Essentially fully anisotropic Orlicz functions and uniqueness to measure data problem

Abstract

Studying elliptic measure data problem with strongly nonlinear operator whose growth is described by the means of fully anisotropic N‐function, we prove the uniqueness for a broad class of measures. In order to provide it, the framework of capacities in fully anisotropic Orlicz–Sobolev spaces is developed and the capacitary characterization of a bounded measure is given. Moreover, we give an example of an anisotropic Young function Φ, such that false|ξfalse|p≲normalΦfalse(ξfalse)≲false|ξfalse|plogαfalse(1+false|ξfalse|false), with arbitrary p ≥ 1, α > 0, but so irregularly growing that the Orlicz–Sobolev‐type space generated by Φ indispensably requires fully anisotropic tools to be handled.

A direct proof of existence of weak solutions to fully anisotropic and inhomogeneous elliptic problems

“…Contributions in this direction were initiated by Donaldson [14] and continued by Gossez [19,20,21] and Mustonen and Tienari [38]. We refer to [2,3,10,26] for analysis of problems in anisotropic Orlicz spaces governed by possibly fully anisotropic modular function, which is independent of the spacial variable. None of these contributions, however, provides a direct proof of existence to PDEs.…”

“…Taking into account [10] and [4,Section 4] one can provide an explicit condition that implies (B) even in the case when the anisotropic function M (x, ξ) does not admit a so-called orthotropic decomposition d i=1 M i (x, ξ i ) even after an affine change of variables.…”

Measure data elliptic problems with generalized Orlicz growth

“…Studies on nonstandard growth problems form a solid stream in the modern nonlinear analysis [9, 20, 23, 25, 29, 32, 34, 39, 40, 50, 51, 61, 68]. The theory of existence of very weak solutions to problems with nonstandard growth and merely integrable data is under intensive investigation [3, 11, 31, 33, 47, 48, 66, 69]. For the study on Musielak–Orlicz growth -data elliptic equations we refer to [47] under growth restrictions on the conjugate of the modular function and to [48, 55], where existence is provided either in (all) reflexive spaces or when the growth of modular function is well-balanced (and the smooth functions are modularly dense, cf.…”