scholarly journals Upper Bounds for the First Eigenvalue of the Laplacian of Hypersurfaces in terms of Anisotropic Mean Curvatures

2013 ◽  
Vol 64 (3-4) ◽  
pp. 383-403 ◽  
Author(s):  
Julien Roth
Keyword(s):  
Upper Bounds ◽  
First Eigenvalue ◽  
The First Eigenvalue

scholarly journals Upper bounds for the first eigenvalue of the operator $L\sb r$ and some applications

2001 ◽  
Vol 45 (3) ◽  
pp. 851-863 ◽  
Author(s):  
Hilário Alencar ◽  
Manfredo do Carmo ◽  
Fernando Marques
Keyword(s):  
Upper Bounds ◽  
First Eigenvalue ◽  
The First Eigenvalue

scholarly journals Isoperimetric Upper Bound for the First Eigenvalue of Discrete Steklov Problems

2020 ◽  
Author(s):  
Hélène Perrin
Keyword(s):  
Upper Bound ◽  
Polynomial Growth ◽  
Upper Bounds ◽  
First Eigenvalue ◽  
Zero Eigenvalue ◽  
Finite Graphs ◽  
Metric Properties ◽  
The First Eigenvalue ◽  
Steklov Eigenvalue ◽  
Groups Of Polynomial Growth

AbstractWe study upper bounds for the first non-zero eigenvalue of the Steklov problem defined on finite graphs with boundary. For finite graphs with boundary included in a Cayley graph associated to a group of polynomial growth, we give an upper bound for the first non-zero Steklov eigenvalue depending on the number of vertices of the graph and of its boundary. As a corollary, if the graph with boundary also satisfies a discrete isoperimetric inequality, we show that the first non-zero Steklov eigenvalue tends to zero as the number of vertices of the graph tends to infinity. This extends recent results of Han and Hua, who obtained a similar result in the case of $$\mathbb {Z}^n$$ Z n . We obtain the result using metric properties of Cayley graphs associated to groups of polynomial growth.

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