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Resolvent Approximations in L2-Norm for Elliptic Operators Acting in a Perforated Space

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Abstract

We study homogenization of a second-order elliptic differential operator Aε = −div a(x/ε) ×∇ acting in an ε-periodically perforated space, where ε is a small parameter. Coefficients of the operator Aε are measurable ε-periodic functions. The simplest case where coefficients of the operator are constant is also interesting for us. We find an approximation for the resolvent (Aε + 1)−1 with remainder term of order ε2 as ε → 0 in operator L2-norm on the perforated space. This approximation turns to be the sum of the resolvent (A0 + 1)−1 of the homogenized operator A0 = −div a0∇, a0 > 0 being a constant matrix, and some correcting operator εCε. The proof of this result is given by the modified method of the first approximation with the usage of the Steklov smoothing operator.

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Authors and Affiliations

  1. Russian Technological University (MIREA), Moscow, Russia

    S. E. Pastukhova

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Correspondence to S. E. Pastukhova.

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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 66, No. 2, Proceedings of the Crimean Autumn Mathematical School-Symposium, 2020.

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Pastukhova, S.E. Resolvent Approximations in L2-Norm for Elliptic Operators Acting in a Perforated Space. J Math Sci 265, 1008–1026 (2022). https://doi.org/10.1007/s10958-022-06097-z

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  • DOI: https://doi.org/10.1007/s10958-022-06097-z

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