I am trying to understand the proof Courant's Nodal Domain Theorem, but I am stuck at a technical issue: Let us consider the simplest case of the Laplace operator on a bounded domain $\Omega\subset\mathbb{R}^n$ with Dirichlet boundary conditions. Then the claim used in the proof is that an eigenfunction restricted to a nodal domain (and set equal to $0$ outside) is again in $H^1_0(\Omega)$. Everybody seems to consider this as obvious, but I am afraid it is not obvious to me.
I was trying to assume that the domain has a nice boundary such that the eigenfunctions are continuous up to the boundary and vanish at the boundary, that is, $u\in C_0(\overline{\Omega})\cap H^1(\Omega)$. Then, if $\Omega'$ is a nodal domain of $u$, the restriction $u' = u \chi_{\Omega'}$ is in $C_0(\overline{\Omega'})\cap H^1(\Omega')$ and hence one would expect it to be in $H^1_0(\Omega')$ (and thus trivially also in $H^1_0(\Omega)$). However, this seems to require some conditions on the boundary of the nodal domain $\partial\Omega'$ (such results seem available, but that would make the whole proof quite complicated). What am I missing here?
It seems a proof is given in Appendix D (Lemma 3) of: P. Bérard and D. Meyer, Inégalités isopérimétriques et applications, Annales scientifiques de l'École Normale Supérieure, Serie 4, Volume 15 (1982) no. 3, pp. 513-541. [https://doi.org/10.24033/asens.1435]