Gradient potential estimates in elliptic obstacle problems with Orlicz growth

TL;DR

This paper develops pointwise and oscillation estimates for solutions to elliptic obstacle problems with Orlicz growth, leading to regularity results using fractional maximal operators and Wolff potentials.

Contribution

It introduces new pointwise estimates and regularity results for elliptic obstacle problems with Orlicz growth involving measure data.

Findings

Pointwise estimates of solutions via fractional maximal operators

Oscillation estimates for gradients using Wolff potentials

Establishment of $C^{1,eta}$ regularity of solutions

Abstract

In this paper,we consider the solutions of the non-homogeneous elliptic obstacle problems with Orlicz growth involving measure data. We first establish the pointwise estimates of the approximable solutions to these problems via fractional maximal operators. As a result, we obtain pointwise and oscillation estimates for the gradients of solutions by the non-linear Wolff potentials, and these yield results on $C^{1,\alpha}$-regularity of solutions.