A note on relaxation with constraints on the determinant
ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 41.

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 ∫ W$$(x, u(x), Du(x)) dx is an upper bound for the relaxation of E and coincides with the relaxation if the quasiconvex envelope W$$ 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.

Reçu le :
Accepté le :
DOI : 10.1051/cocv/2018030
Classification : 49J45, 74B20
Mots-clés : Calculus of variations, nonlinear elasticity
Cicalese, Marco 1 ; Fusco, Nicola 1

1 
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Cicalese, Marco; Fusco, Nicola. A note on relaxation with constraints on the determinant. ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 41. doi : 10.1051/cocv/2018030. http://www.numdam.org/articles/10.1051/cocv/2018030/

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[2] S. Conti and G. Dolzmann, On the theory of relaxation in nonlinear elasticity with constraints on the determinant. Arch. Rational Mech. Anal. 217 (2015) 413–437. | DOI | MR | Zbl

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