11,356 research outputs found
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
An analytical proof of Hardy-like inequalities related to the Dirac operator
We prove some sharp Hardy type inequalities related to the Dirac operator by
elementary, direct methods. Some of these inequalities have been obtained
previously using spectral information about the Dirac-Coulomb operator. Our
results are stated under optimal conditions on the asymptotics of the
potentials near zero and near infinity.Comment: LaTex, 22 page
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
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