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General Solution of the Inhomogeneous Div-Curl System and Consequences

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Abstract

We consider the inhomogeneous div-curl system (i.e. to find a vector field with prescribed div and curl) in a bounded star-shaped domain in \(\mathbb {R}^3\). An explicit general solution is given in terms of classical integral operators, completing previously known results obtained under restrictive conditions. This solution allows us to solve questions related to the quaternionic main Vekua equation \(DW=(Df/f)\overline{W}\) in \(\mathbb {R}^3\), such as finding the vector part when the scalar part is known. In addition, using the general solution to the div-curl system and the known existence of the solution of the inhomogeneous conductivity equation, we prove the existence of solutions of the inhomogeneous double curl equation, and give an explicit solution for the case of static Maxwell’s equations with only variable permeability.

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Authors and Affiliations

  1. Departamento de Matemáticas Cinvestav-Querétaro, Libramiento Norponiente 2000 Fracc. Real de Juriquilla, C.P. 76230, Santiago de Queretaro, Mexico

    Briceyda B. Delgado & R. Michael Porter

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  1. Briceyda B. Delgado
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  2. R. Michael Porter
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Correspondence to R. Michael Porter.

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Communicated by Wolfgang Sprössig.

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Delgado, B.B., Porter, R.M. General Solution of the Inhomogeneous Div-Curl System and Consequences. Adv. Appl. Clifford Algebras 27, 3015–3037 (2017). https://doi.org/10.1007/s00006-017-0805-z

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