to be able to define an open set $\Omega$ whose boundary is a CR submanifold of $\n\times\R^n$ of same CR dimension containing near the origin $\n$.

### Construction of the open set $\Omega$

In $\R^{m+n}$ let $\omega$ be an open set contained in $\R^{m+n} \cap \{t_1>0\}$ and whose boundary (see figure) is the union of two submanifolds $\gamma_1$ and $\gamma_2$, verifying the following properties.

[**(1)**]{} $\gamma_1$ is given by $t_1=0$ for $(s,t)\in B_{2\epsilon}$.

[**(2)**]{} $\gamma_2 \subset B_{3\epsilon}\setminus B_{2\epsilon}$ is such that

\(i) $\gamma_1 \cup \gamma_2$ bounds an open set $\omega$,

\(ii) $\gamma_1 \cup \gamma_2$ is smooth.

Define $\Omega$ by $$\Omega=\{(z,  w(z,s,t): z\in \B_{3\epsilon}, (s,t) \in \omega\}.$$ The boundary of $\Omega$ on which we shall impose the Dirichlet data is defined to be $$\partial \Omega=\{(z ,  w(z,s,t):z\in \B_{3\epsilon} ,(s,t) \in  \gamma_1 \cup \gamma_2
\}.$$

\[l4\] $\partial \Omega$ is a smooth CR submanifold of $\n \times \R^n $ of same CR dimension.

[**Proof of Lemma \[l4\].**]{} The lemma is clear for the part of $\gamma_1$ which contains $\n
\times \R^{n-1}$. Out of $\n
\times \R^{n-1}$ by lemma\[l2\] the generators of the CR vector fields $L_j$ are equal to ${{\partial} \over {\partial \overline z_j}}$, hence the lemma. $\blacksquare$

We have the following picture.

