Regularity results for the minimum time function with Hörmander vector fields

Abstract

Abstract. In a bounded domain of 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 , . . . , X N }, subject to Hörmander'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 , . . . , X N } is a symplectic manifold. We apply our results to Heisenberg's and Martinet's vector fields.

The Aronsson Equation, Lyapunov Functions, and Local Lipschitz Regularity of the Minimum Time Function

“…More recently such results had several extensions in the work by Krastanov and Quincampoix [16] and Marigonda, Rigo and Le [18,19,20]. Our regularity results rather go in the direction of those contained in two recent papers by Albano, Cannarsa and Scarinci [3,4], where they show, by completely different methods, that if a family of smooth vector fields satisfies the Hörmander condition, then the set where the local Lipschitz continuity of the minimum time function fails is the union of singular trajectories, and that it is analytic except on a subset of null measure. Our approach is instead more direct and comes as a consequence of constructing Lyapunov functions as C 1 −supersolutions of the Aronsson equation.…”

Partial regularity for solutions to subelliptic eikonal equations

“…In [3], we investigated the regularity of T . Building on such results, in this paper we analyse the singular support of T .…”

Heat content asymptotics for sub-Riemannian manifolds

“…The assumption on the characteristic set of Ω is crucial to guarantee the smoothness of the distance around the boundary. However, if ∂Ω has characteristic points, δ is not eve Lipschitz in coordinates, see [ACS18] for more details.…”

“…Notice that the existence of a point q ∈ C 2 \ Cut (Σ) is guaranteed by the fact that the cut locus is a nowhere dense set (see [ACS18] and [Agr09], for further details).…”