Nonlocal half-ball vector operators on bounded domains: Poincaré inequality and its applications. (English) Zbl 1537.46028
Summary: 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.
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 26A33 | Fractional derivatives and integrals |
Keywords:
nonlocal models; nonlocal vector calculus; Poincaré inequality; Helmholtz decomposition; bounded domains; nonlocal half-ball gradient operator; Riesz fractional gradient; Marchaud fractional derivative; peridynamics; correspondence modelReferences:
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