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

Abstract

We consider the complement value problem for a class of second order elliptic integro-differential operators. Let $D$ be a bounded Lipschitz domain of ${\mathbb{R}}^d$. Under mild conditions, we show that there exists a unique bounded continuous weak solution to the following equation $$\{\begin{array}{l}(\Delta +a^{\alpha }{\Delta }^{\alpha /2}+b\cdot \nabla +c+\mathrm{div}\hat{b})u+f=0\mathrm{in}D,\\ u=g\mathrm{on}{D}^c.\end{array}$$ Moreover, we give an explicit probabilistic representation of the solution. The recently developed stochastic calculus for Markov processes associated with semi-Dirichlet forms and heat kernel estimates play important roles in our approach.

References in corpus (1)

• The complement value problem for non-local operators