Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations

• Qing Liu ^[1] ; Xiaodan Zhou ^[2]
  1. [1] Fukuoka University

     Fukuoka University

     Hakata Ku, Japón

     
  2. [2] University Okinawa, Japan
• Localización: Annali della Scuola Normale Superiore di Pisa. Classe di scienze, ISSN 0391-173X, Vol. 22, Nº 4, 2021, págs. 2039-2076
• Idioma: inglés
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• Resumen

  • This paper introduces in a natural way a notion of horizontal convex envelopes of continuous functions in the Heisenberg group. We provide a convexification process to find the envelope in a constructive manner. We also apply the convexification process to show h-convexity of viscosity solutions to a class of fully nonlinear elliptic equations in the Heisenberg group satisfying a certain symmetry condition. Our examples show that in general one cannot expect h-convexity of solutions without the symmetry condition.