In this talk I will first give an introduction on the Bernoulli free boundary problems with one and two phases, recalling the main regularity properties. Then, I will present some new results concerning the branching points of the one-phase problem at the boundary of a domain, highlighting its relation with the two-phase problem and the theory of area minimizing currents. In particular we show that, for real analytic domains and in any dimension d of the ambient space, the Hausdorff dimension of the set of one-phase boundary branching points is at most d − 2. Both the analyticity assumption and the dimensional estimate are sharp. This is a joint work with Luca Spolaor and Bozhidar Velichkov.
