# Regularity results for the minimum time function with H\"ormander vector fields

## TL;DR

This paper investigates the regularity of the minimum time function solving the eikonal equation with Hörmander vector fields, characterizing singular trajectories and establishing conditions for Lipschitz continuity and semiconcavity.

## Contribution

It provides a characterization of singular trajectories and links their absence to Lipschitz and semiconcavity properties of the solution, with applications to specific vector fields.

## Key findings

- Singular trajectories are characterized as points where the solution loses Lipschitz continuity.
- Lipschitz continuity, semiconcavity, and absence of singular trajectories are equivalent.
- The absence of singular trajectories holds when the characteristic set is a symplectic manifold.

## Abstract

In a bounded domain of $\mathbb{R}^n$ with smooth boundary, we study the regularity of the viscosity solution, $T$, of the Dirichlet problem for the eikonal equation associated with a family of smooth vector fields $\{X_1,\ldots ,X_N\}$, subject to H\"ormander's bracket generating condition. Due to the presence of characteristic boundary points, singular trajectories may occur in this case. We characterize such trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. We then prove that the local Lipschitz continuity of $T$, the local semiconcavity of $T$, and the absence of singular trajectories are equivalent properties. Finally, we show that the last condition is satisfied when the characteristic set of $\{X_1,\ldots ,X_N\}$ is a symplectic manifold. We apply our results to Heisenberg's and Martinet's vector fields.