𝒜-quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

Jan Krämer; Stefan Krömer; Martin Kružík; Gabriel Pathó

doi:10.1515/acv-2015-0009

Article

𝒜-quasiconvexity at the boundary and weak lower semicontinuity of integral functionals

Jan Krämer , Stefan Krömer , Martin Kružík and Gabriel Pathó

Published/Copyright: December 9, 2015

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From the journal Advances in Calculus of Variations Volume 10 Issue 1

Abstract

We state necessary and sufficient conditions for weak lower semicontinuity of integral functionals of the form u↦∫Ωh⁢(x,u⁢(x))⁢dx, where h is continuous and possesses a positively p-homogeneous recession function, p>1, and u∈Lp⁢(Ω;ℝm) lives in the kernel of a constant-rank first-order differential operator 𝒜 which admits an extension property. In the special case 𝒜=curl, apart from the quasiconvexity of the integrand, the recession function’s quasiconvexity at the boundary in the sense of Ball and Marsden is known to play a crucial role. Our newly defined notions of 𝒜-quasiconvexity at the boundary, generalize this result. Moreover, we give an equivalent condition for the weak lower semicontinuity of the above functional along sequences weakly converging in Lp⁢(Ω;ℝm) and approaching the kernel of 𝒜 even if 𝒜 does not have the extension property.

Keywords: concentrations; oscillations

MSC 2010: 49J45; 35B05

Communicated by Frank Duzaar

Funding statement: We acknowledge the support through the project CZ01-DE03/2013-2014/DAAD-56269992 (PPP program). Moreover, M. Kružík was also partly supported by grants GAČR P201/10/0357 and P107/12/0121. The work of G. Pathó was partially supported by the project at the Faculty of Mathematics and Physics of Charles University no. 260098/2014: Student research in didactics of physics and mathematical and computer modelling.

A Appendix

A.1 DiPerna–Majda measures

DiPerna–Majda measures are generalizations of Young measures; see [13, 23, 26, 27] for their modern treatment and applications. In what follows, ℳ⁢(Ω¯) denotes the space of Radon measures on Ω¯. Let ℛ be a complete (i.e. containing constants, separating points from closed subsets and closed with respect to the supremum norm), separable (i.e. containing a dense countable subset) ring of continuous bounded functions from ℝm into ℝ. It is known that there is a one-to-one correspondence ℛ↦βℛ⁢ℝm between such rings and (metrizable) compactifications βℛ⁢ℝm of ℝm [9]. We only need the special case ℛ=𝒮 with

𝒮:={v0∈C(ℝm):there exist c∈ℝ,v0,0∈C0(ℝm),v0,1∈C(Sm-1) such that

(A.1)v0(s)=c+v0,0(s)+v0,1(s|s|)|s|p1+|s|p if s≠0 and v0(0)=c+v0,0(0)},

where Sm-1 denotes the (m-1)-dimensional unit sphere in ℝm. In this case, β𝒮⁢ℝm is obtained by adding a sphere to ℝm at infinity. More precisely, β𝒮⁢ℝm is homeomorphic to the closed unit ball B⁢(0,1)¯⊂ℝm via the mapping f:ℝm→B⁢(0,1) with f⁢(s):=s/(1+|s|) for all s∈ℝm. Note that f⁢(ℝm) is dense in B⁢(0,1)¯. For simplicity, we will not distinguish between ℝm and its image in β𝒮⁢ℝm.

DiPerna and Majda [8] proved the following theorem:

Theorem A.1

Let Ω be an open domain in Rn with Ln⁢(∂⁡Ω)=0, and let {yk}⊂Lp⁢(Ω;Rm), with 1≤p<+∞, be bounded. Then, there exists a subsequence (not relabeled), a positive Radon measure π∈M⁢(Ω¯) and a family λ:={λx}x∈Ω¯ of probability measures on βS⁢Rm such that for all h0∈C⁢(Ω¯×βS⁢Rm) it holds that

(A.2)limk→∞⁡∫Ωh0⁢(x,yk⁢(x))⁢(1+|yk⁢(x)|p)⁢dx=∫Ω¯∫β𝒮⁢ℝmh0⁢(x,s)⁢dλx⁢(s)⁢dπ⁢(x).

If (A.2) holds we say that {yk} generates (π,λ) and we denote the set of all such pairs of measures generated by some sequence in Lp⁢(Ω;ℝm) by 𝒟⁢ℳ𝒮p⁢(Ω;ℝm).

For any h⁢(x,s):=h0⁢(x,s)⁢(1+|s|p) with h0∈C⁢(Ω¯×β𝒮⁢ℝm) there exists a continuous and positively p-homogeneous function h∞:Ω¯×ℝm→ℝ, i.e., h∞⁢(x,t⁢s)=tp⁢h∞⁢(x,s) for all t≥0, all x∈Ω¯, and s∈ℝm, such that

(A.3)lim|s|→∞⁡h⁢(x,s)-h∞⁢(x,s)|s|p=0 for all x∈Ω¯.

It is already mentioned in [12, formula (A.13)] that if {yk}⊂Lp⁢(Ω;ℝm) is bounded and we have that ℒn({x∈Ω;yk(x)≠0})→0 as k→∞, then it is enough to test (A.2) with recession functions only, i.e., is then equivalent to

(A.4)limk→∞⁡∫Ωh∞⁢(x,yk⁢(x))⁢dx=∫Ω¯∫β𝒮⁢ℝm∖ℝmh∞⁢(x,s)1+|s|p⁢dλx⁢(s)⁢dπ⁢(x),

where (x,s)↦h0⁢(x,s):=h∞⁢(x,s)/(1+|s|p) belongs to C⁢(Ω¯×β𝒮⁢ℝm), which is the closure (in the maximum norm) of the linear hull of {g⊗v/|⋅|p:g∈C(Ω¯),v∈Υp(ℝm)}.

The following theorem is a direct consequence of [12, Theorem 2.2].

Theorem A.2

Let h∈C⁢(Ω¯;Chomp⁢(Rm)) such that h⁢(x,⋅) is A-quasiconvex for all x∈Ω¯ (whence h coincides with its A-quasiconvex envelope QA⁢h, and in particular, QA⁢h⁢(0)=h⁢(0)=0). Suppose that {yk}⊂Lp⁢(Ω;Rm)∩ker⁢A generates (π,λ)∈D⁢MSp⁢(Ω;Rm). Then, for π-almost every x∈Ω,

(A.5)0≤∫β𝒮⁢ℝm∖ℝmh⁢(x,s)1+|s|p⁢dλx⁢(s).

A.2 Uniform continuity properties of the functional

The following lemma essentially allows us to modify sequences inside I as long as the modified sequences approaches the original one in the norm of Lp.

Lemma A.3

Let h∞∈C⁢(Ω¯;Chomp⁢(Rm)). Then, for any pair {uk}, {vk} of bounded sequences in Lp⁢(Ω;Rm) such that uk-vk→0 strongly in Lp, we have that h∞⁢(⋅,uk⁢(⋅))-h∞⁢(⋅,vk⁢(⋅))→0 strongly in L1.

Proof.

For δ>0, let

Ak⁢(δ):={x∈Ω:|uk⁢(x)-vk⁢(x)|≥δ⁢(|uk⁢(x)|+|vk⁢(x)|+1)}.

Since uk-vk→0 in Lp, we see that

(A.6)∫Ak⁢(δ)(|uk⁢(x)|+|vk⁢(x)|+1)p⁢dx→0 as k→∞, for every δ.

In addition, h∞ is uniformly continuous on the compact set Ω¯×B⁢(0,1)¯⊂ℝn×ℝm, with a modulus of continuity μ, whence

∫Ω∖Ak⁢(δ)|h∞(x,uk)-h∞(x,vk)|dx

=∫Ω∖Ak⁢(δ)|h∞⁢(x,uk|uk|+|vk|+1)-h∞⁢(x,vk|uk|+|vk|+1)|⁢(|uk⁢(x)|+|vk⁢(x)|+1)p⁢dx

≤∫Ω∖Ak⁢(δ)μ⁢(δ)⁢(|uk⁢(x)|+|vk⁢(x)|+1)p⁢dx

(A.7)≤μ⁢(δ)⁢C⁢⟶δ→0⁢0 uniformly in k,

where we also used that {uk} and {vk} are bounded in Lp. Combining (A.6) and (A.7), we can verify that ∥h∞⁢(⋅,uk⁢(⋅))-h∞⁢(⋅,vk⁢(⋅))∥L1 can be made arbitrarily small, first choosing δ small enough and then k large, depending on δ. ∎

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Received: 2015-2-27

Revised: 2015-8-31

Accepted: 2015-11-4

Published Online: 2015-12-9

Published in Print: 2017-1-1

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Articles in the same Issue

https://doi.org/10.1515/acv-2015-0009

Keywords for this article

concentrations; oscillations