Harnack Inequality for a Subelliptic PDE in nondivergence form

TL;DR

This paper establishes a scale-invariant Harnack inequality for nonnegative solutions of subelliptic PDEs in nondivergence form involving Grushin vector fields, under bounded eigenvalue ratio conditions.

Contribution

It introduces a novel approach combining weighted Aleksandrov Bakelman Pucci estimates and critical density estimates to prove Harnack's inequality for these PDEs.

Findings

Proved a weighted Aleksandrov Bakelman Pucci estimate.

Established a critical density estimate and the double ball property.

Derived Harnack's inequality from the axiomatic framework.

Abstract

We consider subelliptic equations in non divergence form of the type $Lu = \sum a_{ij} X_jX_iu=0$, 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].