A Survey on the Non Occurence of the Lavrentiev Gap for Convex, Autonomous Multiple Integral Scalar Variational Problems

Abstract

In a recent paper we proved the non occurrence of the Lavrentiev gap between Lipschitz and Sobolev functions for functionals of the form $$\text {I}(u)={\int }_{\Omega }F(u,\nabla u) u|_{\partial {\Omega }} = \phi $$ when $\phi :\mathbb {R}^{n} \rightarrow \mathbb {R}$ is Lipschitz, Ω belongs to a wide class of open bounded sets in $\mathbb {R}^{n}$ containing Lipschitz domains, and the lagrangian F is assumed to be either convex in both variables or a sum of functions F(s, ξ) = a(s)g(ξ) + b(s) with g convex and s ↦ a(s)g(0) + b(s) satisfying a non oscillatory condition at infinity. In this survey we discuss the state of the art on the subject and give a self-contained proof of our result in the simpler case of a (strongly) star-shaped domain, for a lagrangian depending just on the gradient; in particular we point out what are the main difficulties to overcome in order to get the result without assuming growth conditions. We also formulate, and prove, a characterization of a useful class of star-shaped domains in terms of the radii function.