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October 18, 2025

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

Key Points

Operator estimates establish bounds for solutions of elliptic equations with rapid boundary oscillations, emphasizing functional behavior.
The analysis provides $W^1_2$- and $L_2$-estimates, showcasing the applicability of these results in complex boundary settings.
The study considers Dirichlet and Neumann boundary conditions, highlighting their frequent alternation under a rapidly oscillating boundary.
A limiting Dirichlet problem emerges through the process of homogenization, connecting complex oscillations to simpler boundary representations.

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 W¹₂- and L₂-operator estimates. Bibliography: 36 titles.

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