The complement value problem for non-local operators

Abstract

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

References in corpus (3)

• Fractional Laplacians on domains, a development of Hörmander's theory of mu-transmission pseudodifferential operators
• Boundary Harnack inequality for Markov processes with jumps
• Two Theorems on Hunt's Hypothesis (H) for Markov Processes

Cited by in corpus (1)

• The complement value problem for a class of second order elliptic integro-differential operators