I am reading Theorems on Regularity and Singularity of Energy Minimizing Maps by Leon Simon. The problem arises in Section 2.3, the proof of Shoen-Uhlenbeck theorem:
Let $u:\Omega \to N$ is a minimizing maps, $\Omega$ is an open area of $\mathbb{R}^n$, $N$ is a compact analytic submanifold isometrically embedded in $\mathbb{R}^p$.
We have shown in the former section that if $u$ in minimizing, then $u$ must meet the equation:
$$\Delta u =\sum_{j=1}^n - A_u (D_ju,D_ju)$$
in weak sense and $A_u$ is the second fundamental form of $N$.
Theorem 1 Let $\Lambda > 0$, $\theta \in (0,1)$ . There exists $ \epsilon =\epsilon (n, N, \Lambda, \theta) > 0$ such that if $u \in W^{1,2}(\Omega; N)$ is energy minimizing on $B_R(x_0) \subset \Omega$ and if $$R^{2-n} \int_{B_R(x_0)} \vert Du \vert^2 \le \Lambda $$and
$$R^{-n} \int_{B_R(x_0)} \vert u - \lambda_{xo,R} \vert^2 < \epsilon ^2 $$
then there holds
- $u \in C^{\infty}(B_{R/4} (x_0)) $
- for j = 1, 2, ... we have the estimates $$R^j \sup_{B_{\theta R}} \vert D_ju \vert \le C (R^{-n} \int_{B_R(x_0)} \vert u-\lambda_{x_0,R} \vert ^2)^{1/2} $$ where $C$ depends only on $j, \Lambda, N, \epsilon$ and $n$. $\quad \lambda_{x_0,R} = \int_{B_R(x_0)} u $
The author utilises Lemma 3 in Section 1.7 and Technical Lemma in Section 1.8 to prove the theorem. The conclusions are as follows:
Lemma 3 $ \quad \Delta u = f$ in weak sense, $ u \in W^{1,2}(B_R(x_0))$
If $f $ is bounded, then $ u \in C^{1,\alpha}$ for $\forall \alpha < 1$ with estimate: $$ R^{1+\alpha}[Du]_{\alpha,B_{\beta R}} \le C (\vert u \vert _{0;B_{r}(x_0)} + R^2 \vert f \vert _{0;B_R(x_0)} )$$
If $f \in C^{k,\alpha}$, then $ u \in C^{k+2,\alpha} $ with estimate: $$ \sum_{j=1}^{k+2} R^j \vert D^ju \vert _{0;B_{\beta R}(x_0)} + R^{k+2+\alpha}[D^{k+2}u]_{\alpha;B_{\beta R}(x_0)} \le C (\vert u \vert _{0;B_{r}(x_0)} + R^2 \vert f \vert _{0;B_R(x_0)} + R^{2+k+\alpha} [D^kf]_{\alpha;B_R(x_0)} ) $$
where $\beta < 1$ and $\vert \cdot \vert_{k,\Omega} = \Vert \cdot \Vert _{C^{k}(\Omega)}$ , $C$ depends on $\beta, n,\alpha, k$
Technical LemmaThe Conclusion of Technical lemma in section 1.8 is as follows:
$$\vert u \vert _{1;B_{\rho}(x_0)} + [Du]_{\alpha; B_{\rho}(x_0)} \le C (R^{-n} \int_{B_R(x_0)} \vert u-\lambda_{x_0,R} \vert ^2)^{1/2} $$
where $\rho = R/4$
My question occurs in the last part of proof of $\epsilon$ Regularity theorem, the author writes:
Before this we want to prove the theorem 1 in the case that $\theta = 1/8, R=1$, we have proved that $u$ satisfies technical lemma and by lemma 3, $ u \in C^\infty $ as $A_u(D_ju,D_ju)$ has a good property. But how to inductively show this inequality? I couldn't make it since I can only get $$f =\sum_{j=1}^n A_u(D_ju,D_ju) \le C |Du|^2 $$ for $N$ is compact. And I couldn't achieve to show that $$ [D^kf]_{\alpha} \le C (R^{-n} \int_{B_R(x_0)} \vert u-\lambda_{x_0,R} \vert ^2)^{1/2} $$
Could you please help me with that? The book is here if you need more details: Book