Upper bound for the first eigenvalue of algebraic submanifolds

1994 ◽  
Vol 69 (1) ◽  
pp. 199-207 ◽  
Author(s):  
Jean-Pierre Bourguignon ◽  
Peter Li ◽  
Shing Tung Yau
Keyword(s):  
Upper Bound ◽  
First Eigenvalue ◽  
The First Eigenvalue

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

An upper bound for the first eigenvalue of the Dirac operator on compact spin manifolds

1991 ◽  
Vol 206 (1) ◽  
pp. 409-422 ◽  
Author(s):  
Helga Baum
Keyword(s):  
Dirac Operator ◽  
Upper Bound ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Spin Manifolds

scholarly journals On the resonant Lane–Emden problem for the p-Laplacian

2014 ◽  
Vol 16 (04) ◽  
pp. 1350033 ◽  
Author(s):  
Grey Ercole
Keyword(s):  
Asymptotic Behavior ◽  
Positive Solutions ◽  
Upper Bound ◽  
Ground State ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Explicit Estimate ◽  
Best Constant ◽  
The First Eigenvalue ◽  
First Eigenfunction

We study the positive solutions of the Lane–Emden problem -Δpu = λp|u|q-2u in Ω, u = 0 on ∂Ω, where Ω ⊂ ℝN is a bounded and smooth domain, N ≥ 2, λp is the first eigenvalue of the p-Laplacian operator Δp, p > 1, and q is close to p. We prove that any family of positive solutions of this problem converges in [Formula: see text] to the function θpep when q → p, where ep is the positive and L∞-normalized first eigenfunction of the p-Laplacian and [Formula: see text]. A consequence of this result is that the best constant of the immersion [Formula: see text] is differentiable at q = p. Previous results on the asymptotic behavior (as q → p) of the positive solutions of the nonresonant Lane–Emden problem (i.e. with λp replaced by a positive λ ≠ λp) are also generalized to the space [Formula: see text] and to arbitrary families of these solutions. Moreover, if uλ,q denotes a solution of the nonresonant problem for an arbitrarily fixed λ > 0, we show how to obtain the first eigenpair of the p-Laplacian as the limit in [Formula: see text], when q → p, of a suitable scaling of the pair (λ, uλ,q). For computational purposes the advantage of this approach is that λ does not need to be close to λp. Finally, an explicit estimate involving L∞- and L1-norms of uλ,q is also derived using set level techniques. It is applied to any ground state family {vq} in order to produce an explicit upper bound for ‖vq‖∞ which is valid for q ∈ [1, p + ϵ] where [Formula: see text].

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.

An upper bound for the first eigenvalue of a spherical cap

1994 ◽  
Vol 30 (2) ◽  
pp. 171-174 ◽  
Author(s):  
M. A. Pinsky
Keyword(s):  
Upper Bound ◽  
First Eigenvalue ◽  
Spherical Cap ◽  
The First Eigenvalue

scholarly journals An Optimal Bound for Nonlinear Eigenvalues and Torsional Rigidity on Domains with Holes

2020 ◽  
Vol 88 (2) ◽  
pp. 373-384 ◽  
Author(s):  
Francesco Della Pietra ◽  
Gianpaolo Piscitelli
Keyword(s):  
Boundary Value Problem ◽  
Upper Bound ◽  
Torsional Rigidity ◽  
Neumann Boundary ◽  
Laplacian Operator ◽  
First Eigenvalue ◽  
Neumann Boundary Value Problem ◽  
The First Eigenvalue ◽  
Analogous Estimate ◽  
Nonlinear Eigenvalues

Abstract In this paper we prove an optimal upper bound for the first eigenvalue of a Robin-Neumann boundary value problem for the p-Laplacian operator in domains with convex holes. An analogous estimate is obtained for the corresponding torsional rigidity problem.

Both Bounds to Buckling Loads and Fundamental Frequencies

1994 ◽  
Vol 61 (2) ◽  
pp. 479-480 ◽  
Author(s):  
R. Schmidt
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Simple Procedure ◽  
Successive Approximations ◽  
Method Of Successive Approximations ◽  
First Eigenvalue ◽  
Buckling Loads ◽  
The First Eigenvalue ◽  
Fundamental Frequencies ◽  
The Temple

A simple procedure for improving the Krylov-Bogolyubov-Weinstein-Isaacson-Keller (KBWIK) lower bound and the Rayleigh upper bound to the first eigenvalue is described and illustrated by examples. The procedure makes use of the method of successive approximations, the Rayleigh-Schwarz quotients, the Krylov-Bogolyubov enclosure theorem, and the Temple inequality.

scholarly journals Erratum to: Sharp upper bound for the first eigenvalue

2014 ◽  
Vol 174 (1) ◽  
pp. 409-411
Author(s):  
Raveendran Binoy ◽  
G. Santhanam
Keyword(s):  
Upper Bound ◽  
First Eigenvalue ◽  
The First Eigenvalue
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