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 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

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.

scholarly journals Classification of Some Special Types Ruled Surfaces in Simply Isotropic 3-Space

2017 ◽  
Vol 55 (1) ◽  
pp. 87-98 ◽  
Author(s):  
Murat Kemal Karacan ◽  
Dae Won Yoon ◽  
Nural Yuksel
Keyword(s):  
Real Number ◽  
Laplace Operator ◽  
Fundamental Form ◽  
Three Dimensional ◽  
Ruled Surfaces ◽  
Isotropic Space ◽  
Simply Isotropic Space ◽  
The Laplace Operator ◽  
First Fundamental Form

AbstractIn this paper, we classify two types ruled surfaces in the three dimensional simply isotropic space I13under the condition ∆xi= λixiwhere ∆ is the Laplace operator with respect to the first fundamental form and λ is a real number. We also give explicit forms of these surfaces.

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