Horizontally quasiconvex envelope in the Heisenberg group

TL;DR

This paper develops a PDE-based method to characterize and construct the horizontally quasiconvex envelope of functions in the Heisenberg group, using viscosity solutions to a nonlocal Hamilton-Jacobi equation.

Contribution

It introduces a novel PDE approach for h-quasiconvex envelopes in the Heisenberg group, including characterization, construction, and boundary problem solutions.

Findings

Characterization of h-quasiconvex functions via viscosity subsolutions

Construction of the envelope through iterative nonlocal operator

Existence and uniqueness results for boundary value problems

Abstract

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.