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Israel Journal of Mathematics

, Volume 228, Issue 2, pp 835–861 | Cite as

Interior regularity results for zeroth order operators approaching the fractional Laplacian

  • Patricio Felmer
  • Disson dos Prazeres
  • Erwin Topp
Article
  • 16 Downloads

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 Hölder profile for fractional problems, as є → 0. This analysis includes the case when fє deteriorates its modulus of continuity as є → 0.

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Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • Patricio Felmer
    • 1
  • Disson dos Prazeres
    • 2
  • Erwin Topp
    • 3
  1. 1.Departamento de Ingeniería Matemática and CMM (UMI 2807 CNRS)Universidad de ChileSantiagoChile
  2. 2.DMAUniversidade Federal de SergipeAracajuBrazil
  3. 3.Departamento de Matemática y C.C.Universidad de Santiago de ChileSantiagoChile

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