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.     

The parametrix method for parabolic SPDEs

We consider the Cauchy problem for a linear stochastic partial differential equation. By extending the parametrix method for PDEs whose coefficients are only measurable with respect to the time variable, we prove existence, regularity in Hölder classes and estimates from above and below of the fundamental solution. This result is applied to SPDEs by means of the Itô–Wentzell formula, through a random change of variables which transforms the SPDE into a PDE with random coefficients.     

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.     

Improved parametrix in the glancing region for the interior Dirichlet-to-Neumann map

We study the semiclassical microlocal structure of the Dirichlet-to-Neumann map for an arbitrary compact Riemannian manifold with a nonempty smooth boundary. We build a new, improved parametrix in the glancing region compaired with that one built in [Vodev, Commun. Math. Phys. 2015;336(3):1141–1166; Vodev, Asymptotic Anal. 2018;106:147–168]. We also study the way in which the parametrix depends on the refraction index. As a consequence, we improve the transmission eigenvalue-free regions obtained in (Vodev, Asymptotic Anal. 2018;106:147–168) in the isotropic case when the restrictions of the refraction indices on the boundary coincide.     

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.     

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.     

Second order probabilistic parametrix method for unbiased simulation of stochastic differential equations

In this article, following the paradigm of bias–variance trade-off philosophy, we derive parametrix expansions of order two, based on the Euler–Maruyama scheme with random partitions, for the purpose of constructing an unbiased simulation method for multidimensional stochastic differential equations. These formulas lead to Monte Carlo simulation methods which can be easily parallelized. The second order method proposed here requires further regularity of coefficients in comparison with the first order method but achieves finite moments even when Poisson sampling is used for the partitions, in contrast to Andersson and Kohatsu-Higa (2017). Moreover, using an exponential scaling technique one achieves an unbiased simulation method which resembles a space importance sampling technique which significantly improves the efficiency of the proposed method. A hint of how to derive higher order expansions is also presented.