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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Incoming and disappearing solutions for Maxwell's equations


Authors: Ferruccio Colombini, Vesselin Petkov and Jeffrey Rauch
Journal: Proc. Amer. Math. Soc. 139 (2011), 2163-2173
MSC (2010): Primary 35Q61; Secondary 35P25, 35L45
Published electronically: February 4, 2011
MathSciNet review: 2775394
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Abstract: We prove that in contrast to the free wave equation in $ \mathbb{R}^3$ there are no incoming solutions of Maxwell's equations in the form of spherical or modulated spherical waves. We construct solutions which are corrected by lower order incoming waves. With their aid, we construct dissipative boundary conditions and solutions to Maxwell's equations in the exterior of a sphere which decay exponentially as $ t \to +\infty$. They are asymptotically disappearing. Disappearing solutions which are identically zero for $ t \geq T > 0$ are constructed which satisfy maximal dissipative boundary conditions which depend on time $ t$. Both types are invisible in scattering theory.


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Additional Information

Ferruccio Colombini
Affiliation: Dipartimento di Matematica, Università di Pisa, Pisa, Italia
Email: colombini@dm.unipi.it

Vesselin Petkov
Affiliation: Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France
Email: petkov@math.u-bordeaux1.fr

Jeffrey Rauch
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043
Email: rauch@umich.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10885-2
PII: S 0002-9939(2011)10885-2
Keywords: Maxwell equations, disappearing solutions, dissipative boundary conditions.
Received by editor(s): June 12, 2010
Published electronically: February 4, 2011
Additional Notes: The third author’s research was partially supported by the National Science Foundation under grant NSF DMS 0405899
Communicated by: Walter Craig
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.