Analysis of transmission problems on Lipschitz boundaries in stronger norms

We concentrate on a model diffusion equation on a Lipschitz simply connected bounded domain with a small diffusion coef£cient in a Lipschitz simply connected subdomain located strictly inside of the original domain. We study asymptotic properties of the solution with respect to the small diffusion coef£cient vanishing. It is known that the solution asymptotically turns into a solution of a corresponding diffusion equation with Neumann boundary conditions on a part of the boundary. One typical proof technique of this fact utilizes a reduction of the problem to the interface of the 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 £ctitious domain, also called embedding, method, e.g., to prove a classical estimate that guaranties convergence of the solution of the £ctitious domain problem to the solution of the original Neumann boundary value problem. On a continuous level, this analysis is usually performed in an H1/2 norm for second order elliptic equations. This norm appears naturally for Poincaré-Steklov operators, which are convenient to employ to formulate the transmission condition. Using recent advances in regularity theory of PoincaréSteklov operators for Lipschitz domains, we provide, in the present paper, a similar analysis in an H1/2+α norm with α > 0, for a simple model problem. This result leads to a convergence theory of the £ctitious domain method for a second order elliptic PDE in an H 1+α norm, while the classical result is in an H1 norm. Here, α < 1/2 for the case of Lipschitz domains we consider.