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Geometry V pp 239-266 | Cite as

The Minimal Surface Equation

  • Leon Simon
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 90)

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 ⊂ ℝ n 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 \(\frac{\partial }{{\partial {p_j}}}\left( {\frac{p}{{\sqrt {1 + |p{|^2}} }}} \right) = {\left( {1 + |p{|^2}} \right)^{ - 1/2}}\left( {{\delta _{ij}} - \frac{{{p_i}{p_j}}}{{1 + |p{|^2}}}} \right)\).

Keywords

Minimal Surface Dirichlet Problem Quasilinear Elliptic Equation Bernstein Theorem Minimal Surface Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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