Il prossimo incontro del ciclo dei “Seminari di Analisi junior” si terrà venerdì 31 gennaio 2025 alle ore 15.00, nella Sala Riunioni Primo Livello (SR1). Di seguito titoli e abstract.

Luca Barbato: Rigidity and stability results for Torsion functional in the class of multiply connected sets

Abstract: A classical result in Shape Optimization deals with the maximization of the torsional rigidity among the open sets of prescribed measure. Latter the problem was also addressed in the class of multiply connected domains by Pólya, Weinstein (in the plane) and Buonocore (in every dimension). Using symmetrization techniques they proved that the maximizers is the spherical shell, regardless of the number of holes. In this talk I will present a rigidity result in every dimension and a stability result for the planar case, in which the aim is to add an asymmetry index which controls the distance from being an annulus. The key idea of the proof is to use a weighted version of the Brothers and Ziemer theorem for the rigidity part and some suitable propagation of the Fraenkel asymmetry to estimate the distance of a set from the optimal one.

Giuliana Fusco: Variational analysis of discrete Dirichlet problems on periodically perforated domains

Abstract: In this talk we discuss the asymptotic behavior of a family of discrete functionals as the lattice size, ε > 0, tends to zero. We consider pairwise interaction energies defined on discrete functions subjected to Dirichlet conditions on a periodic array of small hypercubes. Our analysis is performed in the framework of  Γ-convergence and we prove that, in the regime where ε vanishes faster then the side of the hypercubes, the effect of the constraints leads to a capacitary term in the limit energy as in the classical theory of periodically perforated domains for local integral functionals. A crucial role in our analysis turns out to be a discrete version of the Gagliardo-Nirenberg-Sobolev inequality, strongly inspired to its non local variant.