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

TL;DR

This paper investigates the behavior of harmonic functions in domains with small holes, revealing that the analytic continuation of solutions depends on the dimension's parity, with implications for understanding perforated domain problems.

Contribution

It establishes the dependence of harmonic function continuation on the dimension's parity in perforated domains, extending previous real analytic family results.

Findings

Continuation depends on the parity of the dimension n.

Analytic extension holds for positive epsilon, but not necessarily for negative epsilon.

Results clarify the structure of harmonic functions in perforated domains.

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.