From PDE Evans, 2nd edition: Chapter 8, Exercise 12:
Assume $u$ is a smooth minimizer of the area integral $$I[w]=\int_U (1+|Dw|^2)^{1/2} \, dx,$$ subject to given boundary conditions $w=g$ on $\partial U$ and the constraint $$J[w] = \int_U w \, dx = 1.$$ Prove that graph of $u$ is a surface of constant mean curvature.
(Hint: Recall Example 4 in ยง8.1.2.)
First, I will print Example 4 from the textbook (page 457) below:
Example 4 (Minimal surfaces). Let $$L(p,z,x)=(1+|p|^2)^{1/2},$$ so that $$I[w] = \int_U (1+|Dw|^2)^{1/2} \, dx$$ is the area of the graph of the function $w : U \to \mathbb{R}$. The associated Euler-Lagrange equation is $$\sum_{i=1}^n \left(\frac{u_{x_i}}{(1+|Du|^2)^{1/2}} \right)_{x_i}=0 \quad \text{in }U.$$
$\quad$ This partial differential equation is the minimal surface equation. The expression $\operatorname{div} \left(\frac{Du}{(1+|Du|^2)^{1/2}} \right)$ on the left side of $(10)$ is $n$ times the mean curvature of the graph of $u$. Thus a minimal surface has zero mean curvature.
I don't have much work started on this, but this question looks interesting. (This is not a homework assignment, as with all my other PDE Evans questions.) But I am asking here because I do not understand fundamentally how mean curvature is applied to Euler-Lagrange equations. In particular, this question uses concepts of differential geometry, which I have not taken any courses in yet in my academic career.
Now, what I do know so far, is that maybe I should show that the graph of $u$ is a minimal surface. This would mean $u$ has zero mean curvature, and hence a constant mean curvature (zero is constant, obviously).
How should I start about this problem?
Best Answer
you don't need to know any differential geometry to confirm the PDE.
The minimal case is this: replace $w$ by $w + t \phi.$ Here $\phi$ refers to a $C^\infty$ function with compact support, the support (closure of the set where $\phi$ is nonzero) being contained in the interior of $\Omega.$ In order to have $I(w)$ a minimizer (or any critical point) it is necessary that $$ \frac{d}{dt} \; I(w+t \phi) $$ be zero when $t=0.$ You write this out carefully. The main lemma you need is this: if a continuous function $F,$ which will be some combination of $w$ and first and second partials of $w,$ satisfies $\int_\Omega F\phi =0$ for all such $\phi,$ then $F$ is constant zero.
For the constant mean curvature case, replace the $\phi$ by functions $\psi$ with compact support in the interior of $\Omega,$ with the additional constraint that $\int_\Omega \psi =0.$ Vary with $w + t \psi.$ This time, the main lemma is that, if continuous $F$ satisfies $\int_\Omega F\psi =0$ for all such $\phi,$ then $F$ is constant, but the constant is allowed to be nonzero if that is how it works out.
The calculations in the above descriptions are simply not difficult; I recommend getting your hands dirty. It will help when later you take differential geometry. See if you can find proofs of the two lemmas. I cannot tell what you know about test functions, so it may be a matter of looking things up, or asking your professor or TA.