The Singular Perturbation Problem for a Class of Generalized Logistic Equations Under Non-classical Mixed Boundary Conditions

Abstract

This paper studies a singular perturbation result for a class of generalized diffusive logistic equa- tions, dLu = uh(u, x), under non-classical mixed boundary conditions, Bu = 0 on ∂Ω. Most of the precursors of this result dealt with Dirichlet boundary conditions and self-adjoint second order elliptic operators. To over- come the new technical difficulties originated by the generality of the new setting, we have characterized the regularity of ∂Ω through the regularity of the associated conormal projections and conormal distances. This seems to be a new result of a huge relevance on its own. It actually complements some classical findings of Serrin, Gilbarg and Trudinger, Krantz and Parks, Foote, and Li and Nirenberg concerning the regularity of the inner distance function to the boundary.