I am reading the proof of Theorem 9.17 in the book Functional Analysis, Sobolev Spaces and Partial Differential Equations, by Haim Brezis. The theorem says:
Suppose that $\Omega$ is of class $C^1$ and $u \in W^{1,p}(\Omega) \bigcap C(\bar{\Omega})$. Then $u \in W^{1,p}_0(\Omega)$ if and only if $u=0 $ on $\partial \Omega $.
I have already seen the proof provided in other books that uses the Trace Operator. It is not the approach given here. I have trouble understanding the implication $\Rightarrow$. The proof starts saying:
Using local charts this is reduced to the following problem: Let $u \in W^{1,p}_0(Q_+) \bigcap C(\bar{Q}_+)$. Prove that $u=0$ on $Q_0$.
Can someone give me some hint on how to understand this statement? This is the definition of a $C^1$ open set:
We define the following sets:
- $R_+ = \{x=(x_1,...,x_n) \in \mathbb{R}^n \ | \ x_n \geq 0\} $
- $ Q = \{x=(x_1,...,x_n) \in \mathbb{R}^n \ | \ (\sum_{i=1}^{n-1} x_i^2)^{1/2} < 1 \ y \ |x_n|<1 \}$
- $ Q_+=R_+ \cap Q $
- $ Q_0=\{(x_1,...,x_{n-1},0) \in \mathbb{R}^n \ | \ (\sum_{i=1}^{n-1} x_i^2)^{1/2} < 1 \}$
An open set $\Omega$ is of class $C^1$ if for every $x \in \partial > \Omega$ there exists a neighborhood $U_x$ of $x$ in $\mathbb{R}^n$ and a bijective map $H: Q \to U_x$ such that:
- $H \in C^1(\overline{Q})$
- $H^{-1} \in C^1(\overline{U_x})$
- $H(Q_+)=U_x \cap Q$
- $H(Q_0)= U_x \cap \partial \Omega$
I don't know how to tackle this. I somehow see I can, by using $H$, send $\partial \Omega$ to $Q_0$. But anyway I don't know how I can transform the problem into the one stated.