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Application of Uniform Distribution to Homogenization of a Thin Obstacle Problem with p-Laplacian
Edinburgh University.
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
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.

Place, publisher, year, edition, pages
2014. Vol. 39, no 10, 1870-1897 p.
Keyword [en]
Capacity, Free boundary, Homogenization, p-Laplacian, Perforated domains, Quasiuniform convergence, Thin obstacle, Uniform distributions
National Category
Mathematics
Identifiers
URN: urn:nbn:se:kth:diva-147634DOI: 10.1080/03605302.2014.895013ISI: 000341003700004Scopus ID: 2-s2.0-84906490732OAI: oai:DiVA.org:kth-147634DiVA: diva2:731200
Note

QC 20140919. Updated from accepted to published.

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