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Homogenization in Perforated Domains
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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: urn:nbn:se:kth:diva-147702ISBN: 978-91-7595-213-0 (print)OAI: oai:DiVA.org:kth-147702DiVA: diva2:731865
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
List of papers
1. Optimal Control of the Obstacle Problem in a Perforated Domain
Open this publication in new window or tab >>Optimal Control of the Obstacle Problem in a Perforated Domain
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.

Keyword
Optimal control, Perforated domains, Obstacle problem, Homogenization
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-103123 (URN)10.1007/s00245-012-9170-4 (DOI)000308229100004 ()2-s2.0-84867278843 (Scopus ID)
Note

QC 20121008

Available from: 2012-10-08 Created: 2012-10-04 Last updated: 2017-12-07Bibliographically approved
2. Highly oscillating thin obstacles
Open this publication in new window or tab >>Highly oscillating thin obstacles
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.

Keyword
Homogenization, Thin obstacle, Ergodicity, Discrepancy, Corrector
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-121108 (URN)10.1016/j.aim.2013.01.007 (DOI)000316512500008 ()2-s2.0-84874437735 (Scopus ID)
Note

QC 20130422

Available from: 2013-04-22 Created: 2013-04-19 Last updated: 2017-12-06Bibliographically approved
3. Gamma-convergence of Oscillating Thin Obstacles
Open this publication in new window or tab >>Gamma-convergence of Oscillating Thin Obstacles
2013 (English)In: Eurasian Mathematical Journal, ISSN 2077-9879, Vol. 4, 88-100 p.Article in journal (Refereed) Published
Keyword
obstacle problem, homogenization theory, Γ-convergence
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-146323 (URN)
Note

QC 20140613

Available from: 2014-06-11 Created: 2014-06-11 Last updated: 2017-12-05Bibliographically approved
4. Application of Uniform Distribution to Homogenization of a Thin Obstacle Problem with p-Laplacian
Open this publication in new window or tab >>Application of Uniform Distribution to Homogenization of a Thin Obstacle Problem with p-Laplacian
2014 (English)In: Communications in Partial Differential Equations, ISSN 0360-5302, E-ISSN 1532-4133, Vol. 39, no 10, 1870-1897 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we study the homogenization of p-Laplacian with thin obstacle in a perforated domain. The obstacle is defined on the intersection between a hyperplane and a periodic perforation. We construct the family of correctors for this problem and show that the solutions for the epsilon-problem converge to a solution of a minimization problem of similar form but with an extra term involving the mean capacity of the obstacle. The novelty of our approach is based on the employment of quasi-uniform convergence. As an application we obtain Poincare's inequality for perforated domains.

Keyword
Capacity, Free boundary, Homogenization, p-Laplacian, Perforated domains, Quasiuniform convergence, Thin obstacle, Uniform distributions
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-147634 (URN)10.1080/03605302.2014.895013 (DOI)000341003700004 ()2-s2.0-84906490732 (Scopus ID)
Note

QC 20140919. Updated from accepted to published.

Available from: 2014-07-01 Created: 2014-07-01 Last updated: 2017-12-05Bibliographically approved

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