A sharp Sobolev trace inequality for the fractional-order derivatives

We establish a sharp Sobolev trace inequality for the fractional-order derivatives. As a close connection with this best estimate, we show a fractional-order logarithmic Sobolev trace inequality with the asymptotically optimal constant, but also sharpen the Poincaré embedding for the conformal invariant energy and BMO spaces.     

An optimal logarithmic Sobolev inequality with Lipschitz constants

In this paper, we give an optimal logarithmic Sobolev inequality on Rn with Lipschitz constants. This inequality is a limit case of the Lp-logarithmic Sobolev inequality of Gentil (2003) [7] as p→∞. As a result of our inequality, we show that if a Lipschitz continuous function f on Rn fulfills some condition, then its Lipschitz constant can be expressed by using the entropy of f. We also show that a hypercontractivity of exponential type occurs in the heat equation on Rn. This is due to the Lipschitz regularizing effect of the heat equation.     

Sharp trace Gagliardo–Nirenberg–Sobolev inequalities for convex cones,and convex domains

We find a new sharp trace Gagliardo–Nirenberg–Sobolev inequality on convex cones, as well as a sharp weighted trace Sobolev inequality on epigraphs of convex functions. This is done by using a generalized Borell–Brascamp–Lieb inequality, coming from the Brunn–Minkowski theory.     

Logarithmic Sobolev trace inequality

A logarithmic Sobolev trace inequality is derived. Bounds on the best constant for this inequality from above and below are investigated using the sharp Sobolev inequality and the sharp logarithmic Sobolev inequality.

     

The Stepanov differentiability theorem in metric measure spaces

We extend Cheeger’s theorem on differentiability of Lipschitz functions in metric measure spaces to the class of functions satisfying Stepanov’s condition. As a consequence, we obtain the analogue of Calderon’s differentiability theorem of Sobolev functions in metric measure spaces satisfying a Poincaré inequality. Communicated by Steven Krantz     

Dirichlet and Neumann problems to critical Emden–Fowler type equations

We describe recent results on attainability of sharp constants in the Sobolev inequality, the Sobolev–Poincaré inequality, the Hardy–Sobolev inequality and related inequalities. This gives us the solvability of boundary value problems to critical Emden–Fowler equations.      

On a Class of Stochastic Semilinear PDEs

Abstract

We consider stochastic semilinear partial differential equations with Lipschitz nonlinear terms. We prove existence and uniqueness of an invariant measure and the existence of a solution for the corresponding Kolmogorov equation in the space L ²(H;ν), where ν is the invariant measure. We also prove the closability of the derivative operator and an integration by parts formula. Finally, under boundness conditions on the nonlinear term, we prove a Poincaré inequality, a logarithmic Sobolev inequality, and the ipercontractivity of the transition semigroup.     

Towards a Fully Nonlinear Sharp Sobolev Trace Inequality

We classify local minimizers of $\int\sigma_2+\oint H_2$ among all conformally flat metrics in the Euclidean $(n+1)$-ball, $n=4$ or $n=5$, for which the boundary has unit volume, subject to an ellipticity assumption. We also classify local minimizers of the analogous functional in the critical dimension $n+1=4$. If minimizers exist, this implies a fully nonlinear sharp Sobolev trace inequality. Our proof is an adaptation of the Frank-Lieb proof of the sharp Sobolev inequality, and in particular does not rely on symmetrization or Obata-type arguments.     

Lipschitz constants and logarithmic Sobolev inequality

We revisit two results of  [3]; they are a logarithmic Sobolev inequality on Rⁿ${\mathbb{R}}^n$ with Lipschitz constants and an expression of Lipschitz constants as the limit of a functional by the entropy. We have two goals in this paper. The first goal is to clarify when the strict inequality holds in this inequality. The second goal is to investigate the asymptotic behavior of this functional by the Abelian and Tauberian theorems of Laplace transforms.     

Non-linear ground state representations and sharp Hardy inequalities

We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces. To do so, we develop a non-linear and non-local version of the ground state representation, which even yields a remainder term. From the sharp Hardy inequality we deduce the sharp constant in a Sobolev embedding which is optimal in the Lorentz scale. In the appendix, we characterize the cases of equality in the rearrangement inequality in fractional Sobolev spaces.     

Sobolev-Lorentz范数约束下的次临界型Adams不等式

本文在Sobolev-Lorentz空间W²L^2,q(R⁴)的范数约束下得到了一个最佳的二阶次临界型Adams不等式.进一步,当次临界指标逼近最佳常数时,得到了Adams泛函的上、下界的估计.本文主要采用了Lam和Lu[A new approach to sharp MoserTrudinger and Adams type inequalities:a rearrangement-free argument,J.Diff Equ.,2013,255(3):298-325]的分割水平集方法.     

Best Constants for Moser-Trudinger Inequalities, Fundamental Solutions and One-Parameter Representation Formulas on Groups of Heisenberg Type

We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ${\Bbb G}We derive the explicit fundamental solutions for a class of degenerate (or singular) one-parameter subelliptic differential operators on groups of Heisenberg (H) type. This extends the results of Kaplan of the sub-Laplacian on H-type groups, which in turn generalizes Folland's result on the Heisenberg group. As an application, we obtain a one-parameter representation formula for Sobolev functions of compact support on H-type groups. By choosing the parameter equal to the homogeneous dimension Q and using the Moser-Trudinger inequality for the convolutional type operator on stratified groups obtained in [18], we get the following theorem which gives the best constant for the Moser-Trudinger inequality for Sobolev functions in H-type groups. Let ? be any group of Heisenberg type whose Lie algebra is g enerated by m left invariant vector fields and with a q-dimensional center. Let and Then, with A _Q as the sharp constant, where ∇_? denotes the subellitpic gradient on ? This continues the research originated in our earlier study of the best constants in Moser-Trudinger inequalities and fundamental solutions for one-parameter subelliptic operators on the Heisenberg group [18]. Received March 15, 2001, Accepted September 21, 2001     

A new modified logarithmic Sobolev inequality for Poisson point processes and several applications

By means of the martingale representation, we establish a new modified logarithmic Sobolev inequality, which covers the previous modified logarithmic Sobolev inequalities of Bobkov-Ledoux and the L ¹-logarithmic Sobolev inequality obtained in our previous work. From it we derive several sharp deviation inequalities of Talagrand's type, by following the powerful Herbst method developed recently by Ledoux and al. Moreover this new modified logarithmic Sobolev inequality is transported on the discontinuous path space with respect to the law of a Lévy process. Received: 16 June 1999 / Revised version: 13 March 2000 / Published online: 12 October 2000     

A proper generalized decomposition approach for optical flow estimation

This paper introduces the use of the proper generalized decomposition (PGD) method for the optical flow (OF) problem in a classical framework of Sobolev spaces, ie, optical flow methods including a robust energy for the data fidelity term together with a quadratic penalizer for the regularization term. A mathematical study of PGD methods is first presented for general regularization problems in the framework of (Hilbert) Sobolev spaces, and their convergence is then illustrated on OF computation. The convergence study is divided in two parts: (a) the weak convergence based on the Brézis-Lieb decomposition and (b) the strong convergence based on a growth result on the sequence of descent directions. A practical PGD-based OF implementation is then proposed and evaluated on freely available OF data sets. The proposed PGD-based OF approach outperforms the corresponding non-PGD implementation in terms of both accuracy and computation time for images containing a weak level of information, namely, low image resolution and/or low signal-to-noise ratio (SNR).     

On the rate of change of the sharp constant in the Sobolev–Poincaré inequality

The rate of change of the sharp constant in the Sobolev–Poincaré or Friedrichs inequality is estimated for a Euclidean domain that moves outward. The key ingredients are a Hadamard variation formula and an inequality that reverses the usual Hölder inequality.     

On the existence of an extremal function in critical Sobolev trace embedding theorem

Sufficient conditions for the existence of extremal functions in the trace Sobolev inequality and the trace Sobolev-Poincaré inequality are established. It is shown that some of these conditions are sharp.     

Poincaré’s inequalities and Talagrand’s concentration phenomenon for the exponential distribution

Summary. We present a simple proof, based on modified logarithmic Sobolev inequalities, of Talagrand’s concentration inequality for the exponential distribution. We actually observe that every measure satisfying a Poincaré inequality shares the same concentration phenomenon. We also discuss exponential integrability under Poincaré inequalities and its consequence to sharp diameter upper bounds on spectral gaps. Received: 10 June 1996 / In revised form: 9 August 1996     

Sensitivity analysis for a system of parametric general quasi-variational-like inequality problems in uniformly smooth Banach spaces

In this paper, using the concept of P-η-proximal-point mapping introduced by Kazmi and Bhat [11], we study the existence and sensitivity analysis of the solution set of a system of parametric general quasi-variational-like inequality problems in uniformly smooth Banach spaces. Further under suitable conditions, we discuss the Lipschitz continuity of the solution set with respect to the parameters. The approach used in this paper may be treated as an extension and unification of approaches for studying sensitivity analysis for various important classes of variational inequalities given in [1,2,4,12,14–16,21–24].     

Balls have the worst best Sobolev inequalities. Part II: variants and extensions

We continue our previous study of sharp Sobolev-type inequalities by means of optimal transport, started in (Maggi and Villani J. Geom. Anal. 15(1), 83–121 (2005)). In the present paper, we extend our results in various directions, including Gagliardo–Nirenberg, Faber–Krahn, logarithmic-Sobolev or Moser–Trudinger inequalities with trace terms. We also identify a class of domains for which there is no need for a trace term to cast the Sobolev inequality.