Organizers: Giovanni Catino and Fabio Cipriani

**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.
**Camilla Nobili**, Universität Hamburg,

*Rigorous bounds on the heat flux in turbulent convection*, Wednesday, Febraury 27, 2019, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
We are interested in thermal convection as described by the Rayleigh-B ?enard convection model. In this model the Navier-Stokes equations for the (divergence-free) velocity u with no-slip bound- ary conditions is coupled to an advection-diffusion equation for the temperature T with inhomo- geneous Dirichlet boundary conditions. The problem of understanding the (average) upward- heat-transport properties is of great interest for the applications and challenging for the rigorous analysis. We show how the PDE theory (in particular, regularity analysis) can contribute to the understanding of the scaling regimes for the heat transport. After reviewing the theory of Constantin& Doering ’99 we will present some recent results and discuss new challenges.
**Guglielmo Feltrin**, Politecnico di Torino,

*Positive solutions to indefinite problems: a topological approach*, Thursday, December 06, 2018, time 15:30, Aula seminari 3° piano

**Abstract:****Abstract:**
In this seminar, we present some recent existence and multiplicity results for positive solutions of boundary value problems associated with second-order nonlinear indefinite differential equations. More precisely, we deal with the ordinary differential equation
u?? + a(t)g(u) = 0,
where a: [0,T] ? R is a Lebesgue integrable sign-changing weight and g: [0,+?[ ? [0,+?[ is a continuous nonlinearity.
We focus on the periodic boundary value problem and on functions g(u) with superlinear growth at zero and at infinity (including the classical superlinear case g(u) = up, with p > 1). Exploiting a new approach based on topological degree theory, we show that there exist 2m ? 1 positive solutions when a(t) has m positive humps separated by negative ones and the negative part of a(t) is sufficiently large. In this manner, we give a complete answer to a question raised by Butler (JDE, 1976) and we solve a conjecture by G ?omez-Ren ?asco and L ?opez-G ?omez (JDE, 2000). The method also applies to Neumann and Dirichlet boundary conditions and, furthermore, provides a topological approach to detect infinitely many subharmonic solutions and globally defined positive solutions with chaotic behaviour.
Thereafter, we illustrate other directions for the research on indefinite problems: super-sublinear problems, models in population genetics, and also problems involving more general differential oper- ators, as the Minkowski-curvature one or the one-dimensional p-Laplacian. Exact multiplicity results and indefinite problems in the PDE setting are also discussed.
The talk is based on joint works with Alberto Boscaggin (University of Torino), Elisa Sovrano (University of Porto) and Fabio Zanolin (University of Udine) and on the book “Positive Solutions to Indefinite Problems. A Topological Approach” (Frontiers in Mathematics, Birkh ?auser/Springer, 2018).
**Luigi Vezzoni**, Università degli Studi di Torino,

*The Quantitative Alexandrov Theorem in Space forms*, Tuesday, November 27, 2018, time 15:15, Aula seminari 3° piano

**Abstract:****Abstract:**
The talk focuses on a recent generalization of a classical result of Alexandrov. The celebrated Alexandrov's Soap Bubble Theorem states that the spheres are the only closed (i.e. compact and without boundary) constant mean curvature hypersurfaces embedded in the Euclidean space. The theorem has been generalized to the hyperbolic space and to the hemisphere and to a large class of curvature operators. The main result of the talk is a quantitative version of Alexandrov's theorem which I've obtained in collaboration with Giulio Ciraolo and Alberto Roncoroni by using a quantitative study of the method of the moving planes. The theorem implies a new pinching Theorem for hypersurfaces in space forms.