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Estimate of the spectrum deviation of the singularly perturbed Steklov problem

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Abstract

A Steklov-type problem with rapidly alternating Dirichlet and Steklov boundary conditions in a bounded n-dimensional domain in considered. The regions on which the Steklov condition is given have diameter of order ε, and the distance between them is larger than or equal to 2ε. It is proved that, as the small parameter tends to zero, the eigenvalues of this problem degenerate, i.e., tend to infinity. It is also proved that the rate of increase to infinity is larger than or equal to |ln ε|δ, δ ∈ (0;2 − 2/n) as ε, tends to zero.

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References

  1. W. Stekloff, Ann. Sci. Ecole Norm. Sup. (3) 19, 191–259 (1902).

    Article  MathSciNet  Google Scholar 

  2. V. A. Steklov, Doctoral Dissertation in Physics and Mathematics (Kharkov Imperial Univ., Kharkov, 1901).

    Google Scholar 

  3. A. G. Chechkina, Probl. Mat. Anal., No. 42, 129–143 (2009).

    MathSciNet  Google Scholar 

  4. V. A. Sadovnichii and A. G. Chechkina, Ufimsk. Mat. Zh. 3 (3), 127–139 (2011).

    Google Scholar 

  5. A. G. Chechkina, Dokl. Math. 84 (2), 695–698 (2011).

    Article  MathSciNet  Google Scholar 

  6. A. G. Chechkina and V. A. Sadovnichy, Eurasian Math. J. 6 (3), 13–29 (2015).

    MathSciNet  Google Scholar 

  7. A. G. Chechkina, Izv.: Math. 81 (1), 199–236 (2017).

    Article  MathSciNet  Google Scholar 

  8. R. R. Gadyl’shin and G. A. Chechkin, Sib. Math. J. 40 (2), 229–244 (1999).

    Google Scholar 

  9. G. A. Chechkin and O. A. Oleinik, Rend. Lincei: Mat. Appl. Ser. 9 7 (1), 5–15 (1996).

    Article  Google Scholar 

  10. O. A. Oleinik and G. A. Chechkin, Russ. Math. Surveys 48 (6), 173–175 (1993).

    Article  Google Scholar 

  11. A. Yu. Belyaev and G. A. Chechkin, Math. Notes 65 (4), 418–429 (1999).

    Article  MathSciNet  Google Scholar 

  12. M. E. Perez, G. A. Chechkin, and E. I. Yablokova (Doronina), Russ. Math. Surveys 57 (6), 1240–1242 (2002).

    Article  Google Scholar 

  13. G. A. Chechkin, M. E. Perez, and E. I. Yablokova, Indiana Univ. Math. J. 54 (2), 321–348 (2005).

    Article  MathSciNet  Google Scholar 

  14. A. L. Pyatnitskii, G. A. Chechkin, and A. S. Shamaev, Homogenization: Methods and Applications (Tamara Rozhkovskaya, Novosibirsk, 2007), Vol. 3 [in Russian].

    Google Scholar 

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

    MATH  Google Scholar 

Download references

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Authors and Affiliations

  1. Mechanics and Mathematics Faculty, Moscow State University, Moscow, 119991, Russia

    A. G. Chechkina

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  1. A. G. Chechkina
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Correspondence to A. G. Chechkina.

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Original Russian Text © A.G. Chechkina, 2017, published in Doklady Akademii Nauk, 2017, Vol. 476, No. 6, pp. 621–624.

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Chechkina, A.G. Estimate of the spectrum deviation of the singularly perturbed Steklov problem. Dokl. Math. 96, 510–513 (2017). https://doi.org/10.1134/S1064562417050301

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