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Sibirsk. Mat. Zh., 2008, Volume 49, Number 1, Pages 101–124 (Mi smj1825)  

This article is cited in 20 scientific papers (total in 20 papers)

Homogenization of degenerate elliptic equations

V. V. Zhikova, S. E. Pastukhovab

a Vladimir State Pedagogical University
b Moscow State Institute of Radio-Engineering, Electronics and Automation (Technical University)

Abstract: We consider the divergent elliptic equations whose weight function and its inverse are assumed locally integrable. The equations of this type exhibit the Lavrentiev phenomenon, the nonuniqueness of weak solutions, as well as other surprising consequences. We classify the weak solutions of degenerate elliptic equations and show the attainability of the so-called $W$-solutions. Investigating the homogenization of arbitrary attainable solutions, we find their different asymptotic behavior. Under the assumption of the higher integrability of the weight function we estimate the difference between the exact solution and certain special approximations.

Keywords: Lavrentiev's phenomenon, attainability, homogenization, approximation solution.

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English version:
Siberian Mathematical Journal, 2008, 49:1, 80–101

Bibliographic databases:

UDC: 517.97
Received: 04.07.2006

Citation: V. V. Zhikov, S. E. Pastukhova, “Homogenization of degenerate elliptic equations”, Sibirsk. Mat. Zh., 49:1 (2008), 101–124; Siberian Math. J., 49:1 (2008), 80–101

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. E. Pastukhova, “Operator Estimates in Nonlinear Problems of Reiterated Homogenization”, Proc. Steklov Inst. Math., 261 (2008), 214–228  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    2. Pastukhova S.E., Tikhomirov R.N., “Estimates of locally periodic and reiterated homogenization for parabolic equations”, Dokl. Math., 80:2 (2009), 674–678  mathnet  crossref  mathscinet  zmath  isi  elib  elib  scopus
    3. Pastukhova S., “Estimates in homogenization of parabolic equations with locally periodic coefficients”, Asymptot. Anal., 66:3-4 (2010), 207–228  crossref  mathscinet  zmath  isi  elib  scopus
    4. Buttazzo G., Kogut P.I., “Weak optimal controls in coefficients for linear elliptic problems”, Rev. Mat. Complut., 24:1 (2011), 83–94  crossref  mathscinet  zmath  isi  elib  scopus
    5. S. E. Pastukhova, A. S. Khripunova, “Several versions of the compensated compactness principle”, Sb. Math., 202:9 (2011), 1387–1412  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. Kogut P.I., Leugering G., “Optimal $L^1$-control in coefficients for Dirichlet elliptic problems: $W$-optimal solutions”, J. Optim. Theory Appl., 150:2 (2011), 205–232  crossref  mathscinet  zmath  isi  elib  scopus
    7. Kogut P.I., Leugering G., “Optimal $L^1$-control in coefficients for Dirichlet elliptic problems: $H$-optimal solutions”, Z. Anal. Anwend., 31:1 (2012), 31–53  crossref  mathscinet  zmath  isi  elib  scopus
    8. S. E. Pastukhova, “Approximation of the Exponential of a Diffusion Operator with Multiscale Coefficients”, Funct. Anal. Appl., 48:3 (2014), 183–197  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    9. Kogut P.I., Leugering G., “Optimal and Approximate Boundary Controls of An Elastic Body With Quasistatic Evolution of Damage”, Math. Meth. Appl. Sci., 38:13 (2015), 2739–2760  crossref  mathscinet  zmath  isi  elib  scopus
    10. Kogut P.I., Kupenko Ol'ga P., Leugering G., “Optimal Control in Matrix-Valued Coefficients For Nonlinear Monotone Problems: Optimality Conditions II”, Z. Anal. ihre. Anwend., 34:2 (2015), 199–219  crossref  mathscinet  zmath  isi  elib  scopus
    11. Kupenko O.P., Leugering G., “on the Existence of Weak Optimal Controls in the Coefficients For a Degenerate Anisotropic P-Laplacian”, Continuous and Distributed Systems II: Theory and Applications, Studies in Systems Decision and Control, 30, eds. Sadovnichiy V., Zgurovsky M., Springer Int Publishing Ag, 2015, 315–337  crossref  mathscinet  zmath  isi  scopus
    12. S. E. Pastukhova, “The Neumann problem for elliptic equations with multiscale coefficients: operator estimates for homogenization”, Sb. Math., 207:3 (2016), 418–443  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    13. V. V. Zhikov, S. E. Pastukhova, “Operator estimates in homogenization theory”, Russian Math. Surveys, 71:3 (2016), 417–511  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    14. B. E. Kanguzhin, N. E. Tokmagambetov, “Resolvents of well-posed problems for finite-rank perturbations of the polyharmonic operator in a punctured domain”, Siberian Math. J., 57:2 (2016), 265–273  mathnet  crossref  crossref  mathscinet  isi  elib
    15. Zhikov V.V., Pastukhova S.E., “Bloch Principle For Elliptic Differential Operators With Periodic Coefficients”, Russ. J. Math. Phys., 23:2 (2016), 257–277  crossref  mathscinet  zmath  isi  scopus
    16. Pastukhova S.E., “Estimates in Homogenization of Higher-Order Elliptic Operators”, Appl. Anal., 95:7, SI (2016), 1449–1466  crossref  mathscinet  zmath  isi  scopus
    17. Horsin T., Kogut P.I., “on Unbounded Optimal Controls in Coefficients For Ill-Posed Elliptic Dirichlet Boundary Value Problems”, Asymptotic Anal., 98:1-2 (2016), 155–188  crossref  mathscinet  zmath  isi  scopus
    18. S. E. Pastukhova, R. N. Tikhomirov, “Operator-type estimates in homogenization of elliptic equations with lower terms”, St. Petersburg Math. J., 29:5 (2018), 841–861  mathnet  crossref  mathscinet  isi  elib
    19. Neukamm S., Schaeffner M., Schloemerkemper A., “Stochastic Homogenization of Nonconvex Discrete Energies With Degenerate Growth”, SIAM J. Math. Anal., 49:3 (2017), 1761–1809  crossref  mathscinet  zmath  isi  scopus
    20. A. M. Meirmanov, S. A. Gritsenko, “Homogenization of the equations of filtration of a viscous fluid in two porous media”, Siberian Math. J., 59:5 (2018), 909–921  mathnet  crossref  crossref  isi
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