Shape optimization for the Steklov problem in higher dimensions

TL;DR

This paper demonstrates that in higher dimensions, the ball does not maximize the first Steklov eigenvalue among contractible domains with fixed boundary volume, contrasting with the 2D case where the disk is optimal.

Contribution

It extends the understanding of Steklov eigenvalues to higher dimensions, showing the ball's non-maximality and the effect of boundary components.

Findings

The ball does not maximize the first Steklov eigenvalue in dimensions n ≥ 3.

Increasing boundary components does not increase the normalized first Steklov eigenvalue in higher dimensions.

Contrasts with 2D results where the disk maximizes the eigenvalue.

Abstract

We show that the ball does not maximize the first nonzero Steklov eigenvalue among all contractible domains of fixed boundary volume in $\mathbb{R}^n$ when $n \geq 3$. This is in contrast to the situation when $n=2$, where a result of Weinstock from 1954 shows that the disk uniquely maximizes the first Steklov eigenvalue among all simply connected domains in the plane having the same boundary length. When $n \geq 3$, we show that increasing the number of boundary components does not increase the normalized (by boundary volume) first Steklov eigenvalue. This is in contrast to recent results which have been obtained for surfaces.