A pocket guide to nonlinear differential equations in the Musielak--Orlicz spaces

TL;DR

This survey provides an overview of the theory of nonlinear PDEs in Musielak--Orlicz spaces, addressing existence issues in complex, inhomogeneous, and anisotropic settings with various growth conditions.

Contribution

It consolidates recent developments in existence theory for PDEs within Musielak--Orlicz spaces, including reflexive and non-reflexive cases, and discusses challenges with data below duality.

Findings

Unified framework for variable exponent, Orlicz, weighted Sobolev, and double-phase spaces.

Analysis of existence of PDE solutions in complex growth and inhomogeneous settings.

Addressing problems with data below natural duality without Lavrentiev's phenomenon.

Abstract

The Musielak--Orlicz setting unifies the variable exponent, Orlicz, weighted Sobolev, and double-phase spaces. They inherit technical difficulties resulting from general growth and inhomogeneity. In this survey we present an overview of developments of the theory of existence of~PDEs in the setting including reflexive and non-reflexive cases, as well as isotropic and anisotropic ones. Particular attention is paid to problems with data below natural duality in absence of Lavrentiev's phenomenon.