[Paper Review] Some isoperimetric inequalities with application to the Stekloff problem

Antoine Henrot, G. A. Philippin+1 more|ArXiv.org|Mar 29, 2008

Point processes and geometric inequalities8 references20 citations

TL;DR

This paper establishes isoperimetric inequalities for the product of moments of inertia in $ N$-dimensional space, proving that ellipsoids symmetric about coordinate planes minimize the product $J( Omega) = \prod_{k=1}^N J_k(\nOmega)$, while the unit ball minimizes $I(\nOmega) = \prod_{k=1}^N I_k(\nOmega)$ for convex domains. As a key application, it derives an isoperimetric inequality for the product of the first $N$ nonzero Stekloff eigenvalues, generalizing a 2D result to higher dimensions and showing that the ball maximizes this product among convex domains.

ABSTRACT

In this paper we establish isoperimetric inequalities for the product of some moments of inertia. As an application, we obtain an isoperimetric inequality for the product of the $N$ first nonzero eigenvalues of the Stekloff problem in $\mathbb{R}^N$.

Motivation & Objective

• To establish isoperimetric inequalities for the product of moments of inertia $J(\nOmega) = \prod_{k=1}^N J_k(\nOmega)$, where $J_k(\nOmega)$ is the moment of inertia of a domain $\nOmega \subset \mathbb{R}^N$ with respect to the plane $x_k = 0$.
• To derive an isoperimetric inequality for the product of boundary moments of inertia $I(\nOmega) = \prod_{k=1}^N I_k(\nOmega)$, where $I_k(\nOmega)$ is the moment of inertia of $\partial\nOmega$ with respect to $x_k = 0$.
• To apply the inequality $I(\nOmega) \geq I(\nOmega^*)$ to obtain a sharp upper bound for the product of the $N$ first nonzero Stekloff eigenvalues in $\mathbb{R}^N$.
• To characterize the minimizers of $I(\nOmega)$ among convex domains in $\mathbb{R}^2$, showing that only the disk achieves equality.
• To extend classical results of Hersch, Payne, and Schiffer from 2D to $\mathbb{R}^N$, proving that the ball maximizes the product of the first $N+1$ Stekloff eigenvalues.

Proposed method

• Uses affine transformations with determinant one to preserve the product $J(\nOmega)$, enabling normalization of individual moments of inertia to the same value.
• Applies the method of Lagrange multipliers in a constrained optimization framework to minimize $J(\nOmega)$ under volume constraint, showing that ellipsoids symmetric about coordinate planes are minimizers.
• Employs Fourier series parametrization of the boundary $\partial\nOmega$ to analyze the condition $\left(\frac{dx}{d\sigma}\right)^2 + \left(\frac{dy}{d\sigma}\right)^2 = \text{const.}$, which ensures constant-speed parametrization.
• Imposes orthogonality and normalization conditions on Fourier coefficients to enforce closed, simple, and regular boundary curves.
• Analyzes the vanishing of harmonic coefficients in the expression for $\left(\frac{dx}{d\sigma}\right)^2 + \left(\frac{dy}{d\sigma}\right)^2$ to deduce that only circular boundaries satisfy the equality case.
• Applies the isoperimetric inequality $I(\nOmega) \geq I(\nOmega^*)$ to the Stekloff eigenvalue problem via spectral theory and variational characterization.

Experimental results

Research questions

• RQ1 What domains minimize the product of moments of inertia $J(\nOmega) = \prod_{k=1}^N J_k(\nOmega)$ among all bounded domains of fixed $N$-volume?
• RQ2 For which domains does equality hold in the inequality $I(\nOmega) \geq I(\nOmega^*)$, where $I_k(\nOmega)$ is the boundary moment of inertia?
• RQ3 Can the 2D result of Hersch, Payne, and Schiffer on the product of Stekloff eigenvalues be generalized to $\mathbb{R}^N$?
• RQ4 What is the geometric structure of a domain that minimizes $I(\nOmega)$ under fixed area and convexity constraints?
• RQ5 Under what conditions on Fourier coefficients does a closed curve in $\mathbb{R}^2$ have constant-speed parametrization and constant curvature?

Key findings

• The product of moments of inertia $J(\nOmega)$ is minimized among all domains of fixed $N$-volume if and only if $\nOmega$ is an $N$-dimensional ellipsoid symmetric with respect to the coordinate planes $x_k = 0$.
• For convex domains $\nOmega \subset \mathbb{R}^N$, the product $I(\nOmega) = \prod_{k=1}^N I_k(\nOmega)$ is minimized when $\nOmega$ is a ball $\nOmega^*$, with equality only for the ball.
• In $\mathbb{R}^2$, the only domain minimizing $I(\nOmega)$ among convex domains of fixed area is the disk centered at the origin.
• The parametric representation of a minimizer $\partial\nOmega^*$ of $I(\nOmega)$ must be of the form $x(\sigma) = a_1\cos\sigma + a_1'\sin\sigma$, $y(\sigma) = b_1\cos\sigma + b_1'\sin\sigma$, which describes a circle.
• The product of the $N$ first nonzero Stekloff eigenvalues satisfies $\prod_{k=2}^{N+1} p_k(\nOmega) \leq \prod_{k=2}^{N+1} p_k(\nOmega^*)$ for all convex domains $\nOmega \subset \mathbb{R}^N$, with equality if and only if $\nOmega = \nOmega^*$.
• The equality case in the isoperimetric inequality $I(\nOmega) \geq I(\nOmega^*)$ implies that the boundary $\partial\nOmega$ must be a circle, confirming the uniqueness of the minimizer in $\mathbb{R}^2$.