On convergence of gradient-dependent integrands

Kružík, Martin

Applications of Mathematics, Tome 52 (2007) no. 6, pp. 529-543.

Résumé

We study convergence properties of $\lbrace v(\nabla u_k)\rbrace _{k\in \mathbb{N}}$ if $v\in C(\mathbb{R}^{m\times n})$, $|v(s)|\le C(1+|s|^p)$, $1$, 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 $m=n=p$.

Détail

DOI : 10.1007/s10492-007-0031-4

Classification : 35B05, 49J45
Keywords: bounded sequences of gradients; concentrations; oscillations; quasiconvexity; weak convergence