Filippo Giuliani, Politecnico di Milano (nuovo rtd-b del DMAT), Some results and open problems in Hamiltonian PDEs, Thursday, March 17, 2022, time 12:15 o'clock, !! CAMBIO D'ORARIO !! Aula Seminari MOX VI Piano, 12:15-13:15
Abstract:Abstract:
We present some recent results concerning the dynamics of Hamiltonian partial differential equations on compact manifolds, especially fluid dynamics models. We talk about both stability and instability issues, like the existence of quasi-periodic in time solutions and transfers of energy between Fourier modes of solutions of nonlinear PDEs. We also discuss some open problems and possible new research directions.

Giulio Tralli, Università di Padova, Heat kernels and intertwining properties in Heisenberg-type groups, Wednesday, Febraury 09, 2022, time 14:15, Aula Seminari III Piano
Abstract:Abstract:
In this talk we discuss the conformal fractional powers of the horizontal Laplacian in groups of Heisenberg type. We present a new approach, based on the heat kernels of suitable extension operators, to the derivation of the fundamental solutions for these nonlocal operators and to the explicit construction of the Aubin-Talenti type functions. Frequent comparisons with the Euclidean case of the powers of the Laplacian will be made, focusing on similarities and main differences. The talk is based on a joint project with N. Garofalo.

Antonio Segatti, Università di Pavia, Evolution of Ginzburg Landau vortices for vector fields on surfaces, Wednesday, January 19, 2022, time 14:15, https://polimi-it.zoom.us/j/86877435731?pwd=aEhvU21QWEh2dGhHU0tQWDVyaE1tUT09
Abstract:Abstract:
In this talk I will report on a joint work with Giacomo Canevari. I will discuss a parabolic Ginzburg-Landau equation for vector fields on a 2 dimensional closed and oriented Riemannian manifold. I will show that in a suitable asymptotic regime the energy of the solutions concentrates on a finite number of points. These points are called vortices and their evolution is governed by gradient flow of the so-called renormalized energy.

Stefania Patrizi, University of Texas at Austin, From the Peierls-Nabarro model to the equation of motion of the dislocation continuum, Monday, December 20, 2021, time 15:15, https://polimi-it.zoom.us/j/89936354344
Abstract:Abstract:
We consider a semi-linear integro-differential equation in dimension one associated to the half-Laplacian. This model describes the evolution of phase transitions associated to dislocations whose solution represents the atom dislocation in a crystal. The equation comprises the evolutive version of the classical Peierls-Nabarro model. We show that for a large number of dislocations, the solution, properly rescaled, converges to the solution of a well known equation called "the equation of motion of the dislocation continuum". The limit equation is a model for the macroscopic crystal plasticity with density of dislocations. In particular, we recover the so called Orowan's law which states that dislocations move at a velocity proportional to the effective stress. This is a joint paper with Tharathep Sangsawang.

Giorgio Tortone, Università di Pisa, EPSILON-REGULARITY FOR THE SOLUTIONS OF FREE BOUNDARY SYSTEMS, Tuesday, December 14, 2021, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
We will present some new results for a class of free boundary systems associated to shape optimization problems (spectral and integral functionals). The new main point of these results is the analysis of the regular part of the free boundary based on a linearization argument that takes care of the vectorial attitude of the problem. This is based on joint works with D. De Silva and with F.P. Maiale and B. Velichkov.

Elisa Sovrano, Università degli Studi di Modena e Reggio Emilia, Sign-indefinite logistic growth models with flux-saturated diffusion, Tuesday, November 30, 2021, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
Reaction-diffusion processes can be based on Fick-Fourier's law. Changing perspective, we deal with a dispersive flux which is a nonlinear bounded function of the gradient. In a bounded domain with a regular boundary, we investigate a Dirichlet problem associated with a quasilinear reaction-diffusion equation where the mean curvature operator drives the diffusion process. As for the reaction, we consider the product of a logistic-type nonlinearity and a sign-indefinite weight function modeling spatial heterogeneities. For this problem, we present some recent results concerning the existence and the multiplicity of positive solutions. Depending on the logistic term's behavior at zero, we prove three qualitatively different bifurcation diagrams by varying the diffusivity parameter. We point out a new multiplicity phenomenon without any similarity with the case of linear-diffusion logistic-growth models. This talk is based on joint works with Pierpaolo Omari (University of Trieste).