Some Results on Subelliptic Equations

In this paper, we consider the principal on R^n, and we find a comparison principle for such semi-linear sub-elliptic equation in the whole R^n and infinitely many positive solutions of the problem. eigenvalue problem for Hormander＇s Laplacian principal eigenvalues. We also study a related prove that, under a suitable condition, we have     

Sharp bounds for the first eigenvalue and the torsional rigidity related to some anisotropic operators

In this paper we prove a sharp upper bound for the first Dirichlet eigenvalue of a class of nonlinear elliptic operators which includes the operator , that is the p‐Laplacian, and , namely the pseudo‐p‐Laplacian. Moreover we prove a stability result by means of a suitable isoperimetric deficit. Finally, we give a sharp lower bound for the anisotropic p‐torsional rigidity.     

An upper bound for nonlinear eigenvalues on convex domains by means of the isoperimetric deficit

We prove an upper bound for the first Dirichlet eigenvalue of the p-Laplacian operator on convex domains. The result implies a sharp inequality where, for any convex set, the Faber-Krahn deficit is dominated by the isoperimetric deficit.     

Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

In this paper we prove general inequalities involving the weighted mean curvature of compact submanifolds immersed in weighted manifolds. As a consequence we obtain a relative linear isoperimetric inequality for such submanifolds. We also prove an extrinsic upper bound to the first non-zero eigenvalue of the drift Laplacian on closed submanifolds of weighted manifolds.     

Resonances on Some Geometrically Finite Hyperbolic Manifolds

Abstract

We first prove the meromorphic extension to ? for the resolvent of the Laplacian on a class of geometrically finite hyperbolic manifolds with infinite volume and we give a polynomial bound on the number of resonances. This class notably contains the quotients Γ\ ⁿ⁺¹ with rational nonmaximal rank cusps previously studied by Froese-Hislop-Perry.     

The distribution of the eigenvalues of the discrete Laplacian

We estimate the distribution of the eigenvalues of the discrete Laplacian on a bounded set inR ⁿ . The proof is based on a variational technique similar to that used by Weyl for the Laplacian. As an application of our estimates we prove stability in the maximum norm for the Crank-Nicolson method for the heat equation on a bounded set.The research of the second author was supported in part by National Science Foundation grant GP-30735X.     

Discrete isoperimetric and Poincaré-type inequalities

We study some discrete isoperimetric and Poincaré-type inequalities for product probability measures μ ⁿ on the discrete cube {0, 1} ⁿ and on the lattice Z ⁿ . In particular we prove sharp lower estimates for the product measures of boundaries of arbitrary sets in the discrete cube. More generally, we characterize those probability distributions μ on Z which satisfy these inequalities on Z ⁿ . The class of these distributions can be described by a certain class of monotone transforms of the two-sided exponential measure. A similar characterization of distributions on R which satisfy Poincaré inequalities on the class of convex functions is proved in terms of variances of suprema of linear processes. Received: 30 April 1997 / Revised version: 5 June 1998     

Estimates for weighted eigenvalues of a fourth-order elliptic operator with variable coefficients

We investigate the Dirichlet weighted eigenvalue problem for a fourth-order elliptic operator with variable coefficients in a bounded domain in \mathbbRⁿ {\mathbb{R}^n} . We establish a sharp inequality for its eigenvalues. It yields an estimate for the upper bound of the (k + 1)th eigenvalue in terms of the first k eigenvalues. Moreover, we also obtain estimates for some special cases of this problem. In particular, our results generalize the Wang–Xia inequality (J. Funct. Anal., 245, No. 1, 334–352 (2007)) for the clamped-plate problem to a fourth-order elliptic operator with variable coefficients.     

A Constant Bound for Geometric Permutations of Disjoint Unit Balls

Abstract. We prove that a set of n disjoint unit balls in R ^d admits at most four distinct geometric permutations, or line transversals, thus settling a long-standing conjecture in combinatorial geometry. The constant bound significantly improves upon the Θ (n ^d-1 ) bound for disjoint balls of unrestricted radii.     

Spherical submanifolds in a Euclidean space

In this paper, we are interested in extending the study of spherical curves in R ³ to the submanifolds in the Euclidean space R ^n+p . More precisely, we are interested in obtaining conditions under which an n-dimensional compact submanifold M of a Euclidean space R ^n+p lies on the hypersphere S ^n+p−1(c) (standardly imbedded sphere in R ^n+p of constant curvature c). As a by-product we also get an estimate on the first nonzero eigenvalue of the Laplacian operator Δ of the submanifold (cf. Theorem 3.5) as well as a characterization for an n-dimensional sphere S ⁿ (c) (cf. Theorem 4.1).     

A Constant Bound for Geometric Permutations of Disjoint Unit Balls

   Abstract. We prove that a set of n disjoint unit balls in R ^d admits at most four distinct geometric permutations, or line transversals, thus settling a long-standing conjecture in combinatorial geometry. The constant bound significantly improves upon the Θ (n ^d-1 ) bound for disjoint balls of unrestricted radii.     

Sharp upper bounds on the number of the scattering poles

We study the scattering poles of a compactly supported “black box” perturbations of the Laplacian in Rⁿ, n odd. We prove a sharp upper bound of the counting function N(r) modulo o(rⁿ) in terms of the counting function of the reference operator in the smallest ball around the black box. In the most interesting cases, we prove a bound of the type N(r)?A_nrⁿ+o(rⁿ) with an explicit A_n. We prove that this bound is sharp in a few special spherically symmetric cases where the bound turns into an asymptotic formula.     

A new Lichnerowicz–Obata estimate in the presence of a parallel p-form

 Let (M ⁿ ,g) be a compact Riemannian manifold with a smooth boundary. In this paper, we give a Lichnerowicz-Obata type lower bound for the first eigenvalue of the Laplacian of (M ⁿ ,g) when M has a parallel p-form (2 ≤p≤n/2). This result follows from a new Bochner-Reilly's formula. Moreover, we give a characterization of the equality case when (M ⁿ ,g) is simply connected. Received: 1 June 2001     

On the isoperimetric problem for the higher eigenvalues of the Robin and Wentzell Laplacians

We consider the problem of minimising the kth eigenvalue, k ≥ 2, of the (p-)Laplacian with Robin boundary conditions with respect to all domains in \mathbbR^N{\mathbb{R}^N} of given volume. When k = 2, we prove that the second eigenvalue of the p-Laplacian is minimised by the domain consisting of the disjoint union of two balls of equal volume, and that this is the unique domain with this property. For p = 2 and k ≥ 3, we prove that in many cases a minimiser cannot be independent of the value of the constant in the boundary condition, or equivalently of the domain’s volume. We obtain similar results for the Laplacian with generalised Wentzell boundary conditions.     

Eigenvalues of the Wentzell–Laplace operator and of the fourth order Steklov problems

We prove a sharp upper bound and a lower bound for the first nonzero eigenvalue of the Wentzell–Laplace operator on compact manifolds with boundary and an isoperimetric inequality for the same eigenvalue in the case where the manifold is a bounded domain in a Euclidean space. We study some fourth order Steklov problems and obtain isoperimetric upper bound for the first eigenvalue of them. We also find all the eigenvalues and eigenfunctions for two kind of fourth order Steklov problems on a Euclidean ball.     

On graphs with the largest Laplacian index

Let G be a connected simple graph on n vertices. The Laplacian index of G, namely, the greatest Laplacian eigenvalue of G, is well known to be bounded above by n. In this paper, we give structural characterizations for graphs G with the largest Laplacian index n. Regular graphs, Hamiltonian graphs and planar graphs with the largest Laplacian index are investigated. We present a necessary and sufficient condition on n and k for the existence of a k-regular graph G of order n with the largest Laplacian index n. We prove that for a graph G of order n ⩾ 3 with the largest Laplacian index n, G is Hamiltonian if G is regular or its maximum vertex degree is Δ(G) = n/2. Moreover, we obtain some useful inequalities concerning the Laplacian index and the algebraic connectivity which produce miscellaneous related results. The first author is supported by NNSF of China (No. 10771080) and SRFDP of China (No. 20070574006). The work was done when Z. Chen was on sabbatical in China.     

On the division by R n

Summary In this paper we prove that, ifS ×R ⁿ is homeomorphic toR ^{n + 1}, thenS is homeomorphic toR.     

Bounds on eigenvalues of Dirichlet Laplacian

In this paper, we investigate an eigenvalue problem of Dirichlet Laplacian on a bounded domain Ω in an n-dimensional Euclidean space R ⁿ . If λ _k+1 is the (k + 1)th eigenvalue of Dirichlet Laplacian on Ω, then, we prove that, for n ≥ 41 and and, for any n and with , where j _p,k denotes the k-th positive zero of the standard Bessel function J _p (x) of the first kind of order p. From the asymptotic formula of Weyl and the partial solution of the conjecture of Pólya, we know that our estimates are optimal in the sense of order of k.Q.-M. Cheng was partially Supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of ScienceH. Yang was partially Supported by Chinese NSF, SF of CAS and NSF of USA     

Extensions of Well-Boundedness and <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>-Scalarity

We consider some extensions of well-boundedness and C^m-scalarity by using fractional calculus, and prove some theorems accordingly. These results are applied to the usual Laplacian on Rⁿ and sub-Laplacians on nilpotent Lie groups. The research of first and second authors has been partly supported by the Project MTM2004-03036 of the M.C.YT.-DGI/F.E.D.E.R., Spain, and the Project E-12/25, D. G. Aragón, Spain. Part of the research of second author was developed in the Christian-Albrechts Universit?t in Kiel, while he was enjoying a HARP-postdoctoral position in the European Harmonic Analysis Network, HARP, IHP 2002-06.     

On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 1. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of n identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch-Payne-Schiffer inequality for n = 2 and show that it is strict in this case.