Gradient continuity estimates for elliptic equations of singular $p$-Laplace type with measure data

TL;DR

This paper establishes new gradient continuity estimates for elliptic equations of singular p-Laplace type with measure data, extending previous results to less regular coefficients and providing pointwise gradient bounds.

Contribution

It introduces a novel comparison estimate and derives gradient pointwise estimates under Dini continuity assumptions on coefficients, generalizing prior work.

Findings

Established interior gradient pointwise estimates using Wolff potential for p≥2.

Derived global gradient estimates via Riesz potential for 1<p<2.

Extended regularity results to equations with less regular coefficients.

Abstract

In this paper, we are concerned with elliptic equations of $p$-Laplace type with measure data, which is given by $-div\big(a(x)(|\nabla u|^2+s^2)^{\frac{p-2}{2}}\nabla u\big)=\mu$ with $p>1$ and $s\geq0$. Under the assumption that the modulus of continuity of the coefficient $a(x)$ in the $L^2$-mean sense satisfies the Dini condition, we prove a new comparison estimate and use it to derive interior and global gradient pointwise estimates by Wolff potential for $p\geq 2$ and Riesz potential for $1<p<2$, respectively. Our interior gradient pointwise estimates can be applied to a class of singular quasilinear elliptic equations with measure data given by $-div(A(x,\nabla u))=\mu$. We generalize the results in the papers of Duzaar and Mingione [Amer. J. Math. 133, 1093-1149 (2011)], Dong and Zhu [J. Eur. Math. Soc. 26, 3939-3985 (2024)], and Nguyen and Phuc [Arch. Rational Mech. Anal. (2023) 247:49], etc., where the coefficient is assumed to be Dini continuous. Moreover, we establish interior and global modulus of continuity estimates of the gradients of solutions.