Geodesic
On convergence of gradient-dependent integrands
Applications of Mathematics, Tome 52 (2007) no. 6, pp. 529-543

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

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 $m=n=p$.
DOI : 10.1007/s10492-007-0031-4
Classification : 35B05, 49J45
Keywords: bounded sequences of gradients; concentrations; oscillations; quasiconvexity; weak convergence 
@article{10_1007_s10492_007_0031_4,
     author = {Kru\v{z}{\'\i}k, Martin},
     title = {On convergence of gradient-dependent integrands},
     journal = {Applications of Mathematics},
     pages = {529--543},
     publisher = {mathdoc},
     volume = {52},
     number = {6},
     year = {2007},
     doi = {10.1007/s10492-007-0031-4},
     mrnumber = {2357579},
     zbl = {1164.49305},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10492-007-0031-4/}
}
TY  - JOUR
AU  - Kružík, Martin
TI  - On convergence of gradient-dependent integrands
JO  - Applications of Mathematics
PY  - 2007
SP  - 529
EP  - 543
VL  - 52
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1007/s10492-007-0031-4/
DO  - 10.1007/s10492-007-0031-4
LA  - en
ID  - 10_1007_s10492_007_0031_4
ER  - 
%0 Journal Article
%A Kružík, Martin
%T On convergence of gradient-dependent integrands
%J Applications of Mathematics
%D 2007
%P 529-543
%V 52
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1007/s10492-007-0031-4/
%R 10.1007/s10492-007-0031-4
%G en
%F 10_1007_s10492_007_0031_4
Kružík, Martin. On convergence of gradient-dependent integrands. Applications of Mathematics, Tome 52 (2007) no. 6, pp. 529-543. doi: 10.1007/s10492-007-0031-4

Cité par Sources :