Oscillations and concentrations in sequences of gradients

Agnieszka Kałamajska; Martin Kružík

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 14, Issue: 1, page 71-104
  • ISSN: 1292-8119

Abstract

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We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, { ∇ u k } , bounded in L p ( Ω ; ℝ m × n ) if p > 1 and Ω ⊂ 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.

How to cite

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Kałamajska, Agnieszka, and Kružík, Martin. "Oscillations and concentrations in sequences of gradients." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2010): 71-104. <http://eudml.org/doc/90866>.

@article{Kałamajska2010,
abstract = { 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 $L^p(\Omega;\{\mathbb R\}^\{m\times n\})$ 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 Ω are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed. },
author = {Kałamajska, Agnieszka, Kružík, Martin},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Sequences of gradients; concentrations; oscillations; quasiconvexity; sequences of gradients},
language = {eng},
month = {3},
number = {1},
pages = {71-104},
publisher = {EDP Sciences},
title = {Oscillations and concentrations in sequences of gradients},
url = {http://eudml.org/doc/90866},
volume = {14},
year = {2010},
}

TY - JOUR
AU - Kałamajska, Agnieszka
AU - Kružík, Martin
TI - Oscillations and concentrations in sequences of gradients
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 14
IS - 1
SP - 71
EP - 104
AB - 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 $L^p(\Omega;{\mathbb R}^{m\times n})$ 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 Ω are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.
LA - eng
KW - Sequences of gradients; concentrations; oscillations; quasiconvexity; sequences of gradients
UR - http://eudml.org/doc/90866
ER -

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