Regularity estimates for nonlocal Schr\"odinger equations

TL;DR

This paper establishes boundary and interior regularity estimates for solutions to nonlocal Schr"odinger equations with kernels satisfying certain bounds, extending known results especially for the fractional Laplacian.

Contribution

It provides new boundary regularity results for nonlocal Schr"odinger equations, including the fractional Laplacian, under mild domain and coefficient regularity assumptions.

Findings

H"older regularity up to the boundary for solutions

Interior $C^{2s+eta}$ regularity under H"older coefficients

New results for fractional Laplacian boundary regularity

Abstract

We prove H\"older regularity estimates up to the boundary for weak solutions $u$ to nonlocal Schr\"odinger equations subject to exterior Dirichlet conditions in an open set $\Omega\subset \mathbb{R}^N$. 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 $s\in (0,1)$. The entries in the equations are in some Morrey spaces and the underline domain $\Omega$ 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 H\"older estimates, up to the boundary of $ \Omega$, for $u$ and the ratio $u/d^s$, with $d(x)=\textrm{dist}(x,\mathbb{R}^N\setminus\Omega)$. If the kernel $K$ defines a nonlocal operator with H\"older continuous coefficients and the entries are H\"older continuous, we obtain interior $C^{2s+\beta}$ regularity estimates of the weak solutions $u$. Our argument is based on blow-up analysis and compact Sobolev embedding.