Seminario de Análisis y Geometría

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:

2016-09-27
16:00 hrs.

Seunghyeok Kim. Pontificia Universidad Católica de Chile
Qualitative properties of multi-bubble solutions for elliptic equations with slightly subcritical exponents
Sala 2, Facultad de Matemáticas UC
Abstract:
The objective of this talk is to deliver qualitative characteristics of solutions of the Lane-Emden equations with slightly subcritical exponents which have multiple blow-up points. By examining the linear problem at ach multi-bubble solution, we will observe that its Morse index can be described in terms of the number of negative eigenvalues of a matrix whose component consists of a combination of the second derivatives of Green’s function and the Robin function. This is a joint work with Woocheol Choi (KIAS) and Ki-Ahm Lee (Seoul National University).

2016-09-13
16:00hrs.

Rémy Rodiac. Pontificia Universidad Católica de Chile
Regularity of limiting vorticities of the Ginzburg-Landau equations
Sala 2, Facultad de Matemáticas UC
Abstract:
Limiting vorticities in the Ginzburg-Landau theory are Radon measures that describe the location of the vortices of the model when the parameter $\epsilon$ is small. Sandier-Serfaty gave some critical conditions that are satisfied by such limiting vorticities. One can then ask about the regularity of these objects. In the case withouth magnetic field the problem is equivalent to the study of stationary harmonic functions whose Laplacian are Radon measures. We will prove that such functions can be written locally as the absolute value of an harmonic function. Thus its Laplacian is concentrated (locally) on the zero set of an harmonic function (a union of smooth curves). We will also give some results in the case with magnetic field.

2016-08-30
16:00hrs.

Mariel Sáez. Pontificia Universidad Católica de Chile
Fractional Mena Curvature Flow
Sala 2 Facultad de Matemáticas UC
Abstract:
In this talk I will discuss a fractional analog to the classical mean curvature flow. Namely, we consider the evolution of surfaces with normal speed equal to the fractional mean curvature and analyze their behavior under suitable assumptions. I will discuss in more depth the evolution of graphical hyper-surfaces, which is an important model in the local case.

This is joint work with Enrico Valdinoci