Regularity of the free boundary for the two-phase Bernoulli problem

I will describe a recent result obtained in collaboration with G. De Philippis and B. Velichkov concerning the regularity of the free boundaries in the two phase Bernoulli problems. The novelty of our work is the analysis of the free boundary at branch points, where we show that it is given by the union of two C1 graphs. This completes the work started by Alt, Caffarelli, and Friedman in the 80’s.