scholarly journals A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-1 symmetric spaces

2007 ◽  
Vol 117 (3) ◽  
pp. 307-315 ◽  
Author(s):  
G. Santhanam
Keyword(s):  
Upper Bound ◽  
Symmetric Spaces ◽  
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 A formula for the first eigenvalue of the Dirac operator on compact spin symmetric spaces

2006 ◽  
Vol 47 (4) ◽  
pp. 043503 ◽  
Author(s):  
Jean-Louis Milhorat
Keyword(s):  
Dirac Operator ◽  
Symmetric Spaces ◽  
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.

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.

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
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