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On the Weyl Formula for Obstacles

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Partial Differential Equations and Mathematical Physics

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 21))

Abstract

In this paper a trace perturbation formula for scattering by obstacles is proved. The main application of this formula concerns high energy asymptotics for scattering phases (or Krein spectral shift functions). In particular we prove a Weyl type asymptotic for obstacle problems with arbitrary geometry, in any dimension, extending a result proved by R. Melrose in odd dimension.

For example, let us consider in the Euclidean plane ℝ2 a compact obstacle K and the Laplace-Dirichlet problem in the domain Ω = ℝ2\K, denoted by P Ω. Then the scattering phase s(λ) for the pair \(({P_\Omega },{P_{{\mathbb{R}^2}}})\) satisfies

$$s(\lambda ) = \frac{{Area(K)}}{{4\pi }}\lambda + o(\lambda )as\lambda \to + \infty $$

, without any smoothness assumption on K. If the boundary of K is smooth then we have the remainder term estimate:

$$s(\lambda ) = \frac{{Area(K)}}{{4\pi }}\lambda + o({\lambda ^{1/2}})as\lambda \to + \infty $$

. More generally, we prove that similar results hold for a large class of perturbations of elliptic operators at infinity which may be degenerate in a bounded set.

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Authors and Affiliations

  1. Department of Mathematics, CNRS-URA 758, University of Nantes, 2, rue de la Houssinière, F-44072, Nantes Cédex 03, France

    Didier Robert

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  1. Didier Robert
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Editors and Affiliations

  1. Department of Mathematics, University of Lund, S-221 00, Lund, Sweden

    Lars Hörmander  & Anders Melin  & 

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© 1996 Springer Science+Business Media New York

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Robert, D. (1996). On the Weyl Formula for Obstacles. In: Hörmander, L., Melin, A. (eds) Partial Differential Equations and Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications, vol 21. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0775-7_18

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  • DOI: https://doi.org/10.1007/978-1-4612-0775-7_18

  • Publisher Name: Birkhäuser, Boston, MA

  • Print ISBN: 978-1-4612-6897-0

  • Online ISBN: 978-1-4612-0775-7

  • eBook Packages: Springer Book Archive

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