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Boundary Harnack principle on uniform domains

Aobo Chen

Abstract

We present a proof of scale-invariant boundary Harnack principle for uniform domain when the underlying space satisfies a scale-invariant elliptic Harnack inequality. Our approach do not assume the underlying space to be geodesic nor the existence of Green function.

Boundary Harnack principle on uniform domains

Abstract

We present a proof of scale-invariant boundary Harnack principle for uniform domain when the underlying space satisfies a scale-invariant elliptic Harnack inequality. Our approach do not assume the underlying space to be geodesic nor the existence of Green function.
Paper Structure (6 sections, 22 theorems, 111 equations, 1 figure)

This paper contains 6 sections, 22 theorems, 111 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Metric measure Dirichlet space and Green functions
  3. Uniform domain
  4. Proof of main result
  5. Representation formula for harmonic functions
  6. Estimates on Green functions

Key Result

Theorem 1.1

Let $(\mathcal{X},d)$ be a complete separable metric doubling space, and let $m$ be a Radon measure on $\mathcal{X}$ with full support. Let $(\mathcal{E},\mathcal{F})$ be a strongly local symmetric regular Dirichlet form on $L^{2}(\mathcal{X},m)$. Suppose that $(\mathcal{X},d,m,\mathcal{E},\mathcal{ where $C$ depends only on $A$, $m$ and the constants that appear in elliptic Harnack inequality; $C

Figures (1)

  • Figure 4.1: Five specified points $x_{\xi}^{\star},y_{\xi}^{\star},z_{\xi}^{\star},\xi_{r}, \xi_{r}^{\prime}$.

Theorems & Definitions (56)

  • Theorem 1.1: BHP
  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5: Local Dirichlet space
  • Remark 2.6
  • Definition 2.7
  • Definition 2.8: Elliptic Harnack inequality
  • Remark 2.9
  • ...and 46 more