A note on relaxation with constraints on the determinant

¹ Zentrum Mathematik – M7, Technische Universität München, Garching, Germany.
² Dipartimento di Matematica e Applicazioni – Università di Napoli “Federico II”, Napoli, Italy.

^* Corresponding author: cicalese@ma.tum.de

Received: 6 November 2017
Accepted: 24 April 2018

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 ∫ W^qc(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^qc 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.