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Luca Battaglia (Università La Sapienza di Roma)

Hora: 12 - 12:45.

Título: A double mean field approach for a curvature prescription problem.

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 (Universidad Carlos III de Madrid).

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Daniel E. Restrepo (University of Texas at Austin)

 

Hora: 12:45 - 13:30.

Título: Uniform stability in the Euclidean Isoperimetric problem for the Allen-Cahn energy.

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.

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José A. Cañizo (Universidad de Granada)

 

Hora: 13:40 - 14:25.

Título: The scaling hypothesis for the Smoluchowski equation: some recent advances.

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.