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 \(\mathbb {R}^n\), 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.
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Notes
Note that, in contrast with other existing nonlocal theories, the fact that both input and output are one-point functions, does not raise any difficulties in applying the same operator multiple times to the same function.
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Funding
The first author (A. H.) was supported by a University of Nebraska-Lincoln UCARE award. The second author (P. R.) was supported by NSF-DMS 1716790.
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Petronela Radu authors contributed equally to this work.
This article is part of the section “Theory of PDEs” edited by Eduardo Teixeira.
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Haar, A., Radu, P. A new nonlocal calculus framework. Helmholtz decompositions, properties, and convergence for nonlocal operators in the limit of the vanishing horizon. Partial Differ. Equ. Appl. 3, 43 (2022). https://doi.org/10.1007/s42985-022-00178-z
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DOI: https://doi.org/10.1007/s42985-022-00178-z