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

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.

References in corpus (4)

• Renormalized solutions of nonlinear parabolic equations with general measure data
• Diffuse measures and nonlinear parabolic equations
• Gradient estimates for problems with Orlicz growth
• Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon

Cited by in corpus (16)

• The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth
• Gradient estimates for problems with Orlicz growth
• Parabolic equation in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• Renormalized solutions to parabolic equations in time and space dependent anisotropic Musielak-Orlicz spaces in absence of Lavrentiev's phenomenon
• Existence of Weak Solutions for $p(.)$-Laplacian Equation via Compact Embeddings of the Double Weighted Variable Exponent Sobolev Spaces
• A fundamental condition for harmonic analysis in anisotropic generalized Orlicz spaces
• Kirchhoff type elliptic equations with double criticality in Musielak-Sobolev spaces
• Parabolic equations in Musielak -- Orlicz spaces with discontinuous in time $N$-function
• Regularity for Orlicz phase problems
• Wolff potentials and local behaviour of solutions to measure data elliptic problems with Orlicz growth
• An extended variational theory for nonlinear evolution equations via modular spaces
• Extension in generalized Orlicz spaces
• Strong solutions of the double phase parabolic equations with variable growth
• A double-phase Neumann problem with $p=1$
• Bloch estimates in non-doubling generalized Orlicz spaces
• Continuity at a boundary point of solutions to quasilinear elliptic equations with generalized Orlicz growth and non-logarithmic conditions