On non quasiconvex problems of the calculus of variations

Bernard Dacorogna; Giovanni Pisante; Ana Margarida Ribeiro

doi:10.3934/dcds.2005.13.961

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2005, Volume 13, Issue 4: 961-983. Doi: 10.3934/dcds.2005.13.961

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On non quasiconvex problems of the calculus of variations

1.
Department of Mathématics, EPFL, 1015 Lausanne
2.
School of Mathematics, Georgia Tech, 30332, Atlanta, GA
3.
Section of Mathematics, EPFL, 1015 Lausanne

Received: August 31, 2004

Revised: February 28, 2005

Published: June 2005

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Abstract

We study existence of minimizers for problems of the type

inf{$\int_\Omega f(Du(x)) dx:u=u_{\xi _0}$ on $\partial\Omega$ }

where $f$ is non quasiconvex and $u_{\xi_0}$ is an affine function. Applying some new results on differential inclusions, we get sufficient conditions. We also study necessary conditions. We then consider some examples.

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Mathematics Subject Classification: Primary: 49K20, 49K24.

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