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Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation

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Abstract

We consider a spectral homogenization problem for the linear elasticity system posed in a domain \(\varOmega \) of the upper half-space \(\mathbb{R}^{3+}\), a part of its boundary \(\varSigma \) being in contact with the plane \(\{x_{3}=0\}\). We assume that the surface \(\varSigma \) is traction-free out of small regions \(T^{\varepsilon }\), where we impose Winkler-Robin boundary conditions. This condition links stresses and displacements by means of a symmetric and positive definite matrix-function \(M(x)\) and a reaction parameter \(\beta (\varepsilon )\) that can be very large when \(\varepsilon \to 0\). The size of the regions \(T^{\varepsilon }\) is \(O(r_{\varepsilon })\), where \(r_{\varepsilon }\ll \varepsilon \), and they are placed at a distance \(\varepsilon \) between them. We provide all the possible spectral homogenized problems depending on the relations between \(\varepsilon \), \(r_{\varepsilon }\) and \(\beta (\varepsilon )\), while we address the convergence, as \(\varepsilon \to 0\), of the eigenpairs in the critical cases where some strange terms arise on the homogenized Robin boundary conditions on \(\varSigma \). New capacity matrices are introduced to define these strange terms.

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References

  1. Agmon, S., Douglas, A., Niremberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II. Commun. Pure Appl. Math. XVII, 35–92 (1964)

    Article  MathSciNet  Google Scholar 

  2. Allaire, G.: Homogenization of the Naviers-Stokes equations in open sets perforated with tiny holes II. Non critical size of the holes for a volume distribution of holes and a surface distribution of holes. Arch. Ration. Mech. Anal. 113, 261–298 (1983)

    Article  Google Scholar 

  3. Attouch, H.: Variational Convergence for Functions and Operators. Applicable Math. Series. Pitman, London (1984)

    MATH  Google Scholar 

  4. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

    MATH  Google Scholar 

  5. Brillard, A., Lobo, M., Pérez, E.: Un probléme d’homogénéisation de frontière en élasticité linéare pour un corps cylindrique. C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers Sci. Terre 311, 15–20 (1990)

    MathSciNet  MATH  Google Scholar 

  6. Brillard, A., Lobo, M., Pérez, E.: Homogénéisation de Frontières par epi-convergence en élasticité linéare. RAIRO Modél. Math. Anal. Numér. 24, 5–26 (1990)

    Article  MathSciNet  Google Scholar 

  7. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications, vol. 4. North-Holland, Amsterdam (1978)

    Book  Google Scholar 

  8. Cioranescu, D., Damlamian, A., Griso, G., Onofrei, D.: The periodic unfolding method for perforated domains and Neumann sieve models. J. Math. Pures Appl. 89, 248–277 (2008)

    Article  MathSciNet  Google Scholar 

  9. Cioranescu, D., Murat, F.: Un terme étrange venu d’ailleurs. In: Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. II&III, Res. Notes in Math., Vol. 60&70, pp. 98–138&154–178, Pitman, Boston (1982). English translation: Topics in the Mathematical Modelling of Composite Materials, Progr. Nonlinear Differential Equations Appl., 31, Birkäuser, Boston, 1997, 45–93

    Google Scholar 

  10. Conca, C.: On the application of the homogenization theory to a class of problems arising in fluid mechanics. J. Math. Pures Appl. 64, 31–75 (1985)

    MathSciNet  MATH  Google Scholar 

  11. Gohberg, I.C., Krein, M.G.: Introduction to the Theory of Linear Nonselfadjoint Operators. Am. Math. Soc., Providence, RI (1969)

    MATH  Google Scholar 

  12. Gómez, D., Lobo, M., Pérez, E., Sanchez-Palencia, E.: Homogenization in perforated domains: a Stokes grill and an adsorption process. Appl. Anal. 97, 2893–2919 (2018)

    Article  MathSciNet  Google Scholar 

  13. Gómez, D., Nazarov, S.A., Pérez, E.: Homogenization of Winkler-Steklov spectral conditions in three-dimensional linear elasticity. Z. Angew. Math. Phys. 69(2), 35 (2018). 23p

    Article  MathSciNet  Google Scholar 

  14. Gómez, D., Nazarov, S.A., Pérez-Martínez, M.-E.: Spectral homogenization problems in linear elasticity with large reaction terms concentrated in small regions of the boundary. In: Computational and Analytic Methods in Science and Engineering, pp. 119–141. Springer, N.Y. (2020). Chap. 7

    Google Scholar 

  15. Gómez, D., Pérez, E., Shaposhnikova, T.A.: On homogenization of nonlinear Robin type boundary conditions for cavities along manifolds and associated spectral problems. Asymptot. Anal. 80, 289–322 (2012)

    Article  MathSciNet  Google Scholar 

  16. Griso, G., Migunova, A., Orlik, J.: Homogenization via unfolding in periodic layer with contact. Asymptot. Anal. 99, 23–52 (2015)

    Article  MathSciNet  Google Scholar 

  17. Ionescu, I., Onofrei, D., Vernescu, B.: \(\varGamma \)-convergence for a fault model with slip-weakening friction and periodic barriers. Q. Appl. Math. 63(4), 747–778 (2005)

    Article  MathSciNet  Google Scholar 

  18. Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs, vol. 85. American Mathematical Society, Providence, RI (2001)

    MATH  Google Scholar 

  19. Landau, L., Lifchitz, E.: Physique Théorique. Tome 7. Théorie de l’Élasticité. Mir, Moscow (1990)

    MATH  Google Scholar 

  20. Lobo, M., Oleinik, O.A., Pérez, M.E., Shaposhnikova, T.A.: On homogenization of solutions of boundary value problems in domains, perforated along manifolds. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) 25, 611–629 (1997)

    MathSciNet  MATH  Google Scholar 

  21. Lobo, M., Pérez, E.: Asymptotic behaviour of an elastic body with a surface having small stuck regions. RAIRO Modél. Math. Anal. Numér. 22, 609–624 (1988)

    Article  MathSciNet  Google Scholar 

  22. Lobo, M., Pérez, E.: On the vibrations of a body with many concentrated masses near the boundary. Math. Models Methods Appl. Sci. 3(2), 249–273 (1993)

    Article  MathSciNet  Google Scholar 

  23. Marchenko, V.A., Khruslov, E.Ya.: Boundary Value Problems in Domains with a Fine-Grained Boundary. Naukova Dumka, Kiev (1974). (in Russian)

    MATH  Google Scholar 

  24. Murat, F.: The Neumann sieve. In: Nonlinear Variational Problems, Isola d’Elba, 1983. Res. Notes in Math., vol. 127, pp. 24–32. Pitman, Boston, MA (1985)

    Google Scholar 

  25. Nazarov, S.A.: Polynomial property of selfadjoint elliptic boundary value problems, and the algebraic description of their attributes. Uspekhi Mat. Nauk 54, 77–142 (1999). English translation: Russian Math. Surveys 54:947–1014 (1999)

    Article  MathSciNet  Google Scholar 

  26. Nazarov, S.A.: Asymptotics of solutions and modeling of the elasticity problems in a domain with the rapidly oscillating boundary. Izv. Math. 72(3), 509–564 (2008)

    Article  MathSciNet  Google Scholar 

  27. Nazarov, S.A., Plamenevsky, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin (1994)

    Book  Google Scholar 

  28. Nazarov, S.A., Sokolowski, J., Specovius-Neugebauer, M.: Polarization matrices in anisotropic heterogeneous elasticity. Asymptot. Anal. 68(4), 189–221 (2010)

    Article  MathSciNet  Google Scholar 

  29. Nguetseng, G., Sanchez-Palencia, E.: Stress concentration for defects distributed near a surface. In: Local Effects in the Analysis of Structures. Stud. Appl. Mech., vol. 12, pp. 55–74. Elsevier, Amsterdam (1985)

    Chapter  Google Scholar 

  30. Oleinik, O.A., Chechkin, G.: On asymptotics of solutions and eigenvalues of the boundary value problem with rapidly alternating boundary conditions for the system of elasticity. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 7, 5–15 (1996)

    MathSciNet  MATH  Google Scholar 

  31. Oleinik, O.A., Shamaev, A.S., Yosifian, G.A.: Mathematical Problems in Elasticity and Homogenization. Studies in Mathematics and Its Applications, vol. 26. North-Holland, Amsterdam (1992)

    MATH  Google Scholar 

  32. Pérez-Martínez, M.-E.: Problemas de homogeneización de fronteras en elasticidad lineal. PhD Thesis, Universidad de Cantabria, Santander (1987)

  33. Pérez-Martínez, M.-E.: Homogenization for alternating boundary conditions with large reaction terms concentrated in small regions. In: Emerging Problems in the Homogenization of Partial Differential Equations, ICIAM2019. SEMA SIMAI Springer Series (2020). To appear

    Google Scholar 

  34. Raviart, P.A., Thomas, J.M.: Introduction à l’Analyse Numérique des Équations aux Dérivées Partielles. Collection Mathématiques Appliquées pour la Maîtrise. Masson, Paris (1983)

    Google Scholar 

  35. Sanchez-Hubert, J., Sanchez-Palencia, E.: Acoustic fluid flow through holes and permeability of perforated walls. J. Math. Anal. Appl. 87, 427–453 (1982)

    Article  MathSciNet  Google Scholar 

  36. Sanchez-Palencia, E.: Boundary value problems in domains containing perforated walls. In: Nonlinear Partial Differential Equations and Their Applications, Collège de France Seminar, Vol. III. Res. Notes in Math., vol. 70, pp. 309–325. Pitman, Boston (1982).

    Google Scholar 

  37. Temam, R.: Problèmes Mathématiques en Plasticité. Méthodes Mathématiques de l’Informatique, vol. 12. Gauthier-Villars, Paris (1983)

    MATH  Google Scholar 

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Acknowledgements

This work has been partially supported by Russian Foundation on Basic Research grant 18-01-00325, Spanish MICINN grant PGC2018-098178-B-I00 and the Convenium Banco Santander - Universidad de Cantabria 2018.

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Correspondence to María-Eugenia Pérez-Martínez.

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Gómez, D., Nazarov, S.A. & Pérez-Martínez, ME. Asymptotics for Spectral Problems with Rapidly Alternating Boundary Conditions on a Strainer Winkler Foundation. J Elast 142, 89–120 (2020). https://doi.org/10.1007/s10659-020-09791-8

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