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 Rn parametrized by a diffeomorphism φo of the unit sphere in Rn into Rn , and we take a point w in the domain I[φo] enclosed by the image of φo, and we consider the ‘hole’ I[w + ξ] enclosed by the image of the hypersurface w + ξ , where ξ is a diffeomorphism as φo with 0 ∈ I[ξ] and is a small positive real parameter. Then we consider the Dirichlet problem for the Laplace equation in the perforated domain I[φo] with the hole I[w + ξ] removed and show real analytic continuation properties of the solution u and of the corresponding energy integral as functionals of the sextuple of w, , ξ , φo, 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 = 0.