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 $N\ge 2$, we show the $C^{1,\alpha }$ regularity of the free boundary outside of a singular set of Hausdorff dimension at most $N-3$. In particular, we prove that the free boundaries are $C^{1,\alpha }$ regular in dimension $N=2$, while in dimension $N=3$ the singular set can contain at most a finite number of points. We use this result to construct singular free boundaries in dimension $N=2$, which are minimizing for one-phase functionals with weight functions in $L^{\mathrm{\infty }}$ that are arbitrarily close to a positive constant.