I need some help to proof the following:

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

What I like to proof is:

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

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

Any ideas?