Singular elliptic measure data problems with irregular obstacles

TL;DR

This paper studies elliptic obstacle problems with irregular data and singular growth conditions, developing new estimates for gradients and solutions in complex measure data scenarios.

Contribution

It introduces novel comparison and potential estimates for elliptic obstacle problems with measure data, especially in strongly singular cases.

Findings

Established gradient potential estimates in an intrinsic form.

Derived zero-order potential estimates applicable to irregular obstacles.

Extended analysis to the strongly singular case $1 < p \,\le\, 2-1/n$.

Abstract

We investigate elliptic irregular obstacle problems with $p$-growth involving measure data. Emphasis is on the strongly singular case $1 < p \le 2-1/n$, and we obtain several new comparison estimates to prove gradient potential estimates in an intrinsic form. Our approach can be also applied to derive zero-order potential estimates.