1,679 research outputs found
Erratum : Existence of 3D Skyrmions. Complete version
This erratum corrects the proof given in \cite{E1,E2} about the existence of
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 and 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
and 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
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
General results on the eigenvalues of operators with gaps, arising from both ends of the gaps. Application to Dirac operators
This paper is concerned with {an extension and reinterpretation} of previous
results on the variational characterization of eigenvalues in gaps of the
essential spectrum of self-adjoint operators. {We state} two general abstract
results on the existence of eigenvalues in the gap and a continuation
principle. Then, these results are applied to Dirac operators in order to
characterize simultaneously eigenvalues corresponding to electronic and
positronic bound states
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