On the boundary Harnack principle in Hölder domains

Daniela De Silva,Ovidiu Savin

doi:10.3934/mine.2022004

Abstract

<abstract> We investigate the boundary Harnack principle for uniformly elliptic operators in divergence form in Hölder domains of exponent $ \alpha > 0 $. We also deal with operators in nondivergence form with coefficient that remain constant in the graph direction. </abstract>

Highlights

In this paper we continue the study of the boundary Harnack principle for solutions to elliptic equations, based on the method developed in [DS]
To state precisely the boundary Harnack principle in Holder domains, first we introduce some notation
The first lemma states that a solution to Lv = 0 in Q1 \ E satisfies the Harnack inequality in measure if E has small capacity

Summary

Introduction

In this paper we continue the study of the boundary Harnack principle for solutions to elliptic equations, based on the method developed in [DS]. Bass and Burdzy [BB1, BB2] and Banuelos, Bass and Burdzy [BBB] provided sharp versions using probabilistic methods. They established the boundary Harnack principle for nondivergence elliptic operators in Holder domains In the present paper we discuss further the case of Holder domains of arbitrary exponent α > 0, and give a proof of Theorem 1.1 using the same ideas from [DS]. We consider some extensions of Theorem 1.1 to non-divergence equations whose coefficients remain constant in the vertical direction.

Two lemmas

Some extensions