  This paper is concerned with a PDE-based approach to the horizontally
quasiconvex (h-quasiconvex for short) envelope of a given continuous function
in the Heisenberg group. We provide a characterization for upper
semicontinuous, h-quasiconvex functions in terms of the viscosity subsolution
to a first-order nonlocal Hamilton-Jacobi equation. We also construct the
corresponding envelope of a continuous function by iterating the nonlocal
operator. One important step in our arguments is to prove the uniqueness and
existence of viscosity solutions to the Dirichlet boundary problems for the
nonlocal Hamilton-Jacobi equation. Applications of our approach to the h-convex
hull of a given set in the Heisenberg group are discussed as well.
