Critical points and level sets in exterior boundary problems

Abstract

We study some geometrical properties of the critical set of the solutions to an exterior boundary problem in R n \Ω, where Ω is a bounded domain with C 2 connected boundary. We prove that this set can be nonempty (in fact, of codimension 3) even when Ω is contractible, thereby settling a question posed by Kawohl. We also obtain new sufficient geometric criteria for the absence of critical points in this problem and analyze the properties of the critical set for generic domains. The proofs rely on a combination of classical potential theory, transversality techniques and the geometry of real analytic sets.