Regularity estimates for nonlocal Schrödinger equations

2017/11/06 by Fall, Mouhamed Moustapha · 1 citation
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1711.02206

Abstract

We prove Hölder regularity estimates up to the boundary for weak solutions u to nonlocal Schrödinger equations subject to exterior Dirichlet conditions in an open set Ω⊂ ℝ^N. The class of nonlocal operators considered here are defined, via Dirichlet forms, by kernels K(x,y) bounded from above and below by |x-y|^N+2s, with s∈ (0,1). The entries in the equations are in some Morrey spaces and the underline domain Ω satisfies some mild regularity assumptions. In the particular case of the fractional Laplacian, our results are new. When K defines a nonlocal operator with sufficiently regular coefficients, we obtain Hölder estimates, up to the boundary of Ω, for u and the ratio u/d^s, with d(x)=\textrmdist(x,ℝ^N∖Ω). If the kernel K defines a nonlocal operator with Hölder continuous coefficients and the entries are Hölder continuous, we obtain interior C^2s+β regularity estimates of the weak solutions u. Our argument is based on blow-up analysis and compact Sobolev embedding.

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• Boundary regularity and a priori estimates for fractional equations on unbounded domains

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