Relaxation for highly discontinuous, possibly unbounded, integral functionals

TL;DR

This paper establishes conditions under which the relaxation of certain integral functionals with highly discontinuous, unbounded Lagrangians avoids the Lavrentiev phenomenon, even when the integrand is non-convex and non-continuous.

Contribution

It proves that for superlinear, weakly assumed integrands, the Lavrentiev phenomenon does not occur, extending results to non-continuous, non-convex, and unbounded cases.

Findings

Lavrentiev phenomenon is absent under weak assumptions

Applicable to non-continuous, non-convex, unbounded Lagrangians

Relaxation results hold for superlinear integrands

Abstract

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.