Essentially fully anisotropic Orlicz functions and uniqueness to measure data problem

TL;DR

This paper develops a framework for analyzing elliptic measure data problems with strongly nonlinear operators governed by fully anisotropic Orlicz functions, establishing measure uniqueness and capacity characterizations in this complex setting.

Contribution

It introduces a new capacity framework in fully anisotropic Orlicz-Sobolev spaces and proves measure uniqueness for nonlinear elliptic problems with irregular anisotropic growth.

Findings

Capacitary characterization of bounded measures

Existence of anisotropic Young functions with irregular growth

Framework applicable to fully anisotropic nonlinear PDEs

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 $\Phi$, such that $|\xi|^p \lesssim\Phi(\xi)\lesssim |\xi|^p\log^\alpha(1+|\xi|)$, with arbitrary $p\geq 1$, $\alpha>0$, but so irregularly growing that % we call it essentially fully anisotropic. In fact, the Orlicz--Sobolev--type space generated by $\Phi$ indispensably requires fully anisotropic tools to be handled.