Organizers: Giovanni Catino and Fabio Cipriani

**Hynek Kovarik**, Università degli studi di Brescia,

*Robin eigenvalues on domains with cusps*, Thursday, June 06, 2019, time 14:00, Aula seminari 3° piano

**Abstract:****Abstract:**
We consider the Laplace operator with the Robin boundary condition with negative coefficient
on bounded domains with cusps. We show that if the cusps is not too strong, then the operator is bounded from below and calculate the leading term in the asymptotic expansion of its negative eigenvalues as the coefficient of the boundary condition tends to infinity. This is a joint work with Konstantin Pankrashkin.
**Fabio Cipriani**, Politecnico di Milano,

*How to hear the shape of a drum*, Tuesday, May 14, 2019, time 15:30, Aula seminari 3° piano

**Abstract:****Abstract:**
In a iconic 1912 paper Hermann Weyl, motivated by problems posed by the physicist H.A. Lorentz about J.H. Jeans's radiation theory, showed that the dimension and the volume of a Euclidean domain may be traced from the asymptotic distribution of the eigenvalues of its Laplace operator.
In a as much famous 1966 paper titled "Can one hear the shape of a drum" Marc Kac popularized this and related problems connecting geometry and spectrum. He noticed that the hope to characterize {\it isometrically}, Euclidean domains or compact Riemannian manifolds by the spectrum of the Laplace operator, is vain: John Milnor in 1964 had showed the existence of non isometric 16 dimensional tori sharing a common (discrete) spectrum.
The aim of the talk is to show how to recognize {\it conformal maps} between Euclidean domains as those homeomorphisms which transform multipliers of the Sobolev-Dirichlet spaces of a domain into multipliers of the other and leave invariant the {\it fundamental tone} or {\it first nonzero eigenvalue} of the Dirichlet integral with respect to the energy measures of any multiplier. Related results hold true for {\it quasiconformal and bounded distortion maps}.
In the opposite direction, we prove that the trace of the Dirichlet integral, with respect to the energy measure of a multiplier, is a Dirichlet space that only depends upon the orbit
of the conformal group of the Euclidean space on the multiplier algebra.
The methods involve potential theory of Dirichlet forms (changing of speed measure, multipliers) and the Li-Yau conformal volume of Riemannian manifolds.
This is a collaboration with Jean-Luc Sauvageot C.N.R.S. France et Universit\'e Paris 7.
**Alberto Tesei**, Università degli Studi di Roma "La Sapienza",

*Soluzioni a valori misure di equazioni di evoluzione nonlineari*, Tuesday, May 14, 2019, time 14:30, Aula seminari 3° piano

**Abstract:****Abstract:**
Soluzioni a valori misure si presentano in modo naturale per importanti classi di equazioni di evoluzione nonlineari (equazione dei mezzi porosi, equazioni "forward-backward", leggi di conservazione). Nel seminario saranno esposti alcuni recenti risultati di esistenza, unicità e comportamento qualitativo di soluzioni entropiche
a valori misure di Radon di leggi di conservazione iperboliche in una dimensione spaziale, con flusso limitato e lipschitziano. Tempo permettendo, sarà discusso il legame fra tali soluzioni e soluzioni viscose discontinue di equazioni di Hamilton-Jacobi. I risultati presentati sono contenuti in alcuni lavori con M. Bertsch, F. Smarrazzo e A. Terracina.
**Andrea Posilicano**, Università degli Studi dell'Insubria,

*Markovian Extensions of Symmetric Second Order Elliptic Differential Operators*, Tuesday, April 30, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
We give a complete classification of the Markovian self-adjoint extensions of the minimal realization of a second order elliptic differential operator on a bounded n-dimensional domain by providing an explicit one-to-one correspondence between such extensions and the class of Dirichlet forms on the boundary which are additively decomposable by the bilinear form of the Dirichlet-to-Neumann operator plus a Markovian form. By such a result two further equivalent classifications are provided: the first one is expressed in terms of a decomposition of the bilinear forms associated to the extensions, the second one uses the decomposition of the resolvents provided by the Krein formula. The Markovian part of the decomposition allows to characterize the operator domain of the corresponding extension in terms of Wentzell-type boundary conditions. Some analogous results hold also in a nonlinear setting.
**Gianmarco Sperone**, Politecnico di Milano,

*Some remarks on the forces exerted by a viscous fluid on a bluff body*, Thursday, March 28, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
The theory of flight developed by Kutta, Lanchester and Zhukovsky is mainly based on the idea that a cambered surface produces lift through its ability to generate a vortex about itself. Since fluid flows around an obstacle generate vortices which are difficult to locate and to describe, in this talk we analyze the Stokes and Navier-Stokes equations for the two-dimensional motion of a viscous fluid in the exterior of a fixed obstacle. Firstly, we discuss nonstandard boundary conditions for the Stokes equations on a smooth obstacle, allowing for the generation of turbulence over the leeward wall of the body. Secondly, after studying the connection between the appearance of lift and the unique solvability of the Navier-Stokes equations, we show some numerical results that compare the aerodynamic response of different non-smooth obstacles, as the inlet velocity is increased until reaching the critical Reynolds number. The talk accounts for results contained in two articles prepared in collaboration with Filippo Gazzola and Andrei Fursikov (Moscow State University).
**Roberto Cominetti**, Universidad Adolfo Ibáñez,

*Stochastic atomic congestion games: Price-of-Anarchy and convergence for large games*, Friday, March 08, 2019, time 11:00, Sala del Consiglio 7° piano

**Abstract:****Abstract:**
We consider atomic congestion games with stochastic demand in which each player participates in the game with probability p, and incurs no cost with probability 1-p. For congestion games with affine costs, we provide a tight upper bound for the Price-of-Anarchy as a function of p, which is monotonically increasing and converges to the well-known bound of 5/2 when p converges 1. On the other extreme, for p? 1/4 the bound is constant and equal to 4/3 independently of the game structure and the number of players. For general costs we also analyze the asymptotic convergence of such games when the number of players n grows to infinity but the probability tends to zero as $p_n=\lambda/n$, in which case we establish the convergence towards a Poisson limit game. In a different approach where the weight of the players tend to zero, we find that the limit yields a Wardrop equilibrium for a corresponding nonatomic game.