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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $U \\subseteq \\R^n$ be open.\n*Laplace's equation* on $U$ is the equation\n$$\\Delta u = \\frac{\\partial^2 u}{\\partial x_1^2} + \\cdots + \\frac{\\partial^2 u}{\\partial x_n^2} = 0 \\quad \\text{in } U$$\nfor an unknown function $u \\colon U \\to \\R$, where $\\Delta$ is the [[Laplacian|0E0B6RA8K0]].\nA classical solution is a function $u$ that is twice continuously differentiable on $U$ and satisfies the equation at every point of $U$.\n\n## Remarks\n\nThe classical solutions are exactly the harmonic functions on $U$.\nEvery classical solution is in fact infinitely differentiable, and even real analytic.\nThe inhomogeneous equation $\\Delta u = f$ is Poisson's equation.\nSome authors write the Laplacian as $\\nabla^2$ or $\\nabla \\cdot \\nabla$, and in physics the equation is often written $\\nabla^2 \\phi = 0$.\n\nExample: on $\\R^3 \\setminus \\set{0}$, the function $u(x) = 1 / \\abs{x}$ is a solution.\nWith $r = \\abs{x}$, one has $\\partial_i u = -x_i / r^3$ and $\\partial_i^2 u = -1/r^3 + 3 x_i^2 / r^5$, and the sum over $i = 1, 2, 3$ is $-3/r^3 + 3 r^2 / r^5 = 0$.\n\nNon-example: on $\\R^2$, the function $u(x, y) = x^2 + y^2$ is not a solution, since $\\Delta u = 2 + 2 = 4 \\ne 0$.\n",
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