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.     

Gagliardo–Nirenberg inequalities in regular Orlicz spaces involving nonlinear expressions

We consider a triple of N-functions (M,H,J) that satisfy the Δ^′-condition, and suppose that an additive variant of interpolation inequality holds

where , is an arbitrary set invariant with respect to external and internal dilations. We show that the above inequality implies its certain nonlinear variant involving the expressions and . Various generalizations of this inequality to the more general class of N-functions, measures and to higher order derivatives are also discussed and the examples are presented.     

Embedding property on rearrangement invariant spaces

Let X be a rearrangement invariant space in R ⁿ and $W_X^{r_1 ,...,r_n } $ be an anisotropic Sobolev space which is a generalization of $W_p^{r_1 ,...,r_n } $ . The main subject of this paper is to prove the embedding theorem for $W_X^{r_1 ,...,r_n } $ .     

Generalized Gagliardo–Nirenberg estimates and differentiability of the solutions to monotone nonlinear parabolic systems

In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo–Nirenberg’s type in generalized Sobolev spaces, we show that u belongs to (see Theorem 3).     

索伯列夫嵌入不等式的证明

We consider the problem about the space embedded by tile space and the embed-ding inequality. With the Holder inequality and interpolation inequality, we give the proof of the space embedding theorem and the space holder embedding theorem.     

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.     

A New Self-Dual Embedding Method for Convex Programming

In this paper we introduce a conic optimization formulation to solve constrained convex programming, and propose a self-dual embedding model for solving the resulting conic optimization problem. The primal and dual cones in this formulation are characterized by the original constraint functions and their corresponding conjugate functions respectively. Hence they are completely symmetric. This allows for a standard primal-dual path following approach for solving the embedded problem. Moreover, there are two immediate logarithmic barrier functions for the primal and dual cones. We show that these two logarithmic barrier functions are conjugate to each other. The explicit form of the conjugate functions are in fact not required to be known in the algorithm. An advantage of the new approach is that there is no need to assume an initial feasible solution to start with. To guarantee the polynomiality of the path-following procedure, we may apply the self-concordant barrier theory of Nesterov and Nemirovski. For this purpose, as one application, we prove that the barrier functions constructed this way are indeed self-concordant when the original constraint functions are convex and quadratic. We pose as an open question to find general conditions under which the constructed barrier functions are self-concordant.     

Resolvent Estimates for Fleming–Viot Operators with Brownian Drift

This article is a supplement to the paper of D. A. Dawson and P. March (J. Funct. Anal.132(1995), 417–472). We define a two-parameter scale of Banach spaces of functions defined on ₁( ^d), the space of probability measures ond-dimensional euclidean space, using weighted sums of the classical Sobolev norms. We prove that the resolvent of the Fleming–Viot operator with constant diffusion coefficient and Brownian drift acts boundedly between certain members of the scale. These estimates gauge the degree of smoothing performed by the resolvent and separate the contribution due to the diffusion coefficient and that due to the drift coefficient.     

Embedding Theorems for a Certain Function Class of Sobolev Type on Metric Spaces

Given a metric space with a Borel measure , we consider a class of functions whose increment is controlled by the measure of a ball containing the corresponding points and a nonnegative function p-summable with respect to . We prove some analogs of the classical theorems on embedding Sobolev function classes into Lebesgue spaces.     

Embedding constants for periodic Sobolev spaces of fractional order

We obtain an explicit expression for the norms of the embedding operators of the periodic Sobolev spaces into the space of continuous functions (the norms of this type are usually called embedding constants). The corresponding formulas for the embedding constants express them in terms of the values of the well-known Epstein zeta function which depends on the smoothness exponent s of the spaces under study and the dimension n of the space of independent variables. We establish that the embeddings under consideration have the embedding functions coinciding up to an additive constant and a scalar factor with the values of the corresponding Epstein zeta function. We find the asymptotics of the embedding constants as s → n/2.     

On Inequalities with Measures of Sobolev Type Embedding Theorems on Open Sets of the Real Axis

We consider some Sobolev-type spaces and obtain a necessary and sufficient condition for their embedding in a Lebesgue space.     

Singular cotangent bundle reduction & spin Calogero–Moser systems

We develop a bundle picture for singular symplectic quotients of cotangent bundles acted upon by cotangent lifted actions for the case that the configuration manifold is of single orbit type. Furthermore, we give a formula for the reduced symplectic form in this setting. As an application of this bundle picture we consider Calogero–Moser systems with spin associated to polar representations of compact Lie groups.     

Estimates for the Sobolev trace constant with critical exponent and applications

In this paper we find estimates for the optimal constant in the critical Sobolev trace inequality that are independent of Ω. This estimates generalized those of Adimurthi and Yadava (Comm Partial Diff Equ 16(11):1733–1760, 1991) for general p. Here p _* : =  p(N  −  1)/(N  −  p) is the critical exponent for the immersion and N is the space dimension. Then we apply our results first to prove existence of positive solutions to a nonlinear elliptic problem with a nonlinear boundary condition with critical growth on the boundary, generalizing the results of Fernández Bonder and Rossi (Bull Lond Math Soc 37:119–125, 2005). Finally, we study an optimal design problem with critical exponent.      

Coarse Embedding into Uniformly Convex Banach Spaces

In this paper, the author studies the coarse embedding into uniformly convex Banach spaces. The author proves that the property of coarse embedding into Banach spaces can be preserved under taking the union of the metric spaces under certain conditions. As an application, for a group G strongly relatively hyperbolic to a subgroup H,the author proves that B(n) = {g ∈ G | |g|S∪H≤ n} admits a coarse embedding into a uniformly convex Banach space if H does.     

A Trudinger–Moser inequality in a weighted Sobolev space and applications

We establish a Trudinger–Moser type inequality in a weighted Sobolev space. The inequality is applied in the study of the elliptic equation where , f has exponential critical growth and h belongs to the dual of an appropriate function space. We prove that the problem has at least two weak solutions provided is small.     

预给二面角的m面凸多胞形嵌入Rd的充分必要条件

本文给出了预给二面角的ｍ面凸多胞形嵌入Ｒ^d的充分必要条件     

Concentration compactness of Moser functionals on manifolds

We will prove a concentration compactness property of the Moser functional on a compact Riemannian manifold.     

Optimizers for the singular Trudinger–Moser inequalities in the critical case in

The main purpose of this paper is to study the existence of extremal functions for the singular Trudinger–Moser inequalities in the critical case in . More precisely, let and denote then we will prove in this article that there exists such that can be achieved for all , .     

Gagliardo-Nirenberg inequalities on manifolds

We prove Gagliardo-Nirenberg inequalities on some classes of manifolds, Lie groups and graphs.     

Random Attractor for Stochastic Partly Dissipative Systems on Unbounded Domains

In this paper, we consider the long time behaviors for the partly dissipative stochastic reaction diffusion equations. The existence of a bounded random absorbing set is firstly discussed for the systems and then an estimate on the solution is derived when the time is sufficiently large. Then, we establish the asymptotic compactness of the solution operator by giving uniform a priori estimates on the tails of solutions when time is large enough. In the last, we finish the proof of existence a pullback random attractor in L²(R^n) × L²(R^n). We also prove the upper semicontinuity of random attractors when the intensity of noise approaches zero. The long time behaviors are discussed to explain the corresponding physical phenomenon.