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G-Convergence and Homogenization of Some Sequences of Monotone Differential Operators

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G-Convergence and Homogenization of Some Sequences of Monotone Differential Operators Thesis for the degree of Doctor of Philosophy Östersund 2009 G-CONVERGENCE AND HOMOGENIZATION OF SOME SEQUENCES OF MONOTONE DIFFERENTIAL OPERATORS Liselott Flodén Supervisors: Associate Professor Anders Holmbom, Mid Sweden University Professor Nils Svanstedt, Göteborg University Professor Mårten Gulliksson, Mid Sweden University Department of Engineering and Sustainable Development Mid Sweden University, SE‐831 25 Östersund, Sweden ISSN 1652‐893X, Mid Sweden University Doctoral Thesis 70 ISBN 978‐91‐86073‐36‐7 i Akademisk avhandling som med tillstånd av Mittuniversitetet framläggs till offentlig granskning för avläggande av filosofie doktorsexamen onsdagen den 3 juni 2009, klockan 10.00 i sal Q221, Mittuniversitetet, Östersund. Seminariet kommer att hållas på svenska. G-CONVERGENCE AND HOMOGENIZATION OF SOME SEQUENCES OF MONOTONE DIFFERENTIAL OPERATORS Liselott Flodén © Liselott Flodén, 2009 Department of Engineering and Sustainable Development Mid Sweden University, SE‐831 25 Östersund Sweden Telephone: +46 (0)771‐97 50 00 Printed by Kopieringen Mittuniversitetet, Sundsvall, Sweden, 2009 ii Tothememoryofmyfather G-convergence and Homogenization of some Sequences of Monotone Differential Operators Liselott Flodén Department of Engineering and Sustainable Development Mid Sweden University, SE-831 25 Östersund, Sweden Abstract This thesis mainly deals with questions concerning the convergence of some sequences of elliptic and parabolic linear and non-linear oper- ators by means of G-convergence and homogenization. In particular, we study operators with oscillations in several spatial and temporal scales. Our main tools are multiscale techniques, developed from the method of two-scale convergence and adapted to the problems stud- ied. For certain classes of parabolic equations we distinguish different cases of homogenization for different relations between the frequen- cies of oscillations in space and time by means of different sets of local problems. The features and fundamental character of two-scale con- vergence are discussed and some of its key properties are investigated. Moreover, results are presented concerning cases when the G-limit can be identified for some linear elliptic and parabolic problems where no periodicity assumptions are made. v Acknowledgements This thesis was completed at the Department of Engineering and Sustainable Development, Mid Sweden University in Östersund. First of all, I would like to express my deep gratitude to my main supervisor Anders Holmbom. An- ders’ constant encouragement and guidance during the process has been ab- solutely crucial to the result. I also want to thank my supervisor Nils Svanst- edt, Göteborg University, for his support and inspirational ideas throughout the work, and my other supervisor Mårten Gulliksson for valuable advice. I would also like to thank all colleagues and friends here in the Q-building. You all make it a pleasure to work here. In particular, I would like to thank Marianne Olsson for her friendship and cooperation and Marie Ohlsson for beingagoodfriend. Jeanette Silfver, I am glad and grateful to have had the privilege to be your colleague and friend. I miss you! Finally, to my family, Göran, Markus and Rickard, thank you for your patience and support, and for just being there! Östersund, April 2009 Liselott Flodén vii Notation For the convenience of the reader, we list some symbols and sets used in this thesis. X : Any linear space X0 : The dual space of X u : The norm of u X,whereX is a normed space k kX ∈ H : Any Hilbert space V : Any Banach space such that the embedding V H is continuous ⊆ V H V 0 : An evolution triple ⊆ ⊆ uh : A sequence of functions uh uh u : uh converges strongly to u © →ª h h u u: ©u ª converges weakly to u uh u∗ : ©uhª converges weakly* to u h h u u : ©u ª two-scale converges to u ε : A sequence ε (h) such that ε = ε (h) 0 as h { } © ª { } → →∞ : Any open bounded subset of RM with O smooth (at least Lipschitz) boundary ∂ : The boundary of O O ¯ : The closure of O O Ω : Any open bounded subset of RN with smooth (at least Lipschitz) boundary ΩT : The set Ω (0,T) × Ω¯ T : The set Ω¯ [0,T] × Y : Unit cube in RM ∗ N Y,Y1,Y2,...,Yn : Unit cubes in R n Y : The set Y1 ... Yn × × n m n,m : The set Y (0, 1) Y × ix a b : The scalar product of two vectors a and b in RN · (u, v)H : The inner product of u and v in a Hilbert space H , : The duality pairing between X0 and X h· ·iX0,X Below is a list of function spaces. All functions u are assumed to be measur- able. F ( ): Any space of functions u : R O O → M M Floc R : All functions u : R R such that their restriction to → any open bounded subset of RM belongs to F ( ) ¡ ¢ O O M F (Y ): All functions in Floc R that are the periodical ∗ repetition of some function in F (Y ) ¡ ¢ ∗ F ( ) /R : All functions in F ( ) with integral mean value zero over O O O Lp ( ): All functions u : R such that O O →p 1/p u Lp( ) = u (x) dx < , p 1 k k O O | | ∞ ≥ p N ¡R N¢ L ( ) : All functions u : R such that O N O → 1/p p u Lp( )N = ui (x) dx < , p 1 k k O | | ∞ ≥ µi=1 O ¶ P R L∞ ( ): All functions u : R such that O O → u L ( ) =ess supx u (x) < k k ∞ O ∈O | | ∞ W 1,p ( ): All functions u in Lp ( ) such that their first-order O distributional derivativesO belong to Lp ( ) 1O/p p p u 1,p = u + u < W ( ) Lp( ) Lp( )M k k O k k O k∇ k O ∞ ³ ´ W 1,p ( ): All functions u in W 1,p ( ) such that u =0on ∂ 0 O O O u W 1,p( ) = u Lp( )M k k 0 O k∇ k O 1,q 1,p 1 1 W − ( ): The dual space of W ( ), + =1 O 0 O p q C ( ): All continuous functions u : R O O → x C ¯ : All continuous functions u : ¯ R O O → u C( ¯) =supx ¯ u (x) ¡ ¢ k k O ∈O | | C∞ ( ): All infinitely differentiable functions u : R O O → D ( ): All infinitely differentiable functions u : R with O compact support in O → O D0 ( ): All distributions in O O M C (Y ): All continuous and Y -periodic functions u : R R ∗ u =sup u∗(y) → C(Y ) y Y k k ∗ ∈ ∗ | | C∞ (Y ): All infinitely differentiable and Y -periodic functions ∗ ∗ u : RM R → 1,2 1,2 M W (Y ): All Y -periodic functions in Wloc R ∗ ∗ 1,2 1,2 ¡ ¢ W (Y ) /R : All functions u in W (Y ) with integral ∗ ∗ mean value zero over Y ∗ u 1,2 = u 2 M W (Y )//R L (Y ) k k ∗ k∇ k ∗ L2 ( ; X): All functions u : X such that O O → 2 1/2 u L2( ;X) =( u (x, ) X dx) < k k O O k · k ∞ R L∞ ( ; X): All functions u : X such that O O → u L ( ;X) =ess supx u (x) X < k k ∞ O ∈O k k ∞ C ¯; X : All continuous functions u : ¯ X O O → u C( ¯,X) =supx ¯ u (x, ) X ¡ ¢ k k O ∈O k · k D ( ; X): All infinitely differentiable functions u : X with O compact support in O → O 1,2 2 2 W (0,T; V,V 0): All u L (0,T; V ) such that ∂tu L (0,T; V 0) ∈ ∈ u 1,2 = u 2 + ∂tu 2 k kW (0,T ;V,V 0) k kL (0,T ;V ) k kL (0T ;V 0) xi Contents 1 Introduction 1 1.1 Convergence for differentialoperators.............. 2 1.2Homogenizationandperiodicmedia............... 3 1.3Outlineofthethesis....................... 8 2 Monotone operators 11 2.1Theconceptofmonotoneoperators............... 11 2.2MonotoneoperatorsonBanachspaces............. 14 2.2.1 Existenceanduniquenessofthesolution........ 15 2.2.2 Elliptic partial differentialequations.......... 18 2.3Monotoneparabolicoperators.................. 21 2.3.1 Existenceanduniquenessofthesolution........ 22 2.3.2 Parabolic partial differentialequations......... 24 2.3.3 Parabolicequationswithmultiplescales........ 25 3 G-convergence 30 3.1 Elliptic G-convergence...................... 31 3.1.1 Linearellipticequations................. 31 3.1.2 Monotoneellipticequations............... 35 3.2 Parabolic G-convergence..................... 36 3.2.1 Linearparabolicequations................ 37 3.2.2 Monotoneparabolicequations.............. 38 4 Multiscale convergence 43 4.1Two-scaleconvergence...................... 43 4.1.1 Thefeaturesoftwo-scaleconvergence.......... 44 4.1.2 An intuitive discussion concerning two-scale convergence 52 4.1.3 Some further notes on the appearance of a second vari- ableinthetwo-scalelimit................ 63 4.2Theconceptofmultiscaleconvergence............. 69 4.2.1 Thenatureofmultiscaleconvergence.......... 70 4.2.2 Thefeaturesofmultiscaleconvergence......... 72 4.2.3 Evolutionmultiscaleconvergence............ 74 xiii 5 Homogenization of periodic operators 85 5.1Homogenizationandmultiplescaleexpansions......... 85 5.1.1 Homogenization by means of two different methods . 86 5.1.2 The asymptotic expansion and two-scale convergence . 89 5.2Homogenizationwithseveralspatialscales........... 98 5.2.1 Linearellipticequations................. 98 5.2.2 Monotoneellipticequations...............101 5.2.3 Monotoneparabolicequations..............102 5.3Homogenizationbymultiscaleconvergence...........106 5.3.1 Linear parabolic equations with one spatial and two temporalmicroscales...................107 5.3.2 Monotone parabolic equations with two spatial microscales andonemicroscaleintime................113 6 G-convergence for some special operators 132 6.1Theellipticcase..........................132 6.2Theparaboliccase........................141 xiv 1 Introduction Heterogeneous materials such as paper, concrete, nylon and plastic exist everywhere around us and new species are constantly being invented. Since they have almost infinitely many methods of application, the demand for a mathematical understanding of these materials is huge. Although we may ex- perience a heterogeneous material as being homogeneous at the macroscopic level, its exact behavior depends on the properties of the component materi- als: how they are arranged and what proportions they have. So, to be able to describe the properties of a heterogeneous material, we must investigate it at the microscopic level.
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