ANALYSIS OF TRANSMISSION PROBLEMS ON LIPSCHITZ BOUNDARIES IN STRONGER NORMS

We consider in [1,2] a model homogeneous Dirichlet problem for a diffusion equation on a Lipschitz simply connected bounded domain with a small diffusion coefficient in a Lipschitz simply connected subdomain located strictly inside of the main domain. We study asymptotic properties of the solution with respect to the small diffusion coefficient vanishing. It is known that the solution asymptotically turns into a solution of a corresponding diffusion equation in a cavity, i.e. with Neumann boundary conditions on the inner part of the boundary. One typical proof technique of this fact utilizes a reduction of the problem to the boundary of the inner subdomain, using a transmission condition. An analogous approach appears in studying domain decomposition methods without overlap, reducing the investigation to the surface that separates the subdomains and in theoretical foundation of a fictitious domain method, e.g., to prove a classical estimate that guaranties convergence of the solution of the fictitious domain problem to the solution in the cavity [3]. On a continuous level, the analysis is usually performed in a trace norm, e.g., the H norm for second order elliptic equations. This norm appears naturally for Poincare-Steklov operators, which are convenient to employ to formulate the transmission condition. A regularity theory of Poincare-Steklov operators for Lipschitz domains is closely related to regularity of transmission and diffraction problems and properties of layer potentials on nonsmooth interfaces, and has been extensively studied. Taking advantage of the recent progres