Mapping Properties of Combined Field Helmholtz Boundary Integral Operators

For the Helmholtz equation (with wavenumber k) and analytic curves or surfaces Γ we analyze the mapping properties of the single layer, double layer as well combined potential boundary integral operators. A k-explicit regularity theory for the single layer and double layer potentials is developed, in which these operators are decomposed into three parts: the first part is the single or double layer potential for the Laplace equation, the second part is an operator with finite shift properties, and the third part is an operator that maps into a space of piecewise analytic functions. For all parts, the k-dependence is made explicit. We also develop a k-explicit regularity theory for the inverse of the combined potential operator A = ±1/2+K− iηV and its adjoint, where V and K are the single layer and double layer operators for the Helmholtz kernel and η ∈ R is a coupling parameter with |η| ∼ |k|. The decomposition of the inverses A and (A) takes the form of a sum of two operators A1, A2 where A1 : Hs(Γ) → Hs(Γ) with bounds independent of k and a smoothing operator A2 that maps into a space of analytic functions on Γ. The k-dependence of the mapping properties of A2 is made explicit.