Weak convergence methods for nonlinear partial differential equations

Conference Board of the Mathematical Sciences REGIONAL CONFERENCE SERIES IN MATHEMATICS supported by the National Science Foundation Number 74

WEAK CONVERGENCE METHODS FOR NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS Lawrence C. Evans

Published for the Conference Board of the Mathematical Sciences by the American Mathematical Society Providence, Rhode Island

Contents 1 Weak Convergence 1.1 Review of Basic Theory . . . . . . . . . . . . 1.2 Convergence of Averages . . . . . . . . . . . . 1.3 Compactness in Sobolev Spaces . . . . . . . . 1.3.1 Embeddings . . . . . . . . . . . . . . . 1.3.2 Compactness theorems . . . . . . . . . 1.3.3 A Refinement of Rellich’s Theorem . . 1.4 Measures of Concentration . . . . . . . . . . . 1.4.1 Generalities . . . . . . . . . . . . . . . 1.4.2 Defect measures . . . . . . . . . . . . . 1.4.3 A refinement of Fatou’s Lemma . . . . 1.4.4 Concentration and Sobolev inequalities 1.5 Measures of Oscillation . . . . . . . . . . . . . 1.5.1 Generalities . . . . . . . . . . . . . . . 1.5.2 Slicing measures . . . . . . . . . . . . . 1.5.3 Young measures . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

. . . . . . . . . . . . . . .

6 6 8 9 9 9 10 12 12 12 14 15 17 17 17 19

2 Convexity 21 2.1 The Calculus of Variations . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Weak lower Semicontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Convergence of Energies and Strong Convergence . . . . . . . . . . . . . . 24 3 Quasiconvexity 3.1 Definitions . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Rank-one convexity. . . . . . . . . . . . . 3.1.2 Quasiconvexity. . . . . . . . . . . . . . . . 3.2 Weak Lower Semicontinuity . . . . . . . . . . . . 3.3 Convergence of Energies and Strong Convergence 3.4 Partial Regularity of Minimizers . . . . . . . . . . 3.5 Examples . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Weak continuity of determinants. . . . . . 1

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

. . . . . . . .

26 26 26 27 28 31 34 36 36

CONTENTS

1

3.5.2

Polyconvexity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4 Concentrated Compactness 4.1 Variational Problems . . . . . . . . . . . . . . . . . . 4.1.1 Minimizers for critical Sobolev nonlinearities. 4.1.2 Strong convergence of minimizing sequences. . 4.2 Concentration-Cancellation . . . . . . . . . . . . . . 4.2.1 Critial gradient growth. . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

39 39 39 42 44 44

5 Compensated Compactness 5.1 Direct Methods . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Harmonic maps into spheres . . . . . . . . . . . 5.1.2 Homogenization of divergence structure PDE’s. 5.1.3 Monotonicity, Minty-Browder method in L2 . . 5.2 Div-Curl Lemma . . . . . . . . . . . . . . . . . . . . . 5.3 Elliptic Systems . . . . . . . . . . . . . . . . . . . . . . 5.4 Conservation Laws . . . . . . . . . . . . . . . . . . . . 5.4.1 Single equations. . . . . . . . . . . . . . . . . . 5.4.2 Systems of two equations. . . . . . . . . . . . . 5.5 Generalization of Div-Curl Lemma . . . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

. . . . . . . . . .

54 54 54 56 58 59 61 63 63 65 70

6 Maximum Principle Methods 6.1 The Maximum Principle for Full Nonlinear PDE . . 6.1.1 Minty-Browder method in L∞ . . . . . . . . . 6.1.2 Viscosity solutions. . . . . . . . . . . . . . . 6.2 Homogenization of Nondivergence Structure PDE’s 6.3 Singular Perturbations . . . . . . . . . . . . . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

72 72 72 76 77 79

7 Appendix

. . . . .

. . . . .

82

2

CONTENTS

Preface

These notes are an expanded version of a series of ten talks I gave at an NSF-CBMS Regional Conference on June 27-July 1, 1988, at Loyola University of Chicago. Additional, advanced lectures were presented by Emilio Acerbi, Mike Crandall, Irene Fonseca, Nicola Fusco, Ron Gariepy, David Kinderlehrer, Fang Hua Lin, Takis Souganidis, and Luc Tartar, all of whom I wish to thank. Let me also thank Nick Barron for initiating, organizing, and managing the conference. Nick is an expert in theoretical stochasitc control theory, and I can attest now to his great practical skills in this field as well. Many thanks are due as well to Tanja Kazan for translating my fully nonlinear handwriting into these typed notes. This work was supported in part by NSF grants DMS-86-10730 and CDR-85-00108. Lawrence C. Evans

3

CONTENTS

Introduction My intension in these notes is to explain systematically, and I hope clearly, many of the most important techniques set forth in recent years for studying nonlinear partial differential equations (PDE’s) via weak convergence methods. The basic issue, in its most abstract, cleanest form, is this: suppose we wish to solve some nonlinear PDE, which we write symbolically as (0.1)

A[u] = f,

A[·] denoting a given nonlinear operator, f a given function, and u the unknown. To establish the existence of a solution u of (0.1), an obvious idea is first to invent an appropriate collection of nicer, approximating problems, which we can in fact solve. These we write abstractly as (0.2)

Ak [uk ] = fk

(k = 1, 2, · · · ),

where Ak [·] represents a nonlinear operator which is somehow close to A[·] for large k, fk is close to f , and uk is a solution. The hope now is that the functions {uk }∞ k=1 will converge to a solution u of (0.1). This proposed procedure is of course very general, too general to be of specific guidance for any particular problem, and consequently practical implementation usually demands great care in the choice of approximations. In practice, the operators Ak [·] may represent finite-dimensional projections, singular regularizations, discretizations, gradients of approximate energy functionals, systems collapsing in the limit to a single equation, etc. Indeed, given a nonlinear PDE like (0.1), it is usually not particularly difficult to dream up some reasonable seeming and solvable approximation: the trick is to demonstrate that solutions of (0.2) really do converge to a solution of (0.1). The overall impediment is of course the nonlinearity. Whereas it is often the case that certain uniform estimates can be had for the family {uk }∞ k=1 , it is equally often also true that these best available bounds are none too strong. With such relatively poor estimates in hand, we can consequently usually show only that the functions {uk }∞ k=1 (or a subsequence) converge weakly in some function space to a limit u: (0.3)

uk * u as k → ∞.

Given the strong nonlinearities we are trying to cope with, this rather feeble mode of convergence is almost always a definite problem. Granted that we have somehow constructed approximate operators Ak [·] which tend to A[·] in some sense, it is in applications by no means certain that the weak convergence (0.3) implies (0.4)

Ak [uk ] → A[u] as k → ∞.

4

CONTENTS

in any way whatsoever. The obstruction, which we will review in Chapter 1, is that weak convergence is terribly behaved with respect to nonlinearities, and yet such weak convergence is apparently the best we can achieve. What saves the game for many interesting PDE’s is the explicit nonlinear structure. Roughly speaking, although nonlinearity generally excludes the derivation of very many really good uniform estimates on the functions {uk }∞ k=1 and their derivatives, in certain settings the very fact that uk solves the appropriately designed approximation (0.2) provides enough extra control to justify passing to limits. This prospect and its diverse implementations and implications are the subject of these notes (We will encounter as well at least two major variants of the general approximation problem. In the first our given PDE (0.1) is variational, which is to say that the nonlinear operator A[·] is the ”derivative” of an appropriate energy functional I[·]. In this setting we will not generally take our approximating sequences {uk }∞ k=1 to be solutions of problems of the form (0.2), but rather as a minimizing sequence for I[·] subject to side constraints. In the second primary variant we will initially be given the approximation scheme (0.2) and must not only pass to the PDE (0.1), but must also guess its form. Such problems come up naturally in theories of homogenization, singular perturbation, etc.) In summary, our working philosophical strategy will be to identify interesting classes of nonlinear PDE’s for which the very nonlinear structure itself yields enough extra information to justify weak convergence techniques for passing to limits. I will concentrate in this exposition primarily upon four fields within the subject which recently have seen very pronounced progress. These areas, utilizing the respective methods of convexity/quasiconvexity, concentrated compactness, compensated compactness, and the maximum principle, are currently rapidly developing and are consequently rather ill-defined: indeed a primary purpose of these notes is to suggest and emphasize some common underlying viewpoints. While there are no hopes for a unified field theory which would somehow neatly coalesce all the diverse techniques and tricks we will see, I do hope to convince readers that there is a substantial common vision which unites the subject and which can be brought into clearer focus. After an initial development of real analysis tools in Chapter 1, the main exposition progresses generally from variational to nonvariational problems. The central measuretheoretic techniques for analyzing concentrations and oscillations are set forth in Chapters 4 and 5. I regard these as comprising the core of the current general theory of weak convergence and nonlinear PDE’s, augmented in Chapters 2, 3 and 6 by additional techniques available for PDE’s with the extra structure of variational and/or maximum principles. Of the several continuing and unifying themes in the exposition two are worth explicit mention here. First, the reader should take note throughout of the primary role of measure theory and real analysis, as opposed to functional analysis. The detailed study of concentration and oscillation phenomena in Chapters 1, 4 and 5 cannot in any obvious way be cast into functional analytic terms. Secondly, note the continual use in quite diverse settings of low amplitude, high frequency periodic test functions to extract useful

CONTENTS

5

information. I have tried to identify throughout the absolutely simplest problems which illustrate various key techniques, and accordingly present in most cases what amount almost to caricatures. I make in particular the standing assumption that, unless otherwise stated, all given functions are smooth. The notation is mostly either self-explanatory or else standard (as in, for instance, Gilbarg-Tudinger [68].) I systematically employ the summation convention and use the letter C to denote various constants. Although these notes are fairly wide ranging, they by no means exhaust our subject. Let me note in particular my total omission of relevant problems in geometry (cf. Aubin [6, 7], Freed-Uhlenbeck [60], Lawson [81], Sacks-Uhlenbeck [105], Taubes [123], etc.), Γ−, G− and H−convergence (Attouch [5], Murat [96], Spagnolo [113], etc.), relaxation of illposed problems (Ball-James [15], Chipot-Kinderlehrer [34], Dacorogna [37], Fonseca [59], Kinderlehrer [75], Kohn-Strang [77], etc.), modification of the Palais-Smale condition (Bahri-Coron [9], [10], Struwe [116], etc.), and new work on the Boltzmann equation (DiPerna-Lions [43, 44]). Various additional references are to be found in the Notes at the end.

Chapter 1 Weak Convergence In this chapter we first recall without proofs the basic facts from functional analysis concerning weak convergence and weak compactness for functions and measures. We next systematically sharpen certain real analysis tools, to be employed later on.

1.1

Review of Basic Theory

Here and hereafter we will denote by U an open, bounded, smooth subset of Rn (n ≥ 2). Assume 1 ≤ q < ∞, q 0 ≡ q/(q − 1). q q Definition 1.1.1 A sequence {fk }∞ k=1 ⊂ L (U ) converges weakly to f ∈ L (U ), written

fk * f

in Lq (U ),

provided Z

(1.1)

fk gdx → U

Z

f gdx as k → ∞ U

0

for each g ∈ Lq (U ).

Theorem 1.1.1 (Boundedness of weakly convergent sequences). Assume fk * f in Lq (U ). Then (i)

{fk }∞ k=1

is bounded in

Lq (U )

and (ii)

kf kLq (U ) ≤ lim inf kfk kLq (U ) . k→∞

6

7

1.1. REVIEW OF BASIC THEORY 0

In view of (i) we see that if fk * f in Lq (U ) and gk → g in Lq (U ), then Z Z fk gk dx → f gdx as. U

U

A refinement of (ii) holds: if 1 < q < ∞, fk * f in Lq (U ) and kf kLq (U ) = lim kfk kLq (U ) , k→∞

then fk → f

strongly in

Lq (U ).

Theorem 1.1.2 (Weak compactness). Assume 1 < q < ∞ and the sequence {fk }∞ k=1 ∞ is bounded in Lq (U ). Then there exists a subsequence {fkj }∞ j=1 ⊂ {fk }k=1 and a function f ∈ Lq (U ) with fkj * f in Lq (U ). ∞ In the case q = ∞ the terminology is slightly different. We say that {fk }∞ k=1 ⊂ L (U ) converges weakly star to f ∈ L∞ (U ), written ∗

in L∞ (U ),

fk * f

provided (1.1) holds for all g ∈ L1 (U ). The analogues of Theorems 1.1.1 and 1.1.2 are valid. The weak compactness theorem is definitely false if q = 1, as simple examples illustrate. This failure of compactness is to a certain extent offset by regarding L1 (U ) in the natural way as a subset of M(U ), the space of signed Radon measures on U with finite mass. Let C0 (U ) denote the space of continuous, real-valued functions on U , with compact support. Definition 1.1.2 A sequence {µk }∞ k=1 ⊂ M(U ) converges weakly to µ ∈ M(U ), written µk * µ in M(U ), provided

for each g ∈ C0 (U ).

Z

gdµk → U

Z

gdµ as k → ∞ U

Theorem 1.1.3 Assume µk * µ weakly in M(U ). Then lim sup µk (K) ≤ µ(K) k→∞

for each compact set K ⊂ U , and µ(V ) ≤ lim inf µk (V ) k→∞

for each open set V ⊂ U .

8

CHAPTER 1. WEAK CONVERGENCE

Theorem 1.1.4 (Weak compactness for measures). Assume that the sequence ∞ ∞ {µk }∞ k=1 is bounded in M(U ). Then there exists a subsequence {µ kj }j=1 ⊂ {µk }k=1 and a measure µ ∈ M(U ) with µkj * µ in M(U ). Remark. We extend the terminology above to the Sobolev space W 1,q (U )(1 ≤ q < ∞) by saying that fk * f weakly in W 1,q (U ) provided fk * f in Lq (U ) and Dfk * Df in Lq (U ; Rn ). The reader should consult, for instance, Billingsley [20] and Yosida [125] for proofs of the foregoing assertions.

1.2

Convergence of Averages

Before proceeding further it may be useful to pause and develop some further insight into the meaning of weak convergence. So assume now 1 < q < ∞ and (1.2)

fk * f

in Lq (U ).

Then if E ⊂ U is a bounded measurable set, we deduce upon setting g = χE in (1.1) that Z Z (1.3) fk dx → f dx. E

E

This implies that the averages of the function {fk }∞ k=1 over the set E converge to the average of f over E. (Conversely, it is straightforward to check that if {fk }∞ k=1 is bounded in Lq (U ) and verifies (1.3) for each bounded,measurable set E ⊂ U , then (1.2) holds.) A problem we will continually confront in PDE applications is that this convergence of averages does not imply norm or even a.e. convergence. It may very well be that the sequence {fk }∞ k=1 effects its weak convergence to f by virtue of perhaps unbounded, very high frequency and quite irregular oscillations. Such behavior utterly excludes any simple analysis of nonlinear functionals of the sequence {fk }∞ k=1 . In particular we note that (1.2) does not imply F (fk ) * F (f ) for any nonlinear, real-valued function F . To see this, select real numbers a < b and 0 < λ < 1 so that F (λa + (1 − λb)) 6= λF (a) + (1 − λ)F (b). Assuming U = (0, 1) ⊂ R, we set ( a if j/k ≤ x ≤ (j + λ)/k, j = 0, · · · , k − 1 fk (x) = b otherwise. ∗

Then fk * f ≡ λa + (1 − λ)b in L∞ , whereas ∗

F (fk ) * F ≡ λF (a) + (1 − λ)F (b) 6= F (f ).

9

1.3. COMPACTNESS IN SOBOLEV SPACES

1.3 1.3.1

Compactness in Sobolev Spaces Embeddings

For later reference we record here the Gagiardo-Nireberg-Sobolev inequality, which asserts that if 1 ≤ q < n and q ∗ ≡ qn/(n − q) is the Sobolev conjugate of q, then (1.4)

kf kLq∗ (Rn ) ≤ Cq kDf kLq (Rn ;Rn )

for all functions f ∈ C01 (Rn ), the optimal constant Cq depending only on q and n. Invoking ∗ usual approximations, we see that this estimate is also valid provided f ∈ Lq , Df ∈ Lq . If U denotes as usual a bounded smooth open subset of Rn , it follows from (1.4) and standard extension theorems that (1.5)

kf kLp (U ) ≤ Ckf kW 1,q (U )

for each 1 ≤ p ≤ q ∗ and f ∈ W 1,q (U ), the constant C depending only on q, n, and U .

1.3.2

Compactness theorems

Our next assertion sharpens the preceding observation, demonstrating that the embedding W 1,q (U ) ⊂ Lp (U ) is in fact compact if 1 ≤ p < q ∗ : Theorem 1.3.1 (Rellich’s Compactness Theorem). Assume that the sequence 1,q p ∗ {fk }∞ (U ). Then {fk }∞ k=1 is bounded in W k=1 is precompact in L (U ) for each 1 ≤ p < q . A proof of this assertion is available in Gilbarg-Trudinger [68]. We will later devote considerable effort to understanding just how compactness fails for the critical case p = q ∗ : see §4 below and Chapter 4, §1. Theorem 1.3.2 (Compactness for measures). Assume the sequence {µk }∞ k=1 is ∞ −1,q ∗ bounded in M(U ). Then {µk }k=1 is precompact in W (U ) for each 1 ≤ q < 1 . ∞ Proof. 1. Using Theorem 1.4 we may extract a subsequence {µkj }∞ j=1 ⊂ {µk }k=1 so that µkj * µ in M(U ), for some measure µ ∈ M(U ). 0

2. Set q = q/(q − 1) and denote by B the closed unit ball in W01,q (U ). Since 0 ¯ ). Thus given ε > 0, there exist 1 ≤ q < 1∗ , we have q > n; and so B is compact in C0 (U N (ε) functions {φi }i=1 ⊂ C0 (U ) so that 0

min kφ − φi kC(U¯ ) < ε

1≤i≤N (ε)

for each φ ∈ B.

10

CHAPTER 1. WEAK CONVERGENCE 3. Thus if φ ∈ B, Z Z Z Z φdµk − φdµ ≤ 2ε sup |µ |(U ) + φ dµ − φ dµ k i k i j j j j U

U

U

U

for some index 1 ≤ i ≤ N (ε). Consequently, Z Z lim sup φdµkj − φdµ = 0, j→∞ φ∈B U

and so µkj → µ in W −1,q (U ).

U



−1,p Corollary 1.3.1 Assume that the sequence {fk }∞ (U ) for some k=1 is bounded in W p > 2. Suppose further that fk = gk + hk (k = 1, . . .), where the sequence {gk }∞ k=1 is −1,2 ∞ precompact in W (U ) and the sequence {hk }k=1 is bounded in M(U ). Then {fk }∞ k=1 is −1,2 precompact in W (U ).

Proof. For k = 1, 2, . . . let uk ∈ W01,2 (U ) be the weak solution of ( −∆uk = fk in U, uk = 0 on ∂U, and then write uk = νk + wk , where ( −∆νk = gk νk = 0 (

−∆ωk = hk ωk = 0

in U, on ∂U, in U, on ∂U,

From standard elliptic theory and Theorem 1.3.2, we see that {νk }∞ k=1 is precompact 1,2 1,q ∞ in W0 (U ) and {wk }k=1 is precompact in W0 (U ) for each 1 ≤ q < 1∗ . Thus {uk }∞ k=1 −1,q ∞ is precompact in W01,q (U ), and so {fk }∞ is precompact in W (U ). As {f } k k=1 is k=1 −1,p bounded in W (U ) for some p > 2, the Corollary follows. 

1.3.3

A Refinement of Rellich’s Theorem

Our intention next is to sharpen Rellich’s Theorem (as well as Egoroff’s Theorem) to show that a sequence bounded in W 1,q (U ) has a subsequence converging uniformly except for a very small set. The idea, following Meyers [92] and Frehse [61], is that the bad set has not only small Lebesgue measure but also small capacity. The usefulness of this capacity

11

1.3. COMPACTNESS IN SOBOLEV SPACES

estimate for studying critical growth nonlinear PDE’s will become apparent in Chapter 4, §2. (See the Appendix for definitions and properties of capacity, and note in particular that we identify each function in the Sobolev space W 1,q (U ) with its precise representative. We will momentarily encounter in §4 an alternative description of small sets in terms of Hausdorff measure.) Suppose 1 < q < n. 1,q Theorem 1.3.3 Assume that the sequence {fk }∞ (U ). Then there k=1 is bounded in W ∞ ∞ 1,q exist a subsequence {fkj }j=1 ⊂ {fk }k=1 and a function f ∈ W (U ) such that for each 1 ≤ p < q and each δ > 0 there exists a relatively closed set E δ ⊂ U with

f kj → f

uniformly on Eδ

and

Capp (U − Eδ ) ≤ δ.

Proof. 1. We may as well suppose fk ∈ W01,q (U ) (k = 1, . . .), since we could otherwise choose V ⊃⊃ U and extend each fk to belong to W01,q (V ). Passing if necessary to a subsequence we may assume fk → f in Lq (U ), for some fk ∈ W01,q (U ). 2 2.Fix δ, ε > 0 and let Eεk ≡ {x ∈ U ; |fk (x) − f (x)| ≥ ε}. Write gεk ≡ max{|fk − f | − ε ε , 0}; so that gεk ∈ W01,q (U ), gεk ≥ 1 on Eεk . Then from the Appendix we see 2 Z k Capp (Eε ) ≤ |Dgεk |p dx U

Z p/q  p  ε 1−p/q 2 n q q L {|fk − f | > } |Dfk | + |Df | dx ≤ ε 2 U ≤ C(ε)kfk − f kLq−p q (U )

(Here Ln denotes n-dimensional Lebesgue measure.) ∞ X ∞ 3. Choose a subsequence {fkj }j=1 so that kfkj − f kLq−p Write Fil ≡ q (U ) < ∞. S∞

kj j=l E1/i .

j=1

Then

Capp (Fil )

≤

∞ X j=l

  kj Capp E1/i

≤ C(i)

∞ X j=l

≤

δ 2i+1

kfkj − f kLq−p q (U )

12

CHAPTER 1. WEAK CONVERGENCE

if l = l(i) is large enough. Since Capp (E) = inf{Capp (V ); V ⊃ E, V open}, we can l(i) find open sets Gli ⊃ Fil with Cap(Gli ) ≤ δ/2i . Finally, set Eδ ≡ U − ∪∞ i=1 Gi . Then ∞ X l(i) Capp (U − Eδ ) ≤ Capp (Gi ) ≤ δ and fkj → f uniformly on Eδ .  i=1

1.4 1.4.1

Measures of Concentration Generalities

We devote this and the next section to the construction of measure-theoretic tools to help us understand the various ways in which a weakly convergent sequence of functions can fail to be strongly convergent. To understand the issues at hand let us assume as always U is a bounded smooth open set, 1 < q < ∞, and (1.6)

fk * f

weakly in Lq (U ),

fk 9 f

strongly in Lq (U ),

but (1.7)

Now there are several quite distinct modes of behavior which can cause this breakdown of strong convergence. First observe that even if we know the functions {fk }∞ k=1 to be bounded in the supremum norm, so that fk converges weakly in f in Lp (U ) for all 1 ≤ p < ∞, we still cannot deduce strong convergence in Lp (U ) for any 1 ≤ p < ∞. The difficulty turns upon the possibility of very rapid fluctuations in the functions fk . This is the problem of oscillation. Secondly, observe that even if we somehow know additionally (1.8)

fk → f

a.e.

in

U,

so that wild oscillations are excluded, we still cannot legitimately deduce strong convergence in Lq (U ). The obstruction is that the mass of |fk − f |q may somehow coalesce onto a set of zero Lebesgue measure. This is the problem of concentration. Finally, of course, both oscillation and concentration effects can transpire simultaneously, creating thereby the general problem of oscillation/concentration. In this section we construct appropriate methodology for characterizing concentration effects. The rather distinct measure-theoretic tools needed for describing oscillation effects will be set forth later in §5.

1.4.2

Defect measures

As was just noted, we intend to introduce certain measures that record the failure of weak convergence in Lq (U ) to imply strong convergence in Lq (U ). So let us suppose now that

13

1.4. MEASURES OF CONCENTRATION (1.6) holds, and consider then the measures θk ≡ |fk − f |q

(1.9) Thus θk (E) =

Z

(k = 1, . . .).

|fk − f |q dx controls how close fk is to f in the Lq -norm restricted to E

the Borel set E, and consequently the limiting behavior of the measures {θk }∞ k=1 should presumably reflect the possible failure of strong convergence. Definition 1.4.1 For each Borel set E ⊂ U , write Z θ(E) ≡ lim sup |fk − f |q dx. k→∞

E

Following DiPerna-Majda [45] we call θ the reduced defect measure associated with the weak convergence (1.6). The idea is that θ encodes information about the extent to which strong convergence fails. Note in particular (1.10)

θ(E) = θ(F )

if

Ln ((E − f ) ∪ (F − E)) = 0

and (1.11)

fk → f

strongly in

Lq (E)

if and only if

θ(E) = 0.

Observe also θ is, strictly speaking, only a finitely-additive outer measure. In as much as the nonvanishing of θ signals a failure of strong convergence, it seems reasonable to analyze the set upon which this measure is concentrated. We utilize the terminology from the Appendix and present two alternative ways to assess the smallness of this set onto which the measure coalesces. Definition 1.4.2 We say that θ is concentrated on a set of p-capacity zero if there exist open sets {Vi }∞ i=1 in U with θ(U − Vi ) = 0

(i = 1, . . .) and Capp (Vi ) → 0.

Definition 1.4.3 We say that θ is concentrated on a set of Hausdroff H s -measure ∞ zero if there exist open sets {Vi }∞ i=1 in U and a sequence {δi }i=1 ⊂ (0, ∞) with θ(U − Vi ) = 0

(i = 1, . . .)

δi → 0, and Hδsi (Vi ) → 0.

14

CHAPTER 1. WEAK CONVERGENCE

The informal idea is that θ is concentrated on the set C ≡ ∩∞ i=1 Vi with either Capp (C) = s 0 or H (C) = 0. Since, however, θ is only finitely subadditive, note carefully that we cannot legitimately deduce θ(U − C) = 0. For instance, let U = (0, 1), f ≡ 0 and   k if 1 − 1 ≤ x ≤ 1 + 1 2 2k 2 2k fk (x) ≡  0 otherwise.   1 Then θ is concentrated on C = , θ(U − V ) = 0 for each open set V ⊃ C, but 2 θ(U − C) = 1. We will employ these notions later in Chapter 4,§2.

1.4.3

A refinement of Fatou’s Lemma

As already explained, concentration phenomena will arise in examples where we have the weak convergence (1.6) simultaneously with the pointwise convergence (1.8). Invoking Fatou’s Lemma, we at once infer from (1.8) that kf kLq (U ) ≤ lim inf kfk kLq (U ) ; but we k→∞

already know this from (1.6) and Theorem 1.1.1. Brezis and Lieb [30] have examined this situation more carefully and established the following sharpened assertion: Theorem 1.4.1 Assume (1.6), (1.8), 1 ≤ q < ∞. Then   (1.12) lim kfk kqLq (U ) − kfk − f kqLq (U ) = kf kqLq (U ) . k→∞

The point is that fk decouples in the limit as measured in the Lq norm into fk − f and f . Note also that the case q = 2 is immediate and does note require (1.8). Proof. Recall first the elementary inequality

(1.13)

||a + b|q − |a|q | ≤ ε|a|q + C(ε)|b|q

(a, b ∈ R, ε > 0)

where C() depends only on  and q. Now set gkε ≡ (| |fk |q − |fk − f |q − |f |q | − ε|fk − f |q |)+ ; so that gkε → 0 a.e.. Inequality (1.13) implies gkε ≤ (| |fk |q − |fk − f |q | + |f |q − ε|fk − f |q |)+ Z Thus the Dominated Convergence Theorem forces lim gkε dx = 0. But then k→∞

lim sup k→∞

Z

U

| |fk |q − |fk − f |q − |f |q |dx U

15

1.4. MEASURES OF CONCENTRATION ≤ ε lim sup k→∞

1.4.4

Z

|fk − f |q dx = O(ε).



U

Concentration and Sobolev inequalities

A particularly important instance where concentration phenomena occur concerns the ∗ lack of compactness of the injection of W 1,q (U ) into Lq (U ) for 1 ≤ q < n. As in P.L. Lions [88] we single out for attention the case U = Rn , for which the following remarkably simple characterization of noncompactness is available. For simplicity we consider only q = 2. Theorem 1.4.2 Assume that n ≥ 3, fk → f

L2loc (Rn ),

strongly in

Dfk * Df

in

L2 (Rn ; Rn )

Suppose further that ∗

|Dfk |2 * µ in M(Rn ),

|fk |2 → ν

M(Rn )

in

(i) Then there exists an at most countable index set J, distinct points {xj }j∈J ⊂ Rn , and nonnegative weights {µj , νj }j∈J such that (1.14)

∗

ν = |f |2 +

X

µ ≥ |Df |2 +

ν j δ xj ,

j∈J

X

µj δ xj .

j∈J

(ii) Furthermore ∗

2∗ /2

νj ≤ C22 µj

(1.15)

(j ∈ J)

∗

(iii) If f ≡ 0 and ν(Rn )1/2 ≥ C2 µ(Rn )1/2 , then ν is concentrated at a single point. (Here C2 is the optimal constant for the Gagliardo-Nirenberg-Sobolev inequality (1.4).) Proof. 1. Assume first that f ≡ 0. Choosing φ ∈ C0∞ (Rn ) we infer from (1.4) that Z

2∗

|φfk | dx Rn

1/2∗

≤ C2

Z

2

|D(φfk )| dx Rn

1/2

Since fk → f ≡ 0 strongly in L2loc , we deduce (1.16)

Z

2∗

|φ| dν Rn

1/2∗

≤ C2

Z

2

|φ| dµ Rn

1/2

16

CHAPTER 1. WEAK CONVERGENCE

By approximation, therefore, ∗

ν(E)1/2 ≤ C2 µ(E)1/2

(1.17)

(E ⊂ Rn , E

Borel).

Now since µ is a finite measure, the set D ≡ {x ∈ Rn |µ({x}) > 0} is at most countable. We can therefore write D = {xj }j∈J , µj ≡ µ({xj })(j ∈ J) so that µ≥

X

µj δ xj .

j∈J

Because (1.17) implies ν  µ, we have Z (1.18) ν(E) = Dµ νdµ

(E

Borel),

E

where ν(B(x, r)) , r→0 µ(B(x, r))

(1.19)

Dµ ν(x) ≡ lim

this limit existing for µ-a.e. x ∈ Rn . (This is a consequence of the theory of symmetric derivatives of Radon measures; see Federer [56, pp. 152-169].) From (1.17), however, ν(B(x, r)) ∗ ≤ C22 µ(B(x, r))2/(n−2) , µ(B(x, r))

(1.20)

provided µ(B(x, r)) 6= 0. We thus infer (1.21)

µ-a.e. on Rn − D.

Dµ ν = 0

Now define νj ≡ Dµ ν(xj )µj . Then (1.18)-(1.21) imply assertions (i) and (ii) (for f ≡ 0). ∗ 2. Next suppose the hypothesis of assertion (iii). Then (1.17) gives ν(Rn )1/2 = C2 µ(Rn )1/2 . Since (1.16) forces Z

2∗

|φ| dν Rn

1/2∗

n 1/n

≤ C2 µ(R )

Z

2∗

|φ| dµ Rn

1/2∗

∗

we deduce ν = C22 µ(Rn )2/(n−2) µ. Consequently (1.16) reads Z

2∗

|φ| dν Rn

1/2∗

n

1/n

(ν(R ))

≤

Z

2

|φ| dν Rn

1/2

17

1.5. MEASURES OF OSCILLATION ∗

and so ν(E)1/2 ν(Rn )1/n ≤ ν(E)1/2 for each Borel set E. This is an impossibility if ν is concentrated at more than one point. 3. Now assume f 6≡ 0 and write gk ≡ fk − f . The calculations in steps 1 and 2 above apply to {gk }∞ k=1 . Moreover |Dgk |2 = |Dfk |2 − 2Dfk · Df + |Df |2 * µ − |Df |2 ∗

in M(Rn )

∗

and |gk |2 * ν − |f |2 in M(Rn ), according to Theorem 1.4.1.  In Chapter 4, §1. 1 we will invoke this precise characterization in studying critical growth nonlinearities in the calculus of variations.

1.5

Measures of Oscillation

1.5.1

Generalities

Continuing our discussion from the beginning of §4 about the possible ways strong convergence breaks down, we turn our attention next to the problem of oscillations. Recall from the earlier exposition that we expect such difficulties to arise when we have weak convergence, but now do not know the convergence is almost everywhere as well. On the other hand, let us now suppose that we have such good estimates that concentration problems per se will not occur. The technical difficulties at hand therefore concern the possibility of extremely wild (but, say, bounded) oscillations which may be present in our weakly convergent sequence. The plan, as in §4, will be to introduce certain measures, which appropriately encode the persistent, limiting structure of such oscillations. For future applications it will be necessary to discuss vector-valued mappings from U into Rm (m ≥ 1).

1.5.2

Slicing measures

In this section let µ be a finite, nonnegative Radon measure on Rn+m . We denote by σ the canonical projection of µ onto Rn ; that is, σ(E) ≡ µ(E × Rm ) for each Borel subset E ⊂ Rn . Theorem 1.5.1 For σ-a.e. point x ∈ Rn there exists a Radon probability measure νx on Rm , such that Z (i) the mapping x 7−→ f (x, y)dνx (y) is σ-measurable and Z

Rm

(ii)

f (x, y)dµ(x, y) = Rn+m

Z

Rn

Z



f (x, y)dνx(y) dσ(x) Rm

for each bounded, continuous function f .

18

CHAPTER 1. WEAK CONVERGENCE

m Proof. 1. Let {fk }∞ k=1 be a countable dense subset of C0 (R ). Define then the signed Radon measures Z

γ k (E) ≡

fk (y)dµ(x, y)

(k = 1, . . .)

E×Rm

for Borel sets E ⊂ Rn . Clearly γ k  σ. Consequently, for σ-a.e. point x ∈ Rn the limits γ k (B(x, r)) r→0 σ(B(x, r))

Dσ γ k (x) ≡ lim

(1.22)

(k = 1, . . .)

exist, the mappings x 7−→ Dσ γ k (x) (k = 1, . . .) are bounded, σ-measurable, and Z Z k (1.23) fk (y)dµ(x, y) = γ (E) = Dσ γ k (x)dσ(x) (k = 1, . . .) E×Rm

E

for each Borel set E ⊂ Rn . (We are again invoking the theory of symmetric derivatives from Federer [56, pp. 152-169].) ∞ 2. Fix any function f ∈ C0 (Rm ), and choose any subsequence {fkj }∞ j=1 ⊂ {fk }k=1 so that fkj → f uniformly on Rm . Then for each point x ∈ Rn for which (1.22) holds, the limit Γx (f ) ≡ lim Dσ γ kj (x) exists (and does not depend upon the choice of subsequence j→∞

approximating f ). As the mapping f 7−→ Γx (f ) is a bounded linear functional on C0 (Rm ), there exists a Radon measure νx on Rm satisfying Z (1.24) Γx (f ) = f (y)dνx (y) (f ∈ C0 (Rm )). Rm

Additionally x 7−→ Γx (f ) is bounded, σ−measurable, and from (1.23) we further deduce Z

(1.25)

f (y)dµ(x, y) = E×Rm

Z Z E



f (y)dνx (y) dσ(x) Rm

for f , E as above. 3. Using approximations we see from (1.25) that Z

g(x)f (y)dµ(x, y) = Rn+m

Z

g(x) Rn

Z



f (y)dνx(y) dσ(x) Rm

for bounded, continuous g, f . Setting f ≡ 1, we deduce νx (Rm ) = 1 σ-a.e. As any bounded continuous function f on Rn+m can be locally uniformly approximated by finite sums of N X the form g i (x)f i (y) (for bounded, continuous g i , f i (1 ≤ i ≤ N )), assertions (i) and i=1

(ii) follow.



19

1.5. MEASURES OF OSCILLATION

1.5.3

Young measures

We next utilize Theorem 1.5.1 to provide us with a concise measure-theoretic characterization of the incompatibility of weak convergence and nonlinear composition. ∞ m Theorem 1.5.2 Assume that the sequence {fk }∞ k=1 is bounded in L (U ; R ). Then there ∞ ∞ exists a subsequence {fkj }j=1 ⊂ {fk }k=1 and for a.e.x ∈ U a Borel probability measure νx on Rm such that for each F ∈ C(Rm ) we have ∗

in L∞ (U ),

F (fkj ) * F

(1.26) where (1.27)

F (x) ≡

Z

F (y)dνx(y)

(a.e. x ∈ U ).

Rm

Definition 1.5.1 We call {νx }x∈U the family of Young measures associated with the subsequence {fkj }∞ j=1 . Proof. 1. Define µk (E) ≡

Z

χE (x, fk (x))dx (k = 1, . . .) for each Borel set E ⊂ Rn × U

∞ Rm . Since sup µk (U × Rm ) = Ln (U ) < ∞, there exist a subsequence {µkj }∞ j=1 ⊂ {µk }k=1 k

and a nonnegative measure µ with µkj * µ in M(U × Rn ). 2. We next claim that the projection of µ onto Rn is σ = Ln |U (= n-dimensional Lebesgue measure restricted to U ). For this, note that if V ⊂ U is open, then Theorem 1.1.3 implies σ(V ) = µ(V × Rm ) ≤ lim inf µkj (V × Rm ) = Ln (V ). j→∞

Thus σ ≤ Ln |U . On the other hand, let K ⊂ U be compact. Since {fkj }∞ j=1 is bounded in ∞ m L (U ; R ), there exists R > 0 such that spt µ, spt µkj ⊂ U × B(0, R). Consequently, σ(K) = µ(K × Rm ) = µ(K × B(0, R)) ≥ lim sup µkj (K × B(0, R)) = Ln (K) j→∞

and so σ ≥ Ln |U . 3. From Theorem 1.5.1, we deduce that there exists for a.e.x ∈ U a Borel probability measure νx such that  Z Z Z f (x, y)dµ(x, y) = f (x, y)dνx(y) dx Rn+m

U

Rm

20

CHAPTER 1. WEAK CONVERGENCE

for each bounded continuous function f . Let f (x, y) = ζ(x)F (y), where ζ ∈ C0 (U ), F ∈ C0 (Rm ). Then Z Z lim ζ(x)F (fkj (x))dx = lim f (x, y)dµkj (x, y) j→∞ U j→∞ Rn+m Z = f (x, y)dµ(x, y) Rn+m Z  Z F (y)dνx (y) dx = ζ(x) U Rm Z = ζ(x)F (x)dx. U

A similar calculation obtains if only F ∈ C(Rm ), since we can as necessary redefine F for large |y| to have compact support.  By making specific choices for F we can read off certain information regarding the structure of the Young measures. For instance,  if there exists a closed set C ⊂ Rm such that (1.28) fk ∈ C a.e. (k = 1, . . .) , then spt νx ⊂ C(a.e. x ∈ U ). To verify this we need only consider functions F vanishing on C. Note also

(1.29)

(

if νx is a unit point mass for a.e. x ∈ U, then (passing if necessary to a further subsequence) we have fkj → f a.e. in U.

Indeed since f (x) = thermore, kfkj k2L2 (U )

Z

ydνx (y), necessarily νx = δf (x) if νx is a unit point mass. FurRm

=

Z

2

|fkj | (x)dx → U

Z Z U

2

Rm



|y| dνx (y) dx = kf k2L2 (U ) ;

so that fkj → f in L2 (U ; Rm ). In Chapter 5 we will use PDE theory to deduce in certain cases much more information about Young measures.

Chapter 2 Convexity Convexity is the simplest structural condition on nonlinearities which is at least partially compatible with weak convergence. This very short chapter reviews the natural role of convexity in the calculus of variations, and will later serve to motivate the exposition in Chapter 3.

2.1

The Calculus of Variations

The Classical calculus of variations provides one of the historically earliest instances where weak convergence methods were successfully applied to nonlinear problems. Let us recall the model problem of finding a minimizer for the functional Z (2.1) I[w] ≡ F (Dw)dx, U

among all candidate functions w lying in some nonempty class A of admissible functions, which for definiteness we take to be A ≡ {w ∈ W 1,q (U ) | w = g

on ∂U }

for 1 < q < ∞, the boundary values assumed in the trace sense for some fixed function g : ∂U → R. we are given the smooth function F : Rn → R, and try to identify conditions ensuring the existence of a minimizer. For this, let {uk }∞ k=1 ⊂ A be a minimizing sequence, (2.2)

I[uk ] → inf I[w], w∈A

and suppose the infimum is finite. Assuming further the coercivity condition F (p) ≥ α|p|q − β (p ∈ Rn ) for constants α > 0, β ≥ 0, we deduce that {uk }∞ k=1 is bounded in ∞ W 1,q (U ); whence there exits a subsequence {ukj }∞ ⊂ {u } and u ∈ A with k k=1 j=1 (2.3)

ukj * u in W 1,q (U ). 21

22

CHAPTER 2. CONVEXITY

To show u is a minimizer, we must prove (2.3) implies (2.4)

I[u] ≤ lim inf I[ukj ], j→∞

that is, I[·] is lower semicontinuous with respect to weak convergence in W 1,q (U ). What structural assumption on the nonlinearity F allows us to deduce (2.4) from (2.3)? For a clue, we temporarily assume that u ∈ A is a smooth minimizer, and then set Z i(t) ≡ I[u + tv] = F (Du + tDv)dx, U

where v is a Lipschitz function with compact support in U and t ∈ R. Since i attains its minimum at 0, Z ∂2F 00 (Du)vxi vxj dx. (2.5) 0 ≤ i (0) = U ∂pi ∂pj Now set (2.6)

v(x) ≡ εζ(x)ρ



x·ξ ε



for ζ ∈ C0∞ (U ), ξ ∈ Rn , and ρ the 2-periodic sawtooth function equaling x on [0,1] and 2 − x on [1,2]. Inserting (2.6) into (2.5) and sending ε to zero, we deduce (2.7)

∂2F D(u(x))ξi ξj ≥ 0 (x ∈ U, ξ ∈ Rn ). ∂pi ∂pj

This necessary inequality strongly suggests it is natural to assume F is convex, that is, (2.8)

2.2

ξ T D 2 F (p)ξ ≥ 0 (p, ξ ∈ Rn ).

Weak lower Semicontinuity

We verify now that convexity is indeed the proper structural hypothesis for our nonlinearity: Theorem 2.2.1 The functional I[·] is lower semicontinous with respect to weak convergence in W 1,q (U ) if and only if F is convex. Proof 1. First we prove necessity. For this fix a vector p ∈ Rn and suppose for simplicity U = Q, the open unit cube in Rn . Fix any v ∈ C0∞ (Q). For each nonnegative 2kn integer k we subdivide Q into disjoint subcubes {Qi }i=1 of side length 1/2k . Define uk (x) ≡

1 v(2k (x − xl )) + p · x (x ∈ Ql ), 2k

23

2.2. WEAK LOWER SEMICONTINUITY xl denoting the center of Ql , and u(x) = p · x (x ∈ Q).

Then uk * u in W 1,q (U ). Assuming I[·] to be weakly lower semicontinuous, we deduce Z n L (Q)F (p) = I[u] ≤ lim inf I[uk ] = F (p + Dv)dx. k→∞

Q

Thus, in particular, the function u(x) = p · x is a minimizer subject to its own boundary values in ∂Q. Consequently, inequality (2.7) asserts that F is convex. 2. Now we establish sufficiency. So suppose uk * u in W 1,q (U ) and assume for the moment F is the maximum of finitely many affine functions: F (p) = max (bj · p + cj ) (p ∈ Rn ).

(2.9)

1≤j≤m

Write Ej ≡ {x ∈ U |F (Du(x)) = bj · Du(x) + cj }. Then U = ∪m j=1 Ej , and we may as m well also assume the sets {Ej }j=1 to be disjoint. Then since weak convergence means the convergence of averages, we have I[u] =

Z

F (Du)dx = U

m Z X j=1

= lim

k→∞

bj · Du + cj dx Ej

m Z X

j=1 Ej m Z X

bj · Duk + cj dx

≤ lim inf k→∞

j=1

F (Duk )dx Ej

= lim inf I[uk ], k→∞

the inequality a consequence of our max representation formula (2.9). This establishes the weak lower semicontinuity should F be the maximum of finitely many planes. In the general case, we can write F (p) = lim F m (p) for F m (p) ≡ max (bj · p + cj ), and apply m→∞

j=1,...,m

the Monotone Convergence Theorem.  The proof of sufficiency clearly illustrates why convex nonlinearities are partially compatible with weak convergence: an affine function is weakly continuous and a convex function is the supremum of affine functions. Recall now that under appropriate growth assumptions on F , we have Z ∂F 0 0 = i (0) = (Du)vxj dx, U ∂pj

24

CHAPTER 2. CONVEXITY

and so our minimizer u is a weak solution of the Euler-Lagrange equation: ( −div(DF (Du)) = 0 in U, u = g on ∂U. This is our first example of a nonlinear PDE solved by weak convergence methods.

2.3

Convergence of Energies and Strong Convergence

For later reference we record here that if q = 2, (2.10)

|F (p)| ≤ C(1 + |p|2 ) (p ∈ Rn ),

and F is uniformly strictly convex, that is, (2.11)

ξ T D 2 F (p)ξ ≥ γ|ξ|2

(γ > 0, p, ξ ∈ Rn ),

then the full minimizing sequence converges strongly to u in W 1,2 (U ). In fact the following assertion holds: Theorem 2.3.1 Assume F satisfies (2.10),(2.11). Suppose also uk * u in W 1,2 (U ) and (2.12)

I[uk ] → I[u].

Then uk → u

strongly in

W 1,2 (U ).

We informally interpret this conclusion to mean that the convergence of energies (2.12) improves weak to strong convergence. The uniform convexity of F somehow damps out wild oscillations in {Duk }∞ k=1 . Proof. From the strict convexity (2.11) we have γ F (q) ≥ F (p) + DF (p) · (q − p) + |q − p|2 2 for all p, q ∈ Rn . Set p = Du, q = Duk and integrate over U to find Z Z γ (2.13) I[uk ] ≥ I[u] + DF (Du) · (Duk − Du)dx + |Duk − Du|2 dx. 2 U U Now the convexity of F and (2.10) imply |DF (p)| ≤ C(1+|p|); thus DF (Du) ∈ L2 (U ; Rn ). Since Duk * Du in L2 (U ; Rn ), estimate (2.13) and hypothesis (2.12) imply Duk → Du strongly in L2 (U ; Rn ). 

2.3. CONVERGENCE OF ENERGIES AND STRONG CONVERGENCE

25

Other results in the spirit of Theorem 2.2 may be found for instance in Visintin [124], See Brezis [25], Ekenland-Temam [48] or Rockafellar [101] for much more on convexity and PDE’s. Having just now finished reviewing the elegant role of convexity in handling weak convergence problems in the calculus of variations, I should emphasize that a major goal of much current PDE reserch is in fact to remove convexity as a key assumption. We will see instances of this in Chapter 3 (quasiconvexity), Chapter 5, § 1.3 (monotonicity), Chapter 5, § 3 (quasilinear elliptic systems), and Chapter 6, § 1.1 (nonconvex, fully nonlinear elliptic equations).

Chapter 3 Quasiconvexity In this chapter we broaden our previous analysis to cover vector-valued calculus of variations problems.

3.1 3.1.1

Definitions Rank-one convexity.

We consider the functional (3.1)

I[w] =

Z

F (Dw)dx, U

defined now for candidate functions w lying in the admissible class A ≡ {w ∈ W 1,q (U ; Rm ) | w = g

on ∂U }

for 1 < q < ∞, g : ∂U → Rm given. In this case we write w = (w 1 , . . . , w m ),   1 1 wN · · · w N n 1   .. Dw =   . m m w N1 · · · w Nn

and assume that F : M m×n → R is a given smooth function, M m×n denoting the space of real m × n matrices. Let us suppose also F (P ) ≥ α|P |q − β (P ∈ M m×n ) for constants α > 0, β ≥ 0 so that the existence problem for a minimizer of I[·] in the class A turns once more upon the weak lower semicontinuity of I[·]. What condition on the nonlinear structure of F will yield this weak lower semicontinuity? To answer, we mimic as much as possible the analysis from Chapter 2 for the case 26

27

3.1. DEFINITIONS

m = 1. So observe that if u is a smooth minimizer and v = (v 1 , . . . , v m ) is a Lipschitz function with compact support in U , then the second variation of Z i(t) ≡ I[u + tv] = F (Du + tDv)dx U

at 0 is (3.2)

00

0 ≤ i (0) =

Z

U

∂2F (Du)vxki vxl j dx. ∂pki ∂plj

Now fix vectors ξ ∈ Rn , and η ∈ Rm choose ζ ∈ C0∞ (U ), and select the sawtooth function ρ as in Chapter 2. We the insert   x·ξ η v(x) ≡ εζ(x)ρ ε into (3.2), and send ε → 0 to deduce ∂2F (Du(x))ηk ηl ξi ξj ≥ 0 ∂pki ∂plj

(3.3)

for each x ∈ U, η ∈ Rm , ξ ∈ Rn . This necessary inequality suggests that concerning F we should assume the Hadamard-Legedre inequality (3.4)

(η ⊗ ξ)T D 2 F (P )(η ⊗ ξ) ≥ 0 (P ∈ M m×n , η ∈ Rm , ξ ∈ Rn ),

where η ⊗ ξ denotes the rank-one m × n matrix whose (k, i)th entry is ηk ξi (1 ≤ k ≤ m, 1 ≤ i ≤ n). A function satisfying (3.4) is called rank-one convex. This means that for each fixed P, η, ξ as above the scalar function (3.5)

f (t) ≡ F (P + t(η ⊗ ξ)) (t ∈ R)

is convex. We will see later that this definitely does not imply F is itself convex.

3.1.2

Quasiconvexity.

The foregoing considerations suggest that rank-one convexity is the proper hypothesis on the nonlinearity F , but it is unknown whether this is truly so. Let us instead, following Morrey [93, 94] and analogous computations in Chapter 2, approach the weak lower semicontinuity issue head on, as follows. Fix any matrix P ∈ M m×n and suppose for simplicity U = Q. We select any v ∈ C0∞ (Q; Rm )and, as in 2kn the previous chapter, for each nonnegative integer k subdivide Q into subcubes {Ql }l=1 .We define 1 uk (x) = k v(2k (x − xl )) + P x (x ∈ Ql ), 2

28

CHAPTER 3. QUASICONVEXITY u(x) = P x.

Then uk * u in W 1,q (U ; Rm ). Thus if I[u] ≤ lim inf I[uk ], we have k→∞

n

L (Q)F (P ) ≤

Z

F (P + Dv)dx. Q

We transform this necessary inequality into a definition: Definition 3.1.1 A function F : M m×n → R is called quasiconvex provided Z Z (3.6) F (P )dx ≤ F (P + Dv)dx Q

Q

for each P ∈ M m×n and v ∈ C0∞ (Q; Rm ). Observe that inequality (3.6) says that the plane u(x) = P x is a minimizer on Q subject to its own boundary values. As such, the necessary inequality (3.3) asserts that a quasiconvex function is automatically rank-one convex. It remains an open question as to whether the converse implication obtains, although Ball [13] has recently constructed some examples which cast serious doubts. It is easy to verify however that ( if F is quadratic, then F is quasiconvex (3.7) if and only if F is rank-one convex. We use Fourier transform to check this (Morrey [94]). Note further that Jensen’s inequality implies each convex function is quasiconvex.

3.2

Weak Lower Semicontinuity

Let us henceforth suppose F satisfies the growth condition (3.8)

0 ≤ F (P ) ≤ C(1 + |P |q ) (P ∈ M m×n )

for some constant C. Lemma 3.2.1 Assume that F is rank-one convex and verifies (3.8). Then (3.9) for some constant C.

|DF (P )| ≤ C(1 + |P |q−1 ) (P ∈ M m×n )

29

3.2. WEAK LOWER SEMICONTINUITY

Proof. Fix P ∈ M m×n , 1 ≤ k ≤ m, 1 ≤ i ≤ n. Choose ηk = 1, ηl = 0 (l 6= k), ξi = 1, ξj = 0 (j 6= i). Define f by (3.5). Then f is convex; whence we have the estimate |f 0 (0)| ≤

C max |f | (r > 0). r B(0,r)

Additionally, max |f | ≤ C(1 + |P |q + r q ),

B(0,r)

according to (3.8). Set r = |P | + 1.  Now we check that quasiconvexity is the proper assumption for the calculus of variations. Theorem 3.2.1 Suppose F satisfies the growth condition (3.8). Then the functional I[·] is lower semicontinuous with respect to weak convergence in W 1,q (U ; Rm ) if and only if F is quasiconvex. Proof. 1. We have already demonstrated the necessity. 2. To prove sufficiency, let us suppose now uk * u in W 1,q (U ; Rm ). Then in particular (3.10)

sup kDuk kLq (U ;M m×n ) < ∞ k

and (3.11)

uk → u strongly in Lq (U ; Rm ).

∞ Pass to a subsequence {ukj }∞ j=1 ⊂ {uk }k=1 so that

(3.12)

lim I[ukj ] = lim inf I[uk ].

j→∞

k→∞

Define then the measures µkj = |Dukj |q + |Du|q + 1 (j = 1, . . .). In view of (3.10), we deduce from Theorem 1.4 in Chapter 1 that, if we pass to a further subsequence as necessary, there exists a measure µ ∈ M(U ) with µkj * µ in M(U ). In particular µ(U ) < ∞. Since therefore µ(U ∩P ) > 0 for at most countably many hyperplanes P ⊂ Rn which are translates of a given coordinate plane, we can assume with no loss of generality that µ(U ∩ P ) = 0 for each hyperplane of the form P = {x|xi = q}, q dyadic, i = 1, . . . , n. Let Qi denote the set of open dyadic cubes in Rn , with edge length 2−i . Then (3.13)

µ(∂Q) = 0 for each Q ∈ Qi

(i = 1, . . .).

(We will see momentarily the utility of condition (3.13), which excludes the concentration of the measures {µkj }∞ j=1 onto the edges of the latticework of cubes employed below. This is our first, simple example using measure theory to control concentrations.)

30

CHAPTER 3. QUASICONVEXITY

3.Now fix ε > 0 and choose V ⊂⊂ U so that Z (3.14) F (Du)dx < ε. U −V

Denote by (Du)i the piecewise constant function which on the cube Ql ∈ Qi takes the value (Du)Ql (-the average of Du over Qi ). Then (Du)i → Du strongly in Lq (U ; M m×n ), and consequently F ((Du)i ) → F (Du) in L1 (U ). Hereafter fix an integer i so large that 1/2i < dist(V, ∂U ), (3.15)

kDu − (Du)i kLq (U ;M m×n ) < ε.

and (3.16)

kF (Du) − F ((Du)i )kL1 (U ) < ε.

Now let {Ql }m l=1 denote the cubes in Qi which intersect V . Choose 0 < σ < 1 and denote ˆ l the open cube concentric with and parallel to Ql , but with edge length σ2−i . Select by Q also smooth cutoff functions {ζl }m l=1 satisfying  ˆl,  0 ≤ ζl ≤ 1, ζl ≡ 1 on Q  C2i  .  ζl ≡ 0 on U − Ql , |Dζl | ≤ (1 − σ) Write vlj ≡ ζl (ukj − u), Al ≡ (Du)Ql (1 ≤ l ≤ m, j = 1, . . .). 4. Then Z I[ukj ] = F (Dukj )dx U

≥

m Z X l=1

= = =

m Z X

E1 ≡ E2 ≡

m Z X

l=1 Ql m Z X l=1

Ql

F (Du + D(ukj − u))dx

l=1 Ql m Z X

l=1 Ql m Z X l=1

for

F (Dukj )dx Ql

Ql

F (Du + Dvlj )dx + E1 F (Al + Dvlj )dx + E1 + E2 ,

F (Du + D(ukj − u)) − F (Du + Dvlj )dx F (Du + Dvlj ) − F (Al + Dvlj )dx.

3.3. CONVERGENCE OF ENERGIES AND STRONG CONVERGENCE

31

Continuing, we use quasiconvexity to compute further that m Z X I[ukj ] ≥ F (Al )dx + E1 + E2 (3.17)

≥

Zl=1

Ql

F ((Du)l )dx + E1 + E2

V

= I[u] + E1 + E2 + E3 , for E3 ≡

Z

F ((Du)l )dx − V

Z

F (Du)dx. U

5.Invoking Lemma 3.1 we estimate m Z X |E1 | ≤ C 1 + |Du|q + |Dukj |q + |Dζl |q |ukj − u|q dx. ˆl Q l −Q

l=1

Consequently, (3.11) implies lim sup |E1 | ≤ Cµ j→∞

m [

!

ˆl) . (Ql − Q

l=1

Additionally, we infer from Lemma 3.1 and (3.14)-(3.16) that lim sup(|E2 | + |E3 |) = O(ε). j→∞

Returning to (3.17), we send σ → 1 to deduce from (3.12) and (3.13) that I[u] ≤ lim inf I[uk ], k→∞

as required.  It is a routine matter now to check that our minimizer u = (u1 , . . . , um ) is a weak solution of the Euler-Lagrange system ( −div(DF (Du)) = 0 in U, (3.18) u = g on ∂U.

3.3

Convergence of Energies and Strong Convergence

Let us now set q = 2 and define F to be uniformly strictly quasiconvex provided Z Z 2 (3.19) F (P ) + γ|Dv| dx ≤ F (P + Dv)dx Q

Q

32

CHAPTER 3. QUASICONVEXITY

for some constant γ > 0 and all P ∈ M m×n , v ∈ C0∞ (Q; Rm ). Then G(P ) ≡ F (P ) − γ|P |2 is quasiconvex. Now suppose additionally that G satisfies the hypothesis of Theorem 3.1 for q = 2. Assume also (3.20)

uk * u in W 1,2 (U ; Rm ).

and furthermore (3.21)

I[uk ] → I[u].

We assert that in fact (3.22)

uk → u strongly in W 1,2 (U ; Rm )

In other words, convergence of the energies improves weak to strong convergence. We have seen this phenomenon before for uniformly convex nonlinearities in Chapter 2, §3. The point now is that uniformly quasiconvex nonlinearities also have the same effect of damping out oscillations in {Duk }∞ k=1 . The proof that (3.20) and (3.21) imply (3.22) is trivial under the stated assumptions. Indeed, since Z J[w] ≡ G(Dw)dx U

is lower semicontinuous with respect to weak convergence in W 1,2 , we have Z Z Z 2 F (Du) − γ|Du| dx = G(Du)dx ≤ lim inf G(Duk )dx k→∞ U U Z U Z = lim F (Duk )dx − γ lim sup |Duk |2 dx. k→∞

k→∞

U

U

But now (3.21) implies lim sup k→∞

Z

2

|Duk | dx ≤ U

Z

|Du|2 dx, U

and (3.22) follows. Now it is not generally the case that G must satisfy the hypothesis of our weak lower semicontinuity Theorem 3.2.1, even if F does. But the basic phenomenon that convergence of energies implies strong convergence still obtains: Theorem 3.3.1 Assume that F is uniformly strictly quasiconvex and satisfies the growth condition (3.8) for q = 2. Then the weak convergence (3.20) and convergence of energies (3.21) imply (3.23)

1,2 uk → u strongly in Wloc (U ; Rm ).

3.3. CONVERGENCE OF ENERGIES AND STRONG CONVERGENCE

33

Outline of proof. The proof is almost precisely like that of Theorem 3.2.1, except that we exploit inequality (3.19) to get extra control on the integral of |Du − Duk |2 . See [54] for details.  As an application we have Theorem 3.3.2 Assume F satisfies (3.8) (for q = 2), (3.19). Suppose furthermore that uk is a minimizer of I[·] for k = 1, . . . , and uk * u in W 1,2 (U ; Rm ). Then 1,2 uk → u strongly in Wloc (U ; Rm ),

and u is a minimizer of I[·]. Proof. 1.Let V ⊂⊂ U be chosen so that (3.24)

µ(∂V ) = 0,

where µ is that measure arising in the proof of Theorem 3.2.1, as the weak limit of the measures {µkj }∞ j=1 . 2. In light of Theorem 3.3.1 it suffices to verify convergence of energies (3.21), where Z we now write I[w] =

F (Dw)dx. Utilizing Theorem 3.2.1, we first note

V

I[u] ≤ lim inf I[uk ]. k→∞

Choose W ⊂⊂ V and select a smooth cutoff function ζ satisfying 0 ≤ ζ ≤ 1, ζ ≡ 1 on W , ζ ≡ 0 on U − V . Then writing wkj ≡ ζu + (1 − ζ)ukj (j = 1, . . .) we conclude from the minimality of ukj that I[ukj ] ≤ I[wkj ] = I[u] + E1 , for E1 ≡

Z

F (dwkj ) − F (Du)dx. V −W

Now lim sup |E1 | ≤ Cµ(V − W ). Thus (3.24) implies lim sup I[uk ] ≤ I[u]. This inequality j→∞

j→∞

is valid for each subsequence {ukj }∞ j=1 ; whence lim sup I[uk ] ≤ I[u].  k→∞

Notice in particular that we have here a sequence of weak solutions {uk }∞ k=1 of the quasilinear Euler-Lagrange system (3.25)

−div(DF (Duk )) = 0 in U

converging weakly in W 1,2 to a function u, which is itself a solution: (3.26)

−div(DF (Du)) = 0 in U

We will return to this phenomenon in Chapter 5, §3.

34

3.4

CHAPTER 3. QUASICONVEXITY

Partial Regularity of Minimizers

As an application of some of the viewpoints set forth above, we consider in this section the problem of the smoothness of minimizers of our energy functional I[·] for a strictly quasiconvex nonlinearity F . The idea is that if the graph of a minimizer u is sufficiently flat in a measure-theoretic sense for a neighborhood of some point x ∈ U , then in fact u is smooth near x. We must set up some technical apparatus to execute this idea. First, assume u ∈ 1,2 W (U ; Rm ) is a minimizer of I[·]. Then for each ball B(x, r) ⊂ U , we introduce the excess Z E(x, r) ≡ − |Du − (Du)x,r |2 dy B(x,r)

to measure the average mean square deviation of Du from its average value (Du) x,r over the ball B(x, r). (The slash through the integral sign denotes average.) The idea is to show that if E(x, r) is sufficiently small for a given ball, then E(x, s) in fact goes to zero like some power of s for 0 < s < r. So let us assume that F is uniformly quasiconvex and satisfies the growth restrictions (3.8) for q = 2. Lemma 3.4.1 For each L > 0 there exist constants ε, τ ∈ (0, 1) such that |(Du)x,r | < L 1 and E(x, r) < ε imply E(x, τ r) ≤ E(x, r). 2 Outline of proof. 1. Were the lemma false there would exist for each 0 < τ < 1 balls B(xk , rk ) ⊂ U such that |(Du)xk ,rk | ≤ L,

E(xk , rk ) ≡ λ2k → 0

but 1 E(xk , τ rk ) > λ2k . 2

(3.27)

We rescale to the unit ball B = U (0, 1) by setting vk (z) ≡

u(xk + rk z) − ak − rk Ak z r k λk

(z ∈ B)

1,2 where ak ≡ (u)xk ,rk , Ak ≡ (Du)xk ,rk . We easily check that {vk }∞ (B), k=1 is bounded in W and so, passing to a subsequence if necessary, we have

(3.28)

vk → v

in W 1,2 (B; Rm ),

Ak → A in M m×n,

35

3.4. PARTIAL REGULARITY OF MINIMIZERS

2. Now since u is a weak solution of the Euler-Lagrange system (3.26), we see that v k is a weak solution of the rescaled system  ∂F  ∂F (A + λ Dv ) − (A ) k k k k  ∂pl ∂plj d   j  = 0 in B (l = 1, . . . , m).  dzj  λk

From this it follows easily that v = (v 1 , . . . , v m ) is a weak solution of the linearized system ! d ∂2F (A)vzkl = 0 (l = 1, . . . , m). dzj ∂pkl ∂plj Now since G(P ) = F (P ) − γ|P |2 is quasiconvex, we see from the rank-one convexity N inequality (3.4) that (η ⊗ ξ)T D 2 F (A)(η ξ) ≥ γ|η|2 |ξ|2 for all η ∈ Rm ,ξ ∈ Rn . Consequently, v solves a constant coefficient, linear, uniformly elliptic system. Thus v is smooth, and in particular Z 1 − |Dv − (Dv)0,τ |2 dz ≤ 4 B(0,τ )

if we initially choose τ small enough. On the other hand, if we rescale the inequalities (3.27) we find Z 1 (k = 1, . . .). − |Dvk − (Dvk )0,τ |2 dz > 2 B(0,τ )

3. We therefore will have the desired contradiction, provided we can somehow improve the weak convergence (3.28) to strong convergence. But the previous section’s analysis suggests that we can exploit the quasiconvexity to this end. Indeed, we observe that vk is a minimizer of the functional Z k Ir [w] ≡ F k (Dw)dx, B(0,r)

for the rescaled energies F k (P ) ≡

F (Ak + λk P ) − F (Ak ) − λk DF (Ak )P λ2k

(P ∈ M m×n ).

But then almost exactly as in the proof of Theorem 3.3.2 we show that lim sup(Irk [vk ] − Irk [v]) ≤ 0 k→∞

for all but at most countably may radii 0 < r < 1. But this estimate in turn implies, owing 1,2 m to the strict quasiconvexity of F and thus {F k }∞ k=1 , that vk → v strongly in Wloc (B; R ). 

36

CHAPTER 3. QUASICONVEXITY We employ Lemma 3.4.1 as follows. In view of Lebesgue’s Differentiation Theorem,lim E(x, r) = r→0

0 for a.e. x ∈ U . Thus Lemma 3.4.1 forces E(x, r) = O(r α ) for some 0 < α < 1 for all x ∈ V , V denoting an open set of full measure in U . This decay estimate then implies Du ∈ C 0,α/2 (V ; M m×n ). Finally, standard elliptic estimates show u has derivatives of all orders in V . See [51, 53] for details. Theorem 3.4.1 Let u ∈ W 1,2 (U ; Rm ) be a minimizer of I[·], where F satisfies (3.8) (for q=2) and (3.19). Then there is an open subset V ⊂ U such that Ln (U − V ) = 0 and u ∈ C ∞ (V ). It is interesting to consider this assertion in light of our guiding principles set forth in the Introduction. I argued there that the lack of good uniform smoothness estimates for solutions of nonlinear PDE’s forces us to investigate weak convergence techniques. Now, on the other hand, we see that enough understanding of weak versus strong convergence can in fact establish (partial) smoothness.

3.5 3.5.1

Examples Weak continuity of determinants.

Following Morrey [94], we introduce some nontrivial examples of quasiconvex integrands by examining first the weak continuity of certain nonlinear functions. Lemma 3.5.1 Assume n < q < ∞ and uk * u in W 1,q (U ; Rn ). Then det Duk * det Du in

Lq/n (U ).

Proof. If A is an n × n matrix, we let cofA denote the cofactor matrix of A (whose (i, j)th entry is (−1)i+j times the (i, j)th minor). Then det Dv = vxi j (cofDv)ij

(1 ≤ i ≤ m)

for any smooth function v. Additionally, direct calculation reveals the functional identity (cofDv)ij,xj = 0 (1 ≤ i ≤ m). Consequently, det Dv = (v i (cofDv)ij )xj

(1 ≤ i ≤ m).

After an approximation by smooth functions we find therefore that Z Z φ det Duk dx = − φxj uik (cofDuk )ij dx. U

U

37

3.5. EXAMPLES Now since we may assume uik → ui uniformly in U , we have for each φ ∈ C0∞ (U ) that Z Z Z i φ det Duk dx → − φxj u (cofDuk )ij dx = φ det Dudx, U

U

U

provided cofDuk * cofDu in Lq/n . However each entry of the matrix cofDuk is a multiple of the determinant of an (n−1)×(n−1) submatrix of Duk . We are therefore in a position to apply an induction argument to complete the proof.  Note carefully that this assertion is not at variance with our comments in Chapter 1, §2 concerning the incompatibility of nonlinearity and weak convergence. The point here is that {Duk }∞ k=1 is a sequence of gradients converging weakly. This extra bit of information is, remarkably, enough to establish the weak continuity of the special nonlinear function determinant. All this suggests the possibility of identifying more general circumstances in which extra structural information available for weakly convergent sequences can be useful. This prospect is a principle goal of the theory of compensated compactness, to be discussed in Chapter 5 (see especially Chapter 5, §2 and §5).

3.5.2

Polyconvexity.

We can now easily identify an interesting class of quasiconvex functions which are not convex. Definition 3.5.1 A function F : M m×n → R is called polyconvex if F (P ) is a convex function of the determinants of square submatrices of P . Theorem 3.5.1 A polyconvex function is quasiconvex. Proof. We write F (P ) = G(. . . , det Pˆ , . . .), G convex and Pˆ denoting a typical k × k submatrix of P (1 ≤ k ≤ min(m, n)). Define uk (k = 1, . . .) and u as in Chapter 3, §1.2. Then ∗ dk * c in L∞ (U ) det Du det Du according to Lemma 3.5.1. As G is convex, we deduce Z I[u] = F (Du)dx ≤ lim inf I[uk ], k→∞

Q

Thus

Z

F (P )dx ≤ Q

Z

F (P + Dv)dx Q

for all P ∈ M m×n , v ∈ C0∞ (Q; Rm ), as required.  We see for instance that F (P ) ≡ |P |q + β(det P ) (P ∈ M m×n )

38

CHAPTER 3. QUASICONVEXITY

is quasiconvex, provided β : R → R is convex. By approximation we can also include the case that β : R → [0, ∞] is convex, with β(s) = +∞ for s ≤ 0. This and related examples arise in nonlinear elasticity: See Ball [11] for a discussion of the physical meaning of quasiconvexity in this context.

Chapter 4 Concentrated Compactness Next we turn our attention to certain classes of nonlinear PDE’s for which we cannot in any obvious way improve weak to strong convergence for our approximations, but can at least demonstrate that the breakdown of strong convergence is somehow isolated within a small set. The operant idea will then be to analyze this concentration set where compactness fails, in the hopes of eventually proving that either the set is really empty after all or is otherwise somehow negligible. Our examples all entail critical growth type nonlinearities.

4.1 4.1.1

Variational Problems Minimizers for critical Sobolev nonlinearities.

Following P.L. Lions [88], we will start with certain variational problems involving critical growth nonlinearities which just fail to satisfy usual compactness criteria. So let us discuss the model problem of minimizing the functional Z (4.1) I[w] = |Dw|2 dx, Rn

n ≥ 3, over the admissible set

Note that

∗ A ≡ {w ∈ L2 (Rn ) kwkL2∗ (Rn ) = 1, Dw ∈ L2 (Rn ; Rn )}. I ≡ inf I[w] = C2−2 , w∈A

C2 denoting the optimal constant in the Gagliardo-Nirenberg-Sobolev inequality (1.4) for q = 2. 39

40

CHAPTER 4. CONCENTRATED COMPACTNESS

We inquire as to whether this infimum is obtained, and so choose a minimizing sequence {uk }∞ k=1 ∈ A, with (4.2)

I[uk ] → I. ∗

We may assume Duk * Du in L2 (Rn ; Rn ), uk * u in L2 (Rn ). As the integrand of I[·] is convex, we infer from Chapter 2 that I[u] ≤ lim inf I[uk ] = inf I[w], thus u is a

(4.3)

w∈A

k→∞

minimizer, provided u ∈ A. Since we have kukL2∗ (Rn ) ≤ 1,

the real question is whether or not kukL2∗ (Rn ) = 1. Now there are two possible failures of compactness here. First of all, it may very well be the case that that strict inequality obtains in (4.3) because some of the mass of approximations ”leaks out at infinity.” That is, it may be that the family of measures 2 {vk }∞ k=1 is not tight, νk = |uk | (k = 1, · · · ). A second and rather more troublesome n prospect is that even if {νk }∞ k=1 is tight, so that we may suppose νk * ν in M(R ) = 1andν(Rn ), then perhaps Z ∗

1 = ν(Rn ) 6=

|u|2 dx.

Rn

This possibility arises if ν has a singular part, which is to say, if the measures {νk }∞ k=1 concentrate in the limit some of their mass onto a set of Lebesgue measure zero. What is worse still is that each of these unpleasant possibilities can definitely happen if we are unfortunate in our choice of minimizing sequence. To see this, observe that if v ∈ A, y ∈ Rn , and s > o, the rescaled function v y,s (x) ≡ s−(n−2)/2 v(

satisfies

Z

y,s 2

|Dv | dx = Rn

Z

2

|Dv| dx, Rn

x−y ) (x ∈ Rn ) s

Z

y,s 2∗

|v | dx = Rn

Z

∗

|v|2 dx = 1. Rn

Consequently, given any minimizing sequence {uk }∞ k=1 , we could consider a new minimizing sequence {ukνk ,sk }∞ for which by appropriate choices of translations {yk }∞ k=1 k=1 and ∞ dilations {sk }k=1 we could effect either of the failures of compactness alluded to above. A far more interesting prospect is designing translations and dilations to exclude the failure of compactness: ∞ Theorem 4.1.1 Let {uk }∞ k=1 ⊂ A satisfy (4.2). Then there exist translations {y k }k=1 ⊂ yk ,sk n ∞ R and dilations {sk }k=1 ⊂ (0, ∞) such that the rescaled family {uk } ⊂ A is strongly ∗ precompact in L2 (Rn ). In particular, there exists a minimizer u ∈ A.

41

4.1. VARIATIONAL PROBLEMS

Given that a minimizer exists, a careful analysis using symmetrization and ODE theory shows that any minimizer is of the form (4.4)

uy,ε (x) =

Cε (ε + |x − y|2 )(n−2)/2

(x ∈ Rn )

for some ε > 0, y ∈ Rn , and an appropriate normalization constant Cε . (We could, by the way, more directly study the problem at hand by replacing each function uk by its spherical symmetrization about the origin. However, the concentration techniques expounded below are rather general and apply to many classes of problems for which symmetrization methods are unavailable: see Lions [87,88].) Outline of proof. 1. Define the Levy concentration functions Z ∗ Qk (t) ≡ sup |uk |2 dx (t > 0, k = 1, · · · ). y∈Rn

B(y,t)

y,s y,s y,t t lim Qk (t) = Then Qy,s k = Qk ( ), where Qk is the concentration function of uk . Since t→∞ s 1 y,s for all y ∈ Rn , k = 1, · · · , This 1 we can choose dilations {sk }∞ k=1 such that Qk (1) = 2 yk ,sk done, it is now possible to select translations {yk }∞ ≡ k=1 so that the measures νk ∗ yk ,sk 2 n |uk | (k = 1, · · · ) are tight in M(R ). The informal idea is that we can shift so that at least half the mass of νkyk ,sk remains in the unit ball, If tightness fails, so that some fraction of the mass escapes to infinity, then our minimization problem ”splits” into two parts, the sum of whose energies turns out to be strictly greater than the energy obtainable were the splitting not to occur. See [88] for full explanation. 2. To simplify notation we henceforth assume that the dilations and translations above 1 (k = 1, · · · ) and the measures {νk }∞ were unnecessary, so that Qk (1) = k=1 are tight. 2 Thus, passing as necessary to a subsequence, we may assume

(4.5)

νk * ν

in M(Rn ), ν(Rn ) = 1.

We may also suppose (4.6)

µk * µ in M(Rn )

for µk ≡ |Duk |2 (k = 1, · · · ). 3. We claim u 6≡ 0. To see this we note that µk (Rn ) → I and also µ(Rn ) ≤ I = C2−2 . But then if u ≡ 0, (4.5) allows us to invoke assertion (iii) of Theorem 1.4.2 in Chapter 1 to find that ν is concentrated at a single point x0 . We thereby obtain the contradiction Z 1 ∗ = Qk (1) ≥ |uk |2 dx → 1. 2 B(x0 ,1)

42

CHAPTER 4. CONCENTRATED COMPACTNESS ∗

4. Next we claim u ∈ A. If not, then kuk2L2∗ (Rn ) = α for some 0 < α < 1. Write Iα ≡ inf I[w] for w∈Aα

Then

∗ ∗ Aα ≡ {w ∈ L2 (Rn ) kwk2L2∗ (R2 ) = α, Dw ∈ L2 (Rn ; Rn )}. ∗

Iα ≡ Iα2/2 .

5. According to assertions (i) and (ii) of Theorem 1.4.2 in Chapter 1, we have X X ∗ ν = |u|2 + νj δxi , µ ≥ |Du|2 + µj δ xi j∈J

j∈J

for some countable set of points {xj }j∈J and positive weights {µj , νj }j∈J , satisfying X 2/2∗ α+ νj = 1, µj ≥ Iνj (j ∈ J). j∈J

Consequently, we arrive at the contradiction Z X n I ≥ µ(R ) ≥ |Du|2 dx + µj Rn

∗

≥ I(a2/2 +

X

≥ Ia +

j∈J

2/2∗

vj

X

µj

j∈J

) > I. 

j∈J

The heuristic principle of the last step in the proof is somewhat like that of the first step: if a fraction of the mass concentrates, our minimization problem ”splits” into two parts, the sum of whose energies strictly exceeds the energy were splitting not to occur.

4.1.2

Strong convergence of minimizing sequences.

Let us now shift our point of view somewhat and scrutinize the problem of the strong convergence of minimizing sequences for a critical growth variational problem set in a bounded smooth domain U ⊂ Rn , n ≥ 3.We will investigate for λ > 0 the functional Z λ (4.7) I [w] ≡ |dw|2 − λw 2 dx, U

for w belonging to the admissible set A ≡ {w ∈ W01,2 (U ) kwkL2∗ (U ) = 1}.

This problem bears obvious resemblance to that just treated, except that now (a) the lower order (and thus compact) perturbation λw 2 occurs in the energy functional and (b)

43

4.1. VARIATIONAL PROBLEMS

we are working on the bounded domain U . The new feature, as established by Aubin [6] and Brezis-Nirenberg [31], will be that (a) and (b) together work to restore strong convergence for minimizing sequences. In other words, we will argue directly, using the exact structure of the nonlinearity, that concentration does not occur (cf. the direct methods presented in Chapter 5,§5.1). So assume {uk }∞ k=1 ⊂ A satisfies (4.8)

I λ [uk ] → inf I λ [w] ≡ Iλ . w∈A

We also write 0

0

I0 ≡ inf I [w] f or I ≡ w∈A

Z

|dw|2 dx. U

Lemma 1. If λ > 0 and n ≥ 4, then Iλ < I0 . Outline of proof. We know from the theory in §4.1.1 that if U = Rn , then the infimum of I 0 [·] over A is obtained by functions uy,ε of the form (4.4). Now return to the situation at hand, where U ⊂ Rn is bounded. Assuming for simplicity that 0 ∈ U , we would guess that the function u0,ε for small ε > 0 is a good candidate for a minimum of I 0 [·], except that u0,ε 6= 0 on ∂U . Let us repair this defect by writing v ε (x) ≡ ζ(x)u0,ε (x) (x ∈ U ), for a cutoff function ζ ∈ C0∞ (U ) which is identically 1 near 0, and readjust the constant Cε in (4.4) so that kv ε kL2∗ (U ) = 1,

v ε ∈ A.

A careful analysis carried out in [31] demonstrates ( I0 + O(ε(n−2)/2 ) − λkε if n ≥ 5, Iλ[v ε ] = I0 + O(ε) − λK| log ε| if n = 4 for some constant K > 0. Thus if we choose ε > 0 small enough, Iλ ≤ I λ [v ε ] < I0 .  Note now that Iλ > 0 if and only if λ < λ1 , λ1 denoting the principal eigenvalue for −∆ on W01,2 (U ). Theorem 4.1.2 Let 0 < λ < λ1 . Assume {uk }∞ k=1 ⊂ A is a minimizing sequence and ∞ n ≥ 4. Then there exists a subsequence {ukj }∞ j=1 ⊂ {uk }k=1 and a function u ∈ A such that ukj → u strongly in W01,2 (U ). In particular, u ∈ A is a minimizer. Proof. 1. We have (4.9)

I λ [uk ] = Iλ + o(1),

||uk ||L2∗ (Rn ) = 1.

Thus there is a subsequence {ukj }∞ j=1 satisfying ( ukj * u in W01,2 (U ), (4.10) ukj → u a.e. and strongly in Lp (U ), 1 ≤ p < 2∗ .

44

CHAPTER 4. CONCENTRATED COMPACTNESS 2. Set vkj ≡ ukj − u (j = 1, · · · ). Then vkj * 0 in W01,2 (U ); whence (4.9) implied I λ [u] + I 0 [vkj ] = Iλ + o(I).

(4.11)

Next we deduce from Theorem 1.4.1 in Chapter 1 that Z Z ∗ ∗ 2∗ 1= |ukj | dx = |u|2 + |vkj |2 dx + o(1). U

U

But then 1 + o(1) ≤ (

Z

∗

U

∗

∗

|u|2 + |vkj |2 dx)2/2 ≤ ||u||2L2∗ (U ) + ||vkj ||2L2∗ (U ) .

from the elementary inequality (a + b)θ ≤ aθ + bθ (0 ≤ θ ≤ 1, a, b > 0). In light of (4.11), we infer (4.12)

I λ [u] + I 0 [vkj ] ≤ Iλ (||u||2L2∗ (U ) + ||vkj ||2L2∗ (U ) ) + o(1).

3. Now by definition, Iλ ||u||2L2∗ (U ) ≤ I λ [u]. Hence (4.12) becomes I 0 [vkj ] ≤ Iλ ||vkj ||2L2∗ (U ) + 0(1). But I0 ||vkj ||2L2∗ (U ) ≤ I 0 [vkj ], and consequently, (I0 − Iλ )I 0 [vkj ] ≤ o(1). Therefore, Lemma 4.2.1 implies ukj → u strongly in W01,2 (U ). 

4.2

Concentration-Cancellation

We have seen in §4.1 several concentration methods centering upon the failure of compact∗ ness of the injection of L2 into W 1,2 . We turn our attention now to some nonvariational problems, where again compactness is lost, or is at least in question, because of other types of critical growth nonlinearities.

4.2.1

Critial gradient growth.

The most common type of criticality for growth conditions of a type not covered in §4.1 concerns systems of quasilinear PDE’s having the form ( − ∆u = b(x, u, Du) in U, (4.13) u = 0 on ∂U,

45

4.2. CONCENTRATION-CANCELLATION with quadratic gradient growth in the nonlinearity: (4.14)

|b(x, z, P )| ≤ C(1 + |P |2 ) (x ∈ U, z ∈ Rm , P ∈ M m×n ).

Following the ideology set forth in the Introduction, let us suppose that for each k = 1, · · · we have, say, a smooth solution uk of the approximate problem ( − ∆ = bk (x, uk , Duk ) in U, (4.15) u = 0 on ∂U, additionally suppose we have the uniform bounds (4.16)

sup ||uk ||W 1,2 (U ;Rm ) , ||uk ||L∞ (U ;Rm ) < ∞. k≥1

0

These are appropriate assumptions in many settings. Assume further that the nonlinearities {bk }∞ k=0 verify uniform quadratic growth assumptions like (4.14) (4.17)

|bk (x, z, P )| ≤ C(1 + |P |2 ) (x ∈ U, z ∈ Rm , P ∈ M m×n )

for some constant C, and (4.18)

bk → b uniformly on compact subsets of U × Rm × M m×n .

We inquire as to whether the bounds (4.16) suffice to let us extract a subsequence converging to a weak solution of (4.13). So assume ukj * u in W 1,2 (U ; Rm ). Is u then a weak solution of (4.13)? Since the nonlinearities entail the gradients, the following compactness assertion for scalar PDE’s will be useful: Theorem 4.2.1 Suppose for k = 1, 2, · · · that uk ∈ W 1,2 (U ) is a weak solution of (4.19)

−∆uk = fk

in U.

1,2 1 Assume further {uk }∞ (U ) and {fk }∞ k=1 is bounded in W k=1 is bounded in L (U ). Then ∞ 1,p {uk }k=1 is precompact in W (U ). for each 1 ≤ p < 2. ∞ Proof. 1. We pass to a subsequence {ukj }∞ k=1 ⊂ {uk }k=1 satisfying

(4.20)

ukj * u in W 1,2 (U ),

ukj → u a.e.and strongly in L2 (U ).

46

CHAPTER 4. CONCENTRATED COMPACTNESS 2. Fix σ, δ > 0 and define  if s ≥ σ  σ if − σ ≤ s ≤ σ, βσ (s) ≡ s   − σ if s ≤ −σ.

According to Egoroff’s Theorem there exists a measurable set Eδ ⊂⊂ U such that L (U − Eδ ) ≤ δ and ukj → u uniformly on Eδ . Let ζ be a cutoff function, ζ ≡ 1 on Eδ . δ Thus if we choose j so large that |ukj − u| ≤ on Eδ . we may calculate using (4.19) that 2 Z Z |Dukj − Du|2 dx ≤ ζD(ukj − u) · Dβσ (ukj − u)dx Eδ U Z = ζfkj βσ (ukj − u)dx U Z − Dζ · D(ukj − u)βσ (ukj − u)dx U Z − ζDu · Dβσ (ukj − u)dx. n

U

Now βα (ukj − u) goes to zero strongly in L2 (U ) and weakly in W 1,2 (U ). Consequently (4.21)

lim sup j→∞

Z

|Dukj − Du|2 dx ≤ σ sup ||fk ||L1 (U ) = O(σ). Eδ

k

3. Hence, passing if need be to a further subsequence, we deduce Dukj → Du a.e. on Eδ . This is true for each δ > 0 and so DuKj → Du a.e. As {Dukj }∞ j=1 is bounded in L2 (U ; Rn ), we conclude that Dukj → Du strongly in Lp (U ; Rn ) for each 1 ≤ p < 2.  If we now apply Theorem 3 to each component of our solution uk of (4.15), we deduce, upon passing as necessary to a further subsequence, that (4.22)

Dukj → Du a.e. and strongly in LP (U ; M m×n )

for each 1 ≤ p < 2. But this is not quite enough to pass to limits and show u solves (4.13): the nonlinearities b, bk grow quadratically in the gradient. a. A nonlinear test function method. Let us turn now to the case of a single equation (m = 1) for which we can indeed improve (4.20) to obtain strong convergence in W01,2 (U ). Theorem 4.2.2 Assume m = 1 and {ukj }∞ j=1 satisfies (4.20). Then ukj → u strongly in 1,2 W0 (U ), and so u is a weak solution of (4.13).

47

4.2. CONCENTRATION-CANCELLATION Proof. 1. By definition u will be a weak solution of (4.13) provided Z Z (4.23) Du · Dvdx = b(x, u, Du)vdx U

for each v ∈ (4.24)

W01,2 (U )

U

∞

∩ L (U ). Since uk solve (4.15) we have as well that Z Z Duk · Dvdx = bk (x, uk , Duk )vdx. U

U

2. Set vk ≡ sinh(λ(uk − u)), where λ > 0 will be selected later. We insert v = vk in (4.24) and compute Z Z λ cosh(λ(uk − u))Duk · D(uk − u)dx = bk (x, uk , Duk ) sinh(λ(uk − u))dx. U

U

We further deduce Z λ cosh(λ(uk − u))|D(uk − u)|2 dx U Z Z −1 ≤ −λ cosh(λ(uk − u)) Du · D(uk − u)dx + λ Du · D(uk − u)dx U Z U + C (1 + |Duk |2 )| sinh(λ(uk − u))|dx, U

C the constant from (4.17). We pass to the subsequence {ukj }∞ k=1 verifying (4.20), set k = kj , and recall cosh s ≥ 1, cosh s ≥ | sinh s|(s ∈ R) to deduce Z (λ − C) |D(uk − u)|2 dx ≤ o(1). U

1,2

The strong convergence in W (U ) follows once we adjust λ to be large enough. Now pass to limits in (4.24) for k = kj to obtain (4.23).  b. Capacity method. This nonlinear test function technique and its many variants do not generally apply to systems of the form (4.13), (4.14) for m > 1 unless additional, often stringent structural hypotheses are made on the nonlinearities. In fact the problem of passing to limits in the general setting discussed here dose not seem to have any really satisfactory solution. Frehse [61,62] has however made the interesting suggestion that capacity techniques may be useful in analyzing the possible failure of strong convergence in W 1,2 and in circumventing the attendant difficulties. The following assertion reacts his ideas, using the terminology of the reduced defect measure Z θ(E) ≡ lim sup |Dukj − Du|2 dx (E a Borel subset of U ). j→∞

E

introduced in Chapter 1, §1.4.2 for a subsequence {ukj }∞ j=1 verifying (4.20). We wish to show that θ is concentrated upon a rather small set, and furthermore if this set is small enough, then u solves (4.13).

48

CHAPTER 4. CONCENTRATED COMPACTNESS

Theorem 4.2.3 Assume m > 1 and {ukj }∞ j=1 satisfies (4.20). Then (i)the reduced defect measure θ is concentrated on a set of p-capacity zero for 1 ≤ p < 2. (ii)Furthermore, if θ is concentrated on a set of 2-capacity zero. then u is a weak solution of (4.13). Proof. 1. Assume 1 ≤ p < 2.According to Theorem 1.3.2 in Chapter 1 there exists for each δ > 0 a relatively closed set Eδ ⊂ U with ukj → u uniformly on Eδ and Capp (U − Eδ ) (1 ≤ p < 2). Fix σ > 0. We then deduce almost exactly as in the proof of Theorem 4.2.1 (except now the cutoff function ζ is not needed) that lim sup |Dukj − Du|2 dx ≤ σ sup kbk kL1 (U ;Rm ) . j→∞

k

Thus (4.16), (4.17) imply θ(Eδ ) = 0 (δ > 0). Now write Vi ≡ U − E1/i (i = 1, · · · ); so that θ(U − Vi ) = 0 (i = 1, · · · ), Capp (Vi ) → 0. Then according to the definitions in Chapter 1, §1.4.2, θ is concentrated on a set of p-capacity zero. 2. Now suppose θ is concentrated on a set of 2-capacity zero; whence there exist open sets {Vi }∞ i=1 such that (4.25)

θ(U − Vi ) = 0 (i = 1, · · · ),

Cap2 (Vi ) → 0.

According to the definition of Cap2 (·), there exist functions {φ}∞ i=1 satisfying ( 0 ≤ φi ≤ 1, φi ≡ 0 on Vi , φi → 1 a.e., (4.26) kDφi kL2 (Rn ) → 0 (i = 1, 2, · · · ). Choose v ∈ W01,2 (U ; Rm ) ∩ L∞ (U ; Rm ) and use φi v as a test function in the definition of weak solution of (4.15) for k = kj : Z Z T (4.27) φi Dukj : Du + Dφi Dukj vdx = φi bkj (x, ukj , Dukj ) · vdx. U

U

Now (4.25) says that Dukj → Du strongly in L2 (U − Vi ; M m×n ). As φi ≡ 0 on Vi we can send kj → ∞ in (4.27) to deduce Z Z T φi Du : Dv + Dφi Duvdx = φi b(x, u, Du) · vdx. U

U

Now let i → ∞ and recall (4.26) to deduce that u is a weak solution of (4.13).  The idea in the second part of the proof is that if the concentration set for the reduced defect measure θ is small enough, then we can build the auxiliary functions {φi }∞ i=1 to

49

4.2. CONCENTRATION-CANCELLATION

“shield” the limiting process from the failure of strong convergence in W 1,2 . This is the technique of concentration-cancellation. 2. Vorticity bounds and Euler’s equations. We investigation next concentration phenomena for a model of incompressible fluid flow in two dimensions, as in DiPernaMajda [45-47]. On the surface, this problem has little in common with those discussed earlier. But in fact by introducing a velocity potential we arrive at very much the same technical impasse as above: the failure of strong convergence is concentrated on a small set, which-if it were slightly smaller-could be ignored. This assertion, theorem 4.2.4 below, is clearly similar to Theorem 4.2.3 above, except that we now measure smallness in terms of Hausdorff measure. We assume for each k = 1, · · · that we have a velocity field uk = (u1k , u2k ) and a scalar pressure pk solving Euler’s equations ( uk,l + uk · Duk = Dpk + fk (4.28) in R2 × (0, T ) div uk = 0 for some known external force fk = (fK1 , fk2 ). We interpret (4.28) in the weak sense by requiring Z TZ Z TZ (4.29) uk · vt + (uk ⊗ uk ) : Dvdxdt = − fk · vdxdt R2

0

R2

0

for all test fields v ∈ C0∞ (R2 × (0, T ); R2 ) with div v = 0. Assuming that fk → f strongly in L2 (R2 × (0, T ); R2 ), we ask if we can extract a subsequence of {uk , pk }∞ k=1 converging somehow to a weak solution {u, p} of ( uk + u · Du = Dp + f (4.30) in R2 × (0, T ), div u = 0 which means

Z

T 0

Z

u · vt + (u ⊗ u) : Dvdxdt = R2

Z

T 0

Z

f · vdxdt R2

for all v as above. To begin with, let us suppose we have bounds, uniform in time, on the kinetic energy: Z (4.31) sup max |uk |2 dx < ∞. k≥1 0≤t≤T

R2

Additionally, motivated by studies of vortex sheet roll up for incompressible flow (see[45]), let us suppose that we have uniform L1 bounds on the vorticity wk ≡ curl uk = u1k,x2 − u2k,x1 .

50

CHAPTER 4. CONCENTRATED COMPACTNESS

We therefore assume (4.32)

sup max

k≥1 0≤t≤T

Z

|wk |dx < ∞. R2

Can we utilize the kinetic energy and vorticity bounds to pass from (4.28) to (4.30)? ∞ First observe from (4.31) that we can secure a subsequence {ukj }∞ j=1 ⊂ {uk }j=1 and a 2 2 2 function u ∈ L (R × (0, T ); R ) so that ukj * u weakly in L2 (R2 × (0, T ); R2 ).

(4.33)

But this alone does not provide enough control to pass to limits in the identity (4.29), because of the nonlinear term uk ⊗uk . As earlier, we introduce the reduced defect measure Z Z θ(E) ≡ lim sup |ukj − u|2 dxdt (E a Borel subset of R2 × (0, T )) j→∞

E

to characterize failure of strong convergence. DiPerna and Majda show. Theorem 4.2.4 Assume {ukj }∞ j=1 satisfies (4.31)-(4.33). Then (i) the reduced defect measure θ is concentrated on a set of ”cylindrical” H s −measure zero for each s > 1. (ii) Furthermore, if θ is concentrated on a set of H s − measure zero for some s < 1, then u is a weak solution of (4.30). (Greengard and Thomann [69] point out that (ii) in fact implies there is no concentration at all.) The full proof in [45] is quite complicated, mainly because uniform control in time is not very good. Let us therefore provide instead some details for the simpler case of the stationary Euler equations and thereby make clearer some connections between capacity and Hausdorff measure for characterizing the concentration set. we assume that uk ∈ W 1,2 (R2 ) is a weak solution of ( uk · Duk = Dpk + fk (4.34) in R2 , div uk = 0 which means (4.35)

Z

(uk ⊗ uk ) : Dvdx = − R2

Z

fk · vdx R2

for all v ∈ C0∞ (R2 ; R2 ) with div v = 0. As above, we suppose fk → f in L2 (R2 ; R2 ). Z Z 2 sup |uk | dx, |wk |dx < ∞, k≥1

R2

R2

51

4.2. CONCENTRATION-CANCELLATION and ukj * u in L2 (R2 ; R2 ).

(4.36)

We want to show u is a weak solution of ( u · Du = Dp + f (4.37) div u = 0

in R2

for some pressure p; that is, Z Z (4.38) (u ⊗ u) : Dvdx = − R2

f · vdx R2

for all v as above. Theorem 4.2.5 Assume {ukj }∞ j=1 satisfies (4.36). Then u is a weak solution of (4.37). We will show that the reduced defect measure is concentrated on such a small set that we can “shield” the PDE from the failure of strong convergence. Proof. 1. The incompressibility condition div uk = 0 implies the existence of a velocity potential ψk satisfying uk = D T ψk ≡ (−ψk,x2 , ψk,x1 ). Then (4.39)

−∆ψk = wk

in R2

1,2 2 in the weak sense, and we may assume {ψk }∞ k=1 is bounded in Wloc (R ). Using Theorem 1.3.2 in Chapter 1 we deduce as in §4.2.1 above (cf. proofs of Theorem 4.2.1 and 4.2.3) that the reduced defect measure θ restricted to any given bounded open set U is concentrated in a set of p-capacity zero for 1 ≤ p < 2. Thus there exist open sets {Vi }∞ i=1 in U with

(4.40)

θ(U − Vi ) = 0 (i = 1, · · · ),

for (4.41)

θ(E) = lim sup j→∞

Z

Capp (Vk ) → 0

|ukj − u|2 dx (E a Borel subset of U ). E

2. We now claim that from (4.40) it follows that for each s > 0 (4.42)

Hδsj (Vi ) → 0

for some sequence {δi }∞ i=1 of positive numbers tending to zero. Indeed write σi ≡ capp (Vi )(i = ∗ 1, · · · ), and choose f ∈ Lp (Rn ),with Df ∈ Lp (Rn ; Rn ), f ≥ 0, f ≥ 1 on Vi and Z |Df |p ≤ 2σi . Rn

52

CHAPTER 4. CONCENTRATED COMPACTNESS

Fix ε > 0, choose any point x0 ∈ Vi , and suppose that Z δi 1 (4.43) |Df |pdx ≤ r ε for each 0 < r ≤ , 2−p r 5 B(x0 ,r) 0 < δ < 1 as below. From (4.43) and Poincar´ e’s inequality we see that Z δi (4.44) − |f − (f )x0 ,r |p ≤ Cr ε (0 ≤ r ≤ ), 5 B(x0 ,r) (f )x0 ,r denoting the average of f over B(x0 , r). Thus 1 ≤ f (x0 ) =

∞ X

[(f )x0 ,δi /5k+1 − (f )x0 ,δi /5k ]

k=1 ∞ X

Z δi ε ≤C +− |f |dx by (4.44) 5k B(x0 ,δi /5) k=1 Z 1/p∗ ε p∗ ≤ Cδi + − |f | dx B(x0 ,δi /5)

≤

Cδiε

+

≤ Cδiε +

C 2/p∗

δi C

2/p∗ δi

Z

p

|Df | dx R2

1/p

1/p

σi

1 ≤ , 2 provided δj is small enough and i is large enough. This contradiction shows (4.43) is untenable. As a result, for each x0 ∈ Vi there exists 0 < rx0 < δi /5 with Z s r x0 < |Df |p dx B(x0 ,rx0 )

for s = 2 − p + ε. Using standard covering arguments we find a collection of disjoint balls {B(xk , rk )}∞ k=1 with Z ∞ [ s Vi ⊂ B(xk , 5rk ), rk ≤ |Df |pdx, B(xk ,rk )

k=1

and 5rk ≤ δi . Then

Hδsi (Vi )

≤C ≤C

∞ X

Zk=1

(5rk )s |Df |pdx

R2

≤ Cσi → 0.

53

4.2. CONCENTRATION-CANCELLATION

This proves (4.42), which in turn, according to our definition from Chapter 1, §4.4.2, says that θ restricted to U is concentrated on a set of H s −measure zero for each s > 0. 3. Now choose a test function v as above, with spt v contained in some bounded open set U . Since div v = 0 we can write v = D t w = (−wx2 , wx1 ) for w ∈ C0∞ (U ). Substituting into (4.38) we find Z Z T (4.45) (uk ⊗ uk ) : DD wdx = − fk · D T wdx U

U

we wish to pass to limits in this identity as k = kj → ∞. Since w can be decomposed into the superposition of plane waves using the Fourier transform, it suffices to consider the case that w itself is a plane wave and so depends only on x · ξ for some fixed ξ ∈ R2 . As Euler’s equations are invariant under rotation, it will be enough to take ξ = (1, 0). Consequently, we may assume w depends only on x1 and substitute into (4.45) to find Z Z d 1 2 00 ). (4.46) uk uk w dx = fk2 w 0 dx (1 − dx1 U U Denote by Wi the projection of Vi onto the x1 −axis, for Vi verifying (4.40). In view of (4.42), L1 (R − Wi ) → 0. Define Z x1 Z s wi (x1 ) ≡ w 00 χR1 −Wi dtds 0

0

so that wi00 = w 00 χR−Wi a.e. Then wi00 = 0 on the set where strong convergence fails. Consequently, we may replace w by wi and pass to limits as k = kj → ∞ in (4.46) to discover Z Z 1 2 00 u u wi dx = f 2 wi0 dx. U

Now

wi00

00

→ w a.e.,

wi0

U

→ wi uniformly. Therefore, sending i → ∞ above we obtain Z Z 1 2 00 u u w dx = f 2 w 0 dx, U

U

as required.  DiPerna and Majda’s concentration-cancellation argument, reproduced in step 3 of the proof, is remarkable in view of the corresponding technique in the demonstration of Theorem 4.2.3. In the earlier proof we needed to assume slightly better control on the concentration set than could really be proved. The proof of Theorem 4.2.5 on the other hand manages to establish initially such good estimates that the “shielding” of the concentration set can be effected under no further assumptions. This technique depends of course upon the explicit structure of Euler’s equations.

Chapter 5 Compensated Compactness We address now nonvariational PDE’s for which oscillations in approximating sequences cause trouble. The overall plan is to analyze, mostly using Young measures, the ways in which oscillations can persist, with a view towards using the PDE itself to prove that either the oscillation did not really occur or else did not matter.

5.1

Direct Methods

Before commencing with the general theory of compensated compactness, it is worth recording here that occasionally passages to weak limits can be justified via ad hoc arguments.

5.1.1

Harmonic maps into spheres

A particularly elegant example is to be found in the resent works of Chen [33] and Shatah [109], which we specialize by considering the system of PDE’s (5.1)

(

− ∆u = |Du|2 u |u| = 1 a.e.

in U.

This is the Euler-Lagrange equation for energy functional Z I[w] = |dw|2 dx, U

defined for the constrained admissible set A ≡ {w ∈ W 1,2 (U ; Rm ) |w| = 1 a.e., w = g on ∂U }. 54

55

5.1. DIRECT METHODS A harmonic mapping is a weak solution u ∈ A of (5.1),so that Z Z (5.2) Du · Dvdx = |Du|2 u · vdx U

W01,2 (U ; Rm )

for each v ∈ a harmonic mapping,

U

∞

m

∩ L (U ; R ).Let us suppose now that for k = 1, · · · , uk ∈ A is (

(5.3)

− ∆uk = |Duk |2 uk |uk | = 1 a.e.

inU,

and additionally u * uinW 1,2 (U ; Rm ) for u ∈ A. Is u harmonic? First observe that our system (5.1) satisfies the quadratic gradient growth discussed in Chapter 4, §4.2.1, although the theory set forth there is not generally good enough to justify passing from (5.3) to (5.1). we are instead able to argue directly, utilizing the explicit nonlinear structure of the PDE: Theorem 5.1.1 The function u is a weak solution of the PDE (5.1). Proof. 1. For each 1 ≤ i, j ≤ m we have Z Z i i Duk · Dv = |Duk |2 uik v i dx U

and

i

j

Z

U

Dujk

j

· Dv dx =

U W01,2 (U ) ∩ L∞ (U ).

Z

U

|Duk |2 ujk v j dx

for k = 1, · · · and v , v ∈ Fix w ∈ C0∞ (U ) and set v i ≡ ukj w, v j ≡ uik w above. Subtracting the resulting expression we obtain Z 0 = Duik · D(ujk w) − Dujk · D(uik w)dx ZU = ujk Duik · dw − uik Dujk · dwdx. U

We deduce upon sending k → ∞ that Z (5.4) 0= uj Dui · dw − ui Duj · dwdx U

for all 1 ≤ i, j ≤ m and w ∈ C0∞ (U ). By approximation, the identity is valid for w ∈ W01,2 (U ) ∩ L∞ (U ). 2. Given v ∈ H01 (U ; Rm ) ∩ L∞ (U ; Rm ), we set w = uj v i in (5.4) and sum on i and j: Z 0= uj Dui · D(uj v i ) − ui Duj · D(uj v i )dx. U

since |u|2 = 1, and thus (Du)u = 0, this expression simplifies to become (5.2). 

56

5.1.2

CHAPTER 5. COMPENSATED COMPACTNESS

Homogenization of divergence structure PDE’s.

An important general question for PDE’s concerns the effect of rapidly oscillating coefficients upon solutions (cf.Bensoussan, Lions, Papanicolaou [19]). We study next a model such problem by considering the divergence structure linear elliptic PDE   − (aij ( x )uε )x = f in U, ε xi j (5.5)  ut = 0 on ∂U.

We assume the uniform ellipticity condition

ξ T A(y)ξ ≥ ν|ξ|2

(y ∈ Y, ξ ∈ Rn )

for ν > 0, A = ((aij )), and suppose also |A(y)| ≤ C, y 7→ A(y) is Y -periodic, Y denoting the unit cube in Rn . W assume uε ∈ W01,2 (U ) to be a weak solution of (5.5), which means Z Z x ε (5.6) aij ( )uxi vxj dx = f vdx ε U U for each v ∈ W01,2 (U ). Setting v = uε we deduce the sup kuε kW 1,2 (U ) < ∞; ε>0

0

so that uεk * u in W01,2 (U ) for some appropriate subsequence. Now we want to find a PDE that u solves. we cannot, however, pass to limits in the term on the left-hand side of (5.6) in any obvious way, so instead we utilize insight of Tartar [117]. To understand the limit problem, we introduce for l = 1, · · · , n the adjoint corrector problems ( − (aij (y)wyl j )yi = (ail (y))yi in Rn , (5.7) w l Y -periodic. As the right-hand side of (5.7) has integral zero over Y , there exists a weak solution w l according to the Fredholm alternative. Define then the averaged coefficients Z a ˜il ≡ aij (y)(δjl + wyl j (y))dy (1 ≤ i, l ≤ n) Y

and the homogenized problem ( (5.8)

− (˜ ail uxi )xl = f u = 0 on ∂U.

in U,

57

5.1. DIRECT METHODS Theorem 5.1.2 The function u is the weak solution of the PDE (5.8).

This assertion is interesting since the limit PDE (5.8) is not at all obvious from a cursory examination of (5.5): the trick is to untangle the effects of the weak convergence of the coefficients and of Duε . Proof. 1. Passing if necessary to a further subsequence we have aij (x/εk )uεxki * ξ j

in L2 (U ),

for some ξ ∈ L2 (U ; Rn ). From (5.6) we deduce Z Z (5.9) ξ · Dvdx = f vdx. U

U

2. Fix 1 ≤ l ≤ m and define the corrector x v ε (x) ≡ xl + εw l ( ) (x ∈ U ). ε Using (5.7) we deduce that v ε is a weak solution of x −(aij ( )vjε )xi = 0 in U. ε

(5.10)

Choose ζ ∈ C0∞ (U ) and set v = ζv ε in (5.6). In light of (5.10), we can simplify to obtain: Z Z x ε ε x ε ε (5.11) aij ( )uxi v ζxj − aij ( )vxj u ζxi dx = f v ε ζdx ε ε U U 3. Note v εk → xl in L2 (U ) and aij (x/εk )vxεkj = aij (x/εk )(δjl + wyl j (x/εk )) * a˜il Thus sending ε = εk → 0 in (5.11), we discover Z Z j ξ xl ζ x j − a ˜il uζxi dx = f xl ζdx. U

But from (5.9) we have

Z

Consequently,

l

U

f xl ζdx = U

Z

Z

ξ l ζ + ξ j xl ζxj dx. U

a ˜il uxi ζdx = U

Z

ξ l ζdx U

for all ζ as above; so that ξ = a ˜il uxi (1 ≤ l ≤ n).  See Chapter 6, §6.2 for more on homogenization.

in L2 (U ).

58

CHAPTER 5. COMPENSATED COMPACTNESS

5.1.3

Monotonicity, Minty-Browder method in L2

One of the earliest techniques systematically applied to circumvent weak convergence difficulties was the monotonicity method of Minty and Browder (cf. J. L. Lions [85]). As an illustrative problem we consider the quasilinear PDE ( − div(E(Du)) = f in U, (5.12) u = 0 on ∂U, where E : Rn → Rn is given. Now if problem (5.12) is variational, that is, if E = DF for some function F : Rn → R, then the methods of the calculus of variations introduced in Chapter 2 are applicable and (5.12) will be the appropriate Euler-Lagrange equation. For this setting the primary structural hypothesis is that F be convex. Note in this situation, we can compute for each p, q ∈ Rn that (E(p) − E(q)) · (p − q) = (DF (p) − DF (q)) · (p − q) ≥ 0, the inequality a consequence of convexity. This calculation suggests that even if E is not in fact the gradient of a convex function, we may still be able to study successfully the PDE (5.12) provided the nonlinearity E is monotone, which means (5.13)

(E(p) − E(q)) · (p − q) ≥ 0 (p, q ∈ Rn ).

We henceforth assume E is monotone and make the growth assumption |E(p)| ≤ C(1 + |p|) (p ∈ Rn ). Let us therefore turn our attention to a sequence of approximating problems ( −div(E(Duk )) = fk in U, (5.14) uk = 0 on ∂U for fk ∈ L2 (U ). Assume uk ∈ W01,2 (U ) is a weak solution, which means Z Z (5.15) E(Duk ) · Dvdx = fk vdx U

U

for all v ∈ W01,2 (U ). Suppose also fk → f in L2 (U ), uk * u in W01,2 (U ). Can we assert that u is a weak solution of (5.12)? The problem, of course, is that we do not know directly that {E(Duk )}∞ k=1 converges to E(Du) in any sense. Theorem 5.1.3 The function u is a weak solution of the PDE (5.12). Proof. Since E is monotone, Z 0 ≤ (E(Dv) − E(Duk )) · (Dv − Duk )dx U

59

5.2. DIV-CURL LEMMA

for all v ∈ W01,2 (U ). Furthermore, the identity (5.15) with v − uk in place of v allows us to rewrite and obtain Z 0≤ E(Dv) · (Dv − Duk ) − fk (v − uk )dx. U

Now pass to limits:

0≤

Z

E(Dv) · (Dv − Du) − f (v − u)dx. U

This inequality holds for each v ∈ W01,2 (U ). So now fix λ > 0, w ∈ W01,2 (U ), and set v ≡ u + λw above. Upon cancelling λ, we discover Z 0≤ E(Du + λdw) · dw − f wdx. U

Send λ to zero to deduce

0≤

Z

E(Du) · dw − f wdx. U

Replacing w by −w, we obtain the desired equality Z 0= E(Du) · dw − f wdx U

W01,2 (U ).

for each w ∈  A generalization of this technique will be presented in Chapter 6,§6.1.

5.2

Div-Curl Lemma

Turning now to the systematic development of compensated compactness theory, we first show that the inner product of two vector fields is weakly continuous, provided we have good enough control on the divergence and curl. The key insight, from Murat [95], is that if we have enough information concerning various combinations of derivatives we can sometimes show certain nonlinear functions are weakly continuous. Notation. If w ∈ L2 (U ; Rn ), w = (w 1 , · · · , w n ), we define curl w ∈ W −1,2 (U ; M n×n ) by setting (curlw)ij ≡ wxi j − wxj i (1 ≤ i, j ≤ n). ∞ Theorem 5.2.1 (Div-Curl Lemma). Assume that {vk }∞ k=1 , {wk }k=1 are two bounded sequences in L2 (U ; Rn ) such that −1,2 (i) {divvk }∞ (U ), k=1 lies in a compact subset of W ∞ −1,2 (ii) {curlwk }k=1 lies in a compact subset of W (U ; M n×n ). Suppose further vk * v, 2 n wk * w in L (U ; R ). Then vk · wk → v · w

in the sense of distributions.

60

CHAPTER 5. COMPENSATED COMPACTNESS

Proof. 1. Consider for each k = 1, · · · the vector field uk ∈ W 2,2 (U ; Rn ) solving ( − ∆uk = wk in U, (5.16) uk = 0 on ∂U, 2 n ∞ in the weak sense. Since {wk }∞ k=1 is bounded in L (U ; R ), {uk }k=1 is bounded in W 2,2 (U ; Rn ). 2. Now set zk ≡ −divuk , yk ≡ wk − Dzk (k = 1, · · · ). Then {z k }∞ k=1 is bounded in 1,2 W (U ). Additionally, if 1 ≤ i ≤ n,

ykl = wkl − zkxi = −ulkxj xj + ujkxi xj

(5.17)

= (ujkxi − ulkxj )xj In view of hypothesis (ii), we infer from (5.16) that {curl uk }∞ k=1 lies in a compact subset of W 1,2 (U ; M n×n ). Thus from (5.17) it follows that {yk }∞ k=1 is contained in a compact 2 n subset of L (U ; R ). 3. Hence we may suppose, upon passing subsequences as necessary, that (5.18)

zk * z in W 1,2 (U ),

yk → y strongly in L2 (U ; Rn ),

where z = −div u, y = w − Dz, for u ∈ W 2,2 (U ; Rn ) solving ( − ∆u = w in U, u = 0 on ∂U. 4. Now if φ ∈ C0∞ (U ), we have Z Z vk · wk φdx = vk · (yk + Dzk )φdx. U

According to (5.18)

U

Z

vk · yk φdx → U

Z

v · yφdx. U

In addition, hypothesis (i) and (5.18) allow us to compute Z Z vk · Dzk φdx = − vk · Dφzk dx − hdivvk , zk φi U U Z Z → − v · Dφzdx − hdivv, zφi = v · Dzφdx. U

U

W01,2 (U ).)

−1,2

(Here h, i is the pairing of W (U ) and Thus Z Z Z vk · wk φdx → v · (y + Dz)φdx = v · wφdx.  U

U

Generalizations may be found in §5.5 below.

U

61

5.3. ELLIPTIC SYSTEMS

5.3

Elliptic Systems

As a first illustration of the use of the Div-Curl Lemma and Young measures, we consider a convergence problem for quasilinear elliptic systems of PDE’s. Assume then that for k = 1, · · · , uk ∈ W 1,2 (U ; Rm ) is a weak solution of the system (5.19)

−div(E(Duk )) = 0 in U,

where E : M m×n → M m×n and |E(P )| ≤ C(1 + |P |) (P ∈ M m×n ). We assume that uk * u in W 1,2 (U ; Rm ), and we inquire as to whether u is a weak solution of (5.20)

−div(E(Du)) = 0 in U.

The difficulty as usual turns upon the nonlinearity E and the weak convergence of {Duk }∞ k=1 . If E is monotone, which in the present context means (E(P ) − E(Q)) : (P − Q) ≥ 0 (P, Q ∈ M m×n ), we can employ the Minty-Browder device as in §5.1.3 to deduce (5.20). In this case we have P T DE(Q)P ≥ 0 (P, Q ∈ M m × n). A strengthened version of this inequality, namely P T DE(Q)P ≥ γ|P |2

(P, Q ∈ M m×n )

for some γ > 0, is the assumption of strict ellipticity for the system (5.20). However, based upon our experience in Chapter 3, §3.1.1 it is more natural to assume instead the strict Hadamard-Legendre condition (5.21)

(η ⊗ ξ)T DE(Q)(η ⊗ ξ) ≥ γ|η|2 |ξ|2

(Q ∈ M m×n , η ∈ Rm , ξ ∈ Rn )

for γ > 0. (This is strict rank-one convexity in the variational case that E=DF.) So we now ask if we can effect the passage to limits from (5.19) to (5.20) under the natural assumption (5.21). A full answer to this question is still unknown, although Theorem 3.3.2 in Chapter 3, §3.3 handles the variational case that E = DF , F strictly quasiconvex, and uk an energy minimizer. We can however establish the following general perturbation assertion, which requires that the magnitude of possible oscillations of {Duk }∞ k=1 be not too large. Theorem 5.3.1 There exists a number ε0 > 0 such that if (5.22)

sup oscU Duk ≤ ε0 , k

then u is a weak solution of the elliptic system (5.20).

62

CHAPTER 5. COMPENSATED COMPACTNESS

Proof. 1. For a.e. x ∈ U let νx be the Young measure on M m×n corresponding to the weak convergence Duk * Du in L∞ (U ; M m×n ). Then for a.e. x ∈ U we deduce from Theorem 1.5.2 in Chapter 1 that Z (5.23) Du(x) = Y dνx ≡ Y . M m×n

We have furthermore E(Duk ) * E in L∞ (U ; M m×n ),

(5.24) for

E(x) ≡

Z

E(Y )dνx . M m×n

in addition we see from the Div-Curl Lemma that Duk : E(Duk ) → Du : E in the sense of distributions. Thus (5.25)

Du(x) : E(x) =

Z

Y : E(Y )dνx . M m×n

Combine (5.23)-(5.25) to find Z (5.26) (Y − Y ) : (E(Y ) − E(Y ))dνx = 0. M m×n

2. Our intention is to exploit the identity (5.26) to show νx is a point mass, necessarily concentrated at Y = Du(x). First we deduce from (5.26) that Z Z T (Y − Y ) DE(Y )(Y − Y )dνx ≤ O(ε0 ) |Y − Y |2 dνx , M m×n

M m×n

since hypothesis (5.22) implies spt νx is contained in a ball of radius ε0 about Y in M m×n . 3. On the other hand, define the quadratic function γ F (P ) ≡ P T DE(Y )P − |P |2 2

(P ∈ M m×n ).

According to (5.21), (η ⊗ ξ)T D 2 F (η ⊗ ξ) ≥ 0 and so F is rank-one convex. Hence remark (3.7) asserts F to be quasiconvex. As a consequence for each open set V ⊂ U , we have Z Z Z F (Du)dx ≤ lim inf F (Duk )dx = F dx, V

k→∞

V

V

63

5.4. CONSERVATION LAWS where F (x) ≡

Z

F (Y )dνx . Thus M m×n

F (Y ) ≤

Z

F (Y )dνx M m×n

for a.e. x ∈ U . Recalling the definition of F , we rewrite this inequality to read Z Z γ 2 (Y − Y )T DE(Y )(Y − Y )dνx . |Y − Y | dνx ≤ 2 M m×n m×n M We invoke step 2 and deduce

Z

|Y − Y |2 dνx = 0 M m×n

if ε0 is small enough. Consequently, νx = δY = δDu(x) , and so Duk → Du a.e. according to (1.29). 

5.4

Conservation Laws

One of the most important applications of compensated compactness has been the analysis using entropy conditions of the Young measures for conservation laws.

5.4.1

Single equations.

Let us begin by considering for each ε > 0 the quasilinear PDE (5.27)

uεl + F (uε)x = εuxx

in R × (0, ∞),

where F : R → R is given. The idea is that the ”εuεxx” term represents a smoothing or viscosity effect, which should vanish in the limit as ε → 0 to give the physically correct solution u of the conservation law (5.28)

ul + F (u) = 0 in R × (0, ∞).

As in Tartar [118], let us assume that the sequence of smooth functions {uε }ε>0 is bounded in L∞ (R × (0, ∞)), so that we may extract a subsequence {uεk }∞ k=1 with ∗

uεk * u in L∞ (R × (0, ∞)). Theorem 5.4.1 The function u is a weak solution of the conservation law (5.28), which means Z ∞Z (5.29) uvl + F (u)vx dxdt = 0 0

for all v ∈ C0∞ (R × (0, ∞)).

R

64

CHAPTER 5. COMPENSATED COMPACTNESS

Proof 1. For a.e. (x, l) ∈ R×(0, ∞) let νx,l be the Young measure on R corresponding to the weak convergence above. Then Z (5.30) u(x, l) = ydνx,l . R

We have further from Theorem 1.5.2 in Chapter 1 that (5.31)

∗

in L∞ (R × (0, ∞)),

F (uεk ) * F

for F (x, t) ≡

Z

F (y)dνx,l . R

2. Choose now any convex function Φ : R → R and define Z y Ψ(y) ≡ Φ0 (s)F 0 (s)ds (y ∈ R). 0

We call (Φ,Ψ) an entropy\entropy flux pair. Then ∗

∗

Φ(uεk ) * Φ, Ψ(uεk ) * Ψ

in L∞ (R × (0, ∞))

for (5.32)

Φ(x, t) ≡

Z

Φ(y)dνx,t ,

Ψ(x, t) ≡

R

Z

Ψ(y)dνx,l R

Multiplying (5.27) by Φ0 (uε ) we discover (5.33)

Φ(uε )l + Ψ(uε ),

εΦ(uε )xx − εΦ00 (uε )(uεx )2 .

Now it is easy to derive the estimate Z ∞Z sup ε(uεx )2 dxdt < ∞. ε>0

0

R

Thus {εΦ0 (uε )uεx }ε>0 is precompact in L2 (R × (0, ∞)) and so {εΦ(uε )xx }ε>0 is precompact in W −1,2 (R × (0, ∞)). Additionally, {εΦ00 (uε )(uεx )2 }ε>0 is bounded in M(R × (0, ∞)). Applying Corollary 1.2.1 from Chapter 1 we see that the right−1,2 hand side of (5.33) is precompact in Wloc (R × (0, ∞)). 3. Employing (5.27) and (5.33), we apply the Div-Curl Lemma to the vector fields vk ≡ (F (uεk ), uεk )

and

wk ≡ (Φ(uεk ), −Ψ(uεk )).

65

5.4. CONSERVATION LAWS ∗

We conclude vk · wk * v · w in L∞ (R × (0, ∞)). for v = (F , u) and w = (Φ, −Ψ). On the other hand, Z ∗ vk · wk * F (y)Φ(y) − yΨ(y)dνx,l. R

Combining the above we deduce Z F (y)Φ(y) − yΨ(y)dνx,l = F (x, t)Φ(x, t) − u(x, t)Ψ(x, t). R

4. The idea is to deduce useful information from this functional relation, which is valid for each entropy\entropy flux pair (Φ, Ψ). First we rearrange to obtain Z (5.34) (F (y) − F (x, t))Φ(y) + (u(x, t) − y)Ψ(y)dνx,l = 0. R

We next set Φ(y) ≡ |y − u(x, t)|; so that Ψ(y) = sgn(y − u(x, t))(F (y) − F (u(x, t))). Substituting in (5.34) we discover (F (u(x, t)) − F (x, t))

Z

|y − u(x, t)|dνx,l = 0. R

Thus either F (u(x, t)) = F (x, t) or νx,l = δu(x,l) . But as the second possibility implies the first, we see that F = F (u) a.e. It is now easy to derive the identity (5.29).  Schonbck [107] has extended the foregoing methods to study competing small diffusion (ε → 0) versus small dispersion (δ → 0) effects for the PDE ε,δ ε,δ ε,δ uε,δ l + F (u )x + δuxxx = εuxx

in R × (0, ∞).

Under various technical assumptions she proves that if δ = O(ε3 ), then uε,δ converges to a weak solution of (5.28). (The zero diffusion, small dispersion case (ε = 0, δ → 0) is completely different, and has been solved by Lax and Levermore [83] using inverse scattering methods.)

5.4.2

Systems of two equations.

Following DiPerna [40] and Tartar [120] we consider next solutions uε = (u1,ε , u2,ε ) of the system (5.35)

uεl + F (uε )x = εuεxx

in R × (0, ∞).

66

CHAPTER 5. COMPENSATED COMPACTNESS

where F : R2 → R2 ,F = (F 1 , F 2 ), is given. We hope to show that as ε → 0 the solutions uε converge somehow to a weak solution u = (u1 , u2 ) of the system of conservation laws (5.36)

ul + F (u)x = 0

in R × (0, ∞)

so that (5.37)

Z

∞ 0

Z

u · vl + F (u) · vx dxdt = 0 R

for all v ∈ C0∞ (R × (0, ∞); R2 ). We will assume the uniform estimates (5.38)

sup ||uε||L∞ , ||ε1/2 uεx ||L2 < ε; ε

and consequently, as usual, there exists a subsequence {uεk }∞ k=1 with ∗

uε k * u

in L∞ (R × (0, ∞); R2 ).

We propose to show u verities (5.37), and for this will need several structural assumptions on the nonlinearity. We say that the system of PDE’s (5.36) is strictly hyperbolic provided for each y ∈ R2 , the 2 × 2 matrix DF (y) has real and distinct eigenvalues λ1 (y) < λ2 (y). Let ri (y) denote a (smooth) right eigenvector corresponding to λi (y): DF (y)ri (y) = λi (y)ri (y)

(i = 1, 2, y ∈ R2 ).

Then F is genuinely nonlinear provided Dλi (y) · ri (y) 6= 0

(i = 1, 2, y ∈ R2 ).

Finally, a smooth function wi : R2 → R is an ith Riemann invariant if dwi (y) · ri (y) = 0

(y ∈ R2 ).

We henceforth assume that (w1 , w2 ) are Riemann invariants and write w = (w1 , w2 ). Then the mapping y 7→ w(y) yields a smooth global change of coordinates on R2 , and we can consequently regard F, λi , ri (i = 1, 2) as being functions of w. The condition of genuine nonlinearity then reads (5.39)

∂λi (w) 6= 0 wj

(w ∈ R2 , i, j = 1, 2, i 6= j).

67

5.4. CONSERVATION LAWS

Theorem 5.4.2 Assume the system of PDE’s is strictly hyperbolic and genuinely nonlinear. Then u is a weak solution of (5.36). Outline of proof. 1. For a.e.(x, t) ∈ R × (0, ∞), let νx,t be the Young measure ∗ on R2 associated with the weak ∗ convergence uεk * u. 2. Let us say that two smooth functions Φ, Ψ : R2 → R comprise an entropy\entropy flux pair for the system (5.36) if Φ is convex and DΦ(y)DF (y) = DΨ(y)

(y ∈ R2 ).

Assuming now that (Φ, Ψ) is such a pair, take the dot product of (5.35) with DΦ(uε ) to discover (5.41)

Φ(uε )l + Ψ(uε )x = εΦ(uε )xx − εulx D 2 Φ(uε )uεx .

Utilizing estimates (5.38) we deduce as in the previous proof that the right−1,2 hand side of (5.41) is precompact in Wloc (R × (0, ∞)). 3. Suppose further that (Φ1 , Ψ1 ) and (Φ2 , Ψ2 ) are two entropy \entropy flux pairs. Recalling the identity (5.41) for each pair, we deduce from the Div-Curl Lemma applied to vk ≡ (Ψ2 (uεk ), Φ2 (uεk ))

and

wk ≡ (Φ1 (uεk ), −Ψ1 (uεk ))

that ∗

vk · wk * v · w

in L∞ (R × (0 × ∞))

for v ≡ (Ψ2 , Φ2 ), w ≡ (Φ1 , −Ψ1 ). On the other hand, ∗

vk · wk * Φ 1 Ψ2 − Φ 2 Ψ1

in L∞ (R × (0, ∞)).

Combining we deduce (5.42)

Φ 1 Ψ2 − Φ 2 Ψ1 = Φ 1 Ψ2 − Φ 2 Ψ1 .

Our intention now is to find enough entropy\entropy flux pairs which can be employed in identity (5.42) to deduce ultimately that the Young measure νx,t is a point mass. 4. Switching to the Riemann invariants w = (w1 , w2 ) as coordinates in R2 , we find after a calculation that (Φ, Ψ) is an entropy\entropy flux pair provided (5.43)

∂Ψ ∂Φ = λ2 , ∂w1 ∂w1

∂Ψ ∂Φ = λ1 ∂w2 ∂w2

(w ∈ R2 );

68

CHAPTER 5. COMPENSATED COMPACTNESS

here we regard Φ, Ψ, λi (i = 1, 2) as functions of w, According to Lax [82] we can find for each nonzero integer k a pair of solutions (Φk , Ψk ) of (5.43) having the asymptotic form  1 1   Φk = ekw1 (A0 + A1 + O( 2 )) k k (k = ±1, ±2, · · · ), (5.44) 1   Ψk = ekw1 (B0 + B1 + O( 1 )) k k2 where A0 , A1 , B0 , B1 are smooth functions of w and do not depend on k. Since Φk is convex, we have

(5.45)

A0 > 0

in R2 .

Inserting Φ = Φk , Ψ = Ψk into (5.43) and matching coefficients of powers of k −1 , we discover (5.46)

B 0 = λ 2 A0 ,

∂B0 ∂A0 = λ1 , ∂w2 ∂w2

B1 +

∂B0 ∂A0 = λ2 (A1 + ). ∂w1 ∂w1

From these relations we further deduce (5.47)

B 1 − λ 2 A1 = −

∂λ2 A0 . ∂w1

5. Define R ≡ {w ∈ R2 | w1− ≤ wi ≤ w1+ (i = 1, 2)} to be the smallest rectangle in R2 containing the support of νx,t . We want to show that R is in fact a point. Let us therefore assume instead, say, w1− < w1+ . 1 Now for large |k|, (5.44) and (5.45) imply Φk = ekw1 (A0 +O( )) is nonnegative k on R. Consequently. Z 1 µk (E) = Φk dνx,t (E a Borel subset of R2 ) Φk E is a probability measure on R2 . Hence there exist subsequences {µkj }∞ j=1 ⊂ −∞ ∞ ∞ + − 2 {µk }k=1 , {µ−kj }j=1 ⊂ {µk }k=−1 and probability measures µ , µ on R such that µkj * µ+ , µ−kj * µ− weakly in M(R2 ). From (5.44) and (5.45) we see that sptµ+ ⊂ R ∩ {w1 = w1+ },

(5.48)

sptµ− ⊂ R ∩ {w1 = w1− }.

Define (5.49)

λ+ 2

≡

Z

+

R

∩ {w1 =

w1+ }

λ2 dµ ,

λ− 2

≡

Z

R

∩ {w1 =

w1+ }

λ2 dµ− .

69

5.4. CONSERVATION LAWS

6. Now suppose (Φ, Ψ) is any entropy\entropy flux pair. Set (Φ1 , Ψ1 ) = (Φ , Ψk ) and (Φ2 , Ψ2 ) = (Φ, Ψ) in (5.42) to obtain k

Φk Ψ − ΦΨk = Φk Ψ − ΦΨk .

(5.50)

Owing to the asymptotic expansion (5.44) and to (5.46) we have Z −1 1 Ψk (Φk ) = λ2 dµk + O( ). k R2 Similarly, Φk Ψ − ΦΨk (Φk )

−1

=

Z

1 Ψ − Φλ2 dµk + O( ). k R2

Thus dividing (5.50) by Φk and sending k = ±kj → ±∞ yields Z (5.51) Ψ − Φλ2 dµ± = Ψ − Φλ± 2. ± R ∩ {w1 = w1 } Next set (Φ1 , Ψ1 ) = (Φk , Ψk ), (Φ2 , Ψ2 ) = (Φ−k , Ψ−k ) in (5.42), and divide by Φk Φ−k : Φk Ψ−k − Φ−k Ψk (Φk Φ−k )−1 = Ψ−k (Φ−k )−1 − Ψk (Φk )−1 . + We send k = ±kj + ±∞. As above, the right-hand side converges to λ− 2 − λ2 . The numerator on the left-hand side is of order kj−1 , and the denominator is of order expkj (w1+ − w1− ) → ∞. Hence the left-hand side goes to zero; whence − λ+ 2 = λ2 . 7. Recalling (5.51) we have Z Z + Ψ − Φλ2 dµ = Ψ − Φλ2 dµ− + − R ∩ {w1 = w1 } R ∩ {w1 = w1 }

for any entropy\entropy flux pair (Φ, Ψ). Set (Φ, Ψ) = (Φk , Ψk ) and recall (5.44), (5.46) to discover Z 1 B1 − λ 2 A1 kw1+ e + O( )dµ+ 2 + k k R ∩ {w1 = w1 } Z B1 − λ 2 A1 1 kw1− =e + O( 2 )dµ− . − k k R ∩ {w1 = w1 } Since w1− < wl+ , it follows that Z R

∩ {w1 =

w1± }

B1 − λ2 A1 dµ± = 0.

70

CHAPTER 5. COMPENSATED COMPACTNESS

However, (5.47) then forces Z

∂λ2 A0 dµ± = 0, ± ∂w 1 R ∩ {w1 = w1 }

an impossibility since A0 > 0 and ∂λ2 /∂w1 does not change sign, according to the genuine nonlinearity condition (5.39). Thus our assumption w1− < w1+ is untenable, and similarly we exclude the possibility w2− < w2+ . Consequently, R is in fact a point, and so νx,t = δu(x,t) . Using (1.29) we now easily deduce (5.37). 

5.5

Generalization of Div-Curl Lemma

In view of the great utility of the Div-Curl Lemma in § 5.3 and §5.4 above, it is natural to try to generalize. As in Tartar [118], let us consider now a sequence {uk }∞ k=1 satisfying ∗

(5.52)

in L∞ (U ; Rm ),

uk * u

and suppose also

∗

F (uk ) * F

in L∞ (U )

for some continuous function F : Rm → R. we ask what further information on {uk }∞ k=1 and on F will allow us to deduce F = F (u). To be specific, let us suppose that we are given the N first-order, constant coefficient differential operators j Ai v ≡ aijl vxl

(i = 1, · · · , N ),

define for v = (v 1 , · · · , v m ). We will then suppose (5.53)

−1,2 {Ai uk }∞ (U ), i = 1, · · · , N. k=1 lies in a compact subset of W

Also assume we have (5.54)

uk ∈ K a.e.

(k = 1, · · · ),

for some given subset K ⊂ Rm . In practice (5.53) corresponds to various conservation laws and (5.54) to nonlinear constitutive relations satisfied by our functions {uk }∞ k=1 . We propose to investigate the extent to which (5.53) and (5.54) allow for or preclude oscillations. consider then the wave cone (5.55) Λ ≡ {λ ∈ Rm | there exists ξ ∈ Rn , ξ 6= 0, with aijl λj ξl = 0, for i = 1, · · · , N }.

5.5. GENERALIZATION OF DIV-CURL LEMMA

71

The idea is that Λ contains information regarding possible large one dimensional oscillations which are compatible with the differential constraints (5.53). Indeed if λ ∈ Λ, λ 6= 0, let us define uk (x) ≡ ρ(k(x · ξ))λ

(x ∈ U, k = 1, · · · ),

where ρ is the sawtooth function introduced in Chapter 2 and ξ 6= 0 is associ∗ ated with λ as in (5.55). Then uk * 0 in L∞ (U ; Rm ), but uk → 0 in Lq (U ; Rm ) for any 1 ≤ q ≤ ∞. On the other hand, for i = 1, · · · , N . Ai uk = aijl (ρ(k(x · ξ))λj )xl = ρi (k(x · ξ))aijl λj ξl = 0 a.e. Consequently, we see that conditions (5.53) do not exclude certain large amplitude, high frequency oscillations, as determined by the structure of the wave cone Λ.Consequently, if we except to deduce F = F (u) from (5.52)-(5.54) we must surely hypothesize some kind of good behavior of F on Λ to compensate for possible oscillations as above. There are no known general conditions to ensure this, but if F is quadratic, the following useful result holds: Theorem 5.5.1 Assume F : Rm → R is quadratic. (i) If F (λ) = 0 for all λ ∈ Λ, then F = F (u), (ii) If F (λ) ≥ 0 for all λ ∈ Λ, then F ≥ F (u). The proof follows from the decomposition into plane waves afforded by the Fourier transform: see Tarter [118] for details. A further generalization, which identifies strong connections with the theory of quasiconvexity set forth in Chapter 3, is to be found in Dacorogna [37]. See also [118] for a counterexample to Theorem 5.5.1 if F is not quadratic. Robbin,Rogers, and Temple [102] exploit the Hodge decomposition to extend the proof of the Div-Curl Lemma to differential forms.

Chapter 6 Maximum Principle Methods In our investigations thus far we have made as yet no use of the maximum principle tool for studying many nonlinear PDE’s. In this chapter we at last invoke the maximum principle and employ it to justify weak convergence techniques (rather than to provide estimate, its more customary role).

6.1 6.1.1

The Maximum Principle for Full Nonlinear PDE Minty-Browder method in L∞ .

To find a clue as to how the maximum principle may be relevant for weak convergence, let us consider as a model problem the fully nonlinear PDE

(6.1)

(

F (D 2 u) = f in U, u = 0 on ∂U.

where F : S n×n → R is given. S n×n denoting the space of real, symmetric n × n matrices. We will assume that problem (6.1) is elliptic, which means F is monotone decreasing with respect to matrix ordering in S n×n , and so (6.2)

F (S) ≤ F (R)

if S ≥ R(R, S ∈ S n×n ).

Now suppose fk → f uniformly. Assume additionally for k = 1, . . . that uk is, say, a smooth solution of

(6.3)

(

F (D 2 uk ) = fk in U, uk = 0 on ∂U. 72

6.1. THE MAXIMUM PRINCIPLE FOR FULL NONLINEAR PDE

73

2,∞ and {uk }∞ (U ). We may thus extract a subsequence k=1 is bounded in W with

(6.4)

u kj → u

uniformly.

D 2 ukj *D ˙ 2 u in L∞ (U ; S n×n )

and as usual ask: does u satisfy (6.1)? (If F is uniformly elliptic and convex, then strong estimate available (cf. Gilbarg-Trudinger [6.8]) and passing to limits is simple. The main interest is consequently for nonconvex F .) Now this is net unlike the convergence problem for divergence structure quasilinear PDE’s solved in Chapter5,§A.3 using the Minty-Browder device, and obviously ellipticity is some kind of monotonicity condition. To get some further insight, let us recast the method of Chapter 5,§A.3 into abstract form, as follows. Suppose we are given a sequence of approximate problems in a Hilbert space H, A[uk ] = fk (k = 1, . . . ), where A[·], a given operator mapping its domain D(A) ⊂ H into H, satisties the monotonicity condition (6.5)

0 ≤ (A[u] − A[v], u − v)

(u, v ∈ D(A)).

Thus putting u = uk we have 0 ≤ (A[v] − fk , v − uk )

(k = 1, . . . ),

and so in the limit 0 ≤ (A[v] − f, v − u), provided fk → f , uk → u in H. We set v − u + λw,divide by λ > 0,and then send λ → 0 to deduce 0 ≤ (A[u] − f, w) for all w ∈ D(A). (We have implicity supposed A[·] to be continuous on finite-dimensional subspace of H. This is a reasonable hypothesis whenever A[·] represents a differential operator.) Then assuming D(A) is dense, we deduce A[u] = f

74

CHAPTER 6. MAXIMUM PRINCIPLE METHODS

Now unfortunately our operator A[u] ≡ F (D 2 u) does not satisfy (6.5) for all smooth function u and v, where (,) is the inner product in H = L2 (U ). On the other hand, solutions of elliptic PDE’s should satisfy maximum and comparison principles, which are pointwise in nature. This suggests that we refocus attention from L2 (U ) to L∞ (U ) or, better, C(U¯ ). We ask whether A[·] is perhaps ”monotone in C(U¯ )”. As there is no inner product in C(¯ u), the very definition presents a problem, however. But observe that if X is any real Banach space, the limit

(6.6)

[f, g]+ ≡ lim+ λ→0

kg + λf k2 − kgk2 2λ

exists for all f, g ∈ X.The pairing [, ]+ so defined is simply the inner product if X = H, a Hilbert space. But more interesting is the possibility of regarding an operator A[·], mapping its domain D(A) ⊂ Xinto X, as being ”monotone in X” provided 6.7

0 ≤ [A[u] − A[v], u − v], (u, v ∈ D(A))

We now examine (6.6), (6.7) for the special case X = C(barU ). Then it turns out that (6.8)

¯ , g(x0 ) = kgkC(U¯ ) }. [f, g]+ = max{f (x0 )g(x0 )|x0 ∈ U

Theorem 6.1.1 The operator A[u] ≡ F (D 2 u), defined for C 2 functions vanish¯ ). ing on ∂U ,satisties the ”monotonicity” inequality (6.7) in C(U Proof. Assume u, v ∈ C 2 (U ), u = v = 0 on ∂U . Pick any point x0 ∈ U where (u − v)(x0 ) = ku − vkC(U¯ ) . Then D 2 (u − v)(x0 ) ≤ 0 and so ellipticity implies F (D 2 u(x0 )) ≥ F (D 2 v(x0 )). Thus (A[u]−A[v])(u−v) ≥ 0 at x0 . The same inequality obtains if (v − u)(x0 ) = ku − vkC(U¯ ) ; and so, in light of (6.8), we have verified (6.7).  Next, let us attempt to modify the abstract Minty-Browder method set forth above to the case at hand. Using (6.3) and (6.7), we have 0 ≤ [fk − A[v], uk − v],

(k = 1, . . . ),

As [, ]+ is upper semicontinuous, we deduce upon sending k = kj → ∞ that (6.9)

0 ≤ [fk − A[v], uk − v], (k = 1, . . . ).

6.1. THE MAXIMUM PRINCIPLE FOR FULL NONLINEAR PDE

75

For all C 2 functions v vanishing on ∂U . But now we encounter another problem, since u is not smooth enough to justify our setting v = u + λw, as above. We proceed differently, as follows: Theorem 6.1.2 The function u solves the PDE(6.1) a.e. Proof. 1. As above we arrive at the inequality (6.9) for all C 2 functions v vanishing on ∂U . The plan is to design test functions v from which we can read off useful information. 2. Since u ∈ W 2,∞ (U ),Rademacher’s Theorem implies u is twice differentiable in the classical sense a.e. Fix any point x0 ∈ U where D 2 (x0 ) exists. We then handcraft a c2 function v having the form 1 v(x) = u(x0 ) + Du(x0 ) × (x − x0 ) + (x − x0 )T D 2 u(x0 ) + ε|x − x0 |2 2 for x near x0 , and such that |u − v| attains its maximum over U¯ only at the point x0 . But then (6.8) and (6.9) say f − A[v] ≥ 0 at x0 , that is, F (D 2 u(x0 )) + 2εI) ≤ f (x0 ) sending ε → 0 we find F (D 2 u(x0 )) ≤ f (x0 ). The opposite inequality follows similarly. Consequently,F (D 2 u(x0 )) = f (x0 ) for a.e. x0 ∈ U .  The preceding proof can be made much simpler by recasting it with the abstract apparatus discarded. For variety, let us consider the problem of justifying the vanishing viscosity method for Hamilton-Jacobi PDE’s. We study the approximating problems (6.10)

uεt + H(D, uε) = ε∆uε

in Rn × (0, ∞)

for ε > 0, H : Rn → R being given. We assume we have the estimates sup kuε kW 1,∞ (0,∞) < ∞, ε>0

so that we can extract a subsequence {uxk }∞ k=1 with uxk → u locally uniformly,

∗

Duxk * Du in L∞ (Rn × (0, ∞); Rn+1 .

Does it follow that (6.11)

ut + H(D, u) = 0

in Rn × (0, ∞)?

76

CHAPTER 6. MAXIMUM PRINCIPLE METHODS

Theorem 6.1.3 The function u solves the Hamilton-Jacobi PDE (6.11) a.e. Proof. 1. Fix a C 2 function v and suppose (6.12) u−v has a local maximum at a nearby point (xεk , xεk ),with (xεk , xεk ) → (x0 , t0 ) as k → ∞. At (xεk , xεk ) we have Dx (uεk − v) = 0,

()(uεk − v)t = 0,

Dx2 (uεk − v) = 0.

Thus from (6.10) we deduce vt + H(D, v) = uεt k + H(D, uεk ) = εk ∆uεk ≤ εk ∆v at the point (xεk , xεk ). Sending k → ∞ we find (6.13)

vt + H(Dx v) ≤ 0

at (x0 , t0 ).

This inequality holds, by approximation, even for C 1 function v satisfying (6.12). 2.Since u is Lipschitz, u is differentiable for a.e. point (x0 , t0 ). A consequence is that we can design for such a point a C 1 function v verifying (6.12); see[50]. Thus (6.13) says ut + H(D, u) ≤ 0 at (x0 , t0 ), and the opposite inequality is similarly proved.  We have now managed to pass to weak limits from (6.3) to (6.1) and from (6.10) to (6.11), deducing that the limit functions solve the proper PDE’s a.e. These facts alone turn out to be unsatisfactory since there can be for instance infinitely many Lipschitz functions with the same initial data solving (6.11) a.e. What we really require is a uniqueness assertion as explained in the next section.

6.1.2

Viscosity solutions.

Crandall and Lions [36] have observed that we can in fact eliminate all mention of derivatives in the limiting arguments in §1 provided we regard u as an appropriately defined weak solution. Definition 6.1.1 Assume u ∈ C(U¯ . We say u is a weak solution of the fully nonlinear elliptic PDE (6.1) provided for all v ∈ C 2 (U ) ( if u − v has a local maximum (minimum) at a point x0 ∈ U then F (D 2 v) ≤ f (≥ 0) at x0

6.2. HOMOGENIZATION OF NONDIVERGENCE STRUCTURE PDE’S

77

Definition 6.1.2 Assume u is bounded and uniformly continuous on Rn × (0, ∞). We say u is a viscosity solution of the Hamilton-Jacibi PDE (6.11) provided for all v ∈ C 2 (Rn × (0, ∞)) ( if u − v has a local maximum (minimum) at a point (x0 , t0 ) ∈ Rn × (0, ∞), then F (D 2 v) ≤ f (≥ 0) at x0 These definition have obvious extension when the PDE’s depend on x or t, or involve lower order terms. What we have really shown above is that (a) the limit of the approximations is a viscosity solution of the appropriate PDE, and (b) if a viscosity solution is appropriately at a point PDE there. The other most important fact is (c) a viscosity solution is unique, subject to appropriate boundary conditions. This very important assertion was established for firstorder equations of Hamilton-Jacobi type by Crandall-lions [36] and for secondorder equations of elliptic and parabolic type by Jensen [73].

6.2

Homogenization of Nondivergence Structure PDE’s

As a further application of maximum principle \ viscosity solution techniques to justify limiting processes let us now address the periodic homogenization problem for nondivergence structure quasilinear elliptic PDE’s of the form

(6.14)

  − aij (x, x , Duε )uε = b(x, x , Duε ) xi xj ε ε  uε = 0 on ∂U

in U

Here aij ,a : R × Y × Rn → R(1 ≤ i, j ≤ n), where Y is the unit cube in Rn . Suppose the ellipticity condition ξ T A(x, y, p)ξ ≥ v|ξ|2

(x ∈ U, y ∈ Y, p, ξ ∈ Rn )

and the growth bounds |A[x, y, p]| ≤ C, |b(x, y, p)| ≤ C(1 + |p|2 ). We suppose additionally the mapping y → A(x, y, p),

b(x, y, p) are Y -periodic.

Now for each fixed x ∈ U and p ∈ Rn , we consider the the linear adjoint problem

78

CHAPTER 6. MAXIMUM PRINCIPLE METHODS (

− (aij (x, y, p)m(p, x, y))yiyj = 0 m Y − periodic

in Rn ,

Z which possesses a positive solution m, unique subject to the normalization m(x, y, p)dy = 1(x ∈ U, p ∈ Rn ). Define the averaged coefficients Y Z e aij (x, p) ≡ aij (x, y, p)m(x, y, p)dy Y Z eb(x, p) ≡ b(x, y, p)m(x, y, p)dy Y

Consider next the homogenized problem (

(6.15)

−e aij (x, Du)uxi ,xj = eb(x, Du) u = 0 on ∂U.

in U,

Theorem 6.2.1 Assume {uε }ε>0 is bounded in L∞ (U ). Then there exists a ε εk subsequence {uεk }∞ → u k=1 ⊂ {u }ε>0 and a function u ∈ C(U ) such that u uniformly on U and u is a weak solution of (6.15). Proof.1. Elliptic estimates available under the quadratic growth condition imply that {uε }ε>0 is bounded in C 0,β (bar(U )) for some β > 0.We can thus secure a uniformly convergent subsequence {uεk }∞ k=1 . 2. Now fix v ∈ C 2 (U ) and suppose u − v has a strict maximum at x0 ∈ U . Applying the Fredholm alternative we find a smooth solution w of the corrector problem

(6.16) ( − aij (x0 , y, Dv(x0 ))wyi yj = [aij (x0 , y, Dv(x0 )) − e aij (x0 , Dv(x0 ))]vxi xj (x0 ) − [b(x0 , y, Dv(x0)) − eb(x0 , D w Y − periodic Define then the perturbed test function

x v ε (x) ≡ v(x) + ε2 w( ) ε ε εk εk since v → v uniformly as ε → 0, u − v has a local maximum at a point xεk , with xεk → x0 as k → ∞. By the classical maximum principle, −aij (xε , for ε = εk . Since

xε xε , Dv ε (xε ))vxε i xj ≤ b(xv arepsilon, , Dv ε(xε )). ε ε

79

6.3. SINGULAR PERTURBATIONS

Dv ε (xv arepsilon) = Dv(xε ) + εdw(

xε ), ε

D 2 v ε (xε ) = D 2 v(xε ) + D 2 w(

xε ) ε

we deduce −aij (x0 , yε , Dv(x0 ))(v( xi xj )(x0 ) + wxi xj (yε )) ≤ b(x0 , yε , Dv(x0 )) + o(1) xε for yε ≡ . We apply the auxiliary PDE (6.16) at the point y = yε and ε simplify to discover in the limit as ε = εk → 0 that −e aij (x0 , Dv(x0 ))vxi xj (x0 ) ≤ eb(x0 , Dv(x0 )).

The opposite inequality is similarly proved should u − v have a strict minimum at x0 . By approximation we remove the requirement that the maximum and minimum be strict. Thus u is a weak solution of (6.15).  The reader should compare this proof with the techniques explained in Chapter 5,§A.2.

6.3

Singular Perturbations

For our last example we investigate the linear perturbation problem of ascertaining the limit as ε → 0 for solutions uε of

(6.17)

  Lε uε ≡ − ε ∆uε − b · Duε = 0 2  uε = g on ∂U.

in U,

Here U is connected. We assume the point 0 ∈ U is the unique attracting point for the vector field b ∈ C 1 (U¯ ; Rn ), in the sense that trajectories of the ODE y(t) ˙ = b(y(t))

(t ∈ R)

converge to zero as t → ∞, andb, v < 0 on ∂U , v denoting the outward unit normal. Assume also g : ∂U → R is continuous. Then clearly the sequence {uε }ε>0 is bounded, but we have absolutely no uniform estimates on {Duε }ε>0 Following Freidlin-Wentzell [6.3] we introduce the quasipotential   Z τ 1 2 V (x) ≡ inf |y(t) ˙ − b(y(t))| dt | y(0) = 0, y(τ ) = 0 2 0

80

CHAPTER 6. MAXIMUM PRINCIPLE METHODS

the infimum taken over all times τ > 0 and function y[·] ∈ W 1,2 ((0, τ ); Rn ) satisfying the stated initial and terminal conditions. The idea is that V (x) measures the least energy in going upstream, that is, against the flow determined by vector field b, of curves connecting 0 to x. Theorem 6.3.1 Assume there exists a unique point x0 ∈ ∂U with V (x0 ) = min V (x) x∈∂U

Then uε → g(x0 ) ¯ as ε → 0. uniformly on U Outline of proof.1.As in Matkowsky-Schuss [91], Kamin [74], and Perthame [100] we introduce the auxiliary function v ε solving  ε   L∗ε v ε ≡ − ∆v ε + div(bv ε ) = o 2 ε ε ∂v   = (b · v)v ε on ∂U 2 ∂v

(6.18)

in U

it turns out that v ε > 0 in U¯ . Furthermore we have

(6.19) 0 =

Z

v

vε L| varepsilonu arwpsilon − u U

ε

L∗ε v ε dx

ε =− 2

Z

Now Kamin [74] has shown ε ∂uε + (g − uε (0))b · 0. 2 ∂v uniformly on ∂U ; and so (6.19) yields

(6.20)

uε (0) =

Z

vε gb · ve ε dH n−1 ∂U

Z

vε b · ve ε dH n−1

+ o(1)

∂U

where V ε ≡ −ε log v ε + ε log v ε (0)

v v arepsilon ∂U

∂uε dH n−1 εv

81

6.3. SINGULAR PERTURBATIONS

Since |uε − uε (0)| → 0 uniformly on U , formula (6.20) yields the required result, once we verify (6.21)

Vε →V

uniformly on ∂ U¯

2 From (6.18) and the definition of V ε we have  |DV ε |2 ε   − ∆V ε + + b · Dv ε = ε ÷ b in U, 2 2 ε  ∂u  + 2b · v = 0 on ∂U, V ε (0) = 0 εv ∞ We utilize the machinery assembled in §A to find a subsequence {V εk }k=1 ⊂ {V k }ε>0 such that V εk → W uniformly, We solving

(6.22)

|DW |2 + b · DW = 0 2

in U.

Additionally,it tuans out that W satisfies the boundary condition

(6.23)

|DW |2 + b · DW > 0 2

on ∂U.

This suitably interpreted in the viscosity sense. From (6.22) (6.23), and uniqueness for viscosity solutions it is possible to deduce W = V , thereby verifying (6.21). The exact calculations are rather subtle and beyond the scope of these notes;see Perthame [100] for details.  Our limiting problem (6.17) has a probabilistic interpretation. We consider the ODE y˙ = b(y) disturbed by ε times ”white noise” and ask about the asymptotic exiting behavior of the resulting random trajectories. The assertion is that for small noise the random trajectories will mostly exit U near the point x0 ∈ ∂U , where the energy required to move against the deterministic flow is least. This is an instance of a large deviation phenomenon in probability theory, here studied by purely PDE techniques.

Chapter 7 Appendix If 0 ≤ s < ∞,0 < δ ≤ ∞,we defined the s − dimensional Hausdorff measure Hδs by setting ) (∞ X diamC j )s |A ⊂ ∪∞ Hδs ≡ inf α(s)( j=1 Cj , diamC ≤ δ 2 j=1 s π2 for each subset A ⊂ Rn , where α(s) ≡ s . Γ( + 1) 2 Then s − dimensional Hausdorff measure H s is given by H s (A) = lim Hδs (A) = sup Hδs (A) δ→0

δ>0

(A ⊂ Rn ).

If 1 ≤ p < n we defined p-capacity Capp by writing

Capp (A) = inf{

Z

n ∗

|Df |pdx | f ∈ Lp (Rn ), Df ∈ Lp (Rn ), A ⊆ {f ≥ 1}0 } R

for each A ⊆ Rn , the superscript0 denoting the interior. If f ∈ L1loc (Rn ), we defined its precise representative to be ( lim fB(x,r) f dy if this limit exists, f ∗ (x) ≡ r→0 0 otherwise 1,p Noting f ∗ = f a.e. Furthermore if f ∈ Wloc (Rn ), then the limit above exists except for points x in a set of p-capacity zero. When we speak of pointwise properties of a function we will always mean the pointwise properties of its precise representative. In particular, it can be shown that

82

83

Capp ({f ≥ 1}) ≤ C ∗

Z

|Df |pdx Rn

for each f ∈ Lp (Rn ) with Df ∈ Lp (Rn mathbbRn ), the constant C depending only on p and n. See Federer-Ziemer [57], Ziemer [126], etc.

Notes Chapter1. See, for instance, Yosida [125]. Billingsley [20] [21], and Kushner [78] for more on weak and weak ∗ convergence, and Gilbarg-Trudinger [68] for Sobolev space. Corollary 1 is due to Murat [98]. Theorem 7 is due to Meyers [92] and to Frechse [61]. DiPerna and Majda [45-47] introduced the Term ”reduced defect measure” Theorem 8 is from Brezis-Lieb [30]. The proof of Theorem 9 is a variant of P.L.Lions [88]; see Federer [56] for the differentiation theorem for Radonmeasure. Theorem 10 is modified from Allard [3,§3.3]. Schulz and Strohmer pointed out to me the connection with regular conditional probabilities in probability theory, cf. Breiman [24,pp.77-78]. Bourbaki in [23,pp.70-72], Ball in [14] and Kinderlehrer (in unpublished notes) have set forth careful function-theoretic proofs. Tartar introduced Young measures into nonlinear PDE theory in [118].

Chapter2. See Brezis [25], Ekeland-Temam [48], and Rockafellar [104] for more on convex analysis. See Ladyzenskaja-Ural’ceva [80] for another proof of the sufficiency in Theorem 1.

Chapter3. See Morrey [93,94], Ball [11,12], Dacorogna [37], Knops-Stuart [76], Giaquinta-Soucek [67], and Stvaloganathan [110] for more on quasiconvexity. Lemma 1 is due to Marcellim [90], Theorem 1 is primarily due to Morrey [93]; the arguments involving the measures are taken from EvanxGariepy [54]; See Acerbi-Fusco [1] for a very general assertion. Theorem 2 and 3 appear in [54]. The proof of Lemma 2 follows Evans-Gariepy [53]. See also Evans [51], Fusco-Hutchinson [64], Giaquinta-Modica [66], Acerbi-Fusco [2], etc. The partial regularity Theorem 4 is strongly reminiscent of Almgren’s proof [4] of partial regularity for elliptic integrands in geometric measure theory. lemma 3 is classical ; cf. Morrey [94]. J.Ball [11] coined the term ”polyconvex.” 84

85

Chapter4. Theorem 1 is due to P.L. Lions [88]; see also his paper [87] for many further application of the ”concentration-compactness” principle. Theorem 2 is from Brezis-Nirenberg [31] and exploits ideas of Aubin [6,7]. Other interesting papers on concentration and critical Sobolev exponents include Bahri-Coron [9,10], Brezis-Coron [28,29], Struwe [114-116]. Theorem 3 and 4 are classical; see, for instance, Ladyzenskaja-Ural’ceva [80], Frehse [62], Bensoussan-Boccarda-Murat [18],Puel [101], etc. Theorem 5 is due to Frehse [62], but I have recast it so as to emphasize similarity with DiPerna-Majda [45], Theorems 6 and 7. Part of the proof of Theorem 7 is based upon Federer-Ziemer [57]. Note also Landies [79].

Chapter5. Theorem 1 is extracted from Chen [33] and Shatah [109], and Theorem 2 is due to Tartar, see also Chapters 1 and 2 of Bensoussan-LionsFapanicolaou [19], and the references Therein, for further information regarding energy methods in homogenization. Note also Avellaneda-Lin [8], Farbu [16], Brezis[25], Leray-Lions [84], and J.L.Lions [85,Chapter 2] provide a good introduction to monotonicity methods for PDE’s. The Div-Curl lemma is due to Murat [95], and extensive generalizations are in Tartar [118], Robbin-Rogers-Temple [102], Rogers-Temple [103], etc. Theorem 5 is a special case of Tartar [120,Theorem 3]. Theorem 6 is from Tartar [118]. Theorem 7 is due to DiPerna [40]; see also Tartar’s exposition in [120], which I have followed closely. Other relevant papers are Dafermos [38], Marcati-Milani-Secchi [89], Murat [99], Serre [108], Slemrod [111], Tartar [121,122], etc.

Chapter6. The extension of the Minty-Browder device to L∞ and C is from Evans [49], which contains the proof of Theorem 2. The proof of Theorem 3 is in Evans [50]. An operator satistying the ”monotonicity” condition (6.7) is called accretive; see Barbu [16] for general theory. The uniqueness proofs are in Crandall-Lions [36] and Jensen [73]. See also Crandall-Evans-Lions [35], Ishii [70], and Ishii-Lions [72]. More on viscosity solutions may be found in Lions [86], Souganidis [112], Ishii [71], BarlesPerthame [17], etc. Evans [52] has the proof of Theorem 4, but see also Bensoussan-BoccardoMurat [18] and Boccardo-Murat [22] for a proof using energy methods. See Perthame [100] for full detail of full detail of The proof of Theorem 5, due originally to Freidlin-Wentzell (cf. [63]). Also relevant are Evans-Ishii [55]

86 and Fleming-Souganidis [58].

CHAPTER 7. APPENDIX