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.

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.

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, $1 < p < N$.

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.  

 

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).

 

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.