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  "body": "\n## Statement\n\nLet $n \\ge 1$ and let $\\Omega \\subseteq \\R^n$ be a nonempty bounded open set.\nLet $\\Omega^*$ be the open ball in $\\R^n$ with the same [[Lebesgue measure|2Z2QVWZ262]] as $\\Omega$.\nLet $\\lambda_1(\\Omega)$ be the first [[Dirichlet eigenvalue|4VD8XXQ4P3]] of $\\Omega$.\nThen\n$$\\lambda_1(\\Omega) \\ge \\lambda_1(\\Omega^*).$$\nEquality holds if and only if $\\Omega$ is a ball up to a set of zero capacity.\n\n## Remarks\n\nProof idea: let $u \\ge 0$ be a first eigenfunction of $\\Omega$ and let $u^*$ be its [[symmetric decreasing rearrangement|2GRPCP5XK9]], which lies in $H^1_0(\\Omega^*)$.\nRearrangement keeps $\\int u^2$ and does not increase $\\int \\abs{\\nabla u}^2$, by the Pólya–Szegő inequality.\nSo the Rayleigh quotient of $u^*$ on $\\Omega^*$ is at most $\\lambda_1(\\Omega)$, which bounds $\\lambda_1(\\Omega^*)$.\n\nExample: in the plane, a square of side $\\sqrt{\\pi}$ has area $\\pi$, the same as the unit disc.\nIts first eigenvalue is $2\\pi^2 / \\pi = 2\\pi \\approx 6.28$.\nThe unit disc has $\\lambda_1 = j_{0,1}^2 \\approx 5.78$, where $j_{0,1}$ is the first zero of the Bessel function $J_0$.\n\nThe 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.\n\n## Explanations\n\n### L1 human\n\nAmong all drums of the same area, the round drum has the lowest fundamental tone.\nThis was conjectured by Lord Rayleigh and proved by Faber and Krahn.\nIt is a spectral analogue of the [[isoperimetric inequality|KBX8RBZ626]], which says the disc has the shortest boundary for a given area.\n\n### L2 human\n\nThe inequality is the prototype of shape optimisation for eigenvalues.\nIts proof combines the variational characterisation of $\\lambda_1$ with symmetrisation, and the same method gives the Pólya–Szegő and Saint-Venant inequalities.\nRelated 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.\n",
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