Where to place a spherical obstacle so as to maximize the first nonzero Steklov eigenvalue

TL;DR

This paper proves that the maximal first nonzero Steklov eigenvalue for doubly connected domains occurs when the inner and outer spheres are concentric, with implications for related eigenvalue problems.

Contribution

It establishes the unique maximizer for the Steklov eigenvalue among doubly connected domains and extends the approach to other boundary condition eigenvalue problems.

Findings

Maximal Steklov eigenvalue occurs for concentric spheres.

The proof technique applies to mixed boundary condition eigenvalue problems.

Provides a characterization of eigenvalue extremizers in geometric domains.

Abstract

We prove that among all doubly connected domains of $\mathbb{R}^n$ of the form $B_1\backslash \overline{B_2}$, where $B_1$ and $B_2$ are open balls of fixed radii such that $\overline{B_2}\subset B_1$, the first nonzero Steklov eigenvalue achieves its maximal value uniquely when the balls are concentric. Furthermore, we show that the ideas of our proof also apply to a mixed boundary conditions eigenvalue problem found in literature.