On the determination of an unknown boundary function in a parabolic equation

We study the determination of an unknown boundary function in an initial-boundary value problem for the heat equation. We prove two uniqueness results. In the first one we consider the case of all measurements on a part of the lateral boundary. The second result concerns the case of a single overdetermined datum. Our tools are differentiation with respect to the boundary data and monotonicity estimates that we derive from the maximum principle.