Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Finite-energy solutions of mixed 3d div-curl systems

Authors: Giles Auchmuty and James C. Alexander
Journal: Quart. Appl. Math. 64 (2006), 335-357
MSC (2000): Primary 35J50, 35Q60, 78A30
DOI: https://doi.org/10.1090/S0033-569X-06-01015-8
Published electronically: April 6, 2006
MathSciNet review: 2243867
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Abstract: This paper describes the existence and representation of certain finite energy $ (L^2$-)solutions of weighted div-curl systems on bounded 3D regions with $ C^2$-boundaries and mixed boundary data. Necessary compatibility conditions on the data for the existence of solutions are described. Subject to natural integrability assumptions on the data, it is then shown that there exist $ L^2$-solutions whenever these compatibility conditions hold. The existence results are proved by using a weighted orthogonal decomposition theorem for $ L^2$-vector fields in terms of scalar and vector potentials. This representation theorem generalizes the classical Hodge-Weyl decomposition. With this special choice of the potentials, the mixed div-curl problem decouples into separate problems for the scalar and vector potentials. Variational principles for the solutions of these problems are described. Existence theorems, and some estimates, for the solutions of these variational principles are obtained. The unique solution of the mixed system that is orthogonal to the null space of the problem is found and the space of all solutions is described.

The second part of the paper treats issues concerning the non-uniqueness of solutions of this problem. Under additional assumptions, this space is shown to be finite dimensional and a lower bound on the dimension is described. Criteria that prescribe the harmonic component of the solution are investigated. Extra conditions that determine a well-posed problem for this system on a simply connected region are given. A number of conjectures regarding the results for bounded regions with handles are stated.

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Additional Information

Giles Auchmuty
Affiliation: Division of Mathematical Sciences, National Science Foundation, 4201 Wilson Blvd., Arlington, Virginia 22230
Email: gauchmut@nsf.gov

James C. Alexander
Affiliation: Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, Ohio 44106-7058
Email: james.alexander@case.edu

DOI: https://doi.org/10.1090/S0033-569X-06-01015-8
Received by editor(s): November 2, 2005
Published electronically: April 6, 2006
Article copyright: © Copyright 2006 Brown University
The copyright for this article reverts to public domain 28 years after publication.