I'm trying to explicit Sobolev spaces $H^1(D)$, where $D$ is the unit disk, i.e. $$D=\{(x,y)\in \mathbb R^2: x^2+y^2\le 1\}\,.$$ In my definition, $H^1(D)$ is defined to be the closure of $C^\infty(D)$ with respect to the norm $||f||^2_{1,2}=||f||_{L^2(D)}^2+||\nabla f||^2_{\vec{L}^2(D)}$. So I tried to explicit this norm.
Let $f\in C^\infty(D)$. Then there exists $(f_n\colon [0,1]\to \mathbb C)_{n\in \mathbb Z}\subseteq C^\infty([0,1])$ such that $$ f(r\cos\theta,r\sin\theta)=\sum_{n\in \mathbb Z}f_n(r)e^{in\theta}$$ for every $r\in [0,1]$ and $\theta\in [0,2\pi]$.
Starting from this observation, I have evalutated $||\nabla f||_{\vec L^2(D)}$ and I have obtained $$ ||f||_{1,2}=2\pi \sum_{n\in \mathbb Z} \int_0^1 \Bigg[|\sqrt{r}f_n(r)|^2+|\sqrt{r}f'_n(r)|^2+\bigg|\frac{nf_n(r)}{\sqrt{r}}\bigg|^2\Bigg]\,\mathrm{d}r\,. $$
From this, if it is correct, is it possibile to define $H^1(D)$?
Thanks in advance.