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arXiv · 2111.01728

Eigenvalue Ratios for vibrating string equations with single-well densities

Jihed Hedhly
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Abstract

In this paper, we prove the optimal upper bound $\frac{λ_n}{λ_m}\leq(\frac{n}{m})^2$ of vibrating string $$-y''=λρ(x) y,$$ with Dirichlet boundary conditions for single-well densities. The proof is based on the inequality $\frac{λ_n(ρ)}{λ_{m}(ρ)}\leq \frac{λ_n(L)}{λ_{m}(L)} ,$ with $L$ must be a stepfunction. We also prove the same result for the Dirichlet Sturm-Liouville problems.

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Jihed Hedhly. 2021-11-02. Eigenvalue Ratios for vibrating string equations with single-well densities. https://doi.org/10.1016/j.jde.2021.11.006

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