Intrinsic Lipschitz graphs in Heisenberg groups and continuous solutions of a balance equation

TL;DR

This paper characterizes intrinsic Lipschitz graphs in Heisenberg groups via distributional gradients and establishes equivalences of continuous weak solutions to a specific balance equation involving these graphs.

Contribution

It introduces a new characterization of intrinsic Lipschitz graphs in the Heisenberg group and links different notions of weak solutions to a related PDE.

Findings

Intrinsic Lipschitz graphs characterized by distributional gradients.

Equivalence of various continuous weak solution concepts for the balance equation.

Connection between geometric structures and PDE solutions in Heisenberg groups.

Abstract

In this paper we provide a characterization of intrinsic Lipschitz graphs in the sub-Riemannian Heisenberg groups in terms of their distributional gradients. Moreover, we prove the equivalence of different notions of continuous weak solutions to the equation \phi_y+ [\phi^{2}/2]_t=w, where w is a bounded function depending on \phi.