Relaxation for highly discontinuous, possibly unbounded, integral functionals

We consider the functional $$F(u)=\int_{\Omega} f(\nabla u)\,dx\qquad u\in\varphi+W^{1,1}_0(\Omega)$$ where $\Omega$ is a Lipschitz bounded open set of $\R^N$, $f:\R^N\to\R\cup \{+\infty\}$ is a superlinear Borel function, $\varphi\in W^{1,\infty}(\Omega)$. 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.