13 Aprile, 2023 15:15

Sezione di Analisi

Unique continuation for the fractional heat operator

Giovanni Siclari, Università degli Studi di Milano-Bicocca

Aula Seminari III piano

abstract

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.

13 Aprile, 2023 16:15

Sezione di Analisi

A classification result for a Gross-Pitaevskii type system

Francesco Esposito, Università della Calabria

Aula seminari III piano

abstract

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).

5 Aprile, 2023 12:15

Seminari di Cultura Matematica

Video killed the radio math?

Roberta Fulci, Giornalista Scientifico Rai3

Sala Consiglio VII piano - tiny.cc/CM2023

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).

30 Marzo, 2023 15:15

Sezione di Analisi

Steady-state Navier-Stokes flow in an obstructed pipe under mixed boundary conditions and with a prescribed transversal flux rate

Gianmarco Sperone, Politecnico di Milano

Aula seminari III piano

abstract

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.

29 Marzo, 2023 12:15

Seminari di Cultura Matematica

La Penisola dei Vulcani

Foresta Martin, Giacomelli, Scandone, Giornalista Scientifico, Vulcanologa, Università Roma Tre

Sala Consiglio VII piano - tiny.cc/CM2023

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.

23 Marzo, 2023 15:30

Sezione di Analisi

Traveling waves of quasilinear reaction-diffusion equations with discontinuous diffusivity

Michaela Zahradníková, University of West Bohemia (Pilsen, Czech Republic)

Aula Seminari III piano

abstract

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.

23 Marzo, 2023 11:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

Double nested Hilbert schemes and stable pair invariants

Sergej Monavari, EPFL Lausanne

Aula seminari III piano

abstract

abstract

Hilbert schemes of points on a smooth projective curve are symmetric powers of the curve itself; they are smooth and we know essentially everything about them. We propose a variation by studying double nested Hilbert schemes of points, which parametrize flags of 0-dimensional subschemes satisfying certain nesting conditions dictated by Young diagrams. These moduli spaces are almost never smooth but admit a virtual structure à la Behrend-Fantechi. We explain how this virtual structure plays a key role in (re)proving the correspondence between Gromov-Witten invariants and stable pair invariants for local curves.

23 Marzo, 2023 16:30

Sezione di Analisi

Lusin and Whitney theorems in Carnot groups: when geometry meets real analysis

Andrea Pinamonti, Università di Trento

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.

22 Marzo, 2023 12:15

Seminari di Cultura Matematica

Entanglement

Marco Genovese, Quantum Optics INRIM

Sala Consiglio VII piano - tiny.cc/CM2023

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.

22 Marzo, 2023 17:00 in punto

Sezione di Geometria, Algebra e loro applicazioni

Discrete octonionic analysis: first ideas and results

Dmitrii Legatiuk, Erfurt University

Online (link: polimi-it.zoom.us/j/98216800053)

abstract

abstract

In recent years, there is a growing interest in the studying octonions, which are 8-dimensional hypercomplex numbers forming the biggest normed division algebras over the real numbers. In particular, various tools of the classical complex function theory have been extended to the octonionic setting in recent years. However not so many results related to a discrete octonionic analysis, which is relevant for various applications in quantum mechanics, have been presented so far. Therefore, in this talk, we present first ideas towards discrete octonionic analysis. In particular, we discuss several approaches to a discretisation of octonionic analysis and present several basic tools of discrete octonionic analysis such as the discrete octonionic Stokes' formulae, Borel-Pompeiu and Cauchy formulae.