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.

scholarly journals Upper bounds for the ground state energy of the Laplacian with zero magnetic field on planar domains

2021 ◽  
Author(s):  
Bruno Colbois ◽  
Alessandro Savo
Keyword(s):  
Magnetic Field ◽  
Upper Bound ◽  
Ground State ◽  
State Energy ◽  
Upper Bounds ◽  
Zero Magnetic Field ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Simply Connected ◽  
Lowest Eigenvalue

AbstractWe obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential 1-form (hence, with zero magnetic field) acting on complex functions of a planar domain $$\Omega $$ Ω , with magnetic Neumann boundary conditions. It is well known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance, the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal $$\epsilon $$ ϵ -net. In the last part, we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one.

scholarly journals Sharp upper bound for the first eigenvalue

2013 ◽  
Vol 169 (1) ◽  
pp. 397-410 ◽  
Author(s):  
Raveendran Binoy ◽  
G. Santhanam
Keyword(s):  
Upper Bound ◽  
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

From Steklov to Neumann and Beyond, via Robin: The Szegő Way

2019 ◽  
Vol 72 (4) ◽  
pp. 1024-1043 ◽  
Author(s):  
Pedro Freitas ◽  
Richard S. Laugesen
Keyword(s):  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Planar Domains ◽  
Steklov Eigenvalue ◽  
Robin Laplacian ◽  
Fixed Area ◽  
Neumann Laplacian ◽  
Simply Connected ◽  
Second Eigenvalue ◽  
Simply Connected Domains

AbstractThe second eigenvalue of the Robin Laplacian is shown to be maximal for the disk among simply-connected planar domains of fixed area when the Robin parameter is scaled by perimeter in the form $\unicode[STIX]{x1D6FC}/L(\unicode[STIX]{x1D6FA})$, and $\unicode[STIX]{x1D6FC}$ lies between $-2\unicode[STIX]{x1D70B}$ and $2\unicode[STIX]{x1D70B}$. Corollaries include Szegő’s sharp upper bound on the second eigenvalue of the Neumann Laplacian under area normalization, and Weinstock’s inequality for the first nonzero Steklov eigenvalue for simply-connected domains of given perimeter.The first Robin eigenvalue is maximal, under the same conditions, for the degenerate rectangle. When area normalization on the domain is changed to conformal mapping normalization and the Robin parameter is positive, the maximiser of the first eigenvalue changes back to the disk.

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