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  "body": "\n## Problem\n\nLet $\\Omega \\subseteq \\R^n$ be a bounded domain with Lipschitz boundary $\\partial\\Omega$ and outer unit normal $\\nu$.\nFind all $\\sigma \\in \\R$ for which some nonzero $u \\in H^1(\\Omega)$ solves\n$$\\Delta u = 0 \\text{ in } \\Omega, \\qquad \\frac{\\partial u}{\\partial \\nu} = \\sigma u \\text{ on } \\partial\\Omega,$$\nin the weak sense.\nHere $\\Delta$ is the [[Laplacian|0E0B6RA8K0]] and $H^1(\\Omega)$ is a [[Sobolev space|M4CEZ5K9HP]].\nSuch numbers $\\sigma$ are the *Steklov eigenvalues* of $\\Omega$, and one asks how they depend on the geometry of $\\Omega$.\n\n## Remarks\n\nThe Steklov eigenvalues are the eigenvalues of the [[Dirichlet-to-Neumann operator|W783KDGT8E]] of $\\Omega$, which maps boundary values $f$ to the normal derivative of the harmonic extension of $f$.\nThey form a sequence $0 = \\sigma_0 \u003c \\sigma_1 \\le \\sigma_2 \\le \\dots \\to \\infty$, and $\\sigma_0 = 0$ belongs to the constant functions.\nExample: for the unit disc, the functions $r^k \\cos(k\\theta)$ and $r^k \\sin(k\\theta)$ are harmonic with $\\partial_r u = k u$ on the circle, so the eigenvalues are $0, 1, 1, 2, 2, 3, 3, \\dots$.\nWeinstock proved that among simply connected planar domains of given perimeter, the disc maximises $\\sigma_1$.\n\n## Explanations\n\n### L1 human\n\nHere the vibration happens only on the boundary, while the inside is in equilibrium.\nThe equation says that the flux across the edge is proportional to the value on the edge.\nIt models, for example, the sloshing of a liquid in a container and heat exchange through a surface.\n",
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      "first_stated": "1902",
      "posed_by": "Vladimir Steklov"
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    "name": "Steklov eigenvalue problem",
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        "url": "https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Steklov_operator"
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