Fine Topology and Fine Trace on the Boundary Associated with a Class of Semilinear Differential Equations

摘要

We investigate the set of all positive solutions of a semilinear equation Lu = ψ(u) where L is a second-order elliptic differential operator in a domain E of R~d or, more generally, in a Riemannian manifold and ψ belongs to a wide class of convex functions that contains ψ(u) = u~α for all α > 1. We define boundary singularities of a solution u in terms of points of rapid growth of the right derivative ψ~+(u), we introduce a fine topology and a fine trace of u on the Martin boundary, and we construct the minimal solution for every possible value of this trace.