A while ago I attempted to solve the classical isoperimetric problem in 3-dimensions, namely "Find the surface that has the smallest surface area for a given volume".
At that time for me to write down the full euler-lagrange equations for such a problem was too tough, so I made the assumption that the solution to this problem was a surface of revolution about an axis.
The functional in question then becomes only a function of $y$ (from the calculus version of Pappus's Centroid Theorem).
Questions:
1) How is such an assumption justified? I remember reading through some results of Antonio Ros, Manuel Ritoré, Fred Almgren, Michael Hutchings et. al on the double bubble conjecture. I didn't really understand them, and I don't even remember the paper I looked at that had at least a useful result within my reach.
2) Upon substituting such a functional into the euler-lagrange equations (for the case of a functional dependent only on $y$ and $y'$), one gets the differential equation \frac{2}{\sqrt{1+y'^2}} + \lambda y = C, where $C$ is a constant and $\lambda$ the lagrange multiplier.
Now if $C = 0$ the equation of a circle is a solution, and one gets $C= 0$ by appplying the boundary conditions $y(a) = y(b)=0$, namely that the endpoints of such a curve (well curve because we are talking of a surface of revolution) lie on the $x-axis$.
What happens if $C$ is not zero? Apparently this would give rise to a different surface (as Delaunay) studied. Are there several solutions to the given differential equation that satisfy the problem?
Thanks.