Minimal surface equation

Remarks

Expanding the divergence gives the equivalent form $$(1 + \abs{\nabla u}^2) \Delta u - \sum_{i,j=1}^n \partial_i u \, \partial_j u \, \partial_i \partial_j u = 0.$$ For $n = 2$ this is $(1 + u_y^2) u_{xx} - 2 u_x u_y u_{xy} + (1 + u_x^2) u_{yy} = 0$. A 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. Example: Scherk's function $u(x, y) = \log(\cos y) - \log(\cos x)$ on the square $\abs{x}, \abs{y} < \pi/2$ is a solution. Here $u_x = \tan x$, $u_y = -\tan y$, $u_{xx} = \sec^2 x$, $u_{yy} = -\sec^2 y$ and $u_{xy} = 0$, so the left side is $\sec^2 y \sec^2 x - \sec^2 x \sec^2 y = 0$. Non-example: $u(x, y) = x^2 + y^2$ is not a solution, since at the origin $\nabla u = 0$ and the left side equals $\Delta u = 4$.