Open Access

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

Boundary Value Problems20072007:048232

Received: 1 August 2006

Accepted: 29 November 2006

Published: 14 March 2007


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.


Authors’ Affiliations

Dipartimento di Matematica, Università di Bologna


  1. Kogoj AE, Lanconelli E: An invariant Harnack inequality for a class of hypoelliptic ultraparabolic equations. Mediterranean Journal of Mathematics 2004,1(1):51–80. 10.1007/s00009-004-0004-8MATHMathSciNetView ArticleGoogle Scholar
  2. Kogoj AE, Lanconelli E: One-side Liouville theorems for a class of hypoelliptic ultraparabolic equations. In Geometric Analysis of PDE and Several Complex Variables, Contemporary Math.. Volume 368. American Mathematical Society, Providence, RI, USA; 2005:305–312.View ArticleGoogle Scholar
  3. Kogoj AE, Lanconelli E: Liouville theorems in halfspaces for parabolic hypoelliptic equations. Ricerche di Matematica 2006,55(2):267–282.MATHMathSciNetView ArticleGoogle Scholar
  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 2005, 29: 243–256.MathSciNetGoogle Scholar
  5. Luo X: Liouville's theorem for homogeneous differential operators. Communications in Partial Differential Equations 1997,22(11–12):1837–1848. 10.1080/03605309708821322MATHMathSciNetView ArticleGoogle Scholar
  6. Lanconelli E, Pascucci A: Superparabolic functions related to second order hypoelliptic operators. Potential Analysis 1999,11(3):303–323. 10.1023/A:1008689803518MATHMathSciNetView ArticleGoogle Scholar
  7. Amano K: Maximum principles for degenerate elliptic-parabolic operators. Indiana University Mathematics Journal 1979,28(4):545–557. 10.1512/iumj.1979.28.28038MATHMathSciNetView ArticleGoogle Scholar
  8. Glagoleva RJa: Liouville theorems for the solution of a second order linear parabolic equation with discontinuous coefficients. Matematicheskie Zametki 1969,5(5):599–606.MATHMathSciNetGoogle Scholar
  9. Bear HS: Liouville theorems for heat functions. Communications in Partial Differential Equations 1986,11(14):1605–1625. 10.1080/03605308608820476MATHMathSciNetView ArticleGoogle Scholar
  10. Bonfiglioli A, Lanconelli E: Liouville-type theorems for real sub-Laplacians. Manuscripta Mathematica 2001,105(1):111–124. 10.1007/PL00005872MATHMathSciNetView ArticleGoogle Scholar
  11. Lanconelli E, Polidoro S: On a class of hypoelliptic evolution operators. Rendiconti Seminario Matematico Università e Politecnico di Torino 1994,52(1):29–63.MATHMathSciNetGoogle Scholar
  12. Priola E, Zabczyk J: Liouville theorems for non-local operators. Journal of Functional Analysis 2004,216(2):455–490. 10.1016/j.jfa.2004.04.001MATHMathSciNetView ArticleGoogle Scholar
  13. Kogoj AE, Lanconelli E: Link of groups and applications to PDE's. to appear in Proceedings of the American Mathematical SocietyGoogle Scholar


© A. E. Kogoj and E. Lanconelli 2007

This article is published under license to BioMed Central Ltd. 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.