On convergence of gradient-dependent integrands

Martin Kružík

Applications of Mathematics (2007)

  • Volume: 52, Issue: 6, page 529-543
  • ISSN: 0862-7940

Abstract

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We study convergence properties of { v ( ∇ u k ) } k ∈ ℕ if v ∈ C ( ℝ m × n ) , | v ( s ) | ≤ C ( 1 + | s | p ) , 1 < p < + ∞ , has a finite quasiconvex envelope, u k → u weakly in W 1 , p ( Ω ; ℝ m ) and for some g ∈ C ( Ω ) it holds that ∫ Ω g ( x ) v ( ∇ u k ( x ) ) d x → ∫ Ω g ( x ) Q v ( ∇ u ( x ) ) d x as k → ∞ . In particular, we give necessary and sufficient conditions for L 1 -weak convergence of { det ∇ u k } k ∈ ℕ to det ∇ u if m = n = p .

How to cite

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Kružík, Martin. "On convergence of gradient-dependent integrands." Applications of Mathematics 52.6 (2007): 529-543. <http://eudml.org/doc/33307>.

@article{Kružík2007,
abstract = {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 $m=n=p$.},
author = {Kružík, Martin},
journal = {Applications of Mathematics},
keywords = {bounded sequences of gradients; concentrations; oscillations; quasiconvexity; weak convergence; bounded sequences of gradients; concentrations; oscillations; quasiconvexity; weak convergence},
language = {eng},
number = {6},
pages = {529-543},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On convergence of gradient-dependent integrands},
url = {http://eudml.org/doc/33307},
volume = {52},
year = {2007},
}

TY - JOUR
AU - Kružík, Martin
TI - On convergence of gradient-dependent integrands
JO - Applications of Mathematics
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 52
IS - 6
SP - 529
EP - 543
AB - 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 $m=n=p$.
LA - eng
KW - bounded sequences of gradients; concentrations; oscillations; quasiconvexity; weak convergence; bounded sequences of gradients; concentrations; oscillations; quasiconvexity; weak convergence
UR - http://eudml.org/doc/33307
ER -

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