Harnack Inequality for a Subelliptic PDE in nondivergence form

2014/06/27 by Annamaria Montanari, Montanari, Annamaria · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP #msc:35J70 #msc:35R05

paper · pdf · doi:10.48550/arxiv.1406.7311

arxiv created 2014/07/01 · arxiv updated 2014/07/02

Abstract

We consider subelliptic equations in non divergence form of the type Lu = ∑ a_ij X_jX_iu=0, where X_j are the Grushin vector fields, and the matrix coefficient is uniformly elliptic. We obtain a scale invariant Harnack's inequality on the X_j's CC balls for nonnegative solutions under the only assumption that the ratio between the maximum and minimum eigenvalues of the coefficient matrix is bounded. In the paper we first prove a weighted Aleksandrov Bakelman Pucci estimate, and then we show a critical density estimate, the double ball property and the power decay property. Once this is established, Harnack's inequality follows directly from the axiomatic theory developed by Di Fazio, Gutierrez and Lanconelli in [6].

Cited by

• Removable singularities and Harnack inequality for nonlinear Hörmander degenerate subelliptic equations

Related

• Abstract approach to non homogeneous Harnack inequality in doubling quasi metric spaces
• A short proof of Boundary Harnack Inequality
• A Subelliptic Analogue of Aronson-Serrin's Harnack Inequality
• The Harnack inequality for a class of degenerate elliptic operators
• Harnack estimates for degenerate parabolic equations modeled on the subelliptic p-Laplacian