Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space

TL;DR

This paper establishes bounds for Steklov eigenvalues of submanifolds with fixed boundary in Euclidean space, revealing geometric dependencies and characterizing extremal shapes like the ball and disk.

Contribution

It provides general upper bounds based on boundary geometry, sharp lower bounds for revolution hypersurfaces, and insights into isospectrality of revolution surfaces.

Findings

Upper bounds depend only on boundary geometry and interior measure.

Balls uniquely minimize Steklov eigenvalues among revolution hypersurfaces.

Revolution surfaces with connected boundary are isospectral to the disk.

Abstract

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: we prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is isospectral to the disk.