Calculus of Variations and Geometric Measure Theory

G. Cupini - E. Mascolo

Existence of minimizers for polyconvex and nonpolyconvex problems

created by cupini on 20 Apr 2005
modified on 26 Oct 2005


Published Paper

Inserted: 20 apr 2005
Last Updated: 26 oct 2005

Journal: SIAM J. Control Optim.
Volume: 44
Number: 4
Pages: 1370-1390
Year: 2005


We study the existence of Lipschitz minimizers of integral functionals $$ \mathcal{I}(u)=\int{\Omega} \varphi(x,\textrm{det}\,Du(x))\,dx$$ where $\Omega$ is an open subset of $\mathbb{R}^N$ with Lipschitz boundary, $\varphi:\Omega\times (0,+\infty)\to [0,+\infty)$ is a continuous function and $u\in W^{1,N}(\Omega, \mathbb{R}^N)$, $u(x)=x$ on $\partial \Omega$. We consider both the cases of $\varphi$ convex and nonconvex with respect to the last variable. The attainment results are obtained passing through the minimization of an auxiliary functional and the solution of a prescribed jacobian equation.

Keywords: Lipschitz regularity, nonpolyconvex functional, existence of minimizers, prescribed jacobian equation