An Exterior Neumann Boundary-Value Problem for the Div-Curl System and Applications

We investigate a generalization of the equation curlw→=g→ to an arbitrary number n of dimensions, which is based on the well-known Moisil–Teodorescu differential operator. Explicit solutions are derived for a particular problem in bounded domains of Rn using classical operators from Clifford analysi...

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Bibliographic Details
Published in:Mathematics (Basel) 2021-07, Vol.9 (14), p.1609
Main Authors: Delgado, Briceyda B., Macías-Díaz, Jorge Eduardo
Format: Article
Language:English
Subjects:
Algebra
Boundary conditions
Boundary value problems
Decomposition
Differential equations
div-curl system
Domains
exterior domains
harmonic functions
Lamé–Navier equation
Laplace equation
Mathematical analysis
Mathematical functions
maximum principle
Neumann boundary-value problem
Neumann problem
Operators (mathematics)
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Summary:We investigate a generalization of the equation curlw→=g→ to an arbitrary number n of dimensions, which is based on the well-known Moisil–Teodorescu differential operator. Explicit solutions are derived for a particular problem in bounded domains of Rn using classical operators from Clifford analysis. In the physically significant case n=3, two explicit solutions to the div-curl system in exterior domains of R3 are obtained following different constructions of hyper-conjugate harmonic pairs. One of the constructions hinges on the use of a radial integral operator introduced recently in the literature. An exterior Neumann boundary-value problem is considered for the div-curl system. That system is conveniently reduced to a Neumann boundary-value problem for the Laplace equation in exterior domains. Some results on its uniqueness and regularity are derived. Finally, some applications to the construction of solutions of the inhomogeneous Lamé–Navier equation in bounded and unbounded domains are discussed.
ISSN:2227-7390
2227-7390
DOI:10.3390/math9141609