On Lacunas in the Spectrum of the Laplacian with the Dirichlet Boundary Condition in a Band with Oscillating Boundary

被引：0
作者：
Borisov D.I. [1 ,2 ,3 ]
机构：
[1] Institute of Mathematics with Computing Center, Ufa Federal Research Center of the Russian Academy of Sciences, Ufa
[2] Bashkir State Pedagogical University named after M. Akmulla, Ufa
[3] University of Hradec Kralove, Hradec Kralove
来源：
Journal of Mathematical Sciences | 2021年 / 257卷 / 3期
关键词：
35B27; 35P05; 47A10; band; Bethe–Sommerfeld hypothesis; Laplacian; oscillating boundary;
D O I：
10.1007/s10958-021-05481-5
中图分类号：
学科分类号：
摘要：
In this paper, we consider the Laplace operator in a flat band whose lower boundary periodically oscillates under the Dirichlet boundary condition. The period and the amplitude of oscillations are two independent small parameters. The main result obtained in the paper is the absence of internal lacunas in the lower part of the spectrum of the operator for sufficiently small period and amplitude. We obtain explicit upper estimates of the period and amplitude in the form of constraints with specific numerical constants. The length of the lower part of the spectrum, in which the absence of lacunas is guaranteed, is also expressed explicitly in terms of the period function and the amplitude. © 2021, Springer Science+Business Media, LLC, part of Springer Nature.
引用
收藏
页码：273 / 285
页数：12
相关论文
共 21 条
  • [1] Barbatis G., Parnovski L., Bethe–Sommerfeld conjecture for pseudo-differential perturbation, Commun. Part. Differ. Equ., 34, 4, pp. 383-418, (2009)
  • [2] Beeken C.B.E., Periodic Schrödinger operators in dimension two: Constant magnetic fields and boundary-value problems, (2002)
  • [3] Borisov D.I., On lacunas in the lower part of the spectrum of a periodic magnetic operator in a band, Sovr. Mat. Fundam. Napravl., 63, 3, pp. 373-391, (2017)
  • [4] Borisov D.I., On lacunas in the spectrum of the Laplacian in a band perturbed by a bounded periodic operator, Ufim. Mat. Zh., 10, 2, pp. 13-29, (2018)
  • [5] Borisov D.I., The absence of lacunas in the lower part of the spectrum of the Laplacian with frequent alternation of boundary conditions in a band, Teor. Mat. Fiz., 195, 2, pp. 225-239, (2018)
  • [6] Borisov D., Bunoiu R., Cardone G., On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition, Ann. Inst. H. Poincaré, 11, 8, pp. 1591-1627, (2010)
  • [7] Borisov D., Bunoiu R., Cardone G., On a waveguide with an infinite number of small windows, Comp. Rend. Math., 349, 1-2, pp. 53-56, (2011)
  • [8] Borisov D., Bunoiu R., Cardone G., Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics, Z. Angew. Math. Phys., 64, 3, pp. 439-472, (2013)
  • [9] Borisov D., Cardone G., Homogenization of the planar waveguide with frequently alternating boundary conditions, J. Phys. A. Math. Gen., 42, 36, (2009)
  • [10] Borisov D., Cardone G., Durante T., Homogenization and uniform resolvent convergence for elliptic operators in a strip perforated along a curve, Proc. Roy. Soc. Edinburgh. Sec. A. Math., 146, 6, pp. 1115-1158, (2016)
123