Fine topology and fine trace on the boundary associated with a class of semilinear differential equations
Eugene B. Dynkin, С. Е. Кузнецов · Communications on Pure and Applied Mathematics · 1998
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 ℝd or, more generally, in a Riemannian manifold and ψ belongs to a wide class of convex functions that contains ψ(u) = uα 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. © 1998 John Wiley & Sons, Inc.