scholarly journals First Eigenvalue of p-Laplacian Along The Normalized Ricci Flow on Bianchi Classes

2020 ◽  
Vol 26 (3) ◽  
pp. 380-392
Author(s):  
Mohammad Javad Habibi Vosta Kolaei ◽  
Shahroud Azami
Keyword(s):  
Laplace Operator ◽  
Lower Bounds ◽  
Ricci Flow ◽  
Homogeneous Manifold ◽  
Upper And Lower Bounds ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Normalized Ricci Flow

Consider M as a 3-homogeneous manifold. In this paper, we are going to study the behavior of the first eigenvalue of p-Laplace operator in a case of Bianchi classes along the normalized Ricci flow also we will give some upper and lower bounds for a such eigenvalue.

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.

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

UPPER AND LOWER BOUNDS FOR THE VOLUME OF A COMPACT SPACELIKE HYPERSURFACE IN A GENERALIZED ROBERTSON–WALKER SPACETIME

2013 ◽  
Vol 11 (01) ◽  
pp. 1450006 ◽  
Author(s):  
JUAN ÁNGEL ALEDO ◽  
ALFONSO ROMERO ◽  
RAFAEL M. RUBIO
Keyword(s):  
Lower Bounds ◽  
Three Dimensional ◽  
Spacelike Hypersurface ◽  
Upper And Lower Bounds ◽  
Warping Function ◽  
First Eigenvalue ◽  
De Sitter ◽  
Sitter Spacetime ◽  
Angle Function ◽  
Spacelike Hypersurfaces

We provide upper and lower bounds for the volume of a compact spacelike hypersurface in an (n + 1)-dimensional Generalized Robertson–Walker (GRW) spacetime in terms of the volume of the fiber, the hyperbolic angle function and the warping function. Under several geometrical and physical assumptions, we characterize the spacelike slices as the only spacelike hypersurfaces where these bounds are attained. As a consequence of these results, we get an upper bound for the first eigenvalue of a compact spacelike surface in a three-dimensional GRW spacetime whose fiber is a topological sphere, which includes the case of the three-dimensional De Sitter spacetime, and show that the bound is attained if and only if M is a spacelike slice.

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.

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