scholarly journals Optimal lower bound for the first eigenvalue of fourth order measure differential equation

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Zhou Lijuan ◽  
Meng Gang
Keyword(s):  
Differential Equation ◽  
Lower Bound ◽  
Fourth Order ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Measure Differential Equation ◽  
Optimal Lower Bound

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 .

scholarly journals On the first frequency of reinforced partially hinged plates

2019 ◽  
pp. 1950074 ◽  
Author(s):  
Elvise Berchio ◽  
Alessio Falocchi ◽  
Alberto Ferrero ◽  
Debdip Ganguly
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Eigenvalue Problem ◽  
Rectangular Plate ◽  
Normal Modes ◽  
Weight Coefficient ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Dynamical Properties ◽  
The Stability

We consider a partially hinged rectangular plate and its normal modes. The dynamical properties of the plate are influenced by the spectrum of the associated eigenvalue problem. In order to improve the stability of the plate, we place a certain amount of denser material in appropriate regions. If we look at the partial differential equation appearing in the model, this corresponds to insert a suitable weight coefficient inside the equation. A possible way to locate such regions is to study the eigenvalue problem associated to the aforementioned weighted equation. In this paper, we focus our attention essentially on the first eigenvalue and on its minimization in terms of the weight. We prove the existence of minimizing weights inside special classes and we try to describe them together with the corresponding eigenfunctions.

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