Gradient estimates for singular elliptic measure data problems with double phase

TL;DR

This paper establishes local Calderón–Zygmund estimates for elliptic measure data problems involving double phase operators in the singular case where the lower exponent p is between 2-1/n and 2.

Contribution

It provides the first Calderón–Zygmund estimates for singular elliptic equations with double phase structure under natural conditions.

Findings

Proved local regularity estimates for singular elliptic measure data problems.

Extended Calderón–Zygmund theory to double phase operators in the singular regime.

Demonstrated estimates hold under minimal assumptions on coefficients.

Abstract

We consider elliptic measure data problems of the type \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = \mu \] in a bounded domain in $\mathbb{R}^n$, where $p<q$ and $a(\cdot) \ge 0$. We prove local Calder\'on--Zygmund estimates in the singular case $2-1/n < p < 2$, under natural assumptions on $p$, $q$ and $a(\cdot)$.