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Article

Kružík, Martin
On convergence of gradient-dependent integrands. (English). Applications of Mathematics, vol. 52 (2007), issue 6, pp. 529-543
MSC: 35B05, 49J45 | MR 2357579 | Zbl 1164.49305 | DOI: 10.1007/s10492-007-0031-4
Keywords:
bounded sequences of gradients; concentrations; oscillations; quasiconvexity; weak convergence
Summary:
We study convergence properties of $\lbrace v(\nabla u_k)\rbrace _{k\in \mathbb{N}}$ if $v\in C(\mathbb{R}^{m\times n})$, $|v(s)|\le C(1+|s|^p)$, $1<p<+\infty $, has a finite quasiconvex envelope, $u_k\rightarrow u$ weakly in $W^{1,p} (\Omega ;\mathbb{R}^m)$ and for some $g\in C(\Omega )$ it holds that $\int _\Omega g(x)v(\nabla u_k(x))\mathrm{d}x\rightarrow \int _\Omega g(x) Qv(\nabla u(x))\mathrm{d}x$ as $k\rightarrow \infty $. In particular, we give necessary and sufficient conditions for $L^1$-weak convergence of $\lbrace \det \nabla u_k\rbrace _{k\in \mathbb{N}}$ to $\det \nabla u$ if $m=n=p$.
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