标题： 紧曲面上Steklov型特征值问题的第一个特征值的精确下界 显示英文标题

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

摘要： 设$\Omega$为具有光滑边界且沿$\partial \Omega$的测地曲率为$k_g \ge {c > 0}$的紧致曲面，其中$c \in \mathbb{R}$为某个常数。 我们证明，如果高斯曲率满足$K \ge -\alpha$对于常数$\alpha \ge 0$，则Steklov型特征值问题的第一个特征值$\sigma_1$满足\[ \sigma_1 + \frac{\alpha}{\sigma_1} \ge c. \]。此外，当且仅当$\Omega$是半径为$\frac{1}{c}$的欧几里得圆盘且$\alpha = 0$时等号成立。 此外，我们得到了在$\Omega$上四阶Steklov型特征值问题的第一个特征值的精确下界。 显示英文摘要

摘要： Let $\Omega$ be a compact surface with smooth boundary and the geodesic curvature $k_g \ge {c > 0}$ along $\partial \Omega$ for some constant $c \in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K \ge -\alpha$ for a constant $\alpha \ge 0$, then the first eigenvalue $\sigma_1$ of the Steklov-type eigenvalue problem satisfies \[ \sigma_1 + \frac{\alpha}{\sigma_1} \ge c. \] Moreover, equality holds if and only if $\Omega$ is a Euclidean disk of radius $\frac{1}{c}$ and $\alpha = 0$. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on $\Omega$.