Pleijel's theorem for a class of degenerate elliptic operators

Abstract

We prove an asymptotic upper bound on the number of nodal domains of eigenfunctions of a class of degenerate elliptic operators. Our proof yields the same constant as in Pleijel's bound for the Dirichlet Laplacian. The operators considered include the Baouendi-Grushin operator and operators with ellipticity degenerating on the boundary.

34 pages; v2: added assumption n\geq 2 in Theorem 2.1