TRANSMISSION PROBLEMS FOR THE LAPLACE AND ELASTICITY OPERATORS: REGULARITY AND BOUNDARY INTEGRAL FORMULATION
Serge Nicaise, Anna‐Margarete Sändig · Mathematical Models and Methods in Applied Sciences · 1999
This paper is devoted to some transmission problems for the Laplace and linear elasticity operators in two- and three-dimensional nonsmooth domains. We investigate the behaviour of harmonic and linear elastic fields near geometrical singularities, especially near corner points or edges where the interface intersects with the boundaries. We give a short overview about the known results for 2-D problems and add new results for 3-D problems. Numerical 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 bundles are also given. Furthermore, we derive boundary integral equations for the solution of the transmission problems, which lead finally to "local" pseudo-differential operator equations with corresponding Steklov–Poincaré operators on the interface. We discuss their solvability and uniqueness. The above regularity results are used in order to characterize the regularity of the solutions of these integral equations.