the minimal-surface equation

Suppose your soap film, instead of doing anything fancy, can be written as a graph: its height above the floor is a function z = u(x, y). Then the geometric condition 'this is a minimal surface' becomes a single, very concrete partial differential equation that u must solve. That equation is the minimal-surface equation, and it is the prototype of a quasilinear elliptic PDE — the first nonlinear equation most students meet where geometry dictates the analysis.

Writing the area of the graph as Area(u) = integral over the domain of sqrt(1 + |grad u|^2) dx dy and demanding that the first variation vanish gives the Euler-Lagrange equation div( grad u / sqrt(1 + |grad u|^2) ) = 0. Expanded, this is (1 + u_y^2) u_xx - 2 u_x u_y u_xy + (1 + u_x^2) u_yy = 0. The quantity grad u / sqrt(1 + |grad u|^2) is the (downward) component of the unit normal, so the equation literally says the normal has zero divergence — a restatement of H = 0. When the gradient is small the equation linearizes to Laplace's equation u_xx + u_yy = 0, which is why minimal graphs behave like harmonic functions in the gently-sloping regime and why so much harmonic-function intuition carries over. The same recipe in higher dimensions or in a curved ambient metric gives the corresponding minimal-graph equation, always quasilinear and elliptic.

This equation is a workhorse: solving it over a domain with prescribed boundary values is the graph version of Plateau's problem, and the regularity theory (a minimal graph is automatically smooth and even real-analytic) is a landmark of elliptic PDE. Two honest points. First, the graph assumption is a genuine restriction — many minimal surfaces are not graphs, and forcing the graph form hides the most interesting global behaviour (necks, self-intersections, multiple sheets). Second, Bernstein's theorem says an entire minimal graph over all of R^2 must be a plane, and the famous surprise is that this fails in high dimensions: entire minimal graphs over R^n are planes only up to n = 7, and the Bombieri-De Giorgi-Giusti counterexample in R^8 shows the equation's solutions are subtler than the two-dimensional intuition suggests.