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This work contributes to nonlocal vector calculus as an indispensable mathematical tool for the study of nonlocal models that arises in a variety of applications. We define the nonlocal half-ball gradient, divergence and curl operators with general kernel functions (integrable or fractional type with finite or infinite supports) and study the associated nonlocal vector identities. We study the nonlocal function space on bounded domains associated with zero Dirichlet boundary conditions and the half-ball gradient operator and show it is a separable Hilbert space with smooth functions dense in it. A major result is the nonlocal Poincaré inequality, based on which a few applications are discussed, and these include applications to nonlocal convection–diffusion, nonlocal correspondence model of linear elasticity and nonlocal Helmholtz decomposition on bounded domains.
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AN INVITATION TO NONLOCAL MODELING, ANALYSIS AND COMPUTATION
This lecture serves as an invitation to further studies on nonlocal models, their mathematics, computation, and applications. We sample our recent attempts in the development of a systematic mathematical framework for nonlocal models, including basic elements of nonlocal vector calculus, well-posedness of nonlocal variational problems, coupling to local models, convergence and compatibility of numerical approximations, and applications to nonlocal mechanics and diffusion. We also draw connections with traditional models and other relevant mathematical subjects.
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- Award ID(s):
- 1719699
- PAR ID:
- 10100540
- Date Published:
- Journal Name:
- Proceedings of the International Congress of Mathematicians (ICM 2018)
- Volume:
- 3
- Page Range / eLocation ID:
- 3523-3552
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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