# 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} }

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…

## 3 Citations

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

- Mathematics
- 2018

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…

The obstacle problem for quasilinear stochastic integral-partial differential equations

- MathematicsStochastics
- 2019

ABSTRACT We prove the existence and uniqueness of solutions to a kind of quasilinear stochastic integral-partial differential equations with obstacles. Our method is based on the probabilistic…

Stochastic partial integral-differential equations with divergence terms

- Mathematics
- 2020

We study a class of stochastic partial integral-differential equations with an asymmetrical non-local operator 1 2 ∆+a∆ α 2 +b ·∇ and a distribution expressed as divergence of a measurable field. For…

## References

SHOWING 1-10 OF 47 REFERENCES

Local and nonlocal boundary conditions for μ-transmission and fractional elliptic pseudodifferential operators

- Mathematics
- 2014

A classical pseudodifferential operator $P$ on $R^n$ satisfies the $\mu$-transmission condition relative to a smooth open subset $\Omega $, when the symbol terms have a certain twisted parity on the…

ON SHAPE OPTIMIZATION PROBLEMS INVOLVING THE FRACTIONAL LAPLACIAN

- Mathematics, Physics
- 2012

Our concern is the computation of optimal shapes in problems involving $\(-\Delta)^{1/2}$. We focus on the energy $J(\Omega)$ associated to the solution $u_\Omega$ of the basic Dirichlet problem…

A priori estimates for integro-differential operators with measurable kernels

- Mathematics
- 2009

The aim of this work is to develop a localization technique and to establish a regularity result for non-local integro-differential operators $${\fancyscript{L}}$$ of order $${\alpha\in (0,2)}$$ .…

Boundary regularity for fully nonlinear integro-differential equations

- Mathematics
- 2016

We study fine boundary regularity properties of solutions to fully nonlinear elliptic integro-differential equations of order 2s, with s ∈ (0,1). We consider the class of nonlocal operators L∗ ⊂L 0,…

Results on Nonlocal Boundary Value Problems

- Mathematics
- 2010

In this article, we provide a variational theory for nonlocal problems where nonlocality arises due to the interaction in a given horizon. With this theory, we prove well-posedness results for the…

The Dirichlet problem for the fractional Laplacian: Regularity up to the boundary

- Physics, Mathematics
- 2012

Abstract We study the regularity up to the boundary of solutions to the Dirichlet problem for the fractional Laplacian. We prove that if u is a solution of ( − Δ ) s u = g in Ω, u ≡ 0 in R n \ Ω ,…

Jump-type Hunt processes generated by lower bounded semi-Dirichlet forms

- Mathematics
- 2012

Let E be a locally compact separable metric space and m be a positive Radon measure on it. Given a nonnegative function k de- fined on E×E off the diagonal whose anti-symmetric part is assumed to be…

Analysis and Approximation of Nonlocal Diffusion Problems with Volume Constraints

- Mathematics, Computer ScienceSIAM Rev.
- 2012

It is shown that fractional Laplacian and fractional derivative models for anomalous diffusion are special cases of the nonlocal model for diffusion that the authors consider.

The Dirichlet problem for stable-like operators and related probabilistic representations

- Mathematics
- 2016

ABSTRACT We study stochastic differential equations with jumps with no diffusion part, governed by a large class of stable-like operators, which may contain a drift term. For this class of operators,…

Heat kernel estimates for Δ+Δα/2 under gradient perturbation

- Mathematics
- 2014

For α∈(0,2) and M>0, we consider a family of nonlocal operators {Δ+aαΔα/2,a∈(0,M]} on Rd under Kato class gradient perturbation. We establish the existence and uniqueness of their fundamental…