SpringerOpen Newsletter

Receive periodic news and updates relating to SpringerOpen.

This article is part of the series Harnack's Estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations.

Open Access Open Badges Research Article

Liouville Theorems for a Class of Linear Second-Order Operators with Nonnegative Characteristic Form

Alessia Elisabetta Kogoj* and Ermanno Lanconelli

Author Affiliations

Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, Bologna 40126, Italy

For all author emails, please log on.

Boundary Value Problems 2007, 2007:048232  doi:10.1155/2007/48232

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/048232

Received:1 August 2006
Revisions received:28 November 2006
Accepted:29 November 2006
Published:14 March 2007

© 2007 Kogoj and Lanconelli

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We report on some Liouville-type theorems for a class of linear second-order partial differential equation with nonnegative characteristic form. The theorems we show improve our previous results.


  1. Kogoj, AE, Lanconelli, E: An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations. Mediterranean Journal of Mathematics. 1(1), 51–80 (2004). Publisher Full Text OpenURL

  2. Kogoj, AE, Lanconelli, E: One-side Liouville theorems for a class of hypoelliptic ultraparabolic equations. Geometric Analysis of PDE and Several Complex Variables, Contemporary Math., pp. 305–312. American Mathematical Society, Providence, RI, USA (2005)

  3. Kogoj, AE, Lanconelli, E: Liouville theorems in halfspaces for parabolic hypoelliptic equations. Ricerche di Matematica. 55(2), 267–282 (2006)

  4. Lanconelli, E: A polynomial one-side Liouville theorems for a class of real second order hypoelliptic operators. Rendiconti della Accademia Nazionale delle Scienze detta dei XL. 29, 243–256 (2005)

  5. Luo, X: Liouville's theorem for homogeneous differential operators. Communications in Partial Differential Equations. 22(11-12), 1837–1848 (1997). Publisher Full Text OpenURL

  6. Lanconelli, E, Pascucci, A: Superparabolic functions related to second order hypoelliptic operators. Potential Analysis. 11(3), 303–323 (1999). Publisher Full Text OpenURL

  7. Amano, K: Maximum principles for degenerate elliptic-parabolic operators. Indiana University Mathematics Journal. 28(4), 545–557 (1979). Publisher Full Text OpenURL

  8. Glagoleva, RJa: Liouville theorems for the solution of a second order linear parabolic equation with discontinuous coefficients. Matematicheskie Zametki. 5(5), 599–606 (1969)

  9. Bear, HS: Liouville theorems for heat functions. Communications in Partial Differential Equations. 11(14), 1605–1625 (1986). Publisher Full Text OpenURL

  10. Bonfiglioli, A, Lanconelli, E: Liouville-type theorems for real sub-Laplacians. Manuscripta Mathematica. 105(1), 111–124 (2001). Publisher Full Text OpenURL

  11. Lanconelli, E, Polidoro, S: On a class of hypoelliptic evolution operators. Rendiconti Seminario Matematico Università e Politecnico di Torino. 52(1), 29–63 (1994)

  12. Priola, E, Zabczyk, J: Liouville theorems for non-local operators. Journal of Functional Analysis. 216(2), 455–490 (2004). Publisher Full Text OpenURL

  13. Kogoj, AE, Lanconelli, E: Link of groups and applications to PDE's.