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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|2Z2QVWZ262]] $\\abs{\\Omega}$.\nLet $\\omega_d$ be the volume of the unit ball of $\\R^d$.\nLet $0 \u003c \\lambda_1 \\le \\lambda_2 \\le \\cdots$ be the [[Dirichlet eigenvalues|4VD8XXQ4P3]] of $-\\Delta$ on $\\Omega$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nIf $\\Omega$ is also connected with Lipschitz boundary, let $0 = \\mu_1 \u003c \\mu_2 \\le \\mu_3 \\le \\cdots$ be its eigenvalues with the [[Neumann boundary condition|GS3KT2H645]].\nAll are repeated according to multiplicity.\nLet $C_d = \\dfrac{4 \\pi^2}{\\omega_d^{2/d}}$.\nThen, for every $k \\ge 1$,\n$$\\lambda_k \\ge C_d \\left( \\frac{k}{\\abs{\\Omega}} \\right)^{2/d} \\qquad \\text{and} \\qquad \\mu_{k+1} \\le C_d \\left( \\frac{k}{\\abs{\\Omega}} \\right)^{2/d}.$$\n\n## Remarks\n\nThe constant is that of [[Weyl's law|2JK9YMZV2B]], $\\lambda_k \\sim C_d (k / \\abs{\\Omega})^{2/d}$ as $k \\to \\infty$, so the conjecture says that the Weyl term is a one-sided bound for every $k$, not only asymptotically.\nFor $d = 2$ the bounds read $\\lambda_k \\ge 4 \\pi k / \\abs{\\Omega}$ and $\\mu_{k+1} \\le 4 \\pi k / \\abs{\\Omega}$.\nThe conjecture is open in general.\nPólya proved the Dirichlet bound for plane domains that tile the plane.\nLi and Yau proved the weaker bound $\\lambda_k \\ge \\frac{d}{d + 2} C_d (k / \\abs{\\Omega})^{2/d}$ for all bounded domains, now called the Berezin–Li–Yau inequality.\nFilonov, Levitin, Polterovich and Sher proved both bounds for the disk.\nFor $k = 1$ the Dirichlet bound follows from the [[Faber–Krahn inequality|E7TY2TPXX7]].\n\n## Explanations\n\n### L1 human\n\nThe eigenvalues of the Laplacian on a region are the squared pitches of a drum of that shape.\nWeyl's law says how the $k$-th pitch grows with $k$, on average.\nPólya guessed that, with the Dirichlet condition, every pitch lies above the Weyl prediction, and with the Neumann condition every pitch lies below it.\n",
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    "history": {
      "first_stated": "1954",
      "posed_by": "George Pólya"
    },
    "kind": "conjecture",
    "msc": [
      "35P15",
      "49R05",
      "35P20"
    ],
    "name": "Pólya's conjecture on Laplacian eigenvalues",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Weyl_law"
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/F68QFEYFY0/",
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  "markdown": "https://tamnd.github.io/math2/node/F68QFEYFY0.md",
  "msc": [
    "35P15",
    "49R05",
    "35P20"
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  "name": "Pólya's conjecture on Laplacian eigenvalues",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/F68/F68QFEYFY0.md",
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  "url": "https://tamnd.github.io/math2/wiki/polyas-conjecture-on-laplacian-eigenvalues/",
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