Otros Seminarios

Jornada de Ecuaciones en Derivadas Parciales

Ponente:  Luca Battaglia, Università di Roma Tre // Daniel E. Restrepo, University of Texas at Austin // José A. Cañizo, Universidad de Granada ()
Fecha:  viernes 22 de abril de 2022 - 12:00-14:30
Lugar:  Aula 520, Módulo 17, Departamento de Matemáticas, UAM


12 - 12:45. "A double mean field approach for a curvature prescription problem", Luca Battaglia (Università di Roma Tre)

Abstract: I will consider a double mean field-type Liouville PDE on a compact surface with boundary, with a nonlinear Neumann condition. This equation is related to the problem of prescribing both the Gaussian curvature and the geodesic curvature on the boundary.
I will discuss blow-up analysis, a Moser-Trudinger inequality for the energy functional, existence of minmax solution when the energy functional is not coercive.
The talk is based on a work with Rafael Lopez-Soriano (Universitat Carlos III de Madrid).

12:45 - 13:30. "Uniform stability in the Euclidean Isoperimetric problem for the Allen-Cahn energy", Daniel E. Restrepo (University of Texas at Austin)

Abstract: The talk will be mainly focused on a version of the Euclidean isoperimetric for the Allen-Cahn energy. In this joint work with Francesco Maggi, we proved the validity of two fundamental properties of the classical isoperimetric problem for this phase transitions approximation: 1) stability, i.e., the difference in energy between competitors and the global minimum controls quantitatively (and uniformly in the length scale of the phase transition)  the distance to minimizers; 2) the only critical points of the associated variational problem (under certain assumptions) are minimizers, i.e., rigidity in the spirit of Alenxandrov's theorem.


13:40 - 14:25. "The scaling hypothesis for the Smoluchowski equation: some recent advances", José A. Cañizo (Universidad de Granada)

Abstract: The Smoluchowski equation is a well known model for coagulation processes. One of its central conjectures, the scaling hypothesis, says that its solutions generally behave in a self-similar way. This has been proved for particular coagulation coefficients, and we will present results (together with B. Lods and S. Throm) which simplify the proof of this. We will also discuss new cases where we prove it for small perturbations of the constant coefficients. We use some typical techniques in kinetic theory such as the use of entropy functionals, and the study of the spectral gap of the linearised operator in several spaces.