We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\left\{\nabla {u}_{k}\right\}$, bounded in ${L}^{p}(\xd8;{\mathbb{R}}^{m\times n})$ if $p>1$ and $\Omega \subset {\mathbb{R}}^{n}$ is a bounded domain with the extension property in ${W}^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $\Omega $ are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.

Keywords: sequences of gradients, concentrations, oscillations, quasiconvexity

@article{COCV_2008__14_1_71_0, author = {Ka{\l}amajska, Agnieszka and Kru\v{z}{\'\i}k, Martin}, title = {Oscillations and concentrations in sequences of gradients}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {71--104}, publisher = {EDP-Sciences}, volume = {14}, number = {1}, year = {2008}, doi = {10.1051/cocv:2007051}, mrnumber = {2375752}, zbl = {1140.49009}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/cocv:2007051/} }

TY - JOUR AU - Kałamajska, Agnieszka AU - Kružík, Martin TI - Oscillations and concentrations in sequences of gradients JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2008 SP - 71 EP - 104 VL - 14 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv:2007051/ DO - 10.1051/cocv:2007051 LA - en ID - COCV_2008__14_1_71_0 ER -

%0 Journal Article %A Kałamajska, Agnieszka %A Kružík, Martin %T Oscillations and concentrations in sequences of gradients %J ESAIM: Control, Optimisation and Calculus of Variations %D 2008 %P 71-104 %V 14 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv:2007051/ %R 10.1051/cocv:2007051 %G en %F COCV_2008__14_1_71_0

Kałamajska, Agnieszka; Kružík, Martin. Oscillations and concentrations in sequences of gradients. ESAIM: Control, Optimisation and Calculus of Variations, Volume 14 (2008) no. 1, pp. 71-104. doi : 10.1051/cocv:2007051. http://archive.numdam.org/articles/10.1051/cocv:2007051/

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