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

Abstract: Let n\ge 3. Let \Omega^i and \Omega^o be open bounded connected subsets of R^n containing the origin. Let \epsilon_0>0 be such that \Omega^o contains the closure of \epsilon\Omega^i for all \epsilon\in]-\epsilon_0,\epsilon_0[. Then, for a fixed \epsilon\in]-\epsilon_0,\epsilon_0[{0} we consider a Dirichlet problem for the Laplace operator in the perforated domain \Omega^o\epsilon\Omega^i. We denote by u_\epsilon the corresponding solution. If p\in\Omega^o and p\neq 0, then we know that under suitable regularity assumptions there exist \epsilon_p>0 and a real analytic operator U_p from ]-\epsilon_p,\epsilon_p[ to R such that u_\epsilon(p)=U_p[\epsilon] for all \epsilon\in]0,\epsilon_p[. Thus it is natural to ask what happens to the equality u_\epsilon(p)=U_p[\epsilon] for \epsilon<0. 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_\epsilon(p)=U_p[\epsilon] for \epsilon<0 depends on the parity of the dimension n.