Consider the problem of minimizing $$ I(u) = \int_a^b F(t,u(t),u'(t)) d t $$ over, say, $W^{1,\infty}(]a,b[)$. Then regularity theory tells us that if $F$ and $F_{\dot q}$ are $C^k$, and in addition $F_{\dot q \dot q} > 0$ in a neighborhood of the minimizing curve, then the minimizer is in fact of class $C^{k+1}$. This can be found in any good book on variational calculus.
If we consider the minimization problem under the additional constraint $$ J(u) = \int_a^b G(t,u(t),u'(t)) d t = \ell, $$ then (assuming that $J$ is differentiable $J'(u)$ is surjective) there exists a Lagrangian multiplier $\lambda$ so that the minimizer $u$ satisfies $\delta I(u) = \lambda J'(u)$. If we now want to apply the regularity theorem to this problem, we have to inspect $$ H = \frac{d^2}{d \dot q^2} \bigg( F(t,q,\dot q) - \lambda G(t,q,\dot q) \bigg). $$ In some cases, the Lagrange multiplier seems to give problems.
For example, consider $F = q \sqrt{ 1 + \dot q^2 }$ and $G = \sqrt{ 1 + \dot q^2 }$. The solution has to be a catenary, but how do we see this without a priori assuming smoothness (once we know that a minimizer is $C^2$, I know how to deal with it)? In this case, the expression $H$ becomes $H = ( q - \lambda ) ( 1 + \dot q^2 )^{-3/2}$. For the regularity theory to kick in, this quantity should be nonzero for all $q$ in a neighborhood of the range of $u$. So $u([a,b])$ has to be bounded away from $\lambda$.
So it seems we would need additional information about $\lambda$, but how to get this? The examples I've seen so far all assume sufficient regularity of minimizers hold from the start... Or maybe there's another way how to tackle this problem?