The complement value problem for non-local operators

Let $D$ be a bounded Lipschitz domain of ${\mathbb{R}}^d$. We consider the complement value problem $$\{\begin{array}{l}(\mathrm{\Delta }+a^{\alpha }{\mathrm{\Delta }}^{\alpha \mathrm{/}2}+b\cdot \mathrm{\nabla }+c)u+f=0\mathrm{i}\mathrm{n}D,\\ u=g\mathrm{o}\mathrm{n}{D}^c\mathrm{.}\end{array}$$ 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 our approach.