A note on relaxation with constraints on the determinant

Marco Cicalese, Nicola Fusco

Publikation: Beitrag in FachzeitschriftArtikelBegutachtung

3 Zitate (Scopus)

Abstract

We consider multiple integrals of the Calculus of Variations of the form E(u) = ∫ W(x, u(x), Du(x)) dx where W is a Carathéodory function finite on matrices satisfying an orientation preserving or an incompressibility constraint of the type, det Du > 0 or det Du = 1, respectively. Under suitable growth and lower semicontinuity assumptions in the u variable we prove that the functional ∫ Wqc(x, u(x), Du(x)) dx is an upper bound for the relaxation of E and coincides with the relaxation if the quasiconvex envelope Wqc of W is polyconvex and satisfies p growth from below for p bigger then the ambient dimension. Our result generalises a previous one by Conti and Dolzmann [Arch. Rational Mech. Anal. 217 (2015) 413-437] relative to the case where W depends only on the gradient variable.

OriginalspracheEnglisch
Aufsatznummer41
FachzeitschriftESAIM - Control, Optimisation and Calculus of Variations
Jahrgang25
DOIs
PublikationsstatusVeröffentlicht - 2019

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