Boundary Harnack principle and gradient estimates for fractional Laplacian perturbed by non-local operators

2015/01/09 by Zhen-Qing Chen, Chen, Zhen-Qing, Yan-Xia Ren +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.PR

paper · pdf · doi:10.48550/arxiv.1501.02023

arxiv created 2016/03/24 · arxiv updated 2016/03/25

Abstract

Suppose d≥ 2 and 0<β<α<2. We consider the non-local operator L^b=Δ^α/2+S^b, where S^bf(x):=lim_{ε→ 0}A(d,-β)∫_|z|>ε(f(x+z)-f(x))\fracb(x,z)|z|^d+β dy. Here b(x,z) is a bounded measurable function on ℝ^d×ℝ^d that is symmetric in z, and A(d,-β) is a normalizing constant so that when b(x, z)≡ 1, S^b becomes the fractional Laplacian Δ^β/2:=-(-Δ)^β/2. In other words, L^bf(x):=lim_{ε→ 0}A(d,-β)∫_|z|>ε(f(x+z)-f(x)) j^b(x, z) dz, where j^b(x, z):= A(d,-α) |z|^-(d+α)+ A(d,-β) b(x, z)|z|^-(d+β). It is recently established in Chen and Wang [arXiv:1312.7594 [math.PR]] that, when j^b(x, z)≥ 0 on ℝ^d× ℝ^d, there is a conservative Feller process X^b having L^b as its infinitesimal generator. In this paper we establish, under certain conditions on b, a uniform boundary Harnack principle for harmonic functions of X^b (or equivalently, of L^b) in any κ-fat open set. We further establish uniform gradient estimates for non-negative harmonic functions of X^b in open sets.

Citations

• Perturbation by non-local operators

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