Publication Cover
Applicable Analysis
An International Journal
Volume 92, 2013 - Issue 9
119
Views
14
CrossRef citations to date
0
Altmetric
Articles

A gap in the spectrum of the Neumann–Laplacian on a periodic waveguide

, &
Pages 1889-1915 | Received 26 Oct 2011, Accepted 10 Jul 2012, Published online: 29 Aug 2012

References

  • Wilcox , CH . 1979 . Scattering Theory for Diffraction Gratings , New York : Springer .
  • Birman , MS and Solomyak , MZ . 1986 . Spectral Theory of Self-adjoint Operators in Hilbert Space , Dordrecht : Reidel Publishing Company .
  • Gelfand , IM . 1950 . Expansions in eigenfunctions of an equation with periodic coefficients . Dokl. Acad. Nauk SSSR , 73 : 1117 – 1120 . (in Russian)
  • Kuchment , P . 1993 . Floquet Theory for Partial Differential Equations , Basel : Birkhäuser Verlag .
  • Nazarov , SA and Plamenevsky , BA . 1991 . Elliptic Problems in Domains With Piecewise Smooth Boundaries , Moscow : Nauka . (English trans.: Walter de Gruyter, Berlin, 1994.)
  • Figotin , A and Kuchment , P . 1996 . Band-gap structure of spectra of periodic dielectric and acoustic media. I. Scalar model . SIAM J. Appl. Math. , 56 : 68 – 88 .
  • Friedlander , L . 2002 . On the density of states of periodic media in the large coupling limit . Comm. Partial Differ. Eqns , 27 : 355 – 380 .
  • Green , EL . 1997 . Spectral theory of Laplace-Beltrami operators with periodic metrics . J. Differ. Eqns , 133 : 15 – 29 .
  • Hempel , R and Lineau , K . 2000 . Spectral properties of the periodic media in large coupling limit . Comm. Partial Differ. Eqns , 25 : 1445 – 1470 .
  • Zhikov , VV . 2004 . On gaps in the spectrum of some divergence elliptic operators with periodic coefficients . Algebra i Analiz , 16 ( 5 ) : 34 – 58 . (in Russian)
  • Filonov , N . 2003 . Gaps in the spectrum of the Maxwell operator with periodic coefficients . Comm. Math. Physics , 240 : 161 – 170 .
  • Nazarov , SA . 2010 . A gap in the essential spectrum of an elliptic formally self-adjoint system of differential equation . Differ. Eqns , 46 ( 5 ) : 730 – 741 .
  • Cardone , G , Nazarov , SA and Perugia , C . 2009 . A gap in the continuous spectrum of a cylindrical waveguide with a periodic perturbation of the surface . Math. Nachr. , 24 : 1222 – 1244 .
  • Friedlander , L and Solomyak , M . 2008 . On the spectrum of narrow periodic waveguides . Russ. J. Math. Phys. , 15 ( 2 ) : 238 – 242 .
  • Nazarov , SA . 2010 . Opening a gap in the continuous spectrum of a periodically perturbed waveguide . Math. Notes , 87 ( 5 ) : 959 – 980 .
  • Yoshitomi , K . 1998 . Band gap of the spectrum in periodically curved quantum waveguides . J. Differ. Eqns , 142 ( 1 ) : 123 – 166 .
  • Nazarov , SA . 2009 . A gap in the essential spectrum of the Neumann problem for an elliptic system in a periodic domain . Funkt. Anal. i Prilozhen. , 43 ( 3 ) : 92 – 95 . (English trans.: Funct. Anal. Appl. 43(3) (2009)
  • Nazarov , SA . 2010 . On the plurality of gaps in the spectrum of a periodic waveguide . Mat. sbornik. , 201 ( 4 ) : 99 – 124 . (English transl.: Sb. Math. 201(4) (2010)
  • Nazarov , SA , Ruotsalainen , K and Taskinen , J . 2010 . Essential spectrum of a periodic elastic waveguide may contain arbitrarily many gaps . Appl. Anal. , 90 ( 1 ) : 109 – 124 .
  • Mazya , VG , Nazarov , SA and Plamenevskii , BA . 2000 . Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains , Basel : Birkhäuser Verlag .
  • Gadyl'shin , RR . 1986 . Asymptotic form of the eigenvalue of a singularly perturbed elliptic problem with a small parameter in the boundary condition . Differents. Uravneniya , 22 : 640 – 652 . (in Russian)
  • Kamotskii , IV and Nazarov , SA . 2000 . Spectral problems in singularly perturbed domains and self-adjoint extensions of differential operators . Trans. Am. Math. Soc. Ser. 2. , 199 : 127 – 181 .
  • Mazya , VG , Nazarov , SA and Plamenevskii , BA . 1984 . Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes . Izv. Akad. Nauk SSSR. Ser. Mat. , 48 ( 2 ) : 347 – 371 . (English trans.: Math. USSR Izvestiya. 24 (1985), pp. 321–345)
  • Mazya , VG and Nazarov , SA . 1989 . Singularities of solutions of the Neumann problem at a conical point . Sibirsk. Mat. Zh. , 30 ( 3 ) : 52 – 63 . (English trans.: Siberian Math. J. 30(3) (1989), pp. 387–396)
  • Nazarov , SA and Sokolowski , J . 2008 . Spectral problems in shape optimization. Singular boundary perturbations . Asymptotic Anal. , 56 ( 3–4 ) : 159 – 196 .
  • Nazarov , SA and Sokolowski , J . 2008 . Shape sensitivity analysis of eigenvalues revisited . Control Cybern. , 37 ( 4 ) : 999 – 1012 .
  • Ozawa , S . 1985 . Asymptotic property of an eigenfunction of the Laplacian under singular variation of domains – The Neumann condition . Osaka J. Math. , 22 ( 4 ) : 39 – 655 .
  • Pólya , G and Szegö , G . 1951 . Isoperimetric Inequalities in Mathematical Physics , NJ : Princeton University Press .
  • Visik , MI and Ljusternik , LA . 1962 . Regular degeneration and boundary layer of linear differential equations with small parameter . Am. Math. Soc. Transl. , 20 : 239 – 364 .
  • Campbell , A and Nazarov , SA . 1998 . An asymptotic study of a plate problem by a rearrangement method. Application to the mechanical impedance . RAIRO Model. Math. Anal. Num. , 32 ( 5 ) : 579 – 610 .

Reprints and Corporate Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

To request a reprint or corporate permissions for this article, please click on the relevant link below:

Academic Permissions

Please note: Selecting permissions does not provide access to the full text of the article, please see our help page How do I view content?

Obtain permissions instantly via Rightslink by clicking on the button below:

If you are unable to obtain permissions via Rightslink, please complete and submit this Permissions form. For more information, please visit our Permissions help page.