Trudinger‐Moser inequalities on the entire Heisenberg group

Continuing our previous work (Cohn, Lam, Lu, Yang, Nonlinear Analysis, 2011), we obtain a class of Trudinger‐Moser inequalities on the entire Heisenberg group, which indicate what the best constants are. All the existing proofs of similar inequalities on unbounded domain of the Euclidean space or the Heisenberg group are based on rearrangement argument. In this note, we propose a new approach to solve this problem. Specifically we get the global Trudinger‐Moser inequality by gluing local estimates with the help of cut‐off functions. Our method still works for similar problems when the Heisenberg group is replaced by the Euclidean space or complete noncompact Riemannian manifolds.     

Refined convex Sobolev inequalities

This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy-entropy production relation improves the known inequalities of this type. The corresponding generalized Poincaré-type inequalities with weights are derived. Optimal constants are compared to the usual Poincaré constant.     

Multivariate inequalities based on Sobolev representations

Here we derive very general multivariate tight integral inequalities of Chebyshev–Grüss, Ostrowski types and of comparison of integral means. These are based on well-known Sobolev integral representation of a function. Our inequalities engage ordinary and weak partial derivatives of the involved functions. We also give their applications. On the way to prove our main results we derive important estimates for the averaged Taylor polynomials and remainders of Sobolev integral representations. Our results expand to all possible directions.     

Characterizations of Sobolev inequalities on metric spaces

We discuss Maz'ya type isocapacitary characterizations of Sobolev inequalities on metric measure spaces.     

General Sobolev type inequalities for symmetric forms

A general Sobolev type inequality is introduced and studied for general symmetric forms by defining a new type of Cheeger's isoperimetric constant. Finally, concentration of measure for the Lp type logarithmic Sobolev inequality is presented.     

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.     

Characterization of the critical Sobolev space on the optimal singularity at the origin

In the present paper, we investigate the optimal singularity at the origin for the functions belonging to the critical Sobolev space , 1<p<∞. With this purpose, we shall show the weighted Gagliardo-Nirenberg type inequality:

(GN)     

Extremal functions for sharp Moser–Trudinger type inequalities in the whole space RN

In this paper, we prove the existence of maximizers for the sharp Moser–Trudinger type inequalities in whole space ${\mathbb{R}}^N$, $N\ge 2$ with more general nonlinearity. The main key in our proof is a precise estimate of the concentrating level of the Moser–Trudinger functional associated with our inequalities on the normalized concentrating sequences. This estimate solves a heavily non-trivial and open problem related to the sharp Moser–Trudinger inequality. Our method gives an alternative proof of the existence of maximizers for the Moser–Trudinger inequality and singular Moser–Trudinger inequality in whole space ${\mathbb{R}}^N$ due to Li and Ruf [30] and Li and Yang [31] without using blow-up analysis argument.     

A general two-scale criteria for logarithmic Sobolev inequalities

We present a general criteria to prove that a probability measure satisfies a logarithmic Sobolev inequality, knowing that some of its marginals and associated conditional laws satisfy a logarithmic Sobolev inequality. This is a generalization of a result by N. Grunewald et al. [N. Grunewald, F. Otto, C. Villani, M.G. Westdickenberg, A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit, Ann. Inst. H. Poincaré Probab. Statist., in press].     

Sobolev type inequalities for general symmetric forms

A general version of the Sobolev type inequality, including both the classical Sobolev inequality and the logarithmic Sobolev one, is studied for general symmetric forms by using isoperimetric constants. Some necessary and sufficient conditions are presented as results. The main results are illustrated by two examples of birth-death processes.

     

Compact embedding results of Sobolev spaces and existence of positive solutions to quasilinear equations

In this paper, we study mainly the existence of multiple positive solutions for a quasilinear elliptic equation of the following form on ${\mathbf{R}}^N$, when $N\ge 2$,

(0.1)$?{\mathrm{\Delta }}_{N}u+V(x){\left|u\right|}^{N?2}u=\lambda {\left|u\right|}^{r?2}u+f(x,u).$

Here, $V(x)>0:{\mathbf{R}}^N\to \mathbf{R}$ is a suitable potential function, $r\in (1,N)$, $f(x,u)$ is a continuous function of N-superlinear and subcritical exponential growth without having the Ambrosetti–Rabinowitz condition, while $\lambda >0$ is a constant. A suitable Moser–Trudinger inequality and the compact embedding $W_{\mathrm{V}}^{1,N}\left({\mathbf{R}}^N\right)?L^r\left({\mathbf{R}}^N\right)$ are proved to study problem (0.1). Moreover, the compact embedding $H_{\mathrm{V}}^1\left({\mathbf{R}}^N\right)?L_{\mathrm{K}}^t\left({\mathbf{R}}^N\right)$ is also analyzed to investigate the existence of a positive ground state to the following nonlinear Schrödinger equation

(0.2)$?\mathrm{\Delta }u+V(x)u=K(x)g(u)$

with potentials vanishing at infinity in a measure-theoretic sense when $N\ge 3$.     

Standing wave solutions for a class of nonhomogeneous systems in dimension two

In this article, minimax procedures and a Trudinger–Moser type inequality in weighted Sobolev spaces are employed to establish sufficient conditions for the existence of solutions for a class of nonhomogeneous elliptic systems involving nonlinear Schrödinger equations with subcritical or critical exponential growth and with potentials which are singular and/or vanishing. The solutions are obtained by suitable control of the size of the perturbations.     

Flows and functional inequalities for fractional operators

This paper collects results concerning global rates and large time asymptotics of a fractional fast diffusion equation on the Euclidean space, which is deeply related with a family of fractional Gagliardo–Nirenberg–Sobolev inequalities. Generically, self-similar solutions are not optimal for the Gagliardo–Nirenberg–Sobolev inequalities, in strong contrast with usual standard fast diffusion equations based on non-fractional operators. Various aspects of the stability of the self-similar solutions and of the entropy methods like carré du champ and Rényi entropy powers methods are investigated and raise a number of open problems.     

New geometric aspects of Moser–Trudinger inequalities on Riemannian manifolds: the non-compact case

In the first part of the paper we investigate some geometric features of Moser–Trudinger inequalities on complete non-compact Riemannian manifolds. By exploring rearrangement arguments, isoperimetric estimates, and gluing local uniform estimates via Gromov's covering lemma, we provide a Coulhon, Saloff-Coste and Varopoulos type characterization concerning the validity of Moser–Trudinger inequalities on complete non-compact n-dimensional Riemannian manifolds $(n\ge 2)$ with Ricci curvature bounded from below. Some sharp consequences are also presented both for non-negatively and non-positively curved Riemannian manifolds, respectively. In the second part, by combining variational arguments and a Lions type symmetrization-compactness principle, we guarantee the existence of a non-zero isometry-invariant solution for an elliptic problem involving the n-Laplace–Beltrami operator and a critical nonlinearity on n-dimensional homogeneous Hadamard manifolds. Our results complement in several directions those of Y. Yang [J. Funct. Anal., 2012].     

扩散过程轨道空间上的对数Sobolev不等式(英语)

设M是连通Riemann流形,Z是M上C′类向量场,L=(△ Z),本文使用Kendall的耦合分析,给出了参考测度为L-扩散过程在t时刻分布的对数Sobolev常数的估计,并由此建立了轨道空间上的对数Sobolev不等式。此外,本文还给出了流形上的对数Sobolev常数的一个上界估计,所获结果,是对文[1],[2]和[3]的相应结果的推广。     

Extremal functions for singular Trudinger–Moser inequalities in the entire Euclidean space

In a previous work (Adimurthi and Yang, 2010 [2]), Adimurthi–Yang proved a singular Trudinger–Moser inequality in the entire Euclidean space ${\mathbb{R}}^N$$(N\ge 2)$. Precisely, if $0\le \beta <1$ and $0<\gamma \le 1?\beta$, then there holds for any $\tau >0$,

$\underset{u\in W^{1,N}({\mathbb{R}}^N),{\int }_{{\mathbb{R}}^N}(|\mathrm{?}u{|}^N+\tau |u{|}^N)dx\le 1}{\sup }?\underset{{\mathbb{R}}^N}{\int }\frac{1}{|x{|}^{N\beta }}(e^{{\alpha }_N\gamma |u{|}^{\frac{N}{N?1}}}?\sum _{k=0}^{N?2}\frac{{{\alpha }_N}^k{\gamma }^k|u{|}^{\frac{kN}{N?1}}}{k!})dx<\infty ,$

where ${\alpha }_N=N{\omega }_{N?1}^{1/(N?1)}$ and ${\omega }_{N?1}$ is the area of the unit sphere in ${\mathbb{R}}^N$. The above inequality is sharp in the sense that if $\gamma >1?\beta$, all integrals are still finite but the supremum is infinity. In this paper, we concern extremal functions for these singular inequalities. The regular case $\beta =0$ has been considered by Li and Ruf (2008) [12] and Ishiwata (2011) [11]. We shall investigate the singular case $0<\beta <1$ and prove that for all $\tau >0$, $0<\beta <1$ and $0<\gamma \le 1?\beta$, extremal functions for the above inequalities exist. The proof is based on blow-up analysis.