scholarly journals p-Laplace Operators for Oriented Hypergraphs

2021 ◽  
Author(s):  
Jürgen Jost ◽  
Raffaella Mulas ◽  
Dong Zhang
Keyword(s):  
Laplace Operator ◽  
Spectral Properties ◽  
General Setting ◽  
Laplace Operators

AbstractThe p-Laplacian for graphs, as well as the vertex Laplace operator and the hyperedge Laplace operator for the general setting of oriented hypergraphs, are generalized. In particular, both a vertex p-Laplacian and a hyperedge p-Laplacian are defined for oriented hypergraphs, for all p ≥ 1. Several spectral properties of these operators are investigated.

scholarly journals On compactness of isospectral conformal, metrics of 4-manifolds

1995 ◽  
Vol 140 ◽  
pp. 77-99 ◽  
Author(s):  
Xingwang Xu
Keyword(s):  
Laplace Operator ◽  
Riemannian Manifolds ◽  
Compact Manifold ◽  
Riemannian Metrics ◽  
Conformal Class ◽  
Conformal Metrics ◽  
Laplace Operators ◽  
Definition Of ◽  
The Laplace Operator ◽  
Isometry Class

In this paper, we are interested in the compactness of isospectral conformal metrics in dimension 4.Let us recall the definition of the isospectral metrics. Two Riemannian metrics g, g′ on a compact manifold are said to be isospectral if their associated Laplace operators on functions have identical spectrum. There are now numeruos examples of compact Riemannian manifolds which admit more than two metrics such that they are isospectral but not isometric. That is to say that the eigenvalues of the Laplace operator Δ on the functions do not necessarily determine the isometry class of (M, g). If we further require the metrics stay in the same conformal class, the spectrum of Laplace operator still does not determine the metric uniquely ([BG], [BPY]).

scholarly journals Spectral properties of the iterated Laplacian with a potential in a punctured domain

Filomat ◽  
2018 ◽  
Vol 32 (8) ◽  
pp. 2897-2900 ◽  
Author(s):  
Gulzat Nalzhupbayeva
Keyword(s):  
Spectral Properties ◽  
Spatial Dimension ◽  
Trace Formulas ◽  
Laplace Operators ◽  
Regularized Trace ◽  
Spatial Dimensions ◽  
Trace Formulae ◽  
Punctured Domain ◽  
Nonharmonic Analysis

In the work we derive regularized trace formulas which were established in papers of Kanguzhin and Tokmagambetov for the Laplace and m-Laplace operators in a punctured domain with the fixed iterating order m 2 N. By using techniques of Sadovnichii and Lyubishkin, the authors in that papers described regularized trace formulae in the spatial dimension d = 2. In this note one claims that the formulas are also true for more general operators in the higher spatial dimensions, namely, 2 ? d ? 2m. Also, we give the further discussions on a development of the analysis associated with the operators in punctured domains. This can be done by using so called ?nonharmonic? analysis.

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.

scholarly journals Fractional Laplace operator in two dimensions, approximating matrices, and related spectral analysis

CALCOLO ◽  
2020 ◽  
Vol 57 (3) ◽  
Author(s):  
Lidia Aceto ◽  
Mariarosa Mazza ◽  
Stefano Serra-Capizzano
Keyword(s):  
Spectral Analysis ◽  
Laplace Operator ◽  
Spectral Properties ◽  
Numerical Experiments ◽  
Two Dimensions ◽  
Spatial Variables ◽  
The Matrix ◽  
Constant Coefficients ◽  
Fractional Laplace Operator ◽  
Concise Description

Abstract In this work we review some proposals to define the fractional Laplace operator in two or more spatial variables and we provide their approximations using finite differences or the so-called Matrix Transfer Technique. We study the structure of the resulting large matrices from the spectral viewpoint. In particular, by considering the matrix-sequences involved, we analyze the extreme eigenvalues, we give estimates on conditioning, and we study the spectral distribution in the Weyl sense using the tools of the theory of Generalized Locally Toeplitz matrix-sequences. Furthermore, we give a concise description of the spectral properties when non-constant coefficients come into play. Several numerical experiments are reported and critically discussed.

scholarly journals Spectral properties of the generalised Samarskii-Ionkin type problems

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 1019-1024
Author(s):  
Nurgissa Yessirkegenov
Keyword(s):  
Boundary Conditions ◽  
Explicit Form ◽  
Laplace Operator ◽  
Spectral Properties ◽  
The Laplace Operator

In this paper, we study spectral properties of the Laplace operator with generalised Samarskii-Ionkin boundary conditions in a disk. The eigenfunctions and eigenvalues of these problems are constructed in the explicit form. Moreover, we prove the completeness of these eigenfunctions

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