# Partial regularity of harmonic maps into spheres

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.

• Schauder estimates and bootstrapping – sharpend Jul 5 at 15:35
• @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. – user136537 Jul 5 at 15:49