Operator estimates for homogenization of the Robin Laplacian in a perforated domain
Result description
Let epsilon > 0 be a small parameter. We consider the domain omega := omega omega epsilon, where omega is an open domain in Rn, and D epsilon is a family of small balls of the radius d epsilon = o(epsilon) distributed periodically with period epsilon. Let ?epsilon be the Laplace operator in ?epsilon subject to the Robin condition partial differential u partial differential n + gamma epsilon u = 0 with gamma epsilon <= 0 on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on d epsilon and gamma epsilon, the operator ?epsilon converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in omega and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of L2 -> L2 and L2 -> H1 operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.(c) 2022 Elsevier Inc. All rights reserved.
Keywords
HomogenizationPerforated domainNorm resolvent convergenceOperator estimatesSpectral convergenceVarying Hilbert spaces
The result's identifiers
Result code in IS VaVaI
Alternative codes found
RIV/62690094:18470/22:50019457
Result on the web
DOI - Digital Object Identifier
Alternative languages
Result language
angličtina
Original language name
Operator estimates for homogenization of the Robin Laplacian in a perforated domain
Original language description
Let epsilon > 0 be a small parameter. We consider the domain omega := omega omega epsilon, where omega is an open domain in Rn, and D epsilon is a family of small balls of the radius d epsilon = o(epsilon) distributed periodically with period epsilon. Let ?epsilon be the Laplace operator in ?epsilon subject to the Robin condition partial differential u partial differential n + gamma epsilon u = 0 with gamma epsilon <= 0 on the boundary of the holes and the Dirichlet condition on the exterior boundary. Kaizu (1985, 1989) and Brillard (1988) have shown that, under appropriate assumptions on d epsilon and gamma epsilon, the operator ?epsilon converges in the strong resolvent sense to the sum of the Dirichlet Laplacian in omega and a constant potential. We improve this result deriving estimates on the rate of convergence in terms of L2 -> L2 and L2 -> H1 operator norms. As a byproduct we establish the estimate on the distance between the spectra of the associated operators.(c) 2022 Elsevier Inc. All rights reserved.
Czech name
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Czech description
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Classification
Type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
—
OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Journal of Differential Equations
ISSN
0022-0396
e-ISSN
1090-2732
Volume of the periodical
338
Issue of the periodical within the volume
NOV
Country of publishing house
US - UNITED STATES
Number of pages
44
Pages from-to
474-517
UT code for WoS article
000859448200002
EID of the result in the Scopus database
2-s2.0-85136559022
Basic information
Result type
Jimp - Article in a specialist periodical, which is included in the Web of Science database
OECD FORD
Pure mathematics
Year of implementation
2022