Upper bounds on eigenvalue multiplicities for spheres and plane domains revisited

We revisit two papers which appeared in 1999: M.~Hoffmann-Ostenhof, T.~Hoffmann-Ostenhof, and N.~Nadirashvili [Ann. Global Anal. Geom. 17 (1999) 43--48] and T.~Hoff\-mann-Ostenhof, P.~Michor, and N.~Nadirashvili [Geom. Funct. Anal. 9 (1999) 1169--1188]. The main result of these papers is that the multiplicity of the $k$th eigenvalue of the Riemannian surface $M$ is bounded from above by $(2k-3)$ provided that $k \ge 3$. In the first paper, $M$ is homeomorphic to a sphere. In the second, $M$ is a plane domain with Dirichlet boundary condition. In both cases, the starting label of eigenvalues is $1$. The proofs given in these papers are not very detailed. The purpose of this monograph is to provide detailed general proofs for the above upper bounds and to extend the results to Robin boundary conditions. We provide a survey of previous results (Chap.~1), as well as proofs of prerequisite theorems (Chap.~2). When $M$ is homeomorphic to a sphere, we provide a complete proof of the upper bound, $\mathrm{mult}(\lambda_k) \le (2k-3)$ for any $k\ge 3$, by introducing and carefully studying the combinatorial type and a labeling of the nodal domains of some eigenfunctions (Chap.~3). When $M$ is a plane domain, we consider the three boundary conditions, Dirichlet, Neumann, Robin, and we also study the combinatorial types and a labeling of the nodal domains. More precisely, we prove the inequality $\mathrm{mult}(\lambda_k) \le (2k-2)$ for general $C^{\infty}$ bounded domains and all $k \ge 3$ (Chap.~4). We prove the inequality $\mathrm{mult}(\lambda_k) \le (2k-3)$ for $k \ge 3$ under the additional assumption that the domain is simply connected (Chap.~5). These chapters rely on Euler's inequality applied to the nodal graph and a careful analysis of eigenfunctions which optimize Euler's inequality. Chap.~6 contains related results (nodal line conjecture; Courant-sharp eigenvalues).