The complement value problem for non-local operators

@article{Sun2018TheCV,
  title={The complement value problem for non-local operators},
  author={Wei Sun},
  journal={arXiv: Probability},
  year={2018}
}
  • W. Sun
  • Published 31 March 2018
  • Mathematics
  • arXiv: Probability
Let $D$ be a bounded Lipschitz domain of $\mathbb{R}^d$. We consider the complement value problem $$ \left\{\begin{array}{l}(\Delta+a^{\alpha}\Delta^{\alpha/2}+b\cdot\nabla+c)u+f=0\ \ {\rm in}\ D,\\ u=g\ \ {\rm on}\ D^c. \end{array}\right.$$ Under mild conditions, we show that there exists a unique bounded continuous weak solution. Moreover, we give an explicit probabilistic representation of the solution. The theory of semi-Dirichlet forms and heat kernel estimates play an important role in… 
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