Maximal positive boundary value problems as limits of singular perturbation problems

Abstract

We study three types of singular perturbations of a symmetric positive system of partial differential equations on a domain Ω ⊂ R n \Omega \subset {{\mathbf {R}}^n} . In all cases the limiting behavior is given by the solution of a maximal positive boundary value problem in the sense of Friedrichs. The perturbation is either a second order elliptic term or a term large on the complement of Ω \Omega . The first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in Ω \Omega and outside Ω \Omega . The results show that in the limit of zero viscosity or infinitely large difference the behavior is described by a maximal positive boundary value problem in Ω \Omega . The boundary condition is determined in a simple way from the system and the singular terms.