Zeitschrift für angewandte Mathematik und Physik

, Volume 66, Issue 5, pp 2173–2196 | Cite as

Spectrum of a diffusion operator with coefficient changing sign over a small inclusion

  • Lucas ChesnelEmail author
  • Xavier Claeys
  • Sergei A. Nazarov


We study a spectral problem \({(\mathscr{P}^{\delta})}\) for a diffusion-like equation in a 3D domain \({\Omega }\). The main originality lies in the presence of a parameter \({\sigma^{\delta}}\), whose sign changes on \({\Omega }\), in the principal part of the operator we consider. More precisely, \({\sigma^{\delta}}\) is positive on \({\Omega }\) except in a small inclusion of size \({\delta > 0}\). Because of the sign change of \({\sigma^{\delta}}\), for all \({\delta > 0}\), the spectrum of \({(\mathscr{P}^{\delta})}\) consists of two sequences converging to \({\pm\infty}\). However, at the limit \({\delta=0}\), the small inclusion vanishes so that there should only remain positive spectrum for \({(\mathscr{P}^{\delta})}\). What happens to the negative spectrum? In this paper, we prove that the positive spectrum of \({(\mathscr{P}^{\delta})}\) tends to the spectrum of the problem without the small inclusion. On the other hand, we establish that each negative eigenvalue of \({(\mathscr{P}^{\delta})}\) behaves like \({\delta^{-2} \mu}\) for some constant \({\mu < 0}\). We also show that the eigenfunctions associated with the negative eigenvalues are localized around the small inclusion. We end the article providing 2D numerical experiments illustrating these results.


Negative materials Small inclusion Plasmonics Metamaterial Sign-changing coefficients Eigenvalues Asymptotics Singular perturbation 

Mathematics Subject Classification

35H99 35P20 35P15 35Q60 65N25 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Lucas Chesnel
    • 1
    Email author
  • Xavier Claeys
    • 2
  • Sergei A. Nazarov
    • 3
    • 4
    • 5
  1. 1.Centre de Mathématiques AppliquéesBureau 2029, École PolytechniquePalaiseau CedexFrance
  2. 2.Laboratory Jacques Louis LionsUniversity Pierre et Marie CurieParisFrance
  3. 3.Faculty of Mathematics and MechanicsSt. Petersburg State UniversitySt. PetersburgRussia
  4. 4.Laboratory for mechanics of new nanomaterialsSt. Petersburg State Polytechnical UniversitySt. PetersburgRussia
  5. 5.Laboratory of mathematical methods in mechanics of materialsInstitute of Problems of Mechanical EngineeringSt. PetersburgRussia

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