A rigorous justification and estimation of the rate of convergence for the partial domain method in two-dimensional eigenvalue problems for the Laplace operator

1990 ◽  
Vol 30 (4) ◽  
pp. 66-75 ◽  
Author(s):  
L.T. Poznyak
Keyword(s):  
Rate Of Convergence ◽  
Laplace Operator ◽  
Eigenvalue Problems ◽  
Two Dimensional ◽  
Domain Method ◽  
Partial Domain ◽  
The Laplace Operator

scholarly journals GEOMETRIC MOMENTUM IN THE MONGE PARAMETRIZATION OF TWO-DIMENSIONAL SPHERE

2013 ◽  
Vol 10 (03) ◽  
pp. 1220031 ◽  
Author(s):  
D. M. XUN ◽  
Q. H. LIU
Keyword(s):  
Quantum Mechanics ◽  
Laplace Operator ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
Constrained Motion ◽  
Geometric Invariant ◽  
Gradient Operator ◽  
The Laplace Operator

A two-dimensional (2D) surface can be considered as three-dimensional (3D) shell whose thickness is negligible in comparison with the dimension of the whole system. The quantum mechanics on surface can be first formulated in the bulk and the limit of vanishing thickness is then taken. The gradient operator and the Laplace operator originally defined in bulk converges to the geometric ones on the surface, and the so-called geometric momentum and geometric potential are obtained. On the surface of 2D sphere the geometric momentum in the Monge parametrization is explicitly explored. Dirac's theory on second-class constrained motion is resorted to for accounting for the commutator [xi, pj] = iℏ(δij - xixj/r2) rather than [xi, pj] = iℏδij that does not hold true anymore. This geometric momentum is geometric invariant under parameters transformation, and self-adjoint.

Sublinear eigenvalue problems associated to the Laplace operator revisited

2011 ◽  
Vol 181 (1) ◽  
pp. 317-326 ◽  
Author(s):  
Mihai Mihăilescu ◽  
Vicentţiu Rădulescu
Keyword(s):  
Laplace Operator ◽  
Eigenvalue Problems ◽  
The Laplace Operator

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 EIGENVALUE PROBLEMS WITH WEIGHT AND VARIABLE EXPONENT FOR THE LAPLACE OPERATOR

2010 ◽  
Vol 08 (03) ◽  
pp. 235-246
Author(s):  
MIHAI MIHĂILESCU ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Eigenvalue Problem ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Variable Exponent ◽  
Eigenvalue Problems ◽  
Smooth Boundary ◽  
The Laplace Operator

This paper deals with an eigenvalue problem for the Laplace operator on a bounded domain with smooth boundary in ℝ N (N ≥ 3). We establish that there exist two positive constants λ* and λ* with λ* ≤ λ* such that any λ ∈ (0, λ*) is not an eigenvalue of the problem while any λ ∈ [λ*, ∞) is an eigenvalue of the problem.

scholarly journals Properties of the resolvent of the Laplace operator on a two-dimensional sphere and a trace formula

2016 ◽  
Vol 8 (3) ◽  
pp. 22-40 ◽  
Author(s):  
Arsen Il'gizovich Atnagulov ◽  
Victor Antonovich Sadovnichii ◽  
Ziganur Yusupovich Fazullin
Keyword(s):  
Laplace Operator ◽  
Trace Formula ◽  
Dimensional Sphere ◽  
Two Dimensional ◽  
The Laplace Operator

scholarly journals Spectral properties of the Neumann–Poincaré operator and cloaking by anomalous localized resonance for the elasto-static system

2017 ◽  
Vol 29 (2) ◽  
pp. 189-225 ◽  
Author(s):  
KAZUNORI ANDO ◽  
YONG-GWAN JI ◽  
HYEONBAE KANG ◽  
KYOUNGSUN KIM ◽  
SANGHYEON YU
Keyword(s):  
Laplace Operator ◽  
Spectral Properties ◽  
Static System ◽  
Two Dimensional ◽  
Eigenvalues And Eigenfunctions ◽  
Lamé System ◽  
Accumulation Points ◽  
Lame System ◽  
The Laplace Operator

We first investigate spectral properties of the Neumann–Poincaré (NP) operator for the Lamé system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for the Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, it is polynomially compact and its spectrum on two-dimensional smooth domains consists of eigenvalues that accumulate to two different points determined by the Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on discs and ellipses. Using these resonances occurring at eigenvalues is considered. We also show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.

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