Abstract
We show that the spectrum of the Dirichlet problem for the Laplace operator \(-\Delta \) in the plane \(\mathbb {R}^2\) perforated by a double-periodic family of holes contains any a priori number of gaps, for sufficiently large holes. While this result was already known in the case of circular holes, we consider here a more general geometric setting with holes of the shape \(\{ |x_1|^\mu + |x_2|^\mu \le r\} \), \(1 < \mu < \infty \).
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Nazarov, S.A., Ruotsalainen, K., Taskinen, J.: Spectral gaps in the Dirichlet and Neumann problems on the plane perforated by a double-periodic family of circular holes. J. Math. Sci. (N. Y.) 181(2), 164–222 (2012)
McIver, P.: Water-wave propagation through an infinite array of cylindrical structures. J. Fluid. Mech. 424, 101–125 (2000)
Nazarov, S.A.: Properties of spectra of boundary value problems in cylindrical and quasicylindrical domains. In: Sobolev Spaces in Mathematics. II, pp. 261–309. International Mathematics Series (N. Y.), Vol. 9. Springer, New York (2009)
Birman, MSh, Solomjak, M.Z.: Mathematics and Its Applications (Soviet Series). Spectral theory of selfadjoint operators in Hilbert space. D. Reidel Publishing Co., Dordrecht (1987)
Kato, T.: Die Grundlehren der Mathematischen Wissenschaften. Perturbation theory for linear operators, vol. 132. Springer-Verlag, New York (1966)
Gel’fand, I.M.: Expansion in characteristic functions of an equation with periodic coefficients. Dokl. Akad. Nauk. SSSR (N.S.) 73, 1117–1120 (1950)
Kuchment, P.: Operator Theory: Advances and Applications, Vol. 60. Floquet theory for partial differential equations. Birkhäuser Verlag, Basel (1993)
Maz’ya, V.G., Plamenevskij, B.A.: On the asymptotic behavior of solutions of differential equations in Hilbert space (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 36(5), 1080–1113 (1972) (letter to edtior, ibid 37(3), 709–710, 1973; English translation, Math. USSR Izv. 6, 1067–1116, 1972)
Maz’ya, V.G., Plamenevskij, B.A.: On the asymptotics of the solution of the Dirichlet problem near the isolated singularity of a boundary (Russian). Vestn. Leningr. Univ. Ser. 1(13), 60–66 (1977)
Maz’ya, V.G., Plamenevskij, B.A.: Estimates in Lp and in Hölder classes and the Miranda–Agmon principle for solutions of elliptic boundary value problems in domains with singular points on the boundary (Russian). Math. Nachr. 81, 25–82 (1978)
Nasarov, S.A., Polyakova, O.P.: Asymptotic expansion of eigenvalues of the Neumann problem in a domain with a thin bridge (Russian). Sib. Mat. Zh. 33(4), 80–96 (1992) (English translation Sib. Math. J. 33(4), 618–633, 1992)
Maz’ya, V.G., Poborchi, S.V.: Differentiable Functions on Bad Domains. World Scientific Publishing, River Edge (1997)
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J. Taskinen was partially supported by the Väisälä Foundation of the Finnish Academy of Sciences and Letters.
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Ferraresso, F., Taskinen, J. Singular perturbation Dirichlet problem in a double-periodic perforated plane. Ann Univ Ferrara 61, 277–290 (2015). https://doi.org/10.1007/s11565-014-0222-3
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DOI: https://doi.org/10.1007/s11565-014-0222-3