The Courant nodal domain theorem (for Neumann boundary conditions) says that the $n$-th eigenfunction has at most $n$ nodal domains (connected components where the eigenfunction has the same sign. However, Chavel in *Eigenvalues in Riemannian Geometry* seems to point out (P23) that Divergence theorem is used, so the regularity of the nodal set matters, whose proof by Cheng in dimension $\ge 3$ is incomplete. My question is: does it really affect the proof of the nodal domain theorem? More generally, does it affect the application of divergence theorem to a nodal domain (say to functions that are restrictions of $C^{\infty}(\mathbb R^n)$.



What's wrong with the Courant nodal domain theorem?
Not an precise answer : 
As far as I remember this is not really an issue. One has to replace by an other version of divergence theorem ( for less regular domain) using geometric measure theory . The key point I think is that the set where the nodal lines are not regular (often called the singular set $\{u=0\}\cap\{\nabla u=0\}$ is of dimension at most $n-2$ so it has measure 0 for the $H^{n-1}$ measure. 
But this is quite specific to solutions of elliptic pde and I don't think that smoothness is enough to guarantee any kind of regularity of the nodal set.


A more precise answer :
Take a look (for example) at the proof of the nice [Sogge-Zelditch formula](http://arxiv.org/abs/1009.3573) which precisely uses this type of div theroem (formula (8) in the paper). 


