Discarding Lavrentiev's Gap in Non-autonomous and Non-Convex Variational Problems

Abstract

We establish that the Lavrentiev gap between Sobolev and Lipschitz maps does not occur for a scalar variational problem of the form: \[ \textrm{to minimize} \qquad u \mapsto \int_Ωf(x,u,\nabla u)\,dx \,, \] under a Dirichlet boundary condition. Here, \(Ω\) is a bounded Lipschitz open set in \(\rn\), \(N\geq 1\) and the function $f$ is required to be measurable with respect to the spatial variable, continuous with respect to the second one, and convex with respect to the last variable. Under these assumptions alone, Lavrentiev gaps may occur, as illustrated by classical examples in the literature. We identify an additional natural condition on $f$ to discard such phenomena, that can be interpreted as a balance between the variations with respect to the first variable and the growth with respect to the last one. This unifies most of the structural assumptions that have been introduced so far to prevent the occurence of Lavrentiev gaps. Remarkably, typical assumptions that are usually imposed on $f$ in this setting are dropped here: we do not require $f$ to be bounded or convex with respect to the second variable, nor impose any condition of ${\Delta }_2$-kind with respect to the last variable.