A geometric approximation of non-local interface and boundary conditions

TL;DR

This paper studies how to approximate Laplacian operators with non-local interface and boundary conditions using Neumann Laplacians on specially structured manifolds, showing convergence of spectra, eigenspaces, and semigroups.

Contribution

It introduces a geometric approximation method for non-local interface conditions, proving resolvent convergence and spectrum stability, and provides explicit examples and extensions to Robin boundary conditions.

Findings

Proves resolvent convergence of approximating operators.

Demonstrates convergence of spectra and eigenspaces.

Provides explicit manifold examples for prescribed kernels.

Abstract

We analyze an approximation of a Laplacian subject to non-local interface conditions of a $\delta'$-type by Neumann Laplacians on a family of Riemannian manifolds with a sieve-like structure. We establish a (kind of) resolvent convergence for such operators, which in turn implies the convergence of spectra and eigenspaces, and demonstrate convergence of the corresponding semigroups. Moreover, we provide an explicit example of a manifold allowing to realize any prescribed integral kernel appearing in that interface conditions. Finally, we extend the discussion to similar approximations for the Laplacian with non-local Robin-type boundary conditions.