Potential Analysis

, Volume 45, Issue 3, pp 545–555 | Cite as

Harnack Inequality for Hypoelliptic Second Order Partial Differential Operators



We consider non-negative solutions \(u:{\Omega }\longrightarrow \mathbb {R}\) of second order hypoelliptic equations
$$ \mathcal {L} u(x) =\sum \limits _{i,j=1}^{n} \partial _{x_{i}} \left (a_{ij}(x)\partial _{x_{j}} u(x) \right ) + \sum \limits _{i=1}^{n} b_{i}(x) \partial _{x_{i}} u(x) =0 $$
where Ω is a bounded open subset of \(\mathbb {R}^{n}\) and x denotes the point of Ω. For any fixed x 0 ∈ Ω, we prove a Harnack inequality of this type
$$ \sup _{K} u \le C_{K} u(x_{0})\qquad \forall \ u \ \text { s.t. } \ \mathcal {L} u=0, u\geq 0, $$
where K is any compact subset of the interior of the \(\mathcal {L}\)-propagation set of x 0 and the constant C K does not depend on u.


Harnack inequality Hypoelliptic operators Potential theory 

Mathematics Subject Classification (2010)

35H10 35K10 31D05 


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© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica ApplicataUniversità degli Studi di SalernoFisciano (SA)Italy
  2. 2.Dipartimento di Scienze Fisiche, Informatiche e MatematicheUniversità di Modena e Reggio EmiliaModenaItaly

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