I need help to understand the proof of the Trace Theorem given in Evans L.C. Partial differential equations (AMS, 1997):
Asume $U$ is a bounded open set and that $\partial U$ is $C^1$. Then there exists a bounded linear operator $T:W^{1,p}(U) \to L^p(\partial U)$ such that:
$\quad$(i) $Tu=u|_{\partial U}$ if $u \in W^{1,p}(U) \cap C(\bar{U})$
and
$\quad$(ii) $\|Tu\|_{L^p(\partial U)} \le C \| U \|_{W^{1,p} (U)},$ for each $u \in W^{1,p}(U)$, with the constant $C$ depending only on $p$ and $U$.
The proof provided starts like this:
Okay, so we first asume that there is a neighborhood $U_{x^0}$ of $x^0$ such that $U_{x^0} \subset \{x \in \mathbb{R}^n \ | \ x_n=0\}$ and prove (1).
Then the proof continues by saying "If $\partial U$ is not flat near $x^0$", we straighten out the boundary near $x^0$ to obtain the setting above. Applying estimate (1) and changing variables... (The proof continues)
Could someone clarify me a little more precisely what straighten out the boundary near $x^0$ means?
Updated: This is the definition of open set of class $C^1 $ which gives us a diffeomorphism to the set $Q_0$ which is "flat".
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:
- $H \in C^1(\overline{Q})$
- $H^{-1} \in C^1(\overline{U_x})$
- $H(Q_+)=U_x \cap Q$
- $H(Q_0)= U_x \cap \partial \Omega$