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

We consider the functional $F_{\infty }\left(u\right)={\int }_{\Omega }f(x,u\left(x\right),\nabla u\left(x\right))dxu\in \phi +W_0^{1,\infty }(\Omega ,R)$where $\Omega$ is an open bounded Lipschitz subset of $R^N$ and $\phi \in W^{1,\infty }\left(\Omega \right)$. 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 $\phi +W^{1,\infty }\left(\Omega \right)$ and in the larger space $\phi +W^{1,p}\left(\Omega \right)$ can be represented by means of the bipolar $f^{\ast \ast }$ of $f$. In particular we can also exclude Lavrentiev Phenomenon between $W^{1,\infty }\left(\Omega \right)$ and $W^{1,1}\left(\Omega \right)$ for autonomous Lagrangians.