Regularity and nondegeneracy for nonlocal Bernoulli problems with variable kernels

Abstract:

We consider a generalization of the Bernoulli free boundary problem where the underlying differential operator is a nonlocal, non-translation-invariant elliptic operator of order $2s\in (0,2)$. Because of the lack of translation invariance, the Caffarelli-Silvestre extension is unavailable, and we must work with the nonlocal problem directly instead of transforming to a thin free boundary problem. We prove global Hölder continuity of minimizers for both the one- and two-phase problems. Next, for the one-phase problem, we show Hölder continuity at the free boundary with the optimal exponent $s$. We also prove matching nondegeneracy estimates. A key novelty of our work is that all our findings hold without requiring any regularity assumptions on the kernel of the nonlocal operator. This characteristic makes them crucial in the development of a universal regularity theory for nonlocal free boundary problems.