Erratum : Existence of 3D Skyrmions. Complete version

This erratum corrects the proof given in \cite{E1,E2} about the existence of $3D$ Skyrmions. This is done by changing the arguments of the proof while remaining in the same framework of concentration-compactness. Note however that the use of this method is here different of most of what has been done with it so far. In that sense, this new proof has some interest by itself. The proof given here is self-contained. I thank F. Lin and Y. Yang for having pointed out to me that there were gaps in my proofs.Comment: 10 page

Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities

We consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities and weighted logarithmic Hardy inequalities which have been obtained recently as a limit case of the first ones. We discuss the ranges of the parameters for which the optimal constants are achieved by extremal functions. The comparison of these optimal constants with the optimal constants of Gagliardo-Nirenberg interpolation inequalities and Gross' logarithmic Sobolev inequality, both without weights, gives a general criterion for such an existence result in some particular cases.Comment: Proc. Edinburgh A (2012) To appea

Dirac-Fock models for atoms and molecules and related topics

An overview on various results concerning the Dirac-Fock model, the various variational characterization of its solutions and its nonrelativistic limit. A notion of ground state for this totally unbounded is also defined.Comment: To appear in Proc. ICMP2003. World Scientif

Extremal functions in some interpolation inequalities: Symmetry, symmetry breaking and estimates of the best constants

This contribution is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg and weighted logarithmic Hardy inequalities. These results have been obtained in a series of papers in collaboration with M. del Pino, S. Filippas, M. Loss, G. Tarantello and A. Tertikas and are presented from a new viewpoint

Ground States for a Stationary Mean-Field Model for a Nucleon

In this paper we consider a variational problem related to a model for a nucleon interacting with the $\omega$ and $\sigma$ mesons in the atomic nucleus. The model is relativistic, and we study it in a nuclear physics nonrelativistic limit, which is of a very different nature than the nonrelativistic limit in the atomic physics. Ground states are shown to exist for a large class of values for the parameters of the problem, which are determined by the values of some physical constants

Symmetric ground states for a stationary relativistic mean-field model for nucleons in the nonrelativistic limit

In this paper we consider a model for a nucleon interacting with the $\omega$ and $\sigma$ mesons in the atomic nucleus. The model is relativistic, but we study it in the nuclear physics nonrelativistic limit, which is of a very different nature from the one of the atomic physics. Ground states with a given angular momentum are shown to exist for a large class of values for the coupling constants and the mesons' masses. Moreover, we show that, for a good choice of parameters, the very striking shapes of mesonic densities inside and outside the nucleus are well described by the solutions of our model

Moral reasoning and judgment in childhood. Relations to mind understanding and peer acceptance

Abstract de póster presentado a First meeting of the SEJyD (Society for the Advancement of Judgment and Decision Making Studies)Children’s moral reasoning on compliance and rules violation, and their moral judgments, are linked to the development of their “theory of mind”. Greater ability to attribute mental states (intentions, knowledge and emotions) enable to base the attribution of responsibility, and judgment on the degree of punishment deserved, not only on the outcome of the action (harmful or not), but also on the intent to cause damage. This effect could vary depending on whether: a) the rule transgressed is a social conventional rule or a moral rule, b) the damage is physical-material or psychological-emotional. Moreover, understanding of other minds and moral reasoning that children make about the actions of others appears to be a key element in their degree of popularity and social impact. The aim of this study is to evaluate the developmental relationship between child moral reasoning, understanding of other minds and degree of acceptance by their peers. Participants were 89 children from 4 to 13 years; they were administered: a battery of stories that assessed moral reasoning abilities, a battery of “theory of mind” tasks, peer-nomination inventory. Results show that by 6 years of age begin differences in mind understanding and moral reasoning and judgment among children of the same age: popular and average distinguish between accidental and deliberate transgression (although up to 8 years all children believe that both deserve to be punished); only rejected children consider that the transgression of conventional norms does not deserve punishment; their moral judgments are not different for physical damage than for psychological-emotional damage. By age 8 differences between popular and rejected children in their mind understanding ability and moral reasoning are increased, especially in situations of accidental damage. Children’s moral reasoning ability may have important implications for their social relationships and positive peer interactions.Universidad de Málaga. Campus de Excelencia Internacional Andalucía Tec

Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces

This paper is motivated by the characterization of the optimal symmetry breaking region in Caffarelli-Kohn-Nirenberg inequalities. As a consequence, optimal functions and sharp constants are computed in the symmetry region. The result solves a longstanding conjecture on the optimal symmetry range. As a byproduct of our method we obtain sharp estimates for the principal eigenvalue of Schr\"odinger operators on some non-flat non-compact manifolds, which to the best of our knowledge are new. The method relies on generalized entropy functionals for nonlinear diffusion equations. It opens a new area of research for approaches related to carr\'e du champ methods on non-compact manifolds. However key estimates depend as much on curvature properties as on purely nonlinear effects. The method is well adapted to functional inequalities involving simple weights and also applies to general cylinders. Beyond results on symmetry and symmetry breaking, and on optimal constants in functional inequalities, rigidity theorems for nonlinear elliptic equations can be deduced in rather general settings.Comment: 33 pages, 1 figur