scholarly journals Riesz means of the counting function of the Laplace operator on compact manifolds of non-positive curvature

2016 ◽  
Vol 6 (3) ◽  
pp. 629-642
Author(s):  
Kamil Mroz ◽  
Alexander Strohmaier
Keyword(s):  
Laplace Operator ◽  
Positive Curvature ◽  
Counting Function ◽  
Compact Manifolds ◽  
Riesz Means ◽  
The Laplace Operator

Eigenvalues of the Laplacian Operator and Neighborhood Enlargement for Compact Manifolds

2020 ◽  
pp. 84-102
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifolds ◽  
Laplacian Operator ◽  
Measure Concentration ◽  
Eigenvalues Of The Laplacian ◽  
The Laplace Operator ◽  
Concentration Phenomenon

In this investigation and under the conception of measure concentration phenomenon we found that the enlargement of the neighborhood for an n – dimensional compact Riemannian manifolds (M,g) relative to the eigenvalues λ of the Laplace operator ∆ on (M,g). And we found that r~1/√λ. تناول هذا البحث تجسيم الجوار لمتعدد الطيات المتراص في البعد n وذلك باستخدام مفهوم تركيز الحجم. كما وجدنا أن نصف القطر r لتجسيم الجوار لمتعدد الطيات يرتبط مع القيم الذاتية λ لمؤثر لابلاس Δ على متعدد الطيات. الكلمات المفتاحية: نصف قطر تجسيم الجوار، متباينات متساوي المقاييس، تركيز الحجم، مؤثر لابلاس، القيم الذاتية لمؤثر لابلاس.

scholarly journals Numerical Optimization of Eigenvalues of the Dirichlet–Laplace Operator on Domains in Surfaces

2014 ◽  
Vol 14 (3) ◽  
pp. 393-409
Author(s):  
Régis Straubhaar
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Laplace Operator ◽  
Numerical Optimization ◽  
Negative Curvature ◽  
Positive Curvature ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
The Finite Element Method ◽  
The Laplace Operator

Abstract.Let (M,g) be a smooth and complete surface, $\Omega \subset M$ be a domain in M, and $\Delta _g$ be the Laplace operator on M. The spectrum of the Dirichlet–Laplace operator on Ω is a sequence $0 < \lambda _1(\Omega ) \le \lambda _2(\Omega ) \le \cdots \nearrow \infty $. A classical question is to ask what is the domain $\Omega ^*$ which minimizes $\lambda _m(\Omega )$ among all domains of a given area, and what is the value of the corresponding $\lambda _m(\Omega _m^*)$. The aim of this article is to present a numerical algorithm using shape optimization and based on the finite element method to find an approximation of a candidate for $\Omega _m^*$. Some verifications with existing numerical results are carried out for the first eigenvalues of domains in ℝ2. Furthermore, some investigations are presented in the two-dimensional sphere to illustrate the case of the positive curvature, in hyperbolic space for the negative curvature and in a hyperboloid for a non-constant curvature.

scholarly journals FINITELY RAMIFIED GRAPH-DIRECTED FRACTALS, SPECTRAL ASYMPTOTICS AND THE MULTIDIMENSIONAL RENEWAL THEOREM

2003 ◽  
Vol 46 (1) ◽  
pp. 1-34 ◽  
Author(s):  
B. M. Hambly ◽  
S. O. G. Nyberg
Keyword(s):  
Fixed Point ◽  
Laplace Operator ◽  
Power Law ◽  
Counting Function ◽  
Dirichlet Forms ◽  
Logarithmic Correction ◽  
Renewal Theorem ◽  
Secondary 28A80 ◽  
Primary 35P20 ◽  
The Laplace Operator

AbstractWe consider the class of graph-directed constructions which are connected and have the property of finite ramification. By assuming the existence of a fixed point for a certain renormalization map, it is possible to construct a Laplace operator on fractals in this class via their Dirichlet forms. Our main aim is to consider the eigenvalues of the Laplace operator and provide a formula for the spectral dimension, the exponent determining the power-law scaling in the eigenvalue counting function, and establish generic constancy for the counting-function asymptotics. In order to do this we prove an extension of the multidimensional renewal theorem. As a result we show that it is possible for the eigenvalue counting function for fractals to require a logarithmic correction to the usual power-law growth.AMS 2000 Mathematics subject classification: Primary 35P20; 58J50. Secondary 28A80; 60K05; 31C25

scholarly journals Evolution of the first eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow

2020 ◽  
Vol 18 (1) ◽  
pp. 1518-1530
Author(s):  
Xuesen Qi ◽  
Ximin Liu
Keyword(s):  
Laplace Operator ◽  
Mean Curvature ◽  
Mean Curvature Flow ◽  
Curvature Flow ◽  
Second Fundamental Form ◽  
Coefficient Function ◽  
First Eigenvalue ◽  
Initial Hypersurface ◽  
The Mean ◽  
The Laplace Operator

Abstract In this paper, we discuss the monotonicity of the first nonzero eigenvalue of the Laplace operator and the p-Laplace operator under a forced mean curvature flow (MCF). By imposing conditions associated with the mean curvature of the initial hypersurface and the coefficient function of the forcing term of a forced MCF, and some special pinching conditions on the second fundamental form of the initial hypersurface, we prove that the first nonzero closed eigenvalues of the Laplace operator and the p-Laplace operator are monotonic under the forced MCF, respectively, which partially generalize Mao and Zhao’s work. Moreover, we give an example to specify applications of conclusions obtained above.

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