On Cooperative Elliptic Systems: Principal Eigenvalue and Characterization of the Maximum Principle
نویسنده
چکیده
The purpose of this set of notes is to present the connection between the classical maximum principle with the principal eigenvalue of the elliptic operator. We will start with the maximum principle for single equations and proceed to the case of cooperative (or weakly-coupled) systems. By adopting an idea due to G. Sweers, we give a characterization of the principal eigenvalue for a cooperative system in terms of the spectral radius of a related positive compact operator, which leads to an eigenvalue comparison criterion. As an application, we present a recent result concerning the vanishing viscosity limit of the principal eigenvalue. For simplicity we only treat the Dirichlet case in this note, and we remark that analogous results in the Neumann and Robin cases follow with minor modifications of the proofs. This is joint work with Y. Lou.
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