Boundary value problems and Heisenberg uniqueness pairs

We describe a general method for constructing Heisenberg uniqueness pairs \((\Gamma ,\Lambda )\) in the euclidean space \(\mathbb {R}^{n}\) based on the study of boundary value problems for partial differential equations. As a result, we show, for instance, that any pair made of the boundary \(\Gamma \) of a bounded convex set \(\Omega \) and a sphere \(\Lambda \) is an Heisenberg uniqueness pair if and only if the square of the radius of \(\Lambda \) is not an eigenvalue of the Laplacian on \(\Omega \). The main ingredients for the proofs are the Paley–Wiener theorem, the uniqueness of a solution to a homogeneous Dirichlet or initial boundary value problem, the continuity of single layer potentials, and some complex analysis in \(\mathbb {C}^{n}\). Denjoy’s theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful.