Eigenfunction Solution for Beam on Elastic Foundation

1969 ◽  
Vol 36 (4) ◽  
pp. 799-802 ◽  
Author(s):  
M. S. Hess
Keyword(s):  
Exact Solution ◽  
Elastic Foundation ◽  
Nontrivial Solution ◽  
Normal Displacement ◽  
First Eigenvalue ◽  
The First Eigenvalue ◽  
Complex Eigenvalues ◽  
Adequate Accuracy ◽  
Beam Foundation ◽  
Beam Theories

A solution for the end problem of a rectangular beam resting on a simple elastic foundation is obtained as a series expansion in the eigenfunctions of the system. For a beam aligned with the ξ-axis, the eigenfunctions are of the form eγξf(y), where γ is one of the complex eigenvalues. The eigenvalue equation is determined by requiring continuity of the normal displacement and pressure at each point of the beam-foundation interface and seeking a nontrivial solution. In order to evaluate the accuracy and limitations of several approximate beam theories, the eigenvalue predicted by each of these theories is compared with the first eigenvalue of the exact solution. It is shown that the approximate theories give adequate accuracy if the beam modulus, E, exceeds the foundation stiffness, k, and that all tend to the same result as E/k → ∞. From a comparison of the first and second eigenvalues of the exact solution, it is found that the first mode (corresponding to beam behavior) ceases to dominate the higher modes for E/k < 1. Thus the approximate theories are necessarily restricted to E/k > 1 since they predict only the first mode.

scholarly journals The existence of nontrivial solutions for a critically coupled Schr\”{o}dinger system in a bounded domain of $\R^3$

2021 ◽  
Author(s):  
Hongyu Ye ◽  
Lina Zhang
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
The First Eigenvalue ◽  
Nirenberg Problem

In this paper, we consider the following coupled Schr\”{o}dinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg problem $$\left\{% \begin{array}{ll} -\Delta u+\lambda_1 u=\mu_1 u^5+ \beta u^2v^3, & \hbox{$x\in \Omega$}, \\ -\Delta v+\lambda_2 v=\mu_2 v^5+ \beta v^2u^3, & \hbox{$x\in \Omega$}, \\ u=v=0,& \hbox{$x\in \partial\Omega$}, \\ \end{array}% \right.$$ where $\Omega$ is a ball in $\R^3,$ $-\lambda_1(\Omega)<\lambda_1,\lambda_2<-\frac14\lambda_1(\Omega)$, $\mu_1,\mu_2>0$ and $\beta>0$. Here $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary condition in $\Omega$. We show that the problem has at least one nontrivial solution for all $\beta>0$.

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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