Analytic proof of dual variational formula for the first eigenvalue in dimension one

1999 ◽  
Vol 42 (8) ◽  
pp. 805-815 ◽  
Author(s):  
Mufa Chen
Keyword(s):  
Variational Formula ◽  
First Eigenvalue ◽  
Analytic Proof ◽  
The First Eigenvalue ◽  
Dimension One

scholarly journals On relations between principal eigenvalue and torsional rigidity

2020 ◽  
pp. 2050093
Author(s):  
Michiel van den Berg ◽  
Giuseppe Buttazzo ◽  
Aldo Pratelli
Keyword(s):  
Optimization Problem ◽  
Torsional Rigidity ◽  
Principal Eigenvalue ◽  
Higher Dimensions ◽  
First Eigenvalue ◽  
Thin Domains ◽  
Convex Domains ◽  
Dirichlet Laplacian ◽  
The First Eigenvalue ◽  
Dimension One

We consider the problem of minimizing or maximizing the quantity [Formula: see text] on the class of open sets of prescribed Lebesgue measure. Here [Formula: see text] is fixed, [Formula: see text] denotes the first eigenvalue of the Dirichlet Laplacian on [Formula: see text], while [Formula: see text] is the torsional rigidity of [Formula: see text]. The optimization problem above is considered in the class of all domains [Formula: see text], in the class of convex domains [Formula: see text], and in the class of thin domains. The full Blaschke–Santaló diagram for [Formula: see text] and [Formula: see text] is obtained in dimension one, while for higher dimensions we provide some bounds.

scholarly journals Lichnerowicz-Obata Estimate, Almost Parallel p-form and Almost Product Manifolds

2021 ◽  
Author(s):  
Masayuki Aino
Keyword(s):  
Riemannian Manifolds ◽  
First Eigenvalue ◽  
Type Estimate ◽  
The First Eigenvalue ◽  
Hausdorff Approximation

AbstractWe show a Lichnerowicz-Obata type estimate for the first eigenvalue of the Laplacian of n-dimensional closed Riemannian manifolds with an almost parallel p-form ($$2\le p \le n/2$$ 2 ≤ p ≤ n / 2 ) in $$L^2$$ L 2 -sense, and give a Gromov-Hausdorff approximation to a product $$S^{n-p}\times X$$ S n - p × X under some pinching conditions when $$2\le p<n/2$$ 2 ≤ p < n / 2 .

scholarly journals First eigenvalue of submanifolds in Euclidean space

2000 ◽  
Vol 24 (1) ◽  
pp. 43-48
Author(s):  
Kairen Cai
Keyword(s):  
Euclidean Space ◽  
Fundamental Form ◽  
Second Fundamental Form ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
The Second Fundamental Form ◽  
Compact Submanifold

We give some estimates of the first eigenvalue of the Laplacian for compact and non-compact submanifold immersed in the Euclidean space by using the square length of the second fundamental form of the submanifold merely. Then some spherical theorems and a nonimmersibility theorem of Chern and Kuiper type can be obtained.

scholarly journals The first eigenvalue of the $$p-$$ p - Laplacian on quantum graphs

2016 ◽  
Vol 6 (4) ◽  
pp. 365-391 ◽  
Author(s):  
Leandro M. Del Pezzo ◽  
Julio D. Rossi
Keyword(s):  
First Eigenvalue ◽  
Quantum Graphs ◽  
The First Eigenvalue

Asymptotic Estimates of the First Eigenvalue of the p-Laplacian

2003 ◽  
Vol 3 (2) ◽  
Author(s):  
Bruno Colbois ◽  
Ana-Maria Matei
Keyword(s):  
Direct Application ◽  
Parameter Family ◽  
Precise Estimate ◽  
First Eigenvalue ◽  
Asymptotic Estimates ◽  
Hyperbolic Surfaces ◽  
The First Eigenvalue

AbstractWe consider a 1-parameter family of hyperbolic surfaces M(t) of genus ν which degenerate as t → 0 and we obtain a precise estimate of λAs a direct application, we obtain that the quotientTo prove our results we use in an essential way the geometry of hyperbolic surfaces which is very well known. We show that an eigenfunction for λ

scholarly journals A variation on the Donsker–Varadhan inequality for the principal eigenvalue

2017 ◽  
Vol 473 (2204) ◽  
pp. 20160877
Author(s):  
Jianfeng Lu ◽  
Stefan Steinerberger
Keyword(s):  
Lower Bound ◽  
Exit Time ◽  
Principal Eigenvalue ◽  
Short Paper ◽  
First Eigenvalue ◽  
Drift Diffusion ◽  
First Exit Time ◽  
The First Eigenvalue ◽  
Order Elliptic Operator ◽  
The Mean

The purpose of this short paper is to give a variation on the classical Donsker–Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain Ω by the largest mean first exit time of the associated drift–diffusion process via λ 1 ≥ 1 sup x ∈ Ω E x τ Ω c . Instead of looking at the mean of the first exit time, we study quantiles: let d p , ∂ Ω : Ω → R ≥ 0 be the smallest time t such that the likelihood of exiting within that time is p , then λ 1 ≥ log ( 1 / p ) sup x ∈ Ω d p , ∂ Ω ( x ) . Moreover, as p → 0 , this lower bound converges to λ 1 .

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