The first problem in the thesis concerns inter-facial effects in homogenization. We study acoustic time harmonic wave propagation in a system of two periodic composite media separated by a plane interface. We construct a full asymptotic expansion to a solution of the transmission problem. The main idea is to introduce special boundary layer correctors. The terms of these correctors depending on a slow variable are defined as solutions of averaged equations for one of the composite media, and the terms depending on a fast variable are obtained from solving special cell problems for the other composite medium.

As a consequence, explicit formulae for calculating averaged reflection and transmission coefficients are obtained. It is shown that the main term of the asymptotic expansion can be written as a solution of the averaged transmission problem for which the transmission conditions are written using the averaged coefficients on each side of the interface. The subsequent terms of the expansion contain different transmission conditions.

The second part of the thesis develops an existence theory of special quasi-exponential solutions for a class of elliptic equations with non-smooth coefficients. Based on this, we prove a new uniqueness result for an inverse problem involving the Schrödinger equation in a magnetic field. The construction of the special solutions is of a general nature and can be applied to a variety of inverse problems for which low regularity of the functions sought is essential.