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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Transmission boundary problems for Dirac operators on Lipschitz domains and applications to Maxwell's and Helmholtz's equations


Authors: Emilio Marmolejo-Olea, Irina Mitrea, Marius Mitrea and Qiang Shi
Journal: Trans. Amer. Math. Soc. 364 (2012), 4369-4424
MSC (2010): Primary 30G35, 35C15, 35F15, 35J56, 42B20, 42B30, 42B37; Secondary 30E20, 31B10, 35F45, 35J25, 45B05, 65N80
Published electronically: March 29, 2012
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Abstract: The transmission boundary value problem for a perturbed Dirac operator on arbitrary bounded Lipschitz domains in $ \mathbb{R}^3$ is formulated and solved in terms of layer potentials of Clifford-Cauchy type. As a byproduct of this analysis, an elliptization procedure for the Maxwell system is devised which allows us to show that the Maxwell and Helmholtz transmission boundary value problems are well-posed as a corollary of the unique solvability of this more general Dirac transmission problem.


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Emilio Marmolejo-Olea
Affiliation: Instituto de Matemáticas Unidad Cuernavaca, Universidad Nacional Autónoma de México, A.P. 273-3 Admon. 3, Cuernavaca, Morelos, 62251, México
Email: emilio@matcuer.unam.mx

Irina Mitrea
Affiliation: Department of Mathematics, Temple University, 1805 N. Broad Street, Philadelphia, Pennsylvania 19122
Email: imitrea@temple.edu

Marius Mitrea
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: mitream@missouri.edu

Qiang Shi
Affiliation: Department of Mathematics, Computer Science and Economics, Emporia State University, Emporia, Kansas 66801
Email: qshi@emporia.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05606-6
PII: S 0002-9947(2012)05606-6
Keywords: Transmission boundary value problems, Lipschitz domains, Dirac operator, Maxwell system, Helmholtz operator, Hardy spaces, Cauchy operator, boundary layer potentials, Clifford algebras, Clifford analysis
Received by editor(s): March 13, 2010
Received by editor(s) in revised form: April 15, 2011
Published electronically: March 29, 2012
Article copyright: © Copyright 2012 American Mathematical Society