Some upper bounds for sums of eigenvalues of the Neumann Laplacian

Let be the th Neumann eigenvalue of a bounded domain with piecewisely smooth boundary in . In 1992, P. Kröger proved that , where the upper bound is sharp in view of Weyl's asymptotic formula. The aim of this paper is twofold. First, we will improve this estimate by multiplying a factor in terms of to its right-hand side which approaches strictly from below to 1 as tends to infinity. Second, we will generalize Kröger's estimate to the case when is a compact Euclidean submanifold.

     

Complex Powers of the Neumann Laplacian in Lipschitz Domains

Let L^q_r(Ω) be the usual scale of Sobolev spaces and let Δ_N be the Neumann Laplacian in an arbitrary Lipschitz domain Ω. We present an interpolation based approach to the following question: for what range of indices does map isomorphically onto L^q_r(Ω)/ℝ?     

Gaps in the spectrum of the Neumann Laplacian generated by a system of periodically distributed traps

The article deals with a convergence of the spectrum of the Neumann Laplacian in a periodic unbounded domain Ω^ϵ depending on a small parameter ϵ > 0. The domain has the form , where S^ϵ is an -periodic family of trap-like screens. We prove that, for an arbitrarily large L, the spectrum has precisely one gap in [0,L] when ϵ is small enough; moreover, when ϵ → 0, this gap converges to some interval whose edges can be controlled by a suitable choice of geometry of the screens. An application to the theory of 2D photonic crystals is discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Extremal metrics for the first eigenvalue of the Laplacian in a conformal class

Let be a compact manifold. First, we give necessary and sufficient conditions for a Riemannian metric on to be extremal for with respect to conformal deformations of fixed volume. In particular, these conditions show that for any lattice of , the flat metric induced on from the standard metric of is extremal (in the previous sense). In the second part, we give, for any , an upper bound of on the conformal class of and exhibit a class of lattices for which the metric maximizes on its conformal class.

     

Sharp norm estimates for a class of integral operators

ABSTRACT

Norm comparison inequalities for two integral operators with radial kernels are established. Sharp norm estimates for operators with monotone and convex/concave kernels are obtained. Integral analogues of Bennett's estimates for summability matrices are given. The exact operator norms with power weights are also obtained for a class of integral operators with radial quasimonotone kernels.     

Two new Weyl-type bounds for the Dirichlet Laplacian

In this paper, we prove two new Weyl-type upper estimates for the eigenvalues of the Dirichlet Laplacian. As a consequence, we obtain the following lower bounds for its counting function. For , one has

and

where

is a constant which depends on , the dimension of the underlying space, and Bessel functions and their zeros.

     

On coefficient estimates for a class of holomorphic mappings

Let X be a complex Banach space with norm ‖ · ‖, B be the unit ball in X, D ⁿ be the unit polydisc in ℂ ⁿ . In this paper, we introduce a class of holomorphic mappings on B or D ⁿ . Let f(x) be a normalized locally biholomorphic mapping on B such that (Df(x))⁻¹ f(x) ∈ and f(x) − x has a zero of order k + 1 at x = 0. We obtain coefficient estimates for f(x). These results unify and generalize many known results. This work was supported by National Natural Science Foundation of China (Grant No. 10571164), Specialized Research Fund for the Doctoral Program of Higher Education (Grant No. 20050358052), the Jiangxi Provincial Natural Science Foundation of China (Grant No. 2007GZS0177) and Specialized Research Fund for the Doctoral Program of Jiangxi Normal University.     

Schatten-von Neumann Properties in the Weyl Calculus, and Calculus of Metrics on Symplectic Vector Spaces

Let s ^w p be the set of all a ∈ ? such that a ^w (x, D) is Schatten p-operator on L ². Then we prove the following:

• $S(m,g)\subseteq s_p^wLet s ^w p be the set of all a ∈ ℓ such that a ^w (x, D) is Schatten p-operator on L ². Then we prove the following:

  •	iff . Furthermore, when . Consequently, when ;
  •	if , then is symplectically invariantly defined. Moreover, if and is slowly varying (and σ-temperate), then the same is true for G;
  •	a generalization of sharp G?rding's inequality.

  Mathematics Subject Classifications (2000) Primary: 35S05, 47B10, 47L15 Secondary: 32F45, 16W80     

Gradient estimates for positive solutions of the Laplacian with drift

Let be a complete Riemannian manifold of dimension without boundary and with Ricci curvature bounded below by where If is a vector field such that and on for some nonnegative constants and then we show that any positive solution of the equation satisfies the estimate

on , for all In particular, for the case when this estimate is advantageous for small values of and when it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201-228).

     

A remark on Schatten-von Neumann properties of resolvent differences of generalized Robin Laplacians on bounded domains

In this note we investigate the asymptotic behavior of the s-numbers of the resolvent difference of two generalized self-adjoint, maximal dissipative or maximal accumulative Robin Laplacians on a bounded domain Ω with smooth boundary ∂Ω. For this we apply the recently introduced abstract notion of quasi boundary triples and Weyl functions from extension theory of symmetric operators together with Krein type resolvent formulae and well-known eigenvalue asymptotics of the Laplace-Beltrami operator on ∂Ω. It is shown that the resolvent difference of two generalized Robin Laplacians belongs to the Schatten-von Neumann class of any order p for which

     

Sharp upper global a posteriori error estimates for nonlinear elliptic variational problems

The paper is devoted to the problem of verification of accuracy of approximate solutions obtained in computer simulations. This problem is strongly related to a posteriori error estimates, giving computable bounds for computational errors and detecting zones in the solution domain where such errors are too large and certain mesh refinements should be performed. A mathematical model embracing nonlinear elliptic variational problems is considered in this work. Based on functional type estimates developed on an abstract level, we present a general technology for constructing computable sharp upper bounds for the global error for various particular classes of elliptic problems. Here the global error is understood as a suitable energy type difference between the true and computed solutions. The estimates obtained are completely independent of the numerical technique used to obtain approximate solutions, and are sharp in the sense that they can be, in principle, made as close to the true error as resources of the used computer allow. The latter can be achieved by suitably tuning the auxiliary parameter functions, involved in the proposed upper error bounds, in the course of the calculations.     

Banach空间中二阶微分方程Neumann边值问题的解

本文在序Banach空间中讨论二阶非线性微分方程的Neumann边值问题 :-u″=f(t,u) ,u′( 0 ) =u′( 1 ) =θ.在上下解反向给定时 ,利用半序理论和新的比较原理 ,证明了此Neumann边值问题最小解和最大解的存在性 ,解的唯一性 ,并给出了唯一解的近似迭代序列的误差估计式 .     

Convergence and decay estimates for a class of second order dissipative equations involving a non-negative potential energy

We estimate the rate of decay of the difference between a solution and its limiting equilibrium for the following abstract second order problem

     

On a refinement of the generalized Catalan numbers for Weyl groups

Let be an irreducible crystallographic root system with Weyl group , coroot lattice and Coxeter number , spanning a Euclidean space , and let be a positive integer. It is known that the set of regions into which the fundamental chamber of is dissected by the hyperplanes in of the form for and is equinumerous to the set of orbits of the action of on the quotient . A bijection between these two sets, as well as a bijection to the set of certain chains of order ideals in the root poset of , are described and are shown to preserve certain natural statistics on these sets. The number of elements of these sets and their corresponding refinements generalize the classical Catalan and Narayana numbers, which occur in the special case and .

     

Weyl almost automorphic solutions for a class of Clifford-valued dynamic equations with delays on time scales

Weyl almost automorphy is a natural generalization of Bochner almost automorphy and Stepanov almost automorphy. However, the space composed of Weyl almost automorphic functions is not a Banach space. Therefore, the results of the existence of Weyl almost automorphic solutions of differential equations are few, and the results of the existence of Weyl almost automorphic solutions of difference equations are rare. Since the study of dynamic equations on time scales can unify the study of differential equations and difference equations. Therefore, in this paper, we first propose a concept of Weyl almost automorphic functions on time scales and then take the Clifford-valued shunt inhibitory cellular neural networks with time-varying delays on time scales as an example of dynamic equations on time scales to study the existence and global exponential stability of their Weyl almost automorphic solutions. We also give a numerical example to illustrate the feasibility of our results.     

Ricci Flow and the Determinant of the Laplacian on Non-Compact Surfaces

On compact surfaces with or without boundary, Osgood, Phillips and Sarnak proved that the maximum of the determinant of the Laplacian within a conformal class of metrics with fixed area occurs at a metric of constant curvature and, for negative Euler characteristic, exhibited a flow from a given metric to a constant curvature metric along which the determinant increases. The aim of this paper is to perform a similar analysis for the determinant of the Laplacian on a non-compact surface whose ends are asymptotic to hyperbolic funnels or cusps. In that context, we show that the Ricci flow converges to a metric of constant curvature and that the determinant increases along this flow.     

Sharp estimates for the identity minus Hardy operator on the cone of decreasing functions

It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions in , then we have the sharp estimate

for In other words,

for each and each integer .

It is also shown, via a connection between the operator and Laguerre functions, that

for all .

     

Bifurcation properties for a class of fractional Laplacian equations in

This paper concerns with the study of some bifurcation properties for the following class of nonlocal problems (P) where , , , is a positive continuous function, is a bounded continuous function and is the fractional Laplacian. The main tools used are the Leray–Shauder degree theory and the global bifurcation result due to Rabinowitz.     

Analytic proof of dual variational formula for the first eigenvalue in dimension one

The first non-zero eigenvalue is the leading term in the spectrum of a self-adjoint operator. It plays a critical role in various applications and is treated in a large number of textbooks. There is a well-known variational formula for it (called the Min-Max Principle) which is especially effective for an upper bound of the eigenvalue. However, for the lower bound of the spectral gap, some dual variational formulas have been obtained only very recently. The original proofs are probabilistic. Some analytic proofs in one-dimensional case are proposed and certain extension is made. Project supported in part by the National Natural Science Foundation of China (Grant No. 19631060), Qiu Shi Science & Technology Foundation, DPFIHE, MCSEC and MCMCAS.