We describe a general method for constructing Heisenberg uniqueness pairs pΓ, Λq in the euclidean space 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 Γ of a bounded convex set Ω and a sphere Λ is an Heisenberg uniqueness pair if and only if the square of the radius of Λ is not an eigenvalue of the Laplacian on Ω. 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 C n . Denjoy's theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful.