Nonlocal vector calculus on the sphere
Analysis of PDEs
2025-05-20 v1
Abstract
We introduce a nonlocal vector calculus on the unit two-sphere using weakly singular integral operators. Within this framework, the operators are diagonalizable in terms of scalar and vector spherical harmonics, a property that facilitates the proof of a nonlocal Stokes theorem. This constitutes the first instance of such a theorem on a curved surface. Furthermore, our analysis demonstrates the strong convergence of these nonlocal operators to the classical differential operators of vector calculus as the interaction range tends to zero.
Cite
@article{arxiv.2505.12372,
title = {Nonlocal vector calculus on the sphere},
author = {Hadrien Montanelli and Richard Mikael Slevinsky and Qiang Du},
journal= {arXiv preprint arXiv:2505.12372},
year = {2025}
}