The Minimal Surface Equation

Abstract

The minimal surface equation (MSE) for functions u: Ω → ℝ, Ω a domain of ℝ², can be written

$$\left( {1 + u_{}^2} \right){u_{xx}} - 2{u_x}{u_y}{u_{xy}} + \left( {1 + u_x^2} \right){u_{yy}} = 0$$

or equivalently ${u_{xx}} + {u_{yy}} - {\left( {1 + |Du{|^2}} \right)^{ - 1}}\left( {u_x^2{u_{xx}} + 2{u_x}{u_y}{u_{xy}} + u_y^2{u_{yy}}} \right) = 0$ where ${u_x} = \frac{{\partial u\left( {x,y} \right)}}{{\partial x}},{u_y} = \frac{{\partial u\left( {x,y} \right)}}{{\partial y}}$. Generally, for domains Ω ⊂ ℝⁿ and functions Ω → ℝ depending on the n variables (x ¹, …, x ⁿ) ∈ Ω, n ≥ 2, the MSE can be written

$$\sum\limits_{i,j = 1}^n {\left( {{\delta _{ij}} - \frac{{{u_i}{u_j}}}{{\left( {1 + |Du{|^2}} \right)}}} \right){u_{ij}} = 0} $$

where ${u_i} = {D_i}u \equiv \frac{{\partial u}}{{\partial {x^i}}}$ and u _ij = D _i D _j u. Notice that this is a quasilinear elliptic equation: that is, it is linear in the second derivatives, and the coefficient matrix $\left( {{\delta _{ij}} - \frac{{{u_i}{u_j}}}{{\left( {1 + |Du{|^2}} \right)}}} \right)$ is positive definite¹ depending only on the derivatives up to first order. The equation can alternatively be written in “divergence form”

$$\sum\limits_{i = 1}^n {{D_i}} \left( {\frac{{{D_i}u}}{{\sqrt {1 + |Du{|^2}} }}} \right) = 0$$

((1))

which is readily checked using the chain rule and the fact that $\frac{\partial }{{\partial {p_j}}}\left( {\frac{p}{{\sqrt {1 + |p{|^2}} }}} \right) = {\left( {1 + |p{|^2}} \right)^{ - 1/2}}\left( {{\delta _{ij}} - \frac{{{p_i}{p_j}}}{{1 + |p{|^2}}}} \right)$.