Eigenvalue Problems

Abstract

For quite general second order elliptic operators onemay use themaximumprinciple and the Kreĭn-Rutman theorem to show that the eigenfunctioncorresponding to the first eigenvalue has a fixed sign. It is then a natural question to ask if a similar result holds for higher order Dirichlet problems where a general maximum principle is not available. A partial answer is that a Kreĭn-Rutman type argument can still be used whenever the boundary value problem is positivity preserving.We will also explain in detail an alternative dual cone approach. Both these methods have their own advantages.The Kreĭn-Rutman approach shows under fairly weak assumptions that there exists a real eigenvalue and, somehow as a byproduct, one finds that the eigenvalue and the corresponding eigenfunction are positive. It applies in particular to non-selfadjoint settings. The dual cone decomposition only applies inaselfadjoint framework in a Hilbert space, where the existence of eigenfunctions is well-known. But in this setting it provides a very simple proof for positivity and simplicity of the first eigenfunction. A further quality of this method is that it applies also to some nonlinear situations as we shall see in Chapter 7.