Existence of Minimizers for Polyconvex and Nonpolyconvex Problems

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