Existence of an optimal domain for minimizing the fundamental tone of a clamped plate of prescribed volume in arbitrary dimension

TL;DR

This paper proves the existence of an optimal domain in higher dimensions that minimizes the fundamental tone of a clamped plate with fixed volume, extending classical two- and three-dimensional results.

Contribution

It establishes the existence of a volume-constrained minimizer for the fundamental tone in dimensions four and higher, using a free boundary problem approach with penalization.

Findings

Existence of an optimal domain in $ abla$-dimensional space for minimizing fundamental tone.

Formulation of the problem as a free boundary value problem with volume penalization.

Convergence of solutions as the penalization parameter tends to zero.

Abstract

In the 19th century, Lord Rayleigh conjectured that among all clamped plates with given area, the disk minimizes the fundamental tone. In the 1990s, N. S. Nadirashvili proved the conjecture in $\mathbb{R}^2$ and M. S. Ashbaugh und R. D. Benguria gave a proof in $\mathbb{R}^2$ and $\mathbb{R}^3$. In the present paper, we prove existence of an optimal domain for minimizing the fundamental tone among all open and bounded subsets of $\mathbb{R}^n$, $n\geq 4$, with given measure. We formulate the minimization of the fundamental tone of a clamped plate as a free boundary value problem with a penalization term for the volume constraint. As the penalization parameter becomes small we show that the optimal shape problem is solved.