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Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface

Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface

来源:Arxiv_logoArxiv
英文摘要

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 $Ω$.

Gunhee Cho、Keomkyo Seo

数学

Gunhee Cho,Keomkyo Seo.Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface[EB/OL].(2025-06-26)[2025-07-16].https://arxiv.org/abs/2506.21376.点此复制

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