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On the Friedrichs inequality in a domain perforated aperiodically along the boundary. Homogenization procedure. Asymptotics for parabolic problems. (English) Zbl 1180.35072

The focus is the investigation of a boundary-value problem posed in a domain perforated aperiodically (with circular holes) along the boundary for the case when the diameters of the circles and the distance between them are of the same order. The authors derive for such a non-periodic scenario a useful Friederichs-type inequality for functions vanishing on the boundary of the perforations. They also prove the convergence of the oscillatory solutions to the homogenized (non-oscillatory) solution. A numerical illustration concludes the paper.

MSC:

35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35K20 Initial-boundary value problems for second-order parabolic equations
35B45 A priori estimates in context of PDEs
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[1] A. G. Belyaev, ”On Singular Perturbations of Boundary-Value Problems,” PhD Thesis (Moscow State University, Moscow, 1990) [in Russian].
[2] L. R. Bikmetov, ”Asymptotics of the Eigenelements of Boundary Value Problems for the Schrödinger Operator with a Large Potential Localized on a Small Set,” Zh. Vychisl. Mat. Mat. Fiz. 46(4), 667–682 (2006) [Comput. Math. Math. Phys. 46 (4), 636–650 (2006)]. · Zbl 1210.81021
[3] G. A. Chechkin, ”On Boundary-Value Problems for a Second-Order Elliptic Equation with Oscillating Boundary Conditions,” in Nonclassical Partial Differential Equations, Ed. by V. N. Vragov (Institute of Mathematics, Siberian Division of the Academy of Sciences of the USSR, Novosibirsk, 1988), pp. 95–104. · Zbl 0825.35038
[4] G. A. Chechkin, ”Asymptotic Expansions of the Eigenvalues and Eigenfunctions of an Elliptic Operator in a Domain with Many ’Light’ Concentrated Masses near the Boundary. The Two-Dimensional Case,” Izv. Ross. Akad. Nauk Ser. Mat. 69(4), 161–204 (2005) [Izv. Math. 69 (4), 805–846 (2005)]. · Zbl 1102.35035 · doi:10.4213/im652
[5] G. A. Chechkin, T.P. Chechkina, C. D’Apice, and U. De Maio, ”Homogenization in Domains Randomly Perforated Along the Boundary,” Discrete Cont. Dyn. Syst. Ser. B 11 (2009). · Zbl 1180.35073
[6] G. A. Chechkin, Yu.O. Koroleva, and L.-E. Persson, ”On the Precise Asymptotics of the Constant in the Friedrich’s Inequality for Functions, Vanishing on the Part of the Boundary with Microinhomogeneous Structure,” J. Inequal. Appl. 2007, Article ID 34138 (2007). · Zbl 1144.35357
[7] G. A. Chechkin and E. L. Ostrovskaya, ”On Behaviour of a Body Randomly Perforated Along the Boundary,” in Book of Abstracts of the International Conference ”Differential Equations and Related Topics” dedicated to the Centenary Anniversary of Ivan G. Petrovskii (May 22–27, 2001), p. 88 (MSU, Moscow, 2001).
[8] G. A. Chechkin, A. L. Piatnitski, and A. S. Shamaev, Homogenization: Methods and Applications, Transl. Math. Monogr. 234 (American Mathematical Society, Providence, 2007). · Zbl 1128.35002
[9] M. I. Cherdantsev, ”Asymptotics of the Eigenvalue of the Laplace Operator in a Domain with a Singularly Perturbed Boundary,” Mat. Zametki 78(2), 299–307 (2005) [Math. Notes 78 (1–2), 270–278 (2005)]. · doi:10.4213/mzm2575
[10] L. C. Evans, Partial Differential Equations, Grad. Stud. Math. 19 (American Mathematical Society, Providence, 1998). · Zbl 0902.35002
[11] R. R. Gadyl’shin, ”Characteristic Frequencies of Bodies with Thin Pipes. I. Convergence and Estimates,” Mat. Zametki 54(6), 10–21 (1993) [Math. Notes 54 (6), 1192–1199 (1993)].
[12] R. R. Gadyl’shin, ”On the Eigenvalue Asymptotics for Periodically Clamped Membranes,” Algebra i Analiz 10(1), 3–19 (1998) [St. Petersburg Math. J. 1 (10), 1–14 (1999)]. · Zbl 0907.35094
[13] R. R. Gadyl’shin, ”Convergence Method of Asymptotic Expansions in a Singularly-Perturbed Boundary-Value Problem for the Laplace Operator,” Itogi Nauki Tekhn. Ser. Sovr. Mat. Prilozh. Temat. Obz. 5, 3–32 (2003) [J. Math. Sci. 125 (5), 579–609 (2005)].
[14] V. V. Jikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functionals (Springer, Berlin, 1994).
[15] M. Lobo, O.A. Oleinik, M.E. Pérez, and T. A. Shaposhnikova, ”On Homogenization of Solutions of Boundary Value Problems in Domains, Perforated along Manifolds,” Ann. Sc. Norm. Super. Pisa Cl. Sci. 4-e série 25(3–4), 611–629 (1997). · Zbl 1170.35320
[16] V. A. Marchenko and E.Ya. Khruslov, Boundary-Value Problems in Domains with Fine-Grained Boundaries (Naukova Dumka, Kiev, 1974) [in Russian].
[17] V. A. Marchenko and E.Ya. Khruslov, Homogenized Models of Microinhomogeneous Media (Naukova Dumka, Kiev, 2005) [in Russian]; Translation: Homogenization of Partial Differential Equations, Progr. Math. Phys. 46 (Birkhäuser, Boston, 2006).
[18] V. G. Mazja, Sobolev Spaces (Leningrad Univ., Leningrad, 1985) [in Russian].
[19] S. Ozawa, ”Approximation of Green’s Function in a Region with Many Obstacles,” in Geometry and Analysis of Manifolds (Katata, Kyoto, 1987), Lect. Notes in Math. 1339 (Springer, Berlin, 1988), pp. 212–225.
[20] M.Yu. Planida, ”On the Convergence of the Solutions of Singularly Perturbed Boundary-Value Problems for the Laplacian,” Mat. Zametki 71(6), 867–877 (2002) [Math Notes 71 (6), 794–803 (2002)]. · Zbl 1130.35306 · doi:10.4213/mzm391
[21] K. Rektorys, The Method of Discretization on Time and Partial Differential Equations, Math. Appl. (East European Series) 4 (D. Reidel, Dordrecht-Boston, 1982). · Zbl 0522.65059
[22] E. Sánchez-Palencia, ”Boundary Value Problems in Domains Containing Perforated Walls,” in Nonlinear Partial Differential Equations and Their Applications. Collège de France Seminar, Vol. III (Paris, 1980/1981), Res. Notes Math. 70 (Pitman, Boston-London, 1982), pp. 309–325.
[23] S. L. Sobolev, Some Applications of Functional Analysis to Mathematical Physics, Transl. Math. Monogr. 90 (American Mathematical Society, Providence, 1991). · Zbl 0732.46001
[24] S. L. Sobolev, Selected Problems in the Theory of Function Spaces and Generalized Functions (Nauka, Moscow, 1989) [in Russian]. · Zbl 0667.46025
[25] V. S. Vladimirov, Equations of Matematical Physics (Nauka, Moscow; 1976) [in Russian].
[26] K. Yosida, Functional Analysis, Reprint of the sixth (1980) edition. Classics in Mathematics (Springer, Berlin, 1995).
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