Seminars

December 4th, 2023, 12:30pm. Location: Aula Gris 2, ICMAT.

Enzo Maria Merino (U. Bologna)

Title: Intrinsic Lipschitz regularity for almost minimizer of a one-phase Bernoulli-type functional in Carnot Groups of step 2.

Abstract: The regularity of minimizers of the classical one-phase Bernoulli functional was deeply studied after the pioneering work of Alt and Caffarelli. More recently, the regularity of almost minimizers was investigated as well. We present a regularity result for almost minimizers for a one-phase Bernoulli-type functional in Carnot Groups of step two. Our approach is inspired by the methods introduced by De Silva and Savin in the Euclidean setting. Moreover, some recent intrinsic gradient estimates have been employed. Some generalizations will be discussed. Some of the results presented are obtained in collaboration with F. Ferrari (University of Bologna) and N. Forcillo (Michigan State University) and will be part of my PhD thesis.


November 23rd, 2023, 3:40pm. Room 520, Modulo 17, Facultad de Ciencias, UAM

Ali Hyder (TIFR-CAM Bangalore)

Title: Blow-up analysis of stationary solutions to a Liouville-type equation in 3-D.

Abstract:  Contrary to the two dimensional case, the Liouvile equation in dimension three and higher is supercritical, and in particular it admits singular solutions. We will talk about partial regularity results of stationary weak solutions. Our approach is based on blow-up analysis and a monotonicity formula. 


November 8th, 2023, 12:30pm. Room 520, Modulo 17, Facultad de Ciencias, UAM

Taehun Lee (Korea Institute for Advanced Study)

Title: Regular solutions to the $L_p$ Minkowski problem

Abstract: A cornerstone of the Brunn--Minkowski theory is the Minkowski problem initiated by Minkowski himself over a century ago. This problem characterizes measures generated by convex bodies and has been generalized to the $L_p$ Minkowski problem. In recent years, much of the interest in the $L_p$ Minkowski problem has migrated to the study of regular solutions. In this talk, we discuss recent developments in this field, focusing on the regularity of convex bodies. We completely describe the range of $p$ for which solutions are regular. This talk is based on joint work with Kyeongsu Choi and Minhyun Kim.



October 9th, 2023. Room "Gris 2", ICMAT

Title: Concentration limit for non-local dissipative convection-diffusion kernels on the hyperbolic space 

Abstract: We study a non-local, non-linear convection-diffusion equation on the hyperbolic space $\mathbb{H}^N$, governed by two kernels, one for each of the diffusion and convection parts. One main novelty is the constucion of the non-symmetric convection kernel defined on the tangent bundle and invariant under the geodesic flow.  Next, we consider the relaxation of this model to a local problem, as the kernels get concentrated near the origin of each tangent space. Under some regularity and integrability conditions, we prove that the solution of the concentrated non-local problem converges to that of the local convection-diffusion equation. We prove and then use in this sense a compactness tool on manifolds inspired by the work of Bourgain-Brezis-Mironescu. 


Title: Asymptotic behavior of solutions for some diffusion problems on metric graphs.

Abstract: In this talk we present some recent result about the long time behavior of the solutions for some diffusion processes on a metric graph. We study  evolution problems on a metric connected finite graph in which some of the edges have infinity length. We show that the asymptotic behaviour of the solutions of the heat equation (or even some nonlocal diffusion problems) is given by the solution of the heat equation, but on a star shaped graph in which there is only one node and as many infinite edges as in the original graph. In this way we obtain that the compact component that consists in all the vertices and all the edges of finite length can be reduced to a single point when looking at the asymptotic behaviour of the solutions. We prove that when time is large the solution behaves like a gaussian profile on the infinite edges. When the nonlinear convective part is present we obtain similar results but only on a star shaped tree. This is a joint work with Cristian Cazacu (University of Bucharest), Ademir Pazoto (Federal University of Rio de Janeiro), Julio D. Rossi (University of Buenos Aires) and  Angel San Antolin (University of Alicante).