The first eigenvalue of the Laplace operator

2016 ◽  
Author(s):  
Baltabek E. Kanguzhin ◽  
Dostilek Dauitbek
Keyword(s):  
Laplace Operator ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
The Laplace Operator

SPECTRAL ASYMPTOTICS OF THE LAPLACE OPERATOR ON MANIFOLDS WITH CYLINDRICAL ENDS

1995 ◽  
Vol 06 (06) ◽  
pp. 911-920 ◽  
Author(s):  
L.B. PARNOVSKI
Keyword(s):  
Laplace Operator ◽  
Scattering Matrix ◽  
Spectral Asymptotics ◽  
First Eigenvalue ◽  
Dimensional Manifold ◽  
The First Eigenvalue ◽  
Counting Functions ◽  
The Laplace Operator

Let M be an n-dimensional manifold with cylindrical ends. We consider the sum of the counting functions of the discrete (Nd(λ)) and continuous spectra of M, the latter beingdefined as [Formula: see text] where T(ν) is the scattering matrix and µ1 is the first eigenvalue of the cylinder’s section. Using the modification of the Colin de Verdière cut-off Laplacian, we prove the followingasymptotic formula: [Formula: see text] where |M0| is the regularized volume of |M|, and Cn is the Weyl constant.

scholarly journals Principal frequency of the p-Laplacian and the inradius of Euclidean domains

2015 ◽  
Vol 07 (03) ◽  
pp. 505-511 ◽  
Author(s):  
Guillaume Poliquin
Keyword(s):  
Lower Bound ◽  
Laplace Operator ◽  
Lower Bounds ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Planar Domains ◽  
Principal Frequency ◽  
The Laplace Operator ◽  
Simply Connected ◽  
Euclidean Domains

We study the lower bounds for the principal frequency of the p-Laplacian on N-dimensional Euclidean domains. For p > N, we obtain a lower bound for the first eigenvalue of the p-Laplacian in terms of its inradius, without any assumptions on the topology of the domain. Moreover, we show that a similar lower bound can be obtained if p > N - 1 assuming the boundary is connected. This result can be viewed as a generalization of the classical bounds for the first eigenvalue of the Laplace operator on simply connected planar domains.

On the first eigenvalue of the Laplace operator for compact spacelike submanifolds in Lorentz–Minkowski spacetime 𝕃 m

2021 ◽  
pp. 1-20
Author(s):  
Francisco J. Palomo ◽  
Alfonso Romero
Keyword(s):  
Euclidean Space ◽  
Laplace Operator ◽  
Minkowski Spacetime ◽  
Upper Bounds ◽  
First Eigenvalue ◽  
Original Result ◽  
The First Eigenvalue ◽  
Spacelike Submanifold ◽  
Compact Submanifold ◽  
The Laplace Operator

By means of a counter-example, we show that the Reilly theorem for the upper bound of the first non-trivial eigenvalue of the Laplace operator of a compact submanifold of Euclidean space (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) does not work for a (codimension ⩾2) compact spacelike submanifold of Lorentz–Minkowski spacetime. In the search of an alternative result, it should be noted that the original technique in (Reilly, 1977, Comment. Mat. Helvetici, 52, 525–533) is not applicable for a compact spacelike submanifold of Lorentz–Minkowski spacetime. In this paper, a new technique, based on an integral formula on a compact spacelike section of the light cone in Lorentz–Minkowski spacetime is developed. The technique is genuine in our setting, that is, it cannot be extended to another semi-Euclidean spaces of higher index. As a consequence, a family of upper bounds for the first eigenvalue of the Laplace operator of a compact spacelike submanifold of Lorentz–Minkowski spacetime is obtained. The equality for one of these inequalities is geometrically characterized. Indeed, the eigenvalue achieves one of these upper bounds if and only if the compact spacelike submanifold lies minimally in a hypersphere of certain spacelike hyperplane. On the way, the Reilly original result is reproved if a compact submanifold of a Euclidean space is naturally seen as a compact spacelike submanifold of Lorentz–Minkowski spacetime through a spacelike hyperplane.

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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