Regularity results for the minimum time function with Hormander vector fields

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:264

Article; Proceedings Paper

Albano, Paolo¹  Cannarsa, Piermarco²  Scarinci, Teresa³ 

[1] Univ Bologna, Dipartimento Matemat, Piazza Porta San Donato 5, I-40127 Bologna, Italy

[2] Univ Roma Tor Vergata, Dipartimento Matemat, Via Ric Sci 1, I-00133 Rome, Italy

[3] Univ Vienna, Dept Stat & Operat Res, Oskar Morgenstern Pl 1, A-1090 Vienna, Austria

关键词: Eikonal equation;    Degenerate equations;    Sub-Riemannian geometry;    Semiconcavity;   

DOI  :  10.1016/j.jde.2017.11.016

来源: Elsevier

【 摘 要 】

In a bounded domain of R-n with boundary given by a smooth (n - 1)-dimensional manifold, we consider the homogeneous Dirichlet problem for the eikonal equation associated with a family of smooth vector fields {X-1,..., X-N} subject to Hormander's bracket generating condition. We investigate the regularity of the viscosity solution Tof such problem. Due to the presence of characteristic boundary points, singular trajectories may occur. First, we characterize these trajectories as the closed set of all points at which the solution loses point-wise Lipschitz continuity. Then, we 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 whenever the characteristic set of {X-1,..., X-N} is a symplectic manifold. We apply our results to several examples. (c) 2017 Elsevier Inc. All rights reserved.

【 授权许可】

Free