Work

An Overdetermined Steklov Eigenvalue Problem on Riemannian Manifolds with Nonnegative Ricci Curvature

Year: 2025

Type: article

Abstract: We consider an overdetermined Steklov eigenvalue problem on a domain in a Riemannian manifold with nonnegative Ricci curvature. We prove that, given a compact connected domain $$\Omega $$ with nonnegative Gaussian curvature with $$C^2$$ boundary, if the first Steklov eigenfunction is a solution to the overdetermined problem, then the domain $$\Omega $$ is flat and the boundary $$\partial \Omega $$ consists of geodesics or geodesic circles. This can be regarded as a generalization of the result by Payne-Philippin [21], where they assumed that $$\Omega $$ is a simply-connected domain in $$\mathbb {R}^2$$ . We also obtain a similar result for the same overdetermined problem on a higher-dimensional compact connected manifold with $$C^2$$ boundary with nonnegative Ricci curvature.

Source: Results in Mathematics

DOI: 10.1007/s00025-025-02422-5

Authors: Eunjoo Lee, Keomkyo Seo

Institutions: Soongsil University, Sookmyung Women's University

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,090

Funders: National Research Foundation, National Research Foundation of Korea