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  "body": "\n## Statement\n\nLet $d \\ge 1$ and let $\\Omega \\subseteq \\R^d$ be a nonempty bounded open set with Lebesgue measure $\\abs{\\Omega}$.\nLet $0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\cdots$ be the [[eigenvalues|0BQAA2BK3Q]] of $-\\Delta$ on $\\Omega$ with the [[Dirichlet boundary condition|EETVA6QYX4]], repeated according to multiplicity, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nFor $\\lambda \u003e 0$ let $N(\\lambda)$ be the number of indices $k$ with $\\lambda_k \\le \\lambda$.\nLet $\\omega_d$ be the volume of the unit ball in $\\R^d$.\nThen\n$$\\lim_{\\lambda \\to \\infty} \\frac{N(\\lambda)}{\\lambda^{d/2}} = \\frac{\\omega_d \\abs{\\Omega}}{(2\\pi)^d}.$$\n\n## Remarks\n\nFor a cube the eigenvalues are explicit, and counting them amounts to counting lattice points in a ball of radius $\\sqrt{\\lambda}$, which gives the formula.\nFor a general set one fills $\\Omega$ from inside and covers it from outside by small cubes.\nThen one compares eigenvalues with Dirichlet and Neumann conditions on the cubes; this is Dirichlet–Neumann bracketing.\nExample: for $\\Omega = (0, L) \\subseteq \\R$, the eigenvalues are $(k\\pi / L)^2$ for $k \\ge 1$, so $N(\\lambda) = \\floor{L \\sqrt{\\lambda} / \\pi}$.\nThis agrees with the formula, since $\\omega_1 = 2$ and $\\frac{2L}{2\\pi} = \\frac{L}{\\pi}$.\nEquivalently, $\\lambda_k \\sim \\frac{(2\\pi)^2}{(\\omega_d \\abs{\\Omega})^{2/d}} \\, k^{2/d}$ as $k \\to \\infty$.\nHermann Weyl proved the law in 1911.\n\n## Explanations\n\n### L1 human\n\nUp to a constant factor, the eigenvalues of the Laplacian on a region are the squares of the pitches of a drum of that shape.\nWeyl's law says that, for high pitches, the number of notes below a given pitch depends only on the dimension and the size of the drum, not on its shape.\n\n### L2 human\n\nWeyl's law shows that the volume of a domain can be read off from its spectrum.\nSo it gives a first positive answer to the question whether one can hear the shape of a drum.\nThe same leading term holds for the Neumann condition when the boundary is regular enough, and an analogous law holds on compact Riemannian manifolds.\n",
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