TITLE: What's wrong with the Courant nodal domain theorem?
QUESTION [9 upvotes]: 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 $C^{\infty}(\mathbb R^n)$.

REPLY [8 votes]: 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  which precisely uses this type of div theroem  (formula (8) in the paper).