Partial Differential Equations in Anisotropic Musielak-Orlicz Spaces

Abstract

This series publishes advanced monographs giving well-written presentations of the "state-of-the-art" in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

Removable sets in non-uniformly elliptic problems

“…There are various approaches how to describe general Musielak-Orlicz spaces, cf. [7,8,26]. We shall specialize them from the very beginning to those related to energy density H(•) given by (1.2).…”

“…In fact, the natural topology for Musielak-Orlicz-Sobolev spaces is the modular one, i.e. the one coming from the notion of modular convergence [1,8,26]. We say that a sequence…”

“…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].…”

Parabolic equations in Musielak -- Orlicz spaces with discontinuous in time $N$-function

“…In this subsection we briefly recall theory of Musielak -Orlicz spaces. For detailed discussion, we refer the reader to the classical book [26] as well as to a modern presentation [9] aimed at applications in PDEs.…”

“…The following form of the Young and the Hölder inequalities are true in Musielak-Orlicz spaces (see e.g. [9,11]):…”