Transmission problems and boundary operator algebras

Abstract

We examine the operator algebra A behind the boundary integral equation method for solving transmission problems. A new type of boundary integral operator, the rotation operator, is introduced, which is more appropriate than operators of double layer type for solving transmission problems for first order elliptic partial differential equations. We give a general invertibility criteria for operators in A by defining a Clifford algebra valued Gelfand transform on A. The general theory is applied to transmission problems with strongly Lipschitz interfaces for the two classical elliptic operators ∂ and Δ. We here use Rellich techniques in a new way to estimate the full complex spectrum of the boundary integral operators. For ∂ we use the associated rotation operator to solve the Hilbert boundary value problem and a Riemann type transmission problem. For the Helmholtz equation, we demonstrate how Rellich estimates give an angular spectral estimate on the rotation operator, which with the general spectral mapping properties in A translates to a hyperbolic spectral estimate for the double layer potential operator.