Eigenvalue Ratios for vibrating string equations with single-well densities
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866916150769614848 |
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| author | Hedhly, Jihed |
| author_facet | Hedhly, Jihed |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_01728 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Eigenvalue Ratios for vibrating string equations with single-well densities Hedhly, Jihed Analysis of PDEs Spectral Theory 34B24, 34L15 F.2.2; I.2.7 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. |
| title | Eigenvalue Ratios for vibrating string equations with single-well densities |
| topic | Analysis of PDEs Spectral Theory 34B24, 34L15 F.2.2; I.2.7 |
| url | https://arxiv.org/abs/2111.01728 |