Abstract:
We consider an elliptic operator in a multi-dimensional domain with frequent alternation of Dirichlet and Robin conditions. We study the case, when the homogenized operator has Robin condition with an additional coefficient generated by the geometry of the alternation. We prove the norm resolvent convergence of the perturbed operator to the homogenized one and obtain the estimate for the rate of convergence. We construct the complete asymptotic expansion for the resolvent in the case, when it acts on sufficiently smooth functions.
Citation:
T. F. Sharapov, “On resolvent of multi-dimensional operators with frequent alternation of boundary conditions: critical case”, Ufa Math. J., 8:2 (2016), 65–94
\Bibitem{Sha16}
\by T.~F.~Sharapov
\paper On resolvent of multi-dimensional operators with frequent alternation of boundary conditions: critical case
\jour Ufa Math. J.
\yr 2016
\vol 8
\issue 2
\pages 65--94
\mathnet{http://mi.mathnet.ru/eng/ufa346}
\crossref{https://doi.org/10.13108/2016-8-2-65}
\elib{https://elibrary.ru/item.asp?id=26255228}
Linking options:
https://www.mathnet.ru/eng/ufa346
https://doi.org/10.13108/2016-8-2-65
https://www.mathnet.ru/eng/ufa/v8/i2/p66
This publication is cited in the following 5 articles:
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, “Operator Estimates in Two-Dimensional Problems with a Frequent Alternation in the Case of Small Parts with the Dirichlet Condition”, Proc. Steklov Inst. Math. (Suppl.), 321, suppl. 1 (2023), S33–S52
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, “Asimptoticheskii analiz kraevykh zadach dlya operatora Laplasa s chastoi smenoi tipa granichnykh uslovii”, Differentsialnye uravneniya s chastnymi proizvodnymi, SMFN, 67, no. 1, Rossiiskii universitet druzhby narodov, M., 2021, 14–129
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