Relaxation for highly discontinuous, possibly unbounded, integral functionals

Abstract

We consider the functional \[ F(u)=\int_Ω f(\nabla u)\,dx\qquad u\inφ+W^{1,1}_0(Ω) \] where $\hat{I}\text{c◯}$ is a Lipschitz bounded open set of ${\mathbb{R}}^N$, $f:{\mathbb{R}}^N\to \mathbb{R}\cup \{+\infty \}$ is a superlinear Borel function, $\ddot{I}†\in W^{1,\infty }(\hat{I}\text{c◯})$. 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.

26 pages