Bull. Korean Math. Soc. 2021; 58(1): 71-111
Online first article January 8, 2021 Printed January 31, 2021
https://doi.org/10.4134/BKMS.b200041
Copyright © The Korean Mathematical Society.
Sung Woo Choi
Duksung Women's University
We consider the boundary value problem for the deflection of a finite beam on an elastic foundation subject to vertical loading. We construct a one-to-one correspondence $\Gamma$ from the set of equivalent well-posed two-point boundary conditions to $\mathrm{gl}(4,\mathbb{C})$. Using $\Gamma$, we derive eigenconditions for the integral operator $\mathcal{K}_\mathbf{M}$ for each well-posed two-point boundary condition represented by $\mathbf{M} \in \mathrm{gl}(4,8,\mathbb{C})$. Special features of our eigenconditions include; (1) they isolate the effect of the boundary condition $\mathbf{M}$ on $\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$, (2) they connect $\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$ to $\mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$ whose structure has been well understood. Using our eigenconditions, we show that, for each nonzero real $\lambda \not \in \mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$, there exists a real well-posed boundary condition $\mathbf{M}$ such that $\lambda \in \mathrm{Spec}\,\mathcal{K}_\mathbf{M}$. This in particular shows that the integral operators $\mathcal{K}_\mathbf{M}$ arising from well-posed boundary conditions, may not be positive nor contractive in general, as opposed to $\mathcal{K}_{l,\alpha,k}$.
Keywords: Beam, deflection, Green's function, eigenvalue, spectrum, integral operator
MSC numbers: 34B09, 47G10, 74K10
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