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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 $\gamma$.

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  • $\begingroup$ It would be helpful if you provided more background and specified what you are having a hard time with. For example, if $\gamma$ were of class $C^2$, would you know the answer? $\endgroup$ – Amitai Yuval Sep 20 '18 at 18:00
  • $\begingroup$ @AmitaiYuval Thank you for your answer! Yes, I think $C^2$ would be enough, since then we could use unicity of the solution of the geodesic differential equation for given initial data, but I have no idea how to prove that $\gamma \in C^2$. $\endgroup$ – tigro Sep 20 '18 at 20:47

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