Probabilistic approach to the Dirichlet problem of second order elliptic PDE

Abstract

In this paper we provide a probabilistic approach to the following Dirichlet problem

$$\left\{ \begin{gathered} \left( {\sum {\frac{\partial }{{\partial x^i }}} \left( {a^{ij} \frac{\partial }{{\partial x^j }}} \right) + \sum {b^i \frac{\partial }{{\partial x^i }} + \xi } } \right)u = 0,inD, \hfill \\ u = g,on\partial D, \hfill \\ \end{gathered} \right.$$

without assuming that the eigenvalues of the operator

$$\sum {\frac{\partial }{{\partial x^i }}} \left( {a^{ij} \frac{\partial }{{\partial x^j }}} \right) + \sum {b^i \frac{\partial }{{\partial x^i }} + \xi }$$

with Dirichlet boundary conditions are all strictly negative. The results of this paper generalized those of Ma^[10].