Harnack Inequality for a Degenerate Elliptic Equation

Harnack inequality for a class of Kolmogorov–Fokker–Planck equations in non-divergence form

“…In this work, we prove Harnack's inequality for non-negative solutions to Ku = 0 under either a Cordes-Landis condition or a continuity assumption on the coefficient matrix A (see subsection 1.1, hypotheses H1 and H2). Similar results have been obtained for other Hörmander type operators, namely for non-divergence form operators structured on Heisenberg vector fields [1,13,32]. The techniques we employ in the present work are inspired by the insightful contributions of Landis from the '60s [24], where he obtained what is nowadays referred to as the growth lemma for nonnegative subsolutions of uniformly elliptic equations, assuming that the eigenvalue ratio is close to 1.…”

Harnack’s inequality for a class of non-divergent equations in the Heisenberg group

“…The Harnack's inequality now follows from the double ball property (see [GT11,T12]) and the results in [DGL08]. In fact, we have the following Theorem.…”

Harnack inequality for a subelliptic PDE in nondivergence form

“…In [15] Gutierrez and Tournier proved this property for elliptic equations on the Heisenberg group H 1 . In [22] Tralli proved that it holds true for a general Carnot group of step two.…”

“…Recently, Gutierrez and Tournier in [15] and Tralli in [23] provided direct proofs (in the Heisenberg group H 1 and in H-type groups, respectively) of the critical density estimates for super solutions using barriers, but under the restrictive assumption that the matrix of the coefficients a ij is a small perturbation of the Identity matrix.…”