Weak Sobolev convergence in Mobius group of disk to a constant.

Question

Let $\Gamma \subset \mathbb{R}^3$ be a Jordan curve, $D^2 \subset \mathbb{R}^2$ be the unit disk and define:

$$\mathcal{F} = \{u \in W^{1,2}(D^2, \mathbb{R}^3) \ \vert \ u \vert_{\partial D} \ \text{weakly monotone parametrization of } \Gamma \}$$.

Let $G $ be the Mobius group of the disc. Let $u \in \mathcal{F}$. Prove that there exists a sequence $f_k \in G$ such that $u \circ f_k$ converges weakly in $W^{1,2}(D^2,\mathbb{R}^3)$ to a constant.

I have a couple questions about the problem.

1. What is $W^{1,2}(D^2, \mathbb{R}^3)$? More specifically, what is the "weak derivative" of a map $u:D^2 \to \mathbb{R}^3$?

2. Is "weak convergence" in this context meaning:

$$\lim_{k \to \infty} \langle u \circ f_k , g \rangle_{W^{1,2}(D^2, \mathbb{R}^3)}= \langle k , g \rangle_{W^{1,2}(D^2, \mathbb{R}^3)} $$

for some constant map $k\in W^{1,2}(D^2, \mathbb{R}^3)$? Is there an equivalent / more practical definition?

3. What ought the $f_k$ be? I have considered something of the form:

$$f_k (z) = \frac{(1-\frac{1}{k})+z}{1+(1-\frac{1}{k})z} $$

satisfying $f_k \in G$ and $\lim_{k \to \infty} f_k(z) = 1$ a constant, but I don't think this particular sequence will work for any arbitrary $u$.