  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 H\"older continuity of minimizers for both
the one- and two-phase problems. Next, for the one-phase problem, we show
H\"older 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.
