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Optimal Control of the Obstacle Problem in a Perforated Domain
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
2012 (English)In: Applied mathematics and optimization, ISSN 0095-4616, E-ISSN 1432-0606, Vol. 66, no 2, 239-255 p.Article in journal (Refereed) Published
Abstract [en]

We study the problem of optimally controlling the solution of the obstacle problem in a domain perforated by small periodically distributed holes. The solution is controlled by the choice of a perforated obstacle which is to be chosen in such a fashion that the solution is close to a given profile and the obstacle is not too irregular. We prove existence, uniqueness and stability of an optimal obstacle and derive necessary and sufficient conditions for optimality. When the number of holes increase indefinitely we determine the limit of the sequence of optimal obstacles and solutions. This limit depends strongly on the rate at which the size of the holes shrink.

Place, publisher, year, edition, pages
2012. Vol. 66, no 2, 239-255 p.
Keyword [en]
Optimal control, Perforated domains, Obstacle problem, Homogenization
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-103123DOI: 10.1007/s00245-012-9170-4ISI: 000308229100004Scopus ID: 2-s2.0-84867278843OAI: oai:DiVA.org:kth-103123DiVA: diva2:559114
Note

QC 20121008

Available from: 2012-10-08 Created: 2012-10-04 Last updated: 2017-12-07Bibliographically 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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