Area-Minimizing Surfaces

[xaer04]

Homework Statement

"If an area-minimizing surface can be given by the graph of a function, that function satisfies the minimal surface equation:
$$(1+{f_y}^2)f_{xx}-2f_xf_yf_{xy}+(1+{f_x}^2)f_{yy} = 0$$

1.)Determine if the graphs of the following functions may be area minimizing:
a.) $$z = 2x+4y+10$$
b.) $$y\tan(z)=x$$
c.) $$\cos(y)e^z=\cos(x)$$
d.) $$\sqrt{x^2+y^2}=\cosh(z)$$

2.)While it is necessary that the graph of a function satisfy the Minimal Surface Equation to be area-minimizing, it may not be sufficient. Enneper's surface satisfies the equation (you do not need to show this), but is not necessarily minimal. Find a surface with the same boundary that has less surface area."

Homework Equations

Minimal Surface Equation
$$(1+{f_y}^2)f_{xx}-2f_xf_yf_{xy}+(1+{f_x}^2)f_{yy} = 0$$

The Attempt at a Solution

I already went through the calculus and found that the first 3 work with the equation, and i started working on the 4th one... and it's extremely messy. I was wondering if anybody knows a way to convert to cylindrical coordinates and use a corresponding version of the minimal surface equation? Converting the function would be easy(i think), I'm just not so sure about if it would still work.

Also, I just don't understand the second half of the problem. I'm not sure what a surface with the same boundary is... I've seen other minimal surfaces, such as the helicoid, or the catenoid: do those work?. I also looked up Enneper's surface and... it's crazy. I read into the article and it describes it with high-degree equations, multi-variable parametrization, and different types of curvature that I've never even heard of... please help.

 

[mighty2000]

I can provide some insights and suggestions for your questions.

1. To determine if the graphs of the given functions may be area-minimizing, we need to plug in the functions into the minimal surface equation and see if it satisfies the equation. I see that you have already done this for the first three functions, and they all satisfy the equation. For the fourth function, it is indeed a messy calculation, but it is possible to convert it into cylindrical coordinates and use the corresponding version of the minimal surface equation. I suggest using the chain rule to convert the derivatives from Cartesian to cylindrical coordinates.

2. The second part of the problem is asking for a surface with the same boundary as Enneper's surface but with less surface area. This means that the surface must have the same edges or boundaries as Enneper's surface, but it can have a different shape and therefore a different surface area. The helicoid and the catenoid are both examples of minimal surfaces, but they do not have the same boundary as Enneper's surface. You can try to find a surface with the same boundary as Enneper's surface by manipulating the equations for Enneper's surface, but it may require some advanced mathematical techniques.

I hope this helps. Keep up the good work in exploring minimal surfaces!