Operator estimates for the Neumann sieve problem

0570282 - ÚJF 2024 RIV DE eng J - Journal Article
Khrabustovskyi, Andrii
Operator estimates for the Neumann sieve problem.
Annali di Matematica Pura ed Applicata. Roč. 202, č. 4 (2023), s. 1955-1990. ISSN 0373-3114. E-ISSN 1618-1891
Institutional support: RVO:61389005
Keywords : Homogenization * Perforated domain * Neumann sieve * Resolvent convergence * Operator estimates * Spectrum
OECD category: Pure mathematics
Impact factor: 1, year: 2022
Method of publishing: Limited access
https://doi.org/10.1007/s10231-023-01308-z

Let omega be a domain in R-n, gamma be a hyperplane intersecting omega, epsilon > 0 be a small parameter, and D-k,D-epsilon,D- k = 1, 2, 3 ... be a family of small holes in gamma n omega, when is an element of -> 0, the number of holes tends to infinity, while their diameters tends to zero. Let AE be the Neumann Laplacian in the perforated domain omega(epsilon) = omega \ gamma(epsilon), where gamma(epsilon) = gamma \ (UkDk,epsilon) ('sieve'). It is well-known that if the sizes of holes are carefully chosen, A(epsilon) converges in the strong resolvent sense to the Laplacian on omega \ gamma subject to the so-called delta'-conditions on gamma & cap, omega. In the current work we improve this result: under rather general assumptions on the shapes and locations of the holes we derive estimates on the rate of convergence in terms of L-2 L-2 and L-2 -> H-1 operator norms. In the latter case a special corrector is required.
Permanent Link: https://hdl.handle.net/11104/0343270