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

We investigate a generalization of the equation curl (w) over right arrow = (g) over right arrow 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 R-3 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 Lame-Navier equation in bounded and unbounded domains are discussed. Data Set License: (CC-BY-NC)