Shape analyticity and singular perturbations for layer potential operators

We study the effect of regular and singular domain perturbations on layer potential operators for the Laplace equation. First, we consider layer potentials supported on a diffeomorphic image phi( partial differential omega) of a reference set partial differential omega and we present some real analyticity results for the dependence upon the map phi. Then we introduce a perforated domain omega(epsilon) with a small hole of size epsilon and we compute power series expansions that describe the layer potentials on partial differential omega(epsilon) when the parameter epsilon approximates the degenerate value epsilon = 0.