A two-dimensional boundary value problem is studied for a scalar elliptic second-order equation of the general form with frequent alternation of boundary conditions. The alternation is defined on small, closely spaced parts of the boundary, on which the Dirichlet boundary condition and the nonlinear third boundary condition are set alternately. The distribution and size of these segments are arbitrary. The case is considered when, upon averaging, the Dirichlet boundary condition completely disappears and only the original nonlinear third boundary condition remains. The main result is estimates for the $W_2^1$- and $L_2$-norms of the difference between the solutions of the perturbed and averaged problems, which are uniform in the $L_2$-norm of the right-hand side and characterize the rate of convergence. It is shown that these estimates are exact in the order of smallness.
Keywords: two-dimensional boundary value problem, elliptic equation, frequent alternation, homogenization, operator estimate
Received January 30, 2023
Revised February 16, 2023
Accepted February 20, 2023
Funding Agency: The work is partially supported by the Grant Agency of Czech Republic (grant no. 22-18739S) and by the Ministry of Education of the Russian Federation in the framework of the state task (agreement No.073-03-2023-010 on 26.01.2023).
Denis Ivanovich Borisov, Dr. Phys.-Math. Sci., Prof., Institute of Mathematics of Ufa Federal Research Center of Russian Academy of Sciences, Ufa, 450008 Russia; Bashkir State Pedagogical University named after M. Akmullah, Ufa, the Republic of Bashkortostan, Russia; University of Hradec Králové, Hradec Kralove, 500 03, Czech Republic e-mail: borisovdi@yandex.ru
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Cite this article as: D.I. Borisov. Operator estimates in two-dimensional problems with a frequent alternation in the case of small parts with the Dirichlet condition. Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 36–55; Proceedings of the Steklov Institute of Mathematics (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S33–S52.
