Andrea Pinamonti, Università di Trento, Lusin and Whitney theorems in Carnot groups: when geometry meets real analysis, Thursday, March 23, 2023, time 16:30, Aula Seminari III piano
Abstract:Abstract:
Whitney extension results characterize when one can extend a mapping from a compact subset to a smooth mapping on a larger space. Lusin approximation results give conditions under which one can approximate a rough map by a smoother map after discarding a set of small measure. We first recall relevant results in the Euclidean setting, then describe recent work extending them to horizontal curves in the Heisenberg group.

Alessio Falocchi, Politecnico di Milano, Some results on the Stokes eigenvalue problem under Navier boundary conditions, Thursday, March 16, 2023, time 15:15, Aula Seminari III piano
Abstract:Abstract:
We study the Stokes eigenvalue problem under Navier boundary conditions in 2D or 3D bounded domains with connected boundary of class $ C^1 $. Differently from the Dirichlet boundary conditions, zero may be the least eigenvalue. We fully characterize the domains where this happens and we show that the ball is the unique domain where the zero eigenvalue is not simple. We apply these results to show the validity/failure of a suitable Poincaré inequality. We then consider the general version of the problem in any space dimension with $ n\geq2 $, characterizing the kernel of the strain tensor for solenoidal vector fields with homogeneous normal trace. We conclude analyzing some similarities and differences with the Laplacian eigenvalue problem. This is based on a joint work with Filippo Gazzola, Politecnico di Milano.

Jacopo Somaglia, Politecnico di Milano, Rotund Gâteaux smooth norms which are not locally uniformly rotund, Thursday, March 02, 2023, time 15:15, Aula seminari III piano
Abstract:Abstract:
In a recent monograph on renorming techniques in Banach spaces, A.J. Guirao, V. Montesinos, and V. Zizler posed the following problem; does every infinite-dimensional separable Banach space admit a norm that is rotund and Gâteaux smooth but not locally uniformly rotund? The aim of the talk is to provide a positive answer to the above question. All the presented results have been obtained in collaboration with C.A. De Bernardi.

Enrico Laeng, Dipartimento di matematica, Politecnico di Milano, Successioni non nulle i cui infiniti momenti sono tutti nulli., Wednesday, March 01, 2023, Aula seminari, III piano, Dipartimento di Matematica, Politecnico di Milano
Abstract:Abstract:
Il teorema di Stone-Weierstrass implica che se tutti i momenti su un intervallo (integrali della funzione per una potenza intera della variabile) di una funzione continua sono nulli, allora anche tale funzione deve essere nulla. E` abbastanza ben noto che questo non e` piu` vero se prendiamo i momenti su una semiretta o su una retta. Noi affrontiamo il caso discreto, ovvero l'esistenza di successioni tali che le serie che hanno per addendi i termini della successione moltiplicati per una qualsiasi potenza intera dell'indice di convergono a zero grazie a un peculiare gioco di cancellazioni. E` un risultato generale che ha anche conseguenze per la teoria delle probabilita` perche` quantita` come media, varianza, curtosi, etc. sono legate ai momenti.

Bruno Volzone, Università degli Studi di Napoli 'Parthenope', Long-time behavior for local and nonlocal porous medium equations with small initial energy, Wednesday, Febraury 08, 2023, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
In the first part of the talk, we will describe some aspects of a study developed in a joint paper with L. Brasco concerning the long-time behavior for the solution of the Porous Medium Equation in an open bounded connected set, with smooth boundary and sign-changing initial datum. Homogeneous Dirichlet boundary conditions are considered. We prove that if the initial datum has sufficiently small energy, then the solution converges to a nontrivial constant-sign solution of a sublinear Lane-Emden equation, once suitably rescaled. We also give a sufficient energetic criterion on the initial datum, which permits to decide whether convergence takes place towards the positive solution or to the negative one. The second part of the talk will be devoted to some new advances obtained in collaboration with G. Franzina, in the spirit of the ones explained above, for the study of the asymptotics of signed solutions for the Fractional Porous Medium Equation.

Simone Dovetta, Politecnico di Torino, Action versus energy ground states in nonlinear Schrödinger equations, Thursday, January 26, 2023, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
The talk investigates the relation between normalized critical points of the nonlinear Schrödinger energy functional and critical points of the corresponding action functional on the associated Nehari manifold. First, we show that the ground state levels are strongly related by the following duality result: the (negative) energy ground state level is the Legendre–Fenchel transform of the action ground state level. Furthermore, whenever an energy ground state exists at a certain frequency, then all action ground states with that frequency have the same mass and are energy ground states too. We see that the converse is in general false and that the action ground state level may fail to be convex. Next we analyze the differentiability of the ground state action level and we provide an explicit expression involving the mass of action ground states. Finally we show that similar results hold also for local minimizers, and we exhibit examples of domains where our results apply. This is a joint work with Enrico Serra and Paolo Tilli.