Sobolev spaces - some kind of extension and regularity theorem

Hello,

I need some help to proof the following:

Let N=2 or 3, $\displaystyle B_r\subset\mathbb{R}^N$ be the ball with raduis $\displaystyle r$ centered at the origin. We say $\displaystyle u\in H^{1/2}(\partial\Omega)$ if u is the restriction of some function $\displaystyle \phi\in H^1(\Omega)$ to $\displaystyle \partial\Omega$.

What I like to proof is:

Let $\displaystyle u\in H^{1/2}(\partial B_2)$. Then there is a function $\displaystyle v\in H^2(B_2)$ with

- $\displaystyle v=0$ on $\displaystyle \partial B_2$, $\displaystyle \frac{\partial v}{\partial n} = u$ on $\displaystyle \partial B_2$
- $\displaystyle \|v\|_{H^2(B_2)} \leq C \|u\|_{H^{1/2}(\partial B_2)}$
- v vanishes inside $\displaystyle B_1$

Any ideas?