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

Let n\ge 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\in]-ε_0,ε_0[. Then, for a fixed ε\in]-ε_0,ε_0[\{0} we consider a Dirichlet problem for the Laplace operator in the perforated domain Ω^o\εΩ^i. We denote by u_εthe corresponding solution. If p\inΩ^o and p\neq 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\in]0,ε_p[. Thus it is natural to ask what happens to the equality u_ε(p)=U_p[ε] for ε<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_ε(p)=U_p[ε] for ε<0 depends on the parity of the dimension n.

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