Homogenization of eigenvalue problem for Laplace-Beltrami operator on Riemannian manifold with complicated ‘bubble-like’ microstructure

2009 ◽  
Vol 32 (16) ◽  
pp. 2123-2137 ◽  
Author(s):  
Andrii Khrabustovskyi
Keyword(s):  
Riemannian Manifold ◽  
Eigenvalue Problem ◽  
Beltrami Operator

scholarly journals Hearing the shape of a compact Riemannian manifold with a finite number of piecewise impedance boundary conditions

1997 ◽  
Vol 20 (2) ◽  
pp. 397-402 ◽  
Author(s):  
E. M. E. Zayed
Keyword(s):  
Riemannian Manifold ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Function ◽  
Compact Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Smooth Functions ◽  
Impedance Boundary ◽  
Impedance Boundary Conditions

The spectral functionΘ(t)=∑i=1∞exp(−tλj), where{λj}j=1∞are the eigenvalues of the negative Laplace-Beltrami operator−Δ, is studied for a compact Riemannian manifoldΩof dimension “k” with a smooth boundary∂Ω, where a finite number of piecewise impedance boundary conditions(∂∂ni+γi)u=0on the parts∂Ωi(i=1,…,m)of the boundary∂Ωcan be considered, such that∂Ω=∪i=1m∂Ωi, andγi(i=1,…,m)are assumed to be smooth functions which are not strictly positive.

scholarly journals Homogenization of the spectral problem on the Riemannian manifold consisting of two domains connected by many tubes

2013 ◽  
Vol 143 (6) ◽  
pp. 1255-1289 ◽  
Author(s):  
Andrii Khrabustovskyi
Keyword(s):  
Riemannian Manifold ◽  
Small Parameter ◽  
Asymptotic Behaviour ◽  
Spectral Problem ◽  
Spectral Parameter ◽  
Beltrami Operator ◽  
Perforated Domain ◽  
Perforated Domains ◽  
Accumulation Points ◽  
Image Position

The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limitsare positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

The Heat Kernel and Green's Function on a Manifold with Heisenberg Group as Boundary

2004 ◽  
Vol 56 (3) ◽  
pp. 590-611
Author(s):  
Yilong Ni
Keyword(s):  
Riemannian Manifold ◽  
Heisenberg Group ◽  
Green’S Function ◽  
Heat Kernel ◽  
Green's Function ◽  
Beltrami Operator ◽  
Small Time ◽  
Time Estimates

AbstractWe study the Riemannian Laplace-Beltrami operator L on a Riemannian manifold with Heisenberg group H1 as boundary. We calculate the heat kernel and Green's function for L, and give global and small time estimates of the heat kernel. A class of hypersurfaces in this manifold can be regarded as approximations of H1. We also restrict L to each hypersurface and calculate the corresponding heat kernel and Green's function. We will see that the heat kernel and Green's function converge to the heat kernel and Green's function on the boundary.

scholarly journals On Eigenfunction Restriction Estimates and L4-Bounds for Compact Surfaces with Nonpositive Curvature

2014 ◽  
Author(s):  
Christopher D. Sogge ◽  
Steve Zelditch
Keyword(s):  
Wave Equation ◽  
Riemannian Manifold ◽  
Wave Equations ◽  
Nonpositive Curvature ◽  
Universal Cover ◽  
Beltrami Operator ◽  
Main Step ◽  
Two Dimensional ◽  
Restriction Theorem ◽  
Wavefront Analysis

This chapter discusses a “restriction theorem,” which is related to certain Littlewood–Paley estimates for eigenfunctions. The main step in proving this theorem is to see that an estimate involving a wave equation associated with an assigned Laplace–Beltrami operator and a bit of microlocal (wavefront) analysis remains valid as well if a certain variable is part of a periodic orbit under a set of curvature assumptions. This can be done by lifting the wave equation for a compact two-dimensional Riemannian manifold without boundary up to the corresponding one for its universal cover. By identifying solutions of wave equations for this Riemannian manifold with “periodic” ones, this chapter is able to obtain the necessary bounds using a bit of wavefront analysis and the Hadamard parametrix for the universal cover.

Existence of the first eigenvalue of the eigenvalue problem for the Laplace-Beltrami operator on the unit sphere

2018 ◽  
Vol 55 (3) ◽  
pp. 374-382
Author(s):  
Mariusz Bodzioch ◽  
Mikhail Borsuk ◽  
Sebastian Jankowski
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Behaviour ◽  
Unit Sphere ◽  
Beltrami Operator ◽  
Positive Eigenvalue ◽  
Conical Point ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Asymptotic Behaviour Of Solutions ◽  
Oblique Derivative Problems

In this paper we formulate and prove that there exists the first positive eigenvalue of the eigenvalue problem with oblique derivative for the Laplace-Beltrami operator on the unit sphere. The firrst eigenvalue plays a major role in studying the asymptotic behaviour of solutions of oblique derivative problems in cone-like domains. Our work is motivated by the fact that the precise solutions decreasing rate near the boundary conical point is dependent on the first eigenvalue.

scholarly journals An integro-differential inequality related to the smallest positive eigenvalue of p(x)-Laplacian Dirichlet problem

2016 ◽  
Vol 15 (1) ◽  
pp. 27-36
Author(s):  
Damian Wiśniewski ◽  
Mariusz Bodzioch
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Unit Sphere ◽  
Differential Inequality ◽  
Beltrami Operator ◽  
Positive Eigenvalue ◽  
Differential Inequalities

AbstractWe consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.

Asymptotics for the spectral heat function and bounds for integrals of Dirichlet eigenfunctions

1999 ◽  
Vol 129 (4) ◽  
pp. 841-854 ◽  
Author(s):  
M. van den Berg ◽  
S. P. Watson
Keyword(s):  
Riemannian Manifold ◽  
Smooth Boundary ◽  
Beltrami Operator ◽  
Heat Function

We introduce the spectral heat function H associated with the Dirichlet-Laplace–Beltrami operator ΔM on a compact smooth Riemannian manifold M with a non-empty smooth boundary. We obtain two-term asymptotics for H without assuming any billiard conditions on M. As a corollary, we obtain estimates for the integral of the k′th Dirichlet eigenfunction of ΔM over M for k→∞.

scholarly journals Clustered solutions for supercritical elliptic equations on Riemannian manifolds

2018 ◽  
Vol 8 (1) ◽  
pp. 1213-1226 ◽  
Author(s):  
Wenjing Chen
Keyword(s):  
Riemannian Manifold ◽  
Positive Integer ◽  
Riemannian Manifolds ◽  
Elliptic Problem ◽  
Elliptic Equations ◽  
Compact Riemannian Manifold ◽  
Blow Up ◽  
Beltrami Operator ◽  
Real Parameter ◽  
Reduction Procedure

Abstract Let {(M,g)} be a smooth compact Riemannian manifold of dimension {n\geq 5} . We are concerned with the following elliptic problem: -\Delta_{g}u+a(x)u=u^{\frac{n+2}{n-2}+\varepsilon},\quad u>0\text{ in }M, where {\Delta_{g}=\mathrm{div}_{g}(\nabla)} is the Laplace–Beltrami operator on M, {a(x)} is a {C^{2}} function on M such that the operator {-\Delta_{g}+a} is coercive, and {\varepsilon>0} is a small real parameter. Using the Lyapunov–Schmidt reduction procedure, we obtain that the problem under consideration has a k-peaks solution for positive integer {k\geq 2} , which blow up and concentrate at one point in M.

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