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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.

References [Enhancements On Off] (What's this?)

  • 1. K. O. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333–418. MR 0100718
  • 2. V. Georgiev, Disappearing solutions for dissipative hyperbolic systems of constant multiplicity, Hokkaido Math. J. 15 (1986), no. 3, 357–385. MR 866953, 10.14492/hokmj/1381518233
  • 3. P. D. Lax and R. S. Phillips, Local boundary conditions for dissipative symmetric linear differential operators, Comm. Pure Appl. Math. 13 (1960), 427–455. MR 0118949
  • 4. Andrew Majda, Disappearing solutions for the dissipative wave equation, Indiana Univ. Math. J. 24 (1974/75), no. 12, 1119–1133. MR 0385330
  • 5. Vesselin Petkov, Scattering theory for hyperbolic operators, Studies in Mathematics and its Applications, vol. 21, North-Holland Publishing Co., Amsterdam, 1989. MR 1028780

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Additional Information

Ferruccio Colombini
Affiliation: Dipartimento di Matematica, Università di Pisa, Pisa, Italia

Vesselin Petkov
Affiliation: Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France

Jeffrey Rauch
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109-1043

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.