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.

  • $\begingroup$ Warning: I don't know the answer to your question at all (indeed I don't even really understand it). Is Brezis good? I've heard good things about it, and I was going to start studying PDEs and Sobolev spaces soon. $\endgroup$ – Chill2Macht May 5 '16 at 2:54
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    $\begingroup$ I consider it a great book for starting studying Sobolev Spaces. It does it in a confortable way, starting with one dimensional Sobolev Spaces (Over an interval $I \subset \mathbb{R}$) followed by the (more complex) n-dimensional Sobolev Spaces. It also provides some examples on how to apply Sobolev Spaces (See Variational Method) for solving PDEs and ODEs. Depending on your level on Functional Analysis, you can also check the first part of the book, that deals with weak topologies, $L^p$ spaces, Hilbert Spaces (With the Lax-Milgram Theorem, the basis of the Variational Method). $\endgroup$ – D1X May 5 '16 at 9:23
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    $\begingroup$ It also provides basic Functional Analysis results like the Open Mapping Theorem, Hahn-Banach and its geometric forms, and results of convex functions... If you are already acquainted with Funtional Analysis, I would skip the first part of the book and read on the go the results you need to understand chapters 8 and 9 (Which deal with Sobolev Spaces). In my opinion Brezis provides a great Functional Analysis basis which doesn't really apply (Not the most part). Note this does not mean that basis is useless for studying PDEs. $\endgroup$ – D1X May 5 '16 at 9:29
  • $\begingroup$ That makes sense thanks. Yeah I am pretty solid on most of basic functional analysis, although not as much with locally convex spaces, semi-norms, and Lax-Milgram. I'm definitely get a copy of it to read! $\endgroup$ – Chill2Macht May 5 '16 at 10:13

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