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

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.
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