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Let $u : B_1(0) \subset \mathbb{R}^n \to S^k$ be an energy minimizing (minimizing in $H^1(B_1(0); S^k)$) harmonic map. I am trying to understand the theory of partial regularity, where the main claim seems to be the following:

Theorem: Any energy minimizing harmonic map is smooth away from a closed subset $\Gamma \subset \mathbb{R}^n$ with $\mathcal{H}^{n - 2}(\Gamma) = 0$.

To my understanding the main steps leading to this conclusion are the following:

  1. Establishing that away from this set $\Gamma$, $u$ is $C^\alpha$ for some $\alpha > 0$. This requires a Caccioppoli-type inequality for $u$, which basically implies that $\int_{B_r} |Du|^2$ controls from above $\theta^{2 - n - \alpha}\int_{B_\theta} |Du|^2$, where $\theta < r/2$, and then an application of a test of Morrey for $C^\alpha$, which can be found in Ladyzhenskaya-Uraltseva, Section 2.4.

  2. The other component in the proof is the geometric measure theory component that basically says that $\limsup_{s \to 0+}s^{2- n}\int_{B_s} |Du|^2$ is zero except on a small set $S$, such that $\mathcal{H}^{n - 2}(S) = 0$ (this is true for all $|Du|^2 \in L^1_{loc}$).

My main question is, how does $C^\alpha$ automatically give $C^\infty$? The two/three sources that I have looked up all say that this is "folklore"/"standard", but give no reference or outline.

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    $\begingroup$ Schauder estimates and bootstrapping $\endgroup$ – sharpend Jul 5 at 15:35
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    $\begingroup$ @sharpend Thanks a lot, I just wrote down the variational PDE satisfied by $u$. So this was very simple! I somehow assumed that it must be complicated since nobody was mentioning the reasons. $\endgroup$ – user136537 Jul 5 at 15:49

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