Paolo Piovano, Politecnico di Milano, Analytical validation of variational models for epitaxially strained thin films, Monday, October 18, 2021, time 14:15, https://polimi-it.zoom.us/j/88014306666?pwd=Vi8rejZPMEFDK0YyUGNJb1pwV25Cdz09
Abstract:Abstract:
The derivation of variational models describing the epitaxial growth of thin films in the framework of the theory of Stress-Driven Rearrangement Instabilities (SDRI) will be presented, and the state of the art of the mathematical results described. By working in the context of both continuum and molecular mechanics, not only free boundary problems, but also atomistic models will be considered, and the discrete-to-continuum passage rigorously investigated in the intent to also provide a microscopical justification of the theory. An overview of the mathematical results achieved through the years with various co-authors for the existence, regularity and evolution of the solutions will be presented.

Stefano Vita, Università di Torino, Strong unique continuation and local asymptotics at the boundary for fractional elliptic equations, Tuesday, October 12, 2021, time 15:15, Aula Seminari III Piano, Dipartimento di Matematica
Abstract:Abstract:
We study local asymptotics of solutions to fractional elliptic equations at boundary points, under some outer homogeneous Dirichlet boundary condition. Our analysis is based on a blow-up procedure which involves some Almgren type monotonicity formulæ and provides a classification of all possible homogeneity degrees of limiting entire profiles. As a consequence, we establish a strong unique continuation principle from boundary points. This is a joint work with A. De Luca and V. Felli.

Francesco Esposito, Universita' della Calabria, Some adaptations of the moving planes method to singular elliptic problems, Wednesday, June 09, 2021, time 15:15 o'clock, https://polimi-it.zoom.us/j/89350397352?pwd=NGd4THV6UFR6VEU1VTVtaWxmOFZiQT09
Abstract:Abstract:
In this talk we will consider positive singular solutions to some quasilinear elliptic problems under zero Dirichlet boundary conditions. We will study qualitative properties of the solutions via the moving plane procedure, that goes back to the celebrated papers of Alexandrov and Serrin. In particular, exploiting a fine adaptation of the moving plane procedure and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions in bounded smooth domains.

Marco Caroccia, Dipartimento di Matematica, Politecnico di Milano, Contact surface of Cheeger sets, Wednesday, May 26, 2021, time 15:15 o'clock, https://polimi-it.zoom.us/j/89745265407
Abstract:Abstract:
Geometrical properties of Cheeger sets have been deeply studied by many authors since their introduction, as a way of bounding from below the first Dirichlet (p)-Laplacian eigenvualue. They represent, in some sense, the first eigenfunction of the Dirichlet (1)-Laplacian of a domain. In this talk we will introduce a recent property, studied in collaboration with Simone Ciani, concerning their contact surface with the ambient space. In particular we will show that the contact surface cannot be too small, with a lower bound on the (Hasudorff) dimension strictly related to the regularity of the ambient space. The talk will focus on the introduction of the problem and on the proof of the dimensional bounds. Functional to the whole argument is the notion of removable singularity, as a tool for extending solutions of pdes under some regularity constraint. Finally examples providing the sharpness of the bounds in the planar case are briefly treated.

Giulia Meglioli, Dipartimento di Matematica Politecnico di Milano, Smoothing effects and infinite time blowup for reaction-diffusion equations: an approach via Sobolev and Poincaré inequalities, Tuesday, May 18, 2021, time 11:15 o'clock, https://polimi-it.zoom.us/j/87010654658?pwd=dURNSzY4WE1VL0M5YWxwMzFLeTJSdz09
Abstract:Abstract:
The talk is concerned with the reaction-diffusion equation $u_t=\Delta(u^m)+u^p$, on a complete noncompact Riemannian manifold $M$. We consider the particularly delicate case when $p$ is less than $m$; moreover, we assume that the Poincaré and the Sobolev inequalities hold on $M$. We prove global existence in time of solutions for $L^m$ initial data. Furthermore, solutions are bounded for all positive times and their $L^\infty$ norm satisfy a certain quantitative bound. We also see that on a special class of Riemannian manifolds, solutions corresponding to sufficiently large $L^m$ data give rise to solutions that blow up in infinite time, a fact that cannot happen in the Euclidean setting. The results have been recently obtained jointly with Gabriele Grillo and Fabio Punzo.

Laura Abatangelo, Dipartimento di Matematica, Politecnico di Milano, Perturbation theory for Dirichlet eigenvalues in perforated domains, Wednesday, May 12, 2021, time 15:15 o'clock, https://polimi-it.zoom.us/j/89595030984?pwd=UG5wWmd1MjJkU1c1UnFRMmNaZUFMQT09
Abstract:Abstract:
In this talk I will present some recent results on asymptotics of eigenvalues of the Dirichlet Laplacian when a small compact set is removed from the initial domain. If the small set is concentrating at a point in some sense, the eigenvalue variation is proved to be strictly related to the vanishing order of one of the relative eigenfunctions at that point. A good understanding of this asymptotics leads to new issues, for instance optimal location or optimal shape of the hole (open problem) as well as possible ramification of multiple eigenvalues.