Solvability of interior transmission problem for the diffusion equation

Gen Nakamura, Haibing Wang

Published 2017-06-15Version 1

Consider the interior transmission problem arising in inverse boundary value problems for the diffusion equation with discontinuous diffusion coefficients. We prove its unique solvability using the method of Green function. The key point is to construct the Green function for the interior transmission problem. We complete this procedure in the following way. First, we construct a local parametrix for the interior transmission problem near the boundary in the Laplace domain, by using the theory of pseudo-differential operators with a large parameter. Second, by carefully analyzing the analyticity of the Green function in the Laplace domain and estimating it there, a local parametrix near the boundary for the original parabolic interior transmission problem is obtained via the inverse Laplace transform. Finally, using a partition of unity, we patch all the local parametrices and the fundamental solution of the diffusion equation to have a parametrix for the parabolic interior transmission problem, and then compensate it to get the Green function. The uniqueness of the Green function is justified by using the duality argument. We would like to emphasize that the Green function for the interior transmission problem is constructed for the first time in this paper. It has many applications. For example, it can be used for the active tomography and diffuse optical tomography modeled by diffusion equations to identify an unknown inclusion and its physical property.