I am stuck at the proof of Theorem 9.25 in Haim Brezis' Sobolev Spaces, Functional Analysis and Partial Differential Equations.
This theorem deals with the regularity for the Dirichlet Problem for the Laplacian. That is, we assume that $u \in H_0^1(\Omega)$ and that verifies: $$\int \nabla u \cdot \nabla \varphi + \int u \varphi = \int f \varphi $$
I have already shown that if $f \in L^2(\Omega)$ then $u \in H^2(\Omega)$ for $\Omega$ of class $C^2$. Now I want to show that if $f \in H^m(\Omega)$ then $u \in H^{m+2}(\Omega)$ for $\Omega$ of class $C^{m+2}$.
The book leaves this to the user and says it can be deduced for any set $\Omega$ of class $C^{m+2}$ as in cases $\Omega= \mathbb{R}^n$ and $\Omega= \mathbb{R}^n_+ =\{x \in \mathbb{R}^n \ | \ x_n>0 \}$.
In these cases this is done by showing that $Du$ verifies the equation: $$\int \nabla (Du) \cdot \nabla \varphi + \int Du \ \varphi = \int (Df) \varphi$$
And then applying the first part for this equation. To do this we have to show that $Du \in H_0^1(\Omega)$. In the case $\Omega= \mathbb{R}^n$ this is obvious since $H^1_0(\mathbb{R}^n)=H^1(\mathbb{R}^n)$.
In the case $\Omega= \mathbb{R}^n_+$ this is done by using the following trick:
He uses tangential translations, that is, $h \in \{x \in \mathbb{R}^n \ | \ x_n=0 \}$, this means $D_hu= \frac{u(x+h)-u(x)}{|h|}$ belongs to $H_0^1(\mathbb{R}^n_+)$, as $ h +\mathbb{R}^n_+ =\mathbb{R}^n_+$.
Here comes the problem. For a general domain, how do I use the trick of translations?
One doesn't use translations for a general domain. One uses a diffeomorphism between a part of the domain, and half-space, and then translates in the half-space. This is what Brezis does later in the text (part C); you quoted some of it in Concerning the proof of regularity of the weak solution for the laplacian problem given in Brezis.