Wolff potentials and local behaviour of solutions to measure data elliptic problems with Orlicz growth

TL;DR

This paper develops sharp pointwise estimates for solutions to measure data elliptic problems with Orlicz growth, using Wolff potentials, leading to new regularity results and a variant of Hedberg--Wolff theorem.

Contribution

It introduces a nonlinear potential framework for Orlicz growth problems, providing sharp bounds and regularity results for solutions with measure data.

Findings

Sharp pointwise bounds for solutions using Wolff potentials

Regularity results including continuity and Hölder continuity

A variant of Hedberg--Wolff theorem for Orlicz spaces

Abstract

We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:\Omega\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls bounds from above and from below. Applying it we provide a bunch of precise regularity results including continuity and H\"older continuity for solutions to problems involving measures that satisfies conditions expressed in the natural scales. Finally, we give a variant of Hedberg--Wolff theorem on characterization of the dual of the Orlicz space.