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

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

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

Characterization of the critical magnetic field in the Dirac-Coulomb equation

We consider a relativistic hydrogenic atom in a strong magnetic field. The ground state level depends on the strength of the magnetic field and reaches the lower end of the spectral gap of the Dirac-Coulomb operator for a certain critical value, the critical magnetic field. We also define a critical magnetic field in a Landau level ansatz. In both cases, when the charge Z of the nucleus is not too small, these critical magnetic fields are huge when measured in Tesla, but not so big when the equation is written in dimensionless form. When computed in the Landau level ansatz, orders of magnitude of the critical field are correct, as well as the dependence in Z. The computed value is however significantly too big for a large Z, and the wave function is not well approximated. Hence, accurate numerical computations involving the Dirac equation cannot systematically rely on the Landau level ansatz. Our approach is based on a scaling property. The critical magnetic field is characterized in terms of an equivalent eigenvalue problem. This is our main analytical result, and also the starting point of our numerical scheme

Kinetic models and quantum effects: A modified Boltzmann equation for Fermi-Dirac particles

We study a modified Boltzmann equation, which takes into account quantum effects for a gas of Fermi-Dirac particles and gives in the classical limit the Boltzmann equation

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