Organizers: Giovanni Catino and Fabio Cipriani
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).
Giacomo E. Sodini, TUM,
A relaxation approach to optimal transport with applications to the unbalanced case, Wednesday, November 24, 2021, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
In this talk, after briefly presenting the classical optimal transport problem, we will discuss a new interpretation of the optimal transport cost as the largest lower semicontinuous convex functional extending the cost between pairs of delta measures.
With this in mind, we will introduce a notion of optimal transport cost for (nonnegative) measures with possibly different masses and discuss its metric and topological properties.
This talk is based on a joint work with Giuseppe Savaré (Bocconi University).
Giovanni Cupini, Università di Bologna,
Formula di media di Gauss: rigidità, stabilità, estensioni, Tuesday, November 16, 2021, time 15:15, Aula Seminari III Piano
Abstract:Abstract:
Il teorema della media di Gauss afferma che la media integrale di funzioni armoniche su una palla è uguale al valore assunto da queste funzioni nel centro della palla. Nel 1972 Kuran ha dimostrato il viceversa: se D è un aperto limitato contenente x, tale che la media integrale delle funzioni armoniche su D eguaglia il valore di queste funzioni in x, allora D è una palla centrata in x. In questo seminario si presenteranno estensioni di questo risultato e si discuterà la questione della stabilità della formula di media.
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.