Abstract: This talk is dedicated to the two-phase Bernoulli free boundary problem introduced by Alt, Caffarelli and Friedman in the 80s.

The focus on the general case in 2D, in which the solution changes sign and has zero set of positive Lebesgue measure.  In this case, thanks to the 2D epiperimetric inequality [Spolaor-Velichkov 2017], we know that the boundaries of the positive and the negative phases are C^{1,\alpha} manifolds, while some explicit examples from [DePhilippis-Spolaor-Velichkov 2023] show that separately the two boundaries are no better than C^{1,1/2}.

In this talk we will discuss the fine regularity of the free boundaries, that is, the structure of the contact set between the positive and the negative phases. Precisely, I will show that the contact set is locally composed of finitely many disjoint smooth arcs. The proof is based on a Weierstrass-type representation formula, inspired by the works of Hélein-Hauswirth-Pacard and Traizet, which allows to rewrite the problem into a geometric free boundary problem for minimal surfaces in the half 3D space. This reformulation in turn allows to reduce the problem to a thin-obstacle problem for a uniformly elliptic operator, for which the structure of the contact set can be obtained via complex-analytic methods. The results presented in this talk have appeared in a series of joint works [Spolaor-Velichkov 2017], [DePhilippis-Spolaor-Velichkov 2023], [Ferreri-Spolaor-Velichkov 2025].  
 

For further information please contact elisur.magrini@unibocconi.it