Nonlocal Boundary-Value Problems with Local Boundary Conditions

Abstract

We study nonlocal integro-differential equations on bounded domains with finite-range nonlocal interactions that are localized at the boundary. Through estimates of boundary localized convolutions, we establish a nonlocal Green's identity, which leads to the well-posedness of these nonlocal problems with various types of classical local boundary conditions. We investigate the regularity properties of the nonlocal problems with rough data and prove the convergence of solutions of these nonlocal boundary-value problems with their classical local counterparts in the local limit. The Poisson data for the local boundary-value problem is permitted to be quite irregular, belonging to the dual of the classical Sobolev space. Heterogeneously mollifying such Poisson data for the local problem on the same length scale as the range of nonlocal interactions, we show that the solutions to the nonlocal problem with the mollified Poisson data converge to the variational solution of the classical Poisson problem with original data.

Locations

• arXiv (Cornell University) - View - PDF