On operator estimates for elliptic equations in two-dimensional domains with fast oscillating boundary and frequent alternation of boundary conditions

D. I. Borisov^ab, R. R. Suleimanov<sup>c

a</sup> Institute of Mathematics with Computing Centre, Ufa Federal Research Centre of the Russian Academy of Sciences, Ufa, Russia
^b Peoples' Friendship University of Russia, Moscow, Russia
^c Ufa University of Science and Technology, Ufa, Russia

Abstract: A second-order semilinear elliptic equation is considered in an arbitrary two-dimensional domain with boundary that is rapidly oscillating with small amplitude. The oscillations are arbitrary, with no assumption of periodicity or local periodicity. Frequently alternating Dirichlet and Neumann boundary conditions are imposed on this boundary. In the case under consideration a Dirichlet problem with the same differential equation arises in the limit under the homogenization. The main results obtained are $W^1_2$- and $L_2$-operator estimates.
Bibliography: 36 titles.

Keywords: oscillating boundary, operator estimate, semilinear elliptic equations, frequently alternating boundary conditions.

MSC: Primary 35J25; Secondary 47A10

Received: 17.08.2024 and 28.10.2024

DOI: 10.4213/sm10171

 English version: Sbornik: Mathematics, 2025, 216:8, 1037–1054

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