= Faber-Krahn inequality
{c}
Among bounded <domains> in $\mathbb R^n$ of a prescribed volume, the ball minimizes the first <Dirichlet eigenvalue> of the negative <Laplace operator>. For sufficiently regular $\Omega$ and a ball $B$ with $|B|=|\Omega|$,
$$
\lambda_1(\Omega)\ge\lambda_1(B),\qquad
\lambda_1(\Omega)=\inf_{0\ne u\in H_0^1(\Omega)}\frac{\int_\Omega|\nabla u|^2}{\int_\Omega|u|^2}.
$$
This is the <Faber-Krahn inequality>. The variational formula shows why rearrangement helps: replacing a function by its symmetric decreasing rearrangement preserves its squared integral while reducing its gradient energy.<Analysis on Manifolds via the Laplacian by Yaiza Canzani>[Section 7.5, Theorem 65]{ref}
For a stretched domain $r\Omega$, the eigenvalue scales as $r^{-2}\lambda_1(\Omega)$, so the volume normalization is essential. For a membrane with fixed wave speed and area, the disk has the lowest fundamental frequency because frequency is proportional to $\sqrt{\lambda_1}$. For diffusion with an absorbing boundary, it instead gives the slowest asymptotic decay among shapes of that volume.
Back to article page
