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Selected Topics in Homogenization
Mid Sweden University, Faculty of Science, Technology and Media, Department of Engineering and Sustainable Development.ORCID iD: 0000-0001-9984-2424
2012 (English)Doctoral thesis, monograph (Other academic)
Abstract [en]

The main focus of the present thesis is on the homogenization of some selected elliptic and parabolic problems. More precisely, we homogenize: non-periodic linear elliptic problems in two dimensions exhibiting a homothetic scaling property; two types of evolution-multiscale linear parabolic problems, one having two spatial and two temporal microscopic scales where the latter ones are given in terms of a two-parameter family, and one having two spatial and three temporal microscopic scales that are fixed power functions; and, finally, evolution-multiscale monotone parabolic problems with one spatial and an arbitrary number of temporal microscopic scales that are not restricted to be given in terms of power functions. In order to achieve homogenization results for these problems we study and enrich the theory of two-scale convergence and its kins. In particular the concept of very weak two-scale convergence and generalizations is developed, and we study an application of this convergence mode where it is employed to detect scales of heterogeneity.

Abstract [sv]

Huvudsakligt fokus i avhandlingen ligger på homogeniseringen av vissa elliptiska och paraboliska problem. Mer precist så homogeniserar vi: ickeperiodiska linjära elliptiska problem i två dimensioner med homotetisk skalning; två typer av evolutionsmultiskaliga linjära paraboliska problem, en med två mikroskopiska skalor i både rum och tid där de senare ges i form av en tvåparameterfamilj, och en med två mikroskopiska skalor i rum och tre i tid som ges i form av fixa potensfunktioner; samt, slutligen, evolutionsmultiskaliga monotona paraboliska problem med en mikroskopisk skala i rum och ett godtyckligt antal i tid som inte är begränsade till att vara givna i form av potensfunktioner. För att kunna uppnå homogeniseringsresultat för dessa problem så studerar och utvecklar vi teorin för tvåskalekonvergens och besläktade begrepp. Speciellt så utvecklar vi begreppet mycket svag tvåskalekonvergens med generaliseringar, och vi studerar en tillämpningav denna konvergenstyp där den används för att detektera förekomsten av heterogenitetsskalor.

Place, publisher, year, edition, pages
Östersund: Mittuniversitetet , 2012. , x + 168 p.
Mid Sweden University doctoral thesis, ISSN 1652-893X ; 127
Keyword [en]
homogenization theory, H-convergence, two-scale convergence, very weak two-scale convergence, multiscale convergence, very weak multiscale convergence, evolution-multiscale convergence, very weak evolution-multiscale convergence, λ-scale convergence, non-periodic linear elliptic problems, evolution-multiscale linear parabolic problems, evolution-multiscale monotone parabolic problems, detection of scales of heterogeneity
National Category
Mathematical Analysis
URN: urn:nbn:se:miun:diva-16230ISBN: 978-91-87103-19-3OAI: diva2:527223
Public defence
2012-06-11, Q221, Akademigatan 1, Östersund, 13:00 (Swedish)
Available from: 2012-05-23 Created: 2012-05-18 Last updated: 2013-11-01Bibliographically approved

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