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Jean Dolbeault

Université Paris Dauphine - PSL, CEREMADE, Faculty Member

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Manuel M Sánchez del Pino

University of Valencia, Spain

Christof Sparber

Davide Ferrario

Delfim F. M. Torres

University of Aveiro

mehdaoui sidaali

Mats Ehrnström

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Papers by Jean Dolbeault

Harnack Inequalities and Discrete—Continuous Error Estimates for a Chain of Atoms with Two—Body Interactions

by Jean Dolbeault and Rafael Benguria

Journal of Statistical Physics, 2009

We consider deformations in R 3 of an infinite linear chain of atoms where each atom interacts wi... more We consider deformations in R 3 of an infinite linear chain of atoms where each atom interacts with all others through a two-body potential. We compute the effect of an external force applied to the chain. At equilibrium, the positions of the particles satisfy an Euler-Lagrange equation. For large classes of potentials, we prove that every solution is well approximated by the solution of a continuous model. We establish an error estimate between the discrete and the continuous solution based on a Harnack lemma of independent interest. Finally we apply our results to some Lennard-Jones potentials.

Oscillating minimizers of a fourth order problem invariant under scaling

by Rafael Benguria and Jean Dolbeault

By variational methods, we prove the inequality: $$ \int_{\mathbb{R}} u''{}^2 dx-\int_{\m... more By variational methods, we prove the inequality: $$ \int_{\mathbb{R}} u''{}^2 dx-\int_{\mathbb{R}} u'' u^2 dx\geq I \int_{\mathbb{R}} u^4 dx\quad \forall u\in L^4({\mathbb{R}}) {such that} u''\in L^2({\mathbb{R}}) $$ for some constant $I\in (-9/64,-1/4)$. This inequality is connected to Lieb-Thirring type problems and has interesting scaling properties. The best constant is achieved by sign changing minimizers of a problem on periodic functions, but

Stationary solutions, intermediate asymptotics and large time behaviour of type II Streater's models

by Jean Dolbeault and Piotr Biler

In this paper, we study type II Streater's models. These models describe the coupled evolutio... more In this paper, we study type II Streater's models. These models describe the coupled evolution of the density of a cloud of particles in an external potential and a temperature, preserving the energy, with eventually a nonlocal Poisson coupling. We introduce an entropy and consider in a bounded domain, or in an unbounded domain with a con- fining external potential,

Large time asymptotics of nonlinear drift-diffusion systems with Poisson coupling

by Jean Dolbeault and Piotr Biler

We study the asymptotic behavior as t →+∞ of a system of den- sities of charged particles satisfy... more

On long time asymptotics of the vlasov—poisson—boltzmann equation

Communications in Partial Differential Equations, 1991

We deal with the long time asymptotics of the Vlasovmdash;Poissonmdash;Boltzmann equation. We pro... more

On Singular Limits of Mean-Field Equations

Variational ProblemIn this section we consider the limit ! 0 of an abstract minimizationproblems ... more Variational ProblemIn this section we consider the limit ! 0 of an abstract minimizationproblems with a class of functionals containing E .Theorem 3. Let (B; k:k) be a Banach space and letC ` Bbe nonvoid, convex and weakly closed in B. LetE; F : C ! R [ f1g; be bounded below with infCE ! 1; infCF ! 1:We setC:=

Entropy Methods for Kinetic Models of Trac Flow

by Jean Dolbeault and Reinhard Illner

In these notes we first introduce logarithmic entropy methods for time-dependent driftdi #usion e... more In these notes we first introduce logarithmic entropy methods for time-dependent driftdi #usion equations and then consider a kinetic model of Vlasov-Fokker-Planck type for tra#c flows. In the spatially homogeneous case the model reduces to a special type of nonlinear drift-di#usion equation which may permit the existence of several stationary states corresponding to the same density. Then we define general convex entropies and prove a convergence result for large times to steady states, even if more than one exist in the considered range of parameters, provided that some entropy estimates are uniformly bounded.

Partial Differential Equations/Functional Analysis Convex Sobolev inequalities and spectral gap

This Note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which ... more 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

Nonlinear diusions, hypercontractivity and the optimal Lp-Euclidean logarithmic Sobolev inequality

The equation ut = p(u1/(p 1)) for p > 1 is a nonlinear generalization of the heat equation whi... more The equation ut = p(u1/(p 1)) for p > 1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal Lp-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focuse on the existence and the uniqueness of the solutions to the Cauchy problem and on

On the One-dimensional Parabolic Obstacle Problem with Variable Coefficients

Progress in Nonlinear Differential Equations and Their Applications, 2005

This note is devoted to continuity results of the time derivative of the solution to the onedimen... more This note is devoted to continuity results of the time derivative of the solution to the onedimensional 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.

Steady States for Streater’s Energy-Transport Models of Self-Gravitating Particles

by Jean Dolbeault and Piotr Biler

The IMA Volumes in Mathematics and its Applications, 2004

We review Streater's energy-transport models which describe the temporal evolution of the density... more We review Streater's energy-transport models which describe the temporal evolution of the density and temperature of a cloud of gravitating particles, coupled to a mean eld Poisson equation. In particular we consider the existence of stationary solutions in a bounded domain with given energy and mass. We discuss the in uence of the dimension and geometry of the domain on existence results.

The flashing ratchet: long time behavior and dynamical systems interpretation

The flashing ratchet is a model for certain types of molecular motors as well as a convenient mod... more The flashing ratchet is a model for certain types of molecular motors as well as a convenient model problem in the more general context of diffusion mediated transport. In this paper we show that it can be derived using a minimum energy dissipation principle for transport in a viscous environment. We then study the long time behavior of the flashing

Time-dependent rescalings and Lyapunov functionals for the Vlasov-Poisson and Euler-Poisson systems, and for related models of kinetic equations, fluid dynamics and quantum physics

We apply the method of time-dependent rescalings which has been developed by G. Rein and the auth... more We apply the method of time-dependent rescalings which has been developed by G. Rein and the author to a model kinetic equation, to the Euler equations for a perfect polytropic gas and to a model with friction and heat transfer. We build Lyapunov functionals which are in the case of the fluid models improved versions of the estimates which have been found by J.-Y. Chemin and D. Serre (see , ).

Computational approaches of relativistic models in quantum chemistry

by Jean Dolbeault, P. Indelicato, and E. S'Er'E

This chapter is a review of some methods used for the computation of relativistic atomic and mole... more This chapter is a review of some methods used for the computation of relativistic atomic and molecular models based on the Dirac equation. In the linear case, we briefly describe finite basis set approaches, including ones that are generated numerically, perturbation theory and effective Hamiltonians procedures, direct variational methods based on nonlinear transformations, min-max formulations and constrained minimizations. In the atomic case, we describe the MCDF method and some ways to solve numerically the homogeneous and inhomogeneous Dirac-Fock equations. Finally, we describe also some numerical methods relevant to the case of molecules.

Fast diffusion equations: Matching large time asymptotics by relative entropy methods

A non self-similar change of coordinates provides improved matching asymptotics of the solutions ... more A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the scaling corresponding to the self-similar Barenblatt solutions, as it is usually done.

About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities

This article is devoted to a review of some recent results on existence, symmetry and symmetry br... more This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) 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. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (CKN) and (WLH).

Existence of sign changing solutions for an equation with a weighted p-Laplace operator

by Carmen Cortazar, Jean Dolbeault, and R. Manasevich

We consider radial solutions of a general elliptic equation involving a weighted p-Laplace operat... more We consider radial solutions of a general elliptic equation involving a weighted p-Laplace operator with a subcritical nonlinearity. By a shooting method we prove the existence of solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane, a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic Hamiltonian problem. Estimating the rate of decay for the energy requires a sub-criticality condition. The method covers the case of solutions which are not compactly supported or which have compact support. In the last case, we show that the size of the support increases with the number of nodes.

Improved interpolation inequalities on the sphere

This paper contains a review of available methods for establishing improved interpolation inequal... more This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.

Nonlinear flows and rigidity results on compact manifolds

This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequa... more This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. This parameter can be used as an estimate for the best constant in the corresponding interpolation inequality. Our approach relies in a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of technical choices of exponents done in earlier works. We also establish two integral criteria for rigidity that improve upon known, pointwise conditions, and hold for general manifolds without positivity conditions on the curvature. Using the flow, we are also able to discuss the optimality of the corresponding constant in the interpolation inequalities.

Spectral estimates on the sphere

In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators ... more In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the d-dimensional unit sphere. These estimates depend on L p norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semi-classical asymptotic regime and discuss how our estimates on the sphere differ from those on the Euclidean space.

Harnack Inequalities and Discrete—Continuous Error Estimates for a Chain of Atoms with Two—Body Interactions

by Jean Dolbeault and Rafael Benguria

Journal of Statistical Physics, 2009

We consider deformations in R 3 of an infinite linear chain of atoms where each atom interacts wi... more We consider deformations in R 3 of an infinite linear chain of atoms where each atom interacts with all others through a two-body potential. We compute the effect of an external force applied to the chain. At equilibrium, the positions of the particles satisfy an Euler-Lagrange equation. For large classes of potentials, we prove that every solution is well approximated by the solution of a continuous model. We establish an error estimate between the discrete and the continuous solution based on a Harnack lemma of independent interest. Finally we apply our results to some Lennard-Jones potentials.

Oscillating minimizers of a fourth order problem invariant under scaling

by Rafael Benguria and Jean Dolbeault

By variational methods, we prove the inequality: $$ \int_{\mathbb{R}} u''{}^2 dx-\int_{\m... more By variational methods, we prove the inequality: $$ \int_{\mathbb{R}} u''{}^2 dx-\int_{\mathbb{R}} u'' u^2 dx\geq I \int_{\mathbb{R}} u^4 dx\quad \forall u\in L^4({\mathbb{R}}) {such that} u''\in L^2({\mathbb{R}}) $$ for some constant $I\in (-9/64,-1/4)$. This inequality is connected to Lieb-Thirring type problems and has interesting scaling properties. The best constant is achieved by sign changing minimizers of a problem on periodic functions, but

Stationary solutions, intermediate asymptotics and large time behaviour of type II Streater's models

by Jean Dolbeault and Piotr Biler

In this paper, we study type II Streater's models. These models describe the coupled evolutio... more In this paper, we study type II Streater's models. These models describe the coupled evolution of the density of a cloud of particles in an external potential and a temperature, preserving the energy, with eventually a nonlocal Poisson coupling. We introduce an entropy and consider in a bounded domain, or in an unbounded domain with a con- fining external potential,

Large time asymptotics of nonlinear drift-diffusion systems with Poisson coupling

by Jean Dolbeault and Piotr Biler

We study the asymptotic behavior as t →+∞ of a system of den- sities of charged particles satisfy... more

On long time asymptotics of the vlasov—poisson—boltzmann equation

Communications in Partial Differential Equations, 1991

We deal with the long time asymptotics of the Vlasovmdash;Poissonmdash;Boltzmann equation. We pro... more

On Singular Limits of Mean-Field Equations

Variational ProblemIn this section we consider the limit ! 0 of an abstract minimizationproblems ... more Variational ProblemIn this section we consider the limit ! 0 of an abstract minimizationproblems with a class of functionals containing E .Theorem 3. Let (B; k:k) be a Banach space and letC ` Bbe nonvoid, convex and weakly closed in B. LetE; F : C ! R [ f1g; be bounded below with infCE ! 1; infCF ! 1:We setC:=

Entropy Methods for Kinetic Models of Trac Flow

by Jean Dolbeault and Reinhard Illner

In these notes we first introduce logarithmic entropy methods for time-dependent driftdi #usion e... more In these notes we first introduce logarithmic entropy methods for time-dependent driftdi #usion equations and then consider a kinetic model of Vlasov-Fokker-Planck type for tra#c flows. In the spatially homogeneous case the model reduces to a special type of nonlinear drift-di#usion equation which may permit the existence of several stationary states corresponding to the same density. Then we define general convex entropies and prove a convergence result for large times to steady states, even if more than one exist in the considered range of parameters, provided that some entropy estimates are uniformly bounded.

Partial Differential Equations/Functional Analysis Convex Sobolev inequalities and spectral gap

This Note is devoted to the proof of convex Sobolev (or generalized Poincaré) inequalities which ... more 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

Nonlinear diusions, hypercontractivity and the optimal Lp-Euclidean logarithmic Sobolev inequality

The equation ut = p(u1/(p 1)) for p > 1 is a nonlinear generalization of the heat equation whi... more The equation ut = p(u1/(p 1)) for p > 1 is a nonlinear generalization of the heat equation which is also homogeneous, of degree 1. For large time asymptotics, its links with the optimal Lp-Euclidean logarithmic Sobolev inequality have recently been investigated. Here we focuse on the existence and the uniqueness of the solutions to the Cauchy problem and on

On the One-dimensional Parabolic Obstacle Problem with Variable Coefficients

Progress in Nonlinear Differential Equations and Their Applications, 2005

This note is devoted to continuity results of the time derivative of the solution to the onedimen... more This note is devoted to continuity results of the time derivative of the solution to the onedimensional 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.

Steady States for Streater’s Energy-Transport Models of Self-Gravitating Particles

by Jean Dolbeault and Piotr Biler

The IMA Volumes in Mathematics and its Applications, 2004

We review Streater's energy-transport models which describe the temporal evolution of the density... more We review Streater's energy-transport models which describe the temporal evolution of the density and temperature of a cloud of gravitating particles, coupled to a mean eld Poisson equation. In particular we consider the existence of stationary solutions in a bounded domain with given energy and mass. We discuss the in uence of the dimension and geometry of the domain on existence results.

The flashing ratchet: long time behavior and dynamical systems interpretation

The flashing ratchet is a model for certain types of molecular motors as well as a convenient mod... more The flashing ratchet is a model for certain types of molecular motors as well as a convenient model problem in the more general context of diffusion mediated transport. In this paper we show that it can be derived using a minimum energy dissipation principle for transport in a viscous environment. We then study the long time behavior of the flashing

Time-dependent rescalings and Lyapunov functionals for the Vlasov-Poisson and Euler-Poisson systems, and for related models of kinetic equations, fluid dynamics and quantum physics

We apply the method of time-dependent rescalings which has been developed by G. Rein and the auth... more We apply the method of time-dependent rescalings which has been developed by G. Rein and the author to a model kinetic equation, to the Euler equations for a perfect polytropic gas and to a model with friction and heat transfer. We build Lyapunov functionals which are in the case of the fluid models improved versions of the estimates which have been found by J.-Y. Chemin and D. Serre (see , ).

Computational approaches of relativistic models in quantum chemistry

by Jean Dolbeault, P. Indelicato, and E. S'Er'E

This chapter is a review of some methods used for the computation of relativistic atomic and mole... more This chapter is a review of some methods used for the computation of relativistic atomic and molecular models based on the Dirac equation. In the linear case, we briefly describe finite basis set approaches, including ones that are generated numerically, perturbation theory and effective Hamiltonians procedures, direct variational methods based on nonlinear transformations, min-max formulations and constrained minimizations. In the atomic case, we describe the MCDF method and some ways to solve numerically the homogeneous and inhomogeneous Dirac-Fock equations. Finally, we describe also some numerical methods relevant to the case of molecules.

Fast diffusion equations: Matching large time asymptotics by relative entropy methods

A non self-similar change of coordinates provides improved matching asymptotics of the solutions ... more A non self-similar change of coordinates provides improved matching asymptotics of the solutions of the fast diffusion equation for large times, compared to already known results, in the range for which Barenblatt solutions have a finite second moment. The method is based on relative entropy estimates and a time-dependent change of variables which is determined by second moments, and not by the scaling corresponding to the self-similar Barenblatt solutions, as it is usually done.

About existence, symmetry and symmetry breaking for extremal functions of some interpolation functional inequalities

This article is devoted to a review of some recent results on existence, symmetry and symmetry br... more This article is devoted to a review of some recent results on existence, symmetry and symmetry breaking of optimal functions for Caffarelli-Kohn-Nirenberg (CKN) and weighted logarithmic Hardy (WLH) 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. Here we put the highlights on a symmetry breaking result: extremals of some inequalities are not radially symmetric in regions where the symmetric extremals are linearly stable. Special attention is paid to the study of the critical cases for (CKN) and (WLH).

Existence of sign changing solutions for an equation with a weighted p-Laplace operator

by Carmen Cortazar, Jean Dolbeault, and R. Manasevich

We consider radial solutions of a general elliptic equation involving a weighted p-Laplace operat... more We consider radial solutions of a general elliptic equation involving a weighted p-Laplace operator with a subcritical nonlinearity. By a shooting method we prove the existence of solutions with any prescribed number of nodes. The method is based on a change of variables in the phase plane, a very general computation of an angular velocity and new estimates for the decay of an energy associated with an asymptotic Hamiltonian problem. Estimating the rate of decay for the energy requires a sub-criticality condition. The method covers the case of solutions which are not compactly supported or which have compact support. In the last case, we show that the size of the support increases with the number of nodes.

Improved interpolation inequalities on the sphere

This paper contains a review of available methods for establishing improved interpolation inequal... more This paper contains a review of available methods for establishing improved interpolation inequalities on the sphere for subcritical exponents. Pushing further these techniques we also establish some new results, clarify the range of applicability of the various existing methods and state several explicit estimates.

Nonlinear flows and rigidity results on compact manifolds

This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequa... more This paper is devoted to rigidity results for some elliptic PDEs and related interpolation inequalities of Sobolev type on smooth compact connected Riemannian manifolds without boundaries. Rigidity means that the PDE has no other solution than the constant one at least when a parameter is in a certain range. This parameter can be used as an estimate for the best constant in the corresponding interpolation inequality. Our approach relies in a nonlinear flow of porous medium / fast diffusion type which gives a clear-cut interpretation of technical choices of exponents done in earlier works. We also establish two integral criteria for rigidity that improve upon known, pointwise conditions, and hold for general manifolds without positivity conditions on the curvature. Using the flow, we are also able to discuss the optimality of the corresponding constant in the interpolation inequalities.

Spectral estimates on the sphere

In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators ... more In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the d-dimensional unit sphere. These estimates depend on L p norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semi-classical asymptotic regime and discuss how our estimates on the sphere differ from those on the Euclidean space.