Acta Mathematica Sinica

, Volume 12, Issue 1, pp 18–32 | Cite as

Dirichlet problems for the quasilinear second order subelliptic equations

  • Xu Chaojiang
Article

Abstract

In this paper, we study the Dirichlet problems for the following quasilinear second order sub-elliptic equation,
$$\left\{ {\begin{array}{*{20}c} {\sum\limits_{i,j = 1}^m {X_i^* (A_{i,j} (x,u)X_j u) + \sum\limits_{j = 1}^m {B_j (x,u)X_j u + C(x,u) = 0in\Omega ,} } } \\ {u = \varphi on\partial \Omega ,} \\ \end{array} } \right.$$
whereX={X1, ...,X m } is a system of real smooth vector fields which satisfies the Hörmander's condition,A i,j ,B j ,CC(\(\bar \Omega\)×R) and (A i,j (x,z)) is a positive definite matrix. We have proved the existence and the maximal regularity of solutions in the “non-isotropic” Hölder space associated with the system of vector fieldsX.

Keywords

Sub-elliptic equation Dirichlet problem A priori estimate 

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Xu Chaojiang
    • 1
  1. 1.Department of MathematicsWuhan UniversityWuhanChina

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