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Regularity estimates for nonlocal Schrödinger equations

The author's work is supported by the Alexander von Humboldt foundation. Part of this work was done while he was visiting the Goethe University in Frankfurt am Main during AugustSeptember 2017 and he thanks the Mathematics department for the kind hospitality. The author is grateful to Xavier Ros-Oton, Tobias Weth and Enrico Valdinoci for their availability and for the many useful discussions during the preparation of this work

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  • 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.

    Mathematics Subject Classification: 35R11, 42B37.


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