  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.
