This article is cited in 17 scientific papers (total in 17 papers)
Uniform convergence and asymptotics for problems in domains finely perforated along a prescribed manifold in the case of the homogenized Dirichlet condition
Abstract:
A boundary value problem for a second-order elliptic equation with variable coefficients is considered in a multidimensional domain which is perforated by small holes along a prescribed manifold. Minimal natural conditions are imposed on the holes. In particular, all of these are assumed to be of approximately the same size and have a prescribed minimal distance to neighbouring holes, which is also a small parameter. The shape of the holes and their distribution along the manifold are arbitrary. The holes are divided between two sets in an arbitrary way. The Dirichlet condition is imposed on the boundaries of holes in the first set and a nonlinear Robin boundary condition is imposed on the boundaries of holes in the second. The sizes and distribution of holes with the Dirichlet condition satisfy a simple and easily verifiable condition which ensures that these holes disappear after homogenization and a Dirichlet condition on the manifold in question arises instead. We prove that the solution of the perturbed problem converges to the solution of the homogenized one in the $W_2^1$-norm uniformly with respect to the right-hand side of the equation, and an estimate for the rate of convergence that is sharp in order is deduced. The full asymptotic solution of the perturbed problem is also constructed in the case when the holes form a periodic set arranged along a prescribed hyperplane.
Bibliography: 32 titles.
Keywords:
perforated domain, boundary value problem, homogenization, uniform convergence, estimate for the rate of convergence, asymptotic.
Citation:
D. I. Borisov, A. I. Mukhametrakhimova, “Uniform convergence and asymptotics for problems in domains finely perforated along a prescribed manifold in the case of the homogenized Dirichlet condition”, Sb. Math., 212:8 (2021), 1068–1121
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\paper Uniform convergence and asymptotics for problems in domains finely perforated along a~prescribed manifold in the case of the homogenized Dirichlet condition
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\yr 2021
\vol 212
\issue 8
\pages 1068--1121
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This publication is cited in the following 17 articles:
D. I. Borisov, “Operator estimates for non-periodically perforated domains: disappearance of cavities”, Applicable Analysis, 103:5 (2024), 859–873
Denis I. Borisov, “Operator estimates for non‐periodically perforated domains with Dirichlet and nonlinear Robin conditions: Strange term”, Math Methods in App Sciences, 47:6 (2024), 4122
J. I. Díaz, T. A. Shaposhnikova, A. V. Podolskiy, “Aperiodical Isoperimetric Planar Homogenization with Critical Diameter: Universal Non-local Strange Term for a Dynamical Unilateral Boundary Condition”, Dokl. Math., 2024
J. I. Diaz, T. A. Shaposhnikova, A. V. Podolskiy, “Aperiodical isoperimetric planar homogenization with critical diameter: universal non-local strange term for a dynamical unilateral boundary condition”, Doklady Rossijskoj akademii nauk. Matematika, informatika, processy upravleniâ, 515:1 (2024), 18
A. I. Mukhametrakhimova, “Operator estimates for non–periodic perforation along boundary: homogenized Dirichlet condition”, Ufa Math. J., 16:4 (2024), 83–93
D. I. Borisov, “Homogenization of Operators with Perturbations of General Form in the Lower-Order Terms”, Math. Notes, 113:1 (2023), 138–142
A. Khrabustovskyi, “Operator estimates for the Neumann sieve problem”, Ann. Mat. Pura Appl. (4), 202:4 (2023), 1955–1990
D. I. Borisov, J. Kříž, “Operator estimates for non-periodically perforated domains with Dirichlet and nonlinear Robin conditions: vanishing limit”, Anal. Math. Phys., 13:1 (2023), 5
D. I. Borisov, “Geometric approximation of point interactions in two-dimensional domains for non-self-adjoint operators”, Mathematics, 11:4 (2023), 947
T. A. Suslina, “Operator-theoretic approach to the homogenization of Schrödinger-type equations with periodic coefficients”, Russian Math. Surveys, 78:6 (2023), 1023–1154
D. I. Borisov, D. M. Polyakov, “Resolvent convergence for differential–difference operators with small variable translations”, Mathematics, 11:20 (2023), 4260
D. I. Borisov, “Homogenization for operators with arbitrary perturbations in coefficients”, Journal of Differential Equations, 369 (2023), 41
D. I. Borisov, A. I. Mukhametrakhimova, “Uniform convergence for problems with perforation alogn a given manifold and with a nonlinear Robin condition on the boundaries of cavities”, St. Petersburg Math. J., 35:4 (2024), 611–652
D. I. Borisov, A. I. Mukhametrakhimova, “Asymptotics for problems in perforated domains with Robin nonlinear condition on the boundaries of cavities”, Sb. Math., 213:10 (2022), 1318–1371
D. I. Borisov, M. N. Konyrkulzhaeva, “Operator $L_2$ -estimates for two-dimensional problems with rapidly alternating boundary conditions”, J. Math. Sci. (N.Y.), 267:3 (2022), 319–337
D. I. Borisov, “Operator estimates for planar domains with irregularly curved boundary. The Dirichlet and Neumann conditions”, J. Math. Sci. (N.Y.), 264:5 (2022), 562–580
D. I. Borisov, “Asymptotic expansion of solution to Dirichlet problem in perforated domain: strange term case”, Ufa Math. J., 14:4 (2022), 26–41
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