# 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.

2019-04-16
16:00hrs.
Duván Henao. Pontificia Universidad Católica de Chile
Aplicando la geometría diferencial para comprender la estructura de los cristales líquidos
sala 2, Facultad de Matemáticas, PUC
Abstract:
Veremos como el teorema de Liouville y los teoremas de Schoen-Uhlenbeck nos permiten demostrar que a bajas temperaturas los defectos de los minimizadores del funcional de Landau-de Gennes son necesariamente biaxiales. (En particular, a bajas temperaturas es falso que sus frustraciones topológicas las resuelvan derritiéndose, como se asume comúnmente.)
2019-04-02
16:00hrshrs.
Ariane Trescases. Cnrs Imt Toulouse
Quaternions in collective motion
Abstract:
We present a model for multi-agent dynamics based on rigid alignment. Each agent is described by its position and body attitude: it travels at a constant speed while trying to coordinate its solid orientation with the solid orientation of the neighboring agents. The body orientations are represented by unitary quaternions. We first introduce an individual based model in the spirit of the Vicsek model, enhanced with the body orientation dynamics. We then derive the corresponding kinetic model. From there we compute the hydrodynamical limit, leading to a self-organized hydrodynamical system based on quaternions.
2019-03-26
16:00hrs.
Satoshi Tanaka. Okayama University of Science, Japan
Uniqueness of positive radial solutions of superlinear elliptic equations in annuli
Sala 2
Abstract:
This is a joint work with Naoki Shioji (Yokohama National University) and

The Dirichlet problem
\begin{equation*}
\left\{
\begin{array}{cl}
\Delta u + f(u) =0 &  \mbox{in} \ x \in A, \\[1ex]
u=0 & \mbox{on} \ \partial A
\end{array}
\right.
\end{equation*}
is considered, where $A:=\{x\in {\bf R}^N : a< |x| <b$,\  $N \in {\bf N}$, $N \ge 2$, $0<a<b<\infty$,
$f \in C^1[0,\infty)$, $f(u)>0$ and $uf'(u) \ge f(u)$ for $u>0$.
Hence the boundary value problem
$$u'' + \frac{N-1}{r} u' + f(u) = 0, \quad r \in (a,b); \qquad u(a) = u(b) = 0$$
is considered.
Uniqueness results of positive solutions are shown.
2019-03-19
16:00hrs.
Azahara de la Torre Pedraza. University of Freiburg
On higher dimensional singularities for the fractional Yamabe problem
Sala 2
Abstract:
We consider the problem of constructing solutions to the fractional Yamabe problem that are singular at a given smooth sub-manifold, for which we establish the classical gluing method of Mazzeo and Pacard for the scalar curvature in the fractional setting. This proof is based on the analysis of the model linearized operator, which amounts to the study of a fractional order ODE,
and thus our main contribution here is the development of new methods coming from conformal geometry and scattering theory for the study of non-local ODEs. Note, however, that no traditional phase-plane analysis is available here. Instead, first, we provide a rigorous construction of radial fast-decaying solutions by a blow-up argument and a bifurcation method. Second, we use conformal geometry to rewrite this non-local ODE, giving a hint of what a non-local phase-plane analysis should be. Third, for the linear theory, we use complex analysis and some non-Euclidean harmonic analysis to  examine a fractional Schrödinger equation with a Hardy type critical potential. We construct its Green's function, deduce Fredholm properties, and analyze its asymptotics at the singular points in the spirit of  Frobenius method. Surprisingly enough, a fractional linear ODE may still have a two-dimensional kernel as in the second order case.
2019-03-12
16:00hrs.
Andrés Larraín-Hubach. University of Dayton, Ohio
Conexiones auto-duales sobre espacios Taub-NUT
Sala 2
Abstract:

2019-01-03
16:00hrs.
Armin Schikorra . University of Pittsburgh
Self-repulsive curvature energies for curves and surfaces: regularity theory and relation to harmonic maps
Sala 2
Abstract:
I will talk about a class of curvature energies for curves, the O'Hara energies, that are nonlocal in nature. In particular, I will present an approach for regularity theory of minimizers and critical points for these curves which is based on a relation to (fractional) harmonic maps. Then I will present some results towards attempts of generalizing this idea to surfaces.
2018-12-19
14:00hrs.
Frank Morgan . Williams College
Isoperimetric Problems with Density
Sala 2
Abstract:
A round soap bubble is the least-perimeter way to enclose a given volume of air. Similarly, the familiar double soap bubble that forms when two soap bubbles come together is the least-perimeter way to enclose and separate two given volumes of air. What if you give space a radial density that weights both perimeter and volume? The presentation will include open questions and recent results, some by undergraduates. Students welcome.

2018-11-20
16:00hrs.
Natham Aquirre Quiñonez. PUC
Funciones p-armónicas con condiciones de Neumann en el borde que involucran medidas
Sala 2
Abstract:
En esta presentación discutiré el problema de encontrar funciones p-armónicas en el semi-espacio superior con condiciones de Neumann en el borde de tipo no lineal y con medidas. Para ello introduciré el concepto de soluciones renormalizadas en dominios acotados y explicaré como usar esta teoría para obtener soluciones de nuestro problema. La idea principal es usar estabilidad local y la simetría del dominio (y el operador). Luego aplicaremos estas ideas para obtener resultados de existencia para diferentes tipos de no linealidades. Las técnicas a utilizar también nos permitirán establecer algunos resultados de no existencia y de eliminación de singularidades.
2018-11-19
16:00hrs.
Natham Aquirre Quiñonez. P. Universidad Católica de Chile
Funciones p-armónicas con condiciones de Neumann en el borde que involucran medidas
Sala 2
Abstract:
En esta presentación discutiré el problema de encontrar funciones p-armónicas en el semi-espacio superior con condiciones de Neumann en el borde de tipo no lineal y con medidas. Para ello introduciré el concepto de soluciones renormalizadas en dominios acotados y explicaré como usar esta teoría para obtener soluciones de nuestro problema. La idea principal es usar estabilidad local y la simetría del dominio (y el operador). Luego aplicaremos estas ideas para obtener resultados de existencia para diferentes tipos de no linealidades. Las técnicas a utilizar también nos permitirán establecer algunos resultados de no existencia y de eliminación de singularidades.
2018-11-13
16:00hrs.
Jose Gabriel Torres. Pontificia Universidad Católica de Chile
Solitones por traslación de una familia de flujos no degenerados
Sala 2
Abstract:
Los flujos por curvatura extrínseca estudian la evolución de superficies en ambientes (semi-)riemannianos, donde la velocidad del flujo está dada por una función en las curvaturas principales sobre la evolución de la superficie. Dicha evolución esta determinada por una ecuación de tipo parabólico no lineal. En esta charla me concentraré en una familia de flujos donde la función velocidad está dada por un cociente entre polinomios simétricos elementales con especial énfasis en soluciones auto-similares donde esta que actúa por traslación. Discutiré resultados conocidos y futuras lineas de trabajo en este contexto.
2018-10-30
16:00hrs hrs.
Michal Kowalczyk. Departamento de Ingeniería Matemática Universidad de Chile
Maximal solution of the Liouville equation in doubly connected domains
Sala 2
Abstract:

In this talk I will discuss a new existence result for the widely  studied Liouville problem $\Delta u+\lambda^2 e^{\,u}=0$ in a bounded, two dimensional, doubly connected  domain with Dirichlet boundary conditions. I will show that for a sequence of   $\lambda_n\to 0$ this equation has solutions that blow-up in  in the whole domain.  Profiles of the blowing-up solutions are related to a free boundary problem which gives a solution to an optimal partition problem for the given domain. I will also describe the role of  the free boundary problem in  other classical equations such as the mean field  model or the prescribed Gaussian curvature equation.

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