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arXiv · 2403.07793

Optimal regularity for nonlocal elliptic equations and free boundary problems

Xavier Ros-Oton·Marvin Weidner
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Abstract

In this article we establish for the first time the $C^s$ boundary regularity of solutions to nonlocal elliptic equations with kernels $K(y)\asymp |y|^{-n-2s}$. This was known to hold only when $K$ is homogeneous, and it is quite surprising that it holds for general inhomogeneous kernels, too. As an application of our results, we also establish the optimal $C^{1+s}$ regularity of solutions to obstacle problems for general nonlocal operators with kernels $K(y)\asymp |y|^{-n-2s}$. Again, this was only known when $K$ is homogeneous, and it solves a long-standing open question in the field. A new key idea is to construct a 1D solution as a minimizer of an appropriate nonlocal one-phase free boundary problem, for which we establish optimal $C^s$ regularity and non-degeneracy estimates.

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Xavier Ros-Oton, Marvin Weidner. 2024-03-12. Optimal regularity for nonlocal elliptic equations and free boundary problems. https://arxiv.org/abs/2403.07793

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