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

Urbanheart surgery - a logic of design alternatives

In 1972 Sir Leslie Martin in his essay “The Grid as Generator”, advocated “a strong theoretical basis for [planning and] urban design” (Carolin P, 2000, p4) by methodically shifting design parameters regarding the way “in which buildings [could be] placed on the land” Martin was able to demonstrate how the generation of alternatives could “allow wider scope for decisions and objectives” to be considered and discussed (Carmona M, & Tiesdell S 2007, p81). Operating within a conventional design studio yet drawing of Sir Leslie Martin’s logic, ie developing an informed understanding of a problem by identifying a finite world of design ‘alternatives’, the following paper outlines a studio based program at the School of Architecture and Building, Deakin University, referred to as the ‘UrbanHeart Surgery’. While most atelier-based courses operate largely on an ad-hoc basis where students often work within self imposed competitive isolation, Urbanheart adopts a more open yet structured approach where students work in design collaboratives to generate a matrix of alternative design scenarios. The program actively integrates postgraduate students from Architecture, Urban Design and Planning into a design research culture and allows them to engage in critical discourse by working on strategic design projects in three areas significant to the future development of the state of Victoria: Metropolitan Urbanism, Urbanism on the Periphery and Regional Urbanism

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