A Nonlinear Eigenvalue Problem

My lectures at the Minicorsi di Analisi Matematica at Padova in June 2000 are written up in these notes1. They are an updated and extended version of my lectures [37] at Jyväskylä in October 1994. In particular, an account of the exciting recent development of the asymptotic case is included, which is called the ∞-eigenvalue problem. I wish to thank the University of Padova for financial support. I am especially grateful to Massimo Lanza de Cristoforis for his kind assistance. I thank Harald Hanche-Olsen for his kind help with final adjustments of the typesetting. These lectures are about a nonlinear eigenvalue problem that has a serious claim to be the right generalization of the linear case. By now I have lectured on four continents about this theme and my reason for sticking to this seemingly very peculiar problem is twofold. First, one can study the interesting questions without any previous knowledge of spectral theory. Second, to the best of my knowledge there are many open problems easy to state. The higher eigenvalues are “mysterious”. The leading example of a linear eigenvalue problem is to find all nontrivial solutions of the equation ∆u+λu = 0 with boundary values zero in a given bounded domain in R. This is the Dirichlet boundary value problem. (In the Neumann boundary value problem the normal derivative is zero at the boundary.) Needless to say, this has been generalized in numerous ways: to Riemann surfaces and manifolds, to equations ∆u + λu + V u = 0 with a potential V , to more general differential operators than the Laplacian, and so on.