scholarly journals An application of the continuous Steiner symmetrization to Blaschke-Santaló diagrams

2021 ◽  
Author(s):  
Giuseppe Buttazzo ◽  
Aldo Pratelli
Keyword(s):  
First Eigenvalue ◽  
Steiner Symmetrization ◽  
The First Eigenvalue ◽  
Continuous Map ◽  
Dense Class

In this paper we consider the so-called procedure of {\it Continuous Steiner Symmetrization}, introduced by Brock in~\cite{bro95,bro00}. It transforms every domain $\Omega\comp\R^d$ into the ball keeping the volume fixed and letting the first eigenvalue and the torsion respectively decrease and increase. While this does not provide, in general, a $\gamma$-continuous map $t\mapsto\O_t$, it can be slightly modified so to obtain the $\gamma$-continuity for a $\gamma$-dense class of domains $\O$, namely, the class of polyedral sets in $\R^d$. This allows to obtain a sharp characterization of the Blaschke-Santaló diagram of torsion and eigenvalue.

scholarly journals A system of local/nonlocal p-Laplacians: The eigenvalue problem and its asymptotic limit as p → ∞

2021 ◽  
pp. 1-33
Author(s):  
S. Buccheri ◽  
J.V. da Silva ◽  
L.H. de Miranda
Keyword(s):  
Eigenvalue Problem ◽  
Geometric Characterization ◽  
Asymptotic Limit ◽  
First Eigenvalue ◽  
Open Domain ◽  
The First Eigenvalue ◽  
Uniformly Convergence ◽  
Viscosity Sense

In this work, given p ∈ ( 1 , ∞ ), we prove the existence and simplicity of the first eigenvalue λ p and its corresponding eigenvector ( u p , v p ), for the following local/nonlocal PDE system (0.1) − Δ p u + ( − Δ ) p r u = 2 α α + β λ | u | α − 2 | v | β u in  Ω − Δ p v + ( − Δ ) p s v = 2 β α + β λ | u | α | v | β − 2 v in  Ω u = 0 on  R N ∖ Ω v = 0 on  R N ∖ Ω , where Ω ⊂ R N is a bounded open domain, 0 < r , s < 1 and α ( p ) + β ( p ) = p. Moreover, we address the asymptotic limit as p → ∞, proving the explicit geometric characterization of the corresponding first ∞-eigenvalue, namely λ ∞ , and the uniformly convergence of the pair ( u p , v p ) to the ∞-eigenvector ( u ∞ , v ∞ ). Finally, the triple ( u ∞ , v ∞ , λ ∞ ) verifies, in the viscosity sense, a limiting PDE system.

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 .

Sign in / Sign up

Export Citation Format

Share Document