Vectorial free boundary problems | 14h00-15h00
Bozhidar Velichkov (Università di Pisa)

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 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.

Regularity of the optimal sets for the second Dirichlet eigenvalue | 15h00-16h00
Dario Mazzoleni (Università di Pavia)

First of all, we recall the basic notions and results concerning shape optimization problems for the eigenvalues of the Dirichlet Laplacian. Then we focus on the study of the regularity of the optimal shapes and on the link with the regularity of related free boundary problems. The main topic of the talk is the regularity of the optimal sets for a "degenerate'" functional, namely the second Dirichlet eigenvalue in a box. Given D⊂R^d an open and bounded set of class C^1,1 we consider the following shape optimization problem, for Λ>0,

(min{λ_2 (A)+Λ|A|:A⊂D," open" },)

where λ_2 (A)denotes the second eigenvalue of the Dirichlet Laplacian on A.

In this talk we show that any optimal set Ω for (1) is equivalent to the union of two disjoint open sets, Ω^±, which are C^(1,α)regular up to a (possibly empty) closed singular set of Hausdorff dimension at most d-5, which is contained in the one-phase free boundaries. In particular, we are able to prove that the set of two-phase points, that is, ∂Ω^+∩∂Ω^-∩D, is contained in the regular set. This is a joint work with Baptiste Trey and Bozhidar Velichkov.

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The Lisbon Webinar in Analysis in Differential Equations is a joint iniciative of CAMGSD, CMAFcIO and GFM, three research centers of the University of Lisbon. It is aimed at filling the absence of face-to-face seminars and wishes to be a meeting point of mathematicians working in the field.