Open AccessResolvent Approximations in L2-Norm for Elliptic Operators Acting in a Perforated Space
Author(s) -
S. E. Pastukhova
Publication year - 2020
Publication title -
sovremennaâ matematika. fundamentalʹnye napravleniâ
Language(s) - English
Resource type - Journals
eISSN - 2949-0618
pISSN - 2413-3639
DOI - 10.22363/2413-3639-2020-66-2-314-334
Subject(s) - resolvent , mathematics , semi elliptic operator , multiplication operator , constant coefficients , operator (biology) , compact operator , operator norm , elliptic operator , mathematical analysis , norm (philosophy) , remainder , resolvent formalism , differential operator , quasinormal operator , finite rank operator , pure mathematics , operator theory , hilbert space , arithmetic , banach space , political science , extension (predicate logic) , gene , programming language , biochemistry , chemistry , repressor , computer science , transcription factor , law
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.