I am using the book "Geometric Measure Theory- An introduction" by Fanghua and Xiaoping. I'm studying the proof of the following Lemma (Lemma 2.1.8 page 38).
This chapter is dealing with the theory of partial regularity of energy minimizing harmonic maps into spheres.
Let $u\in H^1(B_1;S^k)$, where $$B_1:= \{x\in\mathbb{R}^n: \lvert x\rvert<1\}\\ S^k:=\{x\in \mathbb{R}^{k+1}: \lvert x\rvert=1\}. $$ Suppose $u$ is a minimizer for the Dirichlet energy functional $$ I[u]:=\int_{B_1}\lvert Dw\rvert^2\mathrm{d}x $$
Lemma There is a positive number $\varepsilon(n,k)$ such that if $\int_{B_1}|Du|^2\mathrm{d}x\le\varepsilon\le\varepsilon(n,k)$, then there exists $\theta(n,k)\in (0,\frac{1}{4})$ such that $$ \theta^{2-n}\int_{B_{\theta}}\lvert Du\rvert^2\mathrm{d}x\le \frac{1}{2}\int_{B_1}\lvert Du\rvert^2 \mathrm{d}x\;\;\text{for all}\;\, \theta\in(0,\theta(n,k))$$. Proof
By contradiction, suppose there would exist a sequence of energy minimizing maps $u_i$ such that $\int_{B_1}\lvert Du_i\rvert ^2\mathrm{d}x\le\varepsilon_i ^2\to 0$ and $$ \theta^{2-n}\int_{B_{\theta}}\lvert Du_i\rvert^2\mathrm{d}x> \frac{1}{2}\int_{B_1}\lvert Du_i \rvert^2 \mathrm{d}x. $$ Let us define $v_i:=\frac{u_i-\bar{u_i}}{\varepsilon_i}$. Obeserve that $\|v_i\|_{H^1(B_1)}\le 1$, $v_i\rightharpoonup v$ in $H^1$ and $v_i\to v$ in $L^2$ up to a subsequence, where $v\in H^1(B_1).$ Moreover we have that $$ \int_{B_1}\lvert Dv\rvert^2\mathrm{d}x\le 1,\;\;\,\int_{B_1}v\mathrm{d}x=0. $$ Note that $u_i$ are harmonic maps satisfying $$\Delta u_i+u_i\lvert Du_i\rvert^2 =0 $$ hence $$ \Delta v_i+\varepsilon_iu_i\lvert Dv_i\rvert^2 =0. $$ By taking the limit we obtain $\Delta v=0$. Hence $$ \theta^{2-n}\int_{B_{\theta}}\lvert Dv \rvert^2\mathrm{d}x\le c\theta ^2\le \frac{1}{4}.\tag 1 $$ Furthermore $$ \frac{1}{\lvert B_{\theta}\rvert}\int_{B_\theta}\lvert v\rvert ^2\mathrm{d}x\le c\theta^2\\ \lvert\frac{1}{\lvert B_{\theta}\rvert}\int_{B_\theta}(\lvert v_i\rvert^2-\lvert v\rvert^2)\mathrm{d}x \rvert\le C\theta^{2}. $$ This will lead to a contradiction. Indeed, let's recall the following result.
Lemma If u is an energy minimizing map, then $$ 2^{n-2}\int_{B_{1/2}}\lvert Du \rvert^2\mathrm{d}x\le\lambda\int_{B_1}\lvert Du\rvert ^2\mathrm{d}x+\frac{c(n,k)}{\lambda}\int_{B_1}\lvert u-\bar{u}\rvert ^2\mathrm{d}x. $$
Thus, we obtain $$ \theta^{2-n}\int_{B_\theta}\lvert Dv_i\rvert ^2\mathrm{d}x\le \lambda \int_{2\theta}\lvert Dv_i \rvert^2\mathrm{d}x+\frac{c}{\lambda}\frac{1}{\lvert B_{2\theta}\rvert}\int_{B_{2\theta}}\lvert v_i \rvert^2\mathrm{d}x\le\\ \le \lambda^k\int_{B_{2^k\theta}}\lvert Dv_i \rvert^2\mathrm{d}x+\frac{c}{\lambda}\sum_{j=1}^k\frac{1}{\lvert B_{2^j\theta} \rvert}\int_{B_{2^j\theta}}\lvert v_i \rvert^2\mathrm{d}x. $$ Choosing $\lambda,\theta$ properly, we infer that $$ \theta^{2-n}\int_{B_\theta}\lvert Dv_i\rvert^2\mathrm{d}x\le\frac{1}{4} $$
There are some passages that I can't understand. The first one is when they infer $(1)$. Then, when they say "Furthermore...". Any help?