a small hole (cf., *e.g.*, [@DaMu12; @DaMu15], Lanza de Cristoforis [@La07; @La10]), and has later been extended to problems related to the system of equations of the linearized elasticity (cf., *e.g.*, the first-named author and Lanza de Cristoforis [@DaLa10a]) and to the Stokes system (cf., *e.g.*, [@Da13]). Moreover, analyticity results have been obtained in the frame of perturbation problems in spectral theory (cf., *e.g.*, Buoso and Provenzano [@BuPr] and Lamberti and Lanza de Cristoforis [@LaLa04]).

The paper is organized as follows. In Section \[nota\], we introduce some notation and in Section \[for\] we introduce a more general formulation of our problem. In Section \[prel\], we introduce some preliminary results. In Section \[finteq\], we formulate our problem in terms of integral equations. In Section \[fure\], we prove our main result, which answers our questions (i), (ii), (iii) above, and in Section \[asy\] we compute an asymptotic expansion of the solution for $n=2$ and $r_\ast=0$.

Notation {#nota}
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We denote the norm on a normed space ${\mathcal X}$ by $\|\cdot\|_{{\mathcal X}}$. Let ${\mathcal X}$ and ${\mathcal Y}$ be normed spaces. We endow the space ${\mathcal X}\times {\mathcal Y}$ with the norm defined by $\|(x,y)\|_{{\mathcal X}\times {\mathcal Y}}\equiv \|x\|_{{\mathcal X}}+
\|y\|_{{\mathcal Y}}$ for 