# Seminario de Análisis y Geometría

Los seminarios de Análisis y Geometría se llevan a cabo los días martes a las 16:00 en la sala 2 de la Facultad de Matemáticas, Pontificia Universidad Católica de Chile.

2018-10-23
16:00hrs.
Mircea Petrache. Pontificia Universidad Católica de Chile
Desigualdad isoperimétrica en grafos regulares y formas de unos cristales
Sala 2
Abstract:
Voy a presentar unas técnica clásica para demostrar desigualdades isoperimétricas en R^n respecto de varias nociones de perímetro.
Después vamos a ver cómo las mismas técnicas se pueden tal vez transferir a problemas combinatorios en unos grafos periódicos, y permiten predicir la forma de unos cristales.

2018-10-09
15:30hrs.
Mariel Sáez. Pontificia Universidad Católica de Chile
Sobre la unicidad del flujo de curvatura media en graficos de funciones
Sala 2
Abstract:

Voy a discutir resultados recientes con P. Daskalopoulos de condiciones suficientes para la unicidad de soluciones de la ecuación de evolución asociada a gráficos de funciones evolucionando por flujo de curvatura media. Compararemos estos resultados con el comportamiento de soluciones clásicas a la ecuación del calor.

2018-09-25
16:00hrs.
Nikola Kamburov. Pontificia Universidad Católica de Chile
On positive solutions of the Lane-Emden equation in the plane
Sala 2 FMAT
Abstract:
We prove that positive solutions of the Lane-Emden equation in a two-dimensional smooth bounded domain are uniformly bounded for all large exponents. Recent work of De Marchis, Grossi, Ianni and Pacella
provides a fairly complete asymptotic description of such solutions, under a certain integral bound condition. Furthermore, they establish the asymptotic uniqueness of positive solutions satisfying that bound in convex planar domains. We remove this condition by showing that the bound is always satisfied in star-shaped domains.

This is joint work with Boyan Sirakov (PUC-Rio).
2018-09-04
16:00hrs.
Matías Courdurier. Pontificia Universidad Católica de Chile
Construction of Solutions for some Localized Nonlinear Schrodinger Equations
Sala 2
Abstract:
In this talk we will present the constructions of solution of the following reduced non-linear Schrodinger equation: -u''+V(x)h'(|u|^2)u = w u, where V(x)=1 for |x|<1 and 0 otherwise, and where h' is any continuous
strictly increasing function. Reduced non-linear Schrodinger equation are important as mean-field approximations of quantum systems and the constructed solutions characterize bound-states of the dynamic version of the equation.
2018-08-21
16:00hrs.
Dependence on the domain geometry of the Hölder estimates for the Neumann problem applied to void coalescence
Sala 2
Abstract:
We study the dependence on the domain geometry of the Hölder estimates for the Neumann problem, from which, we obtain estimates for a free boundary problem that arise in a nonlinear elasticity problem, more precisely, we start with an incompressible elastic body subject to a multiaxial traction, the position of a fixed number of cavitation points and the final volume of those cavities after the deformation. We want to know under which conditions one can ensure that there is no coalescence. This talk is based on a joint work with Duvan Henao.
2018-08-14
16:00hrs.
Mathew Langford. University of Tennessee Knoxville
Ancient and translating solutions of mean curvature flow
Sala 2
Abstract:
An important result of X.-J. Wang states that a convex ancient solution of mean curvature flow is either entire (sweeps out all of space) or lies in a slab (the region between two fixed parallel hyperplanes). We will describe recent results on the existence and classification of convex ancient solutions and convex translating solutions of mean curvature flow which lie in slab regions, highlighting the connection between the two. All work is joint with Theodora Bourni and Giuseppe Tinaglia.
2018-08-07
16:00hrs.
Karen Corrales. International Centre for Theoretical Physics (Ictp)
Superficies con curvatura media constante en Variedades Riemannianas
Sala 2
Abstract:
Uno de los problemas fundamentales en geometría diferencial es estudiar las superficies con curvatura media constante (CMC) en diversas variedades Riemannianas. Conocer bajo que condiciones topológicas u otras estas existen, son únicas o son soluciones del problema isoperimetrico. Por ejemplo, la clasificación de superficies compactas y embedidas con curvatura media constante en Rn ha sido completamente estudiada por Alexandrov, pero si cambiamos el espacio ambiente o bien, consideramos superficies no necesariamente compactas o embedidas, el panorama cambia completamente.

En esta charla describire algunos resultados recientes sobre la existencia de superficies con CMC en espacios tales como Schwarzschild, anti deSitter o variedades no compactas que sean asintoticamente planas o hiperbólicas. Finalmente, discutiremos un caso hiperbolico particular, variedades asintóticas a cúspides. Este es un trabajo conjunto con Claudio Arezzo.
2018-07-31
16:00hrs.
Bianca Stroffolini. University of Naples
Lipschitz truncations versus regularity
Sala 2
Abstract:
A fundamental important and open problem in the Calculus of Variations is the one of identifying classes of functionals for which everywhere H ?older regularity, or even just continuity, of minimizers occurs. The same problem arises for solutions to systems. So far, the only structure preventing the formation of singularities for minimizers is the one first identified in the fundamental work of K. Uhlenbeck in the 70’s. It prescribes that the dependence of the gradient must occur directly via the modulus |Du|, which makes, in a sense, the functional “less anisotropic” and rules out singularities of minima. The dependance of the gradient was of polynomial type.

Coming to a general vectorial case, partial regularity comes into play. Partial regularity asserts the pointwise regularity of solutions/minimizers, in an open subset whose complement is negligible. The proof of partial regularity compares the original solution u in a ball with the solution h in the same ball of the linearized elliptic system with constant coefficients. The comparison map h is smooth, and enjoys good a-priori estimates. The idea is to establish conditions in order to let u inherit the regularity estimates of h; for example, u and h should be close enough to each other in some integral sense. This is achieved if the original system is “close enough” to the linearized one. Such a linearization idea finds its origins in Geometric Measure Theory, and more precisely in the pioneering work of De Giorgi on minimal surfaces, and of Almgren for minimizing varifolds, and was first implemented by Morrey and Giusti & Miranda for the case of quasilinear systems. Hildebrandt & Kaul & Widman studied partial regularity in the setting of harmonic mappings and related elliptic systems. Another technique is  the “A-approximation method”, once again first introduced in the setting of Geometric Measure Theory by Duzaar & Steffen and applied to partial regularity for elliptic systems and functionals by Duzaar & Grotowski. This method re-exploits the original ideas that De Giorgi introduced in his treatment of minimal surfaces. The linearization is implemented via a suitable variant, for systems with constant coefficients, of the classical “Harmonic approximation lemma” of De Giorgi.

Our revisitation of this approximation is based on Lipschitz approximation of Sobolev functions that was first introduced by Acerbi & Fusco and then revisited by Diening, Malék and Steinhauer1

I will present some variants of this method and applications to regularity for degenerate systems of general growth. I will present also a refinement of the method of Parabolic Lipschitz truncation due to Kinnunen and Lewis, based on a suitable Lipschitz truncation adapted to the parabolic setting. This new approach preserves boundary data and is used to prove the p-caloric approximation. It is the generalization to the parabolic setting of De Giorgi approximation regularity method.
2018-04-24
16:00hrs.
Kirill Cherednichenko. University of Bath
Dispersive effective behaviour of high-contrast periodic media
Abstract:
I will discuss my recent work with Y. Ershova and A. Kiselev, demonstrating that spectral problems for quantum graphs with rapidly oscillating high-contrast weights are asymptotically equivalent to "homogenised'' models with energy-dependent interface conditions. We show that these asymptotically equivalent models are directly related (in the sense of Schur-Frobenius duality) to models for time-dispersive media, which in the time domain involve memory, and we characterise the corresponding time convolution kernels explicitly.
2018-04-10
16:00hrs.
Daniel Alvarez-Gavela. Stanford University
The simplification of caustics
Abstract:
When light is reflected or refracted by a curved object, it accumulates on a caustic curve which typically has isolated semi-cubical cusp singularities. We will describe an h-principle technique that allows for the simplification of more complicated wavefront singularities into superpositions of the familiar semi-cubic cusp.
2017-11-28
16:00hrs.
Peter Veerman. Portland State University
Strange Convex Sets
Abstract:
Given a closed convex set $\Omega \in R^n$ , the metric projection of a given point $x ∈ R^n$ is given by the unique point $\Pi(x) ∈ \Omega$ that minimizes the (Euclidean) distance $\{|y − x| | y ∈ \Omega\}$ between $\Omega$ and $x$. Most mathematicians tend to think of convex sets in R n as very tame objects. It is therefore surprising that it is easy to construct a compact convex set $\omega$ in $R^2$ with the following strange property [Shapiro, 1994]: There is a point $x \notin \Omega$ and a vector $v$ such that the directional derivative
$$\lim_{t\to 0}\frac{ \Pi(x+vt)-?pi(x)}{t}$$
fails to exist. Note that for example convex polygons are not strange in this sense.

We revisit and modify that construction to obtain a convex curve in $R^2$ that is $C^{1,1}$ or differentiable with Lipschitz derivative, and that this curve bounds a convex set that has the property that the directional derivative of the projection is not defined. We also show how this construction can be made $C^n$ for n ≥ 2 except at a single point, and such that directional differentiability still fails.
2017-11-14
16:00hrs.
Eiji Yanagida. Tokyo Institute of Technology
Dynamics of Interfaces in The Fisher-Kpp Equation for Slowly Varying Initial Data
2017-10-31
16:00hrs.
Marie-Françoise Bidaut-Véron. Université de Tours, France
A Priori Estimates and Initial Trace for a Hamilton-Jacobi Equation With Gradient Absorption Terms
2017-10-24
16:00hrs.
Laurent Véron. Université François-Rabelais, Tours, France
Separable P-Harmonic Functions in a Cone, $1 < P \leq \infty$
2017-10-03
16:00hrs.
Ignacio Guerra . Usach
Multiplicty of Solutions for An Elliptic Equation With a Singular Nonlinearity and a Gradient Term
2017-09-05
16:00hrs.
Leonelo Iturriaga. Universidad Técnica Federico Santa María
Teoremas de Liouville para soluciones radiales de ecuaciones elípticas semilinales
Abstract:
En esta charla presentaremos algunos teoremas de Liouville para soluciones positivas radialmente simétricas de la ecuación
$$-\Delta u=f(u) \ \text{ en } \mathbb{R}^n$$
donde $f$ es una función continua en $[0,+\infty)$ que es positiva en $(0,+\infty)$. Nuestro enfoque nos permite considerar problemas mas generales, donde la no linealidad puede ser multiplicada por un peso radialmente simétrico y/o el Laplaciano es reemplazado por el $p$-Laplaciano, $1 < p < N$.
2017-06-06
16:00hrs.
Pilar Herreros. PUC
Mediatrices en superficies
Abstract:
Dados dos puntos en una superficie Riemanniana llamamos mediatriz al conjunto de puntos equidistante a ambos puntos dados.

Hablaremos de la estructura que tiene este conjunto y de su regularidad,  en particular de que sus vertices son radialmente linearizables.

2017-05-30
16:00hrs.
Monica Musso. PUC
Existence, Compactness and Non Compactness for fractional Yamabe Problem
Abstract:
Let $(X^{n+1}, g^+)$ be an $(n+1)$-dimensional asymptotically hyperbolic manifold with a conformal infinity $(M^n, [h])$. The fractional Yamabe problem consists in finding a metric in the conformal class $[h]$ whose fractional scalar curvature is constant.

In this talk, I will present some recent results concerning existence of solutions to the fractional Yamabe problem,  and also properties of compactness and non compactness of its solution set, in comparison with what is known in the classical case.

These results are in collaboration with Seunghyeok Kim and Juncheng Wei.
2017-05-23
16:00hrs.
Duvan Henao. PUC
Biaxiality in liquid crystals at low temperatures