On Lipschitz vector fields and the Cauchy problem in homogeneous groups

2016/06/17 by Valentino Magnani, Dario Trevisan
Mathematics · Physics and Astronomy · #Advanced Differential Geometry Research #Combinatorics #Counterexample #Discrete mathematics #Euclidean geometry #Geometric Analysis and Curvature Flows #Geometry #Homogeneous #Lipschitz continuity #Lipschitz domain #Mathematical analysis #Mathematics #Point processes and geometric inequalities #Pure mathematics #Uniqueness #Vector field #math.CA #msc:34A12 #msc:53C17

paper · pdf · doi:10.1142/s0219199717500572

arxiv created 2016/06/17 · openalex publication_date 2017/06/13 · arxiv updated 2017/07/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Abstract

We introduce a class of “Lipschitz horizontal” vector fields in homogeneous groups, for which we show equivalent descriptions, e.g., in terms of suitable derivations. We then investigate the associated Cauchy problem, providing a uniqueness result both at equilibrium points and for vector fields of an involutive submodule of Lipschitz horizontal vector fields. We finally exhibit a counterexample to the general well-posedness theory for Lipschitz horizontal vector fields, in contrast with the Euclidean theory.

Citations

• Foundations of Differentiable Manifolds and Lie Groups
• Well-posedness of Lagrangian flows and continuity equations in metric measure spaces