Uniqueness and Hölder stability of discontinuous diffusion coefficients in three related inverse problems for the heat equation

Abstract

We consider the heat equation ∂ty − div(c∇y) = H with a discontinuous coefficient in three connected situations. We give the uniqueness and stability results for the diffusion coefficient c(⋅) in the main case from measurements of the solution on an arbitrary part of the boundary and at a fixed time in the whole spatial domain. The diffusion coefficient is assumed to be discontinuous across an unknown interface. The key ingredients are a Carleman-type estimate with non-smooth data near the interface and a stability result for the discontinuous coefficient c(⋅) in an inverse problem associated with the stationary equation − div(c∇u) = f.