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

TL;DR

This work resolves the optimal placement of a spherical obstacle inside a ball to maximize the first nonzero Steklov eigenvalue. By leveraging a test-function framework built from the eigenfunctions of a concentric spherical shell and exploiting symmetry, the authors prove that the maximal eigenvalue is achieved uniquely when the inner and outer spheres are concentric, for doubly connected domains with fixed radii. The approach also yields a simpler, parallel proof for a Dirichlet–Steklov mixed problem and extends to the planar case. Additionally, the paper provides an explicit computation of the first Steklov eigenvalue for spherical shells, including a monotonicity result that identifies the minimizing branch and yields the exact eigenvalue formula. Collectively, these results contribute a sharp, geometrically anchored isoperimetric-type statement in higher dimensions and enhance understanding of obstacle-placement effects in spectral problems.

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.

Figures (1)

• Figure 1: The domains $\Omega_d$ and $\Omega_0$

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Remark 2.2
• Lemma 2.3
• proof
• Lemma 2.4
• proof
• Proposition 3.1
• proof