Skip to content


  • Research Article
  • 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:


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.


  • Differential Equation
  • Partial Differential Equation
  • Ordinary Differential Equation
  • Functional Equation
  • Characteristic Form


Authors’ Affiliations

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


  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