Minimal surface equation

Summary

The minimal surface equation for a function $u$ of $n$ variables says that its graph is a minimal surface. That is, the graph has zero mean curvature. It reads $\operatorname{div}\big(\nabla u / \sqrt{1 + \abs{\nabla u}^2}\big) = 0$. It is the Euler–Lagrange equation of the area of the graph, $\int \sqrt{1 + \abs{\nabla u}^2} \, dx$. It is a model quasilinear elliptic equation, and its study led to Bernstein's problem and to deep results on regularity.