Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908423899054080 |
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| author | Cho, Gunhee Seo, Keomkyo |
| author_facet | Cho, Gunhee Seo, Keomkyo |
| contents | Let $Ω$ be a compact surface with smooth boundary and the geodesic curvature $k_g \ge {c > 0}$ along $\partial Ω$ for some constant $c \in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K \ge -α$ for a constant $α\ge 0$, then the first eigenvalue $σ_1$ of the Steklov-type eigenvalue problem satisfies \[ σ_1 + \fracα{σ_1} \ge c. \] Moreover, equality holds if and only if $Ω$ is a Euclidean disk of radius $\frac{1}{c}$ and $α= 0$. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on $Ω$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_21376 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface Cho, Gunhee Seo, Keomkyo Differential Geometry Let $Ω$ be a compact surface with smooth boundary and the geodesic curvature $k_g \ge {c > 0}$ along $\partial Ω$ for some constant $c \in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K \ge -α$ for a constant $α\ge 0$, then the first eigenvalue $σ_1$ of the Steklov-type eigenvalue problem satisfies \[ σ_1 + \fracα{σ_1} \ge c. \] Moreover, equality holds if and only if $Ω$ is a Euclidean disk of radius $\frac{1}{c}$ and $α= 0$. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on $Ω$. |
| title | Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2506.21376 |