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Highly oscillating thin obstacles
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2013 (English)In: Advances in Mathematics, ISSN 0001-8708, E-ISSN 1090-2082, Vol. 237, 286-315 p.Article in journal (Refereed) Published
Abstract [en]

The focus of this paper is on a thin obstacle problem where the obstacle is defined on the intersection between a hyper-plane Gamma in R-n and a periodic perforation T-epsilon of R-n, depending on a small parameters epsilon > 0. As epsilon -> 0, it is crucial to estimate the frequency of intersections and to determine this number locally. This is done using strong tools from uniform distribution. By employing classical estimates for the discrepancy of sequences of type {k alpha}(k=1)(infinity), alpha is an element of R, we are able to extract rather precise information about the set Gamma boolean AND T-epsilon. As epsilon -> 0, we determine the limit u of the solution u(epsilon) to the obstacle problem in the perforated domain, in terms of a limit equation it solves. We obtain the typical "strange term" behavior for the limit problem, but with a different constant taking into account the contribution of all different intersections, that we call the averaged capacity. Our result depends on the normal direction of the plane, but holds for a.e. normal on the unit sphere in R-n.

Place, publisher, year, edition, pages
2013. Vol. 237, 286-315 p.
Keyword [en]
Homogenization, Thin obstacle, Ergodicity, Discrepancy, Corrector
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-121108DOI: 10.1016/j.aim.2013.01.007ISI: 000316512500008Scopus ID: 2-s2.0-84874437735OAI: oai:DiVA.org:kth-121108DiVA: diva2:617091
Note

QC 20130422

Available from: 2013-04-22 Created: 2013-04-19 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Homogenization in Perforated Domains
Open this publication in new window or tab >>Homogenization in Perforated Domains
2014 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Homogenization theory is the study of the asymptotic behaviour of solutionsto partial differential equations where high frequency oscillations occur.In the case of a perforated domain the oscillations are due to variations in thedomain of the equation. The four articles that constitute this thesis are devotedto obstacle problems in perforated domains. Paper A treats an optimalcontrol problem where the objective is to control the solution to the obstacleproblem by the choice of obstacle. The optimal obstacle in the perforated domain,as well as its homogenized limit, are characterized in terms of certainauxiliary problems they solve. In papers B,C and D the authors solve homogenizationproblems in a perforated domain where the perforation is definedas the intersection between a periodic perforation and a hyper plane. Thetheory of uniform distribution is an indespensible tool in the analysis of theseproblems. Paper B treats the obstacle problem for the Laplace operator andthe authors use correctors to derive a homogenized equation. Paper D is ageneralization of paper B to the p-Laplacian. The authors employ capacitytechniques which are well adapted to the problem. In Paper C the obstaclevaries on the same scale as the perforations. In this setting the authorsemploy the theory of Gamma-convergence to prove a homogenization result.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2014. vii, 22 p.
Series
TRITA-MAT-A, 2014:11
National Category
Natural Sciences
Research subject
Mathematics
Identifiers
urn:nbn:se:kth:diva-147702 (URN)978-91-7595-213-0 (ISBN)
Public defence
2014-09-05, F3, Lindstedtsvägen 25, KTH, Stockholm, 13:00 (English)
Opponent
Supervisors
Note

QC 20140703

Available from: 2014-07-03 Created: 2014-07-02 Last updated: 2014-07-03Bibliographically approved

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