A new nonlocal calculus framework. Helmholtz decompositions, properties, and convergence for nonlocal operators in the limit of the vanishing horizon

Andrew Haar () and Petronela Radu ()
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Andrew Haar: University of Bonn
Petronela Radu: University of Nebraska-Lincoln

Partial Differential Equations and Applications, 2022, vol. 3, issue 3, 1-20

Abstract: Abstract We introduce a new nonlocal calculus framework which parallels (and includes as a limiting case) the differential setting. The integral operators introduced have convolution structures and converge as the horizon of interaction shrinks to zero to the classical gradient, divergence, curl, and Laplacian. Moreover, a Helmholtz-type decomposition holds on the entire $$\mathbb {R}^n$$ R n , so general vector fields can be decomposed into (nonlocal) divergence-free and curl-free components. We also identify the kernels of the nonlocal operators and prove additional properties towards building a nonlocal framework suitable for analysis of integro-differential systems.

Keywords: Nonlocal calculus; Horizon of interaction; Kernels of operators; Helmholtz decomposition; 26A33; 35S30; 41A35; 45A05; 45P05; 46F12; 46N20; 47G10 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s42985-022-00178-z

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