Oscillations and concentrations in sequences of gradients

Kałamajska, Agnieszka  ; Kružík, Martin

ESAIM: Control, Optimisation and Calculus of Variations, Tome 14 (2008) no. 1, pp. 71-104

Résumé

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 $L^p(Ø;{ℝ}^{m\times n})$ if $p>1$ and $\Omega \subset {ℝ}^n$ 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.

Détail

DOI : 10.1051/cocv:2007051

Classification : 49J45, 35B05
Keywords: sequences of gradients, concentrations, oscillations, quasiconvexity