The complement value problem for nonlocal operators
Abstract
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 semiDirichlet forms and heat kernel estimates play an important role in our approach.
 Publication:

arXiv eprints
 Pub Date:
 March 2018
 arXiv:
 arXiv:1804.00212
 Bibcode:
 2018arXiv180400212S
 Keywords:

 Mathematics  Probability