Gossez's approximation theorems in the Musielak-Orlicz-Sobolev spaces

TL;DR

This paper extends Gossez's approximation theorems to Musielak-Orlicz-Sobolev spaces, establishing density of smooth functions under new regularity assumptions and analyzing the absence of Lavrentiev phenomenon in double-phase spaces.

Contribution

It introduces new regularity conditions for density results in Musielak-Orlicz-Sobolev spaces, unifying and improving previous results in related function spaces.

Findings

Density of smooth functions in Musielak-Orlicz-Sobolev spaces established

New regularity assumptions enable unification of approximation results

Absence of Lavrentiev phenomenon in double-phase spaces within sharp parameter range

Abstract

We prove the density of smooth functions in the modular topology in the Musielak-Orlicz-Sobolev spaces essentially extending the results of Gossez \cite{GJP2} obtained in the Orlicz-Sobolev setting. We impose new systematic regularity assumption on $M$ which allows to study the problem of density unifying and improving the known results in the Orlicz-Sobolev spaces, as well as the variable exponent Sobolev spaces. We confirm the precision of the method by showing the lack of the Lavrentiev phenomenon in the double-phase case. Indeed, we get the modular approximation of $W^{1,p}_0(\Omega)$ functions by smooth functions in the double-phase space governed by the modular function $H(x,s)=s^p+a(x)s^q$ with $a\in C^{0,\alpha}(\Omega)$ excluding the Lavrentiev phenomenon within the sharp range $q/p\leq 1+\alpha/N$. See \cite[Theorem~4.1]{min-double-reg1} for the sharpness of the result.