This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimens...This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods(in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality.In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally,a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.展开更多
This paper is devoted to various considerations on a family of sharp interpo- lation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincare, logarithmic Sobolev and critical Sobolev...This paper is devoted to various considerations on a family of sharp interpo- lation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincare, logarithmic Sobolev and critical Sobolev （Onofri in dimension two） inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.展开更多
The authors get on Metivier groups the spectral resolution of a class of operators m（L, -Δ）, the joint functional calculus of the sub-Laplacian and Laplacian on the centre, and then give some restriction theorems t...The authors get on Metivier groups the spectral resolution of a class of operators m（L, -Δ）, the joint functional calculus of the sub-Laplacian and Laplacian on the centre, and then give some restriction theorems together with their asymptotic estimates, asserting the mix-norm boundedness of the spectral projection operators Pμ^m for two classes of functions re（a, b） =（a^α＋b^β）^γ or （1＋a^α＋b^β）^γ,with α,β〉0,γ≠0.展开更多
This paper deals with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each compo...This paper deals with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each component of the system may be controlled in a different subdomain. Assuming that the unperturbed system is controUable--a property that has been recently characterized in terms of a Kalman-like rank condition--the authors give a necessary and sufficient condition for the controllability of the coupled system under the form of a unique continuation property for the corresponding elliptic eigenvalue system. The proof relies on a compactness-uniqueness argument, which is quite unusual in the context of parabolic systems, previously developed for scalar parabolic equations. The general result is illustrated by two simple examples.展开更多
基金supported by the Projects STAB and Kibord of the French National Research Agency(ANR)the Project No NAP of the French National Research Agency(ANR)the ECOS Project(No.C11E07)
文摘This paper is devoted to results on the Moser-Trudinger-Onofri inequality, or the Onofri inequality for brevity. In dimension two this inequality plays a role similar to that of the Sobolev inequality in higher dimensions. After justifying this statement by recovering the Onofri inequality through various limiting procedures and after reviewing some known results, the authors state several elementary remarks.Various new results are also proved in this paper. A proof of the inequality is given by using mass transportation methods(in the radial case), consistently with similar results for Sobolev inequalities. The authors investigate how duality can be used to improve the Onofri inequality, in connection with the logarithmic Hardy-Littlewood-Sobolev inequality.In the framework of fast diffusion equations, it is established that the inequality is an entropy-entropy production inequality, which provides an integral remainder term. Finally,a proof of the inequality based on rigidity methods is given and a related nonlinear flow is introduced.
基金Project supported by ANR grants CBDif and NoNAP,the ECOS project (No. C11E07)the Chileanresearch grants Fondecyt (No. 1090103)Fondo Basal CMM Chile,Project Anillo ACT 125 CAPDEand the National Science Foundation (No.DMS 0901304)
文摘This paper is devoted to various considerations on a family of sharp interpo- lation inequalities on the sphere, which in dimension greater than 1 interpolate between Poincare, logarithmic Sobolev and critical Sobolev （Onofri in dimension two） inequalities. The connection between optimal constants and spectral properties of the Laplace-Beltrami operator on the sphere is emphasized. The authors address a series of related observations and give proofs based on symmetrization and the ultraspherical setting.
基金supported by the National Natural Science Foundation of China(No.11371036)the Specialized Research Fund for the Doctoral Program of Higher Education of China(No.2012000110059)
文摘The authors get on Metivier groups the spectral resolution of a class of operators m（L, -Δ）, the joint functional calculus of the sub-Laplacian and Laplacian on the centre, and then give some restriction theorems together with their asymptotic estimates, asserting the mix-norm boundedness of the spectral projection operators Pμ^m for two classes of functions re（a, b） =（a^α＋b^β）^γ or （1＋a^α＋b^β）^γ,with α,β〉0,γ≠0.
文摘This paper deals with the problem of internal controllability of a system of heat equations posed on a bounded domain with Dirichlet boundary conditions and perturbed with analytic non-local coupling terms. Each component of the system may be controlled in a different subdomain. Assuming that the unperturbed system is controUable--a property that has been recently characterized in terms of a Kalman-like rank condition--the authors give a necessary and sufficient condition for the controllability of the coupled system under the form of a unique continuation property for the corresponding elliptic eigenvalue system. The proof relies on a compactness-uniqueness argument, which is quite unusual in the context of parabolic systems, previously developed for scalar parabolic equations. The general result is illustrated by two simple examples.
