Removable sets in elliptic equations with Musielak–Orlicz growth

Abstract

We characterize, in the terms of intrinsic Hausdorff measures, the size of removable sets for Hölder continuous solutions to elliptic equations with Musielak-Orlicz growth. In the general case we provide a result in the new scale that is more relevant and captures -as special cases -the classical results, slightly refines the ones provided for problems stated in the variable exponent and double phase spaces and essentially improves the known one in the Orlicz case.2010 Mathematics Subject Classification. 35J60, 35J70.

Generalized Superharmonic Functions with Strongly Nonlinear Operator

“…[1, 12-14, 17, 20, 39, 42, 49] concentrating on their existence and gradient estimates. The generalization of studies on removable sets for Hölder continuous solutions provided by [40] to the case of strongly non-uniformly elliptic operators has been carried out lately in [15,16]. There are available various regularity results for related quasiminimizers having Orlicz or generalized Orlicz growth [25,29,30,32,36,37,46].…”

“…We note the following basic information on the existence, the uniqueness, and the Comparison Principle for the obstacle problem are provided in [37] and [16,Section 4]. Then there exists a function u ∈ K ψ,w ( ) being a unique solution to the K ψ,w ( )-obstacle problem (11).…”

“…Theorem 2,[16]) Under Assumption (A) let the obstacle ψ ∈ W 1,ϕ(•) ( ) ∪ {−∞} and the boundary datum w ∈ W 1,ϕ(•) ( ) be such that K ψ,w ( ) = ∅.…”

Barrier Solutions of Elliptic Differential Equations in Musielak-Orlicz-Sobolev Spaces

“…Although they did not require Φ ∈ Δ 2 nor Φ ∈ Δ 2 , condition ( 5) is stronger than the growth condition (3.1) and the coercive condition (3.2) in [2]. In [7], Chlebicka and Karppinen studied removable sets for elliptic equations of the form…”

“…This paper is organized as follows: Section 2 contains some preliminaries and some technical lemmas which will be needed. In Section 3, we first give the sufficient conditions which ensure the existence of weak solutions for (7) in Musielak-Orlicz-Sobolev spaces. Some examples are given for the application of our results.…”