Gradient estimates for singular elliptic measure data problems with double phase

Abstract

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