# Completion of $C_{0,rad}^{\infty}(\Omega)$ with respect to the norm $\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}.$

I have a question that it seems simple but I can not solve it.

Let $$\Omega$$ be the unit ball centered at zero in $$\mathbb{R}^N$$, $$N>4$$. Assume that $$C_{0,rad}^{\infty}(\Omega)$$ is the space of all radial symmetric smooth functions with compact support.

What is the completion of $$C_{0,rad}^{\infty}(\Omega)$$ with respect to the following norm

$$\|u\|= \Bigg(\int_{\Omega} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}.$$

Is it $$W_0^{2,2}(\Omega)$$ or $$W_{0,rad}^{2,2}(\Omega)$$.

I think that the answer is $$W_0^{2,2}(\Omega)$$ but I have no idea to prove it.

I will be thanked to any useful hint or answer.

• I don't see how it can possibly be $W^{2,2}_0(\Omega)$. For instance, the completion certainly can't contain any function whose Laplacian isn't radially symmetric. – Nate Eldredge Jan 11 '16 at 3:56
• Maybe you already know that, but by integration by parts your norm is equal to $\left(\int_\Omega|D^2u|^2dx\right)^{1/2}$ since $\langle \nabla u,\nabla\Delta u\rangle=-|D^2u|^2+\tfrac{1}{2}\Delta|\nabla u|^2$ where $|D^2u|^2$ is the Frobenius norm of the Hessian. – Thomas Richard Jan 11 '16 at 10:30
• Thomas's remark, along with the definitions of the two spaces you're trying to decide between, immediately gives the answer. This is more appropriate for math.stackexchange.com – Deane Yang Jan 11 '16 at 13:51