Horizontally quasiconvex envelope in the Heisenberg group

Antoni Kijowski ^[1] ; Qing Liu ^[1] ; Xiaodan Zhou ^[1]
1. [1] Okinawa Institute of Science and Technology
  
  Okinawa Institute of Science and Technology
  
  Japón
Localización: Revista matemática iberoamericana, ISSN 0213-2230, Vol. 40, Nº 1, 2024, págs. 57-92
Idioma: inglés
DOI: 10.4171/RMI/1417
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Resumen
- 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.