Asymptotic behavior of the solutions of the Dirichlet problem for the Laplace operator in a domain with a small hole. A functional analytic approach

We consider a hypersurface in ${\mathbb{R}}^{n}$ parametrized by a diffeomorphism $\phi^{o}$ of the unit sphere in ${\mathbb{R}}^{n}$ into ${\mathbb{R}}^{n}$, and we take a point $w$ in the domain ${\mathbb{I}}[\phi^{o}]$ enclosed by the image of $\phi^{o}$, and we consider the `hole' ${\mathbb{I}}[w+\epsilon \xi]$ enclosed by the image of the hypersurface $w+\epsilon \xi$, where $\xi$ is a diffeomorphism as $\phi^{o}$ with $0\in {\mathbb{I}}[\xi]$ and $\epsilon$ is a small positive real parameter. Then we consider the Dirichlet problem for the Laplace equation in the perforated domain ${\mathbb{I}}[\phi^{o}]$ with the hole ${\mathbb{I}}[w+\epsilon \xi]$ removed and show real analytic continuation properties of the solution $u$ and of the corresponding energy integral as functionals of the sextuple of $w$, $\epsilon $, $\xi$, $\phi^{o}$, and of the Dirichlet data in the interior and exterior boundaries of the perforated domain, which we think of as a point in an appropriate Banach space, around a degenerate sextuple with $\epsilon=0$.