Vectorial problems with free boundaries

The talk deals with recent results obtained in collaboration with D. De Silva on the regularity of free boundaries in vectorial Bernoulli type problems. The new main point is the analysis of the regular part of the free boundary based on a linearization argument that takes care of the norm of the vector of solutions and that distinguishes appropriately its components. We apply this methodology on the vectorial analogue of the thin free boundary problem introduced by Caffarelli-Roquejoffre-Sire as a realization of a nonlocal version of the classical Bernoulli problem. Time permitting, we discuss the link between this problem and shape optimization problems involving a combination of fractional eigenvalues.