# 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.

Organizadores: Pilar Herreros y Tai Nguyen

2017-06-06
16:00hrs.
Pilar Herreros. PUC
Mediatrices en Superficies
Sala 2, Facultad de Matemática
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
Sala 2, Facultad de Matemática
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
Sala 2, Facultad de Matemática
Abstract:
I will present a joint work with Apala Majumdar and Adriano Pisante where we study the Nobel-winner Landau-de Gennes functional for nematic liquid crystals. We identify the location of defects in the low temperature limit and show the coexistence of biaxial and negative uniaxial points around each defect. We also estimate the size of the biaxial regions.
2017-05-09
16:00hrs.
Suspendido. PUC
Suspendido
Sala 2, Facultad de Matemáticas UC
2017-05-02
16:00hrs.
Hanne Van Den Bosch. PUC
Spectrum Of Dirac Operators Describing Graphene Quantum Dots
Sala 2, Facultad de Matemáticas UC
Abstract:
Low energy electronic excitations in graphene, a two-dimensional lattice of carbon atoms, are described effectively by a two–dimensional Dirac operator. For a bounded flake of graphene (a quantum dot), the choice of boundary conditions determines various properties of the spectrum. Several of these choices appear in the physics literature on graphene. For a simply connected flake and a family of boundary conditions, we obtain an explicit lower bound on the spectral gap around zero. We can also study the effect of the boundary conditions on eigenvalue sums in the semiclassical limit. This is joint work with Rafael Benguria, Søren Fournais and Edgardo Stockmeyer.
2017-04-25
16:00 hrs.
Carmen Cortázar. PUC
Large Time Behavior Of Porous Medium Solutions In Exterior Domains
Sala 2, Facultad de Matemáticas UC
Abstract:
Let  $\mathcal{H}\subset \mathbb{R}^N$ be a non-empty bounded open set. We consider the porous medium equation in the complement of  $\mathcal{H}$ ,  with zero Dirichlet data on  its boundary and nonnegative compactly supported integrable initial data.

Kamin and Vázquez, in 1991, studied the large time behavior of solutions of such  problem  in space dimension 1.    Gilding and Goncerzewicz, in 2007, studied this same problem  dimension 2.  However, their result does not say much about the behavior when the points  are in the so called near field scale. In particular, it does not give a sharp decay rate, neither a nontrivial asymptotic profile, on compact sets.
In this paper we characterize the large time behavior in such scale, thus completing their results.

This a Joint work with Fernando Quiros ( Universidad Autonoma de Madrid, Spain) and Noemí Wolanski ( Universidad de Buenos Aires, Argentina).

2017-04-18
16:00hrs.
Abraham Solar. PUC
Stability Of Semi-Wavefronts For Delayed Reaction-Diffusion Equations
Sala 2, Facultad de Matemáticas UC
Abstract:
Semi-wavefronts are bounded positive solutions of delayed
reaction-diffusion equations such that its shape is not changed in the
time and they move with constant speed en the time. In this talk I
will show the most important results about stability of this solutions
and how they determinate the propagation speed of the a broad class of
solutions.
2017-04-04
16:00hrs.
Erwan Hingant. Universidad del Bio-Bio
The Stochastic Becker-Döring System
Sala 2, Facultad de Matemáticas UC
Abstract:
The Becker-Döring equations might be "one of the simplest kinetic model to describe a number of issues in the dynamics of fase transitions", Penrose (1989). This model describes the evolution of the concentration of clusters (or aggregates) according to their size. The rules are simple, a cluster of size $i$ may encounter a particle (cluster of size $1$) to form a new one of size $i+1$. Conversely, a cluster of size $i$ could release a particle leading to a cluster of size $i-1$. In this talk we will present the stochastic version of this rules when the system consists in a finite number of particles, namely a pure jump Markov process on a finite state space. And we will discuss about some results and issues around the law of large number associate to this problem.

Ref.: E. Hingant and R. Yvinec, Deterministic and Stochastic Becker-Döring equations: Past and Recent Mathematical Developments, Preprint arXiv:1609.00697, 2016.
2017-03-28
16:00hrs.
Sophia Jahns . Tübingen University, Germany
Trapped Light In Stationary Spacetimes
Sala 2, Facultad de Matemáticas UC
Abstract:
Light can circle a massive object (like a black hole or a neutron star) at a "fixed distance", or, more generally, circle the object without falling in or escaping to infinity. This phenomenon is called trapping of light and well understood in static, asymptotically flat (AF) spacetimes. If we drop the requirement of staticity, similar behavior of light is known, but there is no definiton of trapping available. We present some known results about trapping  of light in static AF spacetimes. Using the Kerr spacetime as a model, we then show how trapping can be better understood in the framework of phase space and work towards a definition for photon regions in stationary AF spacetimes.
2017-03-21
16:00hrs.
Mauricio Bogoya. Universidad Nacional de Colombia, Bogota
A Non-Local Diffusion Coupled System Equations In a Bounded Domain
Sala 2, Facultad de Matemáticas UC
2017-03-14
16:00hrs.
Marcos de la Oliva. Universidad Autónoma de Madrid
Relaxation Of a Model For Nematic Elastomers
Sala 2, Facultad de Matemáticas UC
Abstract:
The direct method of the calculus of variations to find minimizers is based on compactness and lower semicontinuity of the energy functional. In the absence of lower semicontinuity, one option is to find the relaxation, i.e., the largest lower semicontinuous functional below a given one. In nonlinear elasticity, computing the relaxation is difficult beacuse of the non-standard growth conditions. In this talk we show that the relaxation for a model in nonlinear elasticity is given by the quasiconvexification of the integrand. We also propose a model for nematic elastomers (a kind of liquid crystals) in which the energy has a part in the reference configuration and a part in the deformed configuration. We show again that the relaxation is given by the quasiconvexification.
2017-01-11
15:00hrs.
Marie-Françoise Bidaut-Véron. Université François Rabelais, Tours, France
A Priori Estimates And Ground States Of Solutions Of An Emden-Fowler Equation With Gradient
Sala 2, Facultad de Matemáticas UC
2017-01-11
16:00hrs.
Laurent Véron. Université François Rabelais, Tours, France
Initial Trace Of Positive Solutions Of Some Nonlinear Diffusion Equations
Sala 2, Facultad de Matemáticas UC
2016-12-06
16:00hrs.
Phan Thanh Nam. Masaryk University, Czech Republic
How Many Electrons That a Nucleus Can Bind?
Sala 2, Facultad de Matemáticas UC
Abstract:
All physicists and chemists know that a neutral atom can bind at most one or two extra electrons. However, justifying this fact rigorously from Schroedinger equation is a long standing open problem, often referred to as the ionization conjecture. I will discuss some recent progress on this problem.
2016-11-29
16:00hrs.
Martin Chuaqui. Pontificia Universidad Católica de Chile
Discos Minimales Embedidos en R^3
Sala 2, Facultad de Matemáticas UC
Abstract:
Se muestra una condicion general que asegura que la parametrizacion de Weierstarss-Enneper de una superficie minima sea inyectiva. Como corolario se deduce un teorema expresado en terminos de la curvatura Gaussiana y el diametro, para que un disco minimal convexo este embedido. El resultado es optimo.
2016-11-22
16:00hrs.
Tai Nguyen. Pontificia Universidad Católica de Chile
Existence And Uniqueness Of Positive Weak Solutions Of Quasilinear Elliptic Equations
Sala 2, Facultad de Matemáticas UC
Abstract:

We study the following quasilinear elliptic equation

$$-\Delta_p u + a(x) u^{p-1} + b(x)g(u)=0 \quad \text{in } \mathbb{R}^N \quad \quad \quad \text{(E)}$$

where $p>1$, $a,b \in L^\infty(\mathbb{R}^N)$, $b\geq 0,b\not\equiv0$ and $g \geq 0$. Under some conditions on $a$ and $g$, we provide a criterion in terms of \textit{generalized principal eigenvalues} for the existence/non-existence of positive weak solutions of (E). We also discuss the uniqueness of positive weak solutions of (E).

2016-11-15
16:00hrs.
Nikola Kamburov. Pontificia Universidad Católica de Chile
The Space Of One-Phase Free Boundary Solutions In The Plane
Sala 2, Facultad de Matemáticas UC
Abstract:
In joint work with David Jerison we study the compactness of the space of solutions to the one-phase free boundary problem in the disk, whose positive phase is of a fixed genus. We describe the local structure of the free boundary and obtain rigidity estimates on its shape. Via a correspondence due to Traizet, our results are direct counterparts to theorems by Colding and Minicozzi for minimal surfaces.
2016-10-18
16:00hrs.
Chulkwang Kwak. Pontificia Universidad Católica de Chile
Fifth-Order Modified Kdv Equation
Sala 2, Facultad de Matemáticas UC
Abstract:
In this talk, I will briefly introduce the basic low regularity well-posedness theory of dispersive equations, we will discuss about the Cauchy problem of the (integrable) fifth-order modified Korteweg-de Vries (modified KdV) equation under the periodic boundary condition. In particular, we will observe the non-trivial resonant phenomena of the Fourier coefficients of the solution and strong high-low interactions in nonlinear interactions. Precisely, non-trivial cubic and quintic resonant interactions do not admit that the nonlinear solution behave as a linear solution, so considering the integrable equation is very useful to study the low regularity Cauchy problem. Moreover, due to the lack of dispersive effect, we encounter the difficulty to control the nonlinearity via the standard way, so I will introduce the short time function space to defeat this enemy. In conclusion, we will prove the local well-posedness of the fifth-order modified KdV in $H^s$ for $s > 2$, via the standard energy method, and it is the first local well-posedness result of the periodic fifth-order KdV equation.
2016-10-11
16:00hrs.
Marta García-Huidobro. Pontificia Universidad Católica de Chile
Singularidades en la Frontera de Soluciones Positivas de $-\delta_P U+|\nabla U|^q=0,\quad X\in\omega\subset\mathbb{R}^n,\quad 0 Sala 2, Facultad de Matemáticas UC Abstract: Estudiamos el comportamiento en la frontera de soluciones positivas de$-\Delta_p u+|\nabla u|^q=0$,$\quad x\in\Omega\subset\mathbb R^N$,$\quad 0<p-1<q<p$Mostramos la existencia de un exponente crítico$q_*<p$de manera que si$p-1<q<q_*$, existen soluciones positivas de esta ecuación que tienen una singularidad aislada en la frontera de$\Omega$, y que si$q_*\le q<p$cualquier singularidad aislada en$\partial\Omega\$ es removible.
2016-10-06
16:00hrs.
Pablo D. Ochoa. Universidad Nacional de Cuyo-Conicet. Argentina
Soluciones Viscosas en Grupos de Carnot
Sala 2, Facultad de Matemáticas UC
Abstract:
En esta charla, discutiremos algunos aspectos esenciales de la teora de soluciones viscosas en grupos de Carnot. Estos aspectos incluyen principios de comparacion, unicidad, existencia de soluciones, estabilidad y regularidad. Comenzaremos con una introduccion a los grupos de Carnot, mostrando aplicaciones de su estructura y su genesis a partir de aproximaciones Riemannianas convenientes. De niremos la nocion de solucion viscosa para un amplio rango de ecuaciones diferenciales, mostrando diversas tecnicas para obtener soluciones (Metodo de Perron, esquemas de aproximacion, etc). En cuanto a unicidad de soluciones,
exhibiremos principios del maximo subRiemannianos necesarios para probar principios de comparacion de soluciones, y pon ende unicidad. Comentarios relacionados a regularidad y estabilidad de soluciones,
y problemas abiertos en la teora, seran tambien discutidos.