Regularity for one-phase Bernoulli problems with discontinuous weights and applications

We study a one-phase Bernoulli free boundary problem with weight function admitting a discontinuity along a smooth jump interface. In any dimension N ≥ 2 N 2, we show the C 1, α C^1, regularity of the free boundary outside of a singular set of Hausdorff dimension at most N − 3 N-3. In particular, we prove that the free boundaries are C 1, α C^1, regular in dimension N = 2 N=2, while in dimension N = 3 N=3 the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension N = 2 N=2, which are minimizing for one-phase functionals with weight functions in L ∞ L^ that are arbitrarily close to a positive constant.