Pleijel nodal domain theorem in non-smooth setting

TL;DR

This paper extends the Pleijel nodal domain theorem to non-smooth spaces, establishing bounds on eigenfunction nodal domains and confirming Courant's theorem holds except for finitely many eigenvalues, even with low boundary regularity.

Contribution

It proves the Pleijel theorem in non-smooth RCD spaces and generalizes the result to Neumann and Dirichlet eigenfunctions on various domains, including low-regularity Euclidean spaces.

Findings

Asymptotic upper bound on nodal domains in non-smooth spaces

Courant's nodal domain theorem holds except finitely often

Extension of Pleijel theorem to low-regularity Euclidean domains

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.