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  "body": "\n## Problem\n\nLet $G = (V, E)$ be a finite connected [[graph|00C0BR7D7H]] with symmetric positive edge weights $c(x, y)$.\nLet $B \\subseteq V$ be a nonempty set of boundary vertices, and let $I = V \\setminus B$ be the set of interior vertices.\nLet $g \\colon B \\to \\R$ be a function.\nFind a function $f \\colon V \\to \\R$ that is [[harmonic|WRY0M7SW59]] at every vertex of $I$ and satisfies $f = g$ on $B$.\n\n## Remarks\n\nThe problem always has exactly one solution.\nUniqueness follows from the maximum principle: the difference $h$ of two solutions is harmonic on $I$ and zero on $B$.\nAt an interior vertex where $h$ is largest, harmonicity forces the same value at all neighbours.\nConnectedness spreads this value to a vertex of $B$, so $\\max h = 0$, and in the same way $\\min h = 0$.\nThe conditions form a square system of $\\abs{I}$ linear equations in the $\\abs{I}$ unknowns $f(x)$, $x \\in I$, so uniqueness gives existence.\nThe solution has the probabilistic form $f(x) = \\mathbb{E}_x\\big[g(X_\\tau)\\big]$, where $(X_n)$ is the random walk with $p(x, y) = c(x, y)/c(x)$ started at $x$ and $\\tau$ is its first hitting time of $B$.\nExample: on the path with vertices $0, 1, \\dots, n$, unit weights and $B = \\set{0, n}$, the solution is the linear interpolation $f(k) = g(0) + \\tfrac{k}{n}\\big(g(n) - g(0)\\big)$.\nConnectedness matters: if a component of the graph does not meet $B$, any constant can be added to $f$ on that component, and the solution is not unique.\n\n## Explanations\n\n### L1 human\n\nFix numbers on some vertices of a graph, the boundary.\nThe task is to fill in the other vertices so that each of them carries the average of its neighbours.\nThink of the boundary numbers as fixed voltages, or fixed temperatures, and of the filled-in numbers as the equilibrium state.\nOn a connected finite graph there is always exactly one way to do it.\nIt can be found by solving a linear system or by running random walks to the boundary.\n",
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      "first_stated": "1923",
      "notes": "Phillips and Wiener studied it in Nets and the Dirichlet problem, J. Math. Phys. 2 (1923). Courant, Friedrichs and Lewy used it in 1928.",
      "posed_by": "H. B. Phillips and Norbert Wiener"
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    "kind": "problem",
    "msc": [
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    "name": "Discrete Dirichlet problem",
    "sources": [
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        "url": "https://en.wikipedia.org/wiki/Discrete_Laplace_operator"
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  "markdown": "https://tamnd.github.io/math2/node/YKETR75ZYW.md",
  "msc": [
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  "name": "Discrete Dirichlet problem",
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/YKE/YKETR75ZYW.md",
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  "url": "https://tamnd.github.io/math2/wiki/discrete-dirichlet-problem/",
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