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

Abstract

Let $\hat{I}\text{c◯}$ be a compact surface with smooth boundary and the geodesic curvature $k_g\ge c>0$ along $\partial \hat{I}\text{c◯}$ for some constant $c\in \mathbb{R}$. We prove that, if the Gaussian curvature satisfies $K\ge -\hat{I}\pm$ for a constant $\hat{I}\pm \ge 0$, then the first eigenvalue $\ddot{I}{ƒ}_1$ of the Steklov-type eigenvalue problem satisfies \[ σ_1 + \fracα{σ_1} \ge c. \] Moreover, equality holds if and only if $\hat{I}\text{c◯}$ is a Euclidean disk of radius $\frac{1}{c}$ and $\hat{I}\pm =0$. Furthermore, we obtain a sharp lower bound for the first eigenvalue of the fourth-order Steklov-type eigenvalue problem on $\hat{I}\text{c◯}$.