Sharp bounds for the first eigenvalue of a fourth order Steklov problem

TL;DR

This paper establishes a sharp, computable lower bound for the first eigenvalue of a biharmonic Steklov problem on Riemannian manifolds, depending on geometric bounds and the domain's inner radius.

Contribution

It provides the first explicit lower bound for the eigenvalue in terms of geometric quantities, along with a comparison theorem for geodesic balls.

Findings

Derived a sharp lower bound depending on Ricci curvature, boundary mean curvature, and inner radius.

Established a comparison theorem for eigenvalues on geodesic balls.

Estimated isoperimetric ratios of subharmonic functions on the domain.

Abstract

We study the biharmonic Steklov eigenvalue problem on a compact Riemannian manifold $\Omega$ with smooth boundary. We give a computable, sharp lower bound of the first eigenvalue of this problem, which depends only on the dimension, a lower bound of the Ricci curvature of the domain, a lower bound of the mean curvature of its boundary and the inner radius. The proof is obtained by estimating the isoperimetric ratio of non-negative subharmonic functions on $\Omega$, which is of independent interest. We also give a comparison theorem for geodesic balls.