Regularity estimates for nonlocal Schrödinger equations

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 $\Omega \subset {\mathbb{R}}^N$. 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 Hölder estimates, up to the boundary of $\Omega$, for $u$ and the ratio $u/d^s$, with $d(x)=\text{dist}(x,{\mathbb{R}}^N\setminus \Omega )$. If the kernel $K$ defines a nonlocal operator with Hölder continuous coefficients and the entries are Hölder 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.

46 pages, minor corrections were made