Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface
- 1Department of Mathematics, Texas State University
- 2Department of Mathematics and Research Institute of Natural Sciences, Sookmyung Women’s University
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 ≥ − α $K \ge -\alpha $ and the geodesic curvature of the boundary satisfies k g ≥ c > 0 kg ≥ 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 ≥ − α $K \ge -\alpha $ and the geodesic curvature satisfies k g ≥ c > 0 kg ≥ c > 0 , then the first eigenvalue σ 1 σ 1 satisfies σ 1 + α σ 1 ≥ 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 α = 0 α = 0 . We also prove that the first eigenvalue of a fourth-order Steklov-type problem is bounded below by 2c under nonnegative Gaussian curvature, with equality again characterizing the Euclidean disk.
Related
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BibTeX
@article{cho2026sharp,
title = {{Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface}},
author = {Gunhee Cho and Keomkyo Seo},
journal = {Boundary Value Problems},
year = {2026},
month = jul,
doi = {10.1186/s13661-026-02331-8},
url = {https://doi.org/10.1186/s13661-026-02331-8},
note = {CC BY},
}
RIS
TY - JOUR
TI - Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface
AU - Gunhee Cho
AU - Keomkyo Seo
PY - 2026
DA - 2026/07/01
JO - Boundary Value Problems
DO - 10.1186/s13661-026-02331-8
UR - https://doi.org/10.1186/s13661-026-02331-8
AB - 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 ≥ − α $K \ge -\alpha $ and the geodesic curvature of the boundary satisfies k g ≥ 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 ≥ − α $K \ge -\alpha $ and the geodesic curvature satisfies k g ≥ c > 0 k_g ≥ c > 0 , then the first eigenvalue σ 1 σ _1 satisfies σ 1 + α σ 1 ≥ 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 α = 0 α = 0 . We also prove that the first eigenvalue of a fourth-order Steklov-type problem is bounded below by 2c under nonnegative Gaussian curvature, with equality again characterizing the Euclidean disk.
SN - 1687-2770
SN - 1687-2762
ER -