I'm trying to understand a paper about minimal surfaces and I came across the following problem: let $N$ be a smooth n-dimensional manifold embedded in an Euclidean space. Let $\gamma$ be a $C^1$ map from $S^1$ to $N$ which is a critical point of the Dirichlet energy $\int_{S^1}\mid\nabla \gamma\mid^2dx$. Does this imply that $\gamma$ is smooth?
Edit: actually my problem was a bit more general: the map $\gamma$ was obtained as the limit in $C^0$ of a minimizing sequence in $C^\infty$ for the Dirichlet energy. I could show that $\gamma \in W^{1,2}$, and since $S^1$ is 1-dimensional, $\gamma$ is a.e. differentiable (but the derivative has not to be continuous). Now I would like to show that $\gamma$ satisfy the geodesic equation $\nabla_\gamma \dot{\gamma}=0$, but for this I need to make sense of the second derivative of $\gamma$.