A New Boundary Harnack Principle (Equations with Right Hand Side)

Mark Allen,Henrik Shahgholian

doi:10.1007/s00205-019-01415-3

Abstract

We introduce a new boundary Harnack principle in Lipschitz domains for equations with a right hand side. Our approach, which uses comparisons and blow-ups, will adapt to more general domains as well as other types of operators. We prove the principle for divergence form elliptic equations with lower order terms including zero order terms. The inclusion of a zero order term appears to be new even in the absence of a right hand side.

Highlights

A further observation is that when the domain D is uniformly C1,Dini a boundary Harnack principle holds between a positive harmonic and a superharmonic functions, vanishing on the boundary (∂ D) ∩ B1
We prove Theorem 1.1 for the Laplace operator, the result will be true for divergence form operators as given in Theorem 4.7
As previously mentioned the boundary Harnack principle has been proven in very general domains as well as for various operators

Summary

Introduction

Let u be the positive homogeneous harmonic function vanishing on the boundary of the cone D, and set v = u − w. We have a boundary Harnack principle u(x) > v(x) where a harmonic function dominates a superharmonic one The difference between this example and the first example above is that the cone is wider. A further observation is that when the domain D is uniformly C1,Dini a boundary Harnack principle holds between a positive harmonic and a superharmonic functions (with bounded right-hand side), vanishing on the boundary (∂ D) ∩ B1. This is an easy consequence of Hopf’s boundary point lemma and. Wk > 0 in D in a small neighbourhood of the ∂ D ∩ B1 and inside D; this is exactly the boundary Harnack we asked for

Main Result