Abstract. In this talk I will discuss some recent results obtained in collaboration with A. Figalli, S. Kim and H. Shahgholian. We study vectorial versions of the obstacle and Bernoulli problems, and we consider minimizers of Dirichlet-type energies among all maps constrained to take values outside a smooth domain in R^m. These problems lie between the theory of harmonic maps and that of scalar free boundary problems, and they have the distinctive feature that the minimizers are in general discontinuous. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundaries.

Abstract. Mechanical models of living tissues can be divided into two main classes. On the one hand, density-based descriptions (which consist of partial differential equations or systems) study the evolution of the population density over time. On the other hand, free boundary problems describe the tissue as a moving patch, whose free boundary evolves in a pressure-dependent way. In this talk, I will present an overview of several results on how we can link these two representations through a singular limit, studying the asymptotic behavior as the pressure law becomes stiffer and stiffer.

Abstract. In this talk, we will consider a class of spectral optimal partition problems with volume constraint for partitions of a given bounded domain. We prove an optimal open partition exists by establishing that the corresponding eigenfunctions are locally Lipschitz continuous. Our proofs are based on a weak formulation involving a minimization problem of a penalized functional where the variables are functions rather than domains, suitable deformations, blowup techniques and a monotonicity formula. This is joint work with Ederson Moreira dos Santos, Makson Santos, and Hugo Tavares.

Abstract. Let Ω be a open, convex set and let E be an almost-minimizer of the relative perimeter in Ω. We focus on the behavior of the boundary of E close to a singular point of the boundary of Ω. If n=3, we show that the boundary of E skips the vertex-type singularities of the boundary of Ω. One of the intermediate results, that for instance allows us to consider a larger class of almost-minimizers, is a boundary Monotonicity Formula valid under some mild, extra assumptions on Ω.