Communications in Mathematical Sciences

Volume 19 (2021)

Number 7

A nonlocal model of elliptic equation with jump coefficients on manifold

Pages: 1881 – 1912

DOI: https://dx.doi.org/10.4310/CMS.2021.v19.n7.a6

Authors

Yajie Zhang (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Zuoqiang Shi (Department of Mathematical Sciences, Tsinghua University, Beijing, China)

Abstract

There has been extensive study for partial differential equations on manifolds in different subjects. In this paper, a nonlocal model of the elliptic transmission problem on manifolds is introduced. In such an elliptic problem, the coefficient of the Laplacian is allowed to have jump across a smooth interface. Additional constraints of solution are imposed on either side of the interface, named transmission conditions. In this paper, the transmission condition is approximated by a nonlocal average of the solution along the interface. The coercivity of the nonlocal model is sustained due to a Poincaré-type inequality. Based on this good property, well-posedness and the convergence rate of the nonlocal model can be proved.

Keywords

elliptic transmission problem, manifold with interface, nonlocal interface model, wellposedness, vanishing nonlocality limit

2010 Mathematics Subject Classification

35A23, 45A05, 45P05, 46E35

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This work was supported by NSFC grant 12071244.

Received 12 July 2020

Accepted 21 February 2021

Published 7 September 2021