A pocket guide to nonlinear differential equations in Musielak–Orlicz spaces

Abstract

The Musielak-Orlicz setting unifies 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.

Hölder continuity of $\omega$-minimizers of functionals with generalized Orlicz growth

“…The basic example of a generalized Orlicz space was introduced by Orlicz [59] in 1931, and a major synthesis is due to Musielak [53] in 1983. Recent monographs on generalized Orlicz spaces are due to Yang, Liang and Ky [64], Lang and Mendez [47], and the first two authors [37] focusing on Hardy-type spaces, functional analysis, and harmonic analysis, respectively; see also the survey article [13]. Generalized Orlicz spaces include as a special case classical Orlicz spaces that are well-known and have been extensively studied, see, e.g., the monograph [62] and references therein.…”

Removable sets in non-uniformly elliptic problems

“…The related variational functionals include the variable-exponent integrand fundamental in modelling electrorheological fluids w → F p(·) (w, Ω) := Ω |Dw| p(x) dx, (1.5) the double phase energy describing strongly inhomogeneous materials w → F H(·) (w, Ω) := Ω |Dw| p + a(x)|Dw| q dx, (1.6) as well as the so-called ϕ-functional, defined by means of an N -functions ϕ cf. [10], and involved in the modelling of non-Newtonian fluids w → F ϕ (w, Ω) := Ω ϕ(|Dw|) dx, (1.7) or more generally w → F ϕ(·) (w, Ω) := Ω ϕ(x, |Dw|) dx, (1.8) see [7,8,26] for more details. In the nonstandard growth framework, the problem of removability of sets have been studied in the case of (1.5) in [24,36] and of (1.7) in [6].…”

Regularizing effect of the lower-order terms in elliptic problems with Orlicz growth

“…Let us mention that all the results involving SOLA naturally concerns only p > 2 − 1/n, since it is necessary to ensure that u ∈ W 1,1 loc (Ω) for arbitrary measure data. See [19] for a survey on problems in the generalized setting with data below duality in various nonstandard growth settings.…”