Transmission problems for the Laplace and elasticity operators: Regularityand boundary integral formulation

This paper is devoted to some transmission problems for the Laplace and lin ear elasticity operators in two- and three-dimensional nonsmooth domains. W e investigate the behaviour of harmonic and linear elastic fields near geom etrical singularities, especially near corner points or edges where the int erface intersects with the boundaries. We give a short overview about the k nown results for 2-D problems and add new results for 3-D problems. Numeric al results for the calculation of the singular exponents in the asymptotic expansion are presented for both two- and three-dimensional problems. Some spectral properties of the corresponding parameter depending operator bundl es are also given. Furthermore, we derive boundary integral equations for t he solution of the transmission problems, which lead finally to "local" pse udo-differential operator equations with corresponding Steklov-Poincare ope rators on the interface. We discuss their solvability and uniqueness. The a bove regularity results are used in order to characterize the regularity of the solutions of these integral equations.