Criteria for Super-and Weak-Poincaré Inequalities of Ergodic Birth-Death Processes

Criteria for the super-Poincaré inequality and the weak-Poincaré inequality about ergodic birth-death processes are presented. Our work further completes ten criteria for birth-death processes presented in Table 1.4 (p. 15) of Prof. Mu-Fa Chen's book "Eigenvalues, Inequalities and Ergodic Theory" (Springer, London, 2005). As a byproduct, we conclude that only ergodic birth-death processes on finite state space satisfy the Nash inequality with index 0 ν≤ 2.     

非正则区域上的加权分数阶Sobolev-Poincaré不等式

本文研究了非正则区域的加权分数阶Sobolev-Poincaré不等式.这里考虑的权重是到边界距离的某次幂,并且这些区域是所谓的s-John区域和β-H?lder区域.我们的主要结果将Hajlasz-Koskela的文[J.Lond.Math.Soc.,1998,58(2):425-450]结果从经典加权Sobolev-Poincare不等式推广到它的分数阶对应式,并且将Guo的文[Chin.Ann.Math.,2017,38B(3):839-856]从分数阶Sobolev-Poincaré不等式推广到其加权情形.     

A Fractional Orlicz–Sobolev Poincaré Inequality in John Domains

Let n ≥ 2, β∈(0, n) and ■ Rnbe a bounded domain. Support that φ : [0, ∞) → [0, ∞)is a Young function which is doubling and satisfies ■If Ω is a John domain, then we show that it supports a(φ~(n/(n-β)), φ)β-Poincaré inequality. Conversely,assume that Ω is simply connected domain when n = 2 or a bounded domain which is quasiconformally equivalent to some uniform domain when n ≥ 3. If Ω supports a((φ~(n/(n-β)), φ)β-Poincaré inequality,then we show that it is a John domain.     

Coupling, convergence rates of Markov processes and weak Poincaré inequalities

Some analytic and probabilistic properties of the weak Poincaré inequality are obtained. In particular, for strong Feller Markov processes the existence of this inequality is equivalent to each of the following: (i)the Liouville property (or the irreducibility); (ii) the existence of successful couplings (or shift-couplings); (iii)the convergence of the Markov process in total variation norm; (iv) the triviality of the tail (or the invariant)σ-field; (v) the convergence of the density. Estimates of the convergence rate in total variation norm of Markov processes are obtained using the weak Poincaré inequality.     

纯应力边界条件下的平面弹性问题的一个二阶非协调矩形元

1引言对于几乎不可压材料,当材料的Lame常数λ→∞时,通常的低阶协调有限元的解不再收敛到原问题的解或达不到最优收敛阶,这就是弹性材料的Locking现象[1,2,3,5,4].为了克服Locking问题,目前已提出许多有效的方法.例如:采用高次元[6,7,8]、非协调元[4,9,10,11]和混合有限元方法[1,2,12,13]等.     

WEIGHTED INEQUALITIES FOR MULTILINEAR POTENTIAL TYPE OPERATORS

The author derives a kind of weighted norm inequalities which relate the multilinear potential type integral operators to the corresponding maximal operators.     

Symmetrization for Fractional Elliptic and Parabolic Equations and an Isoperimetric Application

This paper develops further the theory of symmetrization of fractional Laplacian operators contained in recent works of two of the authors.This theory leads to optimal estimates in the form of concentration comparison inequalities for both elliptic and parabolic equations.The authors extend the theory for the so-called restricted fractional Laplacian defined on a bounded domain Ω of]RN with zero Dirichlet conditions outside of Ω.As an application,an original proof of the corresponding fractional Faber-Krahn inequality is derived.A more classical variational proof of the inequality is also provided.     

The Degree of Proper Holomorphic MappingsBetween Special Domains in C^n

Let Gi be closed Lie groups of U (n), Ω i be bounded Gi-invariant domains in C^n which contains 0, and O(C^n)^Gi = C, for i = 1, 2. It is known that if f : Ω 1 → Ω 2 is a proper holomorphic mapping, and f^-1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f .     

具有p-Laplacian算子的奇异四阶积分边值问题的正解

In this paper, we investigate the existence of positive solutions for singular fourthorder integral boundary-value problem with p-Laplacian operator by using the upper and lower solution method and fixed point theorem. Nonlinear term may be singular at t= 0 and/or t - 1 and x =0.     

Higher-Order Multilinear Poincaré and Sobolev Inequalities in Carnot Groups

The notions of higher-order weighted multilinear Poincaré and Sobolev inequalities in Carnot groups are introduced. As an application, weighted Leibniz-type rules in Campanato-Morrey spaces are established.     

Existence and Multiplicity of Positive Solutions for Fractional Differential Equation with Parameter

In this paper, by using the ?xed point theorem for a cone map, we study the existence and multiplicity of positive solutions for a class of fractional di?erential equation with parameter.     

EXISTENCE AND MULTIPLICITY OF SOLUTIONS TO A CLASS OF DIRICHLET PROBLEMS WITH IMPULSIVE EFFECTS

In this paper, we study the existence and multiplicity of solutions to a class of Dirichlet problems with impulsive effects via variational methods. Under an assumption that the nonlinearity f is superlinear but does not necessarily satisfy the Ambrosetti-Rabinowitz condition, we extend and improve some recent results.     

On Nikol’skii Inequalities for Domains in $${\mathbb {R}}^d$$

Nikol’skii inequalities for various sets of functions, domains, and weights will be discussed. Much of the work is dedicated to the class of algebraic polynomials of total degree n on a bounded convex domain D. That is, we study \(\sigma := \sigma (D,d)\) for which

$$\begin{aligned} \Vert P\Vert _{L_q(D)}\le c n^{\sigma (\frac{1}{p}-\frac{1}{q})}\Vert P\Vert _{L_p(D)},\quad 0<p\le q\le \infty , \end{aligned}$$

where P is a polynomial of total degree n. We use geometric properties of the boundary of D to determine \(\sigma (D,d)\) with the aid of comparison between domains. Computing the asymptotics of the Christoffel function of various domains is crucial in our investigation. The methods will be illustrated by the numerous examples in which the optimal \(\sigma (D,d)\) will be computed explicitly.

     

Exact Boundary Controllability for a Coupled System of Wave Equations with Neumann Boundary Controls

This paper first shows the exact boundary controllability for a coupled system of wave equations with Neumann boundary controls.In order to establish the corresponding observability inequality,the authors introduce a compact perturbation method which does not depend on the Riesz basis property,but depends only on the continuity of projection with respect to a weaker norm,which is obviously true in many cases of application.Next,in the case of fewer Neumann boundary controls,the non-exact boundary controllability for the initial data with the same level of energy is shown.     

VANISHING THEOREMS FOR ACH KHLER MANIFOLDS AND HARMONIC MAPS

We discuss a class of complete Khler manifolds which are asymptotically complex hyperbolic near infinity . The main result is vanishing theorems for the second L2 cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighte...     

Inequalities of Maximum of Partial Sums and Convergence Rates in the Strong Laws for ρ~-mixing Random Variables

In this paper,we establish a Rosenthal-type inequality of partial sums for ρ~mixing random variables.As its applications,we get the complete convergence rates in the strong laws for ρ~-mixing random variables.The result obtained extends the corresponding result.     

Erratum to: “Embeddings of Spaces of Functions of Positive Smoothness on Irregular Domains in Lebesgue Spaces”

Mathematical Notes - An embedding theorem of weighted spaces of functions of positive smoothness defined on irregular domains of n-dimensional Euclidean space in weighted Lebesgue spaces is...     

Sharp Affine and Improved Moser–Trudinger–Adams Type Inequalities on Unbounded Domains in the Spirit of Lions

The purpose of this paper is threefold. First, we prove sharp singular affine Moser–Trudinger inequalities on both bounded and unbounded domains in \({\mathbb {R}}^{n}\). In particular, we will prove the following much sharper affine Moser–Trudinger inequality in the spirit of Lions (Rev Mat Iberoamericana 1(2):45–121, 1985) (see our Theorem 1.4): Let \(\alpha _{n}=n\left( \frac{n\pi ^{\frac{n}{2}}}{\Gamma (\frac{n}{2}+1)}\right) ^{\frac{1}{n-1}}\), \(0\le \beta <n\) and \(\tau >0\). Then there exists a constant \(C=C\left( n,\beta \right) >0\) such that for all \(0\le \alpha \le \left( 1-\frac{\beta }{n}\right) \alpha _{n}\) and \(u\in C_{0}^{\infty }\left( {\mathbb {R}}^{n}\right) \setminus \left\{ 0\right\} \) with the affine energy \(~{\mathcal {E}}_{n}\left( u\right) <1\), we have

$$\begin{aligned} {\displaystyle \int \nolimits _{{\mathbb {R}}^{n}}} \frac{\phi _{n,1}\left( \frac{2^{\frac{1}{n-1}}\alpha }{\left( 1+{\mathcal {E}}_{n}\left( u\right) ^{n}\right) ^{\frac{1}{n-1}}}\left| u\right| ^{\frac{n}{n-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( n,\beta \right) \frac{\left\| u\right\| _{n}^{n-\beta }}{\left| 1-{\mathcal {E}}_{n}\left( u\right) ^{n}\right| ^{1-\frac{\beta }{n}}}. \end{aligned}$$

Moreover, the constant \(\left( 1-\frac{\beta }{n}\right) \alpha _{n}\) is the best possible in the sense that there is no uniform constant \(C(n, \beta )\) independent of u in the above inequality when \(\alpha >\left( 1-\frac{\beta }{n}\right) \alpha _{n}\). Second, we establish the following improved Adams type inequality in the spirit of Lions (Theorem 1.8): Let \(0\le \beta <2m\) and \(\tau >0\). Then there exists a constant \(C=C\left( m,\beta ,\tau \right) >0\) such that

$$\begin{aligned} \underset{u\in W^{2,m}\left( {\mathbb {R}}^{2m}\right) , \int _{ {\mathbb {R}}^{2m}}\left| \Delta u\right| ^{m}+\tau \left| u\right| ^{m} \le 1}{\sup } {\displaystyle \int \nolimits _{{\mathbb {R}}^{2m}}} \frac{\phi _{2m,2}\left( \frac{2^{\frac{1}{m-1}}\alpha }{\left( 1+\left\| \Delta u\right\| _{m}^{m}\right) ^{\frac{1}{m-1}}}\left| u\right| ^{\frac{m}{m-1}}\right) }{\left| x\right| ^{\beta }}dx\le C\left( m,\beta ,\tau \right) , \end{aligned}$$

for all \(0\le \alpha \le \left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\). When \(\alpha >\left( 1-\frac{\beta }{2m}\right) \beta (2m,2)\), the supremum is infinite. In the above, we use

$$\begin{aligned} \phi _{p,q}(t)=e^{t}- {\displaystyle \sum \limits _{j=0}^{j_{\frac{p}{q}}-2}} \frac{t^{j}}{j!},\,\,\,j_{\frac{p}{q}}=\min \left\{ j\in {\mathbb {N}} :j\ge \frac{p}{q}\right\} \ge \frac{p}{q}. \end{aligned}$$

The main difficulties of proving the above results are that the symmetrization method does not work. Therefore, our main ideas are to develop a rearrangement-free argument in the spirit of Lam and Lu (J Differ Equ 255(3):298–325, 2013; Adv Math 231(6): 3259–3287, 2012), Lam et al. (Nonlinear Anal 95: 77–92, 2014) to establish such theorems. Third, as an application, we will study the existence of weak solutions to the biharmonic equation

$$\begin{aligned} \left\{ \begin{array}{l} \Delta ^{2}u+V(x)u=f(x,u)\text { in }{\mathbb {R}}^{4}\\ u\in H^{2}\left( {\mathbb {R}}^{4}\right) ,~u\ge 0 \end{array} \right. , \end{aligned}$$

where the nonlinearity f has the critical exponential growth.

     

Volume of Domains in Symmetric Spaces

The authors derive a formula for the volume of a compact domain in a symmetric space from normal sections through a special submanifold in the symmetric space.This formula generalizes the volume of classical domains as tubes or domains given as motions along the submanifold.Finally,some stereological considerations regarding this formula are provided.     

Some Hardy Type Inequalities in R~n

Some Hardy type inequalities on the ball and its complementary set in the Euclidean space are established by using the Picone type identity and constructing suitable auxiliary functions.