Communications in Mathematical Sciences
Volume 19 (2021)
A nonlocal model of elliptic equation with jump coefficients on manifold
Pages: 1881 – 1912
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.
elliptic transmission problem, manifold with interface, nonlocal interface model, wellposedness, vanishing nonlocality limit
2010 Mathematics Subject Classification
35A23, 45A05, 45P05, 46E35
This work was supported by NSFC grant 12071244.
Received 12 July 2020
Accepted 21 February 2021
Published 7 September 2021