Interior regularity results for zeroth order operators approaching the fractional Laplacian

Abstract

In this article we are interested in interior regularity results for the solution $${\mu _ \in } \in C(\bar \Omega )$$ of the Dirichlet problem $$\{ _{\mu = 0in{\Omega ^c},}^{{I_ \in }(\mu ) = {f_ \in }in\Omega }$$ where Ω is a bounded, open set and $${f_ \in } \in C(\bar \Omega )$$ for all є ∈ (0, 1). For some σ ∈ (0, 2) fixed, the operator $$\mathcal{I}_{\in}$$ is explicitly given by $${I_ \in }(\mu ,x) = \int_{{R^N}} {\frac{{[\mu (x + z) - \mu (x)]dz}}{{{ \in ^{N + \sigma }} + |z{|^{N + \sigma }}}}} ,$$ which is an approximation of the well-known fractional Laplacian of order σ, as є tends to zero. The purpose of this article is to understand how the interior regularity of uє evolves as є approaches zero. We establish that uє has a modulus of continuity which depends on the modulus of fє, which becomes the expected Holder profile for fractional problems, as є → 0. This analysis includes the case when fє deteriorates its modulus of continuity as є → 0.