BV Minimizers of the area functional in the Heisenberg group under the bounded slope condition

TL;DR

This paper investigates the minimizers of the area functional for t-graphs in the Heisenberg group, establishing existence, uniqueness, and Lipschitz regularity under a bounded slope condition, with examples demonstrating the sharpness of regularity results.

Contribution

It proves the existence and uniqueness of Lipschitz continuous minimizers for the area functional in the Heisenberg group under bounded slope conditions, highlighting the optimal regularity.

Findings

Existence and uniqueness of minimizers under bounded slope.

Minimizers are Lipschitz continuous.

Lipschitz regularity is sharp in the first Heisenberg group.

Abstract

We consider the area functional for t-graphs in the sub-Riemannian Heisenberg group and study minimizers of the associated Dirichlet problem. We prove that, under a bounded slope condition on the boundary datum, there exists a unique minimizer and that this minimizer is Lipschitz continuous. We also provide an example showing that, in the first Heisenberg group, Lipschitz regularity is sharp even under the bounded slope condition.