Interior regularity results for zeroth order operators approaching the fractional Laplacian

In this article we are interested in interior regularity results for the solution ({mu _ in } in C(bar Omega )) of the Dirichlet problem

$${ _{mu = 0in{Omega ^c},}^{{I_ in }(mu ) = {f_ in }inOmega }$$

where Ω is a bounded, open set and ({f_ in } in C(bar Omega )) for all ? ∈ (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 ? tends to zero. The purpose of this article is to understand how the interior regularity of u? evolves as ? approaches zero. We establish that u_? has a modulus of continuity which depends on the modulus of f_?, which becomes the expected Hölder profile for fractional problems, as ? → 0. This analysis includes the case when f? deteriorates its modulus of continuity as ? → 0.