Calculus of Variations and Geometric Measure Theory

M. Cicalese - N. Fusco

A note on relaxation with constraints on the determinant

created by cicalese on 29 Jun 2017
modified on 02 Apr 2021


Published Paper

Inserted: 29 jun 2017
Last Updated: 2 apr 2021

Journal: Esaim: COCV
Volume: 25
Number: 41
Pages: 1-15
Year: 2017


We consider multiple integrals of the calculus of variations of the form $E(u)=\int W(x,u(x),Du(x))\, dx$ where $W$ is a Carath\'eodory 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 $\int W^{qc}(x,u(x),Du(x))\, dx$ 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 relative to the case where $W$ depends only on the gradient variable.