Abstract. We introduce a class of integral functionals known as nonlocal
perimeters, which can be thought as interactions between a set and its
complement that are weighted by a positive kernel. In the first part of
the talk, we summarise the main features of these functionals and then
we study the asymptotic behaviour of the family associated with
mass-preserving rescalings of a given kernel. Namely, we prove that when
the scaling parameter approaches $0$, the rescaled non local perimeters
$Gamma$-converge to De Giorgi's perimeter, up to a multiplicative
Abstract. We deduce new uniqueness results for solutions to singular Liouville equations both on spheres and on bounded domains, as well as a new self-contained proof of previously known results. To this end, we derive a singular Sphere Covering Inequality based on the Alexandrov-Bol isoperimetric inequality and symmetric rearrangements. Joint project with D. Bartolucci, C. Gui and A. Moradifam.
Abstract. I will first give a short survey to recall the importance of convexity in the classical Monge problem. I will then introduce the $W_\infty$ Wasserstein distance and discuss the lack of convexity of the underlying problem.
The next step will be to present the dual problem introduced in 2016 by Barron, Bocea and Jensen. The maximum value of this dual problem coincides with the minimum value of the transport problem but it is not clear that non trivial maximizers exist.
Abstract. With applications in the area of biological membranes in mind, we consider the problem of minimizing Willmore’s energy among the class of closed, connected surfaces with given surface area that are confined to a fixed container. Based on a phase field model for Willmore’s energy originally introduced by de Giorgi, we develop a technique to incorporate the connectedness constraint into a diffuse interface model of elastic membranes.
Abstract. In questo seminario affronteremo il problema della regolarità delle frontiere libere associate ai minimi del funzionale di Bernoulli. In un celebre lavoro degli anni ottanta, Alt e Caffarelli dimostrarono che le frontiere libere di Bernoulli si possono decomporre in due insiemi disgiunti: l'insieme dei punti piatti (Reg), localmente dato dal grafico di una funzione liscia, e l'insieme dei punti singolari (Sing), di dimensione di Hausdorff al massimo d-5.
Abstract. We study the small Deborah number limit of the Doi-Onsager equation for the dynamics of nematic liquid crystals. This is a Smoluchowski-type equation that characterizes the evolution of a number density function, depending upon both particle position and its orientation vector, which lies on the unit sphere.
Abstract. We consider networks of curves in the plane moving according to the L^2-gradient flow of a variant of the elastic energy. In this talk we will prove short time existence in the case of networks composed by three curves that are required to meet in one or two triple junctions. As a variation of the result we additionally impose that they form an angle of 120 degrees at the triple junction(s). If time allows we will give some outlook on our expectations concerning the long time behaviour based on numerical work by John Barrett, Harald Garcke and Robert Nürnberg.
Abstract. In this talk we will consider planar networks of three curves minimizing a combination of the elastic energy and the length functional. We will prove existence and regularity of minimizers and we will show some properties of the minimal configurations. In addition to the presentation of the results that are obtained in collaboration with Anna Dall'Acqua and Matteo Novaga, we will give a partial review of the theory of elasticae and discuss about the onset of new phenomena passing from the problem for curves to the one for networks.
Abstract. In this talk we introduce a notion of functions of fractal bounded variation. Here, the sup-norm of test functions as used in the classical definition is replaced by the Hoelder norm with respect to some exponent. Characteristic functions of domains with fractal bounderies are particular examples that belong to this class. Among a characterization in terms of currents, we state some properties that naturally extend those of classical BV functions such as higher integrability, decomposition into Hoelder functions and push forwards.