arXiv subjects
Publications and source records attributed to Jean Dolbeault.
This paper is devoted to the Hartree-Fock model with temperature in the euclidean space. For large classes of free energy functionals, minimizers are obtained as long as the total charge of the system does not exceed a threshold which depends on the temperature. The usual Hartree-Fock model is recovered in the zero temperature limit. An orbital stability result for the Cauchy problem is deduced from the variational approach.
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.
The goal of this paper is to give a non-local sufficient condition for generalized Poincaré inequalities, which extends the well-known Bakry-Emery condition. Such generalized Poincaré inequalities have been introduced by W. Beckner in the gaussian case and provide, along the Ornstein-Uhlenbeck flow, the exponential decay of some generalized entropies which interpolate between the $L^2$ norm and the usual entropy. Our criterion improves on results which, for instance, can be deduced from the Bakry-Emery criterion and Holley-Stroock type perturbation results. In a second step, we apply the same strategy to non-linear equations of porous media type. This provides new interpolation inequalities and decay estimates for the solutions of the evolution problem. The criterion is again a non-local condition based on the positivity of the lowest eigenvalue of a Schrödinger operator. In both cases, we relate the Fisher information with its time derivative. Since the resulting criterion is non-local, it is better adapted to potentials with, for instance, a non-quadratic growth at infinity, or to unbounded perturbations of the potential.
Following Eden and Foias we obtain a matrix version of a generalised Sobolev inequality in one-dimension. This allow us to improve on the known estimates of best constants in Lieb-Thirring inequalities for the sum of the negative eigenvalues for multi-dimensional Schroedinger operators.
We consider non-negative solutions of the fast diffusion equation $u_t=Δu^m$ with $m \in (0,1)$, in the Euclidean space R^d, d?3, and study the asymptotic behavior of a natural class of solutions, in the limit corresponding to $t\to\infty$ for $m\ge m_c=(d-2)/d$, or as t approaches the extinction time when m < mc. For a class of initial data we prove that the solution converges with a polynomial rate to a self-similar solution, for t large enough if $m\ge m_c$, or close enough to the extinction time if m < mc. Such results are new in the range $m\le m_c$ where previous approaches fail. In the range mc < m < 1 we improve on known results.
We first prove a weighted inequality of Moser-Trudinger type depending on a parameter, in the two-dimensional Euclidean space. The inequality holds for radial functions if the parameter is larger than -1. Without symmetry assumption, it holds if and only if the parameter is in the interval (-1,0]. The inequality gives us some insight on the symmetry breaking phenomenon for the extremal functions of the Hardy-Sobolev inequality, as established by Caffarelli-Kohn-Nirenberg, in two space dimensions. In fact, for suitable sets of parameters (asymptotically sharp) we prove symmetry or symmetry breaking by means of a blow-up method. In this way, the weighted Moser-Trudinger inequality appears as a limit case of the Hardy-Sobolev inequality.
Using measure-capacity inequalities we study new functional inequalities, namely L^q-Poincaré inequalities and L^q-logarithmic Sobolev inequalities. As a consequence, we establish the asymptotic behavior of the solutions to the so-called weighted porous media equation for m?1, in terms of L^2-norms and entropies.
In the Dirac operator framework we characterize and estimate the ground state energy of relativistic hydrogenic atoms in a constant magnetic field and describe the asymptotic regime corresponding to a large field strength using relativistic Landau levels. We also define and estimate a critical magnetic field beyond which stability is lost.
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.
We prove a Lieb-Thirring type inequality for potentials such that the associated Schrödinger operator has a pure discrete spectrum made of an unbounded sequence of eigenvalues. This inequality is equivalent to a generalized Gagliardo-Nirenberg inequality for systems. As a special case, we prove a logarithmic Sobolev inequality for infinite systems of mixed states. Optimal constants are determined and free energy estimates in connection with mixed states representations are also investigated.
This note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which interpolate between spectral gap (or Poincaré) inequalities and logarithmic Sobolev inequalities. We extend to the whole family of convex Sobolev inequalities results which have recently been obtained by Cattiaux and Carlen and Loss for logarithmic Sobolev inequalities. Under local conditions on the density of the measure with respect to a reference measure, we prove that spectral gap inequalities imply all convex Sobolev inequalities with constants which are uniformly bounded in the limit approaching the logarithmic Sobolev inequalities. We recover the case of the logarithmic Sobolev inequalities as a special case.
This note is devoted to continuity results of the time derivative of the solution to the one-dimensional parabolic obstacle problem with variable coefficients. It applies to the smooth fit principle in numerical analysis and in financial mathematics. It relies on various tools for the study of free boundary problems: blow-up method, monotonicity formulae, Liouville's results.
We investigate rescaling transformations for the Vlasov-Poisson and Euler-Poisson systems and derive in the plasma physics case Lyapunov functionals which can be used to analyze dispersion effects. The method is also used for studying the long time behaviour of the solutions and can be applied to other models in kinetic theory (2-dimensional symmetric Vlasov-Poisson system with an external magnetic field), in fluid dynamics (Euler system for gases) and in quantum physics (Schrödinger-Poisson system, nonlinear Schrödinger equation).