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Boundary estimates for elliptic operators in divergence form with VMO coefficients

Hongjie Dong, Seongmin Jeon

Abstract

We establish boundary regularity estimates for elliptic systems in divergence form with VMO coefficients. Additionally, we obtain nondegeneracy estimates of the Hopf-Oleinik type lemma for elliptic equations. In both cases, the moduli of continuity are expressed in terms of the $L^p$-mean oscillations of the coefficients and data.

Boundary estimates for elliptic operators in divergence form with VMO coefficients

Abstract

We establish boundary regularity estimates for elliptic systems in divergence form with VMO coefficients. Additionally, we obtain nondegeneracy estimates of the Hopf-Oleinik type lemma for elliptic equations. In both cases, the moduli of continuity are expressed in terms of the -mean oscillations of the coefficients and data.

Paper Structure

This paper contains 9 sections, 8 theorems, 108 equations.

Table of Contents

  1. Introduction
  2. Boundary regularity
  3. Hopf-Oleinik type lemma
  4. Main results
  5. Structure of the paper
  6. Preliminaries
  7. Proofs of main theorems
  8. Proof of Theorem \ref{['thm:reg']}
  9. Proof of Theorem \ref{['thm:Hopf']}

Key Result

Theorem 1.1

Let $u\in H^1({\Omega\cap B_1;\mathbb{R}^m)}$ be a solution of eq:pde, and $p>n$ and $p_0>1$ with $1/2<\frac{1}{p_0}+\frac{1}{p}<1$. Assume that $A^{\alpha\beta}$ and $\nabla_{x'}\gamma_\Omega\in L^\infty(B_1')$ are of VMO, and $A^{\alpha\beta}$ satisfies eq:assump-coeffi. Suppose $\mathbf{f}\in L^\ where $C>0$ is a constant depending only on $n$, $m$, $p$, $p_0$, $\lambda$, $\|\mathbf{f}\|_{L^\in

Theorems & Definitions (18)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.5
  • Lemma 3.1
  • proof
  • ...and 8 more