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

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 $W_0^{1,p}(\Omega )$ 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 $q/p\le 1+\alpha /N$. See \cite[Theorem~4.1]{min-double-reg1} for the sharpness of the result.

References in corpus (3)

• A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces
• Existence of renormalized solutions to elliptic equation in Musielak-Orlicz space
• Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions

Cited by in corpus (8)

• A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces
• Well-posedness of parabolic equations in the non-reflexive and anisotropic Musielak-Orlicz spaces in the class of renormalized solutions
• Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• On a range of exponents for absence of Lavrentiev phenomenon for double phase functionals
• Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces
• An extended variational theory for nonlinear evolution equations via modular spaces
• On the Lavrentiev gap for convex, vectorial integral functionals