  Consider the interior transmission problem arising in inverse boundary value
problems for the diffusion equation with discontinuous diffusion coefficients.
We prove the unique solvability of the interior transmission problem by
constructing its Green function. 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 local parametrix in the Laplace
domain and estimating it there, a local parametrix 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 generate a global
parametrix for the parabolic interior transmission problem, and then compensate
it to get the Green function by the Levi method. The uniqueness of the Green
function is justified by using the duality argument, and then the unique
solvability of the interior transmission problem is concluded. We would like to
emphasize that the Green function for the parabolic interior transmission
problem is constructed for the first time in this paper. It can be applied for
active thermography and diffuse optical tomography modeled by diffusion
equations to identify an unknown inclusion and its physical property.
