Faber-Krahn inequality

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 $|B|=|\Omega |$,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.^[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.