Existence of Minimizers for Polyconvex and Nonpolyconvex Problems
Giovanni Cupini, Elvira Mascolo · SIAM Journal on Control and Optimization · 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.