Publication Cover
Applicable Analysis
An International Journal
Volume 103, 2024 - Issue 3
41
Views
0
CrossRef citations to date
0
Altmetric
Research Article

High-frequency homogenization of nonstationary periodic equations

ORCID Icon
Pages 533-561 | Received 05 Mar 2023, Accepted 30 Mar 2023, Published online: 08 Apr 2023

References

  • Bakhvalov NS, Panasenko GP. Homogenization: averaging processes in periodic media. Mathematical problems in mechanics of composite materials. Dordrecht: Kluwer Acadamic Publishers Group; 1989. (Mathematics and Its Applications (Soviet Series); Vol. 36).
  • Bensoussan A, Lions J-L, Papanicolaou G. Asymptotic analysis for periodic structures. Amsterdam, North-Holland: 1978. (Studies in Mathematics and its Applications; Vol. 5).
  • Zhikov VV, Kozlov SM, Olejnik OA. Homogenization of differential operators. Berlin: Springer-Verlag; 1994.
  • Birman MSh, Suslina TA. Second order periodic differential operators. Threshold properties and homogenization. Algebra Anal. 2003;15(5):1–108. English transl., St Petersburg Math J. 2004;15(5):639–714.
  • Allaire G. Homogenization and two-scale convergence. SIAM J Math Anal. 1992;23(6):1482–1518.
  • Sevost'yanova EV. An asymptotic expansion of the solution of a second order elliptic equation with periodic rapidly oscillating coefficients. Mat Sb. 1981;115(2):204–222. English transl., Math USSR-Sb. 1982;43(2):181–198.
  • Zhikov VV. Spectral approach to asymptotic diffusion problems. Differ Uravn. 1989;25(1):44–50. English transl., Differ Equ. 1989;25(1):33–39.
  • Conca C, Vanninathan M. Homogenization of periodic structures via Bloch decomposition. SIAM J Math Anal. 1997;57(6):1639–1659.
  • Conca C, Orive R, Vanninathan M. Bloch approximation in homogenization and applications. SIAM J Math Anal. 2002;33(5):1166–1198.
  • Conca C, Orive R, Vanninathan M. Bloch approximation in homogenization on bounded domains. Asymptot Anal. 2005;41(1):71–91.
  • Suslina TA. On homogenization of periodic parabolic systems. Funktsional Anal Prilozhen. 2004;38(4):86–90. English transl., Funct Anal Appl. 2004;38(4):309–312.
  • Suslina TA. Homogenization of a periodic parabolic Cauchy problem. Amer Math Soc Transl (2). 2007;220:201–233.
  • Birman MSh, Suslina TA. Homogenization with corrector term for periodic elliptic differential operators. Algebra Anal. 2005;17(6):1–104. English transl., St Petersburg Math J. 2006;17(6):897–973.
  • Birman MSh, Suslina TA. Homogenization with corrector term for periodic differential operators. Approximation of solutions in the Sobolev class H1(Rd). Algebra Anal. 2006;18(6):1–130. English transl., St Petersburg Math J. 2007;18(6):857–955.
  • Vasilevskaya ES. A periodic parabolic Cauchy problem: homogenization with corrector. Algebra Anal. 2009;21(1):3–60. English transl., St Petersburg Math J. 2010;21(1):1–41.
  • Suslina TA. Homogenization of a periodic parabolic Cauchy problem in the Sobolev space H1(Rd). Math Model Nat Phenom. 2010;5(4):390–447.
  • Zhikov VV. On some estimates of homogenization theory. Dokl Ros Akad Nauk. 2006;406(5):597–601. English transl., Dokl Math. 2006;73:96–99.
  • Zhikov VV, Pastukhova SE. On operator estimates for some problems in homogenization theory. Russ J Math Phys. 2005;12(4):515–524.
  • Zhikov VV, Pastukhova SE. Estimates of homogenization for a parabolic equation with periodic coefficients. Russ J Math Phys. 2006;13(2):224–237.
  • Zhikov VV, Pastukhova SE. Operator estimates in homogenization theory. Uspekhi Matem Nauk. 2016;71(3):27–122. English transl., Russ Math Surv. 2016;71(3):417–511.
  • Birman MSh, Suslina TA. Operator error estimates in the homogenization problem for nonstationary periodic equations. Algebra Anal. 2008;20(6):30–107. English transl., St Petersburg Math J. 2009;20(6):873–928.
  • Meshkova YM. On operator error estimates for homogenization of hyperbolic systems with periodic coeffcients. J Spectr Theory. 2021;11(2):587–660.
  • Dorodnyi MA, Suslina TA. Spectral approach to homogenization of hyperbolic equations with periodic coefficients. J Differ Equ. 2018;264(12):7463–7522.
  • Suslina TA. Spectral approach to homogenization of nonstationary Schrödinger-type equations. J Math Anal Appl. 2017;446(2):1466–1523.
  • Dorodnyi MA, Suslina TA. Homogenization of hyperbolic equations with periodic coefficients in Rd: sharpness of the results. Algebra Anal. 2020;32(4):3–136. English transl., St Petersburg Math J. 2021;32(4):605–703.
  • Dorodnyi MA. Operator error estimates for homogenization of the nonstationary Schrödinger-type equations: sharpness of the results. Appl Anal. 2022;101(16):5582–5614.
  • Suslina TA. Homogenization of the Schrödinger-type equations with periodic coefficients: Results with correctors. PDMI Preprint # 04/2022. 2022 [cited 2023 Feb 25]: [121 p.]. Available at https://pdmi.ras.ru/preprint/2022/22-04.html. Russian.
  • Suslina TA. Homogenization of the Schrödinger-type equations: operator estimates with correctors. Funktsional Anal Prilozhen. 2022;56(3):93–99. English transl.: Funct Anal Appl. 2022;56(3): 229–234.
  • Lin F, Shen Z. Uniform boundary controllability and homogenization of wave equations. J Eur Math Soc. 2022;24(9):3031–3053.
  • Craster RV, Kaplunov J, Pichugin AV. High-frequency homogenization for periodic media. Proc R Soc A. 2010;466(2120):2341–2362.
  • Harutyunyan D, Milton GW, Craster RV. High-frequency homogenization for travelling waves in periodic media. Proc R Soc A. 2016;472(2191):20160066.
  • Ceresoli L, Abdeddaim R, Antonakakis T, et al. Dynamic effective anisotropy: asymptotics, simulations, and microwave experiments with dielectric fibers. Phys Rev B. 2015;92(17):174307.
  • Allaire G. Periodic Homogenization and Effective Mass Theorems for the Schrödinger Equation. In: Abdallah NB, Frosali G, editors. Lecture Notes in Mathematics. Quantum Transport. Springer; 2008. p. 1–44.
  • Allaire G, Piatnitski A. Homogenization of the Schrödinger equation and effective mass theorems. Comm Math Phys. 2005;258(1):1–22.
  • Barletti L, Ben Abdallah N. Quantum transport in crystals: effective mass theorem and k⋅p Hamiltonians. Comm Math Phys. 2011;307(3):567–607.
  • Kuchment P, Raich A. Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. spectral edge case. Math Nachr. 2012;285(14-15):1880–1894.
  • Kha M, Kuchment P, Raich A. Green's function asymptotics near the internal edges of spectra of periodic elliptic operators. spectral gap interior. J Spectr Theory. 2017;7(4):1171–1233.
  • Birman MSh. On homogenization procedure for periodic operators near the edge of an internal gap. Algebra Anal. 2003;15(4):61–71. English transl., St Petersburg Math J. 2004;15(4):507–513.
  • Suslina TA, Kharin AA. Homogenization with corrector for a periodic elliptic operator near an edge of inner gap. Problemy Mat Analiza. 2009;41:127–141. English transl., J Math Sci. 2009;159(2):264–280.
  • Mishulovich AA, Sloushch VA, Suslina TA. Homogenization of a one-dimensional periodic elliptic operator at the edge of a spectral gap: operator estimates in the energy norm. Zap Nauchn Sem POMI. 2022;519:114–151. English transl.: J Math Sci. Forthcoming.
  • Akhmatova AR, Aksenova ES, Sloushch VA, et al. Homogenization of the parabolic equation with periodic coefficients at the edge of a spectral gap. Complex Var Elliptic Equ. 2022;67(3):523–555.
  • Birman MSh, Suslina TA. Homogenization of a multidimensional periodic elliptic operator in a neighborhood of the edge of an internal gap. Zap Nauchn Sem POMI. 2004;318:60–74. English transl., J Math Sci. 2006;136(2):3682–3690.
  • Suslina TA, Kharin AA. Homogenization with corrector for a multidimensional periodic elliptic operator near an edge of an inner gap. Problemy Mat Analiza. 2011;59:177–193. English transl., J Math Sci. 2011;177(1):208–227.
  • Mishulovich AA. Homogenization of the multidimensional parabolic equations with periodic coefficients at the edge of a spectral gap. Zap Nauchn Sem POMI. 2022;516:135–175. English transl.: J Math Sci. Forthcoming.
  • Dorodnyi MA. High-energy homogenization of a multidimensional nonstationary Schrödinger equation. arXiv:2301.05907 [Preprint]. 2023 [cited 2023 Feb 25]: [26 p.]. Available at https://arxiv.org/abs/2301.05907.
  • Kirsch W, Simon B. Comparison theorems for the gap of Schrödinger operators. J Funct Anal. 1987;75(2):396–410.
  • Reed M, Simon B. Methods of modern mathematical physics, vol. 4: analysis of operators. New York: Academic Press; 1978.
  • Dunford N, Schwartz JT. Linear operators, part 1: general theory. New York: Interscience Publishers; 1958.

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.