. @éJWeaflfi u «w v-Dv, 5.. 7., 34.... fifflflf nuke. . if. 7va 9 I. n , :9... “ (...; . a «5. «$3 1 begun” 3 n! ‘ 40 .. ft 37.». 11 ... .5. x w- . J. r .. : . Wm. 33... If...) 1 a , . ll... ..i (d: . x x ...:f‘fin‘udw , than“? yang)». savanna ‘ . a. “55;. : fu‘ ...}. 3hr, .I. V Rudd. .. 0.. (L. Tum ~i ( ~ 1’10 [003((9’51/3 This is to certify that the dissertation entitled CONSTRAINED LOWER SEMICONTINUITY PROBLEMS IN THE CALCULUS OF VARIATIONS presented by DANIEL VASILIU has been accepted towards fulfillment of the requirements for the Ph.D. degree in Applied Mathematics WW Major’ Professor’s Signature 05/20/2004 Date MSU is an Aflirmative Action/Equal Opportunity Institution LIBRARY Michigan State University PLACE IN RETURN BOX to remove this checkout from your record. TO AVOID FINES return on or before date due. MAY BE RECALLED with earlier due date if requested. DATE DUE DATE DUE DATE DUE 6/01 cJCIFiCIDateDuepescsz CONSTRAINED LOWER SEMICONTINUITY PROBLEMS IN THE CALCULUS OF VARIATIONS By Daniel Vasiliu A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2004 ABSTRACT CONSTRAINED LOWER SEMICONTINUITY PROBLEMS IN THE CALCULUS OF VARIATIONS By Daniel Vasiliu We study two problems of constrained lower semicontinuity for a functional on Sobolev space. The first problem is motivated by certain models of microstructures and phase transitions which are distinguished by the fact that the associated Young measure is supported on a certain set K. We study the case when K = I; is a lin— ear subspace and we prove that the weak lower semicontinuity of a functional on a Sobolev space restricted to sequences whose gradients approach the linear subspace L satisfying a constant dimension condition is equivalent to a generalized version of quasiconvexity. The second problem is motivated by the Ekeland variational princi- ple. We study a restricted weak lower semicontinuity for a given smooth functional on Sobolev space along all its weakly convergent Palais—Smale sequences. This type of constrained weak lower semicontinuity replaces the usual lower semicontinuity condition required for the direct method in the calculus of variations, and suffices for the existence of minimizers under the usual coercivity assumption. Although, in general, this condition is not equivalent to the usual weak lower semicontinuity con- dition, we show that, in certain cases, these two conditions are equivalent and reduce to the usual convexity or quasiconvexity conditions in the calculus of variations. T0 Oana, for her special presence and support iii ACKNOWLEDGMENTS First I would like to express my deep gratitude and consideration to my advisor Prof. Baisheng Yan for his patient guidance through this research, for sharing his knowledge and insight. Numerous discussions and suggestions lead to the ideas that shaped this research. I would also like to address special thanks to all the members of my committee, Prof. Gang Bao, Prof. Zhengfarig Zhou, Prof. Moran Tang and Prof. Michael Frazier for their professional expertise and insight in Applied Mathematics. iv TABLE OF CONTENTS Introduction 1 Preliminaries and Notations 2 Linear Restrictions with Constant Dimension 2.1 .C-rank one convexity and .C-quasiconvexity ................. 2.2 Examples .......... 2.3 Constant dimension condition ........................ 2.4 .C-weak lower semicontinuity ......................... 2.5 Particular case without the constant dimension condition ......... 3 Nonlinear Restrictions of Palais-Smale Type 3.1 Introduction ......... 0000000000000000000000000 3.2 The (PS)-weak lower semicontinuity .................... 3.3 One dimensional scalar cases ......................... 3.4 Special cases with f = f (6) Bibliography 18 19 22 30 33 40 43 43 45 51 63 74 Introduction A problem of significant importance in the Calculus of Variations is to find among all functions a E W 14’(Q,Rm), with certain prescribed constraints, those which minimize a given functional 1(a) = / f>dz (1) where f : Q x Rm x Mm“ —> R, Q C R" a bounded domain and Du denotes the gradient of u in the sense of distributions. A direct method of proving existence of minimizers is to find minimizing sequences converging in some topology and check that the functional I is lower semicontinuous in that topology; then in this case the limit would be a minimizer. Therefore it is a special interest in finding necessary and sufficient conditions for the function f such that I defined by (1) is weakly lower semicontinuous on certain Sobolev space. One “right” candidate for such condition is the concept of quasiconuerity first introduced by Morrey in the early ’505 [Mo 1]. According to Morrey a function f : Mm“ —> IR is quasiconuea: if / f(A + Duanda: 2 MIA) Q for all A E Mm” and all u E C30(Q,Rm). Acerbi and Fusco [AF] proved that under some proper growth condition the weak lower semicontinuity of the functional I given by (1) is equivalent to the quasiconvexity condition of f with respect to variable 6. The quasiconvexity condition is generally difficult to verify. As a major contri- bution in understanding this condition we distinguish the work of Ball [Ba 1]. He developed the concepts of rank-one convexity and polyconuexity along with the qua- siconvexity emphasizing many interesting facts in the attempt to establish a useful sufficient condition for the weak lower semicontinuity. It turns out that rank-one convexity (see definition below), although easier to check, is the weakest among all three conditions. In general rank-one convexity does not imply quasiconvexity (Sverak [Sv 1]) but vice versa is always true. However there are particular cases when rank—one convexity is equivalent to quasiconvexity, for example when f is a quadratic form. An efficient way to study weakly convergent sequences and the weak lower semi- continuity property for the functional (1) is to use the concept of Young measures developed by Tartar [Ta] following the original idea of L.C.Young [Yo]. Kinder- lehrer and Pedregal [KP] showed that the homogeneous gradient Young measures (i.e. :1: —> V; is the constant map almost every :13) are exactly those probability measures that satisfy Jensen’s inequality for all quasiconvex functions f i.e. / f(/\)dvx(A)2f( / Adz/.0». men men Using the techniques of Young measures, Fonseca and Miiller [FM] studied the so called A—quasiconvexity problem and Miiller [Mu 3] also studied a similar problem without the constant rank condition. For problems relevant to solid-solid phase transitions in the Material Science [B], Mu 2] one can model the so called microstructure through Young measures. In these situations it is very important to study the sequences satisfying dist(Duk(x), K) —-> 0 (2) for almost every :1: E Q where Q C R" and K C Minx", which is a so called a N - energy well of the form K = U]: 1S O(n)H,-. In terms of Young measure this condition (2) is equivalent to the Young measure being supported on the set K. It is as well very useful in practice to study the weak lower semicontinuity of functionals I given by (1) along sequences uk satisfying constraint like (2) for a given set K. In the first part of this thesis we studied this problem with the set K being a linear subspace. In this case we also study the constrained rank-one and quasiconvexity. Let K = .C be a linear subspace of Mm”. We say that a function f : [I —> R is .C-ranlc one convex if for any A 6 [0,1] and A, B E [I such that rank(A — B) S 1 we have le +(1— MB) 3 /\f(A)+(1- MB). Also we say that f is £-quasiconue:r if 1 f(A) 3 I527 q] f(A + Du>dsc for every cube Q C R”, any A E 1.: and every n E W1’°°(Q;Rm), Q-periodic with Du(:1:) E [I for almost every 3:. We remark that if £ =- men we get the usual rank one convexity and quasiconvexity condition and thus the new conditions generalize the classical ones. Let f : Mm” ——> IR and define 1(a) = f“ f(Du)da:. We say I is £-weakly lower semicontinuous on I/V 1,p if 1(a) 3 lim inf [(uk) k—~oo whenever uk —\u and dist(Duk, L) ——> 0 as k —> 00. The main result of the first part is that if the subspace L satisfies the constant dimension condition (see definition below) then L-quasiconvexity is equivalent to the [I—weak lower semicontinuity of the functional I . In the second part of the thesis we assume the functional I defined above is C 1 on W1*”(Q;Rm). This requires that f be C1 in (3,6) and satisfy certain growth conditions. As in many problems in application, I is often also bounded below. When minimizing bounded-below C1 functionals over a Banach space, an important variational principle discovered by Ekeland [Ek] (see also [AE]) can provide more special minimizing sequences. For our functional I minimized over a Dirichlet class Ag in WW6); R"), we can always obtain a minimizing sequence {uk} in A9 which satisfies I’(uk) ——> 0 in W’I’P'(Q;Rm). Here, we assume 1) > 1 and p’ = 5%, and W‘liplm; Rm) denotes the dual space of W01 ‘7’ (Q; Rm). Consequently, the weak limit (if exists) of any such minimizing sequence will be an energy minimizer provided that I (u) only satisfies the condition: uk —\ u in Wl’p(Q;Rm) and I (u) S lim inf I (uk) whenever (3) k—eoo I’(uk) _> 0 in W‘LP'(Q;R’"). Certainly the usual weak lower semicontinuity condition implies the condition (3). We shall say the functional I (u) is restricted weakly lower semicontinuous on Wl’p(Q; Rm) if it satisfies condition (3). If the condition holds only for all uk, u in the Dirichlet class Ag, we then say I is restricted weakly lower semicontinuous on A9. Note that in nonlinear analysis [AE, Ra] the sequences {uk} with bounded 1(uk) satisfying I ’ (uh) —> 0 are usually called the Palais-Smale sequences of the functional I (u) Therefore, in the following, we shall say a sequence {uk} (PS) weakly converges to u (with respect to I) and denote by uk P3 u in W149 if it satisfies uk —\u in WW and I’(uk) —> 0 in W‘I'P'. As we shall see later, this restricted weak lower semicontinuity imposes some intrinsic property on the function f. Such a condition has also been mentioned in [Mn 2] as a point of view to replace Morrey’s quasiconvexity condition. In general, as shown in the paper (see Proposition 3.7), the restricted weak lower semicontinuity is not equivalent to the usual weak lower semicontinuity even for one dimensional scalar problems. However, the main results of the paper deal with certain cases where the restricted weak lower semicontinuity is actually equivalent to the usual weak lower semicontinuity of the functional (hence the convexity or quasiconvexity of f). In general cases, we do not know the necessary and sufficient condition for the restricted weak lower semicontinuity. We point out that the major difficulty in handling this type of restricted weak lower semicontinuity lies in that the test sequences {uk} in the usual techniques [AF, Da, Mo 1] do not satisfy the condition 1'(u,,.) —+ 0 in rv-1»P’(a;1am). A closely related problem to the restricted weak lower semicontinuity of func- tional I is to characterize all the gradient Young measures [KP] generated by weakly convergent (PS) sequences in WWII); Rm). This problem is associated with the the- ory of compensated compactness [CLMS, Ta]. The difficulty lies in that in this case the strong convergence I ’ (uk) —> O in W‘I’P’(Q; Rm) can not be realized by the Young measure of {Duk} in the dimension n 2 2. Recently the weak lower semicon- tinuity of functionals under certain linear differential constraints has been studied using the Young measure theory [FM, Sa]. These linear constraints A(u) are inde- pendent of the functional and usually have large kernel. Then the constrained lower semicontinuity of functionals may be characterized through the Jensen’s inequality of the integrant with the associated Young measures supported on the kernel of .A; this is the so—called A—quasiconvexity [PM]. In this paper, we do not pursue the Young measure method for our restricted weak lower semicontinuity studied here mainly because it does not realize the strong convergence I ’ (u) ——> 0.. To put our restricted weak lower semicontinuity in another perspective similar to the linear constraint cases, one could study the lower semicontinuity of any given functional J (u) under the (PS) weak convergence defined above. For example, one could define J to be restricted weakly lower semicontinuous on WI'P(Q; Rm) (with respect to I) if J(u) 5 lim inf J(uk), ‘v’ uk 38* u (with respect to I), k-aoo and study the relaxation of .1 under this lower semicontinuity if it is not restricted weakly lower semicontinuous. The study in such a direction seems interesting, but difficult in view of the nonlinear constraints. As in the linear constraint case, one might consider the certain convexity property of J on the kernel of I ’ (u) consisting of all critical points of I, which may not be closed under weak convergence. Chapter 1 Preliminaries and Notations Let R" the usual n-dimensional euclidean space with points :1: = (x1,x2,...,:1:,,), at,- E R (real numbers). Let Q be a bounded domain in R" and Q0 = [0, 1]" the unit cube in R". Let Mm” be the set of m x n matrices. For vectors a, b E IR" and matrices g, 77 E Mm”, we define the inner products by m 71 a-b=Zai-bi, £1 77: (6,77) 2225157725 with the corresponding Euclidean norms denoted both by | - [. For vectors q E Rm, a E R", we denote by q®a the rank-one m x n matrix (qiaj) and also define 0 2 0mm where 0mm is the m x n matrix having 0 in all entries. A cube in R" is a set Q={$ERnI$=ZCz‘li,OSCiSI} i=1 where (11,12, ...l,,} is an orthonormal basis of R". Denoting u(Q) or [Q] the Lebesgue measure of a measurable set Q we have that ,a(Q) = [Q] = 1. A function u defined on R" is called Q-periodic if u(a:) = u(:t + Z c,l,~) for any a: E R" and any 6, E Z. Let W 1”’(Q) be the usual Sobolev space of scalar functions on Q, and define Wl’p(Q; R") to be the space of vector functions u: Q —> Rm with each component ui E III/14’ (Q) and we denote by Du the Jacobi matrix of u defined by Du(:r) = (aw/ox,){:,l;j;j;;. Let 1 g p < 00. We make Wl"’(Q; R’") a Banach space with the norm 1 P IIUIIw1-p = (/ (Iulp+ IDundx) . Q Let 08°(Q; Rm) be the set of infinitely differentiable vector functions with compact support in Q, and let Wol’p(Q;Rm) be the closure of C8°(Q;Rm) in Wl’p(Q;lRm). Then W01 ”’ (Q; Rm) is itself a Banach space and has an equivalent norm defined by [I [Dull] Liam). We also recall the following version of Sobolev embedding: Theorem 1.1. If Q is a bounded Lipschitz domain then the embedding Wl’p(Q;Rm) ——> Lp(Q;lRm) is compact for any 1 _<_ p 3 00. By COOK”) we denote the closure of continuous functions on R” with compact support. The dual of 000R") can be identified with the space MGR") of signed Radon measures with finite mass via the pairing W) = 4de A map 1/ : E —> MGR") is called weak* measurable if the functions a: —> (V(x), f) are measurable for all f E COOK"). We shall write Va: instead of 1/(x). Let f : Q X Rm ——+ R a measurable function such that v —> f (x, v) is continuous for all x E Q (a function with this properties is called Carathéodory function). The following result represents the fundamental theorem of Young measures: Theorem 1.2 ( [Ba 2]). Let E C R" be a measurable set of finite measure and let uk : E —> Rm be a sequence of measurable functions. Then there exists a subsequence ukj and a weak* measurable map I/ : E ——> MGR’”) such that the following hold. (i) V3 2 0, [[Vrllem) = fnm duct 3 1, for almost every x E E. (ii) we have IIVIIIM(R'") = 1 if and only if the sequence does not escape to infin- ity,i.e. if lim sup|{|ukj|}| Z r] = O. r—ooo j 10 (iii) Let A C E measurable and f E C(R’"). If [[VxllMakm) : 1 for almost every x E E and if f(ukj) is relatively compact in L1(A) then flukjl—‘(Vx,f)=/ de/x. m (iv) If f is Carathéodory and bounded from below then 11m m, utxxndx = jag... f(rc, ut>>dx < oo n—aoo 9 if and only if {f(-, ukJ(-))} is equi-integrable. The measures (V$)xeg are called the Young measures generated by the sequence {ukj}. The Young measure is said to be homogeneous if there is a Radon measure V0 E MUR’") such that 11x 2 V0 for almost every x E Q. Theorem 1.3 ([Pe]). If {uk} is a sequence of measurable functions with associated Young measure 1! = {whey}, then liminfo(x,uk(x))deL IR'" f(13,)\)dV$(/\)d$, (1.1) k—ooo for every Carathéodory function f, bounded from below, and every measurable subset ECQ. A Young measure (V3) is called a gradient Young measure if it is generated by a sequence of gradients. We say that (V3,) is a W” gradient Young measure if it is 11 generated by {Duk} and uk —\ u in W1'1’(Q, R"). The following result refers to the localization of the gradient Young measures. Theorem 1.4 ([KP]). Let (VI) be a gradient Young measure generated by a se- quence of gradients of functions in W1*p(Q). Then for almost every (1 E Q there exists a sequence of gradients of functions in W’1'p(Q) that generates the homoge— neous Young measure (z/a). We also provide the definitions of convexity, rank one convexity and quasicon- vexity. Definition 1.1. Let h : men -—+ IR. We say that h is convex on men if the inequality hO‘E + (1 - A)77) S MK) + (1 - /\)h(77) (1.2) holds for all 0 < A < 1 and 5, 77 E Mm“. Note also that h is convex if and only if g(t) 2: h(§ + tn) is a convex function of t on R for all 6, 77 E men . For C1 functions h, the convexity condition is equivalent to the condition h(77) 2 12(6) + 05%): (77 - 6), V 77, £6 me“- (13) Furthermore, a C1 function h on R is convex if and only if h’ is nondecreasing, or equivalently, the following condition holds: (h’(a) - h'(b))(a — b) 2 0, V a, b E R. (1.4) 12 Definition 1.2. A function f : men —> R is called rank one convex if f(AA + (1 - MB) _<_ Af(A) + (1 — A)f(B) for all A E [0, 1] and any matrices A and B such that rank(A — B) S 1. Definition 1.3. A function f : R" —-> R is called separately convex if g,-(t) = f(x1,...x,-_1,t,x,-+1, ...xn) is convex in t for all 1 S i S n. Definition 1.4. A function f : M‘“"" —+ R is said to be quasiconvex if Q f(A + Dui($))d:v Z f(A) for any A E Mm“ and u E WOI‘OO(Q0;R"‘). If f is quasiconvex then one can show [Sv 1] that f(A) = inf f(A + Du(x))dx uevr’léfloosm) Qo where Wpléfo(Q0; R") is the class of periodic functions in W1’°°(Qo; Rm). Let A := ZN be the unit lattice, i.e. the additive group of points in IR" with integer coordinates. We say that f : R" —> Rm is A-periodic if f(x+A) = f(x) for all x E IR", A E A. 13 A A — periodic function f may be identified with a function fT on the n-torus Tn :2 {(e2nix‘,e2”i$2, ...,e2m‘r") E C" : (x1,x2, ...x") E R"} through the relation fT(e2”m,e2"ff2, ...,emr") :2 f(x1,x2, ...xn) The space D”(T,,) is identified with LP(Q0) and C (Tn) is the set of A-periodic continuous functions on Q0. We recall some results on Fourier transform for periodic functions. If f E L1(Tn), then its Fourier coefficients are defined as: f(A) := “marriage, A e A. 7% Theorem 1.5. We have the following: (i) The trigonometric polynomials R(x) 2: Z a,\e’2"ix"\, A' all finite subsets of A, a,\ E (C AEA’ are dense in C(Tn) and in LP(T,,) for all 1 S p < 00. (ii) Iff E L2(Tn) then f(r) = Zf(A)€‘2m'A Z Iff(>\)I2 = ”Na AEA AEA 14 Let f: Q x R" x men ——> IR. We say f is Carathéodory if f(x, 3, 6) is measurable in x E Q for all (s, g) E R" x Man and continuous in (s, 6) E R" X men for almost every x E Q. Define the multiple integral functional 1 on W” (Q; Rm) by I(u) = / f(x,u(x),Du(x))dx, u E Wl’p(Q;lRm). Q If f(x, s,§) is measurable in x E Q for all (8,6) E R” x Mm” and is C1 1n (s,£) E IR" x men for almost every x E Q, we shall use the following notation to denote the derivatives of f on s and g: 0 a '— Dsf(:z:, 3.5) = (51- - .a—D Dim, 3. a = (af/ar..>z;:;:::::.. Definition 1.5. A functional I is said to be (sequentially) weakly lower semicon- tinuous on Wl'p(Q; Rm) provided I(u) 3 lim inf I(uk) whenever uk —\ u in Wl’p(Q; Rm). (1.5) k-—+oo The following important result has been proved by Acerbi and Fusco [AF]. Theorem 1.6. Assume f is Carathéodory and satisfies 0 S f(xisié) S €1(|€|” + ISI”) + A(11?), where c1 > 0 and A E L1(Q). Then functional I defined above is weakly lower 15 semicontinuous on Wl’p(Q;lRm) if and only if f (x, s, ) is quasiconvex for almost every x E Q and all s E R"; that is, the inequality f(a: Mgr l—Ql/flx st+Dso IR U {+00} be a lower semicontinuous function which is bounded below. Let 6 > 0 and a E X be given such that e _ < . _. (u) _ 1§f<1> + 2 Then given any A > 0 there exists u,\ E X such that RIM) S (1)01)» (“UAW—t) S A (1-6) (u,\) < (u )+ X,‘d(u u,\) V u 524 uA. (1.7) The following version, which follows from the general Ekeland principle above, is very useful for establishing certain results in chapter 3. Theorem 1.8. Let X be a Banach space and X * its dual space, and let (I): X -—> R be a 01 functional which is bounded below. Then for each 6 > 0 there exists uC E X 16 such that (u€) S Infx (I) + 6 II‘P'WelIIx- S 6- Therefore, there exists a minimizing sequence {uk} in X such that lim (u;,.) = i§f<1>, klim ||’(uk)||X- = 0. kaoo 17 (1.8) (1.9) Chapter 2 Linear Restrictions with Constant Dimension An interesting and motivating problem is to study necessary and sufficient conditions for the weak lower semicontinuity of the operator I restricted only to a class of functions that satisfy certain linear constraints, i.e. their gradients in the sense of distributions approach a preset target linear subspace of Mm“ by means of L2 convergence. When the linear subspace satisfies some special condition we prove that the restricted weak lower semicontinuity is equivalent to a generalized version of quasiconvexity. 18 2.1 L-rank one convexity and L-quasiconvexity Let L be a linear subspace of men and P: mena men the linear map such that PA = 0 if and only if A E L, which is actually the orthogonal projection onto the orthogonal complement of L. Definition 2.1. We say that a function f : L —-+ R is L-rank one convex if for any E [0, 1] and A, B E L such that rank(A — B) 3 1 we have f(AA + (1 - MB) S AHA) + (1 — A)f(B)- Definition 2.2. Given a cube Q C R" we say that a function f : L —> R is Q — L- quasiconvex if A) g Tél/flA + Du(x))dx Q for any A E L and every u E W1’°°(Q; lRm), Q-periodic with Du E L. Definition 2.3. We say that a function f : L —> IR is L-quasiconvex if it is Q-L- quasiconvex for every cube Q, that is gélQ/f (A(+Dux ))d for any cube Q C IR", any A E L and every u E er°°(Q;lRm), Q-periodic with Du E L. Theorem 2.1. If a function f : L —> R is L-quasiconvex then it is also L-rank one 19 convex. Proof. Let A E [0,1] and A, B two elements in the subspace L such that rank(A — B) S 1. Let Q0 = [0, 1]" a unit cube in IR". Since rank(A — B) S 1 there exist two vectors a E R" and b E R" such that A — B = a - bT and it exists a. rotation, a matrix R E lRm‘" such that RRT = In and (RT - b)T = 81 where e1 E R" with el = (1,0,0, ..., 0). Thus (A — B)R = a- (RT - b)T = a®e1. Let Q = RQO. Since f is assumed to be L-quasiconvex we have / f(C + Dr(x))d:c 2 f(C) (2.1) Q for all C E L and p E W1*°°(Q, R") such that is Q periodic and Dtp(x) E L a.e. x. Let f(A) = f(ART) and also denote A 2 AR and @(x) = 99(Rx). Notice that a is Q0 periodic, Dean) 6 Z = {MW = MR, M e L} almost every x and q? E W1'°"(Q0, lRm). By the change of variable under the integral we obtain / f(c? + Brenda: 2 f(é) (2.2) 20 for all C’ E L and all (,5 E W1’°°(Q0, IR’"), Q0 periodic and D(p(~x) E L. Also we have 1. ~ A—B=a®€1. (1 — A) ift E [0,A] Let 77 : [0,1] —> R such that 77’(t) = and let @(x) = n(x1)a —A ift e [A, 1] where x = (x1, x2, ....xn) Thus we obtain that «,5 is Q0 periodic and we can extend by this periodicity to IR" and D95(x) E L a.e. x. Also notice that «p E W1’°°(lR", IR") and (1—A)(A—B) ifxlE 0,,\ 0W): [ ] —/\(.ci — B) if x1 6 [A,1] Thus we have that f f(A/l +(1—— A)B + D95(x))dx = Af(A) + (1 — A)f(B) and Q0 f f(AA + (1 - A)B + Deana: 2 f(A/1+0 —A)B) and obtain f(AA+(1—A)B) 3 Q0 Af(/l) + (1 — NIB) hence f(AA + (1 — A)B) g Af(A) + (1 - A)f(B). 1:] Proposition 2.2. If the subspace L does not contain rank one matrices and a function u E W1’2(Q;lRm), Q-periodic has the property that Du(x) E L almost every x then u = const. Proof. Assume first Q=Q0. Since L does not contain rank one matrices we have that min |P(a (8’) A)[ > 0 (2.3) Ial=1.I/\I=1 21 and it follows that |P(a® A)| > c|a||A| (2.4) for any a E R” \ {0m} and A E IR” \ {On}. We consider now the Fourier transform of PDu which is P(u(A) 8) A). Since L does not contain rank one matrices we have that Pain) a A) = 0 (2.5) for all A E A \ {0"}. Thus, using (2.4) we get that u(A) = 0 for all A E R" \ {On} which proves that u must be a constant. Now, if Q = RQO for a rotation R and u E W 1'2(Q;lR’"), Q-periodic with Du(x) E L we have that u(x) = u(Rx) is in W1’2(Q0;1Rm), QO—periodic. Also D2] = Du(Rx)R so Du E L where L = {A E Mm“ [A 2 AR, A E L}. Since L doesn’t contain rank one matrices it follows that L has the same property. Thus a must be constant and therefore u is constant as well. III 2.2 Examples In this section we are going to discuss particular cases of linear subspaces L and some aspects related to the restricted rank one convexity and quasiconvexity. Example 1. Consider L = {(2 2)] a, b E IR} and let f : L ——> R a L-rank one convex function. We show that f must be Qo-L-quasiconvex. 22 Given u E W1*°°(Q;lRm), u(x,y) = (u1(x, y), u2(x,y) with Du E L it implies Thus it1 and u2 satisfy the wave equation i.e. I) Own“ — Own2 2 0 and we get Wax v) = h(x + y) - g(x - y) 112(33. y) = h(x + y) + g(x - v) where h, g : R —> R, absolutely continuous. If u is assumed to be Q0 periodic it folows that h and g are periodic of period 1. Indeed, u1(x,y) = u1(x + 1,y) so h(x + y + 1) — h(x + y) = g(x — y) — g(x — y +1) for any x,y E R It implies that g(t) — g(t + 1) = 9(0) — g(l) for any t E R since if two absolutely continuous functions a and B verify a(x + y) = B (x — y) for any x, y it follows that they must be constant. Thus we get that 9(1) - 906 +1) = (9(0)- 9(1))k for any positive integer 1:. Since 9 has to be bounded, we get g(O) — 9(1) 2 0 and 23 thus g(t) — g(t + 1) = 0 for any t E R. Let F : IR"2 —> IR defined as F (a, b) = f (if: :2). Since f is L-rank one convex, we have that F is separately convex in each variable and d b —b d —d f((fic)+(:ib:+b))=r(c: +a,c2 +b) (2.6) Now we prove that f is Qo-L-quasiconvex. Making the substitution 5 = x + y and 77 = x — y we get f/OHCI 6:) + Duldxdy = $0] (1:de +h’(£), 0; d +g'(77))d17) dg + f2 ( 72F(C :— d + h'(€). c _2_ d + g’(n))dn) dg 1 2_€ Now using the fact that F is separately convex and Jenssen’s inequality we get: [Oil-d + /( ((EF ((h't - “0‘9(‘€))+(1—oF hence f is QO-L-quasiconvex. Example 2. We show that Qo-L-quasiconvexity might not imply L-rank one con- 24 vexity. Let L = {(3 :‘gfldb E IR} a linear subspace of R2“. If u E W1’°°(Q0;IR"‘) satisfies Du E L it follows that 20,,21 — ayyul = 0 (2.7) which implies that there exist h, g : IR —+ R such that u1(x, y) = h(x + y\/2) + g(x — ZIP/é) (2.8) Also, u1(x,y) is Q0 periodic so we get, by reasoning as before, that h and g are periodic with periods 1 and \/2. Since x/2 is irrational and the set {k\/2+p] k, p E Z} is dense in IR it follows that h and 9 must be constant [La]. Therefore, by definition, every function is QO-L—quasiconvex, but not necessarily L-rank one convex (see Example 1). Example 3. We show that L—rank one convexity does not imply L-quasiconvexity. The following famous example belongs to Sverak [Sv 1]. Let L={ 0 b ,a,b,cElR} (2.9) 25 a linear subspace of M3”. Also let f : L —> IR be defined by and the function f is convex on each rank-one line contained in L. Consider the function u : IR2 —> IR3 given by , and = —abc lo 0 \1 sin(2rrx) \ 1 i “(1331) = '2; [[ sin(27ry) We have that u E W1’°°(Q0;IR3) where Q0 2 [0,1]2, u is Qo-periodic and Du E L since ( cos(27rx) 0 Du(x, y) = 0 cos(27ry) (cos(27r(x + y)) cos(27r(fl? + y») 26 I sin(27r(x + y») 0) 0 1} Thus we get / f(Du(x,y))dxdy = — // (cos(27rx))2(cos(2rry))2dxdy < 0 = f(03x2) (2.11) Q0 Q0 which shows that f is not L-quasiconvex. Now we generalize the Example 3 to the case where some function f : L —> IR which is L-rank one convex but not Qo-L-quasiconvex can be extended to the entire space Mm“ and preserve this property. Theorem 2.3. Let f : L —+ IR be a function which is L-rank one but it is not L-quasiconvex. Also assume that f is (3'2 and for some p Z 2: |f(A)| S C(1+ IAI") (212) ID2f(A)I S C(1+ IAIN). (2-13) forall A E L. Then there exists an function F : Mm“ ——> IR which is rank one convex but not quasiconvex on Mm“. Proof. Since f is not L-quasiconvex it exists a cube Q = RQO and u E W1'°°(Q; IR’"), Q-periodic with Du(x) E L such that f(O) >/Qf(Du(x))dx (2.14) 27 Let Fey, : IVII’“xn —> IR with F,,,,(X) = f(PX) + ele2 + elep+1+ le — PX|2. (215) Here P is the projection onto L. Let A, Y E Mm“ arbitrary such that rankY = 1, [Y] = 1 and let hey, = Fc,k(A + tY). We are going to prove that for every 6 > 0 it exists k such that Fa)c is L-rank one convex. To show this it is enough to prove that H hey. 2 0. Thus, now we prove that: d2 EMA + tY) 2 0 (2.16) for any matrices A, Y E men with rankY = 1, [Y] = 1. We have: [A +tY|p+1=(|A + tY[2)B;_1 = (IA)2 + 2t < Y,A > +9)? (217) d +1 EE'A +2314?“ = (p +1)(|A|2 + 2t < Y,A > +t2)"2—(< Y,A > +0 (218) Thus we get at2 gamer“ =IAr-32+IAIP‘1 (219) t=0 28 and a!2 ——-I~; k(A + tY) = Eli-f(PA + tPY) + 25 + e(p +1)|A[l’"1 dt2 ’ i=0 dt2 ,20 + 6(1) +1)(p —1)|A|”‘3 < Y,A >2 +k|Y — PY|2 Now, from (2.15), we have a3 (PA+tPY) 2—c(1+|A|P-2) (2.20) i=0 and d2 @FCMA + tY) 2 —c(1 + |A|p”2) + 6(p +1)[A[f’—1 + 26 + 2k[Y — PY]2 (2.21) t=0 Assume by contradiction that it exists 60 such that for every positive integer k we get Ak, Yk satisfying d2 12,),(A + tY) (2.22) 0 > — dt2 H, From (2.21) it follows that Ak is bounded and by extracting a subsequence we have Ak —+ A and Y" —9 Y = PY as k —> 00. Thus, passing to the limit in (2.21), d2 —e > — (A + tY) (2.23) a contradiction with the fact that f is L-rank one convex. 29 Now we can also choose 6 small such that F€,),.(O) >LF€,k(Dtt($))d$ (2.24) where u is given in (2.14). Hence Fey, is not L-quasiconvex. CI 2.3 Constant dimension condition Let A E IR” and R2 = {w E lR’”[w (X) A E L}. We notice that R2 is a linear subspace of IR’". Definition 2.4. We say that the subspace L satisfies the constant dimension con- dition if the related subspace R}; has the same dimension for all A E IR" \ {0}. If L satisfies the constant dimension condition we shall prove the equivalence between Qo-L-quasiconvexity and the weak lower semicontinuity of the functional 19(u) :/Qf(Du)dx along sequences satisfying the linear restriction PDuk(x) —+ 0 almost every x. Remark 1. If m = n = 2 and L is the linear subspace of 2 x 2 symmetric matrices then the dimension of R2 is constantly 1 for all A E IR2 \ {02}. Proof. We have that L = {(21 2)] a, b, c E IR} and RA = {1.0 = (w1,w2)E R2] wlAz = ngl}. 30 Clearly the dimension of R2 is 1 for any A E IR2 \ {02} D Lemma 2.4. If L satisfies the constant dimension condition there exists 7 > 0 such that for any a E (R2)i and A E IR" \ {0} we have: |P(A®a)[ Z 7|A®a| (2.25) Proof. Assume by contradiction that min [P(A <8) a)[ = O. (2.26) |A|=L|al=1 Then there exists a minimizing sequence A,- ——> A and aj —+ a. Let k = dim R2. For 6 small enough and any A such that [A — A] < 6 there exists a set w1(A), w2(A), ...wk(A) of linearly independent vectors of R2 and [13:10in = w,(A), for all i, 1 S i S k. Since aj E (R2)f, it implies that (aj,w,-(Aj)) = 0 for all i, 1 S i S k. We get (&,w,-(A)) = 0 so a E (R2)i. Also, since P(A ® a) = 0 it implies a E R2. Thus a = O, in contradiction with |a| = 1. First we shall prove the selection theorem: Theorem 2.5. Let Q a cube in IR" and u E W1’2(Q;IR"‘) a Q-periodic function. If the linear subspace L satisfies the constant dimension condition then for every 6 > 0 there exists a selection v6, v6 E C°° a Q-periodic function such that Dv._(x) E L a.e. 31 xEQand [[Du — DU5[[L2(Q) S [[PDU[[L2(Q) + 6 (2.27) Proof. First we assume that Q = Q0. Let A = Z" be the unit lattice, i.e. the additive group of points in IR" with integer coordinates. Since u is Q-periodic we can expand u as a Fourier series, Thus Du(x) = Z u(A) <8) Ae27’f2x. Let g(x) = P12211091 projection of both real part AeA and imaginary part of ii(A) onto R2. By Riesz-Fischer theorem we have that v(x) = 2 when” (2.28) AEA is a function in W112(Q), Q-periodic and its gradient belong to L almost every x. Applying Lemma (2.4) for a = u(A) — 6(A) we get [[Du — Dv||2 S [[PDU[[2. Now we can consider v€(x) as the real part of E v(A)e2"i2“ Where is A’ is a finite AEA’ subset of A such that [[DvE — Dv[[L2 < e (2.29) since the imaginary part of Z 6(A) ® Aez’rf’“c converges to Omxn as A’ /' A. AeA’ Now if the cube Q is arbitrary then Q = S Q0 for some a E IR" and a rotation 5'. Let L = {A E Mm“ [A 2 AS, A E L} and P the orthogonal projection onto L. 32 Define a : Q0 —> IR’" by u(x) z: u(Sx). Also notice that R2 = R? L and therefore R2 has constant dimension for any A E IR". Thus we can select 6 such that 17 E C °°(Q0), QO-periodic and IIDiz - Dfiellmoo) S IIPDflIILQIQo) + 6 (2-30) For each x E Q there exists a unique :7: E Q0 such that x 2 Sit. Let v : Q —> R” with v€(x) = 56(ST(x)). We notice that v6 satisfies the requirement of the lemma. CI 2.4 L-weak lower semicontinuity Let f : Mm“l -—> IR satisfy the growth condition |f(A)I S C(1+|AI2) (2-31) for any matrix A E Mm“ and consider the integral operator [g(U) = Lf(Du)dx (2.32) where Q is open bounded domain with Lipschitz boundary and u E W1’2(Q; IR’"). 33 In contrast to Example 2 in section 2.2 we show that under the constant dimen- sion condition QO-L—quasiconvexity implies L-rank one convexity. Theorem 2.6. Assume that the linear subspace L satisfies the constant dimension condition. If a continuous function f : men ——> IR satisfies the growth condition (2.31) and is QO-L-quasiconvex then it is also L-rank one convex. Proof. Let A, B E L be such that rank(A -- B) S 1 and A E [0, 1]. For any integer k there exists Q’f, Q; C Q0, Q’f 0 Q5 2 Q) and (pk E W01 ’°°(Q0, IR’") such that i (1—A)(A—B) ifxEQ]c —MA—B) ameog “DWI-[loo S CORSNA, B) L since u(Q0) = 1. (See [Da]). We extend the (pk to be QO-periodic on IR". From these properties we also have that PD<,o)c —> 0 in L2(Q0). Thus, by Theorem 2.5, for any 6 we can find a selection uk,6 E W1’°°(Q0, lRm), Qo-periodic such that Duk,6 E L and ”DUI-s — Dsokllwoo) *" 0 as e —> O and it follows 34 lim inf f(AA + (1 — AB) + Duk’c)dx = lim inf f(AA + (1 — AB) + Dcpk)dx k—»oo,e—10 Q0 k—>oo Q0 = Af(A)+(1- A)f(B) Since f is Qo-L-quasiconvex we have f(AA + (1 — AB) + Duk,e)dx 2 f (AA + (1 — AB)) (2.33) Q0 for any It and 6. Taking lim inf over I: and e for the left hand side of the previous inequality we obtain Af(A) + (1 - A)f(B) Z f(AA + (1 - ABII which proves that f is L-rank one convex. El Definition 2.5. Let f and IQ be defined as above. We say that the functional IQ is L—weakly lower semicontinuous on W1’2(Q; IR’") if for any sequence uk —\u in W1'2(Q;IR"‘) with [[PDUkIIL2m) —> 0 as k —> 00, we have IQ(u) S lign inf 19(uk) (2.34) Theorem 2.7. If the functional IQ is L-weakly lower semicontinuous then the func- tional f is L-quasiconvex. 35 Proof. Let Q = RQO, A E L arbitrary and u E W1’°°(Q;IR"‘), Q-periodic with Du(x) E L for almost every x. We show that LflA + Du(x))dx Z f(A) (2.35) assuming that I is L-weakly lower semicontinuous. For any test function (,9 we have / Du(kx)<,o(x)dx = Du(ka)p(x)dx Q Q0 Thus, by Riemann-Lebesgue theorem, we have that lim Du(kx) A (2.37) and also / f(A + Du(x))dx = k"/ f(A + Du(kx))dx. (2.38) Q %Q For k sufficiently large there exists pk cubes, Q1, Q2, ...ka, which are translates of 36 %Q by multiples of 11:, muttually disjoint, such that U Q. C 9 and um \ U Q.) < e). (2.39) i=1 i=1 Where 6),. —> O as k —> 00. Thus we also get that ff: —> u(Q) as k —2 00. Since I is L-weakly lower semicontinuous it follows: k—aoo liminf [a f(Du),(x))dCD 2 f(A))im) (2.40) Also, from (2.38) we get /f(Duk(x))dx = pk/ f(A+Du(kx))dx+/ f(A+Du(kx))dx . 9 1‘; motile.- = g; / f(A+ Du(x))dx+ekC Q Letting k —> 00 we have u(Q) fQ f(A + Du(x))dx 2 f(A)u(Q) and after dividing by u(Q) we obtain what we had to prove. El Next we show under the constant dimension condition the L-quasiconvexity is always sufficient for the L-weak lower semicontinuity. Theorem 2.8. If the linear subspace L satisfies the constant dimension condition and if the function f is bounded from below, satisfies the growth condition 2.31 and is Qo-L-quasiconvex then functional In is L-weakly lower semicontinuous on W1’2(Q; IR’"). 37 Proof. Let u)c E W1'2(Q,IR"‘) such that uk—Xu in W1:2 and PDu)c —> 0 in L2. We assume that Du)c generates a parametrized Young measure (Vxlxen- Then Du(x): / Adz/AA) Mnixn By Theorem 1.3 we also have that limkian/f(Duk(x))dx2 / / fowl/Andi: (2.41) 9 men For our purpose it would be sufficient to show // “WI/AW 2 / f<0u>dx (2.42) Now we actually prove / fan/M2“ / Adva(/\))=f(Du(a))- (2.43) Man Man for almost every a E Q. By Theorem 1.4 we have that Va is also a gradient Young measure for almost every 0. E Q. Consider a cube Q C Q such that a E Q. There exists wk E W1’2(Q) such that Dw)c generates Va and wk -—> a in L2, by the Sobolev embedding. Also we get that Dwk —\ Du(a) = Di?) and by the fundamental theorem of Young measures PDw)c —> 0 in L2(Q). Let go} E C8°(Q) such that (,0,- /‘ 1 uniformly and v)”- : (OJ-(wk — a). Since 38 wk ——> u? in L2 for each 3' there exists 133- such that _ 1 [ID‘PJ' ‘3 (wk-j — w)[[L2(Q) < :7; Thus we can select a subsequence of vkd- which we can conveniently denote by v)c and we have v)c E III/01 ’2(Q) and [[D’Uk — D('I.Uk — II))[[L2(Q) —> 0 (2.44) By using Theorem 2.5, we can select in, E C °°(Q), Q periodic such that [[ka — ka[[L2(Q) —> O in L2(Q) and ka(x) E L almost every x. So we have limkinf/f(Du(a) + D(wk(x) — u?(x))dx = limkinf / f(Du(a) + ka(x))dx Q Q Also since f is L-quasiconvex fi/flDUW) + ka(x))dx Z f(Du(a))dx Q Thus it follows that 1 . . - — 1/ u a Whmkme f(Du(a) + D(wk(x) — w(x))dx — [mm f(A)d g(A) Z f(D ( I) This completes the proof. Cl 39 2.5 Particular case Without the constant dimen- sion condition Consider the linear subspace L = {(3 [3)] a, b E IR}. We notice that the subspace R2={wEIR2|w®AEL} does not have constant dimension for all A E IR2 \ {0}. Therefore this space L does not satisfy the constant dimension condition defined above. Let f : M2X2 —> IR be a Cl function satisfying 0 s f(E) s c(1+ Isl?) (2.45) IDf(€)| _<_ C(1+ I5I) (2.46) Also, as above, define 19(u) = [Of(Du)dx (2.47) Theorem 2.9. Iff : M2"2 —+ IR satisfies (2.45) and (2.46) and is L-rank one convex then IQ is L-weakly lower semicontinuous on W1*2(Q; R2). The following result by Miiller is going to be essential in the course of the proof. Theorem 2.10 ([Mu 3]). Let f : IR2 —> IR be a separately convex function that satisfies 0 s f(t) s C(1 + kl?)- 40 Let Q C R2 be open and suppose that uk —\ u, vk —\ v in Ll20C(Q) 8,21,. —> Byu, 5)ka —-> dxv, in ngflQ). Then we have liininf/f(uk,vk) )dz> _/f() (u, v) dz. Now we are going to prove the Theorem 2.9. (2.48) (2.49) (250) Proof. Let uk E W1’2(Q; IRz) with uk —Au and PDu)c —1 0 almost everywhere. Thus we have that Byu}, —-> 0 and Bxui —> 0 so 61(8yu2.) —-> 0 and 340.21,?) —> 0 in H‘ 1(9)- Let F : L —+ IR given by F(a, b) = f((a 0)). Since f is L-rank one convex it 0b follows that F is separately convex and satisfies the growth condition from Theorem 2.10. From (2.45) and (2.46) we also have that |f(€) - f(n)! S C(1+ IEI + |nI)(€— 77) 3.10 "k )we get By using this inequality with f = Du)c and r) = (0 a 2 yuk . . 3:111}, 0 llgllglf/Qf( Duk) )dz=lim1)ro1f/f((0 ayui))dz. 41 (2.51) (2.52) From the Theorem 2.10 we obtain [F(Bxul,8yu2)dz S lipiinf/ F(dxuifiyuiflz (2.53) {2 ‘-—+00 Q and since 8 u1 = 0 and 01112 = 0 we finally get y [f() (D)u dz S 11n11nf/f(Duk)dz I] Remark 2. From Theorem 2.9 and Theorem 2.7 for this L, every L-rank one convex function is L-quasiconvex but it may very difficult to prove this directly. 42 Chapter 3 Nonlinear Restrictions of Palais-Smale Type 3.1 Introduction In this chapter we investigate the weak lower semicontinuity for C1 functionals defined as I(u)=/flf(x,u(x),Du(x))dx from the perspective of a nonlinear constraint of Palais—Smale type. This requires that f be C1 in (3,6) and the derivatives satisfy some growth conditions. When minimizing a smooth and bounded-below functional I over a Banach space, an im- portant variational principle was discovered by Ekeland [Ek] in the 1970’s. Applying this principle to the minimization problem for our functional I over a Dirichlet class Ag in WI’P(Q; R"), we can always obtain a minimizing sequence {uk} in A9 which 43 satisfies I’(u),.) -—> 0 in W’I’P'(Q;IR’"). Here, we assume p > 1 and p’ = 5%, and W‘1*P'(Q; IRm') denotes the dual space of I/VO1 ”’ (Q; IR’"). Consequently, the weak limit (if exists) of any such minimizing sequence will be an energy minimizer provided that I (u) only satisfies the condition: uk —> u in IVI’P(Q; IR’") and I(u) S limian(u),.) whenever (3.1) k—too I’(uk) —+ 0 in w-1)P’(n;11m). In this case, we say that the functional I (u) is restricted weakly lower semicontinuous on WI’P(Q;IR’”). If the restricted lower semicontinuity condition (3.1) holds only for all uk,u in the Dirichlet class Ag, we then say I is restricted weakly lower semicontinuous on A9. Since the sequences {uk} with bounded I(uk) satisfying I ’ (uk) ——> 0 are usually called the Palais-Smale sequences [Ra] for the given functional I (u), we shall say that a sequence {uk} (PS) weakly converges to u (with respect to I) and denote by uk flu in WI’P if it satisfies uk —\u in Wl'f’ and I’(uk) —-> 0 in W‘I’P'. As we shall see later, this restricted weak lower semicontinuity imposes some intrinsic property on the function f. Although in the certain cases as presented in this paper the restricted weak lower semicontinuity is equivalent to the usual weak lower semicontinuity of the functional, in general, when f depends on x and s, we also give some examples to show that the restricted weak lower semicontinuity of I is be equivalent to the usual weak lower semicontinuity (see Proposition 3.7). 44 3.2 The (PS)-weak lower semicontinuity In this section, we assume f (x, s, 6) is measurable in x E Q for all (s, 6) E IR" x Mm” and is C1 in (s, 6) E IR" X Mm“ for almost every x E Q. We also assume 1 < p < 00 and f satisfies the growth conditions [f(flrssaéil S 01(I8I” + KI") + A(£15), (3-2) ID.f(x. 3.6)! + IDafur. sail s C2(Islp‘1+lélp‘1)+ B02) (33) for almost every x E Q and for all s E IR", 6 E Mm“, where c1,c2 are positive constants and A, B are positive functions with A E L1(Q), B E LEE—I(Q). From these assumptions, we can obtain the following result: Proposition 3.1. Under the above conditions, the functional I defined above is a C'1 functional on W11P(Q;IR’”) and for each u the Fréchet derivative I'(u) is given by (I'(u), v) = /Q [D,f(x,u, Du) -v + D£f(x,u, Du): Dv] dx for all v E W1*”(Q;IR"’). When minimizing the functional I on a Dirichlet class Ag, one can shift the class to the Banach space X = W01 ’p (Q; R") since inf I(u) = inf (w), (3.4) 116.49 wEX where (w) = I (w + 9). We easily have the following result. 45 Proposition 3.2. Let X = Wol’p(Q;lRm). For any g E WI’P(Q;IR’"), the functional : X ——> IR defined by (w) = I(w +g) is C1 and ’(w) = I’(w +g) as elements in X", the dual space of X. In the following we write X * = W“1’P’(Q;IR’"), where p’ = 513—1. As usual, we define I|1'(U)|Iw—1.p' = SUP{(1'(U)+UI lv 6 WWW), IIUIIwg-P S 1}. (3-5) Note that, given a smooth functional I on X = W'Ol’p (Q; R"), the sequences {uk} in X satisfying 11051311, I’(u)—>0 in W'1~P’(9;Rm) are usually called the Palais-Smale sequences or (PS) sequences for the functional I. Therefore, for simplicity, we use the following definition. Definition 3.1. A sequence {uk} is said to ( PS )-weakly converges to u (with respect to I) in Wl'p(Q; IR”) and denoted by uk EAu provided that u)c —‘- u in Wl'p(Q;lRm) and I’(uk) —> 0 in W’I’P’(Q;IR’”). Define the set of all (PS)-weak limits to be S = {u E Wl’p(Q;lRm) | El u)c E Wl’p(Q;lRm) such that uk 1Au}. (3.6) Let C = {u E WI’P(Q;IR’”) [ [[I’(u)[[W_1,pI = 0}. Then clearly C Q S, and hence S can be viewed as a relaxation of C under the (PS)-weak convergence. However, for certain functionals I the set 8 may be empty. 46 Example 4. Let f (x,6 ) = XE(x)h(6), where XE is the characteristic function of a measurable set E in (0,1) with 0 < [E] < 1 and h(6) = 725 + arctan 6. Define I(u) 2/0 f(x,u’(x))dx, u E W'1’2(0,1). We claim that for the functional I the (PS)-weak' limit set S = 0. Suppose to the contrary u), —p—S>u in W1’2(0, 1). Let g(x) = XE(x)h’(u[.(x)). Then, by Proposition 3.6 below, there exists a subsequence gt]. —-> L strongly in L2(0, 1) for some constant L. We also assume gkj(x) ——+ L for almost every x E (0,1). Hence we must have L = 0 and gkj (x) = h’ (ii;j(x)) ——> 0 for almost every x E E. By Egoroff’s theorem, it follows that [IILJ(£II)[ —+ 00 almost uniformly on E, which implies [[u[cj [| L205) -—> 00, a contradiction. Definition 3.2. Given any nonempty family A (_Z W 1"”(Q;IR""), we say that I is ( PS )-weakly lower semicontinuous on .A provided that I(u) S limian(uk) whenever u, u E A, uk 34w (3.7) k—voo We shall technically assume this property if A H S = (I). The following result shows that if f = f (x, 6) is convex in 6 then the functional I is in fact ( PS )-weakly continuous on all Dirichlet classes. Proposition 3.3. Assume f = f (x,6 ) satisfies the corresponding growth conditions as (3.2) and (3.3) above. Suppose f(x,6) is convex in 6 for almost every x E Q. 47 Then both I and —I are (PS )-weakly lower semicontinuous on all Dirichlet classes A9 with g E WI’P(Q;IR’"). Therefore the functional I is (PS )—weakly continuous on A9 in the sense that I(u) = lim I(uk) V uk, u E Ag, ukfiu. (3.8) k—*OO Proof. For any uk, u E VVI’P(Q;IR’”), by the convexity of f, it follows from (1.3) that f(x, Duk) 2 f(x, Du) + D£f(x, Du): (Du;c —- Du), (3.9) f(x, Du) _>_ f(x, Duk) + D§f(x, Duk): (Du — Duk) (3.10) for almost every x E Q. If uk 36* u, and u — uk E Wol’p (Q; R"), then integrating the above inequalities, we have limian(uk) Z I(u) Z limsupI(uk), k"*°° k—+oo and hence (3.8) follows. CI We show that in general the (PS)—weak lower semicontinuity on all Dirichlet classes does not imply the (PS)-weak lower semicontinuity on the whole space WI’P (Q; IR’") (Without the fixed boundary conditions). Proposition 3.4. Let Q be the unit disc in IR2 and I (u) = — f0 [Du|2dx foru: Q -—> IR. Then I is (PS)-weakly lower semicontinuous on all Dirichlet classes of W1'2(Q) 48 but not (PS )-weakly lower semicontinuous on W1’2(Q). Proof. By the preceding proposition, I is (PS)-weakly lower semicontinuous on all Dirichlet classes of W1’2(Q). We now show it is not (PS)-weakly lower semicontin- uous on W1'2(Q) (without the fixed boundary conditions). We identify IR2 E C1. For 2: = x1+ix2 E Q and k = 1,2, - -- , we define uk(x1,x2) = fiRdzk). Then uk is harmonic in Q and Omit)C — iamu)c = fiz’f‘l. Hence [Duk(x)[ = \félzlk‘1 and thus we have [[Dukllem) = 1. So u)c is bounded in W1’2(Q). It is easy to see uk —1 0 uniformly on Q and hence uk —\0 in W1’2(Q). Since u), is harmonic in Q, it also follows that Duk —> 0 in W‘1’2(Q). Therefore, for functional I (u) = — f0 [Du[2dx, we have uk is‘-0, but I(O) 2: 0 and liminf)c I(uk) = —1. Hence I is not (PS)-weakly lower semicontinuous on W1*2(Q). El As we mentioned in the introduction, the (PS)-weak lower semicontinuity has been motivated by using the Ekeland variational principle in the direct method for the minimization problem. We have the following existence result. Theorem 3.5. Assume f satisfies, in addition to (3.2) and (3.3), the following coercivity condition Colél” - a(x) S f(x, 8,6) S 01(IEI” + ISI”) + A(IE), (311) where co > 0 is a positive constant, a E L1(Q) is a function. Given 9 E Wl'p(Q; IR’"), assume the functional I defined above is (PS )-weakly lower semicontinuous on A9. Then the minimization problem 1an I (u) has at least one solution u E Ag. uE g 49 Proof. The proof uses a standard direct method of the calculus of variations. Let X = Wgrm; Rm). Define <1» X —+ R by (u) = I(u + g) = [fif(x,u(x) + g(x), Du(x) + Dg(x))dx. Then (I) is C1 and bounded below on X, and (F(u) = I’(u + g) in X“. By Theorem 1.8, there exists a sequence {uk} in X such that @(uk) —> 11)}fCI>, [[’(uk)[[Xo —> 0. Let w)c 2 uk + g E Ag. Then I(wkl-r inf I(w), |l1'(wk)|Iw-1.p' —>0- (312) 1126249 Under the condition co > 0 the sequence {wk} determined by (3.12) above is bounded in W140 (Q; R") and, since 1 < p < 00, has a weakly convergence subsequence, relabeled {wk} again. Let u be the weak limit. Then u E Ag and wk 5 u; hence the (PS)-weak lower semicontinuity on A9 implies I(u) S lim I(wk) = inf I(w). k—voo wEAg Hence I(u) = infweAg I(w). Cl Remark 3. Under the growth assumptions (3.2) and (3.3), any minimizer u of I 50 over A9 is a weak solution to the Dirichlet problem of the Euler—Lagrange equation of functional I; that is, —div D€f(x,u,Du) + D,f(x,u, Du) = 0 in Q (3.13) uzg ondQ. 3.3 One dimensional scalar cases In this section we study the (PS)-weak lower semicontinuity in some special one dimensional scalar cases. We first consider the Sobolev space H1(0, 1) = W1’2(0, 1) and functions f (x, 6) satisfying 0 5 flat) 3 C|€|2 + A(x), If£($,§)l s CI€I + 8(8), (3.14) with A E L1(0,1), B E L2(0,1). Define I(u) 2/0 f(x,u’(x))dx, V u E H1(0,1). Proposition 3.6. If uk 33 u in H1(0, 1), then there exists a subsequence {ukj} such that f€(x, ujcj(x)) —> L strongly in L2(0, 1) asj ——1 00, where L is a constant. Proof. Let g(x) = f5(x,u[€(x)) and L)c = folgk(x)dx. Since {9),} is bounded in L2(0,1), we assume for a subsequence gt]. ——=g in L2(0,1) as j -—-> 00, where g E 51 L2(0,1). We define v)c on [0,1] by ”(J/C(33) = Ax(gk(t) — Lk)dt, .’L‘ 6 [0,1]. Then it is easily seen that vk E H6(0, 1) = W01 ’2(0, 1) and v], = gt — L1,. Moreover, {vk} is bounded in HMO, 1) and hence I l grate/1):] f.>v;.da:= / gram—L220 0 0 as k —> 00. Since gkj —\g in L2(0,1), we have ij —> L = fol gdx and 2 1 1 1 / g2(x)dx S liminf/ gzdx = lim inf L2. = (/ g(x)dx) . 0 ]—'*OO O 3 J—mo J 0 This implies g(x) = L a.e. on [0,1] and gt]. ——1 L strongly in L2(0, 1). CI In contrast to the theorem of Acerbi and Fusco (Theorem 1.6), we show below by an example that the (PS)-weak lower semicontinuity of I may not imply f being quasiconvex in 6 even for smooth functions f (x, 6 ) in the scalar case. Proposition 3.7. There exists a C1 function f (x,6) satisfying condition (3.14) above for which the corresponding functional I is ( PS )-weakly, but not (unrestricted) weakly, lower semicontinuous on H1(0, 1). Proof. Assume f(x, 6) = a(x)h(6) with a, h 2 0, both C1 and satisfying the follow- 52 ing conditions: a(zr) = 0 :1: 6 [0,0], a(r) > 0 :1: 6 (9,1], (3.15) h 2 O, (h')'1(0) = {0}, 1]€Tn_}11flll’(€)l > 0., (3.16) where 6 6 (0,1) is a constant. Note that the condition (3.16) implies h(O) < h({) for all 5 E R. Given any uk 354a in H1(0,1), using subsequence if necessary, we assume lim,H00 I (uk) exists. By Proposition 3.6 above, there exists a subsequence {ukj} such that f5(:r, u].) = a(x)h’(u]cj) ——> L strongly in L2(0, 1) for some constant L. Since a = 0 on [0, 6), one must have the limit L = 0; this also implies the whole sequence a(x)h’(u]c) —> 0 strongly in L2(0,1). Therefore h’ (11.1.) ——> 0 strongly in L2(6’, 1) for any 0’ E ((9, 1). Hence, for a subsequence it follows that h’(u]cj(x)) —-> 0 for almost every :1: E (6’, 1). By (3.16), we have that 11.1,),(123) -—> 0 for almost every :1: E (6”, 1). Therefore the weak limit u’ = 0 on (9’, 1) for all 9’ 6 (6,1). This implies u’ = 0 in (6,1). Since h(é) Z h(O) for all g, we have 1 lim I(uk) = lim a(x)h(u;(a:)) d1: 2 f0 a(x)h(0)d:c = I(u). Hence I satisfies the (PS)-weak lower semicontinuity on H1(0,1). Note that the condition (3.16) does not imply that h is convex. (See, e.g., condition (1.4).) Hence I may not be weakly lower semicontinuous on H1(O, 1) by Theorem 1.6 above. C] Remark 4. For the functional I defined by a function f (1r, 5 ) = a(x)h(6) as above, 53 the minimization problem inf I(u) ueH1(o,1) u(0)=a.u(1)=b has as only minimizers any functions u E H1(0, 1) with u(0) = a and u E b on [6,1] and the minimum equals h(O) f01a(:r)d:c, for any constants a,b E R. These minimizers are exactly those functions u in the Dirichlet class for which there exists . ps a sequence {uk} 1n the class such that uk —\ u. Despite of the result above, we shall show that the (PS)-weak lower semicon- tinuity is equivalent to the usual weak lower semicontinuity if f (33,6) satisfies a coercivity condition. In this case, both conditions reduce to the convexity of f in g. For the technical reason of using the following Sard’s theorem [Mi], we assume f is sufficiently smooth in both :1: and 6. Lemma 3.8. Let h: R —> R be C1 andS = {y E R | 32: E R, y = h(x), h’(x) = 0}. Then the Lebesgue measure IS] = 0 and, in particular, the set of regular values of h, R \ S, is dense in R. In the following, for 6 E R, let Wg’p(0, 1) be the Dirichlet class of functions u in Wl’p(0, 1) with u(0) = 0, u(1) = [3. Theorem 3.9. Assume f(x,€) and f€(:r,§) are both C1 on [0,1] x R and satisfy, for some 13 > 1, |€|p S f($,€) S 61(|€|p+ 1), |f:($,€)| S C2(|€|""1 + 1) 54 for all :1: and E. If the functional I defined by f is ( PS )—weakly lower semicontinuous on Wgrm, 1) for all ,8 E R, then f(x,6) is convex in g for all :1: 6 (0,1). We proceed with several lemmas before proving this theorem. First of all, for 6 6 IR, we define 711(6) = inf{I(u) | u E Wfil’p(0, 1)}. It follows easily that WVSWW0S6M66+U- (3”) From Theorem 3.5 above, it follows that, if I is (PS)-wea.kly lower semicontinuous on ngo, 1), then there exists at least one minimizer 11.5 E WANG, 1) such that [(113) = m(fi). Hence I’(Ug) = 0 in W’l’p'(0, 1). This implies f£(:c, u:3(:1:)) is constant in (0, 1). Let 11(6) be this constant. Note that 11(6) depends also on the minimizer ”(b3 . Lemma 3.10. It follows that — 8 limsuplim511pmw+€> m() = +00, (3.18) B—H-oo E—»O+ E 15min111m3nfmw firm“) = —00. (3.19) a—w 6—. _ Proof We only prove (3.18); the other follows similarly. By contradiction, suppose the limit is finite. Then there exist positive constants Bo, 60 and M such that 7M5+0-mm) 6 S M, V5 2 50, 6 6(01€O]° For any positive integer k we get m(fi + k6) — m(,8) S M he and from 3.17 we obtain 55 (5 + he)? s M k6 + m(fi) which is false for k sufficiently large. Cl Lemma 3.11. For any 6 E R, it follows that lim sup m(,B + 6: _ mm) s 11(6) 3 limgnf m(;3 + 6: _ 771(8). e—+O+ 5* " (3.20) Proof. For 0 < (5 < 1 we define w to be the linear function with w(1 — (5) = 0, w(1) = 6. Hence w’(:L‘) = 6/6. Let ug be a minimizer for m0?) and let v(:1:) = 115(33) on [0,1 -— 6] and v(:1:) = ug(:1:) + w($) on [1 — 6,1]. Then v E Wl’p(0, 1) satisfies 11(0) = 0, 12(1) = [3 + 5. Hence mm + e) s M) = 11%) + 1:6”(1‘ v') — f(x,uvldx. Since f(x, v’) — f(x, u'fi) = f5(:1:, 14,)6/5 + 0(5/5) for 5/5 —> 0, we have me + e) 5 mm) + as»: + o<§>6 s m<13>+ was + am, as e —> 0. Fiom this the lemma follows. E] The lemmas above imply limsupuw) = +00, liminfuw) 2 —oo. (3.21) Lemma 3.12. Let h: IR —> R be Cl and h 2 0. Then the following statements are equivalent. 56 (i) h is convex. (ii) For all 0 < A < 1, a, b 6 IR with h'(a) = h'(b), it follows that h(Aa +(1—- A)b) S Ah(a) + (1— A)h(b). (3.22) (iii) There exist no numbers a < 6 satisfying h’(a) = h’(6) # h’(t) for all t E (a, 6). Proof. It is easily seen that (2') implies (ii). To show that (ii) implies (iii), we use a contradiction proof. Suppose (iii) fails. Then there exist numbers a, 6 E R satisfying oz <6, h'(a) = h'(6), h'(t) 79 h'(a) V t E (a, 6). (3.23) Using (3.22) with a = a, b = 6 we have, for all 0 < A < 1 and t,\ = A0 + (1 — A)6, ' S h(t.) — hm). t1 _ a 6 _ a t,\ _ 6 (3.24) Letting /\ —> 1' and 0‘L in (3.24) respectively, we have h’(a) S W S h’(6). Hence h’ (a) = h’ (6) 2 W1. However, by the mean value property, W = h’ (t) for some t E (a, 6), and hence we have arrived at a contradiction with (3.23). Finally, we prove that (iii) implies (2'). Again, by contradiction, suppose h is not convex. Then there exist a < b such that h’(a) > h’ (b). We consider only the case when h’(a) > 0; otherwise, consider h(t) = h(—t), a = —b and b = —a. We claim there exist c < d g a such that h’ (c) < h’ (d) If not, h’ would be nonincreasing 57 on (—00, a] and hence h would be concave on (—oo,a]. Therefore we would have h(t) S h(a) + h’(a)(t — a) for all t < a. Since h’(a) > 0, letting t ——> -—00, we would have h(t) —+ —00, a contradiction with h 2 0. Let c < d S a be any points as above. Let m = maxw] h’. Define S = {t E [c, b] I h’(t) = m}, s“ = min S, and 8+ = maxS. Then s‘,s+ E S and c < s‘ S 3*” < b. Hence h’(c) < m, h’(b) < m. We define a’ < 6’ as follows: If h’(c) = h’(b), define 01’ = c, 6’ = b. If h’(c) > h’(b), then h’ (c) E (lz’(s+),h’(b)) and hence by the intermediate value property of h’, define 6’ E (8+,b) so that h’(6’) = h’(c), and define a’ = c. If h’(c) < h’(b), then h’ (b) E (h’ (c), h’ (3‘)) and hence again by the intermediate value property of h’, we define a’ E (c,s‘) so that h’ (a’ ) : h’ (b), and define 6’ = b. The points 01’ < 6’ defined this way will satisfy a’ < s‘ S s+ < 6’ and h’(a’) = h’(6) < h’(s‘). Let G = {t E (a’,6’) ] h’(t) > h’(a’)}. Then G is an open set and s’ E G. Let (01,6) be the component of G containing 3". Then, for this pair of a, 6, we have (3.23), a contradiction with (iii); hence h is convex. This completes the proof of lemma. Cl Lemma 3.13. For any constant 6 E R, there exists a function Q9 6 LP(0, 1) such that f€(x, q9(x)) = 6 for almost every x 6 (0,1). Proof. In view of (3.21) above, there exist 61 < 62 such that M61) < 6 < 11(62). Hence for almost every x 6 (0,1) we have f€(x, u’fi1(x)) < 6 < f€(x, u],2(x)). Let ((2?) = minfua (17), 1123203)}, (1+0?) = max{utl($),ua(x)}- 58 Then (1i 6 LP(0,1). By the intermediate value property of f5(x, -), there exists q E (q—(x), q+(x)) such that f5(x, q) = 6. Let q9(x) be the infimum of all such q’s. Then f£(x, q9(x)) = 6, q9(x) is lower semicontinuous and q‘(x) S q0(x) S q+(x) at almost every x 6 (0,1) and hence (19 E Lp(0, 1). [:1 Proof of Theorem 3. 9. Given any x0 6 (0, 1), we prove f (x0, ) is convex. By Lemma 3.12, it suffices to show that there exist no numbers 51 < {2 such that f£(-’130,€1) = fact/biz), f£(1150,?5)7é fe($0,€1) V156 (€1,§2)- (3-25) We prove this by contradiction. Suppose £1 < 52 satisfy (3.25). We will derive a contradiction by showing such {i’s must satisfy f(fBO, A51 + (1 - ”‘52) S /\f($0,€1)+(1_ A)f($01€2) (3-26) for all A E (0, 1), which gives a desired contradiction as in the step 2 of the proof of Lemma 3.12. To this end, assume f5(x0, {1) = f€(x0, £2) = 60. Without loss of generality, assume f€(x0,t) > f€(x0,€1) for all t 6 (£1,152). Let [1511?] fawn, ) = 155013015): 5 > 90- To proceed, we need the following lemma, which is the only place we use the smooth assumption of f€(x,€) on (x,£). 59 Lemma 3.14. There exist a sequence 6,, E (60,6) with 6,, —) 60 as n —> oo, a closed interval Jn = [ambn] C (0,1) containing x0, and two continuous functions qfif: J,, —+ (ghgg) such that q;(x) < q,T(x) and f5(x,q,f(x)) = 6,, for all x E J”. Moreover, q;’+(x0) —> €13 as n ——* 00. Proof. The proof is based on a use of Sard’s theorem. By Lemma 3.8 above with h({) = f€(x0,§), the set of regular values of f§(x0, ) is dense. Hence there exists a sequence of regular values 6,, of f5(x0, ) in (60, 6) such that 6,, ——+ 60 as n —-> 00. Since f§(x0,§1,2) = 60 < 6,, < 6 = f€(xo,§), by intermediate value property, there exist 5,: E (51,6) and g: E (5,62) such that f§(x0,{f) = 6,,. The assumption (3.25) implies 6; —) {1 and 5: —> {2 as n —> 00. Since 6,, is a regular value of f5(xo, -), it follows that f55(x0,€,:f) 75 0. By the implicit function theorem, we have interval Jn 2 [am bn] C (0, 1) containing x0 and two differentiable functions qu: Jn —-> (61,5) such that qfif(xo) = i f£(x,qf(x)) = 6,, V x E J,,. (3.27) n, Then the functions qflx) satisfy the requirements of the lemma. C] We continue the proof of the theorem. Let 6,, E (60,6), J,, = [ambn] and qff: J,, —> (€1,52) be given as in the lemma above. Let J = [a, b] C J,, be any interval containing x0. Let q,, E LP(0, 1) be the function (19 determined by Lemma 3.13 with 6 = 6,,. In what follows, we fix n. For each k = 1, 2, - - - , we define function 60 uk(x) by uk(x) = fox wk(t)dt, where wk(t) is defined as follows: q,,(t) t E [0, 1] \ [a,b], wkm _—_ q;(t) t E U§T=1(a + @(b— a), a + (’ —1—+A)(b — a)), (328) 11:;(1) t e 0;: ,a( + ‘1' 1+” (b a),a + ,ga) — a)). It is easily seen that uk E l/V1J’(0, 1) and {11k} is bounded in I'V1*p(0,1). Lemma 3.15. For all continuous functions (x, é), it follows that b b lim (x, u]6(x))dx = / [A(I>(x, q;(x)) + (1 — A)(x, q:(x))]dx. k—>oo a Proof. It is easy to see b k a+-(—‘—”:—1tfl(b—a) / @(x u’ (x))dx = Z/ k (x q‘(x))dx (3 29) a k i=1 n+9%‘l ’ " 2k: +l(b—a) + f . (x,q:{(x))dx (3.30) j=1 Mtg—Ala”) k _ (b-a) = :;<1>(cj,qn(cj)) , (3.31) ’° ))(b—a) + —A)Zq’(d jvqn(d k i (3.32) 1:1 where a+ (lg—”(b—a) S c, S a+ gig—”(b—a) S d,- S a+ [Kb—a) are some points. Hence the sums in (3.31) and (3.32) are Riemann sums; therefore, as k -—> 00, the lemma follows. E] 61 Let 11 E Wl’p(0, 1) be defined by a(x) = for w(t)dt, where q,,(t) t E [0, 1] \ [a, b], w(t) = Aqg+11 — A>q: 0 as k ——> 00. By the definition of 11,, it follows easily that f5(x,u]c(x)) = 6,, for almost every x E (0,1); hence I’(uk) = 0 in W‘I’P'(0, 1). We now modify u}, to a function 11;, E W'gpm, 1) with 6 2 21(1). For 0 < 6 < 1— b to be selected later, we define uk(x) = 111,.(x) for x E [0, 1—15], and 11;,(x) = uk(x)+vk(x) for x E [1 —6, 1], where vk is a linear function on [1 —6, 1] with vk(1—6) = 0, vk(1) = 6,,. Hence in, E W11p(0,1) with 11;,(0) = 0, 11,,(1) 2 11(1) = 6. Note that vfc(x) = 531. Hence we select 6 = (5;, = [6,,.]1/2 for all sufficiently large k. For this choice of 6, it is easily shown that the function 11;, E WE‘WO, 1) satisfies uk—uk —> 0 in W1’P(0, 1), and hence it follows that I’(uk) —> 0 in W’l’p’(0, 1) and I(uk) — 1(1),.) —> 0 as k —-> 00. In particular, 11;, E; 11 in Wfil’p(0, 1). Therefore, by the (PS)-weak lower semicontinuity of I on W’é’p(0, 1), we have 1(a) S liminfk1(uk) = liminf;c I(uk). Using Lemma 3.15, after easy computations, this implies This holds for all intervals [a, b] C Jn containing x0 and hence, letting [a, b] shrink 62 to {x0}, we have f(rro, Miro) + (1 - Ammo» S Af($o,q;(ivo)) + (1 - A)f(fco, (12000))- Finally letting n —+ 00, by Lemma 3.14, we have f(iTO, A451 +(1— A)€2) S Af($0,€1)+ (1 — A)f($o,€2), as desired by (3.26). The proof of the theorem is now completed. CI 3.4 Special cases with f = f (6) In this section, we study some special cases with function f = f (5), where f : men ——+ R is a C1 function satisfying the following growth conditions: Colél" S f(é) S 6106]" +1), (333) |D£f(€)| S 02(|€|”'1 + 1), (3.34) where 1 < p < 00 and co _>_ 0, c1 > 0,62 > 0 are constants. In this case, we shall also use the simplified notation D5 f (g) = D f (5) = f’ (g) As before, let I be the functional associated with f: I(u) = [fif(Du(x))dx u E Wl'p(Q;lRm). 63 We first have the following result when m = 1 (the scalar case) with Co = 0 in (3.33), which is in contrast to Proposition 3.7 above Theorem 3.16. Let m = 1 and let f: IR" ——> R satisfy the conditions (3.33) and (3.34) above with CO = 0. Then the functional I is ( PS )-weakly lower semicontinuous on the Dirichlet classes IVA”) for all A E R" if and only iff is convex on R". Proof. By Theorem 1.6, we only need to show the necessary part of the theorem. Thus assume I is (PS)—weakly lower semicontinuous on the Dirichlet classes Wfi’p for all A E R". We prove that f is convex on R". To this end, let 6, 17 E R" and |17| = 1 be given and let h(t) = f (g + to) We show that h is a convex function of t E R; this implies f is convex on R". By virtue of Lemma 3.12 above, to show h is convex, it suffices to establish the inequality (iii) in that lemma for all a, b E IR with a < b and h’(a) = h’(b). Note that h'(t) = f’(£ + try) -17. Given such a, b, let a = g + an, 6 = 5 + b7). Then h(a) = f(a), h(b) = f(6) and hence h'(a) - h'(b) = (f'(a) - 6(6)) '77 = 0- (3.35) Given any A E (0,1), let 6(t) be the periodic function on R of period 1 satisfying 6 = 0 on [0,A) and 6 = 1 on [A,1). Let p(t) be the Lipschitz function on R with p(0) = 0 and p’(t) = 6(t) for almost every t E R. For k = 1,2,--- , we define functions 11],.(x) = ax + b—i—apch -17) :1: E IR". (3.36) 64 Then Duk(x) = 01 + (b - a)6(kx -17)77 and hence - "+11 01 1fx-17ELJQO GULF) J=-oo Duk(x) = (3.37) 6 ifx-17ELJ90 (113,%1). J=-oo Let {171,172, - -- ,77,,} be an orthonormal basis of IR" with 171 = 77. For each x E R", we write x = 2;, t,r7,- and define jj'l‘)‘ j_ n t1€(k, k )},Bk—{IIJER Let (2’; = Q n (UJ-Ai), 0’“ = Q 0 (UjBi). Then one can easily show that A;={xenn klim log] = AIQI, klim log] = (1 — mm. (3.38) For any 1 S p < 00, the sequence {uk} defined by (3.36) above satisfies uk -—“’L—t in Wl'pm) as k ——) 00, where a(x) 2 [A0 + (1 — /\)6]x. In fact, one can show that uk ——> it uniformly on Q. We leave the proof of these facts to the interested reader. Lemma 3.17. I’(uk) = 0 in W‘I’P’M). Proof. Given any v E WOLP(Q), we extend v to be zero outside 9. Let QN be the cube QN={$ERn]$=Ztinia ltil- ,Dv+f’(fi)- 1912 A1 B] 1(a) - (fpj vdS) (—n) + f’(a)- (fr, vds) 11 +1113)- (A1115) (-11) + 1'61) - (L W) n = f'(a) ~17 (fpw vdS — F1 vdS) . Hence, since F “N lies in R" \ fl, where v = 0, it follows that /Qf'(Duk(x)) -Dv(x) dx = f’(a) - 17 (fpm vdS — F-kN vdS) = 0. This proves I’(uk) = 0 in W‘lrp’m). El To continue the proof of the theorem, we now modify the sequence {uk} above into a sequence in Wj‘pM), where A 2 A01 + (1 —- /\)6. For all sufficiently large j, 66 say j Z jg, consider nonempty open sets Q, = {x E QIdist(x,8§2)>1/j}. Note that the measure IQ \ 52,-] —+ 0 as j —> 00. Let (o,- E C8°(Q) be the cut-off functions such that (,0,- : 1 on Q, and 0 S go,- S 1 in 9. Since 11;, —> 11 uniformly on {_2, we have that, for each j Z jg, there exists k,- > j satisfying _ 1 “(111,- — u)D%||LP(n) < 3- (3-39) Let it,- = gojukj + (1 — 1,0,)1‘1. Then ii,- E Wé’pm) = Wipm) and Du,- = (ojDukj + (1 — 1,0,)D1‘1 + (ukj — @ij. Hence, by (3.39) and also since Dukj, Da are bounded, it follows that 1.1220 HDfljllemmp = 0- (3-40) Therefore a, _1 a in W11P(Q) asj —> 00. Since 11,- = uh]. on Q], by (3.40) and the growth conditions (3.33)-(3.34), it easily follows that “111 ”1’le — I’(ukjlllw—lm'm) = 0, 3.11m |I(1”1,-) — I(u/9)] = 0- j—Aoo 00 Hence 11,-,11 E Wfi’p(fl), and {11-3311 since I’(ukj) = 0. By the (PS)-weak lower semicontinuity of I on W‘i’pGZ), we have [(11) S liminf,- I(ii,~) = liminf,- I(ukj). 67 Using (3.38), we easily see that this implies h(Aa + (1 — A)b) S Ah(a) + (1 — A)h(b). Hence by Lemma 3.12 above, h(t) = f(E + t17) is convex for all 5, 17 with [17] = 1. This proves f is convex on lR". C] We now study the general case with m 2 2. Under the coercivity condition that Co > 0 in (3.33), we have the following result: Theorem 3.18. Let n,m 2 1 and let f: men —> R satisfy the conditions (3.33) and (3.34) above with co > 0. Then the following statements are equivalent: (1) I is weakly lower semicontinuous on Wl'pm; IR”). (11) I is (PS )-weakly lower semicontinuous on erp(f2; lRm). (iii) I is ( PS )-weakly lower semicontinuous on all Wfi’pm; lRm). (iv) f is quasiconvex. Proof. By the theorem of Acerbi-Fusco (Theorem 1.6), (i) 4:) (iv) even when c0 = 0. Moreover, by the definition of quasiconvexity and using approximation, if f is quasiconvex and only satisfies (3.33) with co E IR, then it readily follows that I ( g A) S I (u) for all u E WANG; lR’"). It is also clear that (i) :> (ii) => (iii) in general cases. Therefore, to prove the theorem, it suffices to show that (iii) => (iv). We prove this as separate result in the lemma below. El Lemma 3.19. Under the assumptions (3.33) with co > 0 and (3.34), (iii) => (iv). 68 Proof. Given A E Mm”, by Theorem 3.5 (note that Co > 0 is needed here), there exists 11 E Wj’p(Q;lR"’) which is a minimizer of I (u) on Wfi’p(f2;lRm). We now apply the standard technique of Vitali covering [DM] to construct a sequence {uk} in WX’KQ; lRm) satisfying I(uk) = [(11) = inf I(u); (3.41) uEW’i‘P uk —> g, in Wl’p(f2;lRm) as k —> oo; (3.42) I’(uk) = 0 in W'1~P’(Q;Rm). (3.43) Note that (3.43) will follow from (3.41) since uk E W11,” (9; lRm) is also a minimizer of I (u) on Wj’p(Q;lRm). Once we have constructed such a sequence {uk}, which certainly satisfies uk 35* 9A, the (PS)—weak lower semicontinuity condition (111) will imply [(9/1) S limian(uk) = 1(a) = inf I(u), kfioo uEW/i‘p for all A E Mm“, which is exactly the quasiconvexity condition of f, and hence the result follows. Assume, without loss of generality, 0 E Q and then we use the Vitali covering theorem to decompose Q as follows: Q=U9‘:,Q,-UN; Q,n§2,~=(0 (iaéj), where Q,- =aj+ejQ CC Qwith a,- E Q, 0 < e, < l/k, and |N| =0. Letu =gA+17, 69 where 17 E W01 "’ (Q; R”). We define Ax + 631%?) if x E S2,, 11k(x) = J (3.44) Ax otherwise. Then one can easily check that 11;, belongs to Wfi’pm; IR") and satisfies Amouuxwx: [flammam for all continuous functions 61: men -—> IR satisfying |1/1(€ )| S C ([6 [P +1). Certainly this implies (3.41). Furthermore, it is easy to see 1 _ lluk - 9AllLP(Q) S l1— HU — gAl]LP(Q)- Hence u;C —\ 9,4 as k —> 00. As mentioned above, condition (3.43) follows from (3.41). This completes the construction of {uk} and thus the proof of the lemma. El Remark 5. For any u E Wi'p(§2;lRm), we write u = 9,, + v with v E Wol’p(Q;lRm) and define sequence uk E Wi’p(f2;lRm) as in (3.44) above with 17 = v. Then, if u is not a minimizer of I over W316“); lRm), one only has I ’ (uk) —*0, but not strongly, in W‘IAP'(Q;lRm), as k —> 00, even when I’(u) = 0; hence the (PS)—weak lower semicontinuity can not be applied to this sequence. Finally we show that without the coercivity co > 0 in (3.33) the results of Theorem 3.18 may fail, at least in the case n = 1, m = 2. 70 Theorem 3.20. Let f : IR2 —-> IR be a (3'1 function satisfying the conditions (3.33) and (3.34) with Co = 0. Assume the derivative map D f = f’: IR2 ——> IR2 is one—to-one. Then the functional I defined on X = W1*p((0, 1); IR?) by 1(1):] f(u’(x))drc= f f(ua(:c>.u;>dx. u=ex. is {PS)-weakly lower semicontinuous on X. Proof. Let u E X = Wl‘p((0,1);lR2). Then 1 (I’(ula ’U) 2/ lf€1(ul($))vl + f€2(u’($))v2l (11‘, V ’U : (”11”?) 6 X1 0 and hence it can be shown that “I’(ulllw—lm’ '5 llf€1(u,) — Cl(u)IILP’(0,1) + I|f§,(u’) — 02(ulllLP’(0,l)’ where Cl (u), C2(u) are two constants depending boundedly on u E X. Assume uk flu in X. We also assume lim I (uk) exists as k ——> 00. Then there exists a subsequence {ukj} such that Ilf€1(ulcj) " ClllLP’(0,1)+ IIf€2(uIc,-) — Czlle'(o,1) —’ 0 as j ——> 00, where 01,02 are some constants. We also assume there exists a measur- 71 able set E C (0,1) such that [E] = 1, lim f€u(u]c,(x)) 2 C, V x E E (11 = 1,2). (3.45) J—’00 3 Note that for all M > 0 the measure |{x E E] |u],(x)| > M}| S 170,, for all k, where C is a constant; hence there exists a sufficiently large M > 0 such that |{x E E] |u].(x)| S M}| > % for all k = 1, 2, - -- . Therefore there exists x0 E E such that |u]cj(xo)| S M. By taking another subsequence, assume u11(x0) —> 01 E IR2 as j, —+ 00. Therefore by (3.45), f’(oz) = (C1,C2). 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