . . . . . . .
9.3 Normal Velocity . . . . . . . . . . . . .
9.3.1 Leading Order Dynamics . . . .
9.4 Development of a priori estimates . . .
9.5 Other Estimates . . . . . . . . . . . . .

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LIST OF TABLES

7.1

Definitions of the operators that are used. . . . . . . . . . . . . . . . . . . . . 102

7.2

Definitions of the bilinear forms that are used. . . . . . . . . . . . . . . . . . . 102

vi

LIST OF FIGURES

1.1

2.1

The spontaneous network formation in Ω ⊂ R2 which is typical of (1.0.5).
Reading left to right, the initial data is four circles of “water” (u = 1) represented by the blue color within a background of “polymer” (u = −1) in red.
The boundary between the two domains is given by a front or single-layer solution which it shares with the Cahn-Hilliard equation. In the initial stage
the higher curvature circles grow at the expense of the lower curvature ones,
however, as can be seen in the second frame, the circular domains are unstable
to an antipodal elongation. The elongated circle assumes a dumb-bell shape
which stretches and merges with the adjacent circles in frames 3 and 4, forming
a narrow bi-layer capped by two semi-circular endcaps. In frames 4-8 the bilayer interface narrows and meanders as it is driven to its optimal width, which
is stationary on the O(ε−3 ) time scale. For interpretation of the references to
color in this and all other figures, the reader is referred to the electronic version
of this thesis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

The figure on the left shows an interface Γ ⊂ R2 that does not belong to GK
due to its closeness to self-intersection. The center figure shows an interface
that is not even in G since it is not an embedding. The figure on the right
shows an admissible interface together with its ’whiskered neighborhood’ Γ(ℓ).
For interpretation of the references to color in this and all other figures, the
reader is referred to the electronic version of this dissertation. . . . . . . . . .

27

vii

Chapter 1

Introduction

A central goal of polymer chemistry is to design materials with novel macroscopic properties by
controlling the spontaneous generation of nanoscaled, phase separated networks [4]. A primary
mechanism to generate such networks is through the functionalization of hydrophobic polymer
chains and nanoparticles by the addition of acid or alkaline terminated side-chains. The endgroups interact exothermically with solvents, driving the generation of polymer-solvent or
nanoparticle-solvent interface generating as much as 1000m2 /gram of solvated interface, the
largest density known. Modeling the generation of these polymer networks is a critical need
in chemistry.

A key step is establishing an expression for the ‘free energy’ functional of the system which
decreases along solutions of corresponding gradient flows. The Canham-Helfrich free energy
is a broadly applicable template for sharp-interface energies of an interface Γ ⊂ R3 written in
1

terms of its mean, H, and Gaussian, K, curvatures,

ECH (Γ) =

Γ

σ + κb (H − Hint )2 + κs KdS.

(1.0.1)

The parameter, σ, denotes the energy density per unit surface area, κb and κs , denote the
energy density attributed to extension and splay deformations, and Hint denotes the intrinsic, or zero-energy, value of the mean curvature. In the context of functionalized polymers,
these parameters incorporate the role of solvation, electrostatic energy, and polymer backbone
stiffness into the energy associated with an interface between the solvent and polymer backbone regions. However, the Canham-Helfrich energy is a sharp interface energy and as such is
not useful in describing the merging and pinch-off events that play a crucial role in polymer
network formation. We avoid these difficulties by regularizing the Canham-Helfrich energy
into a finite-width phase field model. To this end the ‘functionalized’ Cahn-Hilliard Energy
(FCH) was introduced in [18]

EF [u] =

1
Ω2

ε
δE 2
− η( |∇u|2 + ε−1 W (u)).
δu
2

(1.0.2)

Here
E [u] =

ε
|∇u|2 + ε−1 W (u)
2
Ω

(1.0.3)

is the well studied Cahn-Hilliard energy, ε ≪ 1 is a small positive parameter, η > 0 represents
the strength of functionalization, and u is a phase function that differentiates between the
solvent and polymer backbone regions (two minima of a symmetric double-well potential W ).

2

Without loss of generality, we assume that the minima occur at u = ±1 and
µ0 := W ′′ (±1) > 0.

(1.0.4)

We have shown that there exist families of functions, {uε (x)}ε>0 , for which in the ε → 0
limit the energy EF recovers a Canham-Helfrich energy of the form (1.0.1) [18]. However,
unlike for the Cahn-Hilliard energy, several distinct forms of the limiting energy are possible.
A conjecture of De Giorgi, concerning the Γ-convergence of (1.0.2) in the case η = −ε2 , when
both terms are positive, has recently been established, [31]; however the nature of the energy
and the dynamics of the associated gradient flows are fundamentally different in the case
η > 0.
The Functionalized Cahn-Hilliard energy, (1.0.2), represents not a perturbation, but a
substantial highly-nonlinear reformulation of the Cahn-Hilliard energy. The Cahn-Hilliard
equation, the H −1 gradient flow on the Cahn-Hilliard Energy, drives the system to minimize
interface area while preserving the volume fraction. As such it is a model for coarsening
processes of binary alloys, see [7], [9], [10]. Network formation in functionalized polymers is
a fundamentally different process, and the FCH energy (1.0.2) takes this into consideration
by balancing the solvation energy released by formation of water-acid interface against the
elastic energy required to bend the interface, and the associated polymer backbones. This
‘functionalization’ of the Cahn-Hilliard functional (1.0.3) through sign inversion and regularization via the square of the variational derivative fundamentally changes the nature of the
energy landscape.
The simplest mass-preserving gradient flow associated with the FCH energy is the non-local
3

L2 flow on Ω ⊂ Rd ,
∂t u = −Π0 F (u),

(1.0.5)

F (u) = (ε2 ∆ − W ′′ (u) + ε2 η)(ε2 ∆u − W ′ (u))

(1.0.6)

where

and the zero-mass projection Π0 subtracts the average value
1
Π0 f = f −
f dx.
|Ω| Ω

(1.0.7)

The evolution presented in Figure 1.1 shows two distinct stages. The initial stage corresponds to the single-layer front solutions that separate the two distinct phases. These class of
quasi-steady solutions have analogues in the Cahn-Hilliard equation. In the second stage the
single-layer fronts come close to merging, but are arrested by the formation of a thin bi-layer,
which subsequently grows and dominates the flow, until it arrives at the quasi-steady equilibrium depicted in the final frame. Understanding of the generation and evolution of these
interfaces is integral to predicting the spontaneous generation of polymer solvent networks.
Equation (1.0.5) has a rich family of approximately steady state solutions that include not
only the single-layer heteroclinic front profile seen in gradient flows of the Cahn-Hilliard energy,
but also a novel one parameter family of homoclinic bi-layer solutions [18]. The single-layer
quasi-steady states Φ(x; Γ, b) are formed by ‘dressing’ an admissible interface Γ (see Section
2.4) and parameterized by a background state b ∈ R (for bi-layers there is an additional
parameter α which represents the width of the homoclinic profile). The class of ‘admissible
4

Figure 1.1: The spontaneous network formation in Ω ⊂ R2 which is typical of (1.0.5). Reading
left to right, the initial data is four circles of “water” (u = 1) represented by the blue color
within a background of “polymer” (u = −1) in red. The boundary between the two domains
is given by a front or single-layer solution which it shares with the Cahn-Hilliard equation.
In the initial stage the higher curvature circles grow at the expense of the lower curvature
ones, however, as can be seen in the second frame, the circular domains are unstable to an
antipodal elongation. The elongated circle assumes a dumb-bell shape which stretches and
merges with the adjacent circles in frames 3 and 4, forming a narrow bi-layer capped by two
semi-circular endcaps. In frames 4-8 the bi-layer interface narrows and meanders as it is driven
to its optimal width, which is stationary on the O(ε−3 ) time scale. For interpretation of the
references to color in this and all other figures, the reader is referred to the electronic version
of this thesis.

5

interface’ GK corresponds to hypersurfaces in Rd which are far from self-intersection and
have curvatures bounded in L∞ (Γ) by a prescribed constant K. In particular, we assume
that the interfaces are free of junctions and provide conditions which insure that no merging
or self-intersections take place.
In the present work we are concerned with co-dimension one interfaces embedded in R2 or
R3 dressed with a single-layer ansatz. The construction of the single-layer ansatz begins with
the choice of an equilibrium of an associated 2nd-order, one-dimensional differential equation.
Via a formal expansion we determine higher-order correction terms that adapt the solution
to the curvature of the interface and the desired background (far-field) value. The resulting
function Φ = Φ(x; Γ, b) has a small residual, indeed F (Φ) = O(ε3 ) in H 4 (Ω). However,
the residual cannot be further reduced since solvability conditions fail at the next order in
the formal expansion. The precise construction of Φ exploits the local normal-tangential
coordinate system about Γ, which we call the ‘whiskered coordinates’ as presented in Chapter
2. The resulting manifold

MK := {Φ(x; Γ, b) : Γ ∈ GK , b ∈ (−¯ 3 , ¯ 3 )},
bε bε

(1.0.8)

is not invariant, however by utilizing the methods from [30] we show that a solution that starts
in a sufficiently small neighborhood of MK may be decomposed as u(x, t) = Φ(x; Γt ; b(t)) +
w(x, t) where the time-dependent interface Γt and the background state b(t) shadow the slow
evolution of u up to a small remainder term w. In particular, we will show in our main result,
Theorem 3.7, that assuming the continued smoothness of the interface, there exists a time
Tb > 0, which depends only upon K, such that if the evolution of the local parametrization
6

γ(s, t) of the interface Γt is described by

∂t γ(s, t) = Vn (s, t)ν(s, t),
γ(s, 0) = γ0 (s),

(1.0.9)
(1.0.10)

for an appropriate choice of normal velocity Vn then the remainder w will remain small, in
an appropriate norm, for all t ∈ [0, Tb ]. Indeed, we show that the appropriate choice of the
normal velocity is given, to leading order, by the expression

Vn = −Π0,Γ

(∆s + η)H −

H3
+ H tr(A2 ) ,
2

(1.0.11)

where H is the mean curvature, A is the Weingarten map whose eigenvalues are the curvatures
of Γ, and Π0,Γ is the zero-mass projection associated to the surface integral over Γ. We also
express the normal velocity as a higher-order Ricci Flow on the fundamental forms associated
to Γ. The reduction of the FCH to a curvature driven flow requires a detailed analysis of the
linearization Lφ of F about each Φ ∈ MK . We show that Lφ has an M -dimensional slow
eigenspace, corresponding to eigenvalues that are asymptotically small, and a remainder that
is uniformly bounded in the left-half complex plane. Moreover the slow-eigenspace, spanned
by Ψ1 , . . . , ΨM , can be well approximated by a convenient basis ZM = {Zi }M . The decomi=1
⊥
position ZM ⊕ ZM breaks L2 (Ω) into two approximately-Lφ invariant subspaces. Moreover,
the dimension M of ZM may be chosen so that the bilinear form which we associated with
⊥
Lφ is uniformly coercive on ZM .
The full program outlined above has not been achieved in this thesis. A key gap in the
7

proof is our inability to produce closed estimates on the smoothness of the interface as it
evolves under the normal flow (1.0.9). There are several obstacles to this effort, not the
least of which is the complexity of the associated Ricci-curvature equations. We outline the
difficulties in the paragraphs below.

The first difficulty is a loss of one derivative of spatial regularity when decomposing the
initial data. More specifically, any initial data u0 which is close to the manifold MK can be
(trivially) written as
u0 = Φ(x; Γ∗ , b∗ ) + w∗ (x),

(1.0.12)

for some Γ∗ ∈ GK with w∗ small in an appropriate norm. To obtain decay of w, we look for
an orthogonal decomposition of the form

u0 = Φ(x; Γ0 , b) + w0 (x),

(1.0.13)

⊥
with the additional condition that w0 lie in the fast space, that is w0 ∈ ZM (Γ0 ). We search
for Γ0 among interfaces Γp near Γ∗ in the form
γp (s) = γ∗ (s) + ν∗ (s)R(s),

(1.0.14)

where ν∗ (s) is the normal to Γ∗ at γ∗ (s). We take candidates for R as among the Galerkin
sums on the first M Laplace-Beltrami eigenmodes, {Θi }M of Γ∗ ,
i=1
M
R=

pi Θi (s),
i=1
8

(1.0.15)

and determine the parameters p = (p1 , . . . , pM ) to impose the orthogonality condition
⊥
w0 := u0 − Φ0 ∈ ZM (Γ0 ).

(1.0.16)

However it is implicit in the construction (1.0.14) that γp loses one derivative compared to
γ∗ , since the normal ν∗ is one derivative less smooth than the γ∗ .
A second, principle difficulty is to understand the smoothness of the interfaces under the
flow (1.0.9). The regularity of the interface is best understood by recasting it as a Riccicurvature driven flow on the first and second fundamental forms of Γ. The analysis of the
resulting PDEs is indeed nontrivial, even for d = 2, when the fundamental forms are scalar,
the Ricci flow reduces, at leading order, to a generalized Kuramoto-Sivashinsky equation,
see (3.3.17), whose regularity is (just) outside the known results. However the situation is
complicated by the fact that we are only able to obtain tight bounds on the higher-order
corrections to the normal velocity, which enter into the Ricci-flow, in relatively weak norms,
such as L2 and L∞ . Moreover, while the leading order terms in the normal velocity are
independent of the remainder, w, the correction terms do depend upon w, thus the smoothness
of the interface couples to the norm bounds obtainable on w. Indeed, we would like the
correction terms to be asymptotically small in H 6 (S), where S ⊂ Rd−1 is the reference set
for the interface parameterization γ. In that case the perturbations to the evolution of the
second fundamental form would be small in H 4 , and hence the corresponding parameterization
γ would reside in H 6 (S).
There are two other obstacles to showing regularity in our orthogonal decomposition of the
solution u. The first is that the interface Γt , which we characterize via its second fundamental
9

form, could generate a self-intersection in finite time, at which point our decomposition would
break down. However, with a strong bound on the L∞ norm of the normal velocity it is
straightforward to obtain a sharp estimate on a possible time to self-intersection. The second
mechanism is more technical, our estimates require that the gap between the M ’th and M +1’st
Laplace-Beltrami eigenvalues for the interface Γt be O(1), where M ≈ ε−2 is the dimension of
the slow eigenspace. This generically holds, according to Weyl’s pioneering estimates, however
for certain degenerate interfaces the gaps between particular eigenvalues may close. We can
prevent this by estimating the rate of motion of the eigenvalues in terms of the normal velocity.
We find that lower bounds on the time to a “spectral-collapse” scale like t = O(ε−1 ). However
we may avoid the spectral collapse at a time t = ts by decomposing the solution u(ts ) as in
(1.0.13) and adjusting the slow dimension M to avoid proximate eigenvalues. However this
too requires smoothness of the interface Γt .
Thus the central theorem of this thesis, Theorem 3.7, is stated under the assumption of
smoothness of the underlying interface Γt for a range of t ∈ 0, Tf ε−4 for some Tf > 0.

10

Chapter 2
Geometry of Interfaces
This chapter lays out the necessary framework on the differential geometry of co-dimension
one interfaces in Rd that is required to state the main results. The cognizant reader my skip
directly to chapter 3 to find the main result, Theorem 3.7.
The main goal of our work is to describe the geometric evolution of the functionalized
polymer-solvent interface. Mathematically, we take such an interface to be represented by
a smooth, closed (compact and without boundary), oriented d − 1 dimensional manifold Γ
embedded in Rd . We denote the family of such manifolds by G. In addition, we consider only
the interfaces that are far from self-intersection, and whose curvatures are O(1) with respect
to ε. We defer the precise definition of admissible interfaces to Chapter 3. We will show that
a large class of solutions u of the FCH gradient flow

∂t u = −Π0 F (u) = −Π0 (ε2 ∆ − W ′′ (u) + ε2 η2 )(ε2 ∆u − W ′ (u)).

(2.0.1)

have an asymptotic limit captured by a higher-order Ricci curvature flow on the fundamental
11

forms associated to Γ. More precisely, equation (2.0.1) has a rich family of quasi-steady states
represented by Γ ∈ G dressed in the normal direction with either single or bi-layer profiles.
In the present work we study the evolution of initial datum in a small enough neighborhood
of a single-layer manifold MK . In particular, for the single-layer solutions the interface is
represented by the u = 0 level sets. In order to make this construction precise we need several
concepts from differential geometry which we review in the next section.

2.1

Weingarten map and fundamental forms

In this section we summarize some key definitions from Differential Geometry (for more details
see [25]).
Let Γ be a d − 1 dimensional smooth manifold embedded in Rd with a chosen orientation.
Let S be an open set in Rd−1 and γ : S → Γ be a local parametrization. Then the tangent
space Tγ(s) Γ is defined as the image of Ts S ∼ Rd−1 under the map Dγ|s . Denote by Sd−1
=
a sphere ||x|| = 1 in Rd .
Definition 2.1. The Gauss map, ν : S → Sd−1 maps points of S into normal unit vectors
ν(s) orthogonal to Tγ(s) Γ.
Definition 2.2. The Weingarten map, A : Tγ(s) Γ → Tγ(s) Γ is defined by
A = −Dν ◦ (Dγ)−1 .

(2.1.1)

Remark 2.3. Letting ei be the standard basis element in Rd−1 , we note that Dν : Ts S ∼
=
Rd−1 → Tγ(s) Γ follows from the fact that ∂ν = (Dν)ei belongs to the tangent space Tγ(s) Γ.
∂si
12

This can be seen from differentiating ν(s) · ν(s) = 1 with respect to si .
Definition 2.4. For X, Y ∈ Tγ(s) Γ the k-th fundamental for of Γ is defined by
Ak−1 X, Y ,

(2.1.2)

where the brackets represent the euclidean inner product in Rd .
Definition 2.5. The principal curvatures k1 , . . . , kd−1 of Γ are defined as the eigenvalues
of the Weingarten map A.
Weingarten Map and Fundamental Forms in Local Coordinates. From definitions of
Dγ and Dν we have
(Dγ)ei =

∂γ
,
∂si

(2.1.3)

(Dν)ei =

∂ν
,
∂si

(2.1.4)

and

where the vector ei contains 1 in the i-th position. It follows from the definition of Γ that Dγ
∂γ
∂γ
,...
∂s1
∂sd−1
Assume that X, Y ∈ Tγ(s) Γ are given by
is full-rank and so the vectors

form a basis for the tangent space Tγ(s) Γ.

d−1
X=

ξi

∂γ
,
∂si

(2.1.5)

ηj

∂γ
.
∂sj

(2.1.6)

i=1
d−1
Y =
j=1
13

Then,
d−1
X, Y =
i=1

ξi

∂γ
,
∂si

d−1

ηj

j=1

d−1

∂γ
∂sj

=

ξ i η j gij ,

(2.1.7)

i,j=1

where
∂γ ∂γ
,
∂si ∂sj

gij =

,

(2.1.8)

is the representation of the first fundamental form in local coordinates. In addition, by
definition, the Weingarten map A maps

∂ν
∂γ
→−
,
∂si
∂si

(2.1.9)

and we have
d−1
AX, Y =

d−1
∂γ
i A ∂γ ,
ξ
ηj
∂si
∂sj
i=1
j=1

d−1
=−

d−1
∂γ
i ∂ν ,
ξ
ηj
∂si
∂sj
i=1
j=1

d−1
=

ξ i η j hij ,

i,j=1
(2.1.10)

where
hij = −

∂ν ∂γ
,
∂si ∂sj

=

ν,

∂ 2γ
∂si ∂sj

,

(2.1.11)

is the representation of the second fundamental form in local coordinates. The last equality
results from differentiating
ν,

∂γ
∂si

14

=0

(2.1.12)

with respect to sj . Similarly,

A2 X, Y

d−1
=

ξ i η j eij ,

(2.1.13)

i,j=1
where
eij =

∂ν ∂ν
,
∂si ∂sj

= − ν,

∂ 2ν
∂si ∂sj

,

(2.1.14)

is the representation of the third fundamental form.

In addition if we write ∂ν in the tangent basis as
∂si
∂ν
=−
∂si

d−1
j=1

∂γ j
h ,
∂sj i

(2.1.15)

then referring to (2.1.11) we see that
d−1
hij =
j=1

∂γ j ∂γ
h ,
∂sj i ∂sj

d−1
=
j=1

j
hi gij ,

(2.1.16)

and
j
hi =

hik g kj ,

(2.1.17)

k

where g kj represents the elements of the inverse matrix of gij . The key relation (2.1.15) shows
j
that hi are the coefficients that express the linear variation of the normal vector in terms of
15

the tangent basis. Moreover, since
d−1
d−1
j ∂γ
i ∂ν =
i A ∂γ = −
,
ξ
ξ i hi
AX =
ξ
∂si
∂si
∂si
i=1
i=1
i=1
d−1

(2.1.18)

j
{hi } is the matrix representation of the linear operator A in the basis

∂γ
∂γ
,...
and
∂s1
∂sd−1
j
from definition (2.1.2), ki is the i-th eigenvalue of the matrix {hi }. The matrix operator norm
j
of hi is then
j
(2.1.19)
||{hi }|| = max ki .
1≤i≤d−1
Finally, defining
j
ei :=

eik g kj ,

(2.1.20)

k

j
j
we use (2.1.15) to express ei in terms of the Weingarten matrix hi
j
ei =

eik g kj =
k

=
k,l,m

k

∂ν ∂ν
,
∂si ∂sk

hl hm glm g kj =
i k

g kj =
k,l,m

hl hkl g kj =
i
l

k,l

hl hm
i k

j
hl h .
i l

∂γ ∂γ
,
∂sl ∂sm

g kj
(2.1.21)

j
Since the principal curvatures ki are the eigenvalues of hi it implies the following estimate
j
for the operator norm of the matrix {ei }
j
||{ei }|| =

max k 2 .
1≤i≤d−1 i

16

(2.1.22)

2.2

Whiskered Coordinates

For Γ ∈ G we introduce the whiskered coordinates in a neighborhood of Γ. Although these
coordinates have been introduced before in the context of geometric evolution, see [12] for
example, their application to the fourth-order problems hinges upon higher order terms which
we make precise in this section.
For x ∈ Rd belonging to a neighborhood of some point γ(s) ∈ Γ, we introduce the following
change of variables ϕ : (s, z) → x, defined by
ϕ(s, z) = γ(s) + εzν(s),

(2.2.1)

where the Gauss map, ν : Γ ⊂ Rd → S d−1 , gives the outward pointing unit normal vector
perpendicular to the tangent plane to Γ at γ(s). This change of variables is a diffeomorphism
in the neighborhood
Γ(ℓ) := {ϕ(s, z)|s ∈ S, −ℓ/ε ≤ z ≤ ℓ/ε},

(2.2.2)

of Γ, so long as ℓ is sufficiently small, as measured against the curvatures of Γ, but independent
of ε. We will call the line segments γ(s) × [−ℓ/ε, ℓ/ε] γ(s) ∈ Γ the whiskers of Γ, and refer
to (s, z) as the whiskered coordinate system.
If we introduce the variables y = (s1 , . . . , sd−1 , z) then x = ϕ(y) and ϕ−1 defines a chart
for Γ(ℓ). The gradient and the Laplace operator in Rd can be written in the y- coordinates as

∇i =
x

d
j=1

Gij

∂
, i = 1, . . . d
∂yj

17

(2.2.3)

and
∆x =

d

1
det(G)

d

i=1 j=1

∂ ij
G
∂yi

det(G)

∂
,
∂yj

(2.2.4)

where G is the metric tensor
Gij =

∂x ∂x
,
∂yi ∂yj

,

(2.2.5)

Rd

and Gij is the ij component of the inverse of the G. Letting J denote the Jacobian matrix
for ϕ = ϕ(s, z), we have,
G = JT J,

(2.2.6)

and consequently, det(G) = J 2 , where J = J(s, z) is the associated Jacobian. The relation
between G and g is clarified by the following Lemma.

Lemma 2.6.



G0 0 
G=

0 ε2

(2.2.7)

G0 (s, 0) = g(s),

(2.2.8)

and

where g = {gij } is the first fundamental form (metric tensor) on Γ.
Proof: The expression (2.2.7) comes from differentiating (2.2.1) with respect to z and si
∂ϕ
= εν,
∂z

(2.2.9)

∂γ
∂ν
∂ϕ
=
+ εz
∈ Tγ(s) Γ,
∂si
∂si
∂si

(2.2.10)

18

(see Remark 2.3). To obtain (2.2.8) we observe that for i, j = 1, . . . , d − 1,
(G0 )ij

= Gij

z=0

z=0

=

∂ϕ ∂ϕ
,
∂si ∂sj

z=0

=

∂γ ∂γ
,
∂si ∂sj

= gij .

(2.2.11)

In particular det g = ε2 det G0 . With this G, in the whiskered variables the gradient and
the Laplacian take the form,

∇i =
x

d−1
j=1

Gij

∂
for i = 1, . . . , d − 1,
∂sj

∇d = ε−2
x

∂
,
∂z

(2.2.12)

and
∆x = ε−2 J −1

∂ ∂
J
+ ∆G ,
∂z ∂z

(2.2.13)

where ∆G is

∆G =

1
det(G)

d−1 d−1
i=1 j=1

∂ ij
G
∂si

∂
det(G)
= J −1
∂sj

d−1 d−1
i=1 j=1

∂ ij ∂
G J
.
∂si
∂sj

(2.2.14)

When z = 0, ∆G is simply the Laplace-Beltrami operator on Γ, which we will denote by ∆s .
The precise relation between ∆G and ∆s will be given in Chapter 3 we define the class of
admissible interfaces.

To simplify the z derivatives in (2.2.13) we derive an expression for J in terms of the
19

d−1
principal curvatures {ki }i=1 of Γ. We first observe that identity (2.1.15) implies,
d−1
∂ϕ
∂γ j
∂γ
∂ν
∂γ
=
+ εz
=
− εz
h .
∂si
∂si
∂si
∂si
∂sj i
j=1

(2.2.15)

From (2.2.15) we see that the Jacobian matrix takes the form,

J=

∂γ
∂s1

∂γ
...
∂s2



j
Id−1 − εzhi 0
∂γ
ν 
,
∂sd−1
0
ε

(2.2.16)

where Id−1 is the (d − 1) × (d − 1) identity matrix. Taking the determinant of the Jacobian
matrix yields

j
J(s, z) = det J = εJ0 (s) det(Id−1 − εzhi )
d−1

= εJ0 (s)
i=1

d

(1 − εzki ) = J0 (s)

εj+1 Kj z j ,

(2.2.17)

j=0

where
K0 = 1,

Ki := (−1)i
j1 <···<ji

20

kj · · · kj ,
1
i

(2.2.18)

and, since

∂γ
∈ Tγ(s) and ν is a unit normal,
∂si

J0 = det
=

=

det

det

∂γ
∂s1

∂γ
...
∂s2

∂γ
∂s1
∂γ
∂s1

∂γ
ν
∂sd−1

∂γ
...
∂s2
∂γ
...
∂s2

∂γ
ν
∂sd−1
∂γ
∂sd−1

∂γ
∂s1
∂γ
∂s1

∂γ
...
∂s2

∂γ
...
∂s2

∂γ
ν
∂sd−1

∂γ
∂sd−1

T

T

1/2

1/2
. (2.2.19)

Therefore, from (2.2.19), we have
det(g),

J0 =

(2.2.20)

where g is the metric tensor for Γ. Taking the z derivative of the product form of the Jacobian
expression in (2.2.17) we obtain the identity,

∂z J = −ε2 J0 (s)

d−1
ki
i=1

(1 − εzkj ) = εκJ,
j=i

(2.2.21)

where the extended curvature
d−1
κ(s, z) := ∂z J/(εJ) = −

i=1

ki
=
1 − εzki

∞

κ j εj z j ,

(2.2.22)

j=0

is expressed in terms of the coefficients
d−1
κj (s) = −

i=1

j+1
ki (s) = −tr(Aj+1 ),

21

(2.2.23)

where A is the Weingarten map (see Section 2.1). In particular,
d−1
H=
i=1

ki = −κ0 ,

(2.2.24)

is the mean curvature.
We remark that while the Jacobian remains smooth, in fact it is a polynomial in z, the
extended curvature becomes singular when the whiskers intersect. Distributing the z derivative
in (2.2.13) and using the identity (2.2.22) yields the central result of this subsection, the local
decomposition of the Laplacian in the whiskered coordinates,

2
∆x = ε−2 ∂z + ε−1 κ(s, z)∂z + ∆G .

(2.2.25)

We call κ the extended mean curvature since κ = ∇x · ν is the Cartesian divergence of the
normal to Γ when the normal is extended off of Γ as a constant along whiskers. We conclude
the chapter with the proof of this fact.

Proposition 2.7. Let ν(x) = ν(s) be the unit normal to Γ extended to the whiskered region
Γ(ℓ) as a constant along whiskers. Then, the following holds for the extended mean curvature
κ defined in (2.2.22).
κ = ∇x · ν.

(2.2.26)

Proof:
d
∇x · ν =

i=1

∂νi
=
∂xi
22

d
i,j=1

∂νi ∂sj
,
∂sj ∂xi

(2.2.27)

and
x = γ(s) + εzν(s),

(2.2.28)

implies
∂ν
=−
∂sj

d−1
j
j ∂γ
=−
h
h
k
k ∂s
k
k=1
k=1
d−1

∂ν
∂x
− εz
∂sk
∂sk

.

(2.2.29)

Introducing matrix notation:
S = (S)ij =

∂si
,
∂xj

(2.2.30)

X = (X)ij =

∂xi
,
∂sj

(2.2.31)

N = (N )ij =

∂νi
,
∂sj

(2.2.32)

we rewrite (2.2.29) in the form,
N = −(X − εzN )B,

(2.2.33)

j
where B = {hi } is the matrix representation of the Weingarten map in the tangent basis (see
(2.1.18)). However,
d
(SX)ij =
k

∂si ∂xk
= δij ,
∂xk ∂sj

(2.2.34)

and (2.2.33) yields after pre-multiplying both sides by S

SN = −(I − εzSN )B.

23

(2.2.35)

Isolating SN on the left-hand side we obtain

SN (I − εzB) = −B,

(2.2.36)

SN = −B(I − εzB)−1 .

(2.2.37)

and

Returning this expression for SN to (2.2.27) and using a Neumann Expansion we obtain
d
∇x · ν =

d
Sji Nij =

i,j=1

j=1

(SN )jj = tr(SN ) = −

∞

εj z j tr(B j+1 ).

(2.2.38)

j=0

j
Finally noting that ki is the i-th eigenvalue of B = {hi } and so tr(B j+1 ) =

d−1 j+1
i=1 ki

yields
∇x · ν = −

∞ d−1
j=1 i=1

j+1
=
εj z j ki

∞

εj z j κj = κ,

(2.2.39)

j=0

where we used definitions of κ and κj in (2.2.22) and (2.2.23) respectively.

2.3

Example: Spherical Coordinates

2
2
Let SR be a sphere of radius R centered at the origin. Parametrizing SR using spherical
coordinates gives









γ1 (φ, Θ) R sin φ cos Θ

 


 

γ(φ, Θ) = γ (φ, Θ) =  R sin φ sin Θ  .

 
 2

 

R cos φ
γ3 (φ, Θ)
24

(2.3.1)

In addition, the normal vector ν takes the form




sin φ cos Θ




ν(φ, Θ) =  sin φ sin Θ  ,




cos φ

(2.3.2)

2
and following (2.2.1) we may introduce the whiskered coordinates in a neighborhood of SR
by





sin φ cos Θ




x = γ(s) + ǫzν(s) = (R + ǫz)  sin φ sin Θ  .




cos φ

(2.3.3)

The metric tensor G takes the form


2
0
0
(R + ǫz)



2 sin2 φ 0  ,
G=

0
(R + ǫz)




2
0
0
ǫ

(2.3.4)

and consequently
J=

det g = ǫ(R + ǫz)2 sin φ.

25

(2.3.5)

Setting y = [φ, Θ, z],

∆ =

=
=

1
det(G)

d

d

i=1 j=1
d

1
(R + ǫz)2 sin φ
1

∂2

(R + ǫz)2 ∂φ2

∂ ij
G
∂yi

i=1
+

det(G)

∂
∂yj

∂
∂ ii
G ((R + ǫz)2 sin φ)
∂yi
∂yj

1
∂
1 2
cos φ
∂
∂2
2
+
.
+ ∂z +
R + ǫz ∂z
(R + ǫz)2 sin φ ∂φ (R + ǫz)2 sin2 φ ∂θ2 ǫ2
∆G
(2.3.6)

Restriction of ∆G to Γ is

∆s =

1 ∂2
1
cos φ ∂
∂2
+
+
R2 ∂φ2 R2 sin φ ∂φ R2 sin2 φ ∂θ2

(2.3.7)

and the eigenfunctions Θi of ∆s are orthonormal with respect to the inner product

(Θi , Θj )Γ =

2.4

Γ

Θi Θj J0 (s)ds =

Γ

Θi Θj R2 sin φds.

(2.3.8)

Admissible Interfaces

Definition 2.8. Denote by G the family of smooth compact oriented (d-1)-dimensional manifolds without boundary embedded in Rd .

Definition 2.9. For K > 0 denote by GK the set of manifolds Γ ∈ G satisfying the following
assumptions:
26

Figure 2.1: The figure on the left shows an interface Γ ⊂ R2 that does not belong to GK due
to its closeness to self-intersection. The center figure shows an interface that is not even in G
since it is not an embedding. The figure on the right shows an admissible interface together
with its ’whiskered neighborhood’ Γ(ℓ). For interpretation of the references to color in this
and all other figures, the reader is referred to the electronic version of this dissertation.

(i) the principal curvatures and their derivatives up to the fourth order are bounded in L∞ (Γ)
norm by K,
(ii) the whiskers of length 1/K (in the unscaled distance) do not intersect each-other,
(iii) the volume vol(Γ) of Γ is bounded by K.
1
In the following we fix K > 0 and choose ℓ ≤ min{K, 4K }.
Lemma 2.10. Let G be the metric tensor for the whiskered coordinates defined in (2.2.5).
Let ∆G be the laplacian in whiskered coordinates given in (2.2.14) and let ∆s be the LaplaceBeltrami operator on the interface Γ. Then the following relationship holds between the operators ∆G and ∆s
∆G = ∆s + εzDs,2 ,
27

(2.4.1)

where
d−1
Ds,2 :=

dij (s, z)
i,j=1

∂2
+
∂si ∂sj

d−1
dj (s, z)
j=1

∂
,
∂sj

(2.4.2)

and
m
m
max ||∂z dij ||L∞ (Γ(2ℓ)) , |||∂z dj ||L∞ (Γ(2ℓ)) ≤ C(K)εm ,
ij

(2.4.3)

where the constant C(K) is independent of ε and Γ ∈ GK .

Proof: We recall the definition of ∆G from (2.2.14)

∆G

= J −1

d−1 d−1
i=1 j=1

∂ ij ∂
G J
∂si
∂sj

(2.4.4)

and observe that (2.2.1) and (2.2.5) imply

Gij =

∂ν(s) ∂γ(s)
∂ν(s)
∂γ(s)
+ εz
,
+ εz
∂si
∂si
∂sj
∂sj

= gij − 2εzhij + ε2 z 2 eij ,

(2.4.5)

where we used the definitions of the fundamental forms in (2.1.8), (2.1.11), and (2.1.14).
In addition, the Jacobian is given in (2.2.17) as
d

εj+1 Kj z j .

J(s, z) = J0 (s)

(2.4.6)

j=0
Writing
1

J −1 =
J0 (s)

d
j=0

εj+1 Kj z j
28

−1
= J0

∞
m=0

am z m ε m ,

(2.4.7)

we deduce
∞

d

Kj am (εz)m+j = 1.

(2.4.8)

am Ki−m (εz)i = 1,

(2.4.9)

j=0 m=0
Reindexing so that m + j = i yields
∞

i

i=0 m=0
and the following formulas for ai
a0 = 1

(2.4.10)

and
i
am Ki−m = 0.

(2.4.11)

m=0
In particular,
i−1
ai = −

am Ki−m ,

(2.4.12)

m=0

and a simple inductive argument shows

|ai | ≤ C(2K)m ,

(2.4.13)

where C depends on d only.

Furthermore, written in matrix notation

˜
G = g(I − 2εzg −1 h + ε2 z 2 g −1 e) = g(I − 2εh + ε2 z 2 e),
˜
29

(2.4.14)

j
j
˜
˜
where h = g −1 h = {hi } and e = g −1 e = {ei } and
˜
(I − 2εh + ε2 z 2 e)−1 =
˜

∞

εm z m Am ,

(2.4.15)

m=0

where
A0 = I,

(2.4.16)

˜
A1 = 2h,

(2.4.17)

˜
Am = 2hAm−1 − eAm+2 .
˜

(2.4.18)

and

˜
Since ||h|| ≤ K and ||˜|| ≤ K 2 (see 2.1.19 and 2.1.22) we again use a recursive argument to
e
establish
||Am || ≤ (3K)m .

(2.4.19)

1
Since, we assumed that εz ≤ ℓ ≤ 4K the series (2.4.7) and (2.4.15) are convergent and the
result of the proposition follows by substituting the expressions for J −1 and Gij into (2.4.4).

Lemma 2.11. Fix K > 0. Let g = {gij } and h = {hij } be the first and second fundamental
forms of the interface Γ ∈ GK . Then the following estimate holds
||(g − εzh)−1 || ≤ C(K).
where C(K) is a positive constant independent of ε and Γ ∈ GK .
30

(2.4.20)

˜
˜
Proof: Letting h := g −1 h, we have from the Neumann expansion of (I − εz h)−1


∞



˜
˜
εm z m hm  g −1 ,
[g(I − εz h)]−1 = 
m=0

(2.4.21)

1
˜
and since ℓ ≤ 4K , (2.4.20) follows from the bound on h in (2.1.19)
1
˜
||εz h|| ≤ ℓK ≤ .
4

(2.4.22)

1
Proposition 2.12. Let Γ ∈ GK . Then, for ε sufficiently small and ℓ ≤ 4K , the following
estimates hold for the metric tensor G, the Jacobian J, and the extended mean curvature κ

m
sup ||∂z Gij ||L∞ (Γ(2ℓ)) ≤ C(K)εm ,
i,j≤d−1

(2.4.23)

m
sup ||∂z Gij ||L∞ (Γ(2ℓ)) ≤ C(K)εm ,
i,j≤d−1

(2.4.24)

ε

3 d−1
1 d−1
≤J ≤ε
,
2
2

(2.4.25)

α m
sup ||Ds ∂z J||L∞ (Γ(2ℓ)) ≤ C(K)εm+1 ,
|α|≤2

(2.4.26)

α m
sup ||Ds ∂z κ||L∞ (Γ(2ℓ)) ≤ C(K)εm .
|α|≤2

(2.4.27)

Moreover, the constants in these estimates depend on K only.
Proof: The result follows from formulas for Gij , Gij and J in the proof of the previous
proposition.
31

A function f (x) which decays exponentially to zero away from Γ at an O(1) rate in z will
be said to be ‘localized on the surface’ or merely ‘localized’. Up to exponentially small terms
localized functions can be written in the whiskered variables, so that f (x) = f (s, z). Functions
h(z) of one variable which decay exponentially to constant values at ±∞ can be extended to
Ω by setting them to their limiting value for z > 2ℓ/ε and smooth for |z| ∈ (ℓ/ε, 2ℓ/ε). The
expression the interface Γ dressed by h refers to the extended function h(x), understood to
mean h(z(x)).

32

Chapter 3
Overview of Results
This chapter provides an overview of the derivation of the curvature driven flow on a singlelayer dressed interface, which we express in our main result, Theorem 3.7.

3.1

Single-layer Dressed Ansatz

Our goal in this section is to construct a family of approximate solutions to the steady-state
equation corresponding to (1.0.5). Recall that the set of admissible interfaces GK consists
of smooth compact oriented (d-1)-dimensional manifolds without boundary embedded in Rd
satisfying the following assumptions:
(i) the principal curvatures and their derivatives up to the fourth order are bounded in L∞ (Γ)
norm by K,
(ii) the whiskers of length 1/K (in the unscaled distance) do not intersect each-other,
(iii) the volume vol(Γ) of Γ is bounded by K.
Fix K > 0 and let Γ ∈ GK be an admissible interface. We use the whiskered coordinates
33

(z, s) defined in Section 2.2 in the whiskered neighborhood of Γ, Γ(2ℓ), of width 2ℓ and the
x coordinates outside of this neighborhood. Recall that the orientation of Γ determines the
direction of the normal vector in the two connected components of Ω. We will denote these
components as Ω± respectively. Therefore z > 0 in Ω+ ∩ Γ(2ℓ) and z < 0 in Ω− ∩ Γ(2ℓ) (see
(2.2.2)).

In order to transition between Cartesian and whiskered coordinates we introduce a smooth
cutoff function η1 which is equal to one near the interface and is zero away from it. Similarly
we introduce another smooth cutoff function η1 (x) which takes the values ±1 away from the
¯
interface and is zero near the interface. More precisely, we define

η0 (z) =



 1,





|z| ≤ ℓ,

 0, |z| ≥ 2ℓ,




 ∈ (0, 1), ℓ < |z| < 2ℓ;



 1, x ∈ Γ(ℓ),




η1 (x) =
 0, x ∈ Ω\Γ(2ℓ),




 η (z), x ∈ Γ(2ℓ)\Γ(ℓ);
0
η1 (x) =
¯





−1 + η1 (x),


 1 − η (x),
1

x ∈ Ω− ,

(3.1.1)

(3.1.2)

(3.1.3)

x ∈ Ω+ .

We remark that η1 is smooth since η1 ≡ 0 in Γ(ℓ).
¯
¯

We associate to each Γ ∈ GK , b ∈ R the corresponding single-layer dressed ansatz Φ ∈
34

H 5 (Ω),
Φ(x; Γ, b) := η1 (z(x))(φ(z(x)) + ε2 φ2 (s(x), z(x))) + b + η1 (z(x)).
¯

(3.1.4)

Here b = ε3 b3 is an O(ε3 ) parameter that incorporates the small, spatially-constant variation
of the background state of P hi away from the limiting values of ±1.
The function φ is the heteroclinic solution to the one-dimensional Allen-Cahn equation,

φzz − W ′ (φ) = 0,

lim φ = ±1,
z→±∞

(3.1.5)

and φ2 incorporates curvature induced shape corrections into the ansatz,

φ2 (s, z) := −

1 2
H + tr(A2 ) (Π1 Lφ )−1 (zφ′ ),
2

(3.1.6)

where Π1 is the orthogonal projection off of the kernel of Lφ ,
Lφ := ∂zz − W ′′ (φ),

(3.1.7)

is the linearization of (3.1.5) about φ, and

H = tr(A) = −κ0 =
tr(A2 ) = −κ1 =

ki ,
2
ki ,

(3.1.8)

(3.1.9)

are the traces of the first two powers of the Weingarten map A (see Chapter 2). As we will
show in Proposition 3.1, the form of φ2 guarantees that the residual F (Φ) defined in (1.0.6)
35

is O(ε3 ). (Π1 Lφ )−1 (zφ′ ) is well defined, since from Lemma 9.1 the operator Lφ , acting on
L2 (R) has a simple kernel, spanned by φ′ , since zφ′ is orthogonal to φ′ , there is a unique
function f , also orthogonal to φ′ such that Lφ f = zφ′ .

Proposition 3.1. Fix K > 0, then there exists C(K) > 0 such that for all Γ ∈ GK , the
single-layer dressed ansatz Φ = Φ(·; Γ, b) defined in (3.1.4) satisfies

||F (Φ)||L∞ (Ω) ≤ C(K)ε3 .

(3.1.10)

Proof: We recall the definition of F from (1.0.6)

F (u) = (ε2 ∆ − W ′′ (u) + ε2 η)(ε2 ∆u − W ′ (u)).

(3.1.11)

Since W ′ (±1) = 0 and as z → ±∞, φ(z) + ε2 φ2 (s, z) approaches ±1 exponentially fast in |z|,
we need only consider ||F (Φ)||L∞ (Γ(ℓ)) . For x ∈ Γ(ℓ),
Φ(x) = φ(z) + ε2 φ2 (s, z) + ε3 b3 ,

(3.1.12)

and we may change to whiskered variables to obtain

2
2
F (Φ) = (∂z + εκ∂z + ε2 ∆G − W ′′ (Φ) + ε2 η) (∂z Φ + εκ∂z Φ + ε2 ∆G Φ − W ′ (Φ)) . (3.1.13)
A

B
36

In addition Taylor expanding W ′ and W ′′ about φ we obtain,
W ′ (Φ) = W ′ (φ) + (ε2 φ2 + ε3 b3 )W ′′ (φ) + O(ε4 ),

(3.1.14)

W ′′ (Φ) = W ′′ (φ) + (ε2 φ2 + ε3 b3 )W ′′′ (φ) + O(ε4 ).

(3.1.15)

and

Here and in the remainder of the proof O(ε4 ) is in L∞ norm. Substituting in the expressions
for W ′ (Φ) and W ′′ (Φ) yields
2
A = ∂z + εκ∂z + ε2 ∆G − W ′′ (Φ) + ε2 η
2
= ∂z + εκ∂z + ε2 ∆G − W ′′ (φ) − (ε2 φ2 + ε3 b3 )W ′′′ (φ) + ε2 η + O(ε4 )
= Lφ + εκ∂z + ε2 ∆G − ε2 φ2 W ′′′ (φ) + ε2 η + O(ε3 ),
Since φ is a function of only z, we have

||ε2 ∆G Φ|| 2
= ||ε4 ∆G φ2 || 2
= O(ε4 )
C (Γ(ℓ))
C (Γ(ℓ))

(3.1.16)

and

2
B = ∂z Φ + εκ∂z Φ + ε2 ∆G Φ − W ′ (Φ)
2
= φ′′ + ε2 ∂z φ2 + εκφ′ + ε3 κ∂z φ2 − W ′ (φ) − (ε2 φ2 + ε3 b3 )W ′′ (φ) + O(ε4 )
= εκφ′ + ε2 Lφ φ2 + ε3 (κ∂z φ2 − b3 W ′′ (φ)) + O(ε4 ),

37

(3.1.17)

where we used φ′′ − W ′ (φ) = 0. Using the expansions for A and B in (3.1.13) and in addition
using (2.2.22) to expand the extended curvature κ in powers of ε we obtain

F (Φ) = εR1 + ε2 R2 + ε3 R3 + O(ε4 )

(3.1.18)

R1 = κ0 Lφ φ′ = 0,

(3.1.19)

where by Lemma 9.1

1 2
R2 = κ1 Lφ (zφ′ ) + L2 φ2 + κ2 φzz = L2 φ2 +
κ + κ1 Lφ (zφ′ ),
0
φ
φ
2 0

(3.1.20)

and

R3 = Lφ (κ2 z 2 φ′ + κ0 ∂z φ2 ) + L2 b3 + κ0 κ1 zφ′′ + κ0 ∂z (κ1 zφ′ + Lφ φ2 )
φ
+ φ′ (∆G + η)κ0 − κ0 W ′′′ (φ)φ2 φ′ .

(3.1.21)

We note that φ2 was selected in (3.1.6) precisely to make
1 2
R2 = L2 φ 2 +
κ + κ1 Lφ (zφ′ ) = 0.
φ
2 0

(3.1.22)

One can easily verify that the constants in all asymptotic estimates depend on K only.
The following corollary follows from Propositon 3.1 and the fact that F (Φ) is localized on
Γ.

Corollary 3.2. Fix K > 0, then there exists C(K) > 0 such that for all Γ ∈ GK , the
38

single-layer dressed ansatz Φ = Φ(·; Γ, b) defined in (3.1.4) satisfies

||Π0 F (Φ)|| 2
≤ C(K)ε7/2 .
L (Ω)

3.2

(3.1.23)

Linearization and the Functionalized Operator

For each K, ¯ > 0, we define the single-layer dressed manifold
b

MK,¯ := {Φ(x; Γ, b) : Γ ∈ GK , b ∈ (−¯ 3 , ¯ 3 )},
bε bε
b

(3.2.1)

where K is the admissible interface parameter (section 2.4), ¯ is a positive constant and Φ
b
is defined in (3.1.4). By Proposition 3.1 the manifold MK,¯ is comprised of small residual
b
functions Φ obtained by dressing an interface Γ ∈ GK with the heteroclinic solution φ to the
one-dimensional Allen-Cahn equation, and then correcting for higher-order effects, such as
interfacial curvature. We also introduce the linearization

Lφ := (−ε2 ∆ + W ′′ (Φ) − ε2 η)(−ε2 ∆ + W ′′ (Φ)) − W ′′′ (Φ)(ε2 ∆Φ − W ′ (Φ)),

(3.2.2)

of the vector field
F (u) = (ε2 ∆ − W ′′ (u) + ε2 η)(ε2 ∆u − W ′ (u)),
39

(3.2.3)

about Φ, see (1.0.6). It will be convenient to write Lφ in the following factored form
Lφ = (ε2 ∆ − W ′′ (Φ) + ε2 η)2 + ε2 η(W ′′ (Φ) − ε2 η) − W ′′′ (Φ)(ε2 ∆Φ − W ′ (Φ))
q
= A2 + ε˜,
φ

(3.2.4)

where
Aφ := ε2 ∆ − W ′′ (Φ) + ε2 η,

(3.2.5)

1
q := εη(W ′′ (Φ) − ε2 η) − W ′′′ (Φ)(ε2 ∆Φ − W ′ (Φ)).
˜
ε

(3.2.6)

is the Allen-Cahn operator and

We remark that the last term on the right-hand side of (3.2.6) is O(1) (see calculations in
Proposition 3.1). In particular, ||˜||L∞ (Γ) ≤ C(K) where C(K) is a positive constant that
q
depends only on the admissible interface parameter K.

In addition, we define the A-norm of u ∈ H 2 (Ω) as
||u||A := ||∆u|| 2
+ ε−2 ||u|| 2
.
L (Ω)
L (Ω)

(3.2.7)

The objective of this section is to characterize the asymptotically small eigenspace of the operator Lφ : H 2 (Ω) → L2 (Ω). To this end, we first construct a convenient ‘separated variables’
basis for this eigenspace. We start with the Laplace-Beltrami operator ∆s : H 2 (Γ) → L2 (Γ)
40

given locally on Γ by

1
∆s = √
det g

d−1 d−1
i=1 j=1

∂ ij
g
∂si

det g

∂
,
∂sj

(3.2.8)

and denote the (nonnegative) eigenvalues of −∆s by {βj,Γ } and the corresponding eigenfunctions by {Θj,Γ },
−∆s Θj,Γ = βj,Γ Θj,Γ .

(3.2.9)

The Laplace-Beltrami eigenvalues {βj,Γ } play an important role in our analysis. We discuss
them in more detail in Remark 3.4 as well as Chapter 5. We introduce the following subspaces
of L2 (Ω).
Definition 3.3. For each M > 0, the M + 1 dimensional slow space associated to Lφ is
defined by
ZM := span {Zi }i=1...M ∪ {1} ,

(3.2.10)

Zi (x) := η1 (x)Θi,Γ (s)φ′ (z);

(3.2.11)

where

⊥
ZM is the orthogonal complement of ZM in L2 (Ω).
⊥
Remark: We will show that ZM corresponds to the fast eigenspace of Lφ , in the sense that
⊥
the bilinear form we associated with Lφ is coercive on ZM .
Recall that η1 is a cutoff function that is identically zero outside the ‘whiskered’ region
Γ(2ℓ) and φ′ is the ground-stable eigenfunction of the one-dimensional Allen-Cahn operator
and corresponds to the zero eigenvalue (see Lemma 9.1). Let [a] represent the greatest integer
41

less than or equal to a. We will show in Chapter 8 that selecting M = [M1 ε−(d−1) ], where
M1 introduced in Theorem 3.5 is a positive constant independent of ε and Γ ∈ GK , guarantees
the coercivity of the bilinear form associated with Lφ on the orthogonal complement to ZM .
The proof uses the following key relation on the asymptotics of the large eigenvalues of the
Laplace-Beltrami operator, first proved by Weyl (see [11]),
d−1 /ω
(d−1)/2 ∼ (2π)
d m,
(βm )
volΓ

as m → ∞

(3.2.12)

to obtain a uniform bound for βM . The Laplace-Beltrami operator and the formula above
are discussed in more detail in Chapter 5.
Remark 3.4. We note that the Weyl asymptotic formula implies that for M = [M1 ε−(d−1) ]

βM ∼

(2π)2
ε−2 .
2/(d−1)
(volΓωd )

(3.2.13)

Moreover, for each admissible interface parameter K > 0 there exists cg > 0 and an interval
[M1− , M1+ ], independent of ε < ε0 , such that for all Γ ∈ GK there exists M1 ∈ [M1− , M1+ ]
for which the corresponding M satisfies

βM +1 − βM > cg .

(3.2.14)

In addition, the sum of the squares of the Laplace-Beltrami eigenmodes satisfy the following
bound (see [19], [21])
M

sup
Θi (s) 2 ≤ C(K, M1 )ε−(d−1) .
s∈Γ i=1
42

(3.2.15)

The following theorem is proved in Chapter 8.
Theorem 3.5. Coercivity Estimates for the Linear Operator. Fix K, ¯ > 0 and ε > 0
b
sufficiently small. Recall the definition of Aφ from (3.2.5). There exists M1− > 0 independent
of ε, such that for all M1 ≥ M1− the following bounds hold for all w from the associated fast
⊥
space, ZM ,

1 4 2
||A2 w|| 2
φ L (Ω) ≥ 32 ε βM +1 ||w||L2 (Ω) ,

(3.2.16)

1
,
||Aφ w|| 2
≥ ε2 βM +1 ||w| 2
L (Ω)
L (Ω) 8

(3.2.17)

1 2
||A2 w|| 2
φ L (Ω) ≥ 8 ε βM +1 ||Aφ w||L2 (Ω) ,

(3.2.18)

||Aφ w|| 2
≥ Cε4 βM +1 ||w|| 2
,
L (Ω)
H (Ω)

(3.2.19)

Ca ε−2 ||Aφ w|| 2
≤ ||w||A ≤ Cb ε−2 ||Aφ w|| 2
,
L (Ω)
L (Ω)

(3.2.20)

where Ca and Cb are constants that depend on M1 , but are independent of ε and Γ ∈ GK .
Moreover, {Θi,Γ }M form an approximate basis of L2 (Γ), in the sense that any function
i=1
f which is orthogonal to this set in L2 (Γ) satisfies the inequality

||f || 2
≤ Cε||∇s f || 2 ,
L (Γ)
L (Γ)

(3.2.21)

where the constant C depends on M1 , but is independent of ε and Γ ∈ GK .
Remark: As will be shown in the proof of Theorem 3.5, the positivity estimates (3.2.16)(3.2.18) are consequences of the coercivity property of the bilinear form associated with Lφ ,
⊥
when restricted to ZM .
43

3.3
3.3.1

Main Results
Statement of the Main Theorem

In this section we state our main result. We wish to decompose solutions u(x, t) to (1.0.5)
which are in a vicinity of the single-layer manifold MK,¯ (see 3.2.1) as
b
u(x, t) = Φ(x; Γt , b) + w(x, t),

(3.3.1)

⊥
where w ∈ ZM , Γt denotes the interface at time t and the dressed ansatz Φ is determined
from Γt ∈ GK and b ∈ (−¯ 3 , ¯ 3 ) via (3.1.4). Moreover we will show that if the evolution of
bε bε
the local parametrization γ(s, t) of Γt is given by

∂t γ(s, t) = Vn (s, t) · ν(s, t)
γ(s, 0) = γ0 (s)

(3.3.2)

for an appropriate choice of normal velocity Vn , then w will remain small in the || · ||A
norm. Moreover we show that the choice of normal velocity depends at (formal) leading order
only upon the second fundamental form of the interface, and not upon the remainder w. In
0
c
0
particular, Vn = ε4 Vn (h) + ε5 Vn (h, w), with Vn given in (3.3.11).
0
In the following theorem we derive an expression for the leading term Vn of Vn for which the
remainder w stays small under the associated flow. An equivalent description of the evolution
of the interface will be given as a Ricci-curvature flow involving the first two fundamental
forms g = gij and h = hij of the interface Γt . In what follows, g ij is the inverse of gij and
44

∇j is the operator of covariant differentiation on the Riemannian manifold Γ:
d−1
i = ∂ Xi +
Γi X k ,
∇j X
jk
∂sj
k=1

(3.3.3)

and Γi are the Christoffel symbols
jk

1
Γk =
ij
2

d−1

∂
∂
∂
g .
gjl +
gil −
∂si
∂sj
∂sl ij
l=1

(3.3.4)

Assumption 3.6. We assume that the Ricci-curvature driven flow (time has been rescaled to
0
c
¯
τ = ε4 t), given in terms of the rescaled normal velocity Vn := ε−4 Vn = Vn (h) + εVn (h, w)
(see (3.3.26) for the defining equation of Vn ),

¯
∂τ gij = −2Vn hij ,
∂τ hij = −∇i ∇j Vn +

¯
Vn hil g lm hmj ,

(3.3.5)

(3.3.6)

l,m

is locally well posed in H 4 (Γ). That is we assume that the second fundamental form h(t) of
the interface Γt remains uniformly bounded in H 4 (S) for all scaled time τ ∈ [0, Tf ].

Theorem 3.7. Let the space dimension d = 2 or 3 and let Assumption 3.6 hold. Fix the
admissible interface parameters ¯ K > 0 and ¯0 < ¯ K0 < K. Fix the slow-space dimension
b,
b
b,
¯
¯
parameter M1 > 0. Then there exist 1 B0 > B > 0 and U, ε0 , Tf > 0 such that for all
2
45

ε ∈ (0, ε0 ) and for all u0 satisfying
¯
u0 − Φ(·, Γ, b) A < B0 εd−3/2 ,
inf
Γ∈GK
0
¯ ε3 ,¯ ε3 )
b∈(−b0 b0

(3.3.7)

then for t < Tf ε−4 (see Assumption 3.6) we have a decomposition of the solution u(x, t)

u(x, t) = Φ(x, Γt , b(t)) + w(x, t)

(3.3.8)

⊥
where b(t) = ε3 b3 (t), w lies in the L2 orthogonal complement of the slow space, w ∈ ZM ,
with the slow dimension M = [M1 ε−(d−1) ] and
¯
||w||A ≤ B0 εd−3/2 e

−U t
2

¯
+ Bε3/2 .

(3.3.9)

0
c
Moreover, the remainder term in the normal velocity Vn = ε4 Vn + ε5 Vn of the interface Γt
satisfies

c
Vn

L2 (Γ)

0
= ε−5 Vn − ε4 Vn

L2 (Γ)

≤ C(K)

M1 ε(1−d)/2 + ||w||A ε(−2−d)/2
(3.3.10)

0
where Vn takes the following form

0
Vn = −

∆s + η H −

H3
+ Htr(A2 ) −
2

46

2
det(g)||φ′ ||2 2
L (R)

µ 2 b3 − ∂ t b3 ,
0

(3.3.11)

and the background state evolves according to

∂ t b3 + ε

4µ2 |Γ|
2ε
0
b3 = −
|Ω|
det(g)||φ′ ||2 2
|Ω|
L (R)
Γ
+O

M1 ε(8−d)/2 ||w||A ,

∆s + η −

H2
+ tr(A2 ) Hds
2

M1 ε(11−d)/2 .

(3.3.12)

Moreover, the evolution equations (3.3.11) and (3.3.12) result in a two time-scale system,
0
which, after an O(ε−1 ) time drives the background to its quasi-equilibrium, and Vn relaxes to
0
Vn = −Π0,Γ

∆s + η −

H2
+ tr(A2 ) H,
2

(3.3.13)

where Π0,Γ denotes the zero-mass projection over the interface Γ.
Remark 3.8. Under the assumptions of Theorem 3.7, we can formally reformulate the evolution in terms of an equivalent (up to rigid body transformations) higher-order Ricci flow
given by the following leading-order closed system of equations for the fundamental forms of
the interface Γt
0
∂τ gij = −2Vn hij ,
0
∂τ hij = −∇i ∇j Vn +

0
Vn hil g lm hmj .

(3.3.14)
(3.3.15)

l,m

In particular,
0
∂τ H = (∆s + tr(A2 ))Vn ,

(3.3.16)

where time has been rescaled to τ = ε4 t to remove all leading-order ε dependence.
In R2 , choosing an arc-length parametrization of the one-dimensional manifold Γt we
47

can write the leading-order evolution equation for the interface as a generalized KuramotoSivashinsky equation

∂τ H − (∂s H)

s
0

0
2
Vn Hds = − ∂s + H 2 Π0,Γ

2
∂s + η +

H2
2

H.

(3.3.17)

Remark 3.9. Theorem 3.7 gives the normal velocity Vn of the interface Γt as an asymptotic
expression (3.3.10) with an O(ε4 ) leading term and asymptotic constants in the remainder
proportional to

M1 . We note that the slow-space dimension parameter M1 (see Remark

3.4) can be chosen arbitrarily small, independent of ε.

3.3.2

The time to exit the Manifold

Before we develop the technical estimates, we first address the salient issue of developing
a uniform bound on the time Tf to exit the Manifold. This happens when the interface
approaches self-intersection, the spectral gap, βM +1 −βM , collapses, or curvature singularity
develops.
The uniform bound on Vn which we obtain in Proposition 3.11 implies a lower bound on the
time to self-intersection which depends only upon the admissible interface parameter K. Since
the distance to self-intersection is O(1) in the unscaled variables, the time to self-intersection
scales like ε−4 .
From the large eigenvalue asymptotics of the Laplace-Beltrami operator (see Remark 3.4),
for d = 3, we know that the gaps in the spectrum are generically O(1), and moreover one
generically expects the eigenvalues to maintain this separation. However in degenerate cases,
the spectral gap may collapse (in squares for example). However we may address this collapse.
48

From the estimate (3.3.9) we see that the norm of the remainder, w A , becomes less than

¯
B0 on an O(1) time scale, while from Proposition 5.6 we see that βM +1 can become close to
βM only on an O(ε−1 ) time-scale. However, if the gap between βM and βM +1 closes at a
time t1 , then since u(t1 ) will be sufficiently close to the manifold MK we may re-decompose
u(t1 ) according to (3.3.1) with the slow manifold dimension M = [M1 ε−(d−1) ] taking a
slightly different value, corresponding to an adjustment of M1 , for which there is a gap of at
least size cg between βM and βM +1 . In this manner we renormalize through any possible
gap-collapse in the Laplace-Beltrami spectrum, by dynamically adjusting the dimension of
the slow-manifold.
Regarding the development of curvature singularities in finite-time, we first observe that
the dominant, -H 5 , term in the Kuramoto-Sivashinsky type equation (3.3.17) is a favorable
term, and one does not anticipate finite-time singularities. Indeed the Kuramoto-Sivashinsky
equation is globally well-posed [34]. However a full analysis of this problem, in either dimension
2 or 3, is beyond the scope of this thesis. We thus make Assumption 3.6, that the higher-order
Ricci curvature flow leads to uniform bounds on the H 4 (S) norm of the fundamental forms.

3.3.3

Interface Evolution

In the remainder of Section 3.3 we give an outline of the proof of Theorem 3.7. We recall that
the dynamics are governed by the FCH gradient flow

∂t u(x, t) = −Π0 F (u),
u(x, 0) = u0 (x),
49

(3.3.18)

where F and the zero-mass projection Π0 are defined in (1.0.6) and (1.0.7) respectively.
By the assumption of Theorem 3.7 there exists Γ0 ∈ GK and b0 ∈ (−¯0 ε3 , ¯0 ε3 ) such
b
b
0
that
¯
u0 − Φ(·, Γ0 , b0 ) A < B0 εd−3/2 .

(3.3.19)

We show in Chapter 6 that for ε sufficiently small we may represent u0 as a sum of an element
⊥
Φ ∈ MK and a perturbation belonging to ZM,Γ , defined in Definition 3.3, which approx0
0
imates the ‘fast’ eigenspaces of the linearization Lφ about Φ(x; Γ0 , b0 ). This decompositon
allows us to separate the evolution into its projections onto the ‘slow’ and ‘fast’ eigenspaces.
We represent a point in the ‘whiskered neighborhood’ Γ(2ℓ) of Γt as (z(x; t), s(x; t)). Differentiating the ansatz (3.1.4) with respect to t, the chain rule yields

∂t Φ(x; Γt ; b) = ∂z Φ(z(x; t), s(x; t))

∂z(x; t)
∂s(x; t)
+ ε 2 Ds φ 2 ·
+ ∂t b.
∂t
∂t

(3.3.20)

We simplify the time derivatives of the whiskered coordinates with the aid of the following
proposition.
Proposition 3.10. Let (z, s) be the whiskered coordinates of a point x ∈ Γt (2ℓ) with respect
to interface Γt . Then, the following relation holds for the derivative of z with respect to time,
1
∂z(x; t)
= − Vn (s(x; t)).
∂t
ε

(3.3.21)

Proof: Recall the evolution equation

∂γ(s, t)
= Vn (s, t) · ν(s, t),
∂t
50

(3.3.22)

given in (3.3.2). Differentiating ν(s, t) · ν(s, t) = 1 with respect to t and si (i = 1, . . . , d − 1)
gives
∂ν(s, t)
· ν(s, t) = 0.
∂si

(3.3.23)

1
x − γ(s(x; t), t) · ν(s(x; t), t).
ε

(3.3.24)

∂ν(s, t)
· ν(s, t) = 0,
∂t
Differentiating
z(x; t) =
with respect to t and using

1
∂z(x; t)
=
∂t
ε

∂γ
· ν = 0, we obtain
∂si

−Ds γ ·

∂s(x; t)
− Vn ν
∂t

·ν+

1
d
x − γ(s(x; t), t) · ν(s(x; t), t)
ε
dt

1
1
d
= − Vn +
x − γ(s(x; t), t) · ν(s(x; t), t).
ε
ε
dt

(3.3.25)

Since 1 (x − γ(s(x), t)) = zν, (3.3.23) implies that the last term in (3.3.25) is zero.
ε
Substituting the decomposition (3.3.1) into (2.0.1), suppressing the dependence of coordinates on x, and using (3.3.20) and Proposition 3.10, we obtain,

1
∂s(x; t)
− Vn (s, t)∂z Φ + ε2 Ds φ2 ·
+ ∂t b + ∂t w = −Π0 F (Φ + w),
ε
∂t
= −Π0 F (Φ) − Π0 Lφ w − Π0 N (w),

(3.3.26)

where
N (w) = F (Φ + w) − F (Φ) − Lφ w,

(3.3.27)

represents the nonlinear terms in w.
To obtain an evolution equation for Γt , we project (3.3.26) onto the ‘slow space’ ZM,Γ
t
51

(see section 3.2). In particular taking the L2 (Ω) inner product of (3.3.26) with Zi = η1 φ′ Θi,Γ
t
and changing to whiskered variables we obtain

−

2ℓ/ε
1
− ∂t b
Zi dx
Vn Θi,Γ
η (∂ Φ)φ′ Jdzds = − Π0 F (Φ), Zi 2
L (Ω)
t −2ℓ/ε 1 z
ε Γ
Ω
II
I

− (Π0 N (w), Zi ) 2
−
ε 2 Ds φ 2 ·
− Π0 Lφ w, Zi 2
L (Ω)
L (Ω)
Ω

∂s(x; t)
Zi dx + ∂t w, Zi 2
.
L (Ω)
∂t
(3.3.28)

We look for Vn of the form

M
Vn =

vi Θi,Γ
i=1

t

(3.3.29)

and substitute (3.3.29) into (3.3.28). We remark, that we can define the Euclidean norm,

|v|2 =

M
i=1

2
vi .

(3.3.30)

for v = (v1 , . . . , vM ) and it follows from orthonormality of eigenfunctions Θi,Γ that
t

||Vn || 2
= |v|.
L (Γ)

(3.3.31)

We show in Chapter 9 that to the leading-order I corresponds to the projection
vi = Vn , Θi,Γ

of the normal velocity Vn onto the ith Laplace-Beltrami mode Θi,Γ .
t
0
0
0
Similarly II corresponds to the projection ε4 vi := ε4 Vn , Θi,Γ
of ε4 Vn onto Θi,Γ .
t L2 (Γ)
t
t L2 (Γ)

Obtaining bounds on the remaining terms in (3.3.28) and thus showing that I and II are
indeed leading-order is somewhat technical and relies on the estimates derived in the following
52

chapters. For this reason we state the result below and defer the proof until Chapter 9 when
all of the required estimates have been established.

Proposition 3.11. Assuming that (3.3.28) holds for each i = 1, . . . , M , and that w satisfies

||w||A ≤ Mb εd−3/2 ,

(3.3.32)

for some constant Mb which depends only on the admissible interface parameter K, then
0
vi − ε4 vi ≤ C(K) ε5 + ε7/2 ||w||A ,

(3.3.33)

||Vn ||L∞ (Γ) ≤ M1 C(K) ε5−(d−1) + ε7/2−(d−1) ||w||A ,

(3.3.34)

0
V n − ε4 V n

L2 (Γ)

M1 ε(11−d)/2 + ε(8−d)/2 ||w||A ,

(3.3.35)

H3
2
+ Htr(A2 ) −
µ 2 b3 − ∂ t b3 .
0
′ ||2
2
||φ
2 (R)
L

(3.3.36)

≤ C(K)

where

0
Vn = −

∆s + η H −

and M1 is the slow-space dimension parameter. In particular,

||Vn || 2
≤ C(K)
L (Γ)

M1 ε(11−d)/2 + ε(8−d)/2 ||w||A .

(3.3.37)

Our objective in this section is to obtain an energy inequality for w, which we will use
53

to show that ||w||A decays to O(ε3/2 ). This, in turn, implies that the right-hand side of
(3.3.35) is C(K)

0
M1 ε(11−d)/2 and Vn is given at leading-order by ε4 Vn . In the following

proposition we use C(K) to denote a constant that depends only on the admissible interface
parameter K.

1¯
Proposition 3.12. Under the assumptions of Proposition 3.11, there exist constants 2 B0 >
¯
B > 0, such that the perturbation w satisfied the following estimate in the || · ||A norm,
−U t ¯
¯
||w||A ≤ B0 εd−3/2 e 2 + Bε3/2 .

(3.3.38)

Proof: We first rewrite equation (3.3.26) in the following form

∂t w = −Π0 Lφ w − Π0 N w + R,

(3.3.39)

R := −∂t Φ − Π0 F (Φ).

(3.3.40)

where

Recall the definition of Aφ in (3.2.5). The idea is to use the coercivity of the operator Lφ
to show that ||w||A (and due to Theorem 3.5 also ε−2 ||Aφ w|| 2
) decay to O(ε3/2 ).
L (Ω)
Similarly to the previous section we defer the more technical estimates to Chapter 9.
q
Multiplying both sides of (3.3.39) by A2 w and writing out Lφ = A2 + ε˜ from (3.2.4), we
φ
φ
54

obtain

2
2
q
(∂t w, A2 w) 2
φ L (Ω) = −(Aφ w + ε˜w, Aφ w)L2 (Ω) +

Lφ w , A2 w 2
φ L (Ω)

2
− (Π0 N (w), A2 w) 2
φ L (Ω) + (R, Aφ w)L2 (Ω)
2
= −(A2 w, A2 w) 2
q
φ
φ L (Ω) − ε(˜w, Aφ w)L2 (Ω) + |Ω| Lφ w
2
− (Π0 N (w), A2 w) 2
φ L (Ω) + (R, Aφ w)L2 (Ω) .

A2 w
φ
(3.3.41)

Integrating by parts on the left-hand side we obtain
2
1
2
2
q
= −(A2 w, A2 w) 2
∂ t Aφ w 2
φ L (Ω) − ε(˜w, Aφ w)L2 (Ω) + |Ω| Lφ w Aφ w
φ
2
L (Ω)
2
− (Π0 N (w), A2 w) 2
φ L (Ω) + (R, Aφ w)L2 (Ω) + ∂t , Aφ w, Aφ w L2 (Ω) .
(3.3.42)

We show in Lemma 9.13 that

≤ C(K)ε1/2 ||Aφ w||2 2
.
∂t , Aφ w, Aφ w 2
L (Ω)
L (Ω)

(3.3.43)

Using coercivity estimates from Theorem 3.5 and βM +1 ∼ ε−2 ,
1
≤ C(K)ε1/2 ||A2 w||2 2
≤ ||A2 w||2 2
∂t , Aφ w, Aφ w 2
φ L (Ω) 16 φ L (Ω)
L (Ω)

(3.3.44)

for ε sufficiently small. Similarly, since q L∞ (Ω) = O(1) and w 2
˜
≤ U1 A2 w 2
φ L (Ω)
L (Ω)

55

we have

q
ε (˜w, A2 w) 2
φ L (Ω)

A2 w 2
≤ ε q L∞ (Ω) w 2
˜
φ L (Ω)
L (Ω)
2
1
2 2
≤ C(K)ε A2 w 2
φ L (Ω) ≤ 16 Aφ w L2 (Ω) .

(3.3.45)

By Young’s Inequality

(Π0 N (w), A2 w) 2
φ L (Ω)

2
1
≤ 2 Π0 N (w) 2 2
+
A2 w 2
φ L (Ω)
L (Ω) 8
2
1
≤ 2 N (w) 2 2
+
A2 w 2
φ L (Ω)
L (Ω) 8

(3.3.46)

and
2
1
≤ 2 R 22
+
(R, A2 w) 2
A2 w 2
φ L (Ω)
φ L (Ω) .
L (Ω) 8

(3.3.47)

In addition, for ε sufficiently small, the following estimate holds and is shown in Proposition
9.4
Lφ w

A2 w
φ

1
≤ ||A2 w||2 2
≤ C(K)ε||A2 w||2 2
φ L (Ω) 8 φ L (Ω) .

(3.3.48)

Substituting (3.3.44)-(3.3.48) into (3.3.42), we obtain
2
2
1
1
2
2
≤ − A2 w 2
∂ t Aφ w 2
φ L (Ω) + 2 N w L2 (Ω) + 2 R L2 (Ω) .
2
2
L (Ω)

(3.3.49)

Coercivity estimate on A2 w from Theorem 3.5 and βM +1 ∼ ε−2 imply
φ
2
2
+ 4 Nw 22
≤ −U1 Aφ w 2
∂ t Aφ w 2
+4 R 22
,
L (Ω)
L (Ω)
L (Ω)
L (Ω)

56

(3.3.50)

where U1 is a positive constant independent of ε and Γ ∈ GK . In addition, we can bound
R = −∂t Φ − Π0 F (Φ) using Lemma 9.12 and Corollary 3.2,
R

L2 (Ω)

= −∂t Φ − Π0 F (Φ) 2
≤ C(K) ε−1/2 ||Vn || 2
+ ε7/2
L (Ω)
L (Γ)

(3.3.51)

and the nonlinear terms using Lemma 9.2

||N (w)|| 2
≤ C(K) ε−2 ||Aφ w||2 2
,
+ ε−4 ||Aφ w||3 2
L (Ω)
L (Ω)
L (Ω)

(3.3.52)

to obtain

∂t ||Aφ w||2 2
≤ −U1 ||Aφ w||2 2
+ Cn (K) ε−4 ||Aφ w||4 2
+ ε−8 ||Aφ w||6 2
L (Ω)
L (Ω)
L (Ω)
L (Ω)
+ ε−1 ||Vn ||2 2
+ ε7 ,
(3.3.53)
L (Γ)
where Cn (K) is a positive constant that depends only on K. From (3.2.20) and (3.3.32)

Aφ w

2
d+1/2 ,
2 (Ω) ≤ C(K)ε ||w||A ≤ Mb C(K)ε
L

(3.3.54)

and it follows that for ε sufficiently small

Cn (K) ε−4 ||Aφ w||4 2
+ ε−8 ||Aφ w||6 2
L (Ω)
L (Ω)

57

≤

U1
||Aφ w||2 2
,
L (Ω)
2

(3.3.55)

and from (3.3.53) we obtain

U
+ Cn (K) ε−1 ||Vn ||2 2
+ ε7 .
∂t ||Aφ w||2 2
≤ − 1 ||Aφ w||2 2
L (Ω)
L (Γ)
L (Ω)
2

(3.3.56)

From Proposition 3.11

||Vn || 2
≤ C(K)
L (Γ)

M1 ε(11−d)/2 + ε(4−d)/2 ||Aφ w|| 2
,
L (Ω)

(3.3.57)

where we used the equivalence of ε2 ||w||A and ||Aφ w|| 2
from Theorem 3.5, and returning
L (Ω)
this estimate to (3.3.56) gives for ε sufficiently small

U
∂t ||Aφ w||2 2
≤ − 1 ||Aφ w||2 2
+ C(K)ε7 ,
L (Ω)
L (Ω)
4

(3.3.58)

Gronwall’s Lemma now implies

||Aφ w||2 2
≤ ||Aφ w0 ||2 2
e−U t + C(K)ε7 ,
L (Ω)
L (Ω)

(3.3.59)

U
where U := 41 .
Using equivalence of ε2 ||w||A and ||Aφ w|| 2
from Theorem 3.5 as well as the initial
L (Ω)
condition (3.3.19) we obtain from (3.3.59) the energy inequality for ||w||A
−U t
¯
||w||A ≤ B0 εd−3/2 e 2 + C(K)ε3/2 .

58

(3.3.60)

b
b
By the assumption of Theorem 3.7 there exists Γ0 ∈ GK and b0 ∈ (−¯0 ε3 , ¯0 ε3 ) such
0
that
¯
w0 A ≤ B0 εd−3/2 .

(3.3.61)

It follows from the decay inequality on ||w||A that (3.3.32) is always satisfied and (3.3.9)
follows.

3.3.4

Background Evolution

Proposition 3.13. The leading order evolution of the background state b = ε3 b3 is given by

∂ t b3 +

4εµ2 |Γ|
2ε
0
b3 = −
|Ω| Γ
||φ′ ||2 2
|Ω|
L (R)

∆s + η H −

H3
+ Htr(A2 ) ds.
2

(3.3.62)

Proof: The evolution of the background state is determined from the total mass conservation.
Projecting (3.3.26) onto Z0 = 1, the right-hand side is identically zero and we obtain
1
∂s(x; t)
− Vn (s, t)Φz + ε2 Ds φ2 ·
+ ∂t b + ∂t wdx = 0.
∂t
Ω ε

(3.3.63)

⊥
It also follows from w ∈ ZM that Ω ∂t wdx = ∂t Ω wdx = 0. Replacing Vn with its leading0
order term ε4 Vn and using Corollary 9.6 to bound ∂s , as well as using [|φ|] = 2 we obtain
∂t
∞
ε3
ε3
0
0
Vn Φz dx + O(ε||Vn || 2 ) = −
Vn
φ′ Jdzds + O(ε||Vn || 2 )
L (Γ)
L (Γ)
|Ω| Ω
|Ω| Γ
−∞
∞
2ε4
ε4
0
Vn
V 0 ds + O(ε||Vn || 2 ).
φ′ dz J 0 ds + O(ε||Vn || 2 ) =
=
L (Γ)
L (Γ)
|Ω| Γ
|Ω| Γ n
−∞

∂t b =

(3.3.64)

59

Substituting the normal velocity, (3.3.36), into this expression and dropping the lower order
terms yields an ordinary differential equation for b3 ,
4εµ2 |Γ|
2ε
0
b3 = −
∂ t b3 +
2
|Ω| Γ
||φ′ || 2
|Ω|
L (R)

∆s + η H −

H3
+ Htr(A2 ) ds.
2

(3.3.65)

We note that the equilibrium value of b3 is given by
||φ′ ||2 2
L (R)
b3 = −
2µ2 |Γ|
Γ
0

∆s + η H −

H3
+ Htr(A2 ) ds.
2

(3.3.66)

Substituting the expression for b3 into the equation for the normal velocity (3.3.36) we obtain
(3.3.13).

3.4

Evolution in R2

Proof of Corollary 3.8: In R2 , the fundamental forms g and h become scalar functions on
a one-dimensional manifold Γ. We have

g = |∂s γ|2 ,

(3.4.1)

2
h = ν, ∂s γ ,

(3.4.2)

H = k = h/g =
60

h
,
|∂s γ|2

(3.4.3)

and the leading-order evolution equations take the following form

0
∂τ |∂s γ|2 = −2Vn H|∂s γ|2 ,
∂τ H =

(3.4.4)

1
1
0
2
2
∂s −
∂ γ, ∂s γ ∂s + H 2 Vn .
2
2 s
|∂s γ|
|∂s γ|

(3.4.5)

To obtain (3.3.17) we choose an arc-length parametrization
s

s1 =

g(ξ, t)dξ.

0

(3.4.6)

Then g(s1 ) ≡ 1. Let H1 (s1 , t) be the mean curvature in new variable s1 ,
H(s, t) = H1 (s1 , t).

(3.4.7)

∂H1 ∂H1 ∂s1
∂H
=
+
.
∂t
∂t
∂s1 ∂t

(3.4.8)

Then

and
s ∂
∂s1
=
(
∂t
0 ∂t
where we used ∂
∂t

g(ξ, t))dξ = −

det(g) = −Vn H

s
0

Vn (ξ, t)H(ξ, t)

g(ξ, t)dξ,

(3.4.9)

det(g) (see Proposition 4.6).

Introducing a new variable
ρ=

ξ
0

g(ξ1 , t)dξ1 ,
61

(3.4.10)

we have
s1
∂s1
=−
Vn (ρ, t)H(ρ, t)dρ.
∂t
0

(3.4.11)

Therefore, rewriting (3.4.5) in terms of the new variable s1 and employing g(s1 ) = 1, we
obtain from (3.4.8) and (3.4.11), that
∂H1 ∂H1 s1
2
0
−
Vn (ρ, t)H(ρ, t)dη = (∂s + H 2 )Vn .
1
∂t
∂s1 0

62

(3.4.12)

Chapter 4
Evolution by Normal Velocity
In this chapter we expand on key differential geometry concepts that relate to interface evolution. In particular we derive equations (3.3.14)-(3.3.16) as well as evolution equations for the
principal curvatures of the interface. These relations are key to the analysis of the spectrum
of Laplace-Beltrami operator in the following chapter.

4.1

Evolution Equations for the Fundamental Forms

We consider a family of interfaces {Γt : t ∈ [0, T ]} evolving by normal velocity Vn (s, t)
according to the evolution equation given by

∂t γ(s, t) = Vn (s, t) · ν(s, t)
γ(s, 0) = γ0 (s),

where γ(s, t) gives a local parametrization of Γt .
63

(4.1.1)

Proposition 4.1. (See [23], [25]) The following equations hold:
i) Weingarten Equation
∂ν
=−
∂si

d−1

∂γ
,
∂sk

(4.1.2)

∂γ
+ hij · ν,
∂sk

(4.1.3)

hk
i

k=1

ii) Gauss Formula
∂ 2γ
=
∂si sj

d−1

Γk
ij

k=1

where Γk is the Christoffel symbol
ij

Γk =
ij

1
2

d−1

∂
∂
∂
g .
gjl +
gil −
∂si
∂sj
∂sl ij
l=1

(4.1.4)

Proposition 4.2. Under the evolution equation given in (4.1.1) the first fundamental form
satisfies
∂
g = −2Vn hij .
∂t ij

(4.1.5)

Proof: Differentiating expression (2.1.8) for the first fundamental form, we obtain

∂
∂
gij =
∂t
∂t

∂γ ∂γ
,
∂si ∂sj

=

Using evolution equation (4.1.1) to replace

∂
g =
∂t ij

∂ 2 γ ∂γ
,
∂t∂si ∂sj

+

∂γ ∂ 2 γ
,
∂si ∂t∂sj

.

(4.1.6)

∂γ
yields
∂t

∂(Vn · ν) ∂γ
,
∂si
∂sj

+

∂γ ∂(Vn · ν)
,
∂si
∂sj

.

(4.1.7)

When the derivatives in the right-hand side of (4.1.7) fall on Vn the resulting inner product
64

is zero since ν is orthogonal to the tangent space and we conclude

∂
g = Vn
∂t ij

∂ν ∂γ
,
∂si ∂sj

∂γ ∂ν
,
∂si ∂sj

+

= −2Vn hij ,

(4.1.8)

where we used the definition of the second fundamental form in (2.1.11).

Proposition 4.3. Under the evolution equation given in (4.1.1) the inverse of the first fundamental form satisfies
∂ ij
g = 2Vn
∂t

d−1
k=1

hi g kj .
k

(4.1.9)

Proof: Differentiating
d−1

g ik gkj = δij ,

(4.1.10)

(∂t g ik )gkj + g ik ∂t (gkj ) = 0.

(4.1.11)

k=1
we obtain
d−1
k=1
Using Proposition 4.2 we substitute the expression for ∂t (gkj ) to obtain
d−1
k=1

(∂t g ik )gkj − 2Vn g ik hkj = 0,

(4.1.12)

and
d−1

(∂t g ik )gkj = 2Vn hi ,
j

(4.1.13)

k=1
where we used (2.1.16) to raise the index on hkj .

Proposition 4.4. Under the evolution equation given in (4.1.1) the unit normal vector ν
65

satisfies
∂
ν=−
∂t

d−1

g ij

i,j=1

∂Vn ∂γ
.
∂si ∂sj

(4.1.14)

Proof: Differentiating the relation ν, ν = 1 with respect to t implies that ∂ν ∈ Tγ(s) Γ and
∂t
we may express ∂ν in terms of the tangent basis as
∂t
∂ν
=
∂t

d−1
aj
j=1

∂γ
.
∂sj

(4.1.15)

∂γ
To find the coefficients aj we take the inner product of (4.1.15) with
to obtain
∂si
∂ν ∂γ
,
∂t ∂si

d−1
=

aj gji .

(4.1.16)

j=1

and
d−1
aj =
i=1
Since

ν,

∂γ
∂si

g ij .

(4.1.17)

=0

d−1
aj = −

∂ν ∂γ
,
∂t ∂si

i=1

∂ 2γ
ν,
∂t ∂si

g ij = −

d−1
i=1

∂(Vn · ν)
ν,
∂si

g ij = −

d−1
i=1

∂Vn ij
g .
∂si

(4.1.18)

Proposition 4.5. Under the evolution equation given in (4.1.1) the second fundamental form
66

satisfies
d−1
∂ 2 Vn
∂Vn
∂
hij =
−
Γk
ij ∂s − eij Vn .
∂t
∂si ∂sj
k
k=1

(4.1.19)

Proof: Differentiating expression (2.1.11) for the second fundamental form and using the
evolution equation (4.1.1) to replace

∂
∂
hij =
∂t
∂t

∂ 2γ
·ν
∂si ∂sj

∂γ
and Proposition 4.4 to replace ∂ν we obtain
∂t
∂t

∂2
=
(Vn · ν) · ν −
∂si ∂sj

d−1
k,l=1

∂ 2 γ kl ∂Vn ∂γ
.
g
∂si ∂sj
∂sk ∂sl

(4.1.20)

∂γ
= 0 we distribute the derivatives in the first term on the right-hand side of
∂si
(4.1.20) to conclude
Using

ν,

∂
∂ 2 Vn
hij =
+ Vn
∂t
∂si ∂sj

∂ 2ν
,ν
∂si ∂sj

d−1
−

k,l=1

∂ 2 γ kl ∂Vn ∂γ
,
g
∂si ∂sj
∂sk ∂sl

(4.1.21)

and
d−1
∂ 2 Vn
∂
∂ 2 γ kl ∂Vn ∂γ
,
hij =
− eij Vn −
g
∂t
∂si ∂sj
∂si ∂sj
∂sk ∂sl
k,l=1

(4.1.22)

where we used the expression for the third fundamental form eij in (2.1.14). To simplify the
second term on the right-hand side of (4.1.22) we use the Gauss Formula of Proposition 4.1
to obtain
d−1
k,l=1

d−1
∂Vn kl
m ∂γ g kl ∂Vn ∂γ =
Γij
Γm
ij ∂s g gml
∂sm
∂sk ∂sl
k
k,l,m=1
k,l,m=1
d−1
∂Vn
(4.1.23)
=
Γk
ij ∂s .
k
k,l=1

∂ 2 γ kl ∂Vn ∂γ
=
g
∂si ∂sj
∂sk ∂sl

d−1

67

Proposition
det(g) =

4.6. Under

the

evolution

equation

given

in

(4.1.1)

the

metric

det({gij }) satisfies
∂
∂t

det(g) = −Vn H

det(g).

(4.1.24)

Proof: To evaluate the time derivative of det(g) we utilize the Leibniz formula for determinant
of g
d−1
sgn(σ)

det(g) =

i=1

σ∈Sn

gi,σ ,
i

(4.1.25)

where the sum is computed over all permutations σ of the set {1, 2, ..., d − 1}. Differentiating
(4.1.25) with respect to t and using the evolution equation for gij given in (4.2) we obtain

∂
det(g) = −2Vn
∂t

d−1

d−1
sgn(σ)hi,σ

σ∈Sn i=1

i

j=1,j=i

gj,σ .
j

(4.1.26)

Using (2.1.16) we write the above expression in terms of the Weingarten matrix {hk }
i
∂
det(g) = −2Vn
∂t

For k = i, the expression

d−1
σ∈Sn i,k=1

σ∈Sn gk,σi

sgn(σ)hk gk,σ
i
i

d−1
gj,σ
j=1,j=i

j

(4.1.27)

d−1
g
is equal to the determinant of a
j=1,j=i j,σj
68

matrix with two identical rows and is identically 0. The remaining terms give

∂
det(g) = −2Vn
∂t

d−1

sgn(σ)hi gi,σ
i
i

d−1
gj,σ

j
σ∈Sn i=1
j=1,j=i
d−1
d−1
i
sgn(σ)
gj,σ = −2Vn H det(g),
hi
= −2Vn
j
j=1
i=1 σ∈Sn

(4.1.28)

j
where we used the definition of H as the trace of the Weingarten matrix {hi }. Relation
(4.1.24) follows easily.

4.2

Evolution of Principal Curvatures

Recall that the principal curvatures k of Γ are defined to be the eigenvalues of the Weingarten
j
map {hi }. Setting g = {gij } and h = {hij } for the matrices of the first two fundamental
j
forms of Γ and using {hi } = g −1 h (see (2.1.16)), the corresponding eigenvalue equation is
hv = kgv,

(4.2.1)

where v is the normalized eigenvector corresponding to the eigenvalue k.

Theorem 4.7. Under the evolution equation given in (4.1.1) the principal curvatures ki satisfy
∂
2 ((∂t h)vi , vi ) ,
ki = 2Vn ki +
∂t
(gvi , vi )
where vi ∈ Rd−1 is the corresponding eigenvector.
69

(4.2.2)

Proof: To obtain an expression for ∂ k we differentiate the eigenvalue equation (4.2.1) with
∂t
respect to t
(∂t h)v + h(∂t v) = (∂t k)gv + k(∂t g)v + kg(∂t v),

(4.2.3)

and take the inner product with v to conclude

((∂t h)v, v) + (h∂t v, v) = ∂t k(gv, v) + k((∂t g)v, v) + k(g∂t v, v).

(4.2.4)

Since g and h are self-adjoint, the eigenvalue equation (4.2.1) implies

(h∂t v, v) = (∂t v, hv) = k(∂t v, gv) = k(g∂t v, v),

(4.2.5)

and we may cancel the corresponding terms in (4.2.4) yielding

((∂t h)v, v) = ∂t k(gv, v) + k((∂t g)v, v).

(4.2.6)

Solving (4.2.6) for ∂t k and using Proposition 4.2 we obtain
((∂t h)v, v) − k((∂t g)v, v)
((∂t h)v, v) + 2Vn k(hv, v)
=
(gv, v)
(gv, v)
((∂t h)v, v) + 2Vn k 2 (gv, v)
((∂t h)v, v)
=
=
+ 2Vn k 2 .
(gv, v)
(gv, v)

∂t k =

70

(4.2.7)

Corollary 4.8. We have the following bound

∂
k (t)
≤ C Vn 2
,
H (Γ)
∂t i
L2 (Γ)

(4.2.8)

where t ∈ [0, T ] and C is a positive constant independent of ε and Γ ∈ GK .
Proof: The estimate follows from Theorem 4.7 and the evolution equation in Proposition 4.5.

71

Chapter 5

Eigenvalues of the Laplace-Beltrami
Operator

5.1

Weyl’s Asymptotic Formula for the Eigenvalues

Recall that we consider a family of interfaces {Γt : t ∈ [0, T ]} evolving by normal velocity
Vn (s, t) according to the evolution equation given by

∂t γ(s, t) = Vn (s, t) · ν(s, t)
γ(s, 0) = γ0 (s),

where γ(s, t) gives a local parametrization of Γt .
72

(5.1.1)

In Chapter 3 we introduced the Laplace-Beltrami operator

1
−∆s = − √
det g

d−1 d−1
i=1 j=1

∂ ij
g
∂si

det g

∂
.
∂sj

(5.1.2)

on Γt and the corresponding eigenvalue/eigenfunction pairs {Θi,Γ , βi,Γ }M . In Chapter 8
t
t i=1
we show that selecting
M := [M1 ε−(d−1) ],

(5.1.3)

where M1 > 0 is a constant independent of ε and Γ ∈ GK and [a] represents the greatest
integer less than or equal to a, guarantees that the bilinear form associated with the linearization Lφ (see (3.2.2)) is coercive on the orthogonal complement to the ‘center space’
ZM = span{1, η1 φ′ Θi,Γ }M . The proof relies on the following asymptotic bound for the
t i=1
Laplace-Beltrami eigenvalue βM,Γ (see [11])
t
(βM )(d−1)/2 ∼

(2π)d−1 /ωd
M,
volΓt

(5.1.4)

as M → +∞, where ωd is the volume of the unit disk in Rd−1 .

5.2

Rayleigh Characterization of Eigenvalues

We start this section with the Rayleigh characterization of eigenvalues of −∆s . The following
can be found in [11] and [14] for example.

Proposition 5.1. The i-th eigenvalue of the weighted Laplace-Beltrami operator −∆s is char73

actarized by the following Rayleigh principle

min (∇s Θ, ∇s Θ)Γ ,
Θ∈Vi⊥
||Θ||2 =1
Γ

βi =

(5.2.1)

where Vi := span{Θ1 , . . . , Θi−1 },
j
∇s =

∇s f, ∇s g

Γ

d−1
i=1

:=

Γ

g ij

∂
∂si

∇s f, ∇s g ξ(s)ds.

(5.2.2)

(5.2.3)

Proposition 5.2. The following identities hold for the tangential derivatives of the eigenfunctions Θi .
∇s Θi,Γ

1/2
= βi,Γ ,
t Γ
t

(5.2.4)

∆s Θi,Γ

= βi,Γ ,
t

(5.2.5)

t Γ

∇s ∆s Θi,Γ

t

3/2
= βi,Γ .
Γ
t

(5.2.6)

Proof: From the definition of Θi,Γ as the normalized eigenfunction of −∆s corresponding
t
to eigenvalue βi,Γ we have
t
2
∆2 Θi,Γ = βi,Γ Θi,Γ ,
s
t
t
t

(5.2.7)

Since ||Θi,Γ ||Γ = 1 taking the L2 (Γ) inner product of (5.2.7) with Θi,Γ yields
t
t
2
(∆2 Θi,Γ , Θi,Γ )Γ = βi,Γ .
s
t
t
t
74

(5.2.8)

Since Γ has no boundary, integrating by parts we obtain

2
(∆s Θi,Γ , ∆s Θi,Γ )Γ = βi,Γ ,
t
t
t

(5.2.9)

||∆s Θi,Γ ||Γ = |βi,Γ | = βi,Γ .
t
t
t

(5.2.10)

and

Similarly we integrate by parts and use (5.2.10) to obtain

||∇s Θi,Γ ||2 = (∇s Θi,Γ , ∇s Θi,Γ )Γ = (−∆s Θi,Γ , Θi,Γ )Γ = βi,Γ ||Θi,Γ ||Γ = βi,Γ ,
t Γ
t
t
t
t
t
t
t
(5.2.11)
and
2
∇s (∆s Θi,Γ )
=
t Γ

= − ∆2 Θi,Γ , ∆s Θi,Γ
∇s (∆s Θi,Γ ), ∇s (∆s Θi,Γ )
s
t
t Γ
t
t Γ
3
3
= βi,Γ Θi,Γ
= βi .
(5.2.12)
t Γ
t

5.3

Smooth Dependence of Eigenfunctions on Time

In order to show that the term involving ∂t w in formula (3.3.28) is lower order we need the
following technical estimate.

⊥
Theorem 5.3. For any differentiable function w which satisfies w(x, t) ∈ ZM we have the
75

estimate

≤ C(K)
(∂t w, Zi ) 2
L (Ω)

1
√ +
ε

√
βi ε
βM +1 − βi

Vn L∞ (Γ) ||w|| 2
,
L (Ω)

(5.3.1)

where Zi = η1 φ′ Θi,Γ span the ‘slow’ space ZM and C(K) is a positive constant that depends
t
only on the admissible interface parameter K.
Proof: First, we note that differentiating the identity (w, Zi ) 2
= 0 with respect to t
L (Ω)
yields
(∂t w, Zi ) 2
+ (w, ∂t Zi ) 2
= 0,
L (Ω)
L (Ω)

(5.3.2)

Using Zi = η1 φ′ Θi,Γ it follows from (5.3.2) that
t
(∂t w, Zi ) 2
= −(w, ∂t Zi ) 2
= −(w, ∂t (η1 φ′ Θi,Γ )) 2
L (Ω)
L (Ω)
t L (Ω)
∂z
= − w, ∂z (η1 φ′ ) Θi,Γ
− w, η1 φ′ ∂t Θi,Γ
t L2 (Ω)
t L2 (Ω)
∂t
=

1
w, ∂z (η1 φ′ )Vn Θi,Γ
− w, η1 φ′ ∂t Θi,Γ
,(5.3.3)
t L2 (Ω)
t L2 (Ω)
ε

where we used Proposition 3.10 in the last step.
We next decompose ∂t Θi,Γ into its projections onto the first M eigenfunctions of −∆s
t
M
∂t Θi,Γ =
t

j=1

aij Θj,Γ + Θ⊥ ,
i
t

(5.3.4)

⊥
where (Θj,Γ , Θ⊥ )Γ = 0. Since w ∈ ZM where ZM = span{η1 φ′ Θj,Γ }, the only cont i
t
comes from the Θ⊥ term in (5.3.4) and after replacing
tribution to w, η1 φ′ ∂t Θi,Γ
i
t L2 (Ω)
76

∂t Θi,Γ with Θ⊥ in (5.3.3) we obtain
i
t
1
w, ∂z (η1 φ′ )Vn Θi,Γ
− w, η1 φ′ Θ⊥ 2
(∂t w, Zi ) 2
=
i L (Ω) .
L (Ω) ε
t L2 (Ω)

(5.3.5)

It follows that

(∂t w, Zi ) 2
L (Ω)

1
||Vn ||L∞ (Γ) ||w|| 2
||∂ (η φ′ )Θi,Γ || 2
L (Ω) z 1
t L (Ω)
ε
+ ||w|| 2
||η φ′ Θ⊥ || 2
i L (Ω) .
L (Ω) 1

≤

(5.3.6)

We estimate the two terms on the right-hand side of (5.3.6) with the aid of the following
lemmas.

Lemma 5.4.
∂z (η1 φ′ )Θi,Γ

√
2 (Ω) ≤ C(K) ε.
t L

(5.3.7)

Proof of Lemma 5.4: We observe that since ||Θi,Γ || 2
= 1 and the Jacobian J is O(ε)
t L (Γ)
in L∞ (Γ(2ℓ)),
∂z (η1 φ′ )Θi,Γ

2
=
t L2 (Ω)

2ℓ/ε

(∂z (η1 φ′ ))2 Jdzds
t −2ℓ/ε
Γ
≤ Cε||Θi,Γ ||2 2
= Cε.
t L (Γ)
Θ2
i,Γ

(5.3.8)

Lemma 5.5. Consider the decomposition of ∂t Θi,Γ given in (5.3.4). Under the assumptions
t
77

of Theorem 5.3 we have the estimate

η1 φ ′ Θ ⊥
i

≤
L2 (Ω)

βi ε
βM +1 −

βi

C(K)||Vn ||L∞ (Γ) .

(5.3.9)

where C(K) is a positive constant independent of ε and Γ ∈ GK .

Proof of Lemma 5.5:
To obtain an expression for the derivative of Θi,Γ with respect to time we differentiate
t
the eigenvalue equation
−∆s Θi = βi Θi

(5.3.10)

−∆s ∂t Θi − [∂t , ∆s ]Θi = (∂t βi )Θi + βi ∂t Θi ,

(5.3.11)

with respect to t to obtain

where (5.1.2) implies
√
1 ∂t det g
[∂t , ∆s ]Θi := ∂t ∆s Θi − ∆s (∂t Θi ) = − √
∆s Θi
2 det g
d−1 d−1
∂Θi
∂
∂
1
det gg kj
.
+ √
∂sk ∂t
∂sj
det g
k=1 j=1

(5.3.12)

Using decomposition (5.3.4) in (5.3.11) we obtain


M

M



aij βj Θj − ∆s Θ⊥ − [∂t , ∆s ]Θi = βi 
aij Θj + Θ⊥  + (∂t βi )Θi (5.3.13)
i
i
j=1
j=1
78

Taking the ( ,

)Γ inner product with Θ⊥ and recalling that (Θj , Θ⊥ )Γ = 0 gives
i
i
−(∆s Θ⊥ , Θ⊥ )Γ − ([∂t , ∆s ]Θi , Θ⊥ )Γ = βi ||Θ⊥ ||2 2 .
i L (Γ)
i
i
i

(5.3.14)

Integrating by parts in s we obtain

⊥ 2
⊥
βi ||Θ⊥ ||2 2
i L (Γ) − ||∇s Θi ||L2 (Γ) = − [∂t , ∆s ]Θi , Θi Γ
and

(5.3.15)

||∇s Θ⊥ ||2 2
i L (Γ)
([∂t , ∆s ]Θi , Θ⊥ )Γ
i
= βi +
.
⊥ ||2
⊥ ||2
||Θi
||Θi
L2 (Γ)
L2 (Γ)

(5.3.16)

We will use the above inequality to obtain an estimate on ||Θ⊥ || 2 . First, we observe that
i L (Γ)
√
∂t det g
√
∆s Θi , Θ⊥
i
det g
Γ

d−1 d−1
∂Θi ⊥ 
1
∂
∂
,Θ
+ √
det gg kj
∂sk ∂t
∂sj i
det g
k=1 j=1
Γ
1
⊥
= − βi Vn HΘi , Θi
Γ
2

d−1 d−1
∂Θ⊥
∂Θi
1
∂
i  , (5.3.17)
− √
,
det gg kj
∂t
∂sj ∂sk
det g
k=1 j=1
Γ

1
([∂t , ∆s ]Θi , Θ⊥ )Γ = −
i
2


√
where we used Proposition 4.6 to evaluate ∂t det g. In addition, applying Proposition 4.3 to

79

estimate the derivative with respect to t of g kj in (5.3.17) we obtain,

([∂t , ∆s ]Θi , Θ⊥ )Γ
i

≤ C(K) βi ||Vn ||L∞ (Γ) ||Θ⊥ || 2
i L (Γ)
+ ||Vn ||L∞ (Γ) ||∇s Θi || 2 ||∇s Θ⊥ || 2
i L (Γ)
L (Γ)
≤ C(K) βi ||Vn ||L∞ (Γ) ||Θ⊥ || 2
i L (Γ)
+

βi ||Vn ||L∞ (Γ) ||∇s Θ⊥ || 2
i L (Γ) ,

(5.3.18)

where we used Proposition 5.2 to bound ||∇s Θi || 2 . Substituting the estimate in (5.3.18)
L (Γ)
into (5.3.16) yields
||∇s Θ⊥ ||2 2
βi C(K)||Vn ||L∞ (Γ)
i L (Γ)
+ C(K)||Vn ||L∞ (Γ)
≤ βi +
||Θ⊥ ||2 2
||Θ⊥ || 2
i L (Γ)
i L (Γ)

||∇s Θ⊥ || 2
i L (Γ)
βi
||Θ⊥ ||2 2
i L (Γ)
(5.3.19)

and

⊥ 2
⊥
||∇s Θ⊥ ||2 2
i L (Γ) ≤ βi ||Θi ||L2 (Γ) + βi C(K)||Vn ||L∞ (Γ) ||Θi ||L2 (Γ)
+ C(K)||Vn ||L∞ (Γ) βi ||∇s Θ⊥ || 2 .
i L (Γ)

(5.3.20)

We rewrite (5.3.20) as

||∇s Θ⊥ || 2
i L (Γ) −

≤

||∇s Θ⊥ || 2
i L (Γ) +

βi ||Θ⊥ || 2
i L (Γ)

βi C(K)||Vn ||L∞ (Γ)

βi ||Θ⊥ || 2
i L (Γ)

⊥
βi ||Θ⊥ || 2
i L (Γ) + ||∇s Θi ||L2 (Γ) .

80

(5.3.21)

It follows that

||∇s Θ⊥ || 2
i L (Γ) −

βi ||Θ⊥ || 2
i L (Γ) ≤

βi C(K)||Vn ||L∞ (Γ)

(5.3.22)

||∇s Θ⊥ || 2
i L (Γ) ≤

βi ||Θ⊥ || 2
i L (Γ) +

βi C(K)||Vn ||L∞ (Γ) .

(5.3.23)

and

Letting Vi := span{Θ1 , . . . , ΘM }, it follows from (5.3.15) and the Rayleigh characterization
in Proposition 5.1 that

||∇s Θ⊥ ||2 2
i L (Γ)
.
βM +1 ≤
||Θ⊥ ||2 2
i L (Γ)

(5.3.24)

Using (5.3.24) in (5.3.23) yields

βM +1 ||Θ⊥ || 2
i L (Γ) ≤

βi ||Θ⊥ || 2
i L (Γ) +

βi C(K)||Vn ||L∞ (Γ)

(5.3.25)

and
||Θ⊥ || 2
i L (Γ) ≤

βi
βM +1 −

βi

C(K)||Vn ||L∞ (Γ) .

(5.3.26)

Finally, since the Jacobian J is O(ε) in L∞ (Γ(2ℓ)),
2
η1 φ ′ Θ ⊥ 2
i L (Ω) =

Γ

(Θ⊥ )2
i

2ℓ/ε
−2ℓ/ε

(η1 φ′ )2 Jdzds

≤ Cε||Θ⊥ ||2 2
i L (Γ) ≤ (
and (5.3.9) follows.

81

βi ε
βM +1 −

βi )2

C(K)||Vn ||2 ∞ (Γ)
(5.3.27)
L

The result of the proposition follows by using the estimates in Lemmas 5.4 and 5.5 in
(5.3.6).

5.4

Smooth Dependence of Eigenvalues on Time

We conclude the chapter with an estimate on the derivative of the Laplace-Beltrami eigenvalues
βi,Γ with respect to time.
t
Proposition 5.6. Under the evolution equation given in (4.1.1) we have the following bound
on the derivative of the Laplace-Beltrami eigenvalues with respect to time

∂t βi ≤ C(K)βi Vn L∞ (Γ) ,

(5.4.1)

where t ∈ [0, T ] and C is a positive constant independent of ε and Γ ∈ GK .
Proof: Taking take the ( ,

)Γ inner product of (5.3.13) with Θi yields

aii βi − (∆s Θ⊥ , Θi )Γ − ([∂t , ∆s ]Θi , Θi )Γ = βi aii + ∂t βi
i

(5.4.2)

where recall that
(u, v)Γ =

Γ

uv

det gds.

(5.4.3)

It follows after integrating by parts in the second term on the left-hand side of (5.4.2) that

∂t βi = −([∂t , ∆s ]Θi , Θi )Γ .
82

(5.4.4)

As in the proof of Lemma 5.5 we have
√
∂t det g
√
∆s Θi , Θi
det g
Γ

d−1 d−1
∂Θi
1
∂
∂
det gg ij
+ √
,Θ 
∂si ∂t
∂sj i
det g
i=1 j=1
Γ
1
= − βi Vn HΘi , Θi Γ
2

d−1 d−1
∂Θi ∂Θi 
1
∂
,
− √
,
det gg ij
∂t
∂sj ∂si
det g
i=1 j=1
Γ

1
([∂t , ∆s ]Θi , Θi )Γ = −
2


and

([∂t , ∆s ]Θi , Θi )Γ

≤ C(K) βi ||Vn ||L∞ (Γ) ||Θi || 2
+ ||Vn ||L∞ (Γ) ||∇s Θi ||2 2
L (Γ)
L (Γ)
≤ C(K)βi ||Vn ||L∞ (Γ) ,

where we used Proposition 5.2 to bound ||∇s Θi || 2 .
L (Γ)

83

(5.4.5)

Chapter 6
Initial Data Decomposition

6.1

Normal Variation

We wish to decompose the initial data as

u0 = Φ(x; Γ0 , b0 ) + w0 (x),

(6.1.1)

where w0 ∈ ZM,Γ . However, first we study normal variations of an interface Γ. More
0
precisely, we make the following definition
Definition 6.1. Fix K > 0, let Γ ∈ GK be given locally by a parametrization γ : S → Γ ⊂ Rd .
For each R ∈ C 2 (Γ) we define the R-variation of Γ as
ΓR := {γ(s) + ν(s)R(s) : s ∈ S},
where ν(s) is the unit normal vector at s.
84

(6.1.2)

If R

C 2 (Γ)

is sufficiently small, then ΓR is locally represented by the diffeomorphism

γR (s) = γ(s) + ν(s)R(s).

(6.1.3)

Indeed, let x = γ(s) + ν(s)R(s) ∈ ΓR and consider the mapping in (6.1.3). We will show that
−1
γR is a chart for ΓR . Let
gR,ij =

∂γR ∂γR
,
∂si ∂sj

(6.1.4)

be the first fundamental form associated with γR . Then substituting (6.1.3) into (6.1.4) and
taking into account that ∂ν ∈ Tγ(s) Γ one obtains
∂sj

gR,ij = gij − 2Rhij + R2 eij +
= gij − 2R
=
k

∂R ∂R
∂si ∂sj
gik ek +
j

gik hk + R2
j
k

k



k
gik δj − 2Rhk + R2 ek +
j
j

k,l
g kl
l

∂R ∂R
∂sl ∂sj


gik g kl

∂R ∂R 
,
∂sl ∂sj

where g ij is the inverse of gij . Bounds (2.1.19) and (2.1.22) for the norms of the second and
third fundamental forms imply that gR,ij is invertible for R with ||R|| 2
less than some
C (Γ)
−1
constant depending on K only. Therefore the Jacobian of γR doesn’t vanish and γR defines
a diffeomorphism of an open neightborhood Mx of x into Rd−1 and so is a chart for ΓR . The
manifold ΓR is compact by compactness of Γ and continuity of the map (6.1.3).
Since parametrizations γ and γR are diffeomorphisms they induce a diffeomorphism
85

γR ◦ γ −1 : γ(s) → γR (s). Since the first, second, third fundamental forms are smooth
on Γ, they will be smooth on ΓR as well. Since Γ is a compact manifold and Γ ∈ GK , there
¯
exists a constant K > 0 such that ΓR ∈ GK for all R ∈ C ∞ (Γ), ||R|| 2
less than some
¯
C (Γ)
¯
constant depending on K.

6.2

Main Theorem

In Chapter 3 we introduced the Laplace-Beltrami operator ∆s and the corresponding eigenvalue/eigenfunction pairs {Θi , βi }M . We constructed the ‘center space’
i=1
ZM = span{1, η1 φ′ Θi } and will show in Chapter 8 that selecting M = [M1 ε−(d−1) ] with a
positive constant M1 independent on ε and Γ ∈ GK , guarantees the coercivity of the bilinear
form associated with L on the orthogonal complement to ZM .
In order to track the evolution of an element of MK (see (3.2.1)), for each p ∈ RM , we
define the normal p-variant Γp of Γ0 given locally by the parametrization
M
γp (s) = γ0 (s) + ν0 (s)
i=1

pi Θi,Γ (s),
0

(6.2.1)

where ν0 (s) is a unit normal vector at γ0 (s) to Γ0 . This corresponds to selecting R(s) =
M pΘ
i=1 i i,Γ0 (s) in Definition 6.1.

Theorem 6.2. Let u0 ∈ H 2 (Ω). Fix M1 > 0 sufficiently small and let M = [M1 ε−(d−1) ].
¯
¯
There exists B0 > 0 such that if u0 is B0 εd−3/2 away from the set of single-layer dressed
86

interfaces, MK in the sense that for some Γ0 ∈ GK and b0 ∈ R
u0 (x) = Φ(x; Γ0 , b0 ) + w∗ ,

(6.2.2)

¯
with ||w∗ || 2
≤ B0 εd−3/2 , then there exist p = p(w∗ ) ∈ RM and b = b(w∗ ) ∈ R such
L (Ω)
that
u0 (x) = Φ(x; Γp , b) + w0 ,

(6.2.3)

⊥
w0 ∈ ZΓ ,M .
p

(6.2.4)

with

Proof: Fix Γ0 ∈ GK . For i = 0, . . . , M let F : U ⊂ L2 (Ω) × RM +1 → RM +1 be defined by
Fi (w∗ , [b, p]) :=

Ω

(Φ(x; Γ0 , b0 ) + w∗ (x) − Φ(x; Γp , b)) Zi dx,

(6.2.5)

:=w0

where Zi = η1 φ′ Θi,Γp span the ‘center space’ ZΓp ,M . Then, the condition that w0 :=
⊥
Φ(x; Γ0 , b0 ) + w∗ − Φ(x; Γp , b) belongs to ZΓ ,M is equivalent to Fi (w∗ , [b, p]) = 0 for i =
p
0, . . . , M . Clearly, Fi (0, [b0 , 0]) = 0, or in other words if w∗ ≡ 0, decomposition (6.2.3) is
achieved with p = 0, b = b0 and w0 ≡ 0.
We will show that if ||w∗ || 2
is sufficiently small, we may select [p(w∗ ), b(w∗ )] ∈ RM +1
L (Ω)
satisfying
Fi (w∗ , [b, p]) = 0 for i = 0, . . . M.

(6.2.6)

In order to use the Implicit Function Theorem we need to calculate the variation of F with
87

respect to p and b. The dependence on p comes from the fact that both the ansatz Φ and
Zi = η1 φ′ Θi,Γp depend on p through the interface Γ. The first fundamental form of Γp
depends analytically on p while classical results [24] show that the Laplace-Beltrami modes
Θi,Γp smoothly depend on p.
The ansatz Φ(x; Γp , b) is defined in terms of the whiskered coordinates s and z (see (3.1.4))
and so to understand the dependence of Φ(x) on p one needs to first analyze the variations of
the coordinates s and z with respect to p.
Recall that in a neighborhood of Γp one can introduce whiskered coordinates (sp , zp ),
where sp ∈ S. In this coordinate system the domain Γ(ℓ) is parametrized by
x = γp (sp ) + εzp νp (sp ),

(6.2.7)

where νp (sp ) is a unit normal vector at γp (sp ).

Lemma 6.3. We have the following formulas
∂γp (sp )
· νp (sp ) = ν0 (sp ) · νp (sp )Θi,Γ (sp ),
0
∂pi

(6.2.8)

∂νp (sp )
· νp (sp ) = 0,
∂pi

(6.2.9)

∂zp
= −ε−1 ν0 (sp ) · νp (sp )Θi,Γ (sp ),
0
∂pi

i = 1, . . . M.

(6.2.10)

Proof: Since
M
γp (sp ) = γ0 (sp ) + ν0 (sp )
i=1
88

pi Θi,Γ (sp ),
0

(6.2.11)

(see (6.2.1)), differentiating (6.2.11) with respect to pi gives
∂sp
∂γp (sp )
= Ds γp (sp ) ·
+ ν0 (sp )Θi,Γ (sp ).
0
∂pi
∂pi

(6.2.12)

Taking the inner product with νp (sp ) gives
∂γp (sp )
· νp (sp ) = ν0 (sp ) · νp (sp )Θi,Γ (sp ),
0
∂pi

(6.2.13)

∂γp (sp )
∂γp (sp )
· νp (sp ) = 0 since
is in the tangent plane of Γp . Identity
∂sj
∂sj
(6.2.9) comes from differentiating

where we used

νp (sp ) · νp (sp ) = 1,

(6.2.14)

with respect to pi . The final formula of the Lemma is obtained by fixing x, taking the
derivative of
zp =

1
x − γp (sp ) · νp (sp )
ε

(6.2.15)

with respect to pi , and using (6.2.8) and (6.2.9).

There is one more identity we need before proceeding to the Implicit Function Theorem.
Lemma 6.4 gives the derivatives with respect to pi of the local coordinates sp,j of the touchdown point γp (sp ).

Lemma 6.4. Let x ∈ Γp (ℓ) be represented locally by the whiskered coordinates zp (x) and
89

sp (x). Then
∂sp,j
∂pi

= O(ε−1 ).

(6.2.16)

Proof: We differentiate
x = γp (sp ) + εzp νp (sp )

(6.2.17)

with respect to pi to obtain

Ds γp (sp ) ·

∂zp
∂sp
∂sp
∂νp
+ ν0 (sp )Θi,Γ (sp ) + ε
νp + εzp Ds νp (sp ) ·
+ εzp
= 0. (6.2.18)
0
∂pi
∂pi
∂pi
∂pi

Expanding the products,
Ds γp (sp ) ·

d−1

∂sp
=
∂pi

k=1

∂γp ∂sp,k
∂sk ∂pi

(6.2.19)

∂νp ∂sp,k
,
∂sk ∂pi

(6.2.20)

and
∂sp
=
Ds νp (sp ) ·
∂pi
multiplying (6.2.18) by

d−1
k=1

∂γp
and using the definitions of the fundamental forms in (2.1.8) and
∂sj

(2.1.11) yields
d−1
k=1

gjk − εzp hjk

To obtain an expression for

∂sp,k

∂γp
∂νp ∂γp
+ ν0 (sp )Θi,Γ (sp )
+ εzp
= 0.
0
∂pi
∂sj
∂pi ∂sj

(6.2.21)

∂νp
, we first express it in terms of the tangent basis
∂pi
∂νp
=
∂pi

d−1
ak
k=1
90

∂γp
.
∂sk

(6.2.22)

Taking the inner product of both sides of the above relation with

∂νp ∂γp
,
∂pi ∂sm

∂γp
and using (2.1.8) yields
∂sm

d−1
=

ak gkm .

(6.2.23)

∂νp ∂γp km
,
g
∂pi ∂sm

(6.2.24)

k=1

Inverting g we solve for a
d−1
ak =
m=1
which when substituted into (6.2.22) yields
d−1
∂ 2 γp ∂γp km
∂νp ∂γp ∂γp km
g
=−
g
,
νp ,
∂pi ∂sm ∂sk
∂pi ∂sm ∂sk
k,m=1
k,m=1
d−1
∂(Θi,Γ ν0 ) ∂γp
0
= −
νp ,
g km
(6.2.25)
∂sm
∂sk
k,m=1

∂νp
=
∂pi

d−1

Substituting this expression for

d−1
k=1

gjk −εzp hjk

∂νp
into (6.2.21) we obtain
∂pi

d−1
∂(Θi,Γ ν0 )
∂γp
0
+ν0 (sp )Θi,Γ (sp )
−εzp
gkj g km = 0,
νp ,
0
∂pi
∂sj
∂sm
k,m=1

∂sp,k

(6.2.26)
and canceling the g, g −1 in the last term,
d−1
k=1

gjk − εzp hjk

∂(Θi,Γ ν0 )
∂γp
0
+ ν0 (sp )Θi,Γ (sp )
− εzp νp ,
= 0.
0
∂pi
∂sj
∂sj

∂sp,k

(6.2.27)

Since |z| ≤ 2ℓ/ε, estimate (6.2.16) follows from uniform invertibility of g − εzh (see Lemma
2.11) and Proposition 5.2 since βM = O(ε−2 ).

91

We return to the proof of Theorem 6.2. For u0 = Φ(x; Γ0 , b0 ) + w∗ (x), the condition
⊥
u0 (x) − Φ(x; Γp , b) ∈ ZM,Γ is equivalent to the equations
p
Fi (w∗ , [b, p]) :=

Ω

Φ(x; Γ0 , b0 ) + w∗ (x) − Φ(x; Γp , b) Zi,Γp dx = 0,

(6.2.28)

for i = 0, . . . , M . Here Z0 := 1, Zi,Γp = η1 φ′ Θi,Γp for i = 1, . . . , M . For w∗ ≡ 0 one solution
is [b, p] = [b0 , 0]. To show invertibility, for i, j = 1 . . . M , we calculate

∂ pj F i =

Ω

−∂pj Φ(x; Γp , b)Zi,Γp + Φ(x; Γ0 , b0 )+w∗ (x)−Φ(x; Γp , b) ∂pj Zi,Γp dx, (6.2.29)

and in particular
∂ pj F i

0,[b0 ,0]

=

Ω

−

∂Φ
(x; Γ0 , b0 ) Zi,Γ dx.
0
∂pj

(6.2.30)

To simplify, we observe that for i, j = 1, . . . , M expressions (6.2.10) and (6.2.16) for the
derivatives of the whiskered coordinates with respect to pj imply

−

2ℓ/ε ∂Φ(z, s)
∂Φ
(x; Γ0 , b0 ) Zi,Γ dx = −
η1 φ′ Θi,Γ Jdzds
0
0
Γ −2ℓ/ε ∂pj
Ω ∂pj

=−
=

2ℓ/ε
Γ −2ℓ/ε

∂s(x)
∂Φ(z, s) ∂z(x)
+ Ds Φ(z, s) ·
∂z
∂pj
∂pj

η1 φ′ Θi,Γ Jdzds
0

2ℓ/ε
η1 · φ′ + ε2 ∂z φ2 φ′ (1 + εK1 z) dzds + O(ε2 )
Θi,Γ (s)Θj,Γ (s)J0 (s)
0
0
−2ℓ/ε
Γ
+ O(ε2 ) = ||φ′ ||2 2
δij + O(ε2 ).
Θi,Γ , Θj,Γ
= ||φ′ ||2 2
0
0 Γ0
L (R)
L (R)

92

(6.2.31)

In addition we obtain with the aid of (3.1.4)

∂ b Fi

0,[b0 ,0]

=

Ω

−

2ℓ/ε
∂Φ
Zi,Γ dx = −
Zi,Γ dx = −
η1 φ′ Θi,Γ Jdzds = O(ε),
0
0
0
∂b
Ω
Γ −2ℓ/ε
(6.2.32)

due to the ε in the Jacobian term. Similarly

∂ pj F 0

=−
=

2ℓ/ε
Γ −2ℓ/ε

0,[b0 ,0]

=−

∂Φ
(x; Γ0 , b0 ) dx,
Ω ∂pj

∂Φ(z, s) ∂z(x)
∂s(x)
+ Ds Φ(z, s) ·
∂z
∂pj
∂pj

(6.2.33)

Jdzds

2ℓ/ε
Θi,Γ (s)J0 (s)
η1 · φ′ + ε2 φ2 + ε3 b (1 + εK1 z) dzds + O(ε2 )
0
Γ
−2ℓ/ε

= [|φ|]

Θi,Γ (s)J0 (s)ds + O(ε2 ) = 2(Θi,Γ , 1)Γ + O(ε2 ) = O(ε2 ).
0
0
0
Γ

(6.2.34)

and
∂ b F0

0,[b0 ,0]

=−

∂Φ
dx = −
dx = −|Ω|.
Ω ∂b
Ω

(6.2.35)

Combining (6.2.29) - (6.2.35) we obtain


O(ǫ2 )




−|Ω|
.
∇b,p F (0, [0, b0 ]) = 

′ ||2
2 )
O(ǫ) ||φ
I
+ O(ε
L2 (R) M ×M

93

(6.2.36)

Since M = [M1 ε−(d−1) ], it follows that for d = 2, 3 and each i
M
j=1,j=i

{∇b,p F (0, [0, b0 ])}ij ≤ Cε2 M1 ε−(d−1) ≤ CM1 ,

(6.2.37)

and for M1 sufficiently small ∇b,p F (0, [0, b0 ]) is diagonally dominant and so invertible. In
addition,

∂ pj pk F i =
+

Ω

−∂pj pk Φ(x; Γp , b)Zi,Γp − ∂pj Φ(x; Γp , b)∂pk Zi,Γp − ∂pk Φ(x; Γp , b)∂pj Zi,Γp

Φ(x; Γ0 , b0 ) + w∗ (x) − Φ(x; Γp , b) ∂pj pk Zi,Γp dx.

(6.2.38)

It follows from (6.2.38) that the second derivatives of Fi with respect to [b, p] are uniformly
bounded. For ||w|| 2
sufficiently small, the Implicit Function Theorem guarantees the
L (Ω)
⊥
existence of smooth functions [b, p] such that u0 − Φ(x; Γp , b) ∈ ZΓ ,M .
p

In particular, y(w∗ ) := [b(w∗ ), p(w∗ )] is given as a fixed point of the map
Gw∗ (y) := y − ∇−1 F (0, [0, b0 ])F (w∗ , y).
y

(6.2.39)

Using the definition above, we obtain for i = 0, (and Z0 = 1)
Gi ∗ (y)
w

≤
=

Gi ∗ ([b0 , 0]) + Gi ∗ (y) − Gi ∗ ([b0 , 0])
w
w
w

w∗ dx + b − b0 − ∇−1 F (0, [0, b0 ]) [F (w∗ , y) − F (w∗ , [b0 , 0])] .
y
i
Ω
(6.2.40)
94

Similarly, for i = 1, . . . M

Gi ∗ (y)
w

≤

w∗ Zi,Γ dx + yi − ∇−1 F (0, [0, b0 ]) [F (w∗ , y) − F (w∗ , [b0 , 0])] .
y
0
i
Ω
(6.2.41)

Taylor expanding F (w∗ , [b, p]), the linear terms cancel and we obtain
Gi ∗ (y)
w





(∂yk ym Fi (˜)yk ym )
y
≤ ||w∗ || 2
||Z ||
+ C1 
L (Ω) i L2 (Ω)
k,m




ε1/2 + 
≤ C(K) ||w∗ || 2
L (Ω)

k,m

(∂yk ym Fi (˜)yk ym ) , (6.2.42)
y

where the extra ε1/2 in the first term came from the fact that Zi is localized on Γ. Similarly




+
G0 ∗ (y) ≤ C(K) ||w∗ || 2
w
L (Ω)

k,m



(∂yk ym Fi (˜)yk ym )
y

(6.2.43)

¯
It follows that there exists M such that for each w∗ satisfying
¯
||w∗ || 2 ≤ M0 εd+1/2
L

(6.2.44)

¯
¯
the function Gw∗ maps {[b, p] : |b| ≤ B1 εd+1/2 , |pi | ≤ B2 εd+1 } into itself and in particular
the fixed point satisfies
¯
|pi | ≤ Bεd+1 , for i = 0, . . . , M.

95

(6.2.45)

We note that the bound above implies
M
||R|| 2
=
H (Γ)

i=1

M
|pi |||Θi || 2
≤C
H (Γ)

i=1

εd+1 |βi | ≤ CM1 ε−(d−1) εd−1 ≤ CM1 ,
(6.2.46)

where we used βi ∼ ε−2 and M ∼ M1 ε−(d−1) . In addition,
w0 = Φ(x; Γ0 , b0 ) + w∗ − Φ(x, Γp , b)

(6.2.47)

and

||w0 || 2
≤ ||w∗ || 2
+ Φ(x; Γ0 , b0 ) − Φ(x, Γp , b) 2
L (Ω)
L (Ω)
L (Ω)
¯
≤ B0 εd+1/2 + C(K)ε1/2 |[b, p]| ≤ C(K)εd+1/2 ,

(6.2.48)

where the extra ε1/2 comes from the Jacobian in the L2 norm of the localized term. Similarly,

||ε2 ∆w0 || 2
≤ ||ε2 ∆w∗ || 2
+ ε2 ∆Φ(x; Γ0 , b0 ) − ε2 ∆Φ(x, Γp , b) 2
L (Ω)
L (Ω)
L (Ω)
2
2
≤ ||ε2 ∆w∗ || 2
+ ∂z Φ(x; Γ0 , b0 ) − ∂z Φ(x, Γp , b) 2
L (Ω)
L (Ω)
+ ε κ∂z Φ(x; Γ0 , b0 ) − κ∂z Φ(x, Γp , b) 2
L (Ω)
+ ε2 ∆s Φ(x; Γ0 , b0 ) − ∆s Φ(x, Γp , b)

L2 (Ω)

¯
≤ B0 εd+1/2 + C(K)ε1/2 |[b, p]|.

(6.2.49)

It follows that
||w||A = ||∆w|| 2
+ ε−2 ||w|| 2
≤ Cεd−3/2 .
L (Ω)
L (Ω)
96

(6.2.50)

97

Chapter 7
Analysis of the Full Linearized
Operator
The goal of the next two chapters is the analysis of the linear operator Lφ defined in (3.2.2),
and in particular the proof of Theorem 3.5. With a slight manipulation we rewrite the operator
as

Lφ = (ε2 ∆ − W ′′ (Φ) + ε2 η)2 + (W ′′ (Φ) − ε2 η)ε2 η − W ′′′ (Φ)(ε2 ∆Φ − W ′′ (Φ)).

(7.0.1)

In this chapter we characterize the small energy eigenspace of a generalization of Lφ , which
we define in the next section. The organization of the chapter is as follows. Section 7.1 is used
to define the needed notation. Next, in Section 7.2 we prove several technical propositions
which follow from basic definitions in Chapter 3, but were moved here for clarity. The last
two sections contain the statement and the proof of the main result of this chapter.
98

7.1

Functionalized Operator: Overview of Results

Fix K > 0 and let Γ ∈ GK be a smooth closed hypersurface as described in Chapter 3. We
consider a linear operator L : L2 (Ω) → L2 (Ω) with domain D(L) = H 2 (Ω) ⊂ L2 (Ω) and
natural boundary conditions having the form

L := ε2 ∆ − q(x; ε)

2

+ ε˜(x; ε),
q

(7.1.1)

where q(x), q (x) are smooth functions. In addition, in Γ(ℓ)
˜

q(x) = q(s, z) = q0 (s, z; ε) + εq1 (s, z; ε),

(7.1.2)

q(x) > q + > 0, in Ω\Γ(ℓ),

(7.1.3)

with

q − q+ and q are localized on Γ, and
˜
α m
sup ||Ds ∂z qi ||L∞ (Γ(2ℓ)) ≤ C(K),
|α|≤2

(7.1.4)

sup ||Dα q||L∞ (Ω\Γ(ℓ)) ≤ C(K),
|α|≤2

(7.1.5)

||qi ||L∞ (Ω) ≤ C(K),

(7.1.6)

||˜||L∞ (Ω) ≤ C(K),
q

(7.1.7)

where C(K) is a constant independent of ε and Γ ∈ GK .
99

Definition 7.1. Let f ∈ C(Ω). If there exist a constant f+ and m > 0 such that
f = f+ on Ω\Γ(2ℓ),

(7.1.8)

sup |f (s, z) − f+ | ≤ Ce−m|z|
s

(7.1.9)

and in Γ(2ℓ)

then we say f − f+ is localized on Γ.
Remark 7.2. Operator Lφ is a special case of L with
q(x) = W ′′ (Φ) − ε2 η = W ′′ (φ) + O(ε2 )

(7.1.10)

and

q = ε−1 (W ′′ (Φ) − ε2 η)ε2 η − W ′′′ (Φ)(ε2 ∆Φ − W ′′ (Φ)) = W ′′′ (φ)κφ′ + O(ε).
˜

(7.1.11)

In particular, q + = W ′′ (±1) and (7.1.4)-(7.1.7) hold for the Functionalized Operator Lφ as
long as Γ ∈ GK .
One can write L in terms of the Allen-Cahn operator A as
L = A2 + ε˜(x),
q

(7.1.12)

A = −ε2 ∆ + q(x).

(7.1.13)

where

100

In Γ(ℓ) we may use (2.2.25) and (7.1.2) to write A in the local coordinates as
2
A = −∂z − εκ(s, z)∂z + q0 (s, z) + εq1 (s, z) − ε2 ∆G .

(7.1.14)

The leading-order approximations are

2
L0 [s] := −∂z + q0 (s, z)

(7.1.15)

and
La [s] := −J −1

∂ ∂
2
J
+ q(s, z) = −∂z − εκ(s, z)∂z + q(s, z).
∂z ∂z

(7.1.16)

Similarly
L := L2 + ε˜(s, z) = (L0 − εκ∂z + εq1 )2 + ε˜,
q
q
a

(7.1.17)

is the leading-order part of L, and in local coordinates
L = A2 + ε˜ = L − ε2 ∆G La − ε2 La ∆G + ε4 ∆2 .
q
G

(7.1.18)

We have a proliferation of operators on different domains, which we summarize in the table
below. Here I denotes the interval (−2ℓ/ε, 2ℓ/ε), and the boundary conditions are natural for
the appropriate bilinear form. The precise definitions of the corresponding bilinear forms will
be given as needed.
Note that the differential operators L0 , La , and L depend upon s as a parameter, and for
a fixed s may be considered as operators acting on H 2 (−2ℓ/ε, 2ℓ/ε). We will sometimes use ′
101

Operator
2
L0 = −∂z + q0 (s, z)
2
La = −∂z − εκ(s, z)∂z + q(s, z)
L = L2 + ε˜(s, z)
q
a
A = −ε2 ∆ + q(x)
L = A2 + ε˜(x)
q

Region

Description

I⊂R

1D Allen-Cahn Operator

I⊂R

1D Allen-Cahn Lead Order

I⊂R
Ω ⊂ Rd

Eigenpairs
0
(ψi , λ0 )
i
a , λa )
(ψi i

1D Functionalized Lead Order
Allen-Cahn Operator

Ω ⊂ Rd

Functionalized Operator

(Ψi , µi )

Table 7.1: Definitions of the operators that are used.

Operator

Inner Product

Bilinear Form

L0

2ℓ/ε
uvdz
−2ℓ/ε

b0 [u, v] =

2ℓ/ε
(u′ v ′ + q0 uv)dz
−2ℓ/ε

La

2ℓ/ε
uvJdz
−2ℓ/ε

ba [u, v] =

2ℓ/ε
(u′ v ′ + quv)Jdz
−2ℓ/ε

L

2ℓ/ε
uvJdz
−2ℓ/ε

A

Ω uvdx

Ba [u, v] = Ω (ε2 ∇u · ∇v + quv)dx

L

Ω uvdx

q
B[u, v] := Ω ((Au)(Av) + ε˜uv)dx

b[u, v] =

2ℓ/ε
((La u)(La v) + ε˜uv) Jdz
q
−2ℓ/ε

Table 7.2: Definitions of the bilinear forms that are used.

102

to represent derivative with respect to the dominant variable z. We make several assumptions
about the structure of the spectrum of L0 .
Assumption (H): There exist C > 0 independent of ε and s such that L0 has r
eigenvalues satisfying sups |λ0 (s)| ≤ Cε for 1 ≤ i ≤ r, and the rest of the spectrum
i
is bounded below by some value ν0 > 0 which is independent of ε and s.
Remark: The leading-order operator L0 corresponds to the negative of the 1D Allen-Cahn
operator defined in (3.1.7) and by Lemma 9.1, Assumption (H) holds for L0 with r = 1.
The following lemma is a classical result.
0
Lemma 7.3. There exists m > 0 such that the eigenfunctions {ψi }r
i=1 of L0 satisfy
n 0
sup sup ∂z ψi (s, ±ℓ/ε) = O(e−m/ε ),
s
n≤4

(7.1.19)

α n 0
sup
sup ||Ds ∂z ψi ||0 = O(1).
|α|≤2,n≤4 s

(7.1.20)

and

7.2

Additional Estimates

We introduce additional notation used in the following chapters. The L2 inner product over
Ω or Γ will be denoted (f, g)Ω or (f, g)Γ respectively. The induced norms will be denoted
by f

L2 (Ω)

and f

L2 (Γ)

. The L2 inner product of a localized function over a particular

whisker will be unadorned,

(f, g)0 (s) =

2ℓ/ε
−2ℓ/ε
103

f (s, z)g(s, z) dz.

(7.2.1)

The corresponding whiskered L2 norm is denoted f 0 . Volume integrals of f in the whiskered
variables take the form
2ℓ/ε
f (x) dx =

f (s, z)J(s, z) dz ds.

(7.2.2)

Γ −2ℓ/ε

Γ(2ℓ)

Consequently, we will also use a J inner product on each whisker denoted by

(f, g)J =

2ℓ/ε
−2ℓ/ε

f (s, z)g(s, z)J(s, z)dz.

(7.2.3)

The norm induced by this inner product will be denoted: || · ||J . The Jacobian J is O(ε) in
L∞ (Γ(2ℓ)) and introduces a factor of ε1/2 into the J − norm of localized functions.

Proposition 7.4. The inverse of the first fundamental form G may be written as

Gij =

d−1
ϑmi ϑmj ,

(7.2.4)

m=1
where
ϑ = (JT )

−1

.

(7.2.5)

Proof: From (2.2.6) and (2.2.7)
Gij = (JT J)ij

(7.2.6)

and
Gij = (J−1 (JT )−1 )ij =

d−1
ϑmi ϑmj ,
m=1

104

(7.2.7)

where
ϑij = ((JT )−1 )ij for i, j ≤ d − 1.

(7.2.8)

Relations (7.2.5) and (2.2.16) also allow us to obtain bounds on the z derivatives of ϑij .

Lemma 7.5. Fix K > 0, then for all m ∈ N there exists C(K) > 0 such that for all Γ ∈ GK
m
sup ||∂z ϑij ||L∞ (Γ(2ℓ)) ≤ C(K)εm .
i,j≤d−1

(7.2.9)

Proposition 7.6. Assume f ∈ L2 (Ω) and f − f+ is localized on Γ. Then
f :=

1
f dx = f+ + O(ε).
Ω Ω

(7.2.10)

Proof: Since f+ is constant, Ω f+ dx = f+ |Ω| and
f =

1
1
1
f dx =
(f − f+ )dx +
f dx = I + f+ ,
|Ω| Ω
|Ω| Ω
|Ω| Ω +

(7.2.11)

where
I :=

2ℓ/ε
1
(f − f+ )Jdzds.
|Ω| Γ −2ℓ/ε

(7.2.12)

However f − f+ is localized on Γ and
|I| ≤

2ℓ/ε
2ℓ/ε
1
C
|f − f+ |Jdzds ≤
e−m|z| Jdzds = O(ε),
|Ω| Γ −2ℓ/ε
|Ω| Γ −2ℓ/ε
105

(7.2.13)

where we used (2.4.25) bound J.
Proposition 7.7. Assume f ∈ L2 (Ω) and f − f+ is localized on Γ. Then
||Π0 f ||2 2
= O(ε).
L (Ω)

(7.2.14)

Proof: Using the definition of Π0
||Π0 f ||2 2
=
L (Ω)
=

Ω

f− f

Ω

f − f+

≤ C2

2
2

2ℓ/ε
Γ −2ℓ/ε

+ |Ω| f+ − f

dx =

Ω

f − f+ + f+ − f

dx + 2 f+ − f

Ω

2

dx

f − f+ dx +

e−2m|z| Jdzds + 2C f+ − f
2

f+ − f

Ω
2ℓ/ε

Γ −2ℓ/ε

dx = O(ε),

2

dx

e−m|z| Jdzds
(7.2.15)

where we used (2.4.25) to bound terms involving J and Proposition 7.6 to bound f+ − f .

As a consequence of (2.2.1) we also have the following inequalities.
Proposition 7.8. For any function f (x) = f (s, z)

||∂z f || 2
≤ ε||f || 1
,
L (Γ(2ℓ))
H (Γ(2ℓ))

(7.2.16)

2
||∂z f || 2
≤ C(K)ε2 ||f || 2
,
L (Γ(2ℓ))
H (Γ(2ℓ))

(7.2.17)

||∂si f || 2
≤ C(K)||f || 1
.
L (Γ(2ℓ))
H (Γ(2ℓ))

(7.2.18)

106

Proof: Since
x = γ(s) + εzν(s),

(7.2.19)

∂z f = εν · ∇x f,

(7.2.20)

we have

and

||∂z f ||2 2
=
L (Γ(2ℓ))

2ℓ/ε
Γ −2ℓ/ε

(∂z f )2 Jdzds ≤ ε2 |ν|2

d

2ℓ/ε

i,j=1 Γ −2ℓ/ε

≤ ε2 ||f ||2 1
.
H (Γ(2ℓ))

|∇x f |2 Jdzds
(7.2.21)

The remaining estimates follow similarly.

7.3

Main Results

We now give definitions analogous to those in Chapter 3, but for the more general operator
L. Consider the bilinear form associated with L of (7.1.12)
B[u, v] := (Au, Av) + ε(˜u, v),
q

(7.3.1)

A = −ε2 ∆ + q(x),

(7.3.2)

in H 2 (Ω), where recall that

107

and A’s leading-order ‘1D’ form on each ‘whisker’ is
La [s] = −J −1

∂ ∂
2
J
+ q(s, z) = −∂z − εκ(s, z)∂z + q(s, z).
∂z ∂z

(7.3.3)

Let Ψ ∈ H 2 (Ω). Let η1 (x) be a smooth cutoff function equal to one in Γ(ℓ) and zero
outside Γ(2ℓ). Set η2 (x) = 1 − η1 (x). Then we have the decomposition
Ψ = Ψ1 + Ψ2 ,

(7.3.4)

where Ψ1 := η1 Ψ is supported inside Γ(2ℓ) and Ψ2 := η2 Ψ is supported inside Ω\Γ(ℓ). For the
remainder of this section we assume that Ψ and consequently Ψ2 satisfies Neumann boundary
conditions on ∂Ω.

7.3.1

Whiskered Operator

For each fixed s ∈ S, we would first like to understand the spectra of the 1D operators
La [s] = L0 [s] − εκ(s, z)∂z + εq1 (s, z),

(7.3.5)

L[s] = L2 [s] + ε˜(s, z),
q
a

(7.3.6)

and

2
as perturbations of L0 = −∂z + q0 (s, z) acting on H 2 (−ℓ/ε, ℓ/ε). We will see that the curvature perturbation εκ(s, z) causes an O(ε2 ) shifting of the asymptotically small eigenvalues.
In addition, εq1 and ε˜ perturbation will result in O(ε) perturbations of the point spectrum
q
108

unless q1 and q satisfy certain orthogonality properties.
˜
Recall that b0 and ba stand for bilinear forms corresponding to L0 and La respectively
(see Table 7.1). Under Neumann boundary conditions ∂z ψ(±2ℓ/ε) = 0, La and L are selfadjoint in the weighted (·, ·)J inner product defined in (7.2.3). Indeed, considering the J-inner
product of La u with v, we integrate by parts and use the relation Jz = εκJ to find,

ba [u, v] = (La u, v)J =

2ℓ/ε
−2ℓ/ε

(−uzz −εκuz +qu)vJdz =

2ℓ/ε
−2ℓ/ε

(uz vz +quv)Jdz = (u, La v),
(7.3.7)

where ba [u, v] = (La u, v)J is the associated bilinear form.

Lemma 7.9. Let (·, ·)J be the weighted inner product defined in (7.2.3). Then
ba [u, v] = b0 [J 1/2 u, J 1/2 v] +

√
( J)′′
√
+ εq1 u, v ,
J
J

(7.3.8)

where the bilinear forms, ba and b0 are given in Table 7.1.

Proof: Simple differentiation establishes the following relation between the bilinear forms

b0 [J 1/2 u, J 1/2 v] =
=
=

2ℓ/ε

((J 1/2 u)′ (J 1/2 v)′ + q0 uvJ)dz

−2ℓ/ε
2ℓ/ε 1
1
(( J −1/2 J ′ u + J 1/2 u′ )( J −1/2 J ′ v + J 1/2 v ′ ) + q0 uvJ)dz
2
−2ℓ/ε 2
2ℓ/ε 1
1
1
( J −1 (J ′ )2 uv + J ′ uv ′ + J ′ u′ v + u′ v ′ J + q0 uvJ)dz.
2
2
−2ℓ/ε 4

(7.3.9)

109

Integrating by parts in the second term on the right-hand side of (7.3.9) we obtain

b0 [u, v] = ba [u, v] +

2ℓ/ε 1
1
[ J −1 (J ′ )2 − J ′′ − εq1 ]uvdz.
2
−2ℓ/ε 4

(7.3.10)

Relation (7.3.8) then follows from the identity
√
( J)′′
−(1/4)J −2 (J ′ )2 + (1/2)J −1 J ′′ = √ .
J

(7.3.11)

Let λ0 (s) be the ground eigenvalue of L0 . The next proposition gives the well-known lower
1
bound estimates on the Allen-Cahn bilinear form in terms of inf s∈Γ λ0 , (see [12]).
1

Proposition 7.10. (The deMottoni-Schatzman Estimates on the Allen-Cahn Form)
Fix K > 0 and let Γ ∈ GK . There exists a positive constant C(K) independent of ε and
Γ ∈ GK such that the following lower bounds hold for the bilinear forms associated with La
and A

(La ψ, ψ)J
≥ inf λ0 (s) − C(K)ε,
2
s 1
||ψ||J

(7.3.12)

(AΨ, Ψ) 2
L (Ω)
≥ inf λ0 (s) − C(K)ε.
2
s 1
||Ψ|| 2
L (Ω)

(7.3.13)

inf
1 (−2ℓ/ε,ℓ/ε)
ψ∈H
∂z ψ(±2ℓ/ε)=0
inf
Ψ∈H 1 (Ω)
∂ν Ψ=0 on ∂Ω

Proof: Fix s ∈ Γ and let ψ(s, z) ∈ H 1 (−2ℓ/ε, 2ℓ/ε) and define
ψ 0 := J 1/2 ψ.
110

(7.3.14)

Then by Lemma 7.3.9

(La ψ, ψ)J = (L0 ψ 0 , ψ 0 )0 +

√
( J)′′
√
+ εq1 ψ, ψ .
J
J

(7.3.15)

Proposition 2.12 implies that there exists a constant C(K) > 0 that depends only on K such
that
√
( J)′′
√
=
J L∞ (Γ(2ℓ))

− (1/4)J −2 (J ′ )2 + (1/2)J −1 J ′′

L∞ (Γ(2ℓ))

≤ C(K)ε2 . (7.3.16)

Using (7.3.16) and ||q1 ||L∞ (Γ(2ℓ)) ≤ C(K) (see (7.1.4)) allows us to bound the second term
on the right-hand side of (7.3.15)
√
( J)′′
√
≤ C(K)ε||ψ||2 .
+ εq1 ψ, ψ
J
J
J

(7.3.17)

Returning this estimate to (7.3.15) and using the minimax characterization for eigenvalues of
L0 gives
(La ψ, ψ)J ≥ (L0 ψ0 , ψ0 )0 − Cε||ψ||2 ≥ λ0 (s)||ψ0 ||2 − C(K)ε||ψ||2 ≥ (inf λ0 (s) − Cε)||ψ||2 ,
1
0
J
J
J
s 1
(7.3.18)
where we used ||ψ0 ||0 = ||J 1/2 ψ||0 = ||ψ||J .

To obtain (7.3.13), we take Ψ ∈ H 1 (Ω) with ∂ν Ψ|∂Ω = 0, form the bilinear form ba [Ψ, Ψ]
111

and integrate by parts using the relation ∂z J = εκJ to find,

(La Ψ, Ψ)J =

2ℓ/ε
−2ℓ/ε

2
(−∂z Ψ − εκ∂z Ψ + qΨ)ΨJdz =

2ℓ/ε
−2ℓ/ε

((∂z Ψ)2 + qΨ2 )Jdz. (7.3.19)

Finally, integrating by parts and using q(x) = q + > 0 in Ω\Γ(2ℓ) gives

(AΨ, Ψ) 2
=
L (Ω)
=
≥

Ω

(ε2 |∇Ψ|2 + qΨ2 )dx ≥

Γ(2ℓ)
Γ

Γ(2ℓ)

(Ψ2 + ε2 |∇G Ψ|2 + qΨ2 )dx
z

(La Ψ, Ψ)J ds ≥

(inf λ0 (s) − C(K)ε)||Ψ||2 ds
1
J
Γ s

≥ (inf λ0 (s) − C(K)ε)||Ψ||2 2
.
s 1
L (Ω)

7.3.2

(ε2 |∇Ψ|2 + qΨ2 )dx

(7.3.20)

Low Energy Function Decomposition

We introduce the following near-field, far-field decomposition

Ψ = Ψ1 + Ψ2 ,

(7.3.21)

Ψ1 = η1 Ψ, Ψ2 = (1 − η1 )Ψ

(7.3.22)

where

0
0
and η1 is given by (3.1.2). Let {ψk }r
k=1 be the normalized (||ψk ||0 = 1) small eigenvalue
2
eigenfunctions of L0 = −∂z + q(s, z) on (−2ℓ/ε, 2ℓ/ε) . Recall that the rest of the eigenvalues
112

0
are bounded below by ν0 > 0 which is independent of ε and s. Denote ψk := J −1/2 ψk and

ck (s) := (Ψ1 , ψk )J .

(7.3.23)

The following theorem gives a ‘separated variables’ decomposition for low energy functions Ψ
in terms of O(ε) eigenvalue eigenfunctins of L0 .

1
2 2
= 1,
Theorem 7.11. Let U0 = 32 min{q+ , ν0 }. Assume Ψ ∈ H 2 (Ω) (with ||Ψ|| 2
L (Ω)
∂ν Ψ = 0 on ∂Ω) satisfies the bound

B[Ψ] ≤ U < U0 ,

(7.3.24)

for the bilinear form B given in (7.3.1). Then ck (s) ∈ H 2 (Γ) and
r
Ψ1 (x) = Ψ1 (s, z) =

ck (s)ψk (s, z) + Ψ⊥ (x),

(7.3.25)

k=1
where
(ψk , Ψ⊥ )J = 0,

(7.3.26)

U
||Ψ⊥ ||2 2
≤
+ O(ε),
L (Γ(2ℓ)) ν 2
0

(7.3.27)

||∆s ck ||2 2
≤ U ε−4 + O(ε−3 ),
L (Γ)

(7.3.28)

U
||Ψ2 ||2 2
+ O(ε),
≤
L (Ω) q 2
+

(7.3.29)

113

r
Γ k=1

7.4

1
c2 ds ≥ .
k
4

(7.3.30)

Proof of Theorem 7.11

We present the proof in a series of lemmas.

Lemma 7.12. Under the conditions of Theorem 7.11, there exists ε0 > 0 and C1 > 0 such
that for ε < ε0
2
q+
4 ||Ψ ||2
2 ||Ψ ||2
2
B[Ψ2 ] ≥ C1 ε
2 H 2 (Ω) + q+ ε
2 H 1 (Ω) + 2 ||Ψ2 ||L2 (Ω) .

(7.4.1)

Proof:

B[Ψ2 ] =

Ω
4
= ε

[(−ε2 ∆Ψ2 + qΨ2 )2 + ε˜Ψ2 ]dx
q 2
Ω

|∆Ψ2 |2 dx − 2ε2

Ω

∆Ψ2 qΨ2 dx +

Ω

q 2 Ψ2 dx + ε
2

Ω

q Ψ2 dx. (7.4.2)
˜ 2

We use Neumann boundary conditions on Ψ (and thus on Ψ2 ) to integrate by parts in the
second term to yield

B[Ψ2 ] = ε4
+

Ω

|∆Ψ2 |2 dx + 2ε2

Ω
q 2 Ψ2 dx + ε
2

Ω

Ω

q|∇Ψ2 |2 dx + 2ε2

q Ψ2 dx.
˜ 2
114

Ω

(∇q · ∇Ψ2 )Ψ2 dx
(7.4.3)

Since, by (7.1.3) , q > q+ in Ω\Γ(ℓ) and Ψ2 is supported inside Ω\Γ(ℓ),
2
B[Ψ2 ] ≥ ε4 ||∆Ψ2 ||2 2
+ 2ε2 q+ ||∇Ψ2 ||2 2
+ q+ ||Ψ2 ||2 2
L (Ω)
L (Ω)
L (Ω)
− 2ε2 ||∇q||L∞ (Ω\Γ(ℓ)) ||∇Ψ2 || 2
||Ψ ||
L (Ω) 2 L2 (Ω)
− ε||˜||L∞ (Ω\Γ(ℓ)) ||Ψ2 ||2 2
q
L (Ω)
2
≥ ε4 ||∆Ψ2 ||2 2
+ 2ε2 q+ ||∇Ψ2 ||2 2
+ q+ ||Ψ2 ||2 2
L (Ω)
L (Ω)
L (Ω)
− ε2 ||∇q||L∞ (Ω\Γ(ℓ)) ε||∇Ψ2 ||2 2
L (Ω)
− ε||˜||L∞ (Ω\Γ(ℓ)) ||Ψ2 ||2 2
q
+ ε−1 ||Ψ2 || 2
L (Ω)
L (Ω)
≥ ε4 ∆Ψ2 2 2
+ ε2 2q+ − ε ∇q L∞ (Ω\Γ(ℓ)) ∇Ψ2 2 2
L (Ω)
L (Ω)
2
+ q+ − ε ∇q L∞ (Ω\Γ(ℓ)) − ε q L∞ (Ω\Γ(ℓ)) Ψ2 2 2
˜
.
L (Ω)
(7.4.4)

We use elliptic regularity theory (Evans, Theorem 4, p.317) to estimate

||Ψ2 ||2 2
≤ C(||∆Ψ2 ||2 2
+ ||Ψ2 ||2 2
)
H (Ω)
L (Ω)
L (Ω)

(7.4.5)

Using (7.4.5) we obtain the bound
ε4
− ε4 ||Ψ2 ||2 2
ε4 ||∆Ψ2 ||2 2
≥ ||Ψ2 ||2 2
H (Ω)
L (Ω)
L (Ω)
C

115

(7.4.6)

Returning this estimate to (7.4.4) gives
ε4
||Ψ2 ||2 2
+ ε2 2q+ − ε||∇q||L∞ (Ω\Γ(ℓ)) ||∇Ψ2 ||2 2
L (Ω)
H (Ω)
C
2
+ q+ − ε||˜||L∞ (Ω\Γ(ℓ)) − ε4 ||Ψ2 ||2 2
q
L (Ω)

B[Ψ2 ] ≥

(7.4.7)

and (7.4.1) follows for ε sufficiently small.

Lemma 7.13. Under the conditions of Theorem 7.11, there exist positive constants
ε0 , C1 , C2 , Cb such that for ε < ε0
||Ψ|| 1
≤ C1 ε−1 ,
H (Ω)

(7.4.8)

||Ψ|| 2
≤ C2 ε−2 ,
H (Ω)

(7.4.9)

B[Ψ1 , Ψ2 ] ≥ −Cb ε2 ,

(7.4.10)

for some C > 0 independent of ε.

Proof: Neumann boundary conditions on Ψ allow us to integrate by parts yielding

(AΨ, Ψ) 2
=
[ε2 |∇Ψ|2 + qΨ2 ]dx.
L (Ω)
Ω

(7.4.11)

Using (7.3.24) and the definition of B in (7.3.1)

||AΨ||2 2
= B[Ψ] − ε(˜Ψ, Ψ) ≤ U0 ,
q
L (Ω)
116

(7.4.12)

for ε sufficiently small, since ||˜||L∞ (Ω) = O(1) and ||Ψ|| 2
q
= 1. Using (7.4.11) and
L (Ω)
(7.4.12) gives

ε2 ||∇Ψ||2 2
= (AΨ, Ψ) 2
− (qΨ, Ψ) 2
L (Ω)
L (Ω)
L (Ω)
≤
||AΨ|| 2
+ ||q||L∞ (Ω) ||Ψ|| 2
L (Ω)
L (Ω)

≤

U0 + O(1) = O(1),
(7.4.13)

since ||q||L∞ (Ω) = O(1) and ||Ψ|| 2
= 1. Estimate (7.4.13) implies (7.4.8). To obtain
L (Ω)
(7.4.9) note that
∆Ψ = −

1
1
AΨ + qΨ2
2
ε
ε2

(7.4.14)

Using elliptic regularity (Evans, Theorem 4, p.317)

≤ C(1+O(ε−2 )) = O(ε−2 ),
||Ψ|| 2
≤ C ||Ψ|| 2
+||−ε−2 AΨ+ε−2 q(x)Ψ2 || 2
H (Ω)
L (Ω)
L (Ω)
(7.4.15)
U0 by (7.4.12) and ||q||L∞ (Ω) = O(1).

since ||AΨ|| 2
≤
L (Ω)

To establish (7.4.10) we write

B[Ψ1 , Ψ2 ] =
=

Ω
Ω

(−ε2 ∆Ψ1 + qΨ1 )(−ε2 ∆Ψ2 + qΨ2 )dx + ε

Ω

q Ψ1 Ψ2 dx
˜

(−ε2 ∆(η1 Ψ) + qη1 Ψ)(−ε2 ∆(η2 Ψ) + qη2 Ψ)dx + ε

= I1 + I2 + I3 + I4 + I5 + I6 + I7

Ω

q η1 η2 Ψ2 dx
˜
(7.4.16)

where
I1 =

Ω

(−ε2 ∆η1 + qη1 )(−ε2 ∆η2 + qη2 )Ψ2 dx,
117

(7.4.17)

I2 = −2ε2

Ω

Ψ((qη2 + ∆η2 )∇η1 + (qη1 + ∆η1 )∇η2 ) · ∇Ψdx,
I3 = 2ε4

I 4 = ε4

Ω

Ω

(∇η1 · ∇Ψ)(∇η2 ∇Ψ)dx,

(η1 ∆η2 + η2 ∆η1 )Ψ∆Ψdx − ε2
I 5 = ε4

Ω

Ω

2q(η1 η2 )Ψ∆Ψdx,

(η1 ∇η2 + η2 ∇η1 ) · Ψ∆Ψdx,

I 6 = ε4
I7 = ε

(7.4.18)
(7.4.19)
(7.4.20)
(7.4.21)

η1 η2 |∆Ψ|2 dx,

Ω
Ω

(7.4.22)

q η1 η2 Ψ2 dx.
˜

(7.4.23)

We use ηi ≥ 0, ||ηi ||L∞ (Ω) , ||∇ηi ||L∞ (Ω) , ||q||L∞ (Ω) , ||∇q||L∞ (Ω\Γ(ℓ)) = O(1) and
||Ψ|| 2
= 1 to estimate I1 through I7 .
L (Ω)
I 1 = ε4
+

Γ(2ℓ)\Γ(ℓ)

Γ(2ℓ)\Γ(ℓ)

∆η1 ∆η2 Ψ2 dx − ε2

Γ(2ℓ)\Γ(ℓ)

q(η1 ∆η2 + η2 ∆η1 )Ψ2 dx

q 2 η1 η2 Ψ2 dx ≥ −Cε2 ,

(7.4.24)

for some C. Similarly, since

1
Ψ (qη2 + ∆η2 )∇η1 + (qη1 + ∆η1 )∇η2 · ∇Ψ =
(qη2 + ∆η2 )∇η1 + (qη1 + ∆η1 )∇η2 · ∇Ψ2 ,
2
(7.4.25)
after integration by parts

I 2 = ε2

Γ(2ℓ)\Γ(ℓ)

∇ · (qη2 + ∆η2 )∇η1 + (qη1 + ∆η1 )∇η2 Ψ2 dx = O(ε2 ).

118

(7.4.26)

Using ||Ψ|| 1
= O(ε−1 ) from (7.4.8) gives
H (Ω)
I3 = 2ε4

Γ(2ℓ)\Γ(ℓ)

(∇η1 · ∇Ψ)(∇η2 · ∇Ψ)dx = O(ε2 ).

(7.4.27)

Using q, η1 , η2 ≥ 0 in Γ(2ℓ)\Γ(ℓ) and ||Ψ|| 2
= O(ε−2 ) from (7.4.9) gives
H (Ω)
I4 = O(ε2 ) + 2ε2
≥ O(ε2 ) − ε2

Γ(2ℓ)\Γ(ℓ)
Γ(2ℓ)\Γ(ℓ)

qη1 η2 |∇Ψ|2 dx + 2ε2

Γ(2ℓ)\Γ(ℓ)

∆(qη1 η2 )Ψ2 dx ≥ −Cε2 ,

∇(qη1 η2 ) · ∇Ψ Ψdx
(7.4.28)

for some C.
I 5 = ε4

d
Γ(2ℓ)\Γ(ℓ) i,j=1

ηi
˜

∂Ψ ∂ 2 Ψ
dx,
∂xi ∂x2
j

(7.4.29)

where η = η1 ∇η2 + η2 ∇η1 . Therefore after integration by parts
˜
d
∂ ηi ∂Ψ ∂Ψ
˜
∂ 2 Ψ ∂Ψ
dx − ε4
dx
Γ(2ℓ)\Γ(ℓ) ∂xi ∂xj ∂xj
Γ(2ℓ)\Γ(ℓ) i,j=1 ∂xj ∂xi ∂xj
d
∂ ∂Ψ 2
ε4
ηi
˜
= −
dx + O(ε2 )
2 Γ(2ℓ)\Γ(ℓ)
∂xi ∂xj
i,j=1
4
ε
(∇ · η )|∇Ψ|2 dx + O(ε2 ) = O(ε2 ).
˜
(7.4.30)
=
2 Γ(2ℓ)\Γ(ℓ)

I5 = −ε4

ηi
˜

Finally,
I6 ≥ 0,

(7.4.31)

I7 = O(ε2 ),

(7.4.32)

and

119

since q is localized.
˜
The following technical integration by parts lemma is key in the proof of Proposition 7.15.

Lemma 7.14. Under the conditions of Theorem 7.11, there exist positive constants ε0 , C such
that for ε < ε0
2ℓ/ε
Γ −2ℓ/ε



d−1

(−La Ψ ) ∆ Ψ −
1 G 1

i,j

∂Ψ ∂Ψ1
∂ 2 Ψ1 ∂ 2 Ψ1
+q 1
∂z∂si ∂z∂sj
∂sj ∂si



Gij  Jdzds < C(K),

(7.4.33)

where La is given in (7.3.3).

Proof: Letting
S :=

2ℓ/ε
Γ −2ℓ/ε

(−La Ψ1 ) ∆G Ψ1 Jdzds,

(7.4.34)

and substituting in the expressions for La from (7.3.3) and ∆G from (2.2.14) we have
d
S=

2ℓ/ε

i,j=1 Γ −2ℓ/ε

∂ ∂Ψ1 −1
J
J
− qΨ1
∂z ∂z

∂Ψ1
∂
Gij J
dzds.
∂si
∂sj

(7.4.35)

Since Ψ1 is compactly supported inside Γ(2ℓ) we may integrate by parts in z
d
S =
i,j=1
+
−

−
2ℓ/ε

2ℓ/ε ∂Ψ ∂ 2
∂Ψ1
1
Gij J
dzds
∂sj
Γ −2ℓ/ε ∂z ∂z∂si

∂Ψ1 ∂z J ∂
∂Ψ1
Gij J
dzds
∂sj
Γ −2ℓ/ε ∂z J 2 ∂si
2ℓ/ε
∂Ψ1
∂
Gij J
dzds .
qΨ1
∂si
∂sj
Γ −2ℓ/ε
J

120

(7.4.36)

Next, integrating by parts with respect to si gives
d
S =
i,j=1
+
+

2ℓ/ε ∂ 2 Ψ ∂
∂Ψ1
1
dzds
Gij J
∂sj
Γ −2ℓ/ε ∂z∂si ∂z

2ℓ/ε ∂z J ∂Ψ ∂
∂Ψ1
1
Gij J
dzds
∂sj
Γ −2ℓ/ε J ∂z ∂si
2ℓ/ε
2ℓ/ε ∂q
∂Ψ
∂Ψ1 ∂Ψ1
qGij J
Ψ1 1 Gij Jdzds +
dzds (7.4.37)
∂sj
∂si ∂sj
Γ −2ℓ/ε
Γ −2ℓ/ε ∂si

Distributing the derivatives and integrating by parts with respect to si in the second term we
obtain
d
S =
i,j=1
+
−
+

2ℓ/ε
2ℓ/ε ∂ 2 Ψ ∂ 2 Ψ
∂ 2 Ψ1 ∂Ψ1
1
1 Gij Jdzds +
Gij ∂z J
dzds
∂z∂si ∂sj
Γ −2ℓ/ε
Γ −2ℓ/ε ∂z∂si ∂z∂sj
2ℓ/ε

Γ −2ℓ/ε
2ℓ/ε

(∂z Gij )
∂ si

2ℓ/ε
∂ 2 Ψ1 ∂Ψ1
∂ 2 Ψ1 ∂Ψ1
Jdzds −
Gij ∂z J
dzds
∂z∂si ∂sj
∂z∂si ∂sj
Γ −2ℓ/ε

∂Ψ1 ∂Ψ1
∂z J
Jdzds
Gij
J
∂z ∂sj

Γ −2ℓ/ε
2ℓ/ε
2ℓ/ε ∂q
∂Ψ
∂Ψ1 ∂Ψ1
qGij J
Ψ1 1 Gij Jdzds +
dzds.
∂sj
∂si ∂sj
Γ −2ℓ/ε
Γ −2ℓ/ε ∂si

121

(7.4.38)

Simplifying
d−1
S =
i,j
+
−
=

2ℓ/ε
Γ −2ℓ/ε
2ℓ/ε

Γ −2ℓ/ε
2ℓ/ε

(∂z Gij )
∂ si

Γ −2ℓ/ε
d−1
2ℓ/ε
i,j

∂Ψ ∂Ψ1
∂ 2 Ψ1 ∂ 2 Ψ1
+q 1
∂z∂si ∂z∂sj
∂sj ∂si

Γ −2ℓ/ε

Gij Jdzds

∂ 2 Ψ1 ∂Ψ1
Jdzds
∂z∂si ∂sj

2ℓ/ε ∂q
∂Ψ
∂Ψ1 ∂Ψ1
Jz
Jdzds +
Ψ1 1 Gij Jdzds
Gij
J
∂z ∂sj
∂sj
Γ −2ℓ/ε ∂si
∂Ψ ∂Ψ1
∂ 2 Ψ1 ∂ 2 Ψ1
+q 1
∂z∂si ∂z∂sj
∂sj ∂si

Gij Jdzds + O(ε−1 ),

where we used (2.4.23) and (2.4.25) as well as estimates (7.4.8) and (7.4.9).

Proposition 7.15. Under the conditions of Theorem 7.11, there exist positive constants ε0 , C
such that for ε < ε0

B[Ψ1 ] =
+

Γ

b[Ψ1 ]ds + 2ε2
2ℓ/ε

Γ −2ℓ/ε

∂ 2 Ψ1 ∂ 2 Ψ1
∂Ψ ∂Ψ1
+q 1
Gij Jdzds
∂z∂si ∂z∂sj
∂sj ∂si
i,j Γ −2ℓ/ε
2ℓ/ε

ε4 (∆G Ψ1 )2 Jdzds + O(ε),

(7.4.39)

where
b[u, v] = (La u, La v)J + ε(˜u, v)J ,
q
is the bilinear form associated with L = L2 + ε˜ on H 2 (−ℓ/ε, ℓ/ε).
q
a
122

(7.4.40)

Proof: Recalling definition (7.3.1) we calculate

B[Ψ1 ] := B[Ψ1 , Ψ1 ] =
= ε4

Γ(2ℓ)

Γ(2ℓ)

(−ε2 ∆Ψ1 + qΨ1 )2 dx +

(∆Ψ1 )2 dx − 2ε2

Ω

q(∆Ψ1 )Ψ1 dx +

Ω

ε˜Ψ2 dx
q 1
Ω

q 2 Ψ2 dx + ε
1

Γ(2ℓ)

= A1 + A2 + A3 + A4 .

q Ψ2 dx
˜ 1
(7.4.41)

We first use (2.2.25) to rewrite A1 and A2 in local coordinates
2
∂Ψ
∂ 2 Ψ1
+ εκ 1 + ε2 ∆G Ψ1 dx
∂z
Γ(2ℓ) ∂z 2
2ℓ/ε
∂ 2 Ψ1
∂Ψ 2
∂Ψ
∂ 2 Ψ1
+ εκ 1 + 2ε2
+ εκ 1 ∆G Ψ1
=
2
2
∂z
∂z
∂z
∂z
Γ −2ℓ/ε
+ ε4 (∆G Ψ1 )2 Jdzds,

A1 =

A2 = −2
= −2
− 2ε2

q

∂ 2 Ψ1
∂Ψ
+ εκ 1 + ε2 ∆G Ψ1 Ψ1 Jdzds
∂z
∂z 2

Γ(2ℓ)
2ℓ/ε

Γ −2ℓ/ε
2ℓ/ε

qΨ1

Γ −2ℓ/ε

∂ 2 Ψ1
∂Ψ
+ εκ 1 Jdzds
∂z
∂z 2

qΨ1 (∆G Ψ1 )dzds.

123

(7.4.42)

Combining the expressions for A1 and A2 we obtain

B[Ψ1 ] =
+
+
=
+
=
+

2ℓ/ε

∂Ψ
∂ 2 Ψ1
+ εκ 1 − qΨ1
∂z
∂z 2

2
Jdzds + ε

2ℓ/ε

q Ψ2 Jdzds
˜ 1

Γ −2ℓ/ε
Γ −2ℓ/ε
2ℓ/ε ∂ 2 Ψ
1 + εκ ∂Ψ1 − qΨ ∆ Ψ Jdzds
2ε2
1
G 1
∂z
Γ −2ℓ/ε ∂z 2
2ℓ/ε
ε4
(∆G Ψ)2 Jdzds
Γ −2ℓ/ε
2ℓ/ε
2ℓ/ε
(∆G Ψ)2 Jdzds
(La Ψ1 )2 + ε˜Ψ2 Jdzds + ε4
q 1
Γ −2ℓ/ε
Γ −2ℓ/ε
2ℓ/ε
2ε2
(−La Ψ1 ) ∆G Ψ1 Jdzds
Γ −2ℓ/ε
2ℓ/ε
(∆G Ψ)2 Jdzds
b[Ψ1 ]ds + ε4
Γ −2ℓ/ε
Γ
2ℓ/ε
2ε2
(−La Ψ1 ) ∆G Ψ1 Jdzds,
Γ −2ℓ/ε

and (7.4.39) follows from (7.4.33).

Lemma 7.16. Under the conditions of Theorem 7.11, there exist positive constants ε0 , C(K)
such that for ε < ε0
d−1

∂ 2 Ψ1 ∂ 2 Ψ1
∂Ψ ∂Ψ1
S :=
+q 1
Gij Jdzds ≥ −C(K)ε−1 .
∂z∂si ∂z∂sj
∂sj ∂si
i,j=1 Γ −2ℓ/ε
2ℓ/ε

(7.4.43)

Proof: The idea is to rewrite S as (La f, f ) for an appropriate function f and use the
deMottoni-Schatzman estimates from Proposition 7.10. Indeed, expanding Gij with the help
124

of Proposition 7.4 yields
d−1

S =

=

=

=

−
−
=

−
−

∂Ψ ∂Ψ1
∂ 2 Ψ1 ∂ 2 Ψ1
+q 1
Gij Jdzds
∂z∂si ∂z∂sj
∂sj ∂si
i,j=1 Γ −2ℓ/ε
d−1
2ℓ/ε
∂ 2 Ψ1
∂Ψ
∂Ψ1
∂ 2 Ψ1
ϑmj
+ qϑmi 1 ϑmj
Jdzds
ϑmi
∂z∂si
∂z∂sj
∂sj
∂si
Γ −2ℓ/ε
m,i,j=1
d−1
d−1
2ℓ/ε d−1
∂ 2 Ψ1 2
∂Ψ 2
ϑmi
+q
ϑmi 1 Jdzds
∂z∂si
∂si
m Γ −2ℓ/ε i=1
i=1
d−1
d−1
2ℓ/ε d−1 ∂
∂Ψ1 2
∂Ψ 2
ϑmi
+q
ϑmi 1 Jdzds
∂si
∂si
m Γ −2ℓ/ε i=1 ∂z
i=1
d−1
2ℓ/ε d−1 ∂ϑ ∂Ψ 2
1 Jdzds
mi
∂z ∂si
m=1 Γ −2ℓ/ε i=1
d−1
d−1
2ℓ/ε d−1 ∂ϑ ∂Ψ
∂ 2 Ψ1
mi
1
2
ϑmj
Jdzds
∂z∂sj
Γ −2ℓ/ε i=1 ∂z ∂si
m=1
j=1
d−1
d−1
d−1
∂Ψ
∂Ψ1
,
ϑmi 1
La
ϑmi
∂si
∂si
J
m Γ
i=1
i=1
d−1
2ℓ/ε d−1 ∂ϑ ∂Ψ 2
1 Jdzds
mi
∂z ∂si
m=1 Γ −2ℓ/ε i=1
d−1
d−1
2ℓ/ε d−1 ∂ϑ ∂Ψ
∂ 2 Ψ1
mi
1
2
ϑmj
Jdzds.
(7.4.44)
∂z∂sj
Γ −2ℓ/ε i=1 ∂z ∂si
m=1
j=1
2ℓ/ε

From Lemma 7.13 we have
∂Ψ1
≤ C(K)ε−1 .
∂si L2 (Ω)

125

(7.4.45)

Lemmas 7.5 and Proposition 7.8 then imply that there exists a constant C(K) > 0 such that
d−1
i=1
d−1
i=1

∂ϑmi ∂Ψ1
∂z ∂si

∂ 2 Ψ1
ϑmi
∂si ∂z

d−1
L2 (Ω)

≤

∂ϑmi
∂Ψ1
≤ C(K),
∂z L∞ (Γ(2ℓ)) ∂si L2 (Ω)
i=1

d−1
L2 (Ω)

≤

i=1

||ϑmi ||L∞ (Γ(2ℓ))

∂ 2 Ψ1
∂si ∂z

||ϑmi ||L∞ (Γ(2ℓ))

∂Ψ1
≤ C(K)ε−1 ,
∂si L2 (Ω)

L2 (Ω)

≤ C(K)ε−1 ,

(7.4.46)

(7.4.47)

and
d−1
ϑmi
i=1

∂Ψ1
∂si

d−1
L2 (Ω)

≤

i=1

(7.4.48)

In addition, it follows from Proposition 7.10 that
d−1
m

d−1
Γ

La
i=1

∂Ψ
ϑmi 1 ,
∂si

d−1
i=1

∂Ψ
ϑmi 1
∂si

2
d−1
d−1
ν0
∂Ψ1
≥
ϑmi
2 m Γ
∂si
J
i=1
J
−1 .
≥ −C(K)ε

Combining (7.4.45)-(7.4.49) into (7.4.44) yields (7.4.43).

Lemma 7.17. Under the conditions of Theorem 7.11, there exist positive constants ε0 , C such
that for ε < ε0 , Ψ1 = η1 Ψ satisfies

−Cε ≤

Γ

b[Ψ1 ]ds ≤ U + O(ε),
126

(7.4.49)

and
||∆G Ψ1 ||2 2
≤ U ε−4 + O(ε−3 ),
L (Γ(2ℓ))

(7.4.50)

b[u, v] = (La u, La v)J + ε(˜u, v)J ,
q

(7.4.51)

where

is the bilinear form associated with L = L2 + ε˜ on H 2 (−ℓ/ε, ℓ/ε).
q
a

Proof: From assumption (7.3.24)

B[Ψ] = B[Ψ1 ] + 2B[Ψ1 , Ψ2 ] + B[Ψ2 ] ≤ U.

(7.4.52)

Lower bounds on B[Ψ2 ] and B[Ψ1 , Ψ2 ] in Lemmas 7.12 and 7.13 imply
B[Ψ1 ] ≤ U + O(ε2 ).

(7.4.53)

Substituting expression (7.4.39) for B[Ψ1 ] gives

Γ

∂Ψ ∂Ψ1
∂ 2 Ψ1 ∂ 2 Ψ1
+q 1
Gij Jdzds
∂z∂si ∂z∂sj
∂sj ∂si
i,j Γ −2ℓ/ε
2ℓ/ε
ε4 (∆G Ψ1 )2 Jdzds + O(ε2 ) ≤ U + O(ε2 ).
(7.4.54)
Γ −2ℓ/ε

b[Ψ1 ]ds + 2ε2
+

2ℓ/ε

Using (7.4.43) in (7.4.54) and dropping the positive (∆G Ψ1 )2 integral yields the upper bound
in (7.4.49). To obtain the lower bound in (7.4.49) we observe that the definition of b in (7.4.51)
127

and ||˜||L∞ (Γ(2ℓ)) = O(1) imply
q

Γ

b[Ψ1 ]ds ≥ ε

Γ

(˜Ψ1 , Ψ1 )J ds ≥ −ε||˜||L∞ (Γ(2ℓ))
q
q

Γ

||Ψ1 ||2 ds ≥ −Cε,
J

(7.4.55)

since
||Ψ1 ||2 ds = ||Ψ1 ||2 2
≤ ||Ψ1 ||2 2
= 1.
J
L (Γ(2ℓ))
L (Ω)
Γ

(7.4.56)

The last estimate (7.4.50) follows from (7.4.54) and Lemma 7.16.

Proof of Theorem 7.11:

By Lemma 7.17, Ψ1 = η1 Ψ satisfies estimate (7.4.49). Substituting the expression for the
bilinear form b (see Table 7.1) into the estimate (7.4.49) yields

Γ

(La Ψ1 , La Ψ1 )J ds + ε

Γ

(˜Ψ1 , Ψ1 )J ds ≤ U + O(ε).
q

(7.4.57)

We use ||˜||L∞ (Ω) = O(1) and ||Ψ|| 2
q
= 1 to estimate the second term on the left-hand
L (Ω)
side of (7.4.57)

ε

Γ

(˜Ψ1 , Ψ1 )J ds = ε
q

q
q Ψ2 dx ≤ ε||˜||L∞ (Ω) ||Ψ1 ||2 2
˜ 1
= O(ε).
L (Ω)
Γ(2ℓ)

(7.4.58)

Returning this estimate to (7.4.57) yields the upper bound

Γ

(La Ψ1 , La Ψ1 )J ds ≤ U + O(ε).
128

(7.4.59)

Substituting the definition of La , from (7.1.16), into (7.4.59) yields

Γ

(L0 + ε(q1 − κ∂z ))Ψ1 , (L0 + ε(q1 − κ∂z ))Ψ1

J

ds ≤ U + O(ε),

(7.4.60)

and expanding the J-inner product we obtain

(L0 Ψ1 , L0 Ψ1 )J + 2ε (q1 − κ∂z )Ψ1 , L0 Ψ1
+ ε2 (q1 − κ∂z )Ψ1 , (q1 − κ∂z )Ψ1 ds
J
J
Γ
≤ U + O(ε).

(7.4.61)

Using estimates (7.2.16) and (7.2.17) of Proposition 7.8 in (7.4.8) and (7.4.9) yields control of
z derivatives of Ψ1
||∂z Ψ1 ||2 2
= O(1),
L (Γ(2ℓ))

(7.4.62)

2
||∂z Ψ1 ||2 2
= O(1).
L (Γ(2ℓ))

(7.4.63)

and

Using these derivative bounds and ||q0 ||L∞ (Γ(2ℓ)) , ||q1 ||L∞ (Γ(2ℓ)) = O(1), we estimate
2
||L0 Ψ1 || 2
= || − ∂z Ψ1 + q0 Ψ1 || 2
L (Γ(2ℓ))
L (Γ(2ℓ))
2
≤ ||∂z Ψ1 || 2
+ ||q0 ||L∞ (Γ(2ℓ)) ||Ψ1 || 2
= O(1),
L (Γ(2ℓ))
L (Γ(2ℓ))
(7.4.64)

129

and

Γ

(q1 − κ∂z )Ψ1 , L0 Ψ1

J

ds

=

2ℓ/ε
Γ −2ℓ/ε

((q1 − κ∂z )Ψ1 )(L0 Ψ1 )Jdzds

≤ ||q1 ||L∞ (Γ(2ℓ)) ||Ψ1 || 2
||L Ψ ||
L (Γ(2ℓ)) 0 1 L2 (Γ(2ℓ))
+ ||κ||L∞ (Γ(2ℓ)) ||∂z Ψ1 || 2
||L Ψ ||
L (Γ(2ℓ)) 0 1 L2 (Γ(2ℓ))
= O(1).

(7.4.65)

Similar estimates imply

Γ

(q1 − κ∂z )Ψ1 , (q1 − κ∂z )Ψ1

J

ds = O(1).

(7.4.66)

Returning (7.4.65) and (7.4.66) to (7.4.61) yields

Γ

(L0 Ψ1 , L0 Ψ1 )J ds ≤ U + O(ε).

(7.4.67)

However, the definition L0 allows us to expand
2
L0 (J 1/2 Ψ1 ) = (−∂z + q0 )(J 1/2 Ψ1 )
= −∂z (1/2)J −1/2 (∂z J)Ψ1 + J 1/2 ∂z (Ψ1 ) + q0 J 1/2 Ψ1
1
1 −3/2
2
J
(∂z J)2 Ψ1 − J −1/2 (∂z J)Ψ1
4
2
2
− J −1/2 (∂z J)(∂z Ψ1 ) − J 1/2 (∂z Ψ1 ) + q0 J 1/2 Ψ1
=

1
1
2
= J 1/2 L0 Ψ1 + J −3/2 (∂z J)2 Ψ1 − J −1/2 (∂z J)Ψ1 − J −1/2 ∂z J∂z Ψ1
4
2

130

to obtain
L0 (J 1/2 Ψ1 ), L0 (J 1/2 Ψ1 )

0

= (L0 Ψ1 , L0 Ψ1 )J + O(ε),

(7.4.68)

where we used (2.4.26) to estimate the terms involving the derivatives of J. Decompose
J 1/2 Ψ1 into tangential and normal modes as in (7.3.25)
r

J 1/2 Ψ1 =

k=1

0
ck (s)ψk (s) + ψ ⊥ (s),

(7.4.69)

where
0
(ψk (s), ψ ⊥ (s))0 = 0,

(7.4.70)

for each s ∈ S and since Γ ||Ψ1 ||2 ds = ||Ψ1 ||2 2
≤ 1,
J
L (Γ(2ℓ))
r
Γ k=1

c2 ds +
k

Γ

||ψ ⊥ ||2 ds = 1.
0

(7.4.71)

Substituting the decomposition (7.4.69) into (7.4.68) yields,
r
(L0 Ψ1 , L0 Ψ1 )J =

L0
r

=

k=1

0
c k ψ k + ψ ⊥ , L0

0
c k λ 0 ψ k + L0 ψ ⊥ ,
k

r

r
l=1

0
cl ψ l + ψ ⊥

0
c l λ 0 ψ l + L0 ψ ⊥
l

l=1
k=1
r
=
c2 (λ0 )2 + (L0 ψ ⊥ , L0 ψ ⊥ )0 + O(ε),
k k
k=1

131

+ O(ε)

0

+ O(ε)

(7.4.72)

and the bound,
r
Γ

k=1

c2 (s)(λ0 (s))2 ds +
k
k

Γ

(L0 ψ ⊥ , L0 ψ ⊥ )ds ≤ U + O(ε),

(7.4.73)

Since the first integral in (7.4.73) is nonnegative,

Γ

(L0 ψ ⊥ , L0 ψ ⊥ ) ≤ U + O(ε).

(7.4.74)

0
0
In addition, choosing V = span{ψ1 , . . . , ψr } in the mini-max characterization for L0 and
recalling the lower bound on the rest of the spectrum in Hypothesis (H) yields
(L ψ ⊥ , ψ ⊥ )0
.
≤ 0
ν 0 ≤ λ0
r+1
||ψ ⊥ ||2
0

(7.4.75)

from which we deduce the bound

ν0 ||ψ ⊥ ||2 ≤ (L0 ψ ⊥ , ψ ⊥ )0 ≤ ||L0 ψ ⊥ ||0 ||ψ ⊥ ||0 .
0

(7.4.76)

Dividing by ||ψ ⊥ ||0 , squaring both sides and integrating with respect to Γ yields the estimate
1
||ψ ⊥ ||2 ds ≤
||L ψ ⊥ ||2 ds ≤
0
0
2 Γ 0
ν0
Γ

U
+ O(ε).
2
ν0

(7.4.77)

Equivalently we may write this as

||J −1/2 ψ ⊥ ||2 2
||ψ ⊥ ||2 ds ≤
=
0
L (Γ(2ℓ))
Γ
132

U
+ O(ε).
2
ν0

(7.4.78)

Setting Ψ⊥ = J −1/2 ψ ⊥ yields (7.3.27).

To address the H 2 bounds of (7.3.28) we take ∆s of (7.3.23), yielding

∆s ck = (∆s Ψ1 , ψk )J + (Ψ1 , ∆s ψk )J + (Ψ1 , ψk J −1 (∆s J))J
+ 2(∇s Ψ1 , ∇s ψk )J + 2(∇s Ψ1 · J −1 ∇s J, ψk )J
+ 2(Ψ1 , ∇s ψk · J −1 ∇s J)J .

(7.4.79)

Utilizing the relation between ∆G and ∆s shown in Lemma 2.10 we further obtain

(∆s Ψ1 , ψk )J

=

2ℓ/ε
−2ℓ/ε

(∆s Ψ1 )ψk Jdz =

2ℓ/ε
−2ℓ/ε

(∆G Ψ1 + εzDΨ1 )ψk Jdz

≤ ||∆G Ψ1 ||J ||ψk ||J + ε||DΨ1 ||J ||zψk ||J ,

(7.4.80)

where D represents a second order differential operator
d−1
D :=

dij
i,j=1

∂2
+
∂si ∂sj

with coefficients bounded by some constant C(K).
133

d−1
dj
j=1

∂
,
∂sj

(7.4.81)

Substituting the bound on the inner product in (7.4.80) into (7.4.79) we deduce

|∆s ck | ≤ ||∆G Ψ1 ||J + ε||DΨ1 ||J ||zψk ||J + ||Ψ1 ||J ||∆s ψk ||J
+ ||J −1 ∆s J||L∞ (Γ(2ℓ)) ||Ψ1 ||J
+ 2||∇s Ψ1 ||J ||∇s ψk ||J + 2||J −1 ∆s J||L∞ (Γ(2ℓ)) ||∇s Ψ1 ||J
+ 2||J −1 ∆s J||L∞ (Γ(2ℓ)) ||Ψ1 ||J ||∇s ψk ||J ,
(7.4.82)

where we used ||ψk ||J = 1. Squaring both sides of (7.4.82) and integrating over Γ, we note
that due to inequalities (7.1.19), (7.1.20), (2.4.26), and (7.4.8) the leading-order contribution
comes from the ∆G Ψ1 term, which yields the desired result after applying (7.4.50),
||∆s ck ||2 2
≤ ||∆G Ψ1 ||2 2
+ O(ε−3 ) ≤ U ε−4 + O(ε−3 ).
L (Γ)
L (Γ(2ℓ))

(7.4.83)

Addressing (7.3.29) we first observe that, taking collectively, the expression (7.4.39) and estimates (7.4.43) and (7.4.49) afford the bound

B[Ψ1 ] ≥ −Cε.

(7.4.84)

Decomposing Ψ into near and far-field parts Ψ = Ψ1 + Ψ2 and using (7.4.10) and (7.4.84) we
find that
B[Ψ2 ] = B[Ψ] − 2B[Ψ1 , Ψ2 ] − B[Ψ1 ] ≤ B[Ψ] + Cε.

134

(7.4.85)

However, inequality (7.4.1) and assumption (7.3.24) imply that
2
q+
2

||Ψ2 ||2 2
≤ B[Ψ2 ] ≤ B[Ψ] + Cε ≤ U + O(ε).
L (Ω)

(7.4.86)

2
q+
Dividing by 2 yields (7.3.29).
To address (7.3.30) we again decompose Ψ as Ψ = Ψ1 + Ψ2 , and recall that ||Ψ1 || 2
=
L (Ω)
||η1 Ψ|| 2
≤ 1. In addition, (7.4.86) implies that
L (Ω)
2
||Ψ2 || 2
≤
B[Ψ2 ] ≤
L (Ω) q 2
+

4
1
U0 ≤ ,
2
4
q+

(7.4.87)

1 2
where we used the assumption that U0 ≤ 32 q+ . This observation leads to the inequalities
||Ψ1 ||2 2
= ||Ψ||2 2
− ||Ψ2 ||2 2
−2
Ψ1 Ψ2 dx
L (Ω)
L (Ω)
L (Ω)
Ω
≥1−

1
1
− 2||Ψ1 || 2
||Ψ2 || 2
> .
L (Ω)
L (Ω) 4
16

(7.4.88)

Rewriting (7.4.71), we isolate the ck terms, and use the estimate above and (7.4.78) to obtain
the lower bound
r

U
1
c2 ds = ||Ψ1 ||2 2
− ||J −1/2 ψ ⊥ ||2 2
− O(ε).
≥ −
k
L (Ω)
L (Ω) 4 ν 2
Γ
0
k=1
1 2
However, by assumption U ≤ 32 ν0 , which yields (7.3.30).

135

(7.4.89)

Chapter 8
Analysis of the Full Linearized
Operator - Coercivity of The Bilinear
Form
The main result of Chapter 7, presented in Theorem 7.11, was the bound on the remainder
Ψ⊥ in the decomposition
r

ck (s)ψk (s, z) + Ψ⊥ (x),

Ψ(x) = Ψ(s, z) =

(8.0.1)

k=1
of ‘low energy’ functions Ψ. It follows that a norm-1 w that is orthogonal to

r
k=1 ck (s)ψk (s, z),

is approximately orthogonal to Ψ. In fact, the size of the projection (w, Ψ) 2
is then less
L (Ω)
than ||Ψ⊥ || 2
.
L (Ω)
In this chapter we use the minimax principle for eigenfunctions Θi of ∆s to control the
136

⊥
size of projections (w, Ψ) 2
for w ∈ ZM , where
L (Ω)
ZM,r := span {η1 Θj (s)Hk (s, z)}j=1...M,k=1...r ∪ {1}

(8.0.2)

is a finite dimensional subspace of L2 (Γ) and Hk (s, z) are any functions supported in Γ(2ℓ)
that satisfy
sup ||Hk − ψk ||J = O(ε).
s

(8.0.3)

Remark: Definition of ZM in (3.3) is a special case and corresponds to r = 1 and H1 =
ε−1/2 φ′ .
⊥
In Proposition 8.1 we establish ‘approximate orthogonality’ of w ∈ ZM,r to the low energy
eigenspace of L. Next, in Proposition 8.2 we show that this implies that the bilinear form
⊥
associated with L is coercive on ZM,r . We conclude the chapter with the proof of Theorem
3.5.

8.1

Orthogonality to Low Energy Eigenspace

The next proposition shows that the functions orthogonal to the ‘slow space’ ZM,r have small
projections onto low energy functions.
1
2
Proposition 8.1. Let U0 = 32 min{q+ , ν}. Assume Ψ ∈ H 2 (Ω) with ||Ψ|| 2
= 1,
L (Ω)
∂ν Ψ = 0 on ∂Ω satisfies the bound

B[Ψ] ≤ U < U0 ,
137

(8.1.1)

for the bilinear form B given in (7.3.1). Recall the definition of ZM,r in (8.0.2). Then for
⊥
any w ∈ ZM,r , satisfying ||w|| 2
= 1, the following holds true
L (Ω)




8 
ε−3
4
U +O
+ +
,ε ,
(w, Ψ)2 2
≤
2
2
2
L (Ω)
βM +1 ν q+
βM +1
4rε−4

(8.1.2)

where βi is the ith eigenvalue of −∆s : H 2 (Γ) → L2 (Γ).

Proof:
We use Theorem 7.11 to decompose Ψ into near and far-field parts

Ψ(x) = Ψ1 (x) + Ψ2 (x) = η1 Ψ(x) + Ψ2 (x) = η1 Ψ1 (x) + Ψ2 (x)
r
=
η1 ck (s)ψk (s, z) + η1 Ψ⊥ (s, z) + Ψ2 (x)
k=1

(8.1.3)

where
U
||Ψ⊥ ||2 2
≤ + O(ε),
L (Γ(2ℓ))
ν

(8.1.4)

U
||Ψ2 ||2 2
+ O(ε),
≤
L (Ω) q 2
+

(8.1.5)

||∆s ck (s)||2 2
≤ U ε−4 + O(ε−3 ).
L (Γ)

(8.1.6)

Since η1 is supported inside Γ(2ℓ) we may write the inner product of (8.1.3) with w as,
r
(w, Ψ) 2
= w,
L (Ω)

η1 c k ψ k
k=1

⊥
2 (Γ(2ℓ)) + w, η1 Ψ L2 (Γ(2ℓ)) + w, Ψ2 L2 (Ω) . (8.1.7)
L
138

With the aid of (8.0.3) we may replace ψk in (8.1.7) with Hk to obtain
r
(w, Ψ) 2
= w,
L (Ω)

η1 c k H k
k=1

L2 (Γ(2ℓ))

+ w, Ψ2 2
+ O(ε).
+ w, η1 Ψ⊥ 2
L (Γ(2ℓ))
L (Ω)
(8.1.8)

In addition, we decompose ck into its projections onto the Laplace-Beltrami eigenfunctions
Θi to obtain

M
ck (s) =
j=1

(ck , Θj )Γ Θj + c⊥ .
k

(8.1.9)

Substituting the expression for ck into (8.1.8) gives

= A1 + A2 + A3 + A4 + O(ε),
(w, Ψ) 2
L (Ω)

(8.1.10)

where
M

r

A1 :=
j=1 k=1

(ck , Θj )Γ (w, η1 Θj Hk ) 2
,
L (Γ(2ℓ))
r

A2 := w,
k=1

η1 c⊥ (s)ψk
k

L2 (Γ(2ℓ))

,

(8.1.11)

(8.1.12)

A3 := (w, η1 Ψ⊥ ) 2
,
L (Γ(2ℓ))

(8.1.13)

A4 := w, Ψ2 2
.
L (Ω)

(8.1.14)

It follows from (8.1.10) that

(w, Ψ)2 2
= (A1 + A2 + A3 + A4 )2 + O(ε2 ) ≤ 4A2 + 4A2 + 4A2 + 4A2 + O(ε2 ). (8.1.15)
4
3
2
1
L (Ω)

139

⊥
We will estimate each term separately. Due to the fact that w ∈ ZM,r ,
A1 = 0.

(8.1.16)

To bound A3 and A4 , we use ||w|| 2
= 1 and (8.1.4) to obtain
L (Ω)
U
A2 = (w, η1 Ψ⊥ )2 2
≤ ||w||2 2
||Ψ⊥ ||2 2
≤ + O(ε),
3
L (Ω)
L (Ω)
L (Ω)
ν

(8.1.17)

and (8.1.5) to bound

2U
A2 = (w, Ψ2 )2 2
≤ 2||w||2 2
||Ψ2 ||2 2
+ O(ε).
≤
4
L (Ω)
L (Ω)
L (Ω) q 2
+

(8.1.18)

It remains to bound A2 . Since (c⊥ , Θj )Γ = 0 for j = 1, . . . , M and Θj is the jth eigenk
2
function of ∆2 corresponding to eigenvalue βj we have from the minimax characterization of
s
eigenvalues of ∆2
s

(∆s c⊥ , ∆s c⊥ )Γ
2
k .
k
βM +1 ≤
⊥ ||2
||c Γ
k

(8.1.19)

In addition, decomposition (8.1.9) yields

∆ 2 ck =
s

M
j=1

(ck , Θj )Γ ∆2 Θj + ∆2 c⊥ =
s
s k

M
j=1

2
βj (ck , Θj )Γ Θj + ∆2 c⊥
s k

(8.1.20)

so that we may expand the inner product as

(∆s ck , ∆s ck )Γ = (∆2 ck , ck )Γ =
s

M
j=1

2
βj (ck , Θj )2 + (∆2 c⊥ , ck )Γ
s k
Γ

140

(8.1.21)

Using (8.1.20) to expand the last term yields
M
(∆s ck , ∆s ck )Γ =
j=1
M
=
j=1

2
βj (ck , Θj )2 +
Γ

M
j=1

2
βj (ck , Θj )Γ (c⊥ , Θj )Γ + (∆s c⊥ , ∆s c⊥ )Γ
k
k
k

2
βj (ck , Θj )2 + (∆s c⊥ , ∆s c⊥ )Γ ,
Γ
k
k

(8.1.22)

2
since (c⊥ , Θj )Γ = 0. However, since βj ≥ 0 for all j we conclude from (8.1.22) that
k
||∆s c⊥ ||2 ≤ ||∆s ck ||2 .
Γ
k Γ

(8.1.23)

||∆s c⊥ ||2 ≤ U ε−4 + O(ε−3 ).
k Γ

(8.1.24)

Using (8.1.6) in (8.1.23) implies

It then follows from (8.1.19) that
U ε−4
ε−3
||c⊥ ||2 ≤
+O
,
k Γ β2
2
βM +1
M +1

(8.1.25)

and
A2 = w,
2

r

2
rU ε−4
ε−3
≤
+O
.
η1 c⊥ (s)ψk 2
k
2
2
L (Γ(2ℓ))
βM +1
βM +1
k=1

Returning the estimates for A1 , A2 , A3 and A4 to (8.1.15) gives (8.1.2).
141

(8.1.26)

8.2

Coercivity of the Bilinear Form

⊥
The previous proposition gives an upper bound on the projection of ZM,r onto the small
eigenspace of L. In this section we show that if this upper bound is sufficiently small, then the
⊥
operator L satisfies a coercivity estimate on ZM,r . Let Ψi be the ith normalized eigenfunction
of L. Fix U0 > 0 independent of ε. For 0 < U < U0 let
σU (L) := σ(L) ∩ {λ < U } = {µ1 , . . . , µN }.
U

(8.2.1)

1
2
Proposition 8.2. Let U0 = 32 min{q+ , ν}. Recall definition of NU from (8.2.1) and ZM,r
from (8.0.2). If U < U0 and M > 0 satisfy



4rε−4

2
βM +1

+



8 
ε−3
4
1
+
U +O
,ε < ,
2
ν q2
2
βM +1
+

(8.2.2)

⊥
then for any w ∈ ZM,r satisfying
∂ν w|∂Ω = 0,

(8.2.3)

U
+ O(ε) ||w||2 2
L (Ω)
2

(8.2.4)

the following coercivity property holds

B[w] ≥

where recall that B is the bilinear form for L defined in (7.3.1). In addition, we have the
bounds

√

U
||Aw|| 2
≥
||w|| 2
,
L (Ω)
L (Ω)
2
142

(8.2.5)

U
,
||A2 w|| 2
≥ ||w|| 2
L (Ω)
L (Ω)
4
√
U
2 w||
||A
2 (Ω) ≥ 2 ||Aw||L2 (Ω) ,
L

(8.2.6)

(8.2.7)

and
√
||Aw|| 2
≥ C U ε2 ||w|| 2
.
L (Ω)
H (Ω)

(8.2.8)

Remark: In particular, if M = [M1 ε−(d−1) ] then βM = O(ε−2 ) and the coercivity estimates above are valid with U sufficiently small, but independent of ε.

Proof: We denote by Ψi the eigenfunction of L corresponding to the eigenvalue µi ∈ σU (L),
defined in (8.2.1). If ZM,r was exactly the eigenspace spanned by eigenfunctions corresponding to σU (L), the coercivity estimates above would follow immediately from the minimax
principle. However, ZM,r only ‘approximates’ the small eigenspace of L. Define the projecNU
⊥
tion of w ∈ ZM,r onto the egeinspace {Ψi }i=1
NU
PU w :=
i=1

(w, Ψi ) 2
Ψ.
L (Ω) i

(8.2.9)

For any w ∈ H 2 (Ω), the minimax characterization of eigenvalues of L gives
B w − PU w ≥ U w − PU w

2
.
L2 (Ω)

(8.2.10)

Our goal is to show that PU w is small in an appropriate norm. Expanding the bilinear term
143

and the norm, and using orthogonality we obtain



B[w] − 2B w, PU w + B PU w ≥ U ||w||2 2
−
L (Ω)

NU
i=1




(w, Ψi )2  .

(8.2.11)

We may simplify the second term of (8.2.11)
NU
B w, PU w

=
i=1
NU
=
i=1

NU
(w, Ψi ) 2
B[w, Ψi ] =
L (Ω)

i=1

(w, Ψi ) 2
µ (w, Ψi ) 2
L (Ω) i
L (Ω)

µi (w, Ψi )2 2
.
L (Ω)

(8.2.12)

Using orthonormality of eigenfunctions Ψi we estimate the third term of (8.2.11)
NU
B PU w = B
i=1

NU
(w, Ψi ) 2
Ψ,
L (Ω) i

NU

j=1

(w, Ψj ) 2
Ψ =
L (Ω) j

i=1

µi (w, Ψi )2 2
. (8.2.13)
L (Ω)

Using (8.2.12) and (8.2.13) in (8.2.11) gives
NU
B[w] ≥

i=1

µi (w, Ψi )2 2
+ U ||w||2 2
−
L (Ω)
L (Ω)

≥ U ||w||2 2
+
L (Ω)

NU
i=1

NU
i=1

(w, Ψi )2 2
L (Ω)

(µi − U )(w, Ψi )2 2
L (Ω)

≥ U ||w||2 2
+ (µ1 − U )
L (Ω)

NU
i=1

144

(w, Ψi )2 2
,
L (Ω)

(8.2.14)

since µ1 ≤ µi . In addition,
||PU w||2 2
=
L (Ω)

NU
i=1

(w, Ψi )2 2
.
L (Ω)

(8.2.15)

and since B[Ψi , Ψj ] = µi δij , (8.2.9) gives
NU
PU w
1
1
B[PU w] =
µi (w, Ψi )2 2
≤ U,
B
=
2
2
L (Ω)
||PU w|| 2
||PU w|| 2
||PU w|| 2
L (Ω)
L (Ω)
L (Ω) i=1
(8.2.16)
PU w
since µi ≤ U for i = 1, . . . , NU . Applying Proposition 8.1 with Ψ =
||PU w|| 2
L (Ω)
NU
i=1

2
PU w
||PU w|| 2
L2 (Ω)
L (Ω)
U
ε−3
rU ε−4 U
+ O(ε) + O
+ +
≤
2
2
2
ν
βM +1
q+
βM +1
1
||w||2 2
,
<
L (Ω)
2

(w, Ψi )2 2
=
L (Ω)

w,

||w||2 2
L (Ω)
(8.2.17)

by assumption (8.2.2). Returning this estimate to (8.2.14) and using −Cε ≤ µ1 ≤ U gives
B[w] ≥ U ||w||2 2
−
L (Ω)

(U − µ1 )
U
||w||2 2
+ O(ε) ||w||2 2
≥
.
L (Ω)
L (Ω)
2
2

(8.2.18)

This gives (8.2.4). To obtain the remaining coercivity estimates note that using the definition
of B in (7.3.1)
||Aw||2 2
= B[w] − ε(˜w, w),
q
L (Ω)

145

(8.2.19)

and if U is independent of ε

U
||Aw||2 2
≥ ||w||2 2
L (Ω)
L (Ω)
4

(8.2.20)

follows from (8.2.4) and ||˜||L∞ (Ω) = O(1). Similarly
q
U
||A2 w|| 2
||w|| 2
≥ (A2 w, w) = ||Aw||2 2
≥ ||w||2 2
,
L (Ω)
L (Ω)
L (Ω)
L (Ω)
4

(8.2.21)

and (8.2.5) follows by dividing by ||w|| 2
. Similarly, (8.2.7) follows from
L (Ω)
4
||Aw||2 = (Aw, Aw) = (A2 w, w) ≤ ||A2 w|| 2
||w|| 2
≤ ||A2 w||2 2
,
L (Ω)
L (Ω) U
L (Ω)

(8.2.22)

To obtain the H 2 coercivity in (8.2.6) we write ∆w in terms of Aw
∆w = −

1
1
Aw + qw,
ε2
ε2

(8.2.23)

and use elliptic regularity (Evans, Theorem 4, p.317) to obtain

.
||w|| 2
≤ C ||w|| 2
+ ε−2 ||Aw|| + ε−2 ||q||∞ ||w|| 2
H (Ω)
L (Ω)
L (Ω)
The H 2 coercivity in (8.2.8) follows from (8.2.5).

146

(8.2.24)

8.3

Proof of Theorem 3.5

In the case of linearization Lφ about a single-layer interface, r = 1.

We choose U =

1 4 2
−(d−1) . Weyl asymptotic formula implies that β
−2
M +1 = O(ε )
16 ε βM +1 and M = [M1 ε
and the assumption of Proposition 8.2 is satisfied. Coercivity estimates (3.2.17)-(3.2.19) follow
from Proposition 8.2 by taking H1 := ε−1/2 φ′ .
In addition, addressing (3.2.20) we observe that since ||W ′′ (Φ)||L∞ (Ω) = O(1),
= ||ε2 ∆w − W ′′ (Φ)w + ε2 ηw|| 2
||Aφ w|| 2
L (Ω)
L (Ω)
+ ε−2 ||W ′′ (Φ) + ε2 η||L∞ (Ω) ||w|| 2
≤ ε2 ||∆w|| 2
L (Ω)
L (Ω)
≤ Cε2 ||w||A .

(8.3.1)

We also deduce from (3.2.17) and (3.2.19) that

||w||A = ||∆w|| 2
+ ε−2 ||w|| 2
≤ ||w|| 2
+ ε−2 ||w|| 2
L (Ω)
L (Ω)
H (Ω)
L (Ω)
1
≤ Cε−4
||A w||
≤ Cb ε−2 ||Aφ w|| 2
,
L (Ω)
βM +1 φ L2 (Ω)
where in the last step we used βM +1 = O(ε−2 ).

147

(8.3.2)

Chapter 9
Interface Evolution - Bounds on Lower
Order Terms

9.1

Linear Operator

The following lemma is established by differentiation of equation (3.1.5).
Lemma 9.1. The spectrum of Lφ near the origin consists of a single eigenvalue at zero
corresponding to the translational eigenfunction φ′ . The remainder of the spectrum of Lφ is
real and uniformly negative. Moreover the following identities hold,

Lφ φ′ = 0,

(9.1.1)

Lφ φzz = W ′′′ (φ)φ′2 ,

(9.1.2)

Lφ (zφ′ ) = 2φzz .

(9.1.3)

148

9.2

Nonlinearity Control

Our main goal in this section is to establish energy inequality (3.3.53). We first need the
following bound on the nonlinearity N (w) defined in (3.3.27).
Lemma 9.2. The nonlinearity N (w) defined in (3.3.27) satisfies
||N (w)|| 2
≤ C(ε−2 ||Aφ w||2 2
+ ε−4 ||Aφ w||3 2
).
L (Ω)
L (Ω)
L (Ω)

(9.2.1)

Proof: Utilizing W (m) (u) = 0 for m > 4, it is straightforward to write N (w) in the following
form

N (w) = a1 w2 + a2 w3 + ε2 a3 |∇w|2 + ε2 a4 w∆w + ε2 a5 w2 ∆w + ε2 a6 w|∇w|2 ,

(9.2.2)

where ai (x) are smooth O(1) functions. We use the coercivity estimates in Proposition 8.2
and the following Sobolev Inequalities (d ≤ 3)
||w||L∞ (Ω) ≤ ||w|| 2
≤ ε−2 ||Aφ w|| 2
,
H (Ω)
L (Ω)
||w||

W 1,4 (Ω)

≤ ||w|| 2
≤ ε−2 ||Aφ w|| 2
,
H (Ω)
L (Ω)

(9.2.3)

(9.2.4)

to estimate

||a1 w2 || 2
≤ ||a1 ||L∞ (Ω) ||w||L∞ (Ω) ||w|| 2
≤ Cε−2 ||Aφ w||2 2
,
L (Ω)
L (Ω)
L (Ω)

(9.2.5)

||a2 w3 || 2
≤ ||a2 ||L∞ (Ω) ||w2 ||L∞ (Ω) ||w|| 2
≤ Cε−4 ||Aφ w||3 2
,
L (Ω)
L (Ω)
L (Ω)

(9.2.6)

149

||ε2 a3 |∇w|2 || 2
≤ ε2 ||a3 ||L∞ (Ω) ||w||2 1,4
≤ Cε−2 ||Aφ w||2 2
,
L (Ω)
L (Ω)
W (Ω)

(9.2.7)

||ε2 a4 w∆w|| 2
≤ ε2 ||a4 ||L∞ (Ω) ||w||L∞ (Ω) ||w|| 2
≤ Cε−2 ||Aφ w||2 2
. (9.2.8)
L (Ω)
H (Ω)
L (Ω)
The rest of the estimates follow similarly and we obtain (9.2.1).
Corollary 9.3. The following estimate holds:

(Π0 N (w), Zi ) 2
≤ C(K)ε5/2 ||w||2 + ||w||3 ,
A
A
L (Ω)

(9.2.9)

where Zi = η1 Θi,Γ φ′ was defined in Definition 3.3 and C(K) > 0 is independent of ε and
t
Γ ∈ GK .
Proof: The result follows from (9.2.1) and

||Aφ w|| 2
= ||ε2 ∆w − W ′′ (Φ)w + ε2 ηw|| 2
L (Ω)
L (Ω)
≤ C ε2 ||∆w|| 2
+ ||w|| 2
L (Ω)
L (Ω)
≤ Cε2 ||w||A

(9.2.10)

and
||Zi ||2 2
=
L (Ω)

2ℓ/ε
Γ −2ℓ/ε

(η1 φ′ Θi )2 Jdzds = O(ε),

(9.2.11)

due to the ε in the Jacobian.
⊥
Proposition 9.4. There exists C > 0 such that for all w ∈ ZM ,
A2 w
φ

≤ C||Aφ w|| 2
ε1/2 .
L (Ω)
150

(9.2.12)

Lw

≤ C||Aφ w|| 2
ε1/2 ,
L (Ω)

(9.2.13)

1
f (x)dx.
|Ω| Ω

(9.2.14)

where
f :=

Proof: Using ∂ν Aφ w = 0 on ∂Ω we obtain

Ω

A2 wdx =
φ

Ω

(−ε2 ∆ + q)Aφ wdx =

Ω

||A w||
,
qAφ wdx ≤ ||q|| 2
L (Ω) φ L2 (Ω)
(9.2.15)

where q is given for the functionalized operator in Remark 7.2.
Since q is localized on Γ, the extra factor of ε that the ||q||2 2
acquires from the Jacobian
L (Ω)
yields (9.2.12). Similarly, since (w, 1) 2
= 0 and consequently Π0 w = w, we obtain
L (Ω)

Ω

q wdx =
˜

Ω

(Π0 q )w +
˜

Ω

q wdx =
˜

Ω

(Π0 q )w,
˜

(9.2.16)

q
and since q is localized on Γ, (9.2.13) follows from Lφ = A2 + ε˜ and coercivity estimate
˜
φ
||w|| 2
≤ U ||Aφ w|| 2
.
L (Ω)
L (Ω)

9.3

Normal Velocity

Proposition 9.5. Fix K > 0. Let z(x; t) ∈ R and s(x; t) ∈ Rd−1 be the whiskered coordinates
of a point x ∈ Γt (2ℓ) with respect to interface Γt . Then, there exists C(K) > 0 such that for
151

all Γ ∈ GK the following relations hold for the derivatives of sj (x, t) with respect to time,
∂sj (x; t)
∂t

L2 (Ω)

≤ C(K) ∇s Vn (s))

L2 (Γ)

.

(9.3.1)

Proof: We differentiate the change of coordinates

x = γ(s(t); t) + εz(t)ν(s(t), t)

(9.3.2)

with respect to t to obtain
d−1
j=1
Multiplying by

∂γ dsj
∂γ
dz
+
+ ε ν + εz(t)
∂sj dt
∂t
dt

d−1

∂ν
∂ν dsj
+ εz
= 0.
∂sj dt
∂t

j=1

(9.3.3)

∂γ
∂γ
= Vn ν yields
and recalling
∂si
∂t
d−1
j=1

gij − εzhij

dsj
dt

+ εz

∂ν ∂γ
= 0.
∂t ∂si

(9.3.4)

∂Vn ∂γ
.
∂sk ∂sm

(9.3.5)

From Proposition 4.4 we have

∂
∂ν
= ν=−
∂t
∂t

d−1
k,m=1

g km

Returning this estimate to (9.3.4)
d−1
j=1

gij − εzhij

dsj
dt

d−1
= εz
k,m=1
152

∂Vn km
g gmi .
∂si

(9.3.6)

Canceling the g, g −1 in the last term yields
d−1

dsj

gij − εzhij

j=1

dt

= εz

∂Vn
,
∂si

(9.3.7)

and (9.3.1) follows from uniform invertibility of g − εzh (see Lemma 2.11).
Corollary 9.6. Let
M
Vn :=
k=1

vi Θi,Γ .
t

(9.3.8)

Then
∂sj (x; t)
∂t

L2 (Ω)

≤ C(K)ε−1 ||Vn || 2
L (Γ)

(9.3.9)

and
Ω

ε 2 Ds φ 2 ·

∂s(x; t)
Zi dx ≤ C(K)ε2 ||Vn || 2 ,
L (Γ)
∂t

(9.3.10)

Proof: We note that Proposition 9.5, as well as

−∆s Θi = βi Θi ,

(9.3.11)

and orthonormality of eigenfunctions Θi with respect to the (·, ·)Γ inner product implies


∂sj (x; t) 2
∂t

M

2
 ds


vi ∇s Θi,Γ
≤ C(K)||∇s Vn ||2 2
= C(K)
t
L (Γ)
Γ i=1
2 (Ω)
L
M
2
= C(K)
vi βi ≤ C(K)ε−2 |v|2 = C(K)ε−2 ||Vn ||2 2 (9.3.12)
,
L (Γ)
i=1

where we used |βi | ≤ C(K)ε−2 (see Remark 3.4) and |v| = ||Vn || 2 . Similarly (9.3.10)
L (Γ)
153

follows from (9.3.12) and J = O(ε).

9.3.1

Leading Order Dynamics

We will denote by C(K) the different constants depending on K only. We start with the
following lemma, which addresses I, the first term on the left-hand side of (3.3.28).

Lemma 9.7. Fix K > 0. Let the normal velocity Vn of interface Γt ∈ GK be given by
(3.3.29). Then the following bound holds
2ℓ/ε
1
.
Vn Θi,Γ
η (∂ Φ)φ′ Jdzds − ||φ′ ||2 2
v ≤ C(K)ε2 ||Vn || 2
L (Ω)
t −2ℓ/ε 1 z
L (R) i
ε Γ

Proof of Lemma 9.7: Due to the expansion of the Jacobian J(s, z) = J0 (s)

d
j=0

(9.3.13)

εj+1 Kj z j

(2.2.17) we obtain

−

2ℓ/ε
1
Vn Θi,Γ
η (z)(∂z Φ)φ′ Jdzds
t −2ℓ/ε 1
ε Γ

d
2ℓ/ε
1
′ + ε2 ∂ φ )φ′
η1 (z)(φ
εj+1 Kj z j dzds. (9.3.14)
Vn Θi,Γ J0 (s)
= −
z 2
t
ε Γ
−2ℓ/ε
j=0
The exponential decay of φ′ when z → ±∞ implies the following asymptotic estimate,
2ℓ/ε

η1 (φ′ )2 dz = −||φ′ ||2 2
+ O(εN ),
L (R)
−2ℓ/ε
where N is any positive integer.
154

(9.3.15)

Therefore, the leading term becomes

−

Γ

Vn Θi,Γ

2ℓ/ε

η (z)(φ′ )2 J 0 dzds =
t −2ℓ/ε 1




M


vk Θk,Γ  Θi,Γ J 0 ds
(−||φ′ ||2 2
+ O(εN ))
t
t
L (R)
Γ k=1
= (−||φ′ ||2 2
+ O(εN ))vi .
L (R)

(9.3.16)

The next term of the asymptotic expansion is zero since η1 (z)(φ′ (z))2 z is an odd function on
the symmetric interval (−2ℓ/ε, 2ℓ/ε)

−ε

2ℓ/ε

K1 Vn Θi,Γ J 0
t
Γ

−2ℓ/ε

η1 (φ′ )2 zdz

ds = 0.

(9.3.17)

Employing again the exponential decay of φ′ (z) and boundness of ∂z φ2 and Kj (s) we obtain
asymptotic estimates uniform with respect to s ∈ Γt with constants depending on K only
2ℓ/ε
−2ℓ/ε

η1 (z)(φ′ )2

d−2
j=0

εj Kj+2 z j+2 dz ≤ C(K),

(9.3.18)

and
2ℓ/ε
−2ℓ/ε

d

η1 (z)φ′ ∂z φ2

j=1

εj dz ≤ C(K).

(9.3.19)

Now we can estimate the sum of the remaining terms in (9.3.14)

ε2 −

d−2
2ℓ/ε
η1 (z)(φ′ )2
εj Kj+2 z j+2 dzds
Vn Θi,Γ J0 (s)
t
−2ℓ/ε
Γ
j=0
155

(9.3.20)

2ℓ/ε

−

Vn Θi,Γ J0 (s)
t
−2ℓ/ε
Γ

η1 (z)φ′ ∂z φ2

d

εj Kj z j dzds

≤ C(K)ε2 ||Vn || 2 . (9.3.21)
L (Γ)

j=0

Referring to (3.3.28), we next estimate II = (Π0 F (Φ), Zi ) 2
+ ∂t b Ω Zi dx.
L (Ω)

Lemma 9.8. Fix K > 0. The following estimate holds

0
Zi dx − ε4 (Vn , Θi,Γ

≤ C(K)ε5 ,

(9.3.22)

2
H3
+ Htr(A2 ) −
µ 2 b3 − ∂ t b3 .
0
′ ||2
2
||φ
2 (R)
L

(9.3.23)

(Π0 F (Φ), Zi ) 2
+ ∂t b
L (Ω)

Ω

t L2 (Γ)

where

0
Vn := −

∆s + η H −

Proof of Lemma 9.8: Since φ′ is exponentially decaying and φ(∞) − φ(−∞) = 2 we first
observe that

Ω

Zi dx =

Γ

= 2ε

Θi,Γ

2ℓ/ε
t −2ℓ/ε

η1 φ′ Jdzds = ε

Θi,Γ J 0 ds + O(ε2 ),
t
Γ

Γ

Θi,Γ

t

∞
−∞

φ′ dz J 0 ds + O(ε2 )
(9.3.24)

and

∂t b

Ω

Zi dx = 2ε∂t b

Θi,Γ ds + O(ε2 |∂t b|) = 2ε4 ∂t b3
Θi,Γ ds + O(ε5 |∂t b3 |), (9.3.25)
t
t
Γ
Γ
156

where we used b = ε3 b3 and (2.2.17) to expand the Jacobian. In addition, letting

F (Φ) :=

1
Φdx,
|Ω| Ω

(9.3.26)

we obtain

(Π0 F (Φ), Zi ) 2
= (F (Φ), Zi ) 2
− F (Φ)
L (Ω)
L (Ω)

Ω

Zi dx

= (F (Φ), Zi ) 2
− 2ε
Θi,Γ ds F (Φ) + O(ε5 ), (9.3.27)
L (Ω)
t
Γ
where we used (9.3.24) and ||F (Φ)||L∞ (Ω) = O(ε3 ).
Again, using the definition of Zi and exponential decay of φ′ away from the interface,
similarly to the previous lemma we have
2ℓ/ε

η φ′ F (Φ)Jdzds
t −2ℓ/ε 1
Γ
∞
φ′ F (Φ)dz J 0 ds + O(ε2 )||F (Φ)||L∞ (Ω) .
(9.3.28)
= ε
Θi,Γ
t
−∞
Γ

(F (Φ), Zi ) 2
=
L (Ω)

Θi,Γ

From the proof of Proposition 3.1 we have

1
F (Φ) = ε3 R3 + O(ε4 ) = ε3 Lφ κ2 z 2 φ′ + κ0 ∂z φ2 + Lφ b3 + κ0 κ1 zφ′
2
+ ε3 κ0 κ1 zφzz + κ0 ∂z (κ1 zφ′ + Lφ φ2 ) + φ′ (∆s + η)κ0 − κ0 W ′′′ (φ)φ2 φ′ + O(ε4 ),
(9.3.29)

where O(ε4 ) is in L∞ (Ω) norm and the asymptotic constant is independent of Γ ∈ GK . We
157

take the L2 (R) inner product of F (Φ) with φ′ and integrate by parts to cancel the κ0 κ1 terms
∞
−∞

φ′ F (Φ)dz = ε3 − κ0 (Lφ φ2 , φ′′ ) 2
+ (∆s + η)(κ0 )||φ′ ||2 2
L (R)
L (R)
+ O(ε4 ).
− κ0 (W ′′′ (φ)(φ′ )2 , φ2 ) 2
L (R)

(9.3.30)

Using the definition (3.1.6) of φ2 and properties of Lφ in Lemma 9.1 we evaluate

(Lφ φ2 , φ′′ ) 2
=−
L (R)

κ2
0

2
1 κ0
+ κ1 (zφ′ , φzz ) 2
+ κ1
=
L (R) 2 2
2

φ′ 2 2
,
L (R)

(9.3.31)

and

(W ′′′ (φ)φ′2 , φ2 ) = −

2
1 κ0
+ κ1 (Lφ φ′′ , L−1 zφ′ ) 2
+ κ1 ||φ′ ||2 2
=
. (9.3.32)
L (R) 2 2
L (R)
2

κ2
0

Combining these results gives the leading-order expression
∞

φ′ F (Φ)dz = ε3 ||φ′ ||2 2
(∆s + η)κ0 −
L (R)
−∞
= −ε3 ||φ′ ||2 2
L (R)

κ3
0

∆s + η H −

158

2

+ κ0 κ1

+ O(ε4 )

H3
+ Htr(A2 ) + O(ε4 ).(9.3.33)
2

and
2ℓ/ε

η φ′ F (Φ)Jdzds
t −2ℓ/ε 1
Γ
∞
φ′ F (Φ)dz J 0 ds + O(ε5 )
= ε
Θi,Γ
t
−∞
Γ
H3
+ Htr(A2 ), Θi,Γ
= −ε4 ||φ′ ||2 2
∆s + η H −
t L2 (Γ)
L (R)
2
+ O(ε5 ).
(9.3.34)

(F (Φ), Zi ) 2
=
L (Ω)

Θi,Γ

In addition, to evaluate F (Φ) Ω we return to (9.3.29) and observe that all terms except those
involving the background perturbation, b, are localized on Γ. We integrate over Ω and change
to the whiskered variables for the localized terms. So long as Γ ∈ GK , we have |Γ| < K, and
the extra factor of ε that the localized terms acquire from the Jacobian renders them lower
order than the far-field terms. At the leading-order we have

F (Φ) Ω = |Ω|

Ω

L2 bdx + O(ε4 ) =
φ

b
(W ′′ (φ))2 dx + O(ε4 ) = bµ2 + O(ε4 ), (9.3.35)
0
|Ω| Ω

where recall from (1.0.4) that µ0 = W ′′ (±1) > 0. Combining (9.3.35) with (9.3.34) and
(9.3.27) yields (9.3.22).

159

9.4

Development of a priori estimates

Proposition 9.9. Assuming that (3.3.28) holds for each i = 1, . . . , M , then the coefficients
of Vn must satisfy

0
vi − ε4 (Vn , Θi,Γ ) 2
t L (Γ)



1
≤ C(K)ε2  √ +
ε

√
βi,Γ ε
t
βM +1,Γ −
t

βi,Γ
t



 Vn ∞ ||w||
A
L (Γ)

ε3 + |v| + ε3/2 ||w||A + ε1/2 ||w||2 + ε1/2 ||w||3
A
A

+

, (9.4.1)

where

0
Vn = −

H3
2
∆s + η H −
+ Htr(A2 ) −
µ 2 b3 − ∂ t b3 ,
0
′ ||2
2
||φ
L2 (R)

(9.4.2)

and C > 0 is independent of ε and Γ ∈ GK .
Proof: The idea is to simplify equation (3.3.28) and isolate the leading-order expression for
vi .
Referring to (3.3.28), we first estimate (∂t w, Zi ) 2
term in the following lemma.
L (Ω)
⊥
Lemma 9.10. For any differentiable function w which satisfies w(x, t) ∈ ZM we have the
estimate


1
≤ C(K)ε2  √ +
(∂t w, Zi ) 2
L (Ω)
ε

√
βi,Γ ε
t
βM +1,Γ −
t

where βi,Γ is the ith eigenvalue of −∆s .
t
160

βi,Γ
t



 Vn ∞ ||w|| ,
A
L (Γ)

(9.4.3)

Proof of Lemma 9.10: The estimate follows from Theorem 5.3 and

||w|| 2
≤ ε2 ||w||A .
L (Ω)

(9.4.4)

In addition, we summarize bounds on lower order terms in (3.3.28) with the aid of the
following table. We also give references to the locations of the corresponding proofs.

∂s(x;t)
2
2
Ω ε Ds φ2 · ∂t Zi dx ≤ C(K)ε ||Vn ||L2 (Ω)

Corollary 9.6

≤ C(K)ε7/2 ||w||A
(Π0 Lφ w, Zi ) 2
L (Ω)

Proposition 9.15

(Π0 N (w), Zi ) 2
≤ C(K)(ε5/2 ||w||2 + ε5/2 ||w||3 )
A
A
L (Ω)

Corollary 9.3

Combining the lemmas with the above estimates as well as leading-order calculations in Lemmas 9.7 and 9.8 we obtain (9.4.1).

Proposition 9.11. Assume that w satisfies

¯
||w||A ≤ 2B0 εd−3/2 .
161

(9.4.5)

Assuming that (3.3.28) holds for each i = 1, . . . , M , then the coefficients of Vn must satisfy

||Vn ||L∞ (Γ) ≤ C(K)ε3 ,

(9.4.6)

and

0
≤ C(K) ε5 + ε7/2 ||w||A .
vi − ε4 (Vn , Θi,Γ ) 2
t L (Γ)
(9.4.7)

Proof: First, utilizing Proposition 9.9 we obtain

Vn − ε4

M

0
(Vn , Θi )Γ Θi
i=1

M
≤

i=1

M
vi Θ i −

0
(ε4 Vn , Θi )Γ Θi

i=1

√
βi ε
Vn L∞ (Γ) ||w||A
βM +1 − βi
i=1
||Vn ||L∞ (Γ) + ε3 + ε3/2 ||w||A + ε1/2 ||w||2
A

≤ C(K)ε2
+

M

+ ε1/2 ||w||3
A

1
√ +
ε

Θi
(9.4.8)

In addition,
ε2

M
i=1



= ε5/2 

M
i=1

(

√
βi ε
βM +1 − βi

βi
βM +1 −

Θi

1/2 
1/2
M


Θi 2 
2
βi )
i=1
162



M

βi (
≤ C(K)ε5/2 
i=1

βM +1 +

1/2 

βi )2 



M



M
i=1

1/2

1/2
Θi 2 

2
Θi 2 
≤ C(K)ε5/2 (M βM )1/2 
i=1
≤ C(K)

M1 ε3/2−d ,

(9.4.9)

where we used M = [M1 ε−(d−1) ], βi ≤ βM ≤ C(K)ε−2 ,
M Θ 2 1/2 ≤ C(K)ε−(d−1)/2 and the fact that the eigenvalue gap β
M +1 −βM > cg
i=1 i
stays O(1). (see Remark 3.4) Similarly, setting

Rw = ||Vn ||L∞ (Γ) + ε3 + ε3/2 ||w||A + ε1/2 ||w||2 + ε1/2 ||w||3
A
A

(9.4.10)

we deduce

ε2



M

M

1/2 

2
Rw 
Rw |Θi | ≤ ε2 
i=1
i=1



M
i=1

1/2
≤ C(K)M1 Rw ε2−(d−1) .
Θi 2 

(9.4.11)

Using (9.4.9) and (9.4.11) in (9.4.8) gives

Vn − ε4

M
i=1

0
(Vn , Θi,Γ )Γ Θi,Γ
t
t

≤ M1 C(K) ε3/2−d ||Vn ||L∞ (Γ) ||w||A
+ ε2−(d−1) (||Vn ||L∞ (Γ) + ε3 + ε3/2 ||w||A
+ ε1/2 ||w||2 + ε1/2 ||w||3 ) .
A
A

163

(9.4.12)

¯
As long as ||w||A ≤ 2B0 εd−3/2 and M1 is sufficiently small, we have
V n − ε4

M
i=1

0
(Vn , Θi,Γ )Γ Θi,Γ ≤ C(K)ε3 ,
t
t

(9.4.13)

and (9.4.6) follows. Using (9.4.6) in the bound in Proposition 9.9 yields (9.4.7).

Proof of Proposition 3.11: The L∞ bound on Vn was shown in the previous proposition.
In addition, we showed that

0
vi − ε4 vi ≤ C(K) ε5 + ε7/2 ||w||A

(9.4.14)

and from (9.4.14)

||Vn ||2 2
= |v|2 =
L (Ω)
≤ ε8

M
i=1

M
i=1

2
vi ≤ ε8

M
i=1

0
(vi )2 + M C(K) ε10 + ε7 ||w||2
A

0
(vi )2 + M1 C(K) ε11−d + ε8−d ||w||2 ,
A

(9.4.15)

where we used M = M1 [ε−(d−1) ]. In addition, the approximate basis property (3.2.21)
implies
M
i=1

0
(vi )2 =

M
i=1

0
0
(Vn , Θi,Γ )2 = ||Vn ||2 2
+ O(ε2 ),
t
L (Γ)

which combined with (9.4.15) yields (3.3.37).
164

(9.4.16)

9.5

Other Estimates

Lemma 9.12. Fix K > 0. Then for all Γ ∈ GK , the following bound holds for the derivative
of the ansatz Φ with respect to t

||∂t Φ|| 2
≤ C(K)ε−1/2 Vn 2
.
L (Ω)
L (Ω)

(9.5.1)

Proof of Lemma 9.12: To bound the time derivative of the ansatz Φ we refer to (3.1.4) and
note that the dependence on t comes from the whiskered coordinates (s(x; t), z(x; t)) and the
background state b. It follows from Proposition 3.10 that

∂z(x; t)
= ε−1 Vn .
∂t

(9.5.2)

Similarly, we show in Proposition 9.5 that
∂sj (x; t)
∂t

L2 (Ω)

≤ C(K) ∇s Vn (s)

L2 (Γ)

≤ C(K)ε−1 ||Vn || 2 .
L (Γ)

(9.5.3)

The evolution of the background state is determined from the total mass conservation. Projecting (3.3.26) onto Z0 = 1, the right-hand side is identically zero and we obtain
∂z(x; t)
∂s(x; t)
Φz + ε2 Ds φ2 ·
+ ∂t bdx = 0,
∂t
Ω ∂t

(9.5.4)

⊥
where we used Ω ∂t wdx = ∂t Ω wdx = 0, since w ∈ ZM . In addition, it follows from (9.5.4)
165

and estimates (9.5.2) and (9.5.3) and J = O(ε) that

|∂t b| =

1
|Ω|

2ℓ/ε
Γ −2ℓ/ε

∂s(x; t)
∂z(x; t)
Φz + ε2 Ds φ2 ·
∂t
∂t

Jdzds ≤ C(K)ε−1/2 Vn

L2 (Γ)

.

(9.5.5)

Recall from (3.3.20) that ∂t Φ is given by

∂t Φ(x; Γt ; b) = ∂z Φ

∂s
∂z
+ ε 2 Ds φ 2 ·
+ ∂t b.
∂t
∂t

(9.5.6)

Combining (9.5.2), (9.5.3), and (9.5.5), we obtain

||∂t Φ|| 2
=
L (Ω)
≤

2ℓ/ε
Γ −2ℓ/ε
Γ

(ε−1 Vn )2

1/2
2
∂z(x; t)
2 D φ · ∂s(x; t) + ∂ b Jdzds
∂z Φ + ε s 2
t
∂t
∂t
2ℓ/ε
−2ℓ/ε

(∂z Φ)2 Jdzds

1/2

1/2
2
2 D φ · ∂s(x; t) + ∂ b Jdzds
ε s 2
+
t
∂t
Γ −2ℓ/ε
≤ C(K)ε−1/2 Vn 2
,
L (Γ)
2ℓ/ε

(9.5.7)

where we again used J = O(ε).

Lemma 9.13. Fix K > 0. Then for all Γ ∈ GK , the following bound holds
≤ C(K)ε1/2 ||Aφ w||2 2
.
∂t , Aφ w, Aφ w 2
L (Ω)
L (Ω)
166

(9.5.8)

Proof: We calculate

∂t Aφ w = ∂t (ε2 ∆w − W ′′ (Φ)w + ε2 ηw) = Aφ ∂t w + (∂t W ′′ (Φ))w,

(9.5.9)

and

[∂t , Aφ ]w

L2 (Ω)

=

(∂t W ′′ (Φ))w

L2 (Ω)

≤ W ′′′ (Φ)

≤ Cε−2 ||∂t Φ|| 2
||A w||
,
L (Ω) φ L2 (Ω)

L∞

∂t Φ

L2 (Ω)

||w||L∞ (Ω)
(9.5.10)

where the estimate of the L∞ norm of w in terms of the L2 norm of Aφ w is a consequence
of the Sobolev Inequality (see Lemma 9.2).
Using the bound on ∂t Φ from Lemma 9.12 in (9.5.10) we obtain
≤ C(K)ε−5/2 Vn 2
||Aφ w||2 2
.
∂t , Aφ w, Aφ w 2
L (Γ)
L (Ω)
L (Ω)

(9.5.11)

In particular, since by Proposition 9.11

Vn

L2 (Γ)

≤ C Vn L∞ (Γ) ≤ C(K)ε3 ,

(9.5.12)

we have (9.5.8).

2
Proposition 9.14. Fix K > 0, let L0 = −∂z + q0 (s, z) and L = L2 + ε˜, where
q
a
2
La = −∂z − εκ(s, z)∂z + q(s, z) = L0 − ε (κ(s, z)∂z + q1 (s, z)). Then there exists C(K) > 0
167

such that for all Γ ∈ GK the following estimates hold
α
sup
sup ||z i Ds La φ′ ||J ≤ C(K)ε3/2 ,
s
|α|≤4,i=0,1

(9.5.13)

α
sup
sup ||z i Ds Lψz ||J ≤ C(K)ε3/2 .
s
|α|≤4,i=0,1

(9.5.14)

Proof: From the definition of La in (7.1.16) we have

La φ′ = L0 φ′ − εκφzz + εq1 φ′ = ε(−κφzz + q1 φ′ ),

(9.5.15)

2
Lφ′ = L2 φ′ + ε˜φ′ = (−∂z − εκ∂z + q)La φ′ + ε˜φ′ .
q
q
a

(9.5.16)

Both terms are localized on Γ and acquire an extra factor of ε1/2 from the Jacobian. The result
follows from bounds on κ in Proposition 2.12 as well as bounds on q and q in (7.1.4)-(7.1.7).
˜

We conclude the chapter with the following proposition which is used in bounding the
lower order terms in Theorem 3.7.
⊥
Proposition 9.15. Let w ∈ ZM and Zi = η1 Θi φ′ ∈ ZM . Then,
.
(Π0 Lφ w, Zi )Ω ≤ C(K)ε3/2 ||w|| 2
L (Ω)

(9.5.17)

Proof: Using the fact that Π0 and Lφ are self-adjoint
(Π0 Lφ w, η1 Θi φ′ ) 2
= (w, Lφ Π0 (η1 Θi φ′ )) 2
.
L (Ω)
L (Ω)
168

(9.5.18)

From the definition of Π0
Π0 (η1 Θi φ′ ) = η1 Θi φ′ − ηΘi φ′

(9.5.19)

and
Lφ Π0 (η1 Θi φ′ ) = Lφ (η1 Θi φ′ ) − Lφ ηΘi φ′ .

(9.5.20)

⊥
However w ∈ ZM and so (w, 1)Ω = 0 implies
w = Π0 w.

(9.5.21)

Using (9.5.20) and (9.5.21) in (9.5.18) gives

(Π0 Lφ w, η1 Θi φ′ )Ω = (w, Lφ (η1 Θi φ′ ))Ω − (w, Lφ η1 Θi φ′ )Ω .

(9.5.22)

We will estimate the two terms on the right-hand side of (9.5.22) separately. Using J = O(ε),

η1 Θ i φ ′ =

2ℓ/ε
1
1
η Θ φ′ Jdzds = O(ε),
η1 Θi φ′ dx =
|Ω| Γ(2ℓ)
|Ω| Γ −2ℓ/ε 1 i

(9.5.23)

In addition, recalling the definition of Lφ in (3.2.2) we obtain
Lφ 1 = (−ε2 ∆ + W ′′ (Φ) − ε2 η) W ′′ (Φ) − W ′′′ (Φ)(ε2 ∆Φ − W ′′ (Φ)),

169

(9.5.24)

2
Since W ′′ (Φ) → q+ as z → ±∞, Lφ 1 − q+ is localized on Γ we apply Proposition 7.7 to get
(w, Π0 Lφ 1)Ω ≤ ||w|| 2
||Π L 1||
≤ Cε1/2 ||w|| 2
.
L (Ω) 0 φ L2 (Ω)
L (Ω)

(9.5.25)

Combining estimates on η1 Θi φ′ and |(w, Π0 Lφ 1)Ω | into (9.5.22) yields
.
(Π0 Lφ w, η1 Θi φ′ )Ω ≤ (w, Lφ (η1 Θi φ′ ))Ω + Cε3/2 ||w|| 2
L (Ω)

(9.5.26)

Since η1 is compactly supported, we use the exponential decay of eigenfunctions φ′ in (7.1.19)
and expansion of Lφ in (7.1.18)
Lφ (η1 Θi φ′ ) = η1 Lφ (Θi φ′ ) + O(e−m/ε ) = η1 (I1 − I2 − I3 + I4 ) + O(e−m/ε ),

(9.5.27)

where
I1 = L(Θi φ′ ),

(9.5.28)

I2 = ε2 ∆G La (Θi φ′ ),

(9.5.29)

I3 = ε2 La ∆G (Θi φ′ ),

(9.5.30)

I4 = ε4 ∆2 (Θi φ′ ).
G

(9.5.31)

We estimate each term separately. Bound on Lφ′ in (9.5.14) give
||I1 || 2
= ||L(Θi φ′ )|| 2
= ||Θi Lφ′ || 2
≤ ||Θi || 2
sup ||Lφ′ ||J ≤ C(K)ε3/2 .
L (Ω)
L (Ω)
L (Ω)
L (Γ) s
(9.5.32)
170

Addressing the next term I2 , we use Lemma 2.10 to write
∆G La (Θi φ′ ) = ∆s La (Θi φ′ ) + εzDs,2 (Θi La φ′ ),

(9.5.33)

where Ds,2 represents a second order differential operator
d−1
Ds,2 :=
i,j=1

∂2
+
dij
∂si ∂sj

d−1
dj
j=1

∂
,
∂sj

(9.5.34)

with coefficients bounded by some constant C(K). In particular,

I2 = ε2 ∆G La (Θi φ′ ) = ε2 ∆s La (Θi φ′ ) + ε3 Ds,2 (Θi zLa φ′ )
= ε2 (∆s Θi )La φ′ + 2ε2 (∇s Θi · ∇s La φ′ ) + ε2 Θi ∆s (La φ′ ) + ε3 Ds,2 (Θi zLa φ′ )
.

(9.5.35)

We note that Proposition 5.2 implies
2ℓ/ε
Γ −2ℓ/ε

((∆s Θi )La φ′ )2 Jdzds =

(∆s Θi )2

2ℓ/ε

(La φ′ )2 Jdzds

−2ℓ/ε
2
2
≤ ||∆s Θi || 2
sup ||La φ′ ||2 = βi sup ||La φ′ ||2
J
J
L (Γ) s
s
2
≤ C(K)ε3 βi .
(9.5.36)
Γ

171

Similarly, one can show that
2ℓ/ε

(∇s Θi · ∇s La φ′ )2 Jdzds ≤ ||∇s Θi ||2 2
sup ||La φ′ ||2
J
L (Γ) s
Γ −2ℓ/ε
≤ βi sup ||La φ′ ||2 ≤ C(K)ε3 βi ,
J
s

(9.5.37)

and
2ℓ/ε

(Θi ∆s (La φ′ ))2 Jdzds ≤ ||Θi ||2 2
sup ||∆s La φ′ ||2
J
L (Γ) s
Γ −2ℓ/ε
≤ C(K)ε3 ,

(9.5.38)

where we used ||Θi || 2
= 1 as well as Proposition 9.14. It follows from estimates (9.5.36)L (Γ)
(9.5.38) that

ε2 (∆s Θi )La φ′ + 2ε2 (∇s Θi · ∇s La φ′ ) + ε2 Θi ∆s (La φ′ )

Γ(2ℓ)

1/2
≤ C(K)ε7/2 (βi + βi + 1).
(9.5.39)

Similar calculation show that

1/2
||ε2 Ds,2 (Θi zLa φ′ )||Γ(2ℓ) ≤ C(K)ε7/2 (βi + βi + 1),

(9.5.40)

from which we deduce the bound

1/2
||I2 ||Γ(2ℓ) ≤ C(K)ε7/2 (βi + βi + 1) ≤ C(K)ε3/2 ,

172

(9.5.41)

where we used βi ≤ βM ≤ C(K)ε−2 (see Remark 3.4). Next, addressing I3 we observe that
I3 = ε2 La ∆G (Θi φ′ ) = ε2 La (φ′ ∆G Θi ) = ε2 (∆s Θi )(La φ′ ) + ε3 (La (zφ′ DΘi ))
= ε2 βi Θi (La φ′ ) + ε3 (La (zφ′ DΘi ))

(9.5.42)

and

||I3 || 2
≤ C(K)ε7/2 βi ≤ C(K)ε3/2 .
L (Ω)

(9.5.43)

Finally,

I4 = ε4 ∆2 (Θi φ′ ) = ε4 φ′ (∆2 Θi ) = ε4 φ′ (∆2 Θi ) + εzφ′ Ds,2 Θi
s
G
G

(9.5.44)

and estimates (5.2.11), (5.2.10), and (5.2.12) yield

3/2
2
2
I4 = ε4 φ′ βi Θi + O(ε5 βi ) = ε4 φ′ βi Θi + O(ε2 ).

(9.5.45)

⊥
However, w ∈ ZM implies
(w, η1 Θi φ′ ) 2
= 0,
L (Ω)

(9.5.46)

(w, I4 ) 2
≤ C(K)ε2
L (Ω)

(9.5.47)

It follows that

and returning the estimates on I1 through I4 to (9.5.26) gives (9.5.17).

173

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