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

TL;DR

This paper defines a new concept of horizontal convex envelopes in the Heisenberg group, provides a constructive convexification process, and applies it to establish h-convexity of solutions to certain nonlinear sub-elliptic equations under symmetry conditions.

Contribution

It introduces a natural notion of horizontal convex envelopes in the Heisenberg group and develops a convexification method to analyze viscosity solutions of sub-elliptic equations.

Findings

Convexification process successfully constructs horizontal convex envelopes.

H-convexity of solutions holds under specific symmetry conditions.

Counterexamples show h-convexity may fail without symmetry.

Abstract

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.