2013/06/24 by Benny Avelin, Luca Capogna, Avelin, Benny +6
Computer Science · Mathematics · #35H20 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #math.AP #math.DG #math.MG #msc:35H20
paper · pdf · doi:10.48550/arxiv.1306.5650
arxiv created 2013/06/24 · openalex publication_date 2013/06/24 · arxiv updated 2013/06/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototype equation* ∂tu= -∑i=1mXi^∗ (|\X u|p-2 Xi u)equation* where p≥ 2, \X = (X1,..., Xm) is a system of Lipschitz vector fields defined on a smooth manifold \M endowed with a Borel measure μ, and Xi^* denotes the adjoint of Xi with respect to μ. Our estimates are derived assuming that (i) the control distance d generated by \X induces the same topology on \M; (ii) a doubling condition for the μ-measure of d-metric balls and (iii) the validity of a Poincaré inequality involving \X and μ. Our results extend the recent work in \citeDiBenedettoGianazzaVespri1, \citeK, to a more general setting including the model cases of (1) metrics generated by Hörmander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.