Fine topology and fine trace on the boundary associated with a class of semilinear differential equations

✍ Scribed by E. B. Dynkin; S. E. Kuznetsov

• Publisher: John Wiley and Sons
• Year: 1998
• Tongue: English
• Weight: 576 KB
• Volume: 51
• Category: Article
• ISSN: 0010-3640
• DOI: 10.1002/(sici)1097-0312(199808)51:8<897::aid-cpa2>3.0.co;2-0

✦ Synopsis

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.