Bozhidar Velichkov

Università di Pisa

Vectorial free boundary problems and regularity of the optimal sets for the eigenvalues of the Dirichlet Laplacian

The vectorial Bernoulli problem is a variational free boundary problem involving the Dirichlet energy of a vector-valued function and the measure of its support. It is the vectorial counterpart of the classical one-phase Bernoulli problem, which was first studied by Alt and Caffarelli in 1981.

In this talk, we will start from the spectral shape optimization problems, which were the initial motivation for the study of the vectorial free boundaries. Then, we will discuss some results on the regularity of the vectorial free boundaries obtained in the last years by Caffarelli–Shahgholian–Yeressian, Kriventsov–Lin, Mazzoleni–Terracini–V., and Spolaor–V.. Finally, we will present some new results on the rectifiability of the singular set obtained in collaboration with Guido De Philippis, Max Engelstein and Luca Spolaor.