Pleijel nodal domain theorem in non-smooth setting

Abstract

We prove the Pleijel theorem in non-collapsed RCD spaces, providing an asymptotic upper bound on the number of nodal domains of Laplacian eigenfunctions. As a consequence, we obtain that the Courant nodal domain theorem holds except at most for a finite number of eigenvalues. More in general, we show that the same result is valid for Neumann (resp. Dirichlet) eigenfunctions on uniform domains (resp. bounded open sets). This is new even in the Euclidean space, where the Pleijel theorem in the Neumann case was open under low boundary-regularity.