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Workshop in honor of Alessio Figalli's Doctor Honoris Causa at UPC

The workshop will take place on Thursday November 21st (from 10 am to 5:30 pm) at sala d'actes of the School of Mathematics and Statistics UPC (C. Pau Gargallo, 14 Barcelona). Talks will be comprehensible to a general audience and students.

Workshop in honor of Alessio Figalli's Doctor Honoris Causa at UPC

Five talks and a Round Table with Prof. Alessio Figalli

#WorkshopFigalliFME

Speakers: Five spanish and catalan mathematicians who have worked (or still working ) with Prof. Alessio Figalli: Xavier Cabré (UPC-ICREA), Xavier Ros-Oton (Universität Zürich), Joaquim Serra (ETH Zürich), Matteo Bonforte (Universidad Autónoma de Madrid- UAM) i Juan Luís Vázquez (Universidad Autónoma de Madrid- UAM)

10h

Welcome from Prof. Francesc Torres, Rector of the Universitat Politècnica de Catalunya (UPC) and Prof. Jaume Franch, Dean of School of Mathematics and Statistics (FME), UPC.

10h15

Elliptic and parabolic equations of fractional nonlocal type

Juan Luís Vázquez (Universidad Autónoma de Madrid- UAM)

11h05

Coffee break served for all participants- Q & R rooms - FME

11h30

Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains

Matteo Bonforte (Universidad Autónoma de Madrid- UAM)

12h20

Generic regularity in free boundary problems

Xavier Ros-Oton (Universität Zürich)

13h10

Informal lunch served for all participants- Q & R rooms - FME

15 h

Round table: Prof. Alessio Figalli and the 5 speakers of the workshop

15h45

Regularity of stable interfaces: from nonlocal to local

Joaquim Serra (ETH Zürich)

16h35

Regularity of stable solutions to semilinear elliptic equations

Xavier Cabré (Universtiat Politècnica de Catalunya-ICREA)

Activity open to the entire mathematical community. Registration is free.

We encourage you to register: registration closed.

Deadline for registration: November 19th, 2019
Workshop poster

With the collaboration of Càtedra Mir-Puig

Elliptic and parabolic equations of fractional nonlocal type

Abstract:

After a light motivation to the world of nonlocal equations of fractional type, placed inside the general theory of PDES and diffusion, the talk will present some recent work on the existence and behaviour of solutions of nonlinear fractional elliptic and parabolic equations, mainly when posed in bounded domains. The boundary behaviour is carelly examined, since it offers many novelties.
Works in collaboration with Figalli, Caffarelli, Bonforte, Sire, and Gomez-Castro,... "

Nonlinear and Nonlocal Degenerate Diffusions on Bounded Domains

Abstract:

Nonlinear diffusion models appear in several real world phenomena, ranging from physics, engineering and information theory to life sciences and finance. This talk will be focussed on a series of recent papers in collaboration with A. Figalli, X. Ros Oton,Y. Sire and J. L. Vázquez. We develop a complete theory for a diffusion model of Porous Medium type, with different nonlocal operators and degenerate nonlinearities. After a brief summary of the main features of such theory, we will focus our attention on (sharp) boundary estimates, showing how the degeneracy of the nonlinearity together with the nonlocal character of the equation, may cause a surprising anomalous boundary behaviour.

Generic regularity in free boundary problems

Abstract:

Free boundary problems are those described by PDE's that exhibit a priori unknown (free) interfaces or boundaries. They appear in Geometry, Physics, Probability, Biology, or Finance, and their study uses tools from PDE, Geometric Measure Theory, and Calculus of Variations. The goal of this talk is to present different free boundary problems, explain the main known results in this context, and give an overview of the current research and open problems. In particular, we will discuss a long-standing open question in the field which concerns the generic regularity of free boundaries, and present some new results in this direction.

Regularity of stable interfaces: from nonlocal to local

Abstract:

I will describe recent results with Alessio Figalli on 1D symmetry of stable critical points of the Ginzburg-Landau functional associated to the half-Laplacian.

As we will discuss, an interesting feature of these functional is that it is asymptotic to the classical (local) perimeter, and thus their minimizers approximate sets of minimal perimeter.

As an application, we obtain the analogue for the half-Laplace of De Giorgi's conjecture in dimension four (a statement that is still open in the case of the Laplacian).

Regularity of stable solutions to semilinear elliptic equations

Abstract:

The regularity of stable solutions to semilinear elliptic PDEs has been studied since the 1970's. In dimensions 10 and higher, there exist singular stable energy solutions. In this talk I will describe a recent work with Figalli, Ros-Oton, and Serra, where we prove that stable solutions are smooth up to the optimal dimension 9. This answers to a famous open problem posed by Brezis in the mid-nineties concerning the regularity of extremal solutions to Gelfand-type problems.