scholarly journals Infinite Kirchhoff plate on a compact elastic foundation may have arbitrary small eigenvalue

2019 ◽  
Vol 488 (4) ◽  
pp. 362-366
Author(s):  
S. A. Nazarov
Keyword(s):  
Continuous Spectrum ◽  
Neumann Problem ◽  
Elastic Foundation ◽  
Laplace Operator ◽  
Small Eigenvalue ◽  
Winkler Foundation ◽  
Infinite Strip ◽  
Compliance Coefficient ◽  
Strip Waveguide ◽  
The Laplace Operator

An inhomogeneous Kirhhoff plate composed from semi-infinite strip-waveguide and a compaсt resonator which is in contact with the Winkler foundation of small compliance, is considered. It is shown that for any 0, it is possible to find the compliance coefficient O(2) such that the described plate possesses the eigenvalue 4embedded into continuous spectrum. This result is quite surprising because in an acoustic waveguide (the spectral Neumann problem for the Laplace operator) a small eigenvalue does not exist for any unsubstantial perturbation. A reason of this dissension is explained as well.

An Equivalence Between the Dirichlet and the Neumann Problem for the Laplace Operator

2015 ◽  
Vol 44 (4) ◽  
pp. 655-672 ◽  
Author(s):  
Lucian Beznea ◽  
Mihai N. Pascu ◽  
Nicolae R. Pascu
Keyword(s):  
Neumann Problem ◽  
Laplace Operator ◽  
The Neumann Problem ◽  
The Laplace Operator

scholarly journals The stiff Neumann problem: Asymptotic specialty and “kissing” domains

2021 ◽  
pp. 1-36
Author(s):  
V. Chiadò Piat ◽  
L. D’Elia ◽  
S.A. Nazarov
Keyword(s):  
Boundary Value Problem ◽  
Neumann Problem ◽  
Bounded Domain ◽  
Laplace Operator ◽  
Mixed Boundary ◽  
Asymptotic Series ◽  
Mixed Boundary Value Problem ◽  
Smooth Bounded Domain ◽  
Dimension 2 ◽  
The Laplace Operator

We study the stiff spectral Neumann problem for the Laplace operator in a smooth bounded domain Ω ⊂ R d which is divided into two subdomains: an annulus Ω 1 and a core Ω 0 . The density and the stiffness constants are of order ε − 2 m and ε − 1 in Ω 0 , while they are of order 1 in Ω 1 . Here m ∈ R is fixed and ε > 0 is small. We provide asymptotics for the eigenvalues and the corresponding eigenfunctions as ε → 0 for any m. In dimension 2 the case when Ω 0 touches the exterior boundary ∂ Ω and Ω 1 gets two cusps at a point O is included into consideration. The possibility to apply the same asymptotic procedure as in the “smooth” case is based on the structure of eigenfunctions in the vicinity of the irregular part. The full asymptotic series as x → O for solutions of the mixed boundary value problem for the Laplace operator in the cuspidal domain is given.

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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