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

Abstract

Let n⩾3. Let Ωi and Ωo be open bounded connected subsets of Rn containing the origin. Let ϵ0>0 be such that Ωo contains the closure of ϵΩi for all ϵ∈]−ϵ0,ϵ0[. Then, for a fixed ϵ∈]−ϵ0,ϵ0[∖{0} we consider a Dirichlet problem for the Laplace operator in the perforated domain Ωo∖ϵΩi. We denote by uϵ the corresponding solution. If p∈Ωo and p≠0, then we know that under suitable regularity assumptions there exist ϵp>0 and a real analytic operator Up from ]−ϵp,ϵp[ to R such that uϵ(p)=Up[ϵ] for all ϵ∈]0,ϵp[. Thus it is natural to ask what happens to the equality uϵ(p)=Up[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 uϵ(p)=Up[ϵ] for ϵ negative depends on the parity of the dimension n.