Gradient estimates for nonlinear elliptic equations with Orlicz growth and measure data

TL;DR

This paper derives gradient estimates for solutions to nonlinear elliptic equations with measure data under Orlicz growth conditions, providing new regularity results including pointwise Wolff potential estimates and Lipschitz regularity.

Contribution

It introduces novel gradient estimates for nonlinear elliptic equations with Orlicz growth, extending known results for p-Laplace equations to more general growth conditions.

Findings

Pointwise Wolff potential estimates in the singular regime

Lipschitz regularity of solutions in the general regime

Recovery of known estimates for p-Laplace equations

Abstract

We establish gradient estimates of solutions to a class of nonlinear elliptic equations with measure data under Orlicz-type growth conditions. The growth is governed by the structural condition \[ 0<i_a\le t g'(t)/g(t)\le s_a<1. \] We obtain two types of regularity results: pointwise Wolff potential estimates for the gradient of solutions in the singular regime $i_a \in \big(\frac{n-1}{2n-1},1\big)$, and Lipschitz regularity of the solutions in the regime $i_a \in (0,1)$. In the power-type case $g(t)=t^{p-1}$, our results recover the known gradient estimates for the singular $p$-Laplace equation.