Regularity estimates for nonlocal Schr\"odinger equations

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.