On elliptic operators with Steklov condition perturbed by Dirichlet condition on a small part of boundary


We consider a boundary value problem for a homogeneous elliptic equation with an inhomogeneous Steklov boundary condition. The problem involves a singular perturbation, which is the Dirichlet condition imposed on a small piece of the boundary. We rewrite such problem to a resolvent equation for a self-adjoint operator in a fractional Sobolev space on the boundary of the domain. We prove the norm convergence of this operator to a limiting one associated with an unperturbed problem involving no Dirichlet condition. We also establish an order sharp estimate for the convergence rate. The established convergence implies the convergence of the spectra and spectral projectors. In the second part of the work we study perturbed eigenvalues converging to limiting simple discrete ones. We construct two-terms asymptotic expansions for such eigenvalues and for the associated eigenfunctions.

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  1. 1.

    Abdullazade, N.N., Chechkin, G.A.: Perturbation of the Steklov problem on a small part of the boundary. J. Math. Sci. 196, 441–450 (2014)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Birman, M.Sh, Suslina, T.A.: Periodic differential operators of second order. Threshold properties and averagings. St. Petersb. Math. J. 15, 639–714 (2004)

    Article  Google Scholar 

  3. 3.

    Besov, O.V., Il’in, V.P., Nikol’skii, S.M.: Integral Representations of Functions and Imbedding Theorems, vol. I. Halsted Press, Washington, DC (1978)

    Google Scholar 

  4. 4.

    Borisov, D., Cardone, G.: Homogenization of the planar waveguide with frequently alternating boundary conditions. J. Phys. A: Math. Gen. 42, 365205 (2009)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Borisov, D., Bunoiu, R., Cardone, G.: On a waveguide with frequently alternating boundary conditions: homogenized Neumann condition. Ann. Henri Poincaré 11, 1591–1627 (2010)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Borisov, D., Cardone, G., Faella, L., Perugia, C.: Uniform resolvent convergence for a strip with fast oscillating boundary. J. Differ. Equ. 255, 4378–4402 (2013)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Borisov, D., Bunoiu, R., Cardone, G.: Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics. Z. Angew. Math. Phys. 64, 439–472 (2013)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Borisov, D., Cardone, G., Durante, T.: Homogenization and uniform resolvent convergence for elliptic operators in a strip perforated along a curve. Proc. R. Soc. Edinb. Sect. A. Math. 146, 1115–1158 (2016)

    Article  Google Scholar 

  9. 9.

    Borisov, D.I., Mukhametrakhimova, A.I.: On norm resolvent convergence for elliptic operators in multi-dimensional domains with small holes. J. Math. Sci. 232, 283–298 (2018)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Borisov, D.: On a \({\cal{PT}}\)-symmetric waveguide with a pair of small holes. Proc. Steklov Inst. Math. Suppl. 281, 5–21 (2013)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Chechkina, A.G.: Convergence of solutions and eigenelements of Steklov type boundary value problems with boundary conditions of rapidly varying type. J. Math. Sci. 162, 443–458 (2009)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Chechkina, A.G.: On singular perturbation of a Steklov-type problem with asymptotically degenerate spectrum. Dokl. Math. 84, 695–698 (2011)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Chechkina, A.G.: The homogenization of spectral problems with singular perturbation of the Steklov condition. Izv. Math. 81, 199–236 (2017)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Chechkina, A.G., D’Apice, C., De Maio, U.: Rate of convergence of eigenvalues to singularly perturbed Steklov-type problem for elasticity system. Appl. Anal. 98, 32–44 (2019)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Chechkina, A.G., D’Apice, C., De Maio, U.: Operator estimates for elliptic problem with rapidly alternating Steklov boundary condition. J. Comput. Appl. Math. 376, 112802 (2020)

    MathSciNet  Article  Google Scholar 

  16. 16.

    Chechkin, G.A., Gadyl’shin, R.R., D’Apice, C., De Maio, U.: On the Steklov problem in a domain perforated along a part of the boundary. ESAIM Math. Model. Numer. Anal. 51, 1317–1342 (2017)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Chiado Piat, V., Nazarov, S., Piatnitski, A.: Steklov problems in perforated domains with a coefficient of indefinite sign. Netw. Heterog. Media 7, 151–178 (2012)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Escobar, J.F.: An isoperimetric inequality and the first Steklov eigenvalue. J. Funct. Anal. 165, 101–116 (1999)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Gadyl’shin, R.R., Piatnitski, A.L., Chechkin, G.A.: On the asymptotic behaviour of eigenvalues of a boundary-value problem in a planar domain of Steklov sieve type. Izv. Math. 82, 1108–1135 (2018)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Griso, G.: Error estimate and unfolding for periodic homogenization. Asymptot. Anal. 40, 269–286 (2004)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Griso, G.: Interior error estimate for periodic homogenization. Anal. Appl. 4, 61–79 (2006)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Jammes, P.: Une inégalité de Cheeger pour le spectre de Steklov. Ann. de l’Inst. Fourier 65, 1381–1385 (2015)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Hassannezhad, A., Laptev, A.: Eigenvalue bounds of mixed Steklov problems. Commun. Contemp. Math. 22, 1950008 (2020)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Il’in, A.M.: Matching of Asymptotic Expansions of Solutions of Boundary Value Problems. American Mathematical Society, Providence, RI (1992)

    Google Scholar 

  25. 25.

    Kenig, C.E., Lin, F., Shen, Z.: Convergence rates in \(L_2\) for elliptic homogenization problems. Arch. Ration. Mech. Anal. 203, 1009–1036 (2012)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Kenig, C.E., Lin, F., Shen, Z.: Periodic homogenization of Green and Neumann functions. Commun. Pure Appl. Math. 67, 1219–1262 (2014)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Levitin, M., Parnovski, L., Polterovich, I., Sher, D.A.: Sloshing, Steklov and corners: asymptotics of sloshing eigenvalues. J. d’Anal. Math. Preprint arXiv:1709.01891

  28. 28.

    Maz’ya, V.G., Nazarov, S.A., Plamenevskii, B.A.: Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes. Math. USSR-Izv. 24, 321–345 (1985)

    Article  Google Scholar 

  29. 29.

    Mel’nyk, T.A.: Asymptotic behavior of eigenvalues and eigenfunctions of the Steklov problem in a thick periodic junction. Nonlinear Oscil. 4, 91–105 (2001)

    MathSciNet  MATH  Google Scholar 

  30. 30.

    Nazarov, S.A., Taskinen, J.: On the spectrum of the Steklov problem in a domain with a peak. Vestn. St. Petersb. Univ. Math. 41, 45–52 (2008)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Nazarov, S.A.: On the spectrum of the Steklov problem in peak-shaped domains. Am. Math. Soc. Trans. Ser. 2(228), 79–132 (2009)

    MathSciNet  MATH  Google Scholar 

  32. 32.

    Nazarov, S.A.: Asymptotic behavior of the Steklov spectral problem in a domain with a blunted peak. Math. Notes 86, 542–555 (2009)

    MathSciNet  Article  Google Scholar 

  33. 33.

    Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam (1992)

    Google Scholar 

  34. 34.

    Pastukhova, S.E., Tikhomirov, R.N.: Operator-type estimates in homogenization of elliptic equations with lower terms. St. Petersb. Math. J. 29, 841–861 (2018)

    Article  Google Scholar 

  35. 35.

    Pérez, M.E.: On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete Contin. Dyn. Syst. Ser. B 7, 859–883 (2007)

    MathSciNet  MATH  Google Scholar 

  36. 36.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics I: Functional Analysis. Academic Press, San Diego (1980)

    Google Scholar 

  37. 37.

    Stekloff, W.: Sur les problèmes fondamentaux de la physique mathématique. Ann. Sci. Ecol. Norm. Super. Ser. 3. 19, 191–259, 455–490 (1902) (in French)

  38. 38.

    Suslina, T.A.: Homogenization of the Dirichlet problem for higher-order elliptic equations with periodic coefficients. St. Petersb. Math. J. 29, 325–362 (2018)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Zhikov, V.V.: Spectral method in homogenization theory. Proc. Steklov Inst. Math. 250, 85–94 (2005)

    MathSciNet  MATH  Google Scholar 

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The authors thank the referee for useful remarks, which allowed them to improve the initial version of the paper. The scientific contribution by B.D.I. to results presented in Sect. 3 was financially supported by Russian Science Foundation (Project No. 20-11-19995). The scientific contribution by C.G.A. to results presented in Sect. 2 was financially supported by Russian Science Foundation (Project No. 20-11-20272). The scientific contribution by G.C. was supported by project GNAMPA-INDAM “Analisi asintotica di problemi stazionari ed evolutivi in materiali compositi e strutture sottili” (2020).

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Borisov, D.I., Cardone, G., Chechkin, G.A. et al. On elliptic operators with Steklov condition perturbed by Dirichlet condition on a small part of boundary. Calc. Var. 60, 48 (2021). https://doi.org/10.1007/s00526-020-01847-w

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Mathematics Subject Classification

  • 35B25
  • 35P05