Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange equation.
The problem occurs, when we are not sure enough in which class of the functions we are going to look for the solution. The typical example is the Dido's problem: we are looking for a function $f: [-1, 1] \longrightarrow \mathbb{R}_{+}$, graph of which encloses the maximum area between $x$ axis and itself under a fixed length. The right answer is $y=\sqrt{1-x^{2}} \in C[-1, 1]$ But we could not recieve this particular result, if we were considering the problem, for example, in class $C^{1}[-1, 1]$ of all continously differentiable functions.
The question is: 1) Does the Euler-Lagrange equation requires twice differentiability from the solutions? 2) How to prove that the Dido's problem has a solution in $C[-1, 1]$ using methods of the calculus of variations or functional analysis?
I was told that it would be reasonable to consider the closure of $C^{1}[-1, 1]$, which should be compact in $C[-1, 1]$, but there are some troubles with proving this and it's not clear enough, which result this fact may lead to.
Any help would be much appreciated.