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

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