Organizers: Giovanni Catino and Fabio Cipriani
, Università degli Studi di Padova, Shape sensitivity and optimization of Grushin eigenvalues
, Thursday, May 11, 2023, time 15:15, Aula Seminari III pianoAbstract:Abstract:
It is well known that among all domains of a fixed volume, the ball minimizes the first eigenvalue of the Dirichlet Laplacian. The counterpart of this result for the degenerate operator known as the Grushin Laplacian is instead an open problem.
In this talk I will present some results in the direction of understanding such a problem. I will first consider the shape sensitivity of Grushin eigenvalues on general domains with the aim of characterizing critical domains under isovolumetric perturbations.
Next I will pass to the simplified case of cartesian product domains, showing that in this class the first eigenvalue admits a unique minimizer and providing some estimates on the minimum. Finally I will discuss some numerical experiments and some open problems.
The talk is based on joint works with Pier Domenico Lamberti (Università degli Studi di Padova), Paolo Musolino (Università Ca' Foscari Venezia) Luigi Provenzano (Sapienza Università di Roma), and Joachim Stubbe (EPFL).
, Università degli Studi di Milano-Bicocca, Unique continuation for the fractional heat operator
, Thursday, April 13, 2023, time 15:15, Aula Seminari III pianoAbstract:Abstract:
We study unique continuation properties and the asymptotic behaviour for a class of equations
involving the fractional heat operator with an Hardy-type potential. Our methods are based on
a Almgren-Poon monotonicity formula combined with a blow-up argument. Since the operator
has a global nature we will also need suitable extension results in the spirit of Caffarelli-Silvestre extension.
Key words: Parabolic partial differential equations, unique continuation, blow-up, asymptotics,
monotonicity formula, Hardy potential.
, Università della Calabria, A classification result for a Gross-Pitaevskii type system
, Thursday, April 13, 2023, time 16:15, Aula seminari III pianoAbstract:Abstract:
This talk will be focused on the study of a family of semilinear elliptic systems defined in $ R^n $, which is doubly critical since it involves Sobolev critical exponents and Hardy-type potentials. We aim to provide qualitative properties of positive solutions for these Gross-Pitaevskii type systems. In particular, we shall deduce that solutions are symmetric about the origin. In order to do it, we apply a suitable version of the moving planes technique for cooperative singular systems. Finally, we are able to provide a classification result for these kind of problems.
This is based on a joint work with Rafael López-Soriano (University of Granada, Spain) and Berardino Sciunzi (University of Calabria, Italy).
, Politecnico di Milano, Steady-state Navier-Stokes flow in an obstructed pipe under mixed boundary conditions and with a prescribed transversal flux rate
, Thursday, March 30, 2023, time 15:15, Aula seminari III pianoAbstract:Abstract:
The steady motion of a viscous incompressible fluid in an obstructed finite pipe is modeled through the Navier-Stokes equations with mixed boundary conditions involving the Bernoulli pressure and the tangential velocity on the inlet and outlet of the tube, while a transversal flux rate F is prescribed along the pipe. Existence of a weak solution to such Navier-Stokes system is proved without any restriction on the data by means of the Leray-Schauder Principle, in which the required a priori estimate is obtained by a contradiction argument based on Bernoulli’s law. Through variational techniques and with the use of an exact flux carrier, an explicit upper bound on F (in terms of the viscosity, diameter and length of the tube) ensuring the uniqueness of such weak solution is given. This upper bound is shown to converge to zero at a given rate as the length of the pipe goes to infinity. In an axially symmetric framework, we also prove the existence of a weak solution displaying rotational symmetry.
, University of West Bohemia (Pilsen, Czech Republic), Traveling waves of quasilinear reaction-diffusion equations with discontinuous diffusivity
, Thursday, March 23, 2023, time 15:30, Aula Seminari III pianoAbstract:Abstract:
We are concerned with traveling wave solutions to a class of quasilinear reaction-diffusion equations on the real line. We consider two types of continuous reaction term frequently found in applications - bistable and monostable. The diffusion coefficient is only piecewise continuous in (0,1) and allows for degenerations as well as singularities at 0 and 1. Under these general assumptions, we establish the existence of non-smooth traveling wave profiles. Uniqueness is shown in the bistable case, while a monostable reaction can give rise to a continuum of admissible wave speeds. Our approach is based on the investigation of an equivalent first order problem in the sense of Carathéodory and provides a broad theoretical background for mathematical treatment of various phenomena in population dynamics, chemistry and physics. Assuming power-type behaviour of the reaction and diffusion terms near the equilibria, we also discuss some asymptotic properties of the solutions.
This is a joint work with Pavel Drábek.
, 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 pianoAbstract: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.