Generalizations and Properties of the Principal Eigenvalue of Elliptic Operators in Unbounded Domains

Using three different notions of the generalized principal eigenvalue of linear second-order elliptic operators in unbounded domains, we derive necessary and sufficient conditions for the validity of the maximum principle, as well as for the existence of positive eigenfunctions for the Dirichlet problem. Relations between these principal eigenvalues, their simplicity, and several other properties are further discussed. (c) 2015 Wiley Periodicals, Inc.