Ridged Domains,Embedding Theorems and Poincaré Inequalities

We study the embeddings E : W(X(Ω), Y(Ω)) ↪ Z(Ω), where X(Ω), Y(Ω) and Z(Ω) are rearrangement–invariant Banach function spaces (BFS) defined on a generalized ridged domain Ω, and W denotes a first–order Sobolev–type space. We obtain two–sided estimates for the measure of non–compactness of E when Z(Ω) = X(Ω) and, in turn, necessary and sufficient conditions for a Poincaré–type inequality to be valid and also for E to be compact. The results are used to analyse the example of a trumpet–shaped domain Ω in Lorentz spaces. We consider the problem of determining the range of possible target spaces Z(Ω), in which case we prove that the problem is equivalent to an analogue on the generalized ridge Γ of Ω. The range of target spaces Z(Ω) is determined amongst a scale of (weighted) Lebesgue spaces for “rooms and passages” and trumpet–shaped domains.     

Schur-Convex Functions and Isoperimetric Inequalities

In this paper, we establish some analytic inequalities for Schur-convex functions that are made of solutions of a second order nonlinear differential equation. We apply these analytic inequalities to obtain some geometric inequalities.

     

Rienmannian Submersions and Isoperimetric Inequalities

Isoperimetric constants of the total spaces of Riemannian submersions are estimated in terms of those of the basis and fibers.     

Isoperimetric Inequalities for Random Walks

In this paper some isoperimetric problems are studied, particularly the extremal property of the mean exit time of the random walk from finite sets. This isoperimetric problem is inserted into the set of equivalent conditions of the diagonal upper estimate of transition probability of random walks on weighted graphs.     

Bonnesen型Ros等周不等式

首先定义了欧氏平面上的Ros等周亏格,得到了Ros等周亏格与经典的等周亏格之间的关系,讨论了Bonnesen型Ros等周不等式.利用前人的一些著名的Bonnesen型不等式,我们得到了一些Bonnesen型Ros等周不等式.     

Isoperimetric Type Inequalities and Hypersurface Flows

New types of hypersurface flows have been introduced recently with goals to establish isoperimetric type inequalities in geometry. These flows serve as efficient paths to achieve the optimal solutions to the problems of calculus of variations in geometric setting. The main idea is to use variational structures to develop hypersurface flows which are monotonic for the corresponding curvature integrals (including volume and surface area). These new geometric flows pose interesting but challenging PDE problems. Resolution of these problems have significant geometric implications.     

体积差的等周不等式

首次提出并建立了凸体的体积差函数的等周不等式,它是经典等周不等式的推广.作为应用,对星体建立了体积差函数的对偶等周不等式和广义对偶等周不等式.     

Isoperimetric Inequalities for Soluble Groups

We approach the question of which soluble groups are automatic. We describe a class of nilpotent-by-Abelian groups which need to be studied in order to answer this question. We show that the nilpotent-by-cyclic groups in this class have exponential isoperimetric inequality and so cannot be automatic.     

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.     

Isoperimetric Inequalities for a Wedge-Like Membrane

For a wedge-like membrane, Payne and Weinberger proved in 1960 an isoperimetric inequality for the fundamental eigenvalue which in some cases improves the classical isoperimetric inequality of Faber–Krahn. In this work, we introduce “relative torsional rigidity” for this type of membrane and prove new isoperimetric inequalities in the spirit of Saint-Venant, Pólya–Szeg?, Payne, Payne–Rayner, Chiti, and Talenti, which link the eigenvalue problem with the boundary value problem in a fundamental way.     

Isodiametric and Isoperimetric Inequalities for Complexes and Groups

It is shown that D. Cohen's inequality bounding the isoperimetricfunction of a group by the double exponential of its isodiametricfunction is valid in the more general context of locally finitesimply connected complexes. It is shown that in this contextthis bound is ‘best possible’. Also studied aresecond-dimensional isoperimetric functions for groups and complexes.It is shown that the second-dimensional isoperimetric functionof a group is bounded by a recursive function. By a similarargument it is shown that the area distortion of a finitelypresented subgroup of a finitely presented group is recursive.Cohen's inequality is extended to second-dimensional isoperimetricand isodiametric functions of 2-connected simplicial complexes.     

Some Isoperimetric Inequalities in Electrochemistry and Hele Shaw Flows

Isoperimetric inequalities are applied to a moving-boundaryproblem for doubly-connected domains. This problem occurs forexample in electrochemistry, in which case the domains in questionare the electrolyte of an electrolytic cell. The two electrodessurrounding the electrolyte are assumed to grow or dissolve,at different rates in general, by electrochemical reaction.We obtain optimal estimates showing, for example, that the leastchange in volume of each electrode always occurs in sphericalsymmetry.     

关于Wulff流情形下的等周不等式的注记

该文主要研究平面上Wulff流情形下的等周不等式.利用凸域的某些量在Wulff流情形下的变化规律(单调性、不变性),得到了Wulff-Gage等周不等式与曲率的Wulff-熵不等式的新的简单证明;进一步地,得到了一个新的Wulff流情形下的不等式.     

Isoperimetric Inequalities for Ramanujan Complexes and Topological Expanders

Expander graphs have been intensively studied in the last four decades (Hoory et al., Bull Am Math Soc, 43(4):439–562, 2006; Lubotzky, Bull Am MathSoc, 49:113–162, 2012). In recent years a high dimensional theory of expanders has emerged, and several variants have been studied. Among them stand out coboundary expansion and topological expansion. It is known that for every d there are unbounded degree simplicial complexes of dimension d with these properties. However, a major open problem, formulated by Gromov (Geom Funct Anal 20(2):416–526, 2010), is whether bounded degree high dimensional expanders exist for \({d \geq 2}\). We present an explicit construction of bounded degree complexes of dimension \({d = 2}\) which are topological expanders, thus answering Gromov’s question in the affirmative. Conditional on a conjecture of Serre on the congruence subgroup property, infinite sub-family of these give also a family of bounded degree coboundary expanders. The main technical tools are new isoperimetric inequalities for Ramanujan Complexes. We prove linear size bounds on \({\mathbb{F}_2}\) systolic invariants of these complexes, which seem to be the first linear \({\mathbb{F}_2}\) systolic bounds. The expansion results are deduced from these isoperimetric inequalities.     

Second Order Isoperimetric Inequalities of Split Extensions

In this paper, we set up the general upper bounds for the second order dehn functions of split extensions.AMS Subject Classification (2000): 20F32This work was partially supported by China State Education Ministry SEYTF Funds and China Guangdong Province GDNSF (000864) Fund.     

Isoperimetric Inequalities for Densities of Lattice-Periodic Sets

 The minimum boundary length density of a lattice-periodic set with given period lattice and area density is determined, together with the extremal sets, and a conjecture on the higher-dimensional analogue is made. This improves previous results of Hadwiger for -periodic d-dimensional sets and of Schnell and Wills on two-dimensional sets with arbitrary period-lattice.     

Isoperimetric Inequalities for Densities of Lattice-Periodic Sets

 The minimum boundary length density of a lattice-periodic set with given period lattice and area density is determined, together with the extremal sets, and a conjecture on the higher-dimensional analogue is made. This improves previous results of Hadwiger for -periodic d-dimensional sets and of Schnell and Wills on two-dimensional sets with arbitrary period-lattice. Received 21 May 1997; in revised form 1 December 1997     

强Z-连续domain和Z-代数domain的刻画定理

给出强Z-连续domain和Z-代数domain的一个刻画及一个范畴性质--余反射性质.     

Local Isoperimetric Inequalities for Sectors on Surfaces and Cones

On surfaces we give conditions under which the solution of a restricted local isoperimetric problem for sectors with small solid angle is the circular sector and we characterize these surfaces. Also we study this problem for general spherical cones on hypersurfaces in higher dimensional Riemannian manifolds.