Volume 25, 2019
|Number of page(s)||15|
|Published online||20 September 2019|
A note on relaxation with constraints on the determinant
Zentrum Mathematik – M7, Technische Universität München,
2 Dipartimento di Matematica e Applicazioni – Università di Napoli “Federico II”, Napoli, Italy.
* Corresponding author: firstname.lastname@example.org
Accepted: 24 April 2018
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.
Mathematics Subject Classification: 49J45 / 74B20
Key words: Calculus of variations / nonlinear elasticity
© EDP Sciences, SMAI 2019
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