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

Abstract Motivated by lower bound estimates of Payne and Escobar for the classical Steklov problem, we study sharp lower bounds for the first eigenvalue of Steklov-type problems on compact surfaces with smooth boundary. We prove that, if the Gaussian curvature satisfies $K \ge -\alpha $ K ≥ − α and the geodesic curvature of the boundary satisfies $k_{g} \ge c > 0$ k g ≥ c > 0 , then the first eigenvalue of the Steklov-type eigenvalue problem satisfies an optimal inequality and equality holds only for the Euclidean disk. In particular, we obtain a sharp lower bound for the first eigenvalue of a Schrödinger-Steklov eigenvalue problem. If the Gaussian curvature satisfies $K \ge -\alpha $ K ≥ − α and the geodesic curvature satisfies $k_{g} \ge c > 0$ k g ≥ c > 0 , then the first eigenvalue $\sigma _{1}$ σ 1 satisfies $$ \sigma _{1} + \frac{\alpha}{\sigma _{1}} \ge c, $$ σ 1 + α σ 1 ≥ c , where equality holds if and only if the surface is a Euclidean disk of radius $1/c$ 1 / c and $\alpha = 0$ α = 0 . We also prove that the first eigenvalue of a fourth-order Steklov-type problem is bounded below by 2 c under nonnegative Gaussian curvature, with equality again characterizing the Euclidean disk.