Integral representations of lower semicontinuous envelopes and Lavrentiev Phenomenon for non continuous Lagrangians

Abstract

We consider the functional $$F_{\infty }(u)={\int }_{\Omega }f(x,u(x),\nabla u(x))dxu\in \phi +W_0^{1,\infty }(\Omega ,\mathbb{R})$$ where $\Omega$ is an open bounded Lipschitz subset of ${\mathbb{R}}^N$ and $\phi \in W^{1,\infty }(\Omega )$. We do not assume neither convexity or continuity of the Lagrangian w.r.t. the last variable. We prove that, under suitable assumptions, the lower semicontinuous envelope of $F_{\infty }$ both in $\phi +W^{1,\infty }(\Omega )$ and in the larger space $\phi +W^{1,p}(\Omega )$ can be represented by means of the bipolar $f^{\ast \ast }$ of $f$. In particular we can also exclude Lavrentiev Phenomenon between $W^{1,\infty }(\Omega )$ and $W^{1,1}(\Omega )$ for autonomous Lagrangians.