Communications in Mathematical Sciences

Volume 21 (2023)

Number 2

Generalized integral equation method for an elliptic nonlocal equation in measure space

Pages: 379 – 404



Chunxiong Zheng (College of Mathematics and Systems Science, Xinjiang University, Urumqi, China; and Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Jia Yin (Department of Mathematical Sciences, Tsinghua University, Beijing, China)


A solution strategy, called generalized integral equation method, is proposed to solve a class of elliptic nonlocal equations in measure space, within which both the continuous and discrete nonlocal problems can be taken as specific instances. By extracting the main ingredients of integral equation method, we develop a generalized integral equation method in an abstract operator framework. As a matter of fact, the classic integral equation method for continuous local partial differential equations can be categorized into this framework. The key ingredient of the proposed method is to derive the generalized boundary integral equations, which can be coupled appropriately with the interior operator equation to obtain a reduced problem. We prove that the resulting system is well-posed by showing that it admits an equivalent formulation with strong coercivity, and the solution of the reduced problem is the same as that of the original one. The proposed method is applied to a nonlocal equation in two-dimensional space discretized by an asymptotically compatible scheme. Numerical experiments validate the effectiveness.


integral equation method, elliptic nonlocal equations, measure space, asymptotic compatibility

2010 Mathematics Subject Classification

31A10, 65N22, 65N80, 65R20

Received 6 July 2021

Received revised 14 February 2022

Accepted 29 May 2022

Published 1 February 2023