Oscillations and concentrations in sequences of gradients

Kałamajska, Agnieszka; Kružík, Martin

doi:10.1051/cocv:2007051

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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

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Résumé

We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, ${\nabla u_{k}}$ , bounded in $L^{p} (Ø; ℝ^{m \times n})$ if $p > 1$ and $Ω \subset ℝ^{n}$ 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 $Ω$ are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.

MR Zbl

DOI : 10.1051/cocv:2007051

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

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     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://geodesic.mathdoc.fr/articles/10.1051/cocv:2007051/}
}

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

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