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 μ∈∈ C(Ω¯) of the Dirichlet problem {μ=0inΩc,I∈(μ)=f∈inΩ where Ω is a bounded, open set and f∈∈ C(Ω¯) for all є ∈ (0, 1). For some σ ∈ (0, 2) fixed, the operator I∈ is explicitly given by I∈(μ,x)=∫RN[μ(x+z)−μ(x)]dz∈N+σ+|z|N+σ, 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 Hölder profile for fractional problems, as є → 0. This analysis includes the case when fє deteriorates its modulus of continuity as є → 0.