Entropy dissipation estimates in a zero-range dynamics

We study the exponential decay of relative entropy functionals for zero-range processes on the complete graph. For the standard model with rates increasing at infinity we prove entropy dissipation estimates, uniformly over the number of particles and the number of vertices.      

Asymptotic Estimates for Gagliardo–Nirenberg Embedding Constants

Sharp asymptotic information is determined for the Gagliardo–Nirenberg embedding constants in high dimension. This analysis is motivated by the earlier observation that the logarithmic Sobolev inequality controls the Nash inequality. Moreover, one sees here that Hardy's inequality can be interpreted as the asymptotic limit of the logarithmic Sobolev inequality.     

Modified Logarithmic Sobolev Inequalities on $mathbb{R}$

We provide a sufficient condition for a measure on the real line to satisfy a modified logarithmic Sobolev inequality, thus extending the criterion of Bobkov and Götze. Under mild assumptions the condition is also necessary. Concentration inequalities are derived. This completes the picture given in recent contributions by Gentil, Guillin and Miclo.     

Best constants for Sobolev inequalities for higher order fractional derivatives

We obtain sharp constants for Sobolev inequalities for higher order fractional derivatives. As an application, we give a new proof of a theorem of W. Beckner concerning conformally invariant higher-order differential operators on the sphere.     

On a class of weighted anisotropic Sobolev inequalities

In this article, motivated by a work of Caffarelli and Cordoba in phase transitions analysis, we prove new weighted anisotropic Sobolev type inequalities where different derivatives have different weight functions. These inequalities are also intimately connected to weighted Sobolev inequalities for Grushin type operators, the weights being not necessarily Muckenhoupt. For example we consider Sobolev inequalities on finite cylinders, the weight being a power of the distance function from the top or the bottom of the cylinder. We also prove similar inequalities in the more general case in which the weight is a power of the distance function from a higher codimension part of the boundary.     

Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux

The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood–Paley theory, in a noncommutative setting, defined via heat flow on surfaces. Received: April 2004 Revision: June 2005 Accepted: July 2005 The first author is partially supported by NSF grant DMS-0070696. The second author is a Clay Prize Fellow and is partially supported by NSF grant DMS-01007791.     

Isoperimetry and symmetrization for logarithmic Sobolev inequalities

Using isoperimetry and symmetrization we provide a unified framework to study the classical and logarithmic Sobolev inequalities. In particular, we obtain new Gaussian symmetrization inequalities and connect them with logarithmic Sobolev inequalities. Our methods are very general and can be easily adapted to more general contexts.     

Weak logarithmic Sobolev inequalities and entropic convergence

In this paper we introduce and study a weakened form of logarithmic Sobolev inequalities in connection with various others functional inequalities (weak Poincaré inequalities, general Beckner inequalities, etc.). We also discuss the quantitative behaviour of relative entropy along a symmetric diffusion semi-group. In particular, we exhibit an example where Poincaré inequality can not be used for deriving entropic convergence whence weak logarithmic Sobolev inequality ensures the result.      

Modified Logarithmic Sobolev Inequalities and Transportation Cost Inequalities in ℝ<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, the modified logarithmic Sobolev inequalities and transportation cost inequalities for measures with density e ^− V in ℝ ⁿ are established. It is proved by using Prékopa–Leindler inequalities following the idea of Bobkov–Ledoux, but a different type of condition is used which recovers Bakry–Emery criterion. As an application, we establish the modified logarithmic Sobolev and transportation cost inequalities for probability measures with p > 1 in ℝ ⁿ , and give out explicit estimates for their constants. This work is supported by NSFC (No. 10721091), 973-Project (No.2006CB805901) and DFMEC (NO. 20070027007).     

The Problem of Prescribed Critical Functions

Let (M,g) be a compact Riemannian manifold on dimension n ≥ 4 not conformally diffeomorphic to the sphere Sⁿ. We prove that a smooth function f on M is a critical function for a metric g conformal to g if and only if there exists x ∈ M such that f(x) > 0.Mathematics Subject Classifications (2000): 53C21, 46E35, 26D10.     

Isoperimetric Inequalities on Compact Manifolds

We prove an isoperimetric inequality on compact Riemannian manifolds corresponding to the limit case of a scale of optimal Sobolev inequalities.     

Gagliardo–Nirenberg inequalities and non-inequalities: The full story

We investigate the validity of the Gagliardo–Nirenberg type inequality

(1)${6f6}_{W^{s,p}(\mathrm{\Omega })}?{6f6}_{W^{s_1,p_1}(\mathrm{\Omega })}^{\theta }{6f6}_{W^{s_2,p_2}(\mathrm{\Omega })}^{1?\theta },$

with $\mathrm{\Omega }?{\mathbb{R}}^N$. Here, $0\le s_1\le s\le s_2$ are non negative numbers (not necessarily integers), $1\le p_1,p,p_2\le \infty$, and we assume the standard relations

$s=\theta s_1+(1?\theta )s_2,1/p=\theta /p_1+(1?\theta )/p_2\text{ for some }\theta \in (0,1).$

By the seminal contributions of E. Gagliardo and L. Nirenberg, (1) holds when $s_1,s_2,s$ are integers. It turns out that (1) holds for “most” of values of $s_1,\dots ,p_2$, but not for all of them. We present an explicit condition on $s_1,s_2,p_1,p_2$ which allows to decide whether (1) holds or fails.     

Global existence for some slightly super-linear parabolic equations with measure data

In this work we study the global existence of a solution to some parabolic problems whose model is

(1)     

Characterization of Talagrand's like transportation-cost inequalities on the real line

In this paper, we give necessary and sufficient conditions for Talagrand's like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimension-free concentration of measure property. Another byproduct is the characterization of modified Log-Sobolev inequalities for log-concave probability measures on .     

Critical dimensions and higher order Sobolev inequalities with remainder terms

Pucci and Serrin [21] conjecture that certain space dimensions behave 'critically' in a semilinear polyharmonic eigenvalue problem. Up to now only a considerably weakened version of this conjecture could be shown. We prove that exactly in these dimensions an embedding inequality for higher order Sobolev spaces on bounded domains with an optimal embedding constant may be improved by adding a 'linear' remainder term, thereby giving further evidence to the conjecture of Pucci and Serrin from a functional analytic point of view. Thanks to Brezis-Lieb [5] this result is already known for the space in dimension n=3; we extend it to the spaces (K>1) in the 'presumably' critical dimensions. Crucial tools are positivity results and a decomposition method with respect to dual cones. Received June 1999     

Best Constants in Sobolev Inequalities on Riemannian Manifolds in the Presence of Symmetries

Let (M,g) be a smooth compact Riemannian manifold, and G a subgroup of the isometry group of (M,g). We compute the value of the best constant in Sobolev inequalities when the functions are G-invariant. Applications to non-linear PDEs of critical or upper critical Sobolev exponent are also presented.