Quasiconvexity at the boundary and concentration effects generated by gradients

DOI:10.1051/cocv/2012028 www.esaim-cocv.org

QUASICONVEXITY AT THE BOUNDARY AND CONCENTRATION EFFECTS GENERATED BY GRADIENTS

Martin Kruˇ z´ık

Abstract.We characterize generalized Young measures, the so-called DiPerna–Majda measures which are generated by sequences of gradients. In particular, we precisely describe these measures at the boundary of the domain in the case of the compactification of R^m×n by the sphere. We show that this characterization is closely related to the notion of quasiconvexity at the boundary introduced by Ball and Marsden [J.M. Ball and J. Marsden, Arch. Ration. Mech. Anal.86 (1984) 251–277]. As a consequence we get new results on weakW^1,2(Ω;R³) sequential continuity ofu→a·[Cof∇u], where Ω⊂R³ has a smooth boundary anda, are certain smooth mappings.

Mathematics Subject Classification. 49J45, 35B05.

Received September 12, 2011. Revised August 2, 2012 Published online May 17, 2013.

1. Introduction

Oscillations and/or concentrations appear in many problems in the calculus of variations, partial differential equations, or optimal control theory, which admit only L^p but not L^∞ apriori estimates. While Young mea- sures [41] successfully capture oscillatory behavior (see e.g. [21,28,31,32]) of sequences they completely miss concentrations. There are several tools how to deal with concentrations. They can be considered as generalization of Young measures, see for example Alibert’s and Bouchitt´e’s approach [1], DiPerna’s and Majda’s treatment of concentrations [6], or Fonseca’s method described in [10]. An overview can be found in [30,38]. Moreover, in many cases, we are interested in oscillation/concentration effects generated by sequences of gradients. A charac- terization of Young measures generated by gradients was completely given by Kinderlehrer and Pedregal [16,18], cf.also [28,29]. The first attempt to characterize both oscillations and concentrations in sequences of gradients is due to Fonsecaet al.[12]. They dealt with a special situation of{gv(∇uk)}k∈Nwherevcoincides with a pos- itively p-homogeneous function at infinity (see (1.35) for a precise statement),uk ∈W^1,p(Ω;R^m),p >1, with g continuous and vanishing on∂Ω. Later on, a characterization of oscillation/concentration effects in terms of DiPerna’s and Majda’s generalization of Young measures was given in [15] for arbitrary integrands and in [11]

for sequences living in the kernel of a first-order differential operator. Recently Kristensen and Rindler [19]

characterized oscillation/concentration effects in the casep= 1. Nevertheless, a complete analysis of boundary

Keywords and phrases.Bounded sequences of gradients, concentrations, oscillations, quasiconvexity at the boundary, weak lower semicontinuity.

∗This work was partly supported by the grants P201/10/0357 and P201/12/0671 (GA ˇCR).

1 Institute of Information Theory and Automation of the ASCR, Pod vod´arenskou vˇeˇz´ı 4, 182 08 Praha 8, Czech Republic.

2 Faculty of Civil Engineering, Czech Technical University, Th´akurova 7, 166 29 Praha 6, Czech Republic.[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2013

M. KRUˇZ´IK

effects generated by gradients is still missing. We refer to [15] for the case whereuk = uon the boundary of the domain. As already observed by Meyers [24], concentration effects at the boundary are closely related to the sequential weak lower semicontinuity of integral functionalsI :W^1,p(Ω;R^m)→R: I(u) =

Ωv(∇u(x)) dx where v : R^m×n → R is continuous and such that |v| ≤ C(1 +| · |^p) for some constant C >0, cf. also [20]

for recent results. Indeed, consider u∈W₀^1,p(B(0,1);R^m), where B(0,1) is the unit ball in Rⁿ centered at 0, and extend it by zero to the whole Rⁿ. Define forx ∈ Rⁿ and k ∈ N uk(x) = k^n/p−1u(kx), i.e., uk 0 in W^1,p(B(0,1);R^m) and consider a smooth convex domainΩ∈Rⁿsuch that 0∈∂Ω,is the outer unit normal to

∂Ω at 0 and let there bex∈Ωsuch that·x <0. Moreover, take a functionv to be positivelyp-homogeneous, i.e., v(αs) =α^pv(s) for allα≥0. Then ifI is weakly lower semicontinuous then

0 =I(0)≤lim inf

k→∞

Ωv(∇uk(x)) dx= lim inf

k→∞

B(0,1)∩Ωv(∇uk(x)) dx= lim inf

k→∞

B(0,1)∩Ωkⁿv(∇(u(kx)) dx

=

B(0,1)∩{x∈Rⁿ; ·x<0}v(∇u(y)) dy. (1.1)

Thus, we see that

0≤

B(0,1)∩{x∈Rⁿ; ·x<0}v(∇u(y)) dy

for all u ∈ W₀^1,p(B(0,1);R^m) forms a necessary condition for weak lower semicontinuity of I. Here we show that the weak lower semicontinuity of the above defined functional I is intimately related to the so-called quasiconvexity at the boundarydefined by Ball and Marsden in [3] and that this notion of quasiconvexity plays a crucial role in the characterization of parametrized measures generated by sequences of gradients. Moreover, we show that if{uk} ⊂W^1,2(Ω;R³),uk u, and h(x, s) := [Cofs]·(a(x)⊗(x)) (“Cof” denotes the cofactor matrix) for somea, ∈C( ¯Ω;R³) such thatcoincides with the outer unit normal to∂Ω on the boundary∂Ω of a smooth bounded domainΩ⊂R³thenh(·,∇uk)→h(·,∇u) weakly* in Radon measures supported in ¯Ω. If, additionally,h(x,∇uk(x))≥0 for allk∈Nand almost allx∈Ωthen the above convergence is even in the weak topology ofL¹(Ω). Hence, there is a continuous functionψ: [0,+∞)→[0,+∞) such that limt→∞ψ(t)/t= +∞

and sup_k∈N

Ωψ(h(x,∇uk(x)) dx <+∞. This result, which can be generalized to higher dimensions, too, is an analogy to the celebrated S. M¨uler’s result on higher integrability of determinants [27]. See also [14,17].

1.1. Basic notation

Let us start with a few definitions and with the explanation of our notation. Having a bounded domain Ω ⊂ Rⁿ we denote by C(Ω) the space of continuous functions: Ω → R. Then C₀(Ω) consists of functions from C(Ω) whose support is contained in Ω. In what follows “rca(S)” denotes the set of regular countably additive set functions on the Borelσ-algebra on a metrizable setS (cf.[7]), its subset, rca⁺₁(S), denotes regular probability measures on a setS. We write “γ-almost all” or “γ-a.e.” if we mean “up to a set with theγ-measure zero”. If γ is then-dimensional Lebesgue measure and M ⊂ Rⁿ we omit writing γ in the notation. Further, W^1,p(Ω;R^m), 1≤p <+∞denotes the usual space of measurable mappings which are together with their first (distributional) derivatives integrable with thep-th power. The support of a measureσ∈ rca(Ω) is a smallest closed set S such that σ(A) = 0 ifS∩A =∅. Finally, ifσ∈rca(S) we write σs anddσ for the singular part and density ofσdefined by the Lebesgue decomposition, respectively. We denote by ‘w-lim’ the weak limit and by B(x₀, r) an open ball in Rⁿ centered at x₀ and the radius r > 0. The dot product on Rⁿ is standardly defined asa·b:=n

i=1aibi and analogously onR^m×n. Finally, ifa∈R^mandb∈Rⁿ thena⊗b∈R^m×n with (a⊗b)ij =aibj, andIdenotes the identity matrix.

If not said otherwise, we will suppose in the sequel thatΩ⊂Rⁿ is a bounded domain with aC¹ boundary.

The same regularity is assumed if we say thatΩ has a smooth boundary.

QUASICONVEXITY AT THE BOUNDARY AND CONCENTRATION EFFECTS GENERATED BY GRADIENTS

1.2. Quasiconvex functions

LetΩ⊂Rⁿ be a bounded Lipschitz domain. We say that a functionv:R^m×n→Ris quasiconvex [26] if for anys₀∈R^m×n and any ϕ∈W₀^1,∞(Ω;R^m)

v(s₀)|Ω| ≤

Ωv(s₀+∇ϕ(x)) dx. (1.2)

Ifv:R^m×n→Ris not quasiconvex we define its quasiconvex envelopeQv:R^m×n→Ras Qv= sup{h≤v; h:R^m×n→Rquasiconvex}

and if the set on the right-hand side is empty we putQv=−∞. Ifv is locally bounded and Borel measurable then for anys₀∈R^m×n (see [5])

Qv(s₀) = inf

ϕ∈W₀^1,∞(Ω;R^m)

1

|Ω|

Ωv(s₀+∇ϕ(x)) dx. (1.3) We will also need the following elementary result. It can be found in a more general forme.g.in [5], Chapter 4, Lemma 2.2, or in [26].

Lemma 1.1. Let v:R^m×n→Rbe quasiconvex with |v(s)| ≤C(1 +|s|^p),C >0, for all s∈R^m×n. Then there is a constant α≥0 such that for every s₁, s₂∈R^m×n it holds

|v(s₁)−v(s₂)| ≤α(1 +|s₁|^p−1+|s₂|^p−1)|s₁−s₂|. (1.4) Following [3,34,36] we define the notion of quasiconvexity at the boundary. In order to proceed, we first define the so-calledstandard boundary domain.

Definition 1.2. Let∈Rⁿ be a unit vector and letΩ be a bounded open Lipschitz domain. We say thatΩ

is a standard boundary domain with the normalif there isa∈Rⁿsuch thatΩ⊂Ha,:={x∈Rⁿ; ·x < a}

and the (n−1)- dimensional interiorΓ of∂Ω∩∂Ha,is nonempty.

We are now ready to define the quasiconvexity at the boundary. We put for 1≤p≤+∞

W_Γ^1,p

(Ω;R^m) :={u∈W^1,p(Ω;R^m); u= 0 on∂Ω\Γ}. (1.5) Definition 1.3. ([3]) Let ∈ Rⁿ be a unit vector. A function v : R^m×n → R is called quasiconvex at the boundary at s₀ ∈ R^m×n with respect to (shortly v is qcb at (s₀, )) if there is q ∈ R^m such that for all u∈W_Γ^1,∞(Ω;R^m) it holds

Γq·u(x) dS+v(s₀)|Ω| ≤

Ωv(s₀+∇u(x)) dx. (1.6) An immediate generalization is the following.

Definition 1.4. Let∈Rⁿ be a unit vector, 1≤p <+∞. A functionv :R^m×n →R,|v| ≤C(1 +| · |^p) for someC >0 is calledW^1,p-quasiconvex at the boundary ats₀∈R^m×n with respect to(shortlyv isp-qcb at (s₀, )) if there is q∈R^msuch that for allu∈W_Γ^1,p(Ω;R^m) it holds

Γq·u(x) dS+v(s₀)|Ω| ≤

Ωv(s₀+∇u(x)) dx. (1.7)

M. KRUˇZ´IK

Remark 1.5.

(i) Ifv is differentiable thenq:= ^∂v_∂s(s₀)is given uniquely;cf.[36]. We denote the set of such of vectorsqfor which (1.6) holds by∂_v^qcb(s₀, ). It may be seen as a notion of a “subdifferential” forv.

(ii) It is clear that ifv is qcb at (s₀, ) it is also quasiconvex ats₀,i.e., (1.2) holds.

(iii) If (1.6) holds for one standard boundary domain it holds for other standard boundary domains, too.

(iv) Ifp >1,v:R^m×n→Ris positivelyp-homogeneous,i.e. v(λs) =λ^pv(s) for alls∈R^m×n, continuous, and p-qcb at (0, ) thenq = 0 in (1.6). Indeed, we havev(0) = 0 and suppose that

Ωv(∇u(x)) dx < 0 for someu∈W_Γ^1,∞(Ω;R^m). By (1.6), we must have for allλ >0

0≤λ^p

Ωv(∇u(x)) dx−λ

Γq·u(x) dS.

However, it is not possible for λ > 0 large enough and therefore for all u ∈ W_Γ^1,∞(Ω;R^m) it holds that

Ωv(∇u(x)) dx≥0. Thus, we can takeq= 0.

The following lemma shows that Definitions1.3and1.4are equivalent for a class of functions whose modulus grows as thep-th power.

Lemma 1.6. Let v:R^m×n→Rbe continuous and such that|v(A)| ≤C(1 +|A|^p)for allA∈R^m×n and some C >0 independent ofA and some1≤p <+∞. If v is qcb at (s₀, )it isp-qcb at(s₀, ).

Proof. Take u ∈ W_Γ^1,p (Ω;R^m) and a sequence {uk}k∈N ⊂ W_Γ^1,∞(Ω;R^m) such that uk → u strongly in W^1,p(Ω;R^m). We get using (1.4) that

Ωv(s₀+∇u(x)) dx= lim

k→∞

Ωv(s₀+∇uk(x)) dx. (1.8)

Asv is qcb at (s₀, ) we have

klim→∞

Ωv(s₀+∇uk(x)) dx≥ |Ω|v(s0) + lim

k→∞

Γq·uk(x) dS=|Ω|v(s0) +

Γq·u(x) dS, (1.9)

which finishes the proof in view of (1.8).

It will be convenient to define the following notion recalling the quasiconvex envelope of v at zero. Here, however, we integrate only over a standard boundary domain with a given normal. If∈Rⁿ has a unit length then put

Qb,v(0) := inf

u∈W_Γ^1,p(Ω;R^m)

1

|Ω|

Ωv(∇u(x)) dx. (1.10)

Remark 1.7. Ifvis positivelyphomogeneous withp >1 then eitherQb,v(0) = 0 orQb,v(0) =−∞. We also have thatQb,v(0)≤Qv(0). Having a ballB(0,1) ={x∈Rⁿ; |x|<1}we putΩ :=B(0,1)∩{x∈Rⁿ; ·x <0}.

In this case, we only integrate over a half-ball in (1.10). Hence, we can use only thoseu∈W_Γ^1,p(Ω;R^m) which are symmetric with respect to the plane{x; ·x= 0},i.e., satisfyingu(x) =u(x−2(·x)) ifx∈Ω.

Quasiconvexity at the boundary was introduced in [3] as a necessary condition for strong local minima of the mixed problem in nonlinear elasticity at boundary points beloging to a free part of the boundary. Contrary to the usual Morrey’s quasiconvexity there are not many papers dealing with this notion. Let me point out several interesting results in this direction. Mielke and Sprenger [25] investigated relation of quasiconvexity at the boundary and Agmon’s condition for quadratic stored energies in nonlinear elasticity and Sprenger [36] in

QUASICONVEXITY AT THE BOUNDARY AND CONCENTRATION EFFECTS GENERATED BY GRADIENTS

his thesis defined the so-called polyconvexity at the boundary. Recently, Grabovsky and Mengesha [13] showed that quasiconvexity at the boundary is a sufficient condition for the so-called W^1,∞-sequential weak* local minima - slight weakening of the notion of strong local minimizers. Here we find another interesting connection, namely the fact that quasiconvexity at the boundary plays a crucial role in the analysis of concentration effects generated by gradients and is essential forW^1,p-sequential weak lower semicontinuity of integral functionals as in (1.1).

We start with the following auxiliary lemma.

Lemma 1.8. Let v : R^m×n → R, |v| ≤ C(1 +| · |^p), C > 0, 1 ≤ p < +∞, be quasiconvex, and such that v(0) =Qb,v(0) = 0 for some∈Rⁿ. LetΩ be a bounded domain inRⁿ with theC¹ boundary, with x₀∈∂Ω, and with the outer unit normal at x₀. Then for every ε > 0 there is δ > 0 and a continuous function f :R→(0+∞),limε→0εf(ε) = 0, such thatΩ∩B(x0, δ)⊂Ωand it holds that for everyU ∈W₀^1,p(B(0, δ);R^m)

that

B(x₀,δ)∩Ω

v(∇U(x)) dx≥ −ε

B(x₀,δ)∩Ω

(|∇U(x)|+f(ε)|∇U(x)|^p) dx, (1.11) Proof. Following [3], we can assume, without loss of generality, thatx₀ = 0 and that= (0,0, . . . ,0,1)∈Rⁿ. Let further

∂Ω∩B(0, r) :={x∈B(0, r); xn =h(x)}, Ω∩B(0, r) :={x∈B(0, r); xn< h(x)},

wherex= (x₁, . . . , xn)∈Rⁿ,x= (x₁, . . . , xn−1), andh∈C¹(Rⁿ⁻¹) is such thath(0) = 0 and∇h(0) = 0. As in [3] we define forξ > 0 Xξ :Rⁿ → Rⁿ byXξ(y) =ξy+h(ξy). Notice that ∇Xξ(y) =ξ(I+⊗ ∇h(ξy)) and that det∇Xξ = ξⁿ because · ∇h = 0. LetU ∈ W₀^1,p(Xξ(B(0, r);R^m). Define u(y) := ¹_ξU(Xξ(y)), i.e.

u ∈ W₀^1,p(B(0, r);R^m). Then ∇u(y) = ¹_ξ∇U(Xξ(y))∇Xξ(y) = ¹_ξ∇U(z)∇Xξ(X_ξ⁻¹(z)) for z :=Xξ(y). Notice thatX_ξ⁻¹(z) =ξ⁻¹(z−h(z)) for allz∈Rⁿ. ForΩ:={x ∈B(0, r); xn <0}, we calculate using Lemma1.1

X_ξ(Ω)v(∇U(z))dz+α

X_ξ(Ω)

1+|∇U(z)|^p−1+ 1

ξ∇U(z)∇Xξ(X_ξ⁻¹(z)) ^p−1

∇U(z)

I−1

ξ∇Xξ(X_ξ⁻¹(z) dz (1.12)

≥

X_ξ(Ω)v 1

ξ∇U(z)∇Xξ(X_ξ⁻¹(z)) dz=ξⁿ

Ωv(∇u(y)) dy≥0.

The last inequality follows from the assumptionQb,v(0)≥0. Hence, exploiting the identityξ⁻¹∇Xξ(X_ξ⁻¹(z)) = I+⊗ ∇h(z), we get

X_ξ(Ω)v(∇U(z))dz≥ −α

X_ξ(Ω)

1+|∇U(z)|^p−1+ 1

ξ∇U(z)∇Xξ(X_ξ⁻¹(z)) ^p−1

∇U(z)

I−1

ξ∇Xξ(X_ξ⁻¹(z)dz (1.13)

=−α

X_ξ(Ω)(|∇U(z)|+|∇U(z)|^p(1 +|I+⊗ ∇h(z)|^p−1))|⊗ ∇h(z)|dz.

However,

0≤

X_ξ(Ω)(|∇U(z)|+|∇U(z)|^p(1 +|I+⊗ ∇h(z)|^p−1))|⊗ ∇h(z)|dz

≤

X_ξ(Ω)(|∇U(z)|+|∇U(z)|^p(1 + 2^p−1(n^(p−1)/2+M(|z|)^p−1)))M(|z|) dz, (1.14)

M. KRUˇZ´IK

where M is the modulus of continuity of the uniformly continuous function z →⊗ ∇h(z) on Xξ(Ω), i.e., lims→0+M(s) = 0. Thus, choosingε > 0, we take ξ >0 so small that sup_z∈Xξ(Ω)M(|z|)< ε/α and define f(ε) := 1 + 2^p−1(n^(p−1)/n+ (ε/α)^p−1). Then we have

0≤α

X_ξ(Ω)(|∇U(z)|+|∇U(z)|^p(1 + 2^p−1(n^(p−1)/2+M(|z|)^p−1)))M(|z|) dz

≤

X_ξ(Ω)(|∇U(z)|+f(ε)|∇U(z)|^p)εdz.

Finally, in view of (1.13) we have

X_ξ(Ω)v(∇U(z)) dz≥ −ε

X_ξ(Ω)(|∇U(z)|+f(ε)|∇U(z)|^p) dz.

We takeδ >0 such thatB(0, δ)∩Ω⊂Xξ(Ω) and take U ∈W₀^1,p(B(0, δ);R^m) extended toRⁿ by zero which is admissible. Then, asv(0) = 0, we have from the previous inequality that

B(0,δ)∩Ωv(∇U(z)) dz≥ −ε

B(0,δ)∩Ω(|∇U(z)|+f(ε)|∇U(z)|^p) dz.

Example 1.9. Ifn=m= 3 then it is shown in [34], Proposition 17.2.4, that the functionv:R^n×n→Rgiven by

v(s) =a·[Cofs]

is quasiconvex at the boundary with the unit normal∈Rⁿ. Herea∈Rⁿ is an arbitrary constant and “Cof” is the cofactor matrix,i.e., Cofsij = (−1)^i+jdets_ij, wheres_ij is the submatrix ofsobtained fromsby removing thei-th row and thej-th column. Hence, v is positively 2-homogeneous. See also [35] for the role of thisv in the definition of the so-called interface polyconvexity.

1.3. Young measures

Forp≥0 we define the following subspace of the spaceC(R^m×n) of all continuous functions onR^m×n: Cp(R^m×n) ={v∈C(R^m×n);v(s) =o(|s|^p) for|s| → ∞}.

The Young measures on a bounded domain Ω ⊂ Rⁿ are weakly* measurable mappings x → νx : Ω → rca(R^m×n) with values in probability measures; and the adjective “weakly* measurable” means that, for any v ∈ C₀(R^m×n), the mapping Ω →R: x→ νx, v=

R^m×nv(λ)νx(dλ) is measurable in the usual sense. Let us remind that, by the Riesz theorem, rca(R^m×n), normed by the total variation, is a Banach space which is isometrically isomorphic with C₀(R^m×n)^∗, where C₀(R^m×n) stands for the space of all continuous functions R^m×n→Rvanishing at infinity. Let us denote the set of all Young measures byY(Ω;R^m×n). It is known that Y(Ω;R^m×n) is a convex subset ofL^∞_w(Ω; rca(R^m×n))∼=L¹(Ω;C₀(R^m×n))^∗, where the subscript “w” indicates the property “weakly* measurable”. A classical result [37,40] is that, for every sequence {yk}k∈N bounded in L^∞(Ω;R^m×n), there exists its subsequence (denoted by the same indices for notational simplicity) and a Young measureν ={νx}x∈Ω ∈ Y(Ω;R^m×n) such that

∀v∈C₀(R^m×n) : lim

k→∞v◦yk =vν weakly* inL^∞(Ω), (1.15) where [v◦yk](x) =v(yk(x)) and

vν(x) =

R^m×nv(λ)νx(dλ). (1.16)

QUASICONVEXITY AT THE BOUNDARY AND CONCENTRATION EFFECTS GENERATED BY GRADIENTS

Let us denote byY^∞(Ω;R^m×n) the set of all Young measures which are created by this way,i.e.by taking all bounded sequences inL^∞(Ω;R^m×n). Note that (1.15) actually holds for anyv:R^m×n→Rcontinuous.

A generalization of this result was formulated by Schonbek [33] (cf. also [2]): if 1 ≤ p < +∞: for every sequence {yk}k∈N bounded in L^p(Ω;R^m×n) there exists its subsequence (denoted by the same indices) and a Young measureν={νx}x∈Ω∈ Y(Ω;R^m×n) such that

∀v ∈Cp(R^m×n) : lim

k→∞v◦yk =vν weakly inL¹(Ω). (1.17) We say that{yk} generatesν if (1.17) holds.

Let us denote byY^p(Ω;R^m×n) the set of all Young measures which are created by this way,i.e. by taking all bounded sequences inL^p(Ω;R^m×n).

The following important lemma was proved in [12].

Lemma 1.10. Let 1 < p < +∞ and Ω ⊂ Rⁿ be an open bounded set and let {uk}k∈N ⊂ W^1,p(Ω;R^m) be bounded. Then there is a subsequence {uj}j∈N and a sequence{zj}j∈N⊂W^1,p(Ω;R^m)such that

jlim→∞|{x∈Ω; zj(x)=uj(x) or∇zj(x)=∇uj(x)}|= 0 (1.18) and {|∇zj|^p}j∈N is relatively weakly compact in L¹(Ω). In particular, {∇uj} and {∇zj} generate the same Young measure.

1.4. DiPerna–Majda measures

Let us take a complete (i.e. containing constants, separating points from closed subsets and closed with respect to the Chebyshev norm) separable ringRof continuous bounded functionsR^m×n→R. It is known [8], Section 3.12.21, that there is a one-to-one correspondenceR → β_RR^m×n between such rings and metrizable compactifications ofR^m×n; by a compactification we mean here a compact set, denoted byβ_RR^m×n, into which R^m×n is embedded homeomorphically and densely. For simplicity, we will not distinguish betweenR^m×n and its image in β_RR^m×n. Similarly, we will not distinguish between elements of Rand their unique continuous extensions onβ_RR^m×n.

Letσ∈rca( ¯Ω) be a positive Radon measure on a bounded domainΩ⊂Rⁿ. A mapping ˆν :x→νˆxbelongs to the spaceL^∞_w( ¯Ω, σ; rca(β_RR^m×n)) if it is weakly* σ-measurable (i.e., for anyv₀∈C₀(R^m×n), the mapping Ω¯ →R:x→

β_RR^m×nv₀(s)ˆνx(ds) isσ-measurable in the usual sense). If additionally ˆνx∈rca⁺₁(β_RR^m×n) for σ-a.a.x∈Ω¯ the collection {ˆνx}x∈Ω¯ is the so-called Young measure on ( ¯Ω, σ) [41], see also [2,30,37,39,40].

DiPerna and Majda [6] shown that having a bounded sequence in L^p(Ω;R^m×n) with 1 ≤p <+∞ and Ω an open domain in Rⁿ, there exists its subsequence (denoted by the same indices), a positive Radon measure σ∈rca( ¯Ω), and a Young measure ˆν :x→νˆxon ( ¯Ω, σ) such that (σ,ν) is attainable by a sequenceˆ {yk}k∈N⊂ L^p(Ω;R^m×n) in the sense that∀g∈C( ¯Ω)∀v₀∈R:

klim→∞

Ωg(x)v(yk(x))dx=

Ω¯

β_RR^m×ng(x)v₀(s)ˆνx(ds)σ(dx), (1.19) where

v∈Υ_R^p(R^m×n) :={v₀(1 +| · |^p); v₀∈ R}.

In particular, puttingv₀= 1∈ Rin (1.19) we can see that

klim→∞(1 +|yk|^p) = σ weakly* in rca( ¯Ω). (1.20) If (1.19) holds, we say that {yk}∈N generates (σ,νˆ). Let us denote by DM^p_R(Ω;R^m×n) the set of all pairs (σ,ν)ˆ ∈rca( ¯Ω)×L^∞_w( ¯Ω, σ; rca(β_RR^m×n)) attainable by sequences fromL^p(Ω;R^m×n); note that, takingv₀= 1

M. KRUˇZ´IK

in (1.19), one can see that these sequences must be inevitably bounded in L^p(Ω;R^m×n). We also denote by GDM^p_R(Ω;R^m×n) measures from DM^p_R(Ω;R^m×n) generated by a sequence of gradients of some bounded sequence in W^1,p(Ω;R^m). The explicit description of the elements from DM^p_R(Ω;R^m×n), called DiPerna–

Majda measures, for unconstrained sequences was given in [22], Theorem 2. In fact, it is easy to see that (1.19) can be also written in the form

klim→∞

Ωh(x, yk(x))dx=

¯ Ω

β_RR^m×nh₀(x, s)ˆνx(ds)σ(dx), (1.21) whereh(x, s) :=h₀(x, s)(1 +|s|^p) andh₀∈C( ¯Ω⊗β_RR^m×n).

We say that{yk}generates (σ,νˆ) if (1.19) holds. Moreover, we denotedσ∈L¹(Ω) the absolutely continuous (with respect to the Lebesgue measure) part ofσin the Lebesgue decomposition of σ.

Let us recall that for any (σ,ν)ˆ ∈ DM^p_R(Ω;R^m×n) there is precisely one (σ^◦,νˆ^◦) ∈ DM^p_R(Ω;R^m×n) such

that

Ω

R^m×nv₀(s)ˆνx(ds)g(x)σ(dx) =

Ω¯

R^m×nv₀(s)ˆν_x^◦(ds)g(x)σ^◦(dx) (1.22) for anyv₀ ∈C₀(R^m×n) and anyg ∈C( ¯Ω) and (σ^◦,νˆ^◦) is attainable by a sequence{yk}k∈Nsuch that the set {|yk|^p; k∈N}is relatively weakly compact inL¹(Ω); see [22,30] for details. We call (σ^◦,νˆ^◦) the nonconcentrating modification of (σ,ν). We call (σ,ˆ ν)ˆ ∈ DM^p_R(Ω;R^m×n) nonconcentrating if

¯ Ω

β_RR^m×n\R^m×nνˆx(ds)σ(dx) = 0.

There is a one-to-one correspondence between nonconcentrating DiPerna–Majda measures and Young measures;

cf.[30].

We wish to emphasize the following fact: if{yk} ∈L^p(Ω;R^m×n) generates (σ,νˆ)∈ DM^p_R(Ω;R^m×n) andσis absolutely continuous with respect to the Lebesgue measure it generallydoes not mean that{|yk|^p}is weakly relatively compact inL¹(Ω). A simple examples can be founde.g.in [23,30].

Having a sequence bounded inL^p(Ω;R^m×n) generating a DiPerna−Majda measure (σ,νˆ)∈ DM^p_R(Ω;R^m×n) it also generates anL^p-Young measureν ∈ Y^p(Ω;R^m×n). It easily follows from [30], Theorem 3.2.13, that

νx(ds) =dσ^◦(x)νˆ_x^◦(ds)

1 +|s|^p for a.a.x∈Ω. (1.23)

Note that (1.23) is well-defined as ˆν_x^◦ is supported on R^m×n. As pointed out (in [22], Rem. 2) for almost all x∈Ω

dσ(x) =

R^m×n

ˆ νx(ds) 1 +|s|^p

−1· (1.24)

In fact, that (1.22) can be even improved to

Ω

R^m×nv₀(s)ˆνx(ds)g(x)σ(dx) =

Ω¯

R^m×nv₀(s)ˆν_x^◦(ds)g(x)σ^◦(dx) (1.25) for any v₀ ∈ R and any g ∈ C( ¯Ω). The one-to-one correspondence between Young and DiPerna−Majda measures, in particular (see (1.23) and (1.25))

R^m×nv(s)νx(ds) =dσ(x)

R^m×nv₀(s)ˆνx(ds)

QUASICONVEXITY AT THE BOUNDARY AND CONCENTRATION EFFECTS GENERATED BY GRADIENTS

wheneverv∈Υ_R^p(R^m×n), finally yields that ∀g∈C( ¯Ω)∀v∈Υ_R^p(R^m×n):

klim→∞

Ωg(x)v(yk(x))dx=

Ω

R^m×nv(s)g(x)νx(ds) dx+

Ω¯

β_RR^m×n\R^m×n

v(s)

1 +|s|^pνˆx(ds)g(x)σ(dx), (1.26) where ν ∈ Y^p(Ω;R^m×n) and (σ,νˆ)∈ DM^p_R(Ω;R^m×n) are Young and DiPerna–Majda measures generated by {yk}k∈N, respectively. We will denote elements from DM^p_R(Ω;R^m×n) which are generated by {∇uk}k∈N for some bounded {uk} ⊂W^1,p(Ω;R^m) byGDM^p_R(Ω;R^m×n).

We will also use the following result, whose proof can be found in several places in various contexts (see [22], Lem. 1, Th. 1,2, [30], Prop. 3.2.17, or for a compactification ofR^m×nby the sphere see [1], Prop. 4.1, part (iii)).

Lemma 1.11. Let Ω⊂Rⁿ be a bounded open domain such that |∂Ω|= 0,Rbe a separable complete subring of the ring of all continuous bounded functions onR^m×n and(σ,νˆ)∈ DM^p_R(Ω;R^m×n). Then for σs- almost all x∈Ω¯ we have

ˆ

νx(R^m×n) = 0. (1.27)

The following two theorems were proved in [15].

Theorem 1.12. Let Ω ⊂ Rⁿ be a bounded domain with Lipschitz boundary, 1 < p < +∞ and (σ,νˆ) ∈ DM^p_R(Ω;R^m×n). Then then there is u∈W^1,p(Ω;R^m) and a bounded sequence{uk −u}k∈N ⊂W₀^1,p(Ω;R^m) such that {∇uk}k∈Ngenerates (σ,νˆ)if and only if the following three conditions hold

for a.a. x∈Ω:∇u(x) =dσ(x)

β_RR^m×n

s

1 +|s|^pνˆx(ds), (1.28) for almost allx∈Ω and for allv∈Υ_R^p(R^m×n)the following inequality is fulfilled

Qv(∇u(x))≤dσ(x)

β_RR^m×n

v(s)

1 +|s|^pνˆx(ds), (1.29)

for σ-almost all x∈Ω¯ and allv∈Υ_R^p(R^m×n)withQv >−∞it holds that 0≤

β_RR^m×n\R^m×n

v(s)

1 +|s|^pνˆx(ds). (1.30)

The next theorem addresses DiPerna−Majda measures generated by gradients of maps with possibly different traces.

Theorem 1.13. Let Ω be an arbitrary bounded domain, 1 < p < +∞ and (σ,νˆ) ∈ GDM^p_R(Ω;R^m×n) be generated by {∇uk}k∈N such that w-limk→∞uk =uin W^1,p(Ω;R^m). Then the conditions (1.28), (1.29) hold, and(1.30)is satisfied forσ-a.a. x∈Ω.

Remark 1.14. (i) It can happen that under the assumptions of Theorem1.13formula (1.30) does not hold on

∂Ω. See an example in [4] showing the violation of weak sequential continuity of W^1,2(Ω;R²)→L¹(Ω) :u→ det∇uifΩ= (−1,1)².

(ii) In terms of Young measures, conditions (1.28) and (1.29) read, respectively: there isu∈W^1,p(Ω;R^m):

∇u(x) =

R^m×nsνx(ds), (1.31)

M. KRUˇZ´IK

for allv:R^m×n→R,|v| ≤C(1 +| · |^p):

Qv(∇u(x))≤

R^m×nv(s)νx(ds). (1.32)

Finally, we have the following result from [15].

Theorem 1.15. LetΩ⊂Rⁿ be a bounded Lipschitz domain. Let0≤g∈C( ¯Ω),v∈C(R^m×n),|v| ≤C(1+|·|^p), C >0, quasiconvex, and1< p <+∞. Then the functional I:W^1,p(Ω;R^m)→Rdefined as

I(u) :=

Ωg(x)v(∇u(x)) dx (1.33)

is sequentially weakly lower semicontinuous in W^1,p(Ω;R^m) if and only if for any bounded sequence {wk} ⊂ W^1,p(Ω;R^m)such that∇wk →0 in measure we have lim infk→∞I(wk)≥I(0).

1.4.1. Compactification ofR^m×n by the sphere

In what follows we will work mostly with a particular compactification ofR^m×n, namely, with the compact- ification by the sphere. We will consider the following ring of continuous bounded functions

S :=

v₀∈C(R^m×n) : there existc∈R^m×n, v₀,0∈C₀(R^m×n), andv₀,1∈C(S^(m×n)−1) s.t.

v₀(s) =c+v₀,0(s) +v₀,1

s

|s|

|s|^p

1 +|s|^p ifs= 0 andv₀(0) =v₀,0(0)

, (1.34)

whereS^m×n−1denotes the (mn−1)-dimensional unit sphere inR^m×n. Thenβ_SR^m×n is homeomorphic to the unit ballB(0,1)⊂R^m×n viathe mapping d:R^m×n→B(0,1),d(s) :=s/(1 +|s|) for alls∈R^m×n. Note that d(R^m×n) is dense inB(0,1).

For any v ∈ Υ_S^p(R^m×n) there exists a continuous and positively p-homogeneous function v_∞ : R^m×n → R (i.e. v_∞(αs) =α^pv_∞(s) for allα≥0 ands∈R^m) such that

|slim|→∞

v(s)−v_∞(s)

|s|^p = 0. (1.35)

Indeed, if v₀is as in (1.34) andv=v₀(1 +| · |^p) then set v_∞(s) :=

c+v₀,1

s

|s|

|s|^p fors∈R^m×n\ {0}.

By continuity we define v_∞(0) := 0. It is easy to see that v_∞ satisfies (1.35). Such v_∞ is called the recession function ofv. It will be useful to denote

v_S(s) := (c+v₀,1) s

|s| · (1.36)

The following lemma can be found in [11,12].

Lemma 1.16. Letv∈C(R^m×n)be Lipschitz continuous on the unit sphereS^m×n−1andp-homogeneous,p≥1.

Then v isp-Lipschitz,i.e., there is a constant α >0 such that for anys₁, s₂∈R^m×n it holds

|v(s₁)−v(s₂)| ≤α(|s₁|^p−1+|s₂|^p−1)|s₁−s₂|. (1.37)