Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Bounds and representations of solutions of planar div-curl problems


Author: Giles Auchmuty
Journal: Quart. Appl. Math. 75 (2017), 505-524
MSC (2010): Primary 35J05; Secondary 35J56, 35Q60, 35C99
DOI: https://doi.org/10.1090/qam/1463
Published electronically: March 15, 2017
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Abstract: Estimates and representations of solutions of div-curl systems for planar vector fields are described. Potentials are used to represent solutions as the sum of fields that depend on the source terms and harmonic fields dependent on the boundary data. Sharp 2-norm (energy) bounds for the least energy solutions on bounded regions with Lipschitz boundary are found. Prescribed flux, tangential or mixed flux and tangential boundary conditions require different potentials. The harmonic fields are represented and estimated using Steklov eigenfunctions. Some regularity results are obtained.


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Additional Information

Giles Auchmuty
Affiliation: Department of Mathematics, University of Houston, Houston, Texas 77204-3008
Email: auchmuty@uh.edu

DOI: https://doi.org/10.1090/qam/1463
Keywords: Div-curl, planar vector fields, stream function, potential representations, harmonic fields
Received by editor(s): January 3, 2017
Published electronically: March 15, 2017
Additional Notes: The author gratefully acknowledges research support by NSF award DMS 11008754
Article copyright: © Copyright 2017 Brown University


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