Intrinsic Lipschitz maps vs. Lagrangian type solutions in Carnot groups of step 2

TL;DR

This paper investigates the relationship between intrinsic Lipschitz graphs and Lagrangian solutions in certain step 2 Carnot groups, establishing an equivalence that generalizes known solutions in Heisenberg groups.

Contribution

It proves the equivalence between intrinsic Lipschitz maps and weak solutions to a nonlinear PDE system in a broad class of step 2 Carnot groups, extending previous results.

Findings

Intrinsic Lipschitz maps are equivalent to weak solutions of a nonlinear PDE system.

The results include corank 1 Carnot groups, Heisenberg groups, and complexified Heisenberg groups.

Generalizes Lagrangian solutions in the context of these groups.

Abstract

We focus our attention on the notion of intrinsic Lipschitz graphs, inside a subclass of Carnot groups of step 2 which includes a corank 1 Carnot groups (and so the Heisenberg groups), Free groups of step 2 and the complexified Heisenberg group. More precisely, we prove the equivalence between intrinsic Lipschitz map and a weak solution to a suitable non linear first order PDE system, which generalizes Lagrangian solution in the context of Heisenberg groups.