Boundary value problems and Heisenberg uniqueness pairs
We describe a general method for constructing Heisenberg uniqueness pairs $(Γ,Λ)$ 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 $Γ$ 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 $\mathbb{C}^{n}$. Denjoy's theorem on topological conjugacy of circle diffeomorphisms with irrational rotation numbers is also useful.