It is easy to see that the equation for a minimal surface is a partial differential equation. In this section, an attempt will be made to find its two famous special solutions: the catenoid and the helicoid.
After that, we will give two other examples of minimal surfaces: Enneper’s surface and Scherk’s surface.
Given the minimal surface equation:
$$ f_{xx}(1 + f_y^2) - 2f_x f_y f_{xy} + f_{yy}(1 + f_x^2) = 0, $$
where the function $f(x, y)$ has continuous second partial derivatives. Let $(r, \theta)$ be polar coordinates with $x = r \cos \theta$ and $y = r \sin \theta$, and define $h(r, \theta) = f(r \cos \theta, r \sin \theta)$. If $h(r, \theta) = h(r)$ is a function of $r$ only, we can attempt to reduce the minimal surface equation into an ordinary differential equation (ODE). By solving the ODE, we can obtain one particular solution.
Change of Variables
First, we need to compute $\theta_x$, $\theta_y$, $r_x$, $r_y$, $x_\theta$, $x_r$, $y_\theta$, and $y_r$. It is straightforward to find:
$$ \theta_x = -\frac{1}{r} \sin \theta, \quad \theta_y = \frac{1}{r} \cos \theta, \quad r_x = \cos \theta, \quad r_y = \sin \theta, $$
$$ x_r = \cos \theta, \quad y_r = \sin \theta, \quad x_\theta = -r \sin \theta, \quad y_\theta = r \cos \theta. $$
Next, calculate $h_r$ and $h_\theta$:
$$ h_r = f_x \cos \theta + f_y \sin \theta, \quad h_\theta = -f_x r \sin \theta + f_y r \cos \theta. $$
This forms a linear system of equations. Solving for $f_x$ and $f_y$:
$$ f_x = \cos \theta h_r - \frac{\sin \theta}{r} h_\theta, \quad f_y = \sin \theta h_r + \frac{\cos \theta}{r} h_\theta. $$
Reduction to ODE
By the chain rule, compute $f_{xx}$, $f_{yy}$, and $f_{xy}$:
$$ f_{xx} = h_{rr} \cos^2 \theta - \frac{2}{r} \sin \theta \cos \theta h_{r \theta} + \frac{2}{r^2} \sin \theta \cos \theta h_\theta + \frac{1}{r^2} \sin^2 \theta h_{\theta \theta} + \frac{1}{r} \sin^2 \theta h_r, $$
$$ f_{yy} = h_{rr} \sin^2 \theta + \frac{2}{r} \sin \theta \cos \theta h_{r \theta} - \frac{2}{r^2} \sin \theta \cos \theta h_\theta + \frac{1}{r^2} \cos^2 \theta h_{\theta \theta} + \frac{1}{r} \cos^2 \theta h_r, $$
$$ f_{xy} = \sin \theta \cos \theta h_{rr} + \frac{\sin^2 \theta}{r^2} h_\theta - \frac{\sin^2 \theta}{r} h_{r \theta} - \frac{\sin \theta \cos \theta}{r} h_r + \frac{\cos^2 \theta}{r} h_{r \theta} - \frac{\cos^2 \theta}{r^2} h_{\theta \theta} - \frac{\cos^2 \theta}{r^2} h_\theta. $$
If $h(r, \theta)$ depends only on $r$, we simplify the minimal surface equation to:
$$ 2r h_{rr} + h_r^3 + h_r = 0. $$
Solution for the Catenoid
Solving the simplified ODE gives one particular solution, known as the catenoid:
$$ r = c \cosh \left(\frac{h - d}{c}\right), $$
where $c$ and $d$ are constants.
Solution for the Helicoid
Similarly, if $h(r, \theta) = h(\theta)$, substituting into the minimal surface equation gives another particular solution, known as the helicoid:
$$ h = c \theta + d, $$
where $c$ and $d$ are constants.