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 imageϕ(∂Ω) of a reference set ∂Ω and we present some real analyticity results for the dependence upon the mapϕ. Then we introduce a perforated domain Ω(ε) with a small hole of sizeεand we compute power series expansions that describe the layer potentials on ∂Ω(ε) when the parameterεapproximates the degenerate valueε = 0.