Uniform convergence for problems with perforation along a given manifold and with a nonlinear Robin condition on the boundaries of cavities

Borisov, D.; Mukhametrakhimova, A. R.

doi:10.1090/spmj/1819

Uniform convergence for problems with perforation along a given manifold and with a nonlinear Robin condition on the boundaries of cavities
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by D. I. Borisov and A. R. Mukhametrakhimova;
Translated by: S. V. Kislyakov

St. Petersburg Math. J. 35 (2024), 611-652

DOI: https://doi.org/10.1090/spmj/1819

Published electronically: October 4, 2024

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Abstract:

A boundary value problem is treated for a second order elliptic equation with variable coefficients in a multidimensional domain perforated by small cavities closely spaced along a given manifold. The sizes of the cavities are assumed to be of the same smallness order, while their shapes and the distribution along the manifold are arbitrary. A nonlinear Robin condition is imposed on the boundaries of the cavities. It is proved that the solution of the perturbed problem converges to that of the homogenized problem in the $L_2$- and $W_2^1$-norms uniformly with respect to the $L_2$-norm of the right-hand side of the equation. Estimates of the convergence rates are also obtained.

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