Resolvent Estimates for the Laplacian on Asymptotically Hyperbolic Manifolds

Combining results of Cardoso-Vodev [6] and Froese-Hislop [9], we use Mourre’s theory to prove high energy estimates for the boundary values of the weighted resolvent of the Laplacian on an asymptotically hyperbolic manifold. We derive estimates involving a class of pseudo-differential weights which are more natural in the asymptotically hyperbolic geometry than the weights used in [6]. submitted 28/04/05, accepted 26/09/05     

Estimates on Eigenvalues of Laplacian

In this paper, we study eigenvalues of Laplacian on either a bounded connected domain in an n-dimensional unit sphere Sⁿ(1), or a compact homogeneous Riemannian manifold, or an n-dimensional compact minimal submanifold in an N-dimensional unit sphere S^N(1). We estimate the k+1-th eigenvalue by the first k eigenvalues. As a corollary, we obtain an estimate of difference between consecutive eigenvlaues. Our results are sharper than ones of P. C. Yang and Yau [25], Leung [19], Li [20] and Harrel II and Stubbe [12], respectively. From Weyls asymptotical formula, we know that our estimates are optimal in the sense of the order of k for eigenvalues of Laplacian on a bounded connected domain in an n-dimensional unit sphere Sⁿ(1).Mathematics Subject Classification (2000): 35P15, 58G25, 53C42Research was partially supported by a Grant-in-Aid for Scientific Research from the Japan Society for the Promotion of Science.Research was partially Supported by SF of CAS, Chinese NSF and NSF of USA.     

Some Estimates for the Symmetrized First Eigenfunction of the Laplacian

In this paper a method is developed to study the first eigenfunction u>0 of the Laplacian. It is based on a study of the distribution function for u. The distribution function satisfies an integro–differential inequality, and by introducing a maximal solution Z of the corresponding equation, bounds obtained for Z are then used to estimate u. These bounds come from a detailed study of Z, especially the basic identity derived in Theorem 3.1.     

On Sublevel Set Estimates and the Laplacian

Potential Analysis - Carbery proved that if $u:\mathbb {R}^{n} \rightarrow \mathbb {R}$ is a positive, strictly convex function satisfying $\det D^{2}u \geq 1$ , then we have the estimate $$ \left|...     

Estimates for eigenvalues of Laplacian operator with any order

Let D be a bounded domain in an n-dimensional Euclidean space Rn. Assume that 0 ＜ λ1 ≤λ2 ≤ … ≤ λκ ≤ … are the eigenvalues of the Dirichlet Laplacian operator with any order l{(-△)lu=λu, in D u=(δ)u/(δ)(→n)=…(δ)l-1u/(δ)(→n)l-1=0,on (δ)D.Then we obtain an upper bound of the (k 1)-th eigenvalue λκ 1 in terms of the first k eigenvalues.k∑i=1(λκ 1-λi) ≤ 1/n[4l(n 2l-2)]1/2{k∑i=1(λκ 1-λi)1/2λil-1/l k∑i=1(λκ 1-λi)1/2λ1/li}1/2.This ineguality is independent of the domain D. Furthermore, for any l ≥ 3 the above inequality is better than all the known results. Our rusults are the natural generalization of inequalities corresponding to the case l = 2 considered by Qing-Ming Cheng and Hong-Cang Yang. When l = 1, our inequalities imply a weaker form of Yang inequalities. We aslo reprove an implication claimed by Cheng and Yang.     

Estimates for lower order eigenvalues of quadratic polynomials of the Laplacian

Extending the results of Cheng et al. [8], we study eigenvalues of lower order of quadratic polynomial of the Laplacian on a bounded domain in a complete Riemannian manifold and obtain sharp universal inequalities for them.     

Estimates of the Green Function for the Fractional Laplacian Perturbed by Gradient

The Green function of the fractional Laplacian of the differential order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth multidimensional open sets if the drift function is in an appropriate Kato class.     

含无穷拉普拉斯算子的渗流问题的李普希兹估计

本文研究含无穷拉普拉斯算子的渗流问题.运用改进的Bernstein方法和光滑逼近,分别建立了该问题严格正粘性解和非负粘性解关于空间变量的李普希兹估计.     

Estimates of eigenvalues of the Laplacian by a reduced number of subsets

Chung–Grigor’yan–Yau’s inequality describes upper bounds of eigenvalues of the Laplacian in terms of subsets (“input”) and their volumes. In this paper we will show that we can reduce “input” in Chung–Grigor’yan–Yau’s inequality in the setting of Alexandrov spaces satisfying CD(0,∞). We will also discuss a related conjecture for some universal inequality among eigenvalues of the Laplacian.     

A parametrix for the linear elastic equation with diffractive boundary and its application

In this paper, we construct parametrices near diffractive points for the boundary value problems for the linear elastic equation with free boundary condition or Dirichlet boundary condition. Naturally, our construction is similar to that for the wave equation case. However, since the linear elastic equation is a second order system, our method is more complicated. As an application to the existence of the parametrices, we prove the theorem on propagation of singularities for solutions of the boundary value problem.     

Variational Inequalities for the Fractional Laplacian

In this paper we study the obstacle problems for the fractional Lapalcian of order s ∈ (0, 1) in a bounded domain \({\Omega }\subset \mathbb {R}^{n}\), under mild assumptions on the data.     

A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces

In this paper we construct a parametrix for the forward fundamental solution of the wave and Klein-Gordon equations on asymptotically de Sitter spaces without caustics. We use this parametrix to obtain asymptotic expansions for solutions of (□−λ)u=f and to obtain a uniform Lp estimate for a family of bump functions traveling to infinity.     

Bounds for the signless Laplacian energy

The energy of a graph G is the sum of the absolute values of the eigenvalues of the adjacency matrix of G. The Laplacian (respectively, the signless Laplacian) energy of G is the sum of the absolute values of the differences between the eigenvalues of the Laplacian (respectively, signless Laplacian) matrix and the arithmetic mean of the vertex degrees of the graph. In this paper, among some results which relate these energies, we point out some bounds to them using the energy of the line graph of G. Most of these bounds are valid for both energies, Laplacian and signless Laplacian. However, we present two new upper bounds on the signless Laplacian which are not upper bounds for the Laplacian energy.     

Generalization for Laplacian energy

Let G be a simple graph with n vertices and m edges. Let λ1, λ2,…, λn, be the adjacency spectrum of G, and let μ1, μ2,…, μn be the Laplacian spectrum of G. The energy of G is E（G） = n∑i=1｜λi｜, while the Laplacian energy of G is defined as LE（G） = n∑i=1｜μi-2m/n｜ Let γ1, γ2, ~ …, γn be the eigenvalues of Hermite matrix A. The energy of Hermite matrix as HE（A） = n∑i=1｜γi-tr（A）/n｜ is defined and investigated in this paper. It is a natural generalization of E（G） and LE（G）. Thus all properties about energy in unity can be handled by HE（A）.