Lower semicontinuity of <span class="mathjax-formula">$L^\infty $</span> functionals

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LOWER SEMICONTINUITY OF L

^∞

FUNCTIONALS

E.N. BARRONâ,1, R.R. JENSENâ,1, C.Y. WANGâ,b,2

aDepartment of Mathematical & Computer Sciences, Loyola University Chicago, Chicago, IL 60626, USA

bDepartment of Mathematics, University of Kentucky, Lexington, KY 40506, USA Received 28 August 2000

ABSTRACT. – We study the lower semicontinuity properties and existence of a minimizer of the functional

F (u)=ess sup

x∈

f (x, u(x), Du(x))

onW^1,^∞(;R^m). We introduce the notions of Morrey quasiconvexity, polyquasiconvexity, and rank-one quasiconvexity, all stemming from the notion of quasiconvexity (= convex level sets) of f in the last variable. We also formally derive the Aronsson–Euler equation for such problems.

2001 Éditions scientifiques et médicales Elsevier SAS AMS classification: 49A50; 49A51

Keywords: Lower semicontinuity; Morrey convex; Morrey quasiconvex; Polyquasiconvex;

Rank-one quasiconvex

RÉSUMÉ. – On examine les propriétés de semi-continuité inférieure et l’existence du minimizeur de la fonctionnelle

F (u)=ess sup

x∈

f (x, u(x), Du(x))

surW^1,^∞(;R^m). On introduit les idées du quasi-convexité de Morrey, du polyquasi-convexité, et du quasi-convexité du rang-un , qui suivent tous de l’idée de quasi-convexité ( = les ensembles à niveau convexes) def à la dernière variable. En plus, on en déduit dans les formes l’équation d’Aronsson–Euler pour de tels problèmes. 2001 Éditions scientifiques et médicales Elsevier SAS

1. Introduction

The major area of study in calculus of variations in the last thirty years has been the study of variational problems with vector valued functions and the associated necessary

E-mail addresses: enb@math.luc.edu (E.N. Barron), rrj@math.luc.edu (R.R. Jensen), wang@math.luc.edu (C.Y. Wang).

1Supported in part by grants DMS-9532030 and DMS-9972043 from the National Science Foundation.

2Supported in part by NSF grant DMS-9970549 and an AMS Centennial Fellowship.

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and sufficient conditions for lower semicontinuity of integral functionals. It is our goal to consider this problem for essential supremum functionals.

It is well known that for integral problems these necessary and sufficient conditions involve notions of convexity in some form. This has led to concepts of quasiconvexity, polyconvexity, and rank-one convexity. In this paper we change the terminology slightly to account for multiple uses of the word quasiconvexity. In place of quasiconvex when it relates to vector valued variational problems we use the term Morrey convex in honor of the founder of this condition. Here is the definition of Morrey convexity which is well known (see Dacorogna [9] and the references there) to be a necessary and sufficient for weakW^1,plower semicontinuity of the integral functional

F (u)=

f (Du)dx.

DEFINITION 1.1. – LetR^nmdenote the class of alln×mmatrices with real entries.

A functionf:R^nm→Ris Morrey convex if for eachA∈R^nm 1

||

fA+Dϕ(x)dxf (A),

for each, a bounded domain inRⁿ, andϕ∈W₀^1,^∞(;R^m).Equivalently, f (A)=min

1

||

fA+Dϕ(x)dx|ϕ∈W₀^1,^∞;R^m.

Observe the connection with Jensen’s inequality for convex functions. Indeed, this inequality has been fundamental to the existence theory in variational problems. In Barron, Jensen, Liu [7] we have derived an extended Jensen inequality which is just as fundamental for variational problems inL^∞. It applies to quasiconvex functions, i.e., functions with convex level sets. In symbols,f is quasiconvex ifEγ= {x∈Rⁿ|f (x) γ}is convex for anyγ ∈R;equivalently,

fλx+(1−λ)yf (x)∨f (y), ∀x, y∈Rⁿ, λ∈(0,1).

Here is our extended Jensen inequality and its short proof.

THEOREM 1.2. – Let f:⊂Rⁿ→R be lower semicontinuous and quasiconvex and letµbe a probability measure onRⁿsupported on. Letϕ∈L¹(;µ)be a given function. Then

f

ϕdµ

µ−ess sup

x∈

fϕ(x). Theµ-essential supremum means we exclude sets ofµ-measure zero.

Proof. – Defineγ =µ−ess sup_x_∈f (ϕ(x)) and Eγ = {q: f (q)γ}. Then forµ- a.e.x∈,ϕ(x)∈Eγ. Sincef is lsc and quasiconvex,Eγ is a closed convex set. Hence, sinceµis a probability measure,ϕdµ∈Eγ and the theorem is proved. ✷

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Notice that a quasiconvex function may be neither continuous nor even lower semicontinuous.

We now introduce a condition similar to Morrey convexity which we will see is (almost) necessary and sufficient for weak-* lower semicontinuity of functionals of the form

F (u)=F (u, )=ess sup

x∈

fDu(x).

DEFINITION 1.3. –f: R^nm→Ris (weak) Morrey quasiconvex if for eachA∈R^nm, f (A)ess sup

x∈Q

f (A+Dϕ), ∀ϕ∈W₀^1,^∞Q;R^m. Equivalently,

f (A)=min ess sup

x∈Q

f (A+Dϕ):ϕ∈W₀^1,^∞Q;R^m, whereQ= [0,1]ⁿ⊂Rⁿthroughout this paper denotes the standard unit cube.

The space of test functions in the classical definition of Morrey convexity may be replaced byϕ∈C^∞₀ (Q;R^m).It is clear that one cannot replace the class of test functions in this definition byC^∞₀ since we do not have uniform approximation of the derivatives.

The following lemma shows that our definition of Morrey quasiconvexity satisfies the analogue of the quasiconvexity definition in convex analysis, i.e., that the level sets of a quasiconvex function must be convex.

LEMMA 1.4. – f is Morrey quasiconvex iffEc= {A∈R^nm: f (A)c} is Morrey convex for everyc∈R. That is,

δ(A|Ec)=

0, if A∈Ec, +∞, if A /∈Ec, is a Morrey convex function for everyc∈R.

Proof. – Suppose that δ(A|Ec) is Morrey convex for every c ∈ R. If f is not Morrey quasiconvex at A∈ R^nm, there is an ε >0 and ϕ ∈W₀^1,^∞(Q;R^m) so that f (A)−εess sup_x_∈_Qf (A+Dϕ). Choosec=f (A)−ε. Thenδ(A|Ec)= +∞. Since δ(A|Ec)is Morrey convex, we conclude thatδ(A+Dϕ|Ec)= +∞on a subset ofQ with positive measure. Hence, cess sup_x_∈_Qf (A+Dϕ) > c, a contradiction and f must be Morrey quasiconvex.

Conversely, suppose f is Morrey quasiconvex at A∈R^nm and c∈R is arbitrary.

Suppose that f (A)c. Thenδ(A|Ec)=0 and immediately, sinceδ0,δ(A|Ec)is Morrey convex. Iff (A) > c thenδ(A|Ec)= +∞. Ifδ(A|Ec)is not Morrey convex, there is aϕ∈W₀^1,^∞(Q;R^m)so that_Qδ(A+Dϕ|Ec)dx=0.But then,f (A+Dϕ)c a.e. onQ, i.e., ess sup_x_∈_Qf (A+Dϕ)c < f (A), a contradiction to the assumption that f is Morrey quasiconvex. ✷

For lower semicontinuity of functionals onL^∞Morrey quasiconvexity is the analogue of Morrey convexity, but it turns out that for vector valued problems we need to modify

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the notion to weak and strong Morrey quasiconvexity. We refer to the definition given above as the weak version. The strong version permits some play in the test functionsϕ on the boundary of the domain and this is what we need to get the lower semicontinuity result we are after. In the scalar case, i.e., m=1 or n=1, weak and strong Morrey quasiconvexity are equivalent. It is an important open problem to determine if they are also equivalent in the vector case. We conjecture that they are not equivalent.

We also introduce the notions of polyquasiconvex and rank-one quasiconvex as generalizations of the classical notions polyconvex and rank-one convex. In the classical case we know that we have the implications

f convex⇒f polyconvex⇒f Morrey convex⇒f rank-one convex.

It is known that f Morrey convex does not imply f is polyconvex. The famous counterexample of Sverak [14] shows that

f rank-one convex⇒f Morrey convex,

at least when m3, n2. Obviously the question arises as to whether this can be extended to L^∞ but counterexamples, which are not easy to come by in the classical cases, are no easier in L^∞. As is usual in a paper of this type, more questions are raised than are answered but we hope to return to these questions in a future paper.

In particular we hope to resolve the relaxation question and the use of Young measures in such relaxations. Excellent references for these and many other considerations are Dacorogna [9] and Pedregal [13].

The main difficulty in dealing with L^∞ functionals is the fact that we do not have available to us the major tool used in L^p functionals, namely, the use of piecewise affine functions. Approximation of aW^1,^∞function on a domainby piecewise affine functions leaves a piece of small measure left over. In integral problems this portion is controlled since it is a small measure set, but in L^∞ small measure sets cannot be ignored, and functions on them cannot be controlled easily. Another idea which one might think of is the use of approximating L^∞ by L^p. In lower semicontinuity considerations this leads to then studying an iterated limit limk→∞limp→∞ – in which there is no reason to believe that the limits can be reversed.

Variational problems inL^∞were first studied systematically by Aronsson [1–3]. Then Jensen in [12] considered the uniqueness question for the Aronsson equation arising in the minimization of ess sup_x_∈|Du|²or ess sup_x_∈|Du|. Barron and Ishii [6] initiated the study of optimal control via viscosity solutions inL^∞and Barron and Liu [8] studied calculus of variations inL^∞in the scalar case from the point of view of relaxation and duality. A survey of the foregoing and many other results is in Barron [5].

Finally, the motivation to consider L^∞ variational problems is provided by simple examples without simple solutions. A typical vector valued problem arising in elasticity (see for example [9]) developed by Ball [4] is to minimize the integral

I (u)=

Wx, Du(x)+ψx, u(x)dx,

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where is the reference configuration of a given elastic material and u is the deformation of the body. Usually it is assumed that det(Du(x)) >0, x∈.The stored energy of the configuration is measured in the function W =W (x, Du(x)) and the function ψ measures the body force per unit volume. It would seem to make sense that one should replace the integral by the essential supremum in certain bodies in which cracks are a primary consideration since in fact cracks occur due to pointwise excessive energy. Pointwise considerations arise in many practical applications including temperature distribution, chemotherapy, risk management, etc., and of course it is a fundamental problem in Chebychev approximation of functions.

2. Morrey quasiconvexity and lower semicontinuity inL^∞

We begin with a precise definition of what we mean by Morrey quasiconvexity. Then we show that this condition is necessary and sufficient for lower semicontinuity.

DEFINITION 2.1. – A measurable functionf:R^nm→Ris said to be (strong) Morrey quasiconvex, if for any ε >0, for any A∈R^nm, and any K >0, there exists a δ= δ(ε, K, A) >0 such that ifϕ∈W^1,^∞(Q;R^m)satisfies

DϕL^∞(Q)K, max

x∈∂Q|ϕ(x)|δ, then,

f (A)ess sup

x∈Q

fA+Dϕ(x)+ε. (2.1)

DEFINITION 2.2. – A measurable function f:R^nm→Ris said to be weak Morrey quasiconvex, or(0,0)Morrey quasiconvex, if for anyA∈R^nm, andϕ∈W₀^1,^∞(Q;R^m), we have

f (A)ess sup

x∈Q

fA+Dϕ(x). (2.2)

Remark 2.3. – One can easily check from the definitions that if f:R^nm → R is Morrey quasiconvex then it is automatically weak Morrey quasiconvex. In Section 3 we prove that for either n=1 orm=1 weak Morrey quasiconvexity also implies Morrey quasiconvexity. However, it is an open question whether Morrey quasiconvexity and weak Morrey quasiconvexity are equivalent form, n >1.

Throughout this paper Morrey quasiconvexity will mean strong Morrey quasiconvex- ity.

Recall from Definition 1.1 that if a measurable function f:R^nm →R is Morrey convex, then for anyA∈R^nm,

f (A) 1

|Q|

Q

f (A+Dϕ)dx, ∀ϕ∈W₀^1,^∞Q;R^m. (2.3) Just as a convex function is always quasiconvex, a Morrey convex function with appropriate growth conditions is always Morrey quasiconvex. In fact, we have

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PROPOSITION 2.4. – Letf:R^nm→Rbe Morrey convex. For anyε >0 there exists a δ >0, depending only on ε and the Lipschitz constant of f on Q, such that if ϕ∈W^1,^∞(Q;R^m)satisfies maxx∈∂Q|ϕ(x)|δthen

f (A) 1

|Q|

Q

f (A+Dϕ)dx+εess sup

x∈Q

fA+Dϕ(x)+ε. (2.4)

Proof. – Let δ > 0 be chosen later. Let ϕ ∈W^1,^∞(Q;R^m) with maxx∈∂Q|ϕ(x)| δ. Then there exists η= η(δ,DϕL^∞(Q)) >0 such that |ϕ(y)|2δ for any y = (y1, . . . , yn)∈Q with |yi|1−η for some 1in. Letξ ∈C₀¹(Rⁿ) be such that ξ ≡1 fory∈(1−η)Q,ξ≡0 fory /∈Q, and|Dξ|4/η. Then the Morrey convexity condition implies

f (A) 1

|Q|

Q

fA+D(ξ φ)dx.

However, 1

|Q|

Q

fA+D(ξ ϕ)dx

= 1

|Q|

Q

f (A+Dϕ)dx+ 1

|Q|

Q\(1−η)Q

f (A+D(ξ ϕ))−f (A+Dϕ)dx

=I+II.

Sincef is Morrey convex and hence convex in each component variable as well as locally Lipschitz, it follows thatDfL^∞(Q)C. Therefore,

|II|C

Q\(1−η)Q

|Dϕ| + |Dξ||ϕ|dxCDϕL^∞(Q)η+δ.

Choose now δ ε/2C and η=η(ε,DϕL^∞(Q)) sufficiently small so that |II|ε.

Hence,

f (A)I+II 1

|Q|

Q

f (A+Dϕ)dx+ε.

Since

1

|Q|

Q

f (A+Dϕ)dxess sup

x∈Q

fA+Dϕ(x), we conclude that

f (A) 1

|Q|

Q

f (A+Dϕ)dx+εess sup

x∈Q

fA+Dϕ(x)+ε. ✷

One key property we need for Morrey quasiconvex functions is the following.

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PROPOSITION 2.5. – Let f:R^nm → R be Morrey quasiconvex. Let {uk} ⊂ W^1,^∞(Q;R^m)be a sequence converging to zero, weak* inW^1,^∞(Q;R^m). Then,

f (A)lim inf

k→∞ ess sup

x∈Q

fA+Duk(x), ∀A∈R^nm. (2.5) Proof. – Sinceuk converges to 0 weak-* inW^1,^∞(Q;R^m), we have the existence of a finite constant K0=sup_kDukL^∞(Q), anduk→0 uniformly onQ. For anyε >0 and A∈R^nm, it then follows from the definition of Morrey quasiconvexity that there exists a δ=δ(ε, K0, A)such that

f (A)ess sup

x∈Q

fA+Dϕ(x)+ε, (2.6)

for anyϕ∈W^1,^∞(Q;R^m)satisfyingDϕL^∞(Q)K0and maxx∈∂Q|ϕ(x)|δ. On the other hand, we know that there exists a k0=k0(δ) >0 so that maxx∈Q|uk|δ for all kk0. Therefore, we have

f (A)ess sup

x∈Q

fA+Duk(x)+ε, ∀kk0. This implies

f (A)lim inf

k→∞ ess sup

x∈Q

fA+Duk(x)+ε. (2.7)

Sinceε >0 is arbitrary, (2.7) gives (2.5). ✷

Now, we are ready to prove the first main result of our paper. It says that Morrey quasiconvexity gives a sufficient condition for lower semicontinuity.

THEOREM 2.6 (Sufficient condition). – Letf:Rⁿ×R^m×R^nm→Rsatisfy:

(i) For any(x, s)∈Rⁿ×R^m,f (x, s,·):R^nm→Ris Morrey quasiconvex;

(ii) There exists a functionω:R₊×R₊→R₊, which is continuous in its first variable and non-decreasing in its second variable, such that

f (x1, s1, A)−f (x2, s2, A)ω|x1−x2| + |s1−s2|,|A|, (2.8) for any(x1, s1), (x2, s2)∈Rⁿ×R^mandA∈R^nm.

Then for any bounded domain⊂Rⁿthe functional F (u, )=ess sup

x∈

fx, u(x), Du(x) is sequentially weak* lower semicontinuous onW^1,^∞(;R^m).

Proof. – Letϕk converge to 0 weak* inW^1,^∞(;R^m). In particular, we may assume that ϕk→0 uniformly onand Dϕk→0 weak* inL^∞(,R^m). Setuk =u+ϕk. We need to prove

ess sup

x∈

fx, u(x), Du(x)lim inf

k→∞ ess sup

x∈

fx, uk(x), Duk(x). (2.9)

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Let{uk}denote a subsequence so that

klim→∞ess sup

x∈

fx, uk(x), Duk(x)=lim inf

k→∞ ess sup

x∈

fx, uk(x), Duk(x). (2.10) Since u is Lipschitz continuous on , it follows from Rademacher’s theorem and the Lebesgue density theorem that there exists an0⊂, with|\0| =0, such thatuis differentiable at anyx0∈0andx0is a Lebesgue point ofDu, namely

xlim→x0

|u(x)−u(x0)−Du(x0)(x−x0)|

|x−x0| =0, (2.11)

and

limr↓0

1

|Qr(x0)|

Qr(x₀)

Du−Du(x0)dx=0. (2.12)

HereQr(x0)= {x0+rx: x∈Q}denotes the cube with side lengthr and center atx0. Forr >0 small, we define the rescaling maps:

v_x^r

0(x)=1

r

u(x0+rx)−u(x0):Q→R^m, and

ϕ_k,x^r

0(x)=1

r

ϕk(x0+rx)−ϕk(x0):Q→R^m. Then (2.11) and (2.12) imply

limr↓0max

x∈Q

v^r_x

0(x)−Du(x0)x=0 and lim

r↓0

Dv_x^r

0−Du(x0)_L1(Q)=0. (2.13) Notice also that v_x^r

0 is bounded inW^1,^∞(Q;R^m). Hence (2.13) implies that v_x^r

0(x)→

Du(x0)xweak* inW^1,^∞(Q;R^m).

For any fixedr >0, we observe that sup

k1

maxx∈Q

ϕ_k,x^r ₀(x)sup

k1DϕkL^∞()<∞, and

sup

k1

{x,y∈supQ,x=y}

|ϕ_k,x^r ₀(x)−ϕ_k,x^r ₀(y)|

|x−y| sup

k1

DϕkL^∞()<∞.

Hence, by the Arzela–Ascoli theorem, we may assume that, for anyr >0 small,ϕ_k,x^r ₀ converges to ϕuniformly onQ. Moreover, since Dϕ_k,x^r

0(x)=Dϕk(x0+rx)forx∈Q

and Dϕk →0 weak* in L^∞(,R^m), we have that, for any r >0 small, Dϕ_k,x^r ₀ →0 weak* inL^∞(Q,R^m). ThereforeDϕ≡0 andϕ≡constant onQ. Sinceϕ_k,x^r ₀(0)=0 we haveϕ≡0 onQ.

In particular, we obtain that, for any r >0 small, ϕ_k,x^r ₀ converges to 0 weak* in W^1,^∞(Q;R^m). By the Cauchy diagonal process, we have that for anyri↓0 there exists

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ki↑ ∞such that

v_x^rⁱ₀(x)→Du(x0)x and ϕ_k^rⁱ_i_,x₀(x)→0, (2.14) weak* inW^1,^∞(Q;R^m). Applying Proposition 2.5, we can conclude

fx0, u(x0), Du(x0)lim inf

k_i↑∞ ess sup

x∈Q

fx0, u(x0), Dv_x^rⁱ₀+ϕ_k^rⁱ_i_,x₀(x) (2.15) On the other hand, by (2.8), we have

ess sup

x∈Q_ri(x₀)

fx, u(x), Duki(x)

=ess sup

y∈Q

fx0+riy, u(x0+riy), Duki(x0+riy) ess sup

y∈Q

fx0, u(x0), Duki(x0+riy) +max

y∈Qω|riy| + |u(x0+riy)−u(x0)|,|Duki(x0+riy)|

=I +II.

It is easy to see that

I=ess sup

x∈Q

fx0, u(x0), Dv_x^rⁱ

0+ϕ_k^r_iⁱ_,x₀(x). Since

maxx∈Q|Duki(x0+riy)|DukiL^∞()C <∞,∀ki, whereC=sup_k_iDuk_iL^∞(), we have

|II|max

y∈Qω(|riy| + |u(x0+riy)−u(x0)|, C).

Combined with the continuity ofuandωin its first variable, yields limri↓0II=0.

Therefore, we obtain lim inf

k→∞ ess sup

x∈

fx, uk(x), Duk(x)

= lim

k→∞ess sup

x∈

fx, uk(x), Duk(x) lim inf

ki↑∞ ess sup

x∈

fx, uki(x), Duki(x) lim inf

r_i↓0 ess sup

x∈Q_ri(x₀)

fx, uki(x), Duki(x)

=lim inf

ri↓0 ess sup

y∈Q

fx0+riy, u(x0+riy), Duki(x0+riy)

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=lim inf

ri↓0 ess sup

y∈Q

fx0, u(x0), Dv_x^rⁱ₀+ϕ_k^rⁱ_i_,x₀(y) fx0, u(x0), Du(x0) by (2.15).

Since this holds for anyx0∈0 and|\0| =0, we have proved (2.9) and the proof of Theorem 2.6 is complete. ✷

Now we prove that Morrey quasiconvexity is also a necessary condition for the sequential weak* lower semicontinuity of the functionalF (u, )onW^1,^∞(;R^m).

THEOREM 2.7 (Necessary condition). – Let f:Rⁿ ×R^m×R^nm →R satisfy the condition (2.8). For any bounded⊂Rⁿ, letF (u, )=ess sup_x_∈f (x, u(x), Du(x)).

Assume thatF (·, )is sequentially weak* lower semicontinuous onW^1,^∞(;R^m). Then p→f (x, u, p)is Morrey quasiconvex for any(x, u)∈Rⁿ×R^m.

To prove Theorem 2.7 we first need the following lemma. The second part of this lemma exhibits the importance of the extended Jensen inequality.

LEMMA 2.8. – Letf:Rⁿ×R^m×R^nm→Rsatisfy (2.8). IfF (·, )is sequentially weak* lower semicontinuous onW^1,^∞(;R^m), then

(i) For any(x0, s0)∈Rⁿ×R^m,f (x0, s0,·):R^nm→Ris lower semicontinuous.

(ii) For any(x0, s0)∈Rⁿ×R^m, we have f

x0, s0, 1

|Q|

Q

Du(x)dx

ess sup

x∈Q

fx0, s0, Du(x) (2.16) for anyu∈W^1,^∞(Q;R^m)withDu Q-periodic.

(iii) For any (x0, s0)∈Rⁿ×R^m, f (x0, s0,·):R^nm →R satisfies the weak Morrey quasiconvexity property:

f (x0, s0, A)=inf ess sup

x∈Q

fx0, s0, A+Dϕ(x): ϕ∈W₀^1,^∞Q;R^m. (2.17) Proof. – (i) For(x0, s0)∈Rⁿ×R^m. Let{Aj} ⊂R^nm satisfy Aj→A. Thenuj(x)= s0+Aj(x−x0)converges tou(x)=s0+A(x−x0)inW^1,^∞(Br(x0);R^m)for anyr >0.

Hence we have ess sup

x∈Br(x₀)

fx, u(x), Du(x)=ess sup

x∈Br(x₀)

fx, s0+A(x−x0), A lim inf

j→∞ ess sup

x∈Br(x0)

fx, uj(x), Duj(x)

=lim inf

j→∞ ess sup

x∈Br(x0)

fx, s0+Aj(x−x0), Aj

. (2.18) On the other hand, sincef satisfies (2.8) we have

rlim→0ess sup

x∈Br(x₀)

fx, s0+A(x−x0), A=f (x0, s0, A), and

rlim→0ess sup

x∈B_r(x₀)

fx, s0+Aj(x−x0), Aj

=f (x0, s0, Aj),

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uniformly inj. Hence, sendingr to zero, (2.18) implies f (x0, s0, A)lim inf

j→∞ f (x0, s0, Aj).

(ii) For(x0, s0)∈Rⁿ×R^m. We first extendu∈W^1,^∞(Q;R^m)toRⁿwithDuas aQ- periodic function. Defineuj(x)=u(j (x−x0))/j+s0:Rⁿ→R^m. Then we have

uj(x)→s0+ 1

|Q|

Q

Dudx

(x−x0),

weak* in W^1,^∞(Br(x0);R^m), for anyr >0. In particular, Duj→ _|_Q¹_|_QDudx weak*

inL^∞(Br(x0),R^m). Hence we have ess sup

x∈Br(x₀)

f

x, s0+ 1

|Q|

Q

Dudx

(x−x0), 1

|Q|

Q

Dudx

lim inf

j→∞ ess sup

x∈B_r(x₀)

fx, uj(x), Duj(x)

=lim inf

j→∞ ess sup

x∈Br(x₀)

f

x, s0+u(j (x−x0))

j , Duj (x−x0). (2.19) Sincef satisfies (2.8) we know, settingA=_|_Q¹_|QDudx

rlim→0ess sup

x∈Br(x0)

fx, s0+A(x−x0), A=f (x0, s0, A), and

rlim→0ess sup

x∈Br(x0)

f

x, s0+u(j (x−x0))

j , Duj (x−x0)

=lim

r→0ess sup

x∈B_r(x₀)

f

x0, s0+u(0)

j , Duj (x−x0), uniformly inj. Here we have used the fact that

xlim→x₀sup

j

|u(j (x−x0)−u(0)|

j =0.

On the other hand, sinceDuisQ- periodic we have

jlim→∞ess sup

x∈Br(x₀)

f

x0, s0+u(0)

j , Duj (x−x0)=ess sup

x∈Q

fx0, s0, Du(x). Putting these together and sendingr to zero in (2.19), we obtain

f

x0, s0, 1

|Q|

Q

Dudx

ess sup

x∈Q

fx0, s0, Du(x).

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Observe that if we were to assume the condition f (x0, s0, p)=f (x0, s0,−p)we could drop the assumption thatDuisQ-periodic.

(iii) Notice we can rewrite (2.16) as f (x0, s0, A)=inf

ess sup

y∈Q

fx0, s0, Dϕ(y): ϕ∈W^1,^∞Rⁿ;R^m, DϕisQ-periodic, 1

|Q|

Q

Dϕdx=A

=inf ess sup

y∈Q

fx0, s0, A+Dψ(y): ψ∈W₀^1,^∞Q;R^m. (2.20) In fact, given ψ∈W₀^1,^∞(Q;R^m) setϕ(y)=Ay+ψ(y). This shows the first infimum in (2.20) is not greater than the second. For the reverse, given anyϕ∈W^1,^∞(Rⁿ;R^m) withDϕ Q-periodic and 1/|Q|_QDϕdx=A, takeψ≡0. Then, by (ii), we have

inf ess sup

y∈Q

fx0, s0, A+Dψ(y): ψ∈W₀^1,^∞Q;R^m f (x0, s0, A)ess sup

y∈Q

fx0, s0, Dϕ(y).

This shows that the second infimum in (2.20) is not greater than the first and hence they are equal. This gives the weak Morrey quasiconvexity off in its last variable. ✷

Now we can return to the proof of Theorem 2.7.

Proof of Theorem 2.7. – We prove it by contradiction. Suppose thatf is not Morrey quasiconvex in its last variable. Then there exist (x0, s0)∈Rⁿ×R^m, ε0>0, K0>0, A0∈R^nm, and sequences{ϕk} ⊂W^1,^∞(Q;R^m)such that

sup

k DφkL^∞(Q)K0, max

x∈∂Q|ϕk(x)| =δk→0 ask→ ∞, (2.21) but

f (x0, s0, A0) >ess sup

x∈Q

fx0, s0, A0+Dϕk(x)+ε0, (2.22) for all k1. It follows from (2.21) that we may assume that there exists a ϕ ∈ W₀^1,^∞(Q;R^m)such thatϕk→ϕ weak* inW^1,^∞(Q;R^m). For anyr >0, we see that

s0+A0(x−x0)+rϕk

x−x0

r

→s0+A0(x−x0)+rϕ

x−x0

r

,

weak* in W^1,^∞(Qr(x0);R^m), as k → ∞. Since F (·, Qr(x0)) is sequentially weak*

lower semicontinuous onW^1,^∞, we have ess sup

y∈Q

fx0+ry, s0+rA0y+rϕ(y), A0+Dϕ(y)

=ess sup

x∈Q_r(x₀)

f

x, s0+A0(x−x0)+rϕ

x−x0

r

, A0+Dϕ

x−x0

r

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lim inf

k→∞ ess sup

x∈Qr(x0)

f

x, s0+A0(x−x0)+rϕk

x−x0

r

, A0+Dϕk

x−x0

r

=lim inf

k→∞ ess sup

y∈Q

fx0+ry, s0+rA0y+rϕk(y), A0+Dϕk(y). Notice that

limr↓0max

y∈Q|x0+ry−x0| =0, limr↓0max

y∈Q|s0+rA0y+rϕ(y)−s0| =0, and

limr↓0sup

k

maxy∈Q|s0+rA0y+rϕk(y)−s0| =0.

Sincef satisfies (ii) of Theorem 2.6, we have, by sendingr to zero, ess sup

y∈Q

fx0, s0, A0+Dϕ(y)lim inf

k→ı ess sup

y∈Q

fx0, s0, A0+Dϕk(y). (2.23) This, combined with (2.22), implies

ess sup

y∈Q

fx0, s0, A0+Dϕ(y)+ε0< f (x0, s0, A0). (2.24) On the other hand, sinceϕ∈W₀^1,^∞(Q;R^m), it follows from (iii) of Lemma 2.8 that

f (x0, s0, A0)ess sup

y∈Q

fx0, s0, A0+Dϕ(y). This contradicts (2.24). The proof is complete. ✷

We finish this section with an existence theorem for minimizing problems inL^∞. THEOREM 2.9. – If, in addition to the conditions in Theorem 2.6, letf:Rⁿ×R^m× R^nm→Rsatisfy the following coercivity condition:

f (x, s, A)C1|A|^p−C2, ∀(x, s, A)∈Rⁿ×R^m×R^nm (2.25) for someC1>0,C2>0, andp >0. Then for any bounded domain ⊂Rⁿ and ψ∈ W^1,^∞(;R^m)there exists at least one functionu∈W^1,^∞(;R^m), withu|∂=ψ|∂, so that

ess sup

x∈

fx, u(x), Du(x)

=inf ess sup

x∈

fx, v(x), Dv(x): v∈W^1,^∞;R^m, v|∂=ψ|∂

. Proof. – First notice that

c≡inf ess sup

x∈

fx, v(x), Dv(x): v∈W^1,^∞;R^m, v|∂=ψ|∂

ess sup

x∈

fx, ψ(x), Dψ(x)<∞.

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Moreover,c−C2>−∞.

Let{uk} ⊂W^1,^∞(;R^m)be a minimizing sequence withuk|∂=ψ|∂, and

klim→∞ess sup

x∈

fx, uk(x), Duk(x)=c.

It follows from the coercivity condition (2.25) that DukL^∞() is bounded. By the Poincare inequality we see that ukψ + Duk in the L^∞() norm and so {uk} is bounded in W^1,^∞(;R^m). Therefore, we may assume that there exists u∈ W^1,^∞(;R^m)withu|∂=ψ|∂so thatuk→uweak* inW^1,^∞(;R^m). It then follows from Theorem 2.6 thatuis a minimizer and ess sup_x_∈f (x, u(x), Du(x))=c. ✷

Remark 2.10. – The coercivity condition of the theorem is essential for the existence of a minimizer. Indeed, considerF (u)=ess sup_x_∈[_0,1_]|x u(x)|, withu(0)=1, u(1)=0.

Clearly this is not coercive. Consider the sequence un(x)=

1, if 0x1/n,

−logx/logn, if 1/nx1.

Thenun∈W^1,^∞([0,1])and x u_n(x)=

0, if 0x1/n,

−1/logn, if 1/nx1.

Hence F (un)=1/logn→ 0. The infimum of F is therefore zero, but clearly no Lipschitz function assuming the boundary data can giveF (u)=0.

On the other hand if we consider instead F (u)=ess sup_x_∈[_0,1_]|u(x)| with u(0)= 1, u(1)=0 there is a unique minimizer and it isu^∗(x)=1−x. Indeed,u(1)−u(0)= ₁

0 u(x)dx implies that F (u)1 for any Lipschitz u with u(0)=1, u(1)=0, and F (u^∗)=1.The uniqueness of the minimizer follows from the fact that

ess sup

x∈[x₀,x₁]

(u^∗)(x)=1 for any 0x0< x11.

This is Theorem 1 of [1].

Remark 2.11. – This theorem establishes the existence of a minimizer but not an absolute minimizer, i.e., a function which minimizesF on any subdomain. This is what we need to use the Aronsson–Euler equation derived in the last section. In a companion paper The Euler equation and absolute minimizers of L^∞ functionals (to appear in Archives Rat. Mech. Anal.), we do establish the existence of an absolute minimizer for problems withu:Rⁿ→R.

3. Various classes of quasiconvex functions

In this section, we introduce various classes of functions, which are natural extensions toL^∞ of the well-known concepts of convexity, polyconvexity, Morrey convexity, and rank one convexity. For convenience we rewrite the definition of quasiconvexity.