Abstract

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.

Highlights

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  • One of the main results presented in this work (Theorem 5) establishes that a weak solution of the div-curl system in exterior domains star-shaped with respect to (w.r.t.)

  • We present the construction of hyper-conjugate harmonic pairs in unbounded domains in terms of certain layer potentials and give explicit formulas for a solution of the div-curl system without boundary conditions for exterior domains satisfying the strong local Lipschitz condition. We derive another explicit solution of the div-curl system in exterior domains under the geometric condition that Ω− is star-shaped w.r.t. infinity. The construction of this second solution relies on the properties of a family of radial integral operators restricted to a family of harmonic functions with good behavior at infinity

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Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. One of the main results presented in this work (Theorem 5) establishes that a weak solution of the div-curl system in exterior domains star-shaped with respect to (w.r.t.). The construction of this second solution relies on the properties of a family of radial integral operators restricted to a family of harmonic functions with good behavior at infinity. We close the section showing that these weak solutions are strong solutions through an embedding argument

Clifford Algebras
Geometric Properties of the Domain
Clifford Integral Operators
An N-Dimensional Generalization of the Div-Curl System
Unbounded Domains
Teodorescu Transform over Unbounded Domains
The Div-Curl System over Unbounded Domains
Div-Curl System in Exterior Domains
Neumann Boundary-Value Problems
Lamé–Navier Equation
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