# Sobolev spaces - some kind of extension and regularity theorem

• Dec 8th 2010, 04:24 PM
brandon86
Sobolev spaces - some kind of extension and regularity theorem
Hello,

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?