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Scattering Problems for Symmetric Systems with Dissipative Boundary Conditions

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Studies in Phase Space Analysis with Applications to PDEs

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 84))

Abstract

We study symmetric systems with dissipative boundary conditions. The solutions of the mixed problems for such systems are given by a contraction semigroup \(V (t)f = {e}^{tG_{b}}f,\:t \geq 0\), and the solutions \(u = {e}^{tG_{b}}f\) with eigenfunctions f of the generator G b with eigenvalues λ, Reλ < 0, are called asymptotically disappearing (ADS). We prove that the wave operators are not complete if there exist (ADS). This is the case for Maxwell system with special boundary conditions in the exterior of the sphere. We obtain a representation of the scattering kernel, and we examine the inverse backscattering problem related to the leading term of the scattering kernel.

2010 Mathematics Subject Classification: Primary: 35P25; Secondary: 47A40, 35L50, 81U40

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Authors and Affiliations

  1. Institut de Mathématiques de Bordeaux, 351 Cours de la Libération, 33405, Talence, France

    Vesselin Petkov

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  1. Vesselin Petkov
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Correspondence to Vesselin Petkov .

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Editors and Affiliations

  1. , Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato, 5, Bologna, I-40126, Italy

    Massimo Cicognani

  2. , Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo, 5, Pisa, I-56127, Italy

    Ferruccio Colombini

  3. , Dipartimento di Matematica e Geoscienze, Università di Trieste, Via Valerio, 12/1 12/1, Trieste, I-34127, Italy

    Daniele Del Santo

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Petkov, V. (2013). Scattering Problems for Symmetric Systems with Dissipative Boundary Conditions. In: Cicognani, M., Colombini, F., Del Santo, D. (eds) Studies in Phase Space Analysis with Applications to PDEs. Progress in Nonlinear Differential Equations and Their Applications, vol 84. Birkhäuser, New York, NY. https://doi.org/10.1007/978-1-4614-6348-1_15

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