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 $\Omega \subset {{\mathbf {R}}^n}$. 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 $\Omega$. The first corresponds to a sort of viscosity and the second to physical systems with vastly different properties in $\Omega$ and outside $\Omega$. 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 $\Omega$. The boundary condition is determined in a simple way from the system and the singular terms.