Corentin Léna: Pleijel's nodal theorem and its extensions

Abstract: Courant's nodal theorem tells us that an eigenfunction of the Laplacian associated with eigenvalue number k has at most k nodal domains. Å.Pleijel showed in 1956 that for a given planar domain with a Dirichlet boundary condition, equality is reached for a finite number of k's only.

Pleijel's proof actually gives an asymptotic upper bound of the number of nodal domains. It has been extended afterwards to other geometric settings and boundary conditions. In the last decade, several generalizations and refined versions have been obtained, and a number of special cases analyzed. I will describe some of these recent developments and present new results which I have obtained with K. Gittins.