Speaker: David Stolnicki

La Sapienza University of Rome, Italy

Date: Tuesday, October 27, 2022 at 2 p.m. Santiago time

Abstract: In joint work with F. Pacella, we study the existence and asymptotic behavior of radial positive solutions of some fully nonlinear equations involving Pucci’s extremal operators in dimension two and higher. In particular we prove the existence of a positive solution of a fully nonlinear version of the Liouville equation in the plane. Moreover, for the (negative) Pucci P^- operator, we show the existence of a critical exponent and give bounds for it. The same technique is then applied in higher dimensions to improve the previously known bounds.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Gabrielle Nornberg

Speaker: Satoshi Tanaka

Tohoku University, Japan

Date: Tuesday, October 25, 2022 at 12 Santiago time

Abstract: In this talk, we consider the Dirichlet problem for the scalar-field equation in a large annulus in the three-dimensional sphere. We obtain the existence, uniqueness, and multiplicity results of the positive solutions depending only on the latitude. This is joint work with Noaki Shioji (Yokohama National University) and Kohtaro Watanabe (National Defense Academy).

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Hanne Van Den Bosch

Speaker: Gunther Uhlmann

University of Washington

Date: Tuesday, October 6, 2022 at 2 p.m. Santiago time

Abstract: In inverse scattering ione attempts to find the properties of a medium
by making remote observations. It has applications in physics,
geophysics, medical imaging, non-destructive evaluation of materials.
Radar and sonar are examples of inverse scattering methods that are
used routinely nowadays. In this case we consider the inverse problem
of determining the nonlinearity for  critical semilinear wave
equations. This is joint work with A, Sa Barreto and Y. Wang.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Claudio Muñoz

Speaker: Pilar Herreros

Pontificia Universidad Católica de Chile

Date: Tuesday, October 4, 2022 at 12 Santiago time

Abstract: Estudiaremos las soluciones radialmente simétricas del problema $$ \Delta u+f(u)=0,\quad x\in \mathbb{R}^N, N> 2,   \lim_{|x|\to \infty} u(x)=0. $$ Veremos que podemos generar nuevas soluciones del problema si introducimos cambios bruscos en la magnitud de la función f. Usando esto construiremos funciones f, definidas por partes, tales que el problema tiene cualquier número pre-determinado de soluciones.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Gabrielle Nornberg

Speaker: Claudia García

Universitat de Barcelona, Spain

Date: Tuesday, September 27, 2022 at 12 Santiago time

Abstract: The aim of this talk is to study time periodic solutions for 3D inviscid quasigeostrophic model. We show the existence of non trivial simply-connected rotating patches by suitable perturbation of stationary solutions given by generic revolution shapes around the vertical axis. The construction of those special solutions are done through bifurcation theory. In general, the spectral problem is very delicate and strongly depends on the shape of the initial stationary solutions. More specifically, the spectral study can be related to an eigenvalue problem of a self-adjoint compact operator and we are able to implement the bifurcation only from the largest eigenvalues of such operator which are simple. At the end of the talk, we will speak also about the doubly-connected case. This is a joint work with T. Hmidi and J. Mateu.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Claudio Muñoz

Speaker: David Poyato

University of Granada, Spain

Date: Tuesday, September 20, 2022 at 12 Santiago time

Abstract: The Fisher infinitesimal model is a widely used statistical model in quantitative genetics that describes the propagation of a quantitative trait along generations of a population subjected to sexual reproduction. Recently, this model has pulled the attention of the mathematical community and some integro-differential equations have been proposed to study the precise dynamics of traits under the coupled effect of sexual reproduction and natural selection. Whilst some partial results have already been obtained, the complete understanding of the long-time behavior is essentially unknown when selection is not necessarily weak. In this talk, I will introduce a simplified time-discrete version inspired in the previous time-continuous models, and I will present two novel results on the long-time behavior of solutions to such a model. First, when selection has quadratic shape, we find quantitative convergence rates toward a unique equilibrium for generic initial data. Second, when selection is any strongly convex function, we recover similar convergence rates toward a locally-unique equilibrium for initial data sufficiently close to such an equilibrium. Our method of proof relies on a novel Caffarelli-type maximum principle for the Monge-Ampère equation, which provides a sharp contraction factor on a L^\infty version of the Fisher information. This is a joint work with Vincent Calvez, Filippo Santambrogio and Thomas Lepoutre.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Claudio Muñoz

Speaker: Rodolfo Viera 

Pontificia Universidad Católica de Chile

Date: Tuesday, September 6, 2022 at 12 Santiago time

Abstract: En 1994, Gromov preguntó si toda red separada del plano \( X\subset\mathbb{R}^2 \) (i.e, un conjunto discreto y denso de una manera uniforme) es bi-Lipschitz equivalente al lattice est\’andar \(\mathbb{Z}^2 \) (i.e si X es bi-Lipschitz rectificable). Esto fue respondido de manera negativa por Burago y Kleiner, e independientemente por McMullen. Su demostración se basa en la existencia de una función de densidad \(\rho:[0,1]^2\to\mathbb{R}\) tal que \( 0<\inf\rho<\sup\rho<\infty \) y para la cual la ecuación

$$
Jac(f)=\rho\qquad a.e
$$

no tiene solución bi-Lipschitz \( f:[0,1]^2\to\mathbb{R}^2\). En esta charla veremos algunos resultados en esta línea, por ejemplo condiciones suficientes para asegurar la rectificabilidad de una red separada como consecuencia de la existencia de soluciones bi-Lipschitz para ciertas ecuaciones que involucran un jacobiano. También intentaremos pasar por otros resultados de no-rectificabilidad bajo condiciones más débiles que bi-Lipchitz.

Venue: Online via Zoom / Sala de seminarios John Von Neumann, 7th floor, Beauchef 851
Chair: Hanne Van Den Bosch

Speaker: Erwin Topp

Universidad de Santiago, Chile

Date: Thursday, August 25, 2022 at 16 p.m. Santiago time

Abstract: In this talk I will report some multiplicity results for large solutions of fractional Hamilton-Jacobi equations posed on a bounded domain, subject to exterior Dirichlet conditions. We construct large solutions using the method of sub and supersolutions, following the classical approach of J.M. Lasry and P.L. Lions for second-order equations with subquadratic gradient growth. We identify two classes of solutions: the one coming from the natural scaling of the problem; and a one-parameter family of solutions, different from the previous, which can be formally described as a lower-order perturbation of blow-up fractional harmonic functions. Joint work with Alexander Quaas and Gonzalo Dávila (UTFSM-Chile).

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

Speaker: Nicolas Torres Escorza

Université Claude Bernard Lyon 1, France

Date: Tuesday August 23, 2022 at 12 Santiago time

Abstract: We introduce and study an extension of the classical elapsed time equation in the context of neuron populations that are described by the elapsed time since last discharge. In this extension we incorporate the elapsed since the penultimate discharge and we obtain a more complex system of integro-differential equations. For this new system we prove convergence to stationary state by means of Doeblin’s theory in the case of weak non-linearities in an appropriate functional setting, inspired by the case of the classical elapsed time equation. Moreover, we present some numerical simulations to observe how different firing rates can give different types of behaviors and to contrast them with theoretical results of both classical and extended models.

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

Speaker: Argenis Mendez 

Pontificia Universidad Católica de Valparaíso

Date: August 16, 2022 at 12 Santiago time

Abstract: In this talk, we will present some new results related to the regularity properties of the initial value problem (IVP) for the equation
\begin{equation}\label{eq1}
\left\{
\begin{array}{ll}
\partial_{t}u-\partial_{x_{1}}(-\Delta)^{\alpha/2} u+u\partial_{x_{1}}u=0, \quad 0< \alpha< 2, & \\
u(x,0)=u_{0}(x),x=(x_{1},x_{2},\dots,x_{n})\in \mathbb{R}^{n},n\geq 2,& t\in\mathbb{R}, \\
\end{array}
\right.
\end{equation}
where $(-\Delta)^{\alpha/2}$ denotes the $n-$dimensional fractional Laplacian.

In the case that \(\alpha=2,\) the equation is known as the Zakharov-Kuznetsov-(ZK) equation, Zakharov and Kuznetsov proposed it as a model to describe the propagation of ion-sound waves in magnetic fields in dimension n=3.

A property that enjoys the solutions of the ZK equation is Kato’s smoothing effect. Roughly speaking, the solution to the initial value problem is, locally, one derivative smoother (in all directions) in comparison to the initial data.

The goal of this talk is to show that despite the non-local character of the operator \((-\Delta)^{\frac{\alpha}{2}}\), the solution of the equation (IVP) is locally smoother. It becomes \(\frac{\alpha}{2}-\) smoother in all directions.

As a byproduct, we show the applicability of this result in establishing the propagation of localized regularity of the solutions of (IVP) in a suitable Sobolev space.

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