Boundary Harnack inequalities for regional fractional Laplacian

TL;DR

This paper establishes boundary Harnack inequalities for regional fractional Laplacians on various domains, extending known results to less regular sets and providing decay rates for harmonic functions near boundaries.

Contribution

It generalizes boundary Harnack inequalities to C^{1,eta-1} and Lipschitz domains for regional fractional Laplacians, including non-homogeneous cases.

Findings

Proved decay estimates for harmonic functions near boundaries.

Extended inequalities to less regular domains like Lipschitz sets.

Generalized known results from C^{1,1} to broader classes of domains.

Abstract

We consider boundary Harnack inequalities for regional fractional Laplacian which are generators of censored stable-like processes on G taking \kappa(x,y)/|x-y|^{n+\alpha}dxdy, x,y\in G as the jumping measure. When G is a C^{1,\beta-1} open set, 1<\alpha<\beta\leq 2 and \kappa\in C^{1}(\overline{G}\times \overline{G}) bounded between two positive numbers, we prove a boundary Harnack inequality giving dist(x,\partial G)^{\alpha-1} order decay for harmonic functions near the boundary. For a C^{1,\beta-1} open set D\subset \overline{D}\subset G, 0<\alpha\leq (1\vee\alpha)<\beta\leq 2, we prove a boundary Harnack inequality giving dist(x,\partial D)^{\alpha/2} order decay for harmonic functions near the boundary. These inequalities are generalizations of the known results for the homogeneous case on C^{1,1} open sets. We also prove the boundary Harnack inequality for regional fractional Laplacian on Lipschitz domain.