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Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation

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Abstract

We prove a statement on the averaging of a hyperbolic initial-boundary-value problem in which the coefficient of the Laplace operator depends on the space L 2-norm of the gradient of the solution. The existence of the solution of this problem was studied by Pokhozhaev. In a space domain in ℝn, n ≥ 3, we consider an arbitrary perforation whose asymptotic behavior in a sense of capacities is described by the Cioranesku-Murat hypothesis. The possibility of averaging is proved under the assumption of certain additional smoothness of the solutions of the limiting hyperbolic problem with a certain stationary capacitory potential.

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References

  1. S. I. Pokhozhaev, “On a class of quasilinear hyperbolic equations,” Mat. Sb., 96(138), No. 1, 152–166 (1975).

    MathSciNet  Google Scholar 

  2. S. I. Pokhozhaev, “On a quasilinear hyperbolic Kirchhoff equation,” Differents. Uravn., 21, No. 1, 101–108 (1985).

    MATH  MathSciNet  Google Scholar 

  3. S. I. Pokhozhaev, “Quasilinear hyperbolic Kirchhoff equations and conservation laws,” Tr. Mosk. Énerg. Inst., Issue 201, 118–126 (1974).

  4. D. Cioranesku and F. Murat, “Un terme étrange venu d’ailleurs,” Res. Notes Math., 60, 93–138 (1982).

    Google Scholar 

  5. D. Cioranesku, P. Donato, F. Murat, and E. Zuazua, “Homogenization and corrector for the wave equation in domains with small holes,” Ann. Scuola Norm. Super. Pisa. Sci. Fis. Mat., 18, No. 2, 251–293 (1991).

    Google Scholar 

  6. I. V. Skrypnik, Methods for the Investigation of Nonlinear Elliptic Boundary-Value Problems [in Russian], Nauka, Moscow (1990).

    Google Scholar 

  7. G. Dal Maso and I. V. Skrypnik, Asymptotic Behavior of Nonlinear Dirichlet Problems in Perforated Domains, Ref. SISSA 162 95 M (December 95), Trieste (1995).

  8. J.-L. Lions and E. Magenes, Problémes aux Limites non Homogénes et Applications, Dunod, Paris (1968).

    Google Scholar 

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    N. R. Sidenko

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  1. N. R. Sidenko
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 2, pp. 236–249, February, 2006.

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Sidenko, N.R. Averaging of the Dirichlet problem for a special hyperbolic Kirchhoff equation. Ukr Math J 58, 263–279 (2006). https://doi.org/10.1007/s11253-006-0065-x

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  • DOI: https://doi.org/10.1007/s11253-006-0065-x

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