The $C^{\alpha}$ regularity of weak solutions of ultraparabolic equations

Pages: 1261 - 1275, Volume 29, Issue 3, March 2011

doi:10.3934/dcds.2011.29.1261       Abstract        References        Full Text (399.1K)       Related Articles

Wendong Wang - Institute of Mathematics, AMSS, Academia Sinica, Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, China (email)
Liqun Zhang - Institute of Mathematics, AMSS, Academia Sinica, Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, 100190, China (email)

Abstract: We obtained the $C^{\alpha}$ continuity of weak solutions for a class of ultraparabolic equations with measurable coefficients of the form

$\partial_t \ u= \sum_{i,j=1}^{m_0}X_i(a_{ij}(x,t)X_j\ u )+X_0 u.$

By choosing a new cut-off function, we simplified and generalized the earlier arguments and proved the $C^{\alpha}$ regularity of weak solutions for more general ultraparabolic equations.

Keywords:  Hypoelliptic, ultraparabolic, $C^{\alpha}$ regularity.
Mathematics Subject Classification:  Primary: 35K70, 35D10; Secondary: 35B65.

Received: November 2009;      Revised: May 2010;      Published: November 2010.

 References