Noticias en español
Existence of infinitely many solutions for the Einstein-Lichnerowicz system
Abstract: “We will consider in this talk the Einstein-Lichnerowicz system of equations. It originates in General Relativity as a way to determine initial-data sets for the evolution problem. This system takes the form of a strongly coupled, supercritical, nonlinear system of elliptic PDEs. We will investigate its blow-up properties and show that, under some assumptions on the physics data, it possesses a non-compact family of solutions. This family of solutions will be constructed by combining toplogical methods with a finite-dimensional reduction approach; due to the non-variational...
Read MoreThe variation of maximal functions on Euclidean spaces and spheres.
Abstract: Maximal operators are central objects of study in harmonic analysis. The most classical example is the Hardy-Littlewood maximal operator (in both its centered and uncentered form) but there are also other interesting ones, for instance the ones associated to a convolution with a smooth kernel. In this introductory talk we will introduce the sub-area of harmonic analysis that studies how the variation of a maximal function behaves with respect to the initial data. We will discuss the classical results in this topic (most of them dealing with the one dimensional case) and...
Read MoreUnique continuation for some nonlinear dispersive models
ABSTRACT : This talk is concerned with unique continuation properties (UCP) for solutions to some time evolution equations. We shall study two types of UCP (1) local and (2) asymptotic at infinity. Roughly, (1) local means : If u, v are solutions of the equation which agree in an open set D, then they are identical in the whole domain of definition. Roughly, (2) asymptotic at infinity means if u, v are solutions such that ||| u(t)-v(t)|||<\Infty for t=t_1,and t=t_2, then they are identical in the whole domain of definition. The class of dispersive model to be considered includes the...
Read MoreFunciones p-armónicas con condiciones de Neumann en el borde que involucran medidas
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. Las técnicas a utilizar también nos permitirán establecer algunos resultados de no existencia y de eliminación de singularidades.
Read MorePartially localized solutions of some elliptic equations on $R^{N+1}
Abstract: It is well known that, under quite general assumptions, fully localized solutions of a homogeneous semilinear elliptic equation must be radially symmetric. Many authors have exhibited the complexity of solutions which are only partially localized (that is, only decaying in some variables). In this talk I consider solutions which are quasiperiodic in one variable, decaying in all the others. In the first part of the talk I develop a framework for finding such solutions in the nonhomogeneous case; in the second part I show the existence of nonlinearities for which these solutions...
Read MoreDecay of small odd solutions of the long range Schrödinger and Hartree equations in one dimension.
Abstract: We consider the long time asymptotics of (not necessarily small) odd solutions to the nonlinear Schrödinger equation with semilinear and nonlocal Hartree nonlinearities, in one dimension space. We assume data in the energy space only and we prove decay to zero in compact regions of space as time tends to infinity. We give three different results were decay holds: NLS without potential, NLS with potential and Hartree (defocusing case). The proof is based in the use of suitable virial identities and covers all range of scattering sub, critical and supercritical (long range)...
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