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

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 $\mathbb{R}<sup>2$ and M. S. Ashbaugh und R. D. Benguria gave a proof in $\mathbb{R}<sup>2$ and $\mathbb{R}<sup>3$. In the present paper, we prove existence of an optimal domain for minimizing the fundamental tone among all open and bounded subsets of $\mathbb{R}<sup>n$, $n\geq 4$, 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.