I'm trying to prove that the space region that contains maximum volume for a given fixed surface area is a sphere. I've done this for the analogous 2d isoperimetric problem using calculus of variations.
Here's what I have:
Let our region of space be V and the enclosing surface . We want to minimize the volume
subject to the constraint .
We parametrize with parameters u and v. Then , where and .
Let us define . Then , and .
Thus, by the divergence theorem,
where is the outward unit normal vector. For our parametrization of , we have and thus
, and thus we want to find the extremum of:
And we put these into the formula for M to get a double integral over dudv of a function .
Were there only one independent variable, I could find the Euler-Lagrange equations for x, y and z from L; or more specifically, I'd use the lack of explicit dependence on the independent variable to apply the Beltrami Identity to find lower order differential equations for x, y, z. However, this has two independent variables. How do I set up and solve the Euler-Lagrange equations for this sort of problem? Is there a Beltrami Identity equivalent for multiple independent variables when the integrand has no explicit dependence on both of the independent variables?