Sobolev spaces ring domain

Question

Let $0<r_i<r_e$, the ring domain $\Omega=\{(r\cos(\theta),r\sin(\theta)), r_i<r<r_e;\ 0<\theta<2\pi \}$ and the subespace of $2\pi$-periodic functions in the second variable $\mathcal{C}_{per}^\infty(\overline{\Omega})=\{f\in\mathcal{C}^\infty(\overline{\Omega});f(r,0)=f(r,2\pi),\ r_i\leq r\leq r_e\}$. Consider the norm in polar coordinates $$\Vert f\Vert_\mathcal{V}^2=\int_\Omega \left(f^2+f_r^2+\frac{1}{r^2}f_\theta\right)rdrd\theta,\qquad\forall f\in \mathcal{C}_{\operatorname{ per} }^\infty(\overline{\Omega})$$ The space $$\overline{\mathcal{C}_{\operatorname{per}} ^\infty(\overline{\Omega})}^{\Vert\cdot\Vert_\mathcal{V}}$$ have some caracterization? If you have some references it'd help me a lot.

Answer 1

(I assume that your $f$ is also given in polar coordinates).

Your norm is equivalent to the norm $$\|f\|^2 = \int_\Omega \Big( f^2 + (r f_r)^2 + (\frac1r f_\theta)^2 \Big) \, r \, \mathrm{d}r$$ and this norm equals the usual $H^1$ norm. Hence, your space should just be the usual $H^1$.