Faber–Krahn inequality

Remarks

Proof idea: let $u \ge 0$ be a first eigenfunction of $\Omega$ and let $u^*$ be its symmetric decreasing rearrangement, which lies in $H^1_0(\Omega^*)$. Rearrangement keeps $\int u^2$ and does not increase $\int \abs{\nabla u}^2$, by the Pólya–Szegő inequality. So the Rayleigh quotient of $u^*$ on $\Omega^*$ is at most $\lambda_1(\Omega)$, which bounds $\lambda_1(\Omega^*)$.

Example: in the plane, a square of side $\sqrt{\pi}$ has area $\pi$, the same as the unit disc. Its first eigenvalue is $2\pi^2 / \pi = 2\pi \approx 6.28$. The unit disc has $\lambda_1 = j_{0,1}^2 \approx 5.78$, where $j_{0,1}$ is the first zero of the Bessel function $J_0$.

The volume constraint is needed: dilating a ball by a factor $s$ divides $\lambda_1$ by $s^2$, so without fixing the volume there is no lower bound.

Going deeper

For a student

The inequality is the prototype of shape optimisation for eigenvalues. Its proof combines the variational characterisation of $\lambda_1$ with symmetrisation, and the same method gives the Pólya–Szegő and Saint-Venant inequalities. Related results include the Szegő–Weinberger inequality, where the ball maximises the first nonzero Neumann eigenvalue, and the Hong–Krahn–Szegő inequality for $\lambda_2$, where two equal disjoint balls are optimal.