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  "body": "\n## Definition\n\nLet $n \\ge 1$ and let $\\Omega \\subseteq \\R^n$ be an open set.\nConsider a twice continuously differentiable function $u \\colon \\Omega \\to \\R$.\nThe *minimal surface equation* on $\\Omega$ is the [[quasilinear|7X9N9H3RK6]] equation\n$$\\operatorname{div} \\left( \\frac{\\nabla u}{\\sqrt{1 + \\abs{\\nabla u}^2}} \\right) = 0.$$\nHere $\\nabla u$ is the [[gradient|0ABK9V9Y64]] and $\\operatorname{div}$ is the [[divergence|0MWV6ZTK6S]].\n\n## Remarks\n\nExpanding the divergence gives the equivalent form\n$$(1 + \\abs{\\nabla u}^2) \\Delta u - \\sum_{i,j=1}^n \\partial_i u \\, \\partial_j u \\, \\partial_i \\partial_j u = 0.$$\nFor $n = 2$ this is $(1 + u_y^2) u_{xx} - 2 u_x u_y u_{xy} + (1 + u_x^2) u_{yy} = 0$.\nA function solves it exactly when the mean curvature of its graph in $\\R^{n+1}$ is zero, that is, when the graph is a [[minimal surface|41X0YW2R9N]].\nExample: Scherk's function $u(x, y) = \\log(\\cos y) - \\log(\\cos x)$ on the square $\\abs{x}, \\abs{y} \u003c \\pi/2$ is a solution.\nHere $u_x = \\tan x$, $u_y = -\\tan y$, $u_{xx} = \\sec^2 x$, $u_{yy} = -\\sec^2 y$ and $u_{xy} = 0$, so the left side is $\\sec^2 y \\sec^2 x - \\sec^2 x \\sec^2 y = 0$.\nNon-example: $u(x, y) = x^2 + y^2$ is not a solution, since at the origin $\\nabla u = 0$ and the left side equals $\\Delta u = 4$.\n",
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