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

Ying Li, Chao Zhang

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.

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

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 We obtain two types of regularity results: pointwise Wolff potential estimates for the gradient of solutions in the singular regime , and Lipschitz regularity of the solutions in the regime . In the power-type case , our results recover the known gradient estimates for the singular -Laplace equation.
Paper Structure (4 sections, 16 theorems, 183 equations)

This paper contains 4 sections, 16 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Preliminart Lemmas
  3. Pointwise gradient estimates
  4. Interior Lipschitz continuity of the gradient

Key Result

Theorem 1.1

Let $i_a\in (\frac{n-1}{2n-1},1)$. If $u\in W^{1,G}_{\mathrm{loc}}(\Omega)$ is a solution to eq-main, and assume that eq:struct3--eq-0 hold. Then there exists a constant such that for every Lebesgue point $x\in\Omega$ of the vector--valued function $\nabla u$ and for any $B_R(x)\subset\Omega$ with $R\in(0,1]$, the pointwise estimate holds almost everywhere. Here the truncated Wolff potential is

Theorems & Definitions (25)

  • Theorem 1.1: Interior pointwise gradient estimate
  • Theorem 1.2: Interior Lipschitz estimate
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4: Theorem 4.1. A-26-ar
  • Corollary 2.5
  • Lemma 2.6
  • ...and 15 more