Horizontally quasiconvex envelope in the Heisenberg group

  • Antoni Kijowski

    Okinawa Institute of Science and Technology Graduate University, Japan
  • Qing Liu

    Okinawa Institute of Science and Technology Graduate University, Japan
  • Xiaodan Zhou

    Okinawa Institute of Science and Technology Graduate University, Japan
Horizontally quasiconvex envelope in the Heisenberg group cover
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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 problem 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.

Cite this article

Antoni Kijowski, Qing Liu, Xiaodan Zhou, Horizontally quasiconvex envelope in the Heisenberg group. Rev. Mat. Iberoam. 40 (2024), no. 1, pp. 57–92

DOI 10.4171/RMI/1417