Musielak-Orlicz-Sobolev embeddings: Necessary and Sufficient Conditions

TL;DR

This paper establishes the precise conditions on the domain for Musielak-Orlicz-Sobolev embeddings involving variable exponent and logarithmic growth functions, advancing the understanding of functional analysis in variable exponent spaces.

Contribution

It provides the necessary and sufficient domain conditions for embeddings of Musielak-Orlicz-Sobolev spaces with specific growth functions involving variable exponents and logarithmic factors.

Findings

Characterization of domain conditions for embeddings

Extension of classical Sobolev embedding results

Identification of sharp criteria for Musielak-Orlicz spaces

Abstract

In this paper we study the necessary and sufficient conditions on domain for Musielak-Orlicz-Sobolev embedding of the space $W^{1,\Phi(\cdot,\cdot)}(\Omega)$ where $\Phi(x,t):=t^{p(x)}{(\log(e+t))}^{q(x)}.$