Abstract
We consider a system of second-order semilinear elliptic equations in a multidimensional domain with an arbitrarily curved boundary contained in a narrow layer along the unperturbed boundary. The Dirichlet or Neumann condition is imposed on the curved boundary. In the case of the Neumann condition, rather natural and weak conditions are additionally imposed on the structure of the curving. Under these conditions, we show that the homogenized problem is one for the same system of equations in the unperturbed problem with a boundary condition of the same kind as on the perturbed boundary. The main result is operator \(W_{2}^{1}\)- and L2- estimates.
Similar content being viewed by others
REFERENCES
E. Sanchez-Palencia, Non-homogeneous Media and Vibration Theory (Springer-Verlag, New York, 1980).
O. A. Oleinik, G. A. Iosifyan, and A. S. Shamaev, Mathematical Problems in the Theory of Highly Inhomogeneous Elastic Media (Mosk. Gos. Univ., Moscow, 1990) [in Russian].
A. G. Belyaev, A. G. Mikheev, and A. S. Shamaev, Comput. Math. Math. Phys. 32, 1121–1133 (1992).
G. A. Chechkin, E. A. Akimova, and S. A. Nazarov, Dokl. Math. 37 (9), 1276–1283 (2001).
V. V. Grushin and S. Yu. Dobrokhotov, Math. Notes 95 (3), 324–337 (2014).
V. A. Kozlov and S. A. Nazarov, St. Petersburg Math. J. 22 (6), 941–983 (2011).
S. E. Pastukhova, Differ. Equations 37 (9), 1276–1283 (2001).
Y. Amirat, O. Bodart, G. A. Chechkin, and A. L. Piatnitski, Stoch. Process. Appl. 121 (1), 1–23 (2011).
J. Arrieta and S. Brushi, Discret. Cont. Dyn. Syst. Ser. B 14 (2), 327–351 (2010).
G. A. Chechkin, A. Friedman, and A. L. Piatnitski, J. Math. Anal. Appl. 231 (1), 213–234 (1999).
W. Jäger and A. Mikelić, Commun. Math. Phys. 232 (3), 429–455 (2003).
Myong-Hwan Ri, Preprint (2013). arXiv:1311.0977.
N. Neuss, M. Neuss-Radu, and A. Mikelić, Appl. Anal. 85 (5), 479–502 (2006).
D. Borisov, G. Cardone, L. Faella, and C. Perugia, J. Differ. Equations 255 (12), 4378–4402 (2013).
D. I. Borisov, J. Math. Sci. 264 (5), 562–580 (2022).
Funding
This work was supported by the Russian Science Foundation, project no. 23-11-00009, https://rscf.ru/en/project/23-11-00009/.
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The authors of this work declare that they have no conflicts of interest.
Additional information
Translated by I. Ruzanova
Publisher’s Note.
Pleiades Publishing remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Borisov, D.I., Suleimanov, R.R. Operator Estimates for Problems in Domains with Singularly Curved Boundary: Dirichlet and Neumann Conditions. Dokl. Math. (2024). https://doi.org/10.1134/S1064562424701758
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1134/S1064562424701758