Category Archives: Past seminar

p-harmonic functions with Neumann conditions and measure data

Speaker: Natham Aguirre

Universidad de Chile

Date: August 9, 2022 at 12 Santiago time

Abstract: In this talk I will discuss the problem of finding solutions to some nonlinear elliptic equations with measure data. To this end I will introduce the concept of Renormalized Solutions, which is a very useful tool to solve both Dirichlet and Neumann problems. I will present some results in this area and also discuss some open problems.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Hanne Van Den Bosch

 

A nonlocal isoperimetric problem: density perimeter

Speaker: Andrés Zúñiga

Universidad de O’Higgins

Date: July 6, 2022 at 12 Santiago time

Abstract: We will discuss a variant of a classical geometric minimization problem, known as the “nonlocal isoperimetric problem”, which arises from studies in Nuclear Physics by Gamow in the 1930’s. By introducing a density in the perimeter functional, we obtain features that differ substantially from existing results in the framework of the classical problem without densities. In the regime of “small” non-local contribution, we completely characterize the minimizer, in the case the density is a monomial radial weight. This work is a collaboration with Stan Alama and Lia Bronsard (McMaster University) and Ihsan Topaloglu (Virginia Commonwealth University), as part of the project QUALITATIVE PROPERTIES OF WEIGHTED AND ANISOTROPIC VARIATIONAL PROBLEMS financed by ANID CHILE FONDECYT INICIACION Nº 11201259.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Claudio Muñoz

 

 

 

The search of finite-time singularity solutions of the Euler equations for incompressible and inviscid fluids

Speaker: Sergio Rica

Instituto de Física, Pontificia Universidad Católica de Chile

Date: June 28, 2022 at 12 Santiago time

Abstract: Despite 250 years of history, the nature of solutions of the Euler equations remains an open problem. To date, it is not known if general smooth initial conditions of the Euler equations with finite energy do or do not blow-up in finite time. I will review the approach initiated by Leray of self-similar blow-up solutions. Lastly, I will show that under some conditions an axisymmetric incompressible and inviscid flow presents a finite-time singularity. This singularity appears to be generic and robust for a wide number of finite energy initial conditions.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Michal Kowalczyk

 

 

Sobre ecuaciones tipo Cummins y convertidores de energía de olas

Speaker: Gastón Vergara

Maynooth University, Ireland

Date: June 22, 2022 at 12 Santiago time

Abstract: En esta charla comenzaremos abordando algunas formulaciones recientes de las ecuaciones water-waves, para luego tomar ventaja de ellas y establecer algunos problemas de transmisión explícitos que describen interacciones fluido-estructuras. En una segunda parte estudiaremos el cómo bajo ciertas restricciones es posible obtener algunas generalizaciones de la ecuación de Cummins. Por último, mostraremos métodos con los cuales la comunidad que estudia la extracción de energía a partir de las olas del mar utiliza dichas ecuaciones integro-diferenciales para obtener energías limpias e infinitamente renovables.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Gabrielle Nornberg

 

 

Bianchi cosmologies with massless Vlasov matter

Speaker: Hamed Barzegar 

University of Vienna, Austria

Date: June 14, 2022 at 12 Santiago time

Abstract: In this talk, I will give a short introduction to “mathematical cosmology” with a focus on the application of the kinetic theory in cosmology. As such, I will talk about Bianchi cosmologies, i.e., spatially homogeneous spacetimes that are governed by the Einstein equations which are coupled to massless collisionless (Vlasov) matter. Then, I will discuss their future attractors and show future stability of such models within Bianchi types I, II, and V symmetry class. The proof turns out to be more challenging compared to the corresponding massive case where the cosmological constant is absent, since the massless particles indicate less decay rates in the course of the expansion of the universe. The proof is based on an energy method for small initial data.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Paola Rioseco

 

 

Integrability and the singular manifold method: a toolkit to determine soliton solutions

Speaker: Paz Albares

Universidad de Salamanca, Spain

Date: June 7, 2022 at 12 Santiago time

Abstract: The Painlevé Property has proved to be a fruitful tool when it comes to identifying the integrability of nonlinear PDEs. The combination of this technique with the so-called singular manifold method offers an ideal framework to approach nonlinear integrable systems: it provides a systematic methodology to obtain the associated spectral problem, as well as a recursive procedure to determine soliton-like solutions. In this talk, we review the main characteristics of this setting, with applications on several examples related to Nonlinear Schrödinger equations, in which solutions as solitons and lumps are thoroughly discussed.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Claudio Muñoz

 

 

Modeling chemotaxis with a nonlinear Schrödinger equation: solitary waves

Speaker: Miguel Alejo

University of Córdoba, Spain

Date: May 31, 2022 at 12 Santiago time

Abstract: In this talk I will show how chemotaxis can be modeled by using a nonlinear Schrödinger equation with  well-known quantum dissipative mechanisms. This relation will allow us to find explicit new solitary wave solutions.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Claudio Muñoz

 

 

Fractional Sobolev regularity for fully nonlinear elliptic equations

Speaker: Makson Santos

Instituto Superior Técnico, Lisbon

Date: May 24, 2022 at 12 Santiago time

Abstract: We study high-order fractional Sobolev regularity for fully nonlinear, uniformly elliptic equations, in the presence of unbounded source terms. Our techniques are based on touching the solution with C1,α cone-like functions to produce a decay rate of the measure of certain sets.

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Gabrielle Nornberg

YouTube video (in English)

 

Una nueva visión del Laplaciano fraccionario vía redes neuronales profundas

Speaker: Nicolás Valenzuela

DIM – Universidad de Chile

Date:  May 10, 2022 at 12 Santiago time

Venue: Online via Zoom / Sala de seminarios John Von Neumann, Beauchef 851,  piso 7
Chair: Jessica Trespalacios

Abstract: El Laplaciano fraccionario ha sido fuertemente estudiado durante las últimas décadas. En esta charla presentamos un nuevo enfoque al problema de Dirichlet asociado, usando técnicas recientes de aprendizaje profundo. En efecto, últimamente se ha demostrado que las soluciones aciertas ecuaciones en derivadas parciales se pueden representar de manera estocástica, y aproximar dicha representación mediante redes neuronales profundas, superando la llamada maldición de la dimensionalidad. Entre estas ecuaciones se encuentran las de tipo parabólicas sobre el espacio R^d, y las de tipo elípticas sobre un dominio acotado.

 

 

Nonlocal truncated Laplacians: representation formulas and Liouville results

Speaker: Giulio Galise 

La Sapienza Università di Roma, Italy

Date:  May 3, 2022 at 12 Santiago time

Venue: Online via Zoom / Sala de seminarios DIM, Beauchef 851, piso 5
Chair: Erwin Topp

Abstract: We consider some nonlinear extremal integral operators that approximate the, so called, degenerate truncated Laplacians. For these operators we obtain representation formulas that lead to the construction of “fundamental solutions” and  to Liouville type results. Differences with respect to both the local case and the uniformly elliptic framework will be emphasized.