Regularity estimates for nonlocal Schrödinger equations
Mouhamed Moustapha Fall · Discrete and Continuous Dynamical Systems · 2018
We are concerned with Hölder regularity estimates 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 symmetric 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 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.