Wolff potential estimates for elliptic obstacle problems with generalized Orlicz growth

TL;DR

This paper develops Wolff potential estimates for elliptic obstacle problems with generalized Orlicz growth, establishing existence, regularity, and gradient bounds for solutions under minimal assumptions.

Contribution

It introduces new Wolff potential estimates for solutions to obstacle problems with generalized Orlicz growth, extending regularity theory in this setting.

Findings

Existence of solutions in Musielak-Orlicz spaces.

Pointwise gradient estimates via Wolff potentials.

Solutions exhibit $C^{1,\alpha}$ regularity.

Abstract

This paper investigates elliptic obstacle problems with generalized Orlicz growth involving measure data, which includes Orlicz growth, variable exponent growth, and double-phase growth as specific cases of this setting. First, we establish the existence of solutions in the Musielak-Orlicz space. Then, we derive pointwise and oscillation gradient estimates for solutions in terms of the non-linear Wolff potentials, assuming minimal conditions on the obstacle. These estimates subsequently lead to $C^{1,\alpha}$-regularity results for the solutions.