Communications in Mathematical Sciences

Volume 17 (2019)

Number 3

Nonlocal approximation of elliptic operators with anisotropic coefficients on manifold

Pages: 859 – 882



Zuoqiang Shi (Department of Mathematical Sciences and the Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


In this paper, we give an integral approximation for the elliptic operators with anisotropic coefficients on smooth manifold. Using the integral approximation, the elliptic equation is transformed to an integral equation. The integral approximation preserves the symmetry and coercivity of the original elliptic operator. Based on these good properties, we prove the convergence between the solutions of the integral equation and the original elliptic equation.


nonlocal approximation, elliptic operator, anisotropic coefficients, point integral method

2010 Mathematics Subject Classification

35A23, 45A05, 45P05

Received 15 January 2018

Accepted 8 March 2019

Published 30 August 2019