Abstract
The asymptotic behavior as h→0 of the solution of a mixed boundary value problem is investigated for an elliptic (in the sense of Petrovskii) system of second-order differential equations in the n-dimensional cylinder Q h =Ω×(−h/2, h/2) of small altitude h; Ω is a domain in R n −1. The limit problem in Ω contains a small parameter ε=h θ,θ ∈ θ (0, 1), for higher-order derivatives and degenerates regularly, as ε→ 0, into an elliptic problem of a lower order. It is shown that the limit problem and its corresponding degenerate problem (ε=0) are uniquely solvable. An estimate for the difference of solutions of the original and the limit problem in the energy norm is established. As an example, a problem on the deformation of a thin plate in the framework of the Cosserat continuum is considered.
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References
S. N. Leora, S. A. Nazarov, and A. V. Proskura, “Derivation of limit equations for elliptic problems in thin domains using a computer,” Zh. Vychisl. Mat. i Mat. Fiz.,26, No. 7, 1032–1048 (1986).
J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vols. I-III, Springer, New York (1972–1973).
M. G. Dzhavadov, “The asymptotic behavior of the solutions of a boundary value problem for a second-order elliptic equation in thin domains,” Differents. Uravn.,4, No. 10, 1901–1909 (1968).
I. E. Zino and É. A. Tropp, Asymptotic Methods in Problems of the Theory of Heat Conduction and Thermoelasticity [in Russian], Leningr. Gos. Univ., Leningrad (1978).
S. A. Nazarov, “Structure of the solutions of elliptic boundary value problems in thin domains,” Vestnik Leningrad. Univ., Mat. Mekh. Astronom., No. 7, Issue 2, 65–68 (1982).
M. I. Vishik and L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with a small parameter,” Uspekhi Mat. Nauk,12, No. 5, 3–122 (1957).
S. A. Nazarov, “The Vishik-Lyusternik method for elliptic boundary value problems in regions with conic points. I. Problem in a cone,” Sibirsk. Mat. Zh.,22, No. 4, 142–163 (1981).
Yu. A. Romashev, “On the regularity of the degeneracy of a boundary value problem of the Cosserat continuum theory at an angular point,” Manuscript deposited at VINITI, August 15, 1985, No. 6031 Dep., Moscow (1985).
S. Nazarov, “On the index of an elliptic boundary value problem with a small parameter at the highest derivatives,” Differents. Uravn. i Primenen. Trudy Sem. Processy Optimal. Upravl., I Sekts., No. 24, 75–85 (1979).
M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations, Noordhoff, Leyden (1974).
V. A. Dudnikov and S. A. Nazarov, “Asymptotically exact equations of thin plates on the basis of Cosserat's theory,” Dokl. Akad. Nauk SSSR,262, No. 2, 306–309 (1982)
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Translated from Problemy Matematicheskogo Analiza, No. 11, pp. 191–208, 1990.
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Nazarov, S.A. Asymptotic behavior of the solution of an elliptic boundary value problem in a thin domain. J Math Sci 64, 1351–1362 (1993). https://doi.org/10.1007/BF01098027
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DOI: https://doi.org/10.1007/BF01098027