A one-sided two phase Bernoulli free boundary problem

We study a two-phase free boundary problem in which the two-phases satisfy an impenetrability condition. Precisely, we have two ordered positive functions, which are harmonic in their supports, satisfy a Bernoulli condition on the one-phase part of the free boundary and a two-phase condition on the collapsed part of the free boundary. For this two-membrane type problem, we prove an epsilon-regularity theorem with sharp modulus of continuity. Precisely, we show that at flat points each of the two boundaries is $C^{1,1\mathrm{/}2}$ regular surface. Moreover, we show that the remaining singular set has Hausdorff dimension at most $N-5$ as in the case of the classical one-phase problem, $N$ being the dimension of the space.