Relaxation for highly discontinuous, possibly unbounded, integral functionals

We consider the functional $$F(u)={\int }_{\mathrm{\Omega }}f(\mathrm{\nabla }u)dxu\in \phi +W_0^{1,1}(\mathrm{\Omega })$$ where $\mathrm{\Omega }$ is a Lipschitz bounded open set of ${\mathbb{R}}^N$, $f:{\mathbb{R}}^N\to \mathbb{R}\cup \{+\mathrm{\infty }\}$ is a superlinear Borel function, $\phi \in W^{1,\mathrm{\infty }}(\mathrm{\Omega })$. We prove that, if $f$ is superlinear and satisfies very weak assumptions, then the Lavrentiev phenomenon does not occur. We underline that our assumptions include the case of non continuous, non convex, and unbounded Lagrangians.