Real analytic families of harmonic functions in a domain with a small hole

Let n≥3. Let Ω i and Ω o be open bounded connected subsets of ℝ n containing the origin. Let ε 0>0 be such that Ω o contains the closure of εΩ i for all ε∈]-ε 0, ε 0[. Then, for a fixed ε∈]-ε 0, ε 0[{set minus}{0} we consider a Dirichlet problem for the Laplace operator in the perforated domain Ω o{set minus}εΩ i. We denote by u ε the corresponding solution. If p∈Ω o and p≠0, then we know that under suitable regularity assumptions there exist ε p>0 and a real analytic operator U p from ]-ε p, ε p[ to R such that u ε(p)=U p[ε] for all ε∈]0, ε p[. Thus it is natural to ask what happens to the equality u ε(p)=U p[ε] for ε negative. We show a general result on continuation properties of some particular real analytic families of harmonic functions in domains with a small hole and we prove that the validity of the equality u ε(p)=U p[ε] for ε negative depends on the parity of the dimension n. © 2012 Elsevier Inc.