The reconstruction of a source and a potential from boundary measurements

We are concerned with the reconstruction of both the heat coefficient and the source appearing in a parabolic equation from observations of the solution on the boundary. We propose two methods. The first uses a single point on the boundary and a final time overdetermination, while the second uses only the observations over a small region on the boundary only. These methods are based on properties of the principal Dirichlet eigenfunction and Holmgren's theorem.