I am studying the following article : http://hal.archives-ouvertes.fr/docs/00/12/87/60/PDF/fbpLaplacian.pdf
In this article the authors considers $K \subset \{ x \in R^n ; x_1 = 0 \}$ a smooth, bounded and convex set (when I say smooth I mean $C^\infty$).
The authors define $\mathcal{C} = \{ \Omega \subset R^{n}_+; \Omega \text{ is smooth, bounded, convex, and open such that } K \subset \partial \Omega \}$. Here $\mathbb{R}^n_+ = \{x = (x_1,...,x_n) \in \mathbb{R}^n ; x_1>0\}$.
For $\Omega \in \mathcal{C}$ the problem
$$ \left\{ \begin{array}{cl} \Delta u = 0 &\text{ in } \Omega \text{ in the weak sense} \\ u = 1 &\text{ in } K \text{ in the trace sense} \\ u = 0 &\text{ on } \partial \Omega\setminus K \text{ in the trace sense} \\ \end{array} \right. $$
has a unique solution $u_{\Omega} \in H^{1}(\Omega)$. Now define the classes
$$ \mathcal{B} = \{ \Omega \in \mathcal{C} ; \limsup_{y \rightarrow x} |\nabla u(y)| \leq 1 \quad \forall x \in \partial \Omega - K \}$$
$$\mathcal{A}_0 = \{ \Omega \in \mathcal{C} ; \limsup_{y \rightarrow x} |\nabla u(y)| > 1 \quad \forall x \in \partial \Omega - K \} $$ (the classical gradient exists in $\Omega$ by regularity theory for the p-Laplacian).
In page five of the article the authors prove : if $\Omega_1 , \Omega_2 \in \mathcal{B} $ then $\Omega_1 \cap \Omega_2 \in \mathcal{B}$.
My first question is: Intersection of smooth domains is not smooth. Then when the authors says smooth, does he means piecewise smooth?
After this he states the following theorem: Let $(\Omega_n )$ be a decreasing sequence of convex domains of $\mathcal{B}$ such that
$$ \Omega = \text{int}(\overline{\cap \Omega_n}) \in \mathcal{C}.$$
Then $\Omega \in \mathcal{B}$.
After this he states and proves the following proposition:
Proposition: Assume that there exist two domains $\Omega_0 \in \mathcal{A}_0$ and $\Omega_1 \in \mathcal{B}$. We denote by $\mathcal{S}$ the class of domains $D$ such that $\Omega_0 \subset D$ and $D \in \mathcal{B}$. Then there exists a minimal element in the class $\mathcal{S}$ for the inclusion.
In the proof of this proposition the author constructs a decreasing sequence $(\Omega_n) $ of bounded, convex , open and piecewise smooth domains (they are piecewise smooth because each element of this sequence is finite intersection of smooth domains). After the construction the authors use the previous result for the sequence $(\Omega_n)$. Then they have to use that $\Omega = \text{int} (\overline{\cap \Omega_n})$ is smooth (or piecewise smooth depending of the meaning of smooth - I believe that it means piecewise smooth).
This proof raises the following question (I believe the author uses what I will ask):
Question 2: Is this claim true: Let $(\Omega_n)$ be a sequence of open, bounded, convex piecewise smooth domains such that $K \subset \partial \Omega_n $ ($K$ as in the beginning of this text). Then $\Omega = \text{int}(\overline{\cap \Omega_n})$ is piecewise smooth.
Can someone help me with these questions? Honestly I don't know the answers of my questions. Any commentary will be appreciated. Sorry for the long text and sorry for the English. My English is terrible...