Gradient estimates for degenerate elliptic measure data problems with double phase

TL;DR

This paper establishes local gradient estimates for solutions to nonlinear elliptic equations with degenerate measure data, extending Calderón–Zygmund theory under new assumptions on the parameters and coefficients.

Contribution

It introduces novel assumptions enabling Calderón–Zygmund type gradient estimates for SOLA in degenerate elliptic measure data problems with double phase structure.

Findings

Proved gradient estimates for solutions with measure data.

Extended Calderón–Zygmund theory to degenerate elliptic equations.

Identified new conditions on parameters and coefficients for regularity.

Abstract

We study nonlinear elliptic equations modeled on \[ -\mathrm{div}\,(|Du|^{p-2}Du+a(x)|Du|^{q-2}Du) = \mu, \] where $2\le p<q<\infty$, $a(\cdot) \ge 0$, and $\mu$ is a signed Borel measure with finite total mass. We prove local Calder\'on--Zygmund type gradient estimates for SOLA (Solutions Obtained as Limits of Approximations) by finding new and natural assumptions on $p$, $q$ and $a(\cdot)$.