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  "body": "\n## Statement\n\nLet $n \\ge 2$ and let $\\Omega \\subseteq \\R^n$ be a bounded, simply connected domain with smooth boundary $\\partial\\Omega$ and outer unit normal $\\nu$.\nSuppose that for some $\\lambda \u003e 0$ and some constant $c$ there is a nonzero function $u$ with\n$$\\Delta u + \\lambda u = 0 \\text{ in } \\Omega, \\qquad \\frac{\\partial u}{\\partial \\nu} = 0 \\text{ on } \\partial\\Omega, \\qquad u = c \\text{ on } \\partial\\Omega,$$\nwhere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nThe conjecture states that $\\Omega$ is then a ball.\n\n## Remarks\n\nThe boundary value problem is overdetermined: $u$ is a Neumann eigenfunction that is also constant on the boundary.\nEvery ball admits such a solution: a radial Neumann eigenfunction is constant on each sphere centred at the centre of the ball.\nBy a result of Williams (1976), for these domains the conjecture is equivalent to the [[Pompeiu problem|M1KYAQMF9W]], that only balls fail the Pompeiu property.\nConsider the analogous problem for a Dirichlet eigenfunction with constant normal derivative.\nBy Serrin's theorem on overdetermined problems, when the Dirichlet eigenfunction is positive, the domain must be a ball.\n\n## Explanations\n\n### L1 human\n\nTake a membrane with a free edge and look at one of its vibration patterns.\nSuppose that along the whole edge the membrane sits at one common height.\nSchiffer guessed that this can only happen when the membrane is round.\n",
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      "posed_by": "Menahem Max Schiffer"
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    "kind": "conjecture",
    "msc": [
      "35N25",
      "35P05",
      "35R30"
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    "name": "Schiffer's conjecture",
    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Pompeiu_problem"
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  "kind": "conjecture",
  "link": "https://tamnd.github.io/math2/node/J05VKG4XQB/",
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  "msc": [
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    "35P05",
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  "name": "Schiffer's conjecture",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/J05/J05VKG4XQB.md",
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  "uri": "m2://J05VKG4XQB",
  "url": "https://tamnd.github.io/math2/wiki/schiffers-conjecture/",
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    "tag": "J05VKG4XQB",
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