The Strong Maximum Principle and the Harnack inequality for a class of hypoelliptic divergence-form operators

2014/07/07 by Battaglia, Erika, Biagi, Stefano, Bonfiglioli, Andrea
#35B45 #35B50 #35H20 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1407.1669

Abstract

In this paper we consider a class of hypoelliptic second-order partial differential operators L in divergence form on ℝ^N, arising from CR geometry and Lie group theory, and we prove the Strong and Weak Maximum Principles and the Harnack Inequality for L. The involved operators are not assumed to belong to the Hörmander hypoellipticity class, nor to satisfy subelliptic estimates, nor Muckenhoupt-type estimates on the degeneracy of the second order part; indeed our results hold true in the infinitely-degenerate case and for operators which are not necessarily sums of squares. We use a Control Theory result on hypoellipticity in order to recover a meaningful geometric information on connectivity and maxima propagation, yet in the absence of any Hörmander condition. For operators L with C^ω coefficients, this control-theoretic result will also imply a Unique Continuation property for the L-harmonic functions. The (Strong) Harnack Inequality is obtained via the Weak Harnack Inequality by means of a Potential Theory argument, and by a crucial use of the Strong Maximum Principle and the solvability of the Dirichlet problem for L on a basis of the Euclidean topology.

Related

• Boundary weak Harnack estimates and quantitative strong maximum principles for uniformly elliptic PDE
• Semi-local behaviour of non-local hypoelliptic equations: divergence form
• Fundamental solution and Harnack inequality for subelliptic evolution operators on Carnot groups
• Harnack inequality for hypoelliptic second order partial differential\n operators
• Harnack Inequality for a Subelliptic PDE in nondivergence form