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Technische Universität Dresden
Herausgeber: Der Rektor
G-convergence of linear differentialequations.
Marcus Waurick
Institut für AnalysisMATH-AN-03-2013 -convergence of linear differentialequations.
Marcus WaurickInstitut für Analysis, Fachrichtung MathematikTechnische Universität [email protected] 3, 2018
Abstract.
We discuss G -convergence of linear integro-differential-algebaricequations in Hilbert spaces. We show under which assumptions it is generic for thelimit equation to exhibit memory effects. Moreover, we investigate which classesof equations are closed under the process of G -convergence. The results have ap-plications to the theory of homogenization. As an example we treat Maxwell’sequation with the Drude-Born-Fedorov constitutive relation. Keywords and phrases: G -convergence, integro-differential-algebraic equations, homogenization, integralequations, Maxwell’s equations Mathematics subject classification 2010: ontents Contents Introduction
We discuss some issues occuring in the homogenization of linear integro-differential equationsin Hilbert spaces. Similar to [19, 18, 20, 21] we understand homogenization theory as the studyof limits of sequences of equations in the sense of G -convergence. Whereas in [19, 18] (non-linear) ordinary differential equations in finite-dimensional space are considered, we choosethe perspective given in [14, 15, 20, 21, 1, 10, 11]. The abstract setting is the following. Definition 1.1 ( G -convergence, [28, p. 74], [25]) . Let H be a Hilbert space. Let ( A n : D ( A n ) j H → H ) n be a sequence of continuously invertible linear operators onto H and let B : D ( B ) j H → H be linear and one-to-one. We say that ( A n ) n G -converges to B if (cid:0) A − n (cid:1) n converges in the weak operator topology to B − , i.e., for all f ∈ H the sequence ( A − n ( f )) n converges weakly to some u , which satisfies u ∈ D ( B ) and B ( u ) = f . B is called the G -limit of ( A n ) n and we write A n G −→ B .Our starting point will be equations of the form ∂ M u + N u = f, where M , N are suitable operators in space-time and ∂ is the time-derivative established ina Hilbert space setting to be specified below (see also [16, 8]). In the usual framework ofhomogenization theory, one assumes M and N to be multiplication operators in space-time,i.e., there are mappings a and b such that M = a ( · ) and N = b ( · ) . Assuming well-posedness ofthe above equation, i.e., existence, uniqueness and continuous dependence on the right-handside f in a suitable (Hilbert space) framework, one is interested in the sequence of equations ∂ M n u n + N n u n = f (1)with M n = a ( n · ) and correspondingly for N n yielding a sequence of solutions ( u n ) n . Thequestion arises, whether the sequence ( u n ) n converges and if so whether the respective limit u satisfies an equation of similar form. A formal computation in (1) reveals that u n = ( ∂ M n + N n ) − f. Thus, if we show the convergence of ( ∂ M n + N n ) − in the weak opertor topology to someone-to-one mapping C =: B − , we deduce the weak convergence of ( u n ) n the limit of whichdenoted by u satisfies Bu = f. In other words, ( ∂ M n + N n ) G -converges to B .In this article we think of ( M n ) n and ( N n ) n to be bounded sequences of bounded linearoperators in space-time. We want to discuss assumptions on these sequences guaranteeing acompactness result with respect to G -convergence. Moreover, we outline possible assumptionsyielding the closedness under G -convergence and give examples for equations, where the asso-ciated sequences of differential operators itself are G -convergent. We exemplify our findingswith examples from the literature [14, 20, 21, 10, 11], highlight possible connections and give Note that the G -limit is uniquely determined, cf. [25, Proposition 4.1]. Setting and main theorems an example for a Drude-Born-Fedorov model in electro-magentism (see [7] and Example 3.11below), where homogenization theorems are – to the best of the author’s knowledge – not yetavailable in the literature. We will also underscore the reason of the limit equation to exhibitmemory effects. An heuristic explanation is the lack of continuity of computing the inversewith respect to the weak operator topology.In Section 2 we introduce the functional analytic setting used for discussing integro-differential-algebraic equations and state our main Theorems. We successively apply the results fromSection 2 to time-independent coefficients (Section 3), time-translation invariant coefficients(Section 4) and time-dependent coefficients (Section 5). In each of the Sections 3, 4 and 5we give examples and discuss whether particular classes of equations are closed under limitswith respect to G -convergence. In Section 6 we prove the main theorems of Section 2. Therespective proofs rely on elementary Hilbert space theory. The key fact giving way for computations is the possibility of establishing the time-derivativeas a continuously invertible normal operator in an exponentially weighted Hilbert space. For ν > we define the operator ∂ : H ν, ( R ) j L ν ( R ) → L ν ( R ) , f f ′ , where L ν ( R ) := L ( R , exp( − ν · ) λ ) is the space of square-integrable functions with respect tothe weighted Lebesgue measure exp( − ν · ) λ and H ν, ( R ) is the space of L ν ( R ) -functions withdistributional derivative in L ν ( R ) . We denote the scalar-product on L ν ( R ) by h· , ·i ν and theinduced norm by |·| ν . Of course the operator ∂ depends on the scalar ν . However, since it willbe obvious from the context, which value of ν is chosen, we will omit the explicit reference toit in the notation of ∂ . It can be shown that ∂ is continuously invertible ([16, Example 2.3]or [8, Corollary 2.5]). The norm bound of the inverse is /ν . Of course the latter constructioncan be extended to the Hilbert-space-valued case of L ν ( R ; H ) -functions . We will use thesame notation for the time-derivative. In order to formulate our main theorems related to thetheory of homogenization of ordinary differential equations, we need to introduce the followingnotion. Definition 2.1.
Let H , H be Hilbert spaces, ν > . We call a linear mapping M : D ( M ) j \ ν> L ν ( R ; H ) → \ ν ≧ ν L ν ( R ; H ) (2) evolutionary (at ν > ) if D ( M ) is dense in L ν ( R ; H ) for all ν ≧ ν , if M extends to abounded linear operator from L ν ( R ; H ) to L ν ( R ; H ) for all ν ≧ ν and is such that lim sup ν →∞ k M k L ( L ν ( R ; H ) ,L ν ( R ; H )) < ∞ . We will also use the notation h· , ·i ν and |·| ν for the scalar product and norm in L ν ( R ; H ) , respectively. The notion “evolutionary” is inspired by the considerations in [16, Definition 3.1.14, p. 91], where polynomialexpressions in partial differential operators are considered. For a linear operator A from L ν ( R ; H ) to L ν ( R ; H ) we denote its operator norm by k A k L ( L ν ( R ; H ) ,L ν ( R ; H )) .If the spaces H and H are clear from the context, we shortly write k A k L ( L ν ) . M to some L ν will also be denoted by M . In particular, wewill not distinguish notationally between the different realizations of M as a bounded linearoperator for different ν as these realizations coincide on a dense subset. We define the set L ev ,ν ( H , H ) := { M ; M is as in (2) and is evolutionary at ν } . We abbreviate L ev ,ν ( H ) := L ev ,ν ( H , H ) . A subset M j L ev ,ν ( H , H ) is called bounded if lim sup ν →∞ sup M ∈ M k M k L ( L ν ) < ∞ . A family ( M ι ) ι ∈ I in L ev ,ν ( H , H ) is called bounded if { M ι ; ι ∈ I } is bounded.Note that L ev ,ν ( H , H ) j L ev ,ν ( H , H ) for all ν ≦ ν . We give some examples of evolu-tionary mappings.
Example 2.2.
Let H be a Hilbert space and M ∈ L ( H ) . Then there is a canonical extension M of M to L ν ( R ; H ) -functions such that ( M φ ) ( t ) := ( M φ ( t )) for all φ ∈ L ν ( R ; H ) and a.e. t ∈ R . In that way M ∈ T ν> L ev ,ν ( H ) . Henceforth, we shall not distinguish notationallybetween M and M . Example 2.3.
Let H be a Hilbert space and let L ∞ s ( R ; L ( H )) be the space of bounded stronglymeasurable functions from R to L ( H ) . For A ∈ L ∞ s ( R ; L ( H )) we denote the associated mul-tiplication operator on L ν ( R ; H ) by A ( m ) . Thus, also in this case, A ( m ) ∈ T ν> L ev ,ν ( H ) . Example 2.4.
For ν > let g ∈ L ν ( R > ) := { g ∈ L loc ( R ); g = 0 on R < , R R | g ( t ) | e − νt dt < ∞} . By Young’s inequality or by Example 4.3 below, we deduce that g ∗ ∈ L ev ,ν ( C ) , where g ∗ f denotes the convolution of some function f with g .To formulate our main theorems, we denote the weak operator topology by τ w . Convergencewithin this topology is denoted by τ w → . Limits within this topology are written as τ w - lim .We will extensively use the fact that for a separable Hilbert space H bounded subsets of L ( H ) , which are τ w -closed, are τ w -sequentially compact. Our main theorems concerning the G -convergence of differential equations read as follows. Theorem 2.5.
Let H be a separable Hilbert space, ν > . Let ( M n ) n , ( N n ) n be boundedsequences in L ev ,ν ( H ) . Assume there exists c > such that for all n ∈ N and ν ≧ ν Re hM n φ, φ i ν ≧ c h φ, φ i ν ( φ ∈ L ν ( R ; H )) . Then there exists ν ≧ ν and a subsequence ( n k ) k of ( n ) n such that ∂ M n k + N n k G −→ ∂ M − hom, + ∂ ∞ X j =1 − ∞ X ℓ =1 M − hom, M hom,ℓ ! j M − hom, , as k → ∞ in L ν ( R ; H ) , where M hom, = τ w - lim k →∞ M − n k Setting and main theorems and M hom,ℓ = τ w - lim k →∞ M − n (cid:0) − ∂ − N n M − n (cid:1) ℓ . Remark . (a) It should be noted that the positive-definiteness condition in Theorem 2.7 isa well-posedness condition, i.e., a condition for ∂ M n k + N n k to be continuously invertible forall ν sufficiently large. Indeed, for f ∈ L ν ( R ; H ) and u ∈ L ν ( R ; H ) with ( ∂ M n + N n ) u = f we multiply by ∂ − and get (cid:0) M n + ∂ − N n (cid:1) u = ∂ − f. The positive definiteness condition yields, see also Lemma 6.1, the invertibility of M n . Hence,we arrive at (cid:0) M − n ∂ − N n (cid:1) u = M − n ∂ − f. Choosing ν > sufficiently large, we deduce that the operator (cid:0) M − n ∂ − N n (cid:1) is continu-ously invertible with a Neumann series expression.(b) If N = 0 in Theorem 2.5, then we deduce that equations of the form ∂ M u = f are closedunder the process of G -convergence. If N 6 = 0 , then the above theorem suggests that this isnot true for equations of the form ( ∂ M + N ) u = f . However, if we consider ∂ M + N as ∂ (cid:0) M + ∂ − N (cid:1) , the equations under consideration in Theorem 2.5 are closed under G -limits.Indeed, the limit may be represented by ∂ M − hom, + ∞ X k =1 − ∞ X ℓ =1 M − hom, M hom,ℓ ! k M − hom, . In the forthcoming sections we will further elaborate the aspect of closedness under G -limits.In system or control theory one is interested in differential-algebraic systems, see e.g. [9]. We,thus, formulate the analogous statement for (integro-differential-)algebraic systems. Theorem 2.7.
Let H , H be separable Hilbert spaces, ν > . Let ( M n ) n , (cid:16) N ijn (cid:17) n be boundedsequences in L ev ,ν ( H ) and L ev ,ν ( H j , H i ) , respectively ( i, j ∈ { , } ). Assume there exists c > such that for all n ∈ N and ν ≧ ν we have for all ( φ, ψ ) ∈ L ν ( R ; H ⊕ H ) Re hM n φ, φ i ν ≧ c | φ | ν , Re hN n ψ, ψ i ν ≧ c | ψ | ν . Then there exists ν ≧ ν and a subsequence ( n k ) k of ( n ) n such that ∂ (cid:18) M n k
00 0 (cid:19) + (cid:18) N n k N n k N n k N n k (cid:19) G −→ (cid:18) ∂
00 1 (cid:19) M − hom, , N − hom, − , ! + ∞ X ℓ =1 − M − hom, , N − hom, − , ! M (1) ! ℓ M − hom, , N − hom, − , ! , here we put N n := N n − N n (cid:0) N n (cid:1) − N n ( n ∈ N ) as well as M (1) := (cid:18) P ∞ ℓ =1 M hom,ℓ, P ∞ ℓ =0 M hom,ℓ, P ∞ ℓ =0 M hom,ℓ, P ∞ ℓ =0 M hom,ℓ, (cid:19) and M hom,ℓ, = τ w - lim k →∞ M − n k (cid:0) − ∂ − N n k M − n k (cid:1) ℓ , M hom,ℓ, = τ w - lim k →∞ −M − n k (cid:0) − ∂ − N n k M − n k (cid:1) ℓ ∂ − N n k (cid:0) N n k (cid:1) − , M hom,ℓ, = τ w - lim k →∞ − (cid:0) N n k (cid:1) − N n k M − n k (cid:0) − ∂ − N n k M − n k (cid:1) ℓ , M hom,ℓ, = τ w - lim k →∞ (cid:0) N n k (cid:1) − N n k M − n k (cid:0) − ∂ − N n k M − n k (cid:1) ℓ ∂ − N n k (cid:0) N n k (cid:1) − , N hom, − , = τ w - lim k →∞ (cid:0) N n k (cid:1) − . Remark . (a) As in Theorem 2.5 the positive definiteness conditions in Theorem 2.7 serveas well-posed conditions for the respective (integro-differential-algebraic) equations. We willcompute the respetive inverse in the proof of Theorem 2.7.(b) Note that, by the definition of G -convergence, both the Theorems 2.5 and 2.7 implicitlyassert that the limit equations are well-posed, i.e., that the limit operator is continuouslyinvertible. In fact it will be the strategy of the respective proofs to compute the limit ofthe respective solution operators, which will be continuous linear operators and afterwardsinverting the limit.(c) Assume that in Theorem 2.7 the expressions M hom,ℓ, , M hom,ℓ, , M hom,ℓ, , M hom,ℓ, , N hom, − , can be computed without choosing subsequences. Then the sequence (cid:18) ∂ (cid:18) M n
00 0 (cid:19) + (cid:18) N n N n N n N n (cid:19)(cid:19) n is G -convergent. Indeed, the latter follows with a subsequence argument.(c) Assuming H = { } and, as a consequence, N ij = 0 for all i, j ∈ { , } except i = j = 0 ,we see that Theorem 2.7 is more general than Theorem 2.5. The generalization in Theorem2.7 is also needed in the theory of homogenization of partial differential equations, see e.g. [25,Theorem 4.4] for a more restrictive case. We give an example in the forthcoming sections.For convenience, we include easy examples that show that the assumptions in the abovetheorems are reasonable. Example 2.9 (Uniform positive definiteness condition does not hold, [25]) . Let H = C , ν > and, for n ∈ N , let M n = ∂ −
10 1 n , f ∈ L ν ( R ) \ { } . For n ∈ N , let u n ∈ L ν ( R ) be defined by ∂ M n u n = 1 n u n = f. Then ( u n ) n is not relatively weakly compact and contains no weakly convergent subsequence.9 Time-independent coefficients
Example 2.10 (Boundedness assumption does not hold) . Let H = C , ν > and, for n ∈ N ,let M n = ∂ − n , f ∈ L ν ( R ) . For n ∈ N , let u n ∈ L ν ( R ) be defined by ∂ M n u n = nu n = f. Then ( u n ) n converges to . Thus, a limit “equation” would be in fact the relation { }× L ν ( R ) j L ν ( R ) ⊕ L ν ( R ) .We will now apply our main theorems to particular situations. In this section, we treat time-independent coefficients. That is to say, we assume that theoperators in the sequences under consideration only act on the “spatial” Hilbert spaces H and H in Theorem 2.7 or H in Theorem 2.5. More precisely and similar to Example 2.2, fora bounded linear operator M ∈ L ( H , H ) there is a (canonical) extension to L ν -functions inthe way that ( M φ )( t ) := M ( φ ( t )) for φ ∈ L ν ( R ; H ) and a.e. t ∈ R . Thus M is evolutionary(Example 2.2). We only state the specialization of this situation for Theorem 2.5. The resultreads as follows. Theorem 3.1.
Let H be a separable Hilbert space, ν > . Let ( M n ) n , ( N n ) n be boundedsequences in L ( H ) . Assume there exists c > such that for all n ∈ N Re h M n φ, φ i H ≥ c h φ, φ i H ( φ ∈ H ) . Then there exists ν ≧ ν and a subsequence ( n k ) k of ( n ) n such that ∂ M n k + N n k G −→ ∂ M − hom, + ∂ ∞ X j =1 − ∞ X ℓ =1 M − hom, M hom,ℓ (cid:0) − ∂ − (cid:1) ℓ ! j M − hom, , as k → ∞ in L ν ( R ; H ) , where M hom, = τ w - lim k →∞ M − n k and M hom,ℓ = τ w - lim k →∞ M − n k (cid:0) N n K M − n K (cid:1) ℓ . Proof.
At first observe that
M ∂ − = ∂ − M for all bounded linear operators M ∈ L ( H ) .Moreover, the estimate Re h M φ, φ i H ≧ c h φ, φ i H for φ ∈ H also carries over to the analogousone for φ ∈ L ν ( R ; H ) and the extended M . Hence, the result follows from Theorem 2.5. Remark . As it has already been observed in [14, 21], the class of equations treated inTheorem 3.1 is not closed under G -convergence in general. The next example shows that thiseffect only occurs if the Hilbert space H is infinite-dimensional and the convergence of ( M n ) n and ( N n ) n is “weak enough” in a sense to be specified below.10 xample 3.3. Assume that H is finite-dimensional. Then ( M n ) n and ( N n ) n are a merebounded sequences of matrices with constant coefficients. In particular, the weak operatortopology coincides with the topology induced by the operator norm. Hence, the processes ofcomputing the inverse and computing the limit interchange and multiplication is a continuousprocess as well. Thus, assuming ( M n ) n and ( N n ) n to be convergent with the respective limits M and N , we compute ∂ M − hom, + ∂ ∞ X j =1 − ∞ X ℓ =1 M − hom, M hom,ℓ (cid:0) − ∂ − (cid:1) ℓ ! j M − hom, = ∂ (cid:16) τ w - lim k →∞ M − n k (cid:17) − + ∂ ∞ X j =1 − ∞ X ℓ =1 (cid:16) τ w - lim k →∞ M − n k (cid:17) − (cid:16) τ w - lim k →∞ M − n k (cid:0) N n k M − n k (cid:1) ℓ (cid:17) (cid:0) − ∂ − (cid:1) ℓ ! j (cid:16) τ w - lim k →∞ M − n k (cid:17) − = ∂ (cid:16) lim k →∞ M − n k (cid:17) − + ∂ ∞ X j =1 − ∞ X ℓ =1 (cid:16) lim k →∞ M − n k (cid:17) − (cid:16) lim k →∞ M − n k (cid:0) N n k M − n k (cid:1) ℓ (cid:17) (cid:0) − ∂ − (cid:1) ℓ ! j (cid:16) lim k →∞ M − n k (cid:17) − = ∂ M + ∂ ∞ X j =1 − ∞ X ℓ =1 M (cid:16) M − (cid:0) NM − (cid:1) ℓ (cid:17) (cid:0) − ∂ − (cid:1) ℓ ! j M = ∂ M + ∂ ∞ X j =1 − ∞ X ℓ =1 (cid:0) NM − (cid:1) ℓ (cid:0) − ∂ − (cid:1) ℓ ! j M = ∂ ∞ X j =0 − ∞ X ℓ =1 (cid:0) NM − (cid:1) ℓ (cid:0) − ∂ − (cid:1) ℓ ! j M = ∂ ∞ X ℓ =1 (cid:0) NM − (cid:1) ℓ (cid:0) − ∂ − (cid:1) ℓ ! − M = ∂ ∞ X ℓ =0 (cid:0) − NM − ∂ − (cid:1) ℓ ! − M = ∂ (cid:16)(cid:0) NM − ∂ − (cid:1) − (cid:17) − M = ∂ (cid:0) M + N∂ − (cid:1) = ∂ M + N. Thus, in finite-dimensional spaces, the above theorem restates the continuous dependence ofthe solution on the coefficients. Note that we only used that multiplication and computingthe inverse are continuous operations. Hence, the above calculation literally expresses the factof continuous dependence on the coefficients if H is infinite-dimensional and the sequences ( M n ) n and ( N n ) n converge in the strong operator topology. Thus, one can only expect thatthe limit expression differs from the one, which one might expect, if the actual convergence ofthe operators involved is strictly weaker than in the strong operator topology.We will turn to a more sophisticated example. For this we recall the concept of periodicity in R n , see e.g. [4]. Definition 3.4.
Let a : R n → C m × m be bounded and measurable. a is called ]0 , n -periodic ,if for all x ∈ R n and k ∈ Z n we have a ( x + k ) = a ( x ) .Moreover, recall the following well-known convergence result on periodic mappings, cf. e.g. [4,Theorem 2.6]. Theorem 3.5.
Let a : R n → C m × m be bounded and measurable and ]0 , n -periodic. Then ( a ( k · )) k converges in L ∞ ( R n ) m × m ∗ -weakly to the integral mean R [0 , n a ( y ) dy . 11 Time-independent coefficients
Remark . For any bounded measurable function a : R n → C m × m one can associate thecorresponding multiplication operator in L ( R n ) m . Hence, Theorem 3.5 states the fact thatin case of periodic a the sequence of associated multiplication operators of a ( k · ) converges inthe weak operator topology to the operator of multiplying with the respective integral mean.Indeed, this follows easily from L ( R n ) · L ( R n ) = L ( R n ) . See also [5, 8.10]. Example 3.7.
Let H = L ( R n ) m and let a, b : R n → C m × m be bounded, measurable and ]0 , n -periodic. We assume Re a ( x ) ≧ c for all x ∈ R n . Observe that any polynomial in a and b is ]0 , n -periodic and so is a − := (cid:0) x a ( x ) − (cid:1) . Thus, by Theorem 3.1, we deduce that ∂ a ( k · ) + b ( k · ) G −→ ∂ Z [0 , n a ( y ) − dy ! − + ∂ ∞ X j =1 − ∞ X ℓ =1 Z [0 , n a ( y ) − dy ! − Z [0 , n a ( y ) − (cid:0) b ( y ) a ( y ) − (cid:1) ℓ dy (cid:0) − ∂ − (cid:1) ℓ ! j Z [0 , n a ( y ) − dy ! − , as k → ∞ in L ν ( R ; H ) . Remark . In [20, Theorem 1.2] or [21] the author considers the equation ( ∂ + b k ( · )) u k = f with ( b k ) k being a [ α, β ] -valued (for some α, β ∈ R ) sequence of bounded, measurable mappingsdepending on one spatial variable. Also in that exposition a memory effect is derived. However,the method uses the concept of Young measures. The reason for that is the representation ofthe solution being a function of the oscillating coefficent. More precisely, the convergence ofthe sequence (cid:0) e tb ( k · ) (cid:1) k is addressed. In order to let k tend to infinity in this expression oneneeds a result on the (weak- ∗ ) convergence of (continuous) functions of bounded functions.This is where the Young-measures come into play, see e.g. [2, Section 2] and the referencestherein or [21, p. 930]. The result used is the following. There exists a family of probabiltymeasures ( ν x ) x supported on [ α, β ] such that for (a subsequence of) ( k ) k and all real continuousfunctions G we have G ◦ b k ( · ) → R ∋ x Z [ α,β ] G ( λ ) dν x ( λ ) ! as k → ∞ in L ∞ ( R ) ∗ -weakly. The family ( ν x ) x is also called the Young-measure associatedto ( b k ) k . With the help of the family ( ν x ) x a convolution kernel is computed such that therespective limit equation can be written as ∂ u ( t, x ) + b ( x ) u − Z t K ( x, t − s ) u ( x, s ) ds = f ( x, t ) , where b is a weak- ∗ -limit of a subsequence of ( b k ) k and K ( x, t ) = R R > e − λt dν x ( λ ) for a.e. t ∈ R > and x ∈ R . The relation to our above considerations is as follows. The resultinglimit equation within our approach can also be considered as an ordinary differential equationperturbed by a convolution term. Denoting limits with respect to the σ ( L ∞ , L ) -topology by ∗ - lim , we realize that Theorem 3.1 in this particular situation states that the limit equationadmits the form ∂ + ∂ ∞ X k =1 − ∞ X ℓ =1 ∗ - lim k →∞ ( b k ) ℓ (cid:0) − ∂ − (cid:1) ℓ ! k ∂ + b + ∞ X ℓ =2 ∗ - lim k →∞ ( b k ) ℓ (cid:0) − ∂ − (cid:1) ℓ − + ∂ ∞ X k =2 − ∞ X ℓ =1 ∗ - lim k →∞ ( b k ) ℓ (cid:0) − ∂ − (cid:1) ℓ ! k as k → ∞ in L ν ( R ; L ( R )) . Indeed, using [16, 6.2.6. Memory Problems, (b) p. 448] or [22,Theorem 1.5.6 and Remark 1.5.7], we deduce that the term ∞ X ℓ =2 ∗ - lim k →∞ ( b k ) ℓ (cid:0) − ∂ − (cid:1) ℓ − + ∂ ∞ X k =2 − ∞ X ℓ =1 ∗ - lim k →∞ ( b k ) ℓ (cid:0) − ∂ − (cid:1) ℓ ! k can be represented as a (temporal) convolution. Moreover, note that the choice of subsequencesis the same. Indeed, in the above rationale with the Young measure approach, by a densityargument, it suffices to choose a subsequence of ( b k ) k such that any polynomial of ( b k ) k converges ∗ -weakly. This choice of subsequences also suffices to deduce G -convergence of therespective equations within the operator-theoretic perspective treated in this exposition.In the next example, we consider a partial differential equation, which can be reformulatedas ordinary differential equation in an infinite-dimensional Hilbert space. More precisely, wetreat Maxwell’s equations with the Drude-Born-Fedorov material model, see e.g. [7]. In orderto discuss this equation properly, we need to introduce several operators from vector analysis. Definition 3.9.
Let Ω j R be open. Then we define curl c : C ∞ ,c (Ω) j L (Ω) → L (Ω) φ − ∂ ∂ ∂ − ∂ − ∂ ∂ φ, where we denote by ∂ i the partial derivative with respect to the i ’th variable, i ∈ { , , } .Moreover, introduce div c : C ∞ ,c (Ω) j L (Ω) → L (Ω)( φ , φ , φ ) X i =1 ∂ i φ i . We define curl := curl c , div := div c . The serves as a reminder for (the generalization of)the electric and the Neumann boundary condition, respectively. If Ω is simply connected, wealso introduce curl ⋄ : D (curl ⋄ ) j L (Ω) → L (Ω) φ curl φ, where D (curl ⋄ ) := { φ ∈ D (curl); curl φ ∈ D (div ) } . Remark . It can be shown that if Ω is simply connected with finite measure, then curl ⋄ is a selfadjoint operator, see [6, 7]. In that reference it is also stated that curl ⋄ has, except ,only discrete spectrum. In particular, this means that the intersection of the resolvent set of We denote by C ∞ ,c (Ω) the set of arbitrarily often differentiable functions with compact support in Ω . Time-independent coefficients curl ⋄ with R is non-empty. For other geometric properties of Ω resulting in the selfadjointnessof curl ⋄ , we refer to [17].We now treat a homogenization problem of the Drude-Born-Fedorov model as treated in [7]. Example 3.11.
Assume that Ω j R is open, simply connected and has bounded Lebesguemeasure. Invoking Remark 3.10 and following [7, Theorem 2.1], the equation (cid:18) ∂ (1 + η curl ⋄ ) (cid:18) ε µ (cid:19) + (cid:18) − curl ⋄ curl ⋄ (cid:19)(cid:19) (cid:18) EH (cid:19) = (cid:18) J (cid:19) (3)for η ∈ R such that − η ∈ ̺ (curl ⋄ ) , J ∈ L ν ( R ; L (Ω) ) and given ε, µ ∈ L ( L (Ω) ) beingstrictly positive selfadjoint operators, admits a unique solution ( E, H ) ∈ H ν, ( R ; L (Ω) ) . Indeed, multiplying (3) by (1 + η curl ⋄ ) − , we get that (cid:18) ∂ (cid:18) ε µ (cid:19) + curl ⋄ (1 + η curl ⋄ ) − (cid:18) −
11 0 (cid:19)(cid:19) (cid:18) EH (cid:19) = (1 + η curl ⋄ ) − (cid:18) J (cid:19) . Realizing that curl ⋄ (1 + η curl ⋄ ) − is a bounded linear operator by the spectral theorem forthe selfadjoint operator curl ⋄ , we get that ( E, H ) ∈ H ν, ( R ; L (Ω) ) solves the above equation.Note that the equation derived from (3) is a mere ordinary differential equation in an infinite-dimensional Hilbert space. Assume we are given bounded sequences of selfadjoint operators ( ε n ) n and ( µ n ) n satisfying ε n ≧ c and µ n ≧ c for some c > and all n ∈ N . For n ∈ N weconsider the problem (cid:18) ∂ (cid:18) ε n µ n (cid:19) + curl ⋄ (1 + η curl ⋄ ) − (cid:18) −
11 0 (cid:19)(cid:19) (cid:18) E n H n (cid:19) = (1 + η curl ⋄ ) − (cid:18) J (cid:19) and address the question of G -convergence of (a subsequence of) ( DBF n ) n := (cid:18) ∂ (cid:18) ε n µ n (cid:19) + curl ⋄ (1 + η curl ⋄ ) − (cid:18) −
11 0 (cid:19)(cid:19) n . Clearly, Theorem 3.1 applies and we get that (a subsequence of) ( DBF n ) n G -converges to ∂ M − hom, + ∂ ∞ X k =1 − ∞ X ℓ =1 M − hom, M hom,ℓ (cid:0) − ∂ − (cid:1) ℓ ! k M − hom, , as k → ∞ in L ν ( R ; H ) , where M hom, = τ w - lim k →∞ (cid:18) ε − n k µ − n k (cid:19) and M hom,ℓ = τ w - lim k →∞ (cid:18) ε − n k µ − n k (cid:19) (cid:18) curl ⋄ (1 + η curl ⋄ ) − (cid:18) − µ − n k ε − n k (cid:19)(cid:19) ℓ . Note that for ( E, H ) ∈ H ν, ( R ; L (Ω) ) being a solution of (3) can only be true in the distributional sense,which can be made more precise with the help of the extrapolation spaces of curl ⋄ . We shall, however, notfollow this reasoning here in more details and refer again to [7] or [16, Chapter 2].
14e have seen that the class of problems discussed in Theorem 3.1 in this section is not closedunder the G -convergence, unless N = 0 . In Theorem 3.1, we have seen that the limit equation can be described as a power series expres-sion in ∂ − . A possible way to generalize this is the introduction of holomorphic functions in ∂ − , see [16, Section 6.1, page 427]. To make this precise, we need the spectral representationfor ∂ − , the Fourier-Laplace transform L ν , which is given as the unitary operator being theclosure of C ∞ ,c ( R ) j L ν ( R ) → L ν ( R ) φ (cid:18) x √ π Z R e − ixy − νy φ ( y ) dy (cid:19) . Denoting by m : D ( m ) j L ( R ) → L ( R ) , f ( x xf ( x )) the multiplication-by-argument-operator with maximal domain D ( m ) , we arrive at the representation ∂ − = L ∗ ν (cid:18) im + ν (cid:19) L ν . Thus, for bounded and analytic functions M : B ( r, r ) → C with r > ν we define M (cid:0) ∂ − (cid:1) := L ∗ ν M (cid:18) im + ν (cid:19) L ν , where (cid:16) M (cid:16) im + ν (cid:17) φ (cid:17) ( t ) := M (cid:16) it + ν (cid:17) φ ( t ) for φ ∈ L ( R ) and a.e. t ∈ R . We canonicallyextend the above definitions to the case of vector-valued functions L ν ( R ; H ) with values in aHilbert space H . In this way, the definition of M (cid:0) ∂ − (cid:1) can be generalized to bounded andoperator-valued functions M : B ( r, r ) → L ( H , H ) for Hilbert spaces H and H . We denote H ∞ ( B ( r, r ); L ( H , H )) := { M : B ( r, r ) → L ( H , H ); M bounded, analytic } . A subset M j H ∞ ( B ( r, r ); L ( H , H )) is called bounded , if sup {k M ( z ) k ; z ∈ B ( r, r ) , M ∈ M } < ∞ . A family ( M ι ) ι ∈ I in H ∞ ( B ( r, r ); L ( H , H )) is bounded , if { M ι ; ι ∈ I } is bounded.We will treat some examples for H ∞ -functions of ∂ − below, see also [23]. In this reference,a homogenization theorem of problems of the kind treated in Theorem 2.5 with ( M n ) n = (cid:0) M n (cid:0) ∂ − (cid:1)(cid:1) n for a bounded sequence ( M n ) n in H ∞ has been presented, see [23, Theorem 5.2].Moreover, in [25, Theorem 4.4] a special case of an analogous result of Theorem 2.7 has beenpresented and used. In order to state a G -convergence theorem in a more general situation,note that (cid:8) M (cid:0) ∂ − (cid:1) ; M ∈ H ∞ ( B ( r, r ); L ( H , H )) (cid:9) j \ r <ν L ev ,ν ( H , H ) . Time-translation invariant coeffcients
The theorem reads as follows.
Theorem 4.1.
Let H , H be separable Hilbert spaces, ν > , r > ν . Let ( M n ) n , (cid:16) N ijn (cid:17) n be bounded sequences in H ∞ ( B ( r, r ); L ( H )) and H ∞ ( B ( r, r ); L ( H j , H i )) , respectively ( i, j ∈{ , } ). Assume there exists c > such that for all n ∈ N we have for all ( φ, ψ ) ∈ H ⊕ H and z ∈ B ( r, r ) Re h M n ( z ) φ, φ i H ≧ c | φ | H , Re h N n ( z ) ψ, ψ i H ≧ c | ψ | H . Then there exists ν > ν and a subsequence ( n k ) k of ( n ) n such that ∂ (cid:18) M n k (cid:0) ∂ − (cid:1)
00 0 (cid:19) + (cid:18) N n k (cid:0) ∂ − (cid:1) N n k (cid:0) ∂ − (cid:1) N n k (cid:0) ∂ − (cid:1) N n k (cid:0) ∂ − (cid:1) (cid:19) G −→ (cid:18) ∂
00 1 (cid:19) M hom, , (cid:0) ∂ − (cid:1) − N hom, − , (cid:0) ∂ − (cid:1) − ! + ∞ X ℓ =1 − M hom, , (cid:0) ∂ − (cid:1) − N hom, − , (cid:0) ∂ − (cid:1) − ! M (1) (cid:0) ∂ − (cid:1)! ℓ M hom, , (cid:0) ∂ − (cid:1) − N hom, − , (cid:0) ∂ − (cid:1) − ! , where we put N n := N n − N n (cid:0) N n (cid:1) − N n ( n ∈ N ) as well as M (1) (cid:0) ∂ − (cid:1) := P ∞ ℓ =1 M hom,ℓ, (cid:0) ∂ − (cid:1) (cid:0) ∂ − (cid:1) ℓ P ∞ ℓ =0 M hom,ℓ, (cid:0) ∂ − (cid:1) (cid:0) ∂ − (cid:1) ℓ +1 P ∞ ℓ =0 M hom,ℓ, (cid:0) ∂ − (cid:1) (cid:0) ∂ − (cid:1) ℓ P ∞ ℓ =0 M hom,ℓ, (cid:0) ∂ − (cid:1) (cid:0) ∂ − (cid:1) ℓ +1 ! and M hom,ℓ, ( z ) = τ w - lim k →∞ M n k ( z ) − (cid:0) − N n k ( z ) M n k ( z ) − (cid:1) ℓ ,M hom,ℓ, ( z ) = τ w - lim k →∞ − M n k ( z ) − (cid:0) − N n k ( z ) M n k ( z ) − (cid:1) ℓ N n k ( z ) (cid:0) N n k ( z ) (cid:1) − ,M hom,ℓ, ( z ) = τ w - lim k →∞ − (cid:0) N n k ( z ) (cid:1) − N n k ( z ) M n k ( z ) − (cid:0) − N n k ( z ) M n k ( z ) − (cid:1) ℓ ,M hom,ℓ, ( z ) = τ w - lim k →∞ (cid:0) N n k ( z ) (cid:1) − N n k ( z ) M n k ( z ) − (cid:0) − N n k ( z ) M n k ( z ) − (cid:1) ℓ N n k ( z ) (cid:0) N n k ( z ) (cid:1) − ,N hom, − , ( z ) = τ w - lim k →∞ (cid:0) N n k ( z ) (cid:1) − , for all z ∈ B (cid:16) ν , ν (cid:17) for some ν > ν ≧ ν . Proof.
Observe that bounded and analytic functions of ∂ − commute with ∂ − . Note thatthe only thing left to prove is that the operator-valued functions involved are indeed analyticfunctions of ∂ − . For this we need to introduce a topology on H ∞ ( B ( r, r ); L ( H , H )) . Let τ be the topology induced by the mappings H ∞ ( B ( r, r ); L ( H , H )) → H ( B ( r, r )) M
7→ h φ, M ( · ) ψ i , ( φ, ψ ) ∈ H ⊕ H , where H ( B ( r, r )) is the set of analytic functions endowed with thecompact open topology. In [23, Theorem 3.4] or [26, Theorem 4.3] it is shown that closedand bounded subsets of H ∞ ( B ( r, r ); L ( H , H )) are sequentially compact with respect to the τ -topology. Furthermore, by [23, Lemma 3.5], we have that if a bounded sequence ( T n ) n in H ∞ ( B ( r, r ); L ( H , H )) converges in the τ -topology then the operator sequence (cid:0) T n (cid:0) ∂ − (cid:1)(cid:1) n converges in the weak operator topology of L (cid:0) L ν ( R ; H ⊕ H ) (cid:1) . Putting all this together, wededuce that the assertion follows from Theorem 2.7. Remark . (a) Theorem 4.1 asserts that the time-translation invariant equations under con-sideration are closed under G -convergence. Though the formulas may become a bit cluttered,in principle, an iterated homogenization procedure is possible.(b) In [25, Theorem 4.4] operator-valued functions that are analytic at were treated. Thisassumption can be lifted. Indeed, we only require analyticity of the operator-valued functionsunder consideration on the open ball B ( r, r ) for some radius r > and do not assume thatany of these functions have holomorphic extensions to .We give several examples. Example 4.3.
Let ν > . In this example we treat integral equations of convolution type. Let ( g n ) n be a bounded sequence in L ν ( R > ) such that there is h ∈ L ν ( R > ) with k g ( t ) k ≦ h ( t ) for all n ∈ N and a.e. t ∈ R . For f ∈ C ∞ ,c ( R ) consider the equation u n + g n ∗ u n = f. (4)The latter equation fits into the scheme of Theorem 4.1 for H = C . Indeed, using that theFourier transform F translates convolutions into multiplication, we get for any g ∈ L ν ( R > ) and u ∈ L ν ( R ) for some ν > ν that g ∗ u = √ π L ∗ ν L ν g ( m ) L ν u = √ π L ∗ ν ( F g ) ( m − iν ) L ν u = √ π L ∗ ν ( F g ) (cid:18) − i im + ν ) − (cid:19) L ν u. The support and integrability condition of g implies analyticity of M g := √ π ( F g ) (cid:16) − i · ) (cid:17) on B ( r, r ) for < r < ν . The computation also shows that | g ∗ u | ν = (cid:12)(cid:12)(cid:12)(cid:12) √ π ( F g ) (cid:18) − i im + ν ) − (cid:19) L ν u (cid:12)(cid:12)(cid:12)(cid:12) L . ≦ π (cid:12)(cid:12)(cid:12)(cid:12) ( F g ) (cid:18) − i i ( · ) + ν ) − (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ∞ |L ν u | L ≦ π | ( F g ) (( · ) − iν ) | ∞ | u | ν , Time-translation invariant coeffcients where π | ( F g ) (( · ) − iν ) | ∞ := sup t ∈ R π | ( F g ) ( t − iν ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z R e − i ( t − iν ) y g ( y ) dy (cid:12)(cid:12)(cid:12)(cid:12) ≦ (cid:18)Z R e − νy | g ( y ) | dy (cid:19) , which tends to zero, if ν → ∞ . Thus, by our assumption on the sequence ( g n ) n having theuniform majorizing function h , there exists ν > such that we have ε := sup n ∈ N k g n ∗ k L ( L ν ( R )) < . Hence, we can reformulate (4) as follows (cid:0) M g n (cid:0) ∂ − (cid:1)(cid:1) u n = f, Thus with H = { } , H = H and N = (1 + M g n ) n Theorem 4.1 is applicable. (Note that Re N n ≧ − ε > for all n ∈ N ). The assertion states that, for a suitable subsequence forwhich we will use the same notation, we have (cid:0) M g n (cid:0) ∂ − (cid:1)(cid:1) G −→ N hom, − , (cid:0) ∂ − (cid:1) − with N hom, − , ( z ) = τ w - lim n →∞ (1 + M g n ( z )) − = τ w - lim n →∞ ∞ X ℓ =1 M g n ( z ) ℓ = τ w - lim n →∞ ∞ X ℓ =1 M ( g n ) ∗ ℓ ( z )= τ w - lim n →∞ M P ∞ ℓ =1 ( g n ) ∗ ℓ ( z ) for all z ∈ B (cid:16) ν , ν (cid:17) for some ν > ν ≧ ν , where we denoted the ℓ -fold convolution with afunction g by g ∗ ℓ , ℓ ∈ N .In [27] we discussed the following variant of Example 3.7. Example 4.4.
In the situation of Example 3.7, we let ( h k ) k be a convergent sequence ofpositive real numbers with limit h . Then Theorem 4.1 gives ∂ a ( k · ) + τ − h k b ( k · ) G −→ ∂ Z [0 , n a ( y ) − dy ! − + ∂ ∞ X k =1 − ∞ X ℓ =1 Z [0 , n a ( y ) − dy ! − Z [0 , n a ( y ) − (cid:0) τ − h b ( y ) a ( y ) − (cid:1) ℓ dy (cid:0) − ∂ − (cid:1) ℓ ! k Z [0 , n a ( y ) − dy ! − . τ − h = L ∗ ν e − h ( im + ν ) L ν .Fractional differential equations are also admissible as the following example shows. Example 4.5.
Again in the situation of Example 3.7, let ( α k ) k and ( β k ) k be convergentsequences in ]0 , and [ − , with limits α and β , resp. Then Theorem 4.1 gives ∂ α k a ( k · ) + ∂ β k b ( k · ) = ∂ ∂ α k − a ( k · ) + ∂ β k b ( k · ) G −→ ∂ α Z [0 , n a ( y ) − dy ! − + ∂ α ∞ X k =1 − ∞ X ℓ =1 ∂ α − Z [0 , n a ( y ) − dy ! − Z [0 , n a ( y ) − ∂ − α (cid:16) ∂ β − α b ( y ) a ( y ) − (cid:17) ℓ dy (cid:0) − ∂ − (cid:1) ℓ ! k · Z [0 , n a ( y ) − dy ! − . Remark . Note that all the above theorems on homogenization of differential equationsstraightforwardly apply to higher order equations. For example the equation n X k =0 ∂ k a k u = f can be reformulated as a first order system in the standard way. Another way is to integrate n − times, to get that n X k =0 ∂ k − n a k u = ∂ − ( n − f, which is by setting M ( ∂ − ) = a n and N ( ∂ − ) = P n − k =0 ∂ k − n a k of the form treated inTheorem 4.1. In this section we treat operators depending on temporal and spatial variables, which are,in contrast to the previous section, not time-translation invariant. Thus, the structural hy-pothesis of being analytic functions of ∂ − has to be lifted. Consequently, the expressionsfor the limit equations do not simplify in the manner as they did in the Theorems 3.1 and4.1. Particular ((non-)linear) equations have been considered in [14, 20, 10, 11, 15]. The mainobjective of this section is to give a sufficient criterion under which the choice of subsequencesin Theorem 2.7 is not required. We introduce the following notion. Definition 5.1.
Let H be a Hilbert space. A family (( T n,ι ) n ∈ N ) ι ∈ I of sequences of linearoperators in L ( H ) is said to have the product-convergence property , if for all k ∈ N and ( ι , . . . , ι k ) ∈ I k the sequence (cid:16)Q ki =1 T n,ι i (cid:17) n converges in the weak operator topology of L ( H ) .19 Time-dependent coefficients
Example 5.2.
Let
N, M ∈ N and denote P := { a : R N → C M × M ; a is [0 , N -periodic } .Theorem 3.5 asserts that the family (cid:0) ( a ( k · )) k ∈ N (cid:1) a ∈ P has the product-convergence property in L ( L ( R N ) M ) .We refer to the notion of homogenization algebras for other examples, see e.g. [12, 13]. Themain theorem of this section reads as follows. Recall from Example 2.3 the space L ∞ s ( R ; L ( H )) of strongly measurable bounded functions with values in L ( H ) endowed with the sup-norm.Moreover, recall that for A ∈ L ∞ s ( R ; L ( H )) the associated multiplication operator A ( m ) isevolutionary at ν for every ν > . Theorem 5.3.
Let H be a Hilbert space, ν > . Let (cid:0) ( A ι,n ) n (cid:1) ι be a family of bounded se-quences in L ∞ s ( R ; L ( H )) . Assume that the family (cid:0) ( A ι,n ( t )) n (cid:1) ι,t ∈ R has the product-convergenceproperty. Then (cid:0) ( A ι,n ( m )) n , (cid:0) ∂ − (cid:1) n (cid:1) ι has the product-convergence property.Remark . (a) With the latter result, it is possible to deduce that the choice of subsequencesin Theorem 2.7 is not needed. Indeed, assume that (cid:18) M n
00 0 (cid:19) + (cid:18) N n N n N n N n (cid:19) = (cid:18) M n ( m ) 00 0 (cid:19) + (cid:18) N n ( m ) N n ( m ) N n ( m ) N n ( m ) (cid:19) for some strongly measurable and bounded M n , N n , N n , N n , N n and assume that the family (cid:18)(cid:18) M n ( t ) 00 0 (cid:19) n , (cid:18) M n ( t ) −
00 0 (cid:19) n , (cid:18) N n ( t ) 00 0 (cid:19) n , (cid:18) N n ( t )0 0 (cid:19) n , (cid:18) N n ( t ) 0 (cid:19) n , (cid:18) N n ( t ) (cid:19) n , (cid:18) N n ( t ) − (cid:19) n (cid:19) t ∈ R satisfies the product-convergence property. Then Theorem 5.3 ensures that the limit expres-sions in Theorem 2.7 converge without choosing subsequences.(b) The crucial fact in Theorem 5.3 is that powers of ∂ − are involved. Indeed, let H bea Hilbert space, ν > . Let (cid:0) ( A ι,n ) n (cid:1) ι be a family of bounded sequences in L ∞ s ( R ; L ( H )) .Assume that, for every t ∈ R , the family (cid:0) ( A ι,n ( t )) n (cid:1) ι has the product-convergence property.Then (cid:0) ( A ι,n ( m )) n (cid:1) ι has the product-convergence property. Showing the assertion for two se-quences ( A ,n ) n and ( A ,n ) n and using the boundedness of the sequence ( A ,n ( m ) A ,n ( m )) n ,we deduce that it suffices to show weak convergence on a dense subset. For this to show let K, L j R be bounded and measurable and φ, ψ ∈ H . We get for n ∈ N and ν > that h χ K φ, A ,n ( m ) A ,n ( m ) χ L ψ i ν = Z K ∩ L h φ, A ,n ( t ) A ,n ( t ) ψ i e − νt dt → Z K ∩ L lim n →∞ h φ, A ,n ( t ) A ,n ( t ) ψ i e − νt dt, by dominated convergence. Lemma 5.5.
Let H be a Hilbert space, ν > . Let (cid:0) ( A ι,n ) n (cid:1) ι ∈ I be a family of bounded se-quences in L ∞ s ( R ; L ( H )) . Assume that the family (cid:0) ( A ι,n ( t )) n (cid:1) ι,t ∈ R has the product-convergence roperty. Then Q kj =1 (cid:0) A ι j ,n ( m ) , ∂ − (cid:1) ℓ j converges in the weak operator toplogogy for all k ∈ N , ℓ , . . . , ℓ k ∈ { , } × N and ι , . . . , ι k ∈ I .Proof. Let k ∈ N , ℓ , . . . , ℓ k ∈ { , } × N and ι , . . . , ι k ∈ I . Moreover, take φ, ψ ∈ H and K, L j R be bounded and measurable. For n ∈ N and ν > we compute * χ K φ, k Y j =1 (cid:0) A ι j ,n ( m ) , ∂ − (cid:1) ℓ j χ L ψ + ν = Z K * φ, A ι ,n ( s ) ℓ , Z s −∞ Z s ℓ , − −∞ · · · Z s −∞ A ι ,n ( s ) ℓ , Z s −∞ Z s ℓ , − −∞ · · · Z s −∞ · · · A ι k ,n ( s k − ) ℓ k, Z s k − −∞ Z s kℓk, − −∞ · · · Z s k −∞ χ L ( s k ) ψ + ds k · · · ds kℓ k, − ds kℓ k, − · · · ds · · · ds ℓ , − ds ℓ , − ds · · · ds ℓ , − ds ℓ , − e − νs ds = Z K Z s −∞ Z s ℓ , − −∞ · · · Z s −∞ Z s −∞ Z s ℓ , − −∞ · · · Z s −∞ · · · Z s k − −∞ Z s kℓk, − −∞ · · · Z s k −∞ D φ, A ι ,n ( t ) ℓ , A ι ,n ( s ) ℓ , · · · χ L ( s k ) ψ E ds k · · · ds kℓ k, − ds kℓ k, − · · · ds · · · ds ℓ , − ds ℓ , − ds · · · ds ℓ , − ds ℓ , − e − νs ds . Using dominated convergence, we deduce the convergence of the latter expression.
Proof of Theorem 5.3.
The proof follows easily with Lemma 5.5.Theorem 5.3 serves as a possibility to deduce G -convergence of differential operators, wherethe coefficients take values in, for example, periodic mappings as in Example 5.2. Anotherinstance is given in the following example. Example 5.6.
Let
A, B ∈ L ∞ ( R ) be -periodic, f ∈ C ∞ ,c ( R ) . Assume that A ≧ c for some c > . For n ∈ N and ν > consider ( ∂ A ( n · m ) + B ( n · m )) u n = f. Recall that from Theorem 2.5, in order to compute the limit equation, we have to computeexpressions of the form M hom,ℓ = τ w - lim n →∞ M − n (cid:0) − ∂ − N n M − n (cid:1) ℓ , where M n = A ( n · m ) and N n = B ( n · m ) , ℓ ∈ N . In what follows we adopt multiindex notation: For two operators A , B and k = ( k , k ) ∈ N we denote ( A, B ) k := A k B k . If k j is a multiindex in N , we denote its first and second component respectively by k j, and k j, . Time-dependent coefficients
In order to deduce G -convergence in the latter example we need the following theorem. Theorem 5.7.
Let A , . . . , A k ∈ L ∞ ( R ) be -periodic. Then for every ν > we have A n := A ( n · m ) k − Y j =1 ∂ − A j +1 ( n · m ) τ w ,n →∞ −−−−→ (cid:0) ∂ − (cid:1) k − k Y j =1 Z A j ( y ) dy ∈ L (cid:0) L ν ( R ) (cid:1) . Proof.
For n ∈ N and K, L j R bounded, measurable we compute h χ K , A n χ L i ν = Z K A ( nt ) Z t −∞ A ( nt ) Z t −∞ · · · Z t k − −∞ A k ( nt k ) χ L ( t k ) dt k · · · dt e − νt dt = Z K Z t −∞ Z t −∞ · · · Z t k − −∞ k Y j =1 A j ( nt j ) χ L ( t k ) e − νt dt k · · · dt = Z R · · · Z R | {z } k -times k Y j =1 A j ( nt j ) χ K ( t ) k Y j =2 χ R > ( t j − − t j ) χ L ( t k ) e − νt dt k · · · dt . Now, observe that ( t , · · · , t k ) χ K ( t ) (cid:16)Q kj =2 χ R > ( t j − − t j ) (cid:17) χ L ( t k ) e − νt ∈ L ( R k ) . More-over, the mapping ( t , · · · , t k ) Q kj =1 A j ( t j ) is [0 , k -periodic. Thus, by Theorem 3.5, weconclude that h χ K , A n χ L i ν → * χ K , (cid:0) ∂ − (cid:1) k − k Y j =1 Z A j ( y ) dyχ L + as n → ∞ for all K, L j R bounded and measurable. A density argument concludes theproof. Example 5.8 (Example 5.6 continued) . Thus, with the Theorems 5.7 and 2.5, we concludethat ( ∂ A ( n · m ) + B ( n · m )) G -converges to ∂ (cid:18)Z A ( y ) dy (cid:19) − + ∂ ∞ X k =1 − ∞ X ℓ =1 (cid:18)Z A ( y ) dy (cid:19) − Z A ( y ) dy (cid:18) − ∂ − Z B ( y ) A ( y ) dy (cid:19) ℓ ! k (cid:18)Z A ( y ) dy (cid:19) − = ∂ (cid:18)Z A ( y ) dy (cid:19) − ∞ X k =1 − ∞ X ℓ =1 (cid:18) − ∂ − Z B ( y ) A ( y ) dy (cid:19) ℓ ! k = ∂ (cid:18)Z A ( y ) dy (cid:19) − ∞ X k =0 − ∞ X ℓ =1 (cid:18) − ∂ − Z B ( y ) A ( y ) dy (cid:19) ℓ ! k = ∂ (cid:18)Z A ( y ) dy (cid:19) − ∞ X ℓ =1 (cid:18) − ∂ − Z B ( y ) A ( y ) dy (cid:19) ℓ ! − ∂ (cid:18)Z A ( y ) dy (cid:19) − ∞ X ℓ =0 (cid:18) − ∂ − Z B ( y ) A ( y ) dy (cid:19) ℓ ! − = ∂ (cid:18)Z A ( y ) dy (cid:19) − (cid:18) ∂ − Z B ( y ) A ( y ) dy (cid:19) = ∂ (cid:18)Z A ( y ) dy (cid:19) − + (cid:18)Z A ( y ) dy (cid:19) − Z B ( y ) A ( y ) dy. Remark . In [15], the authors consider an equation of the form ( ∂ + a n ( m )) u n = f inthe space L ( R ; L ( R )) with ( a n ) n being a bounded sequence in L ∞ ( R × R ) . Assuming weak- ∗ -convergence of ( a n ) n , the author shows weak convergence of ( u n ) n . The limit equation is aconvolution equation involving the Young-measure associated to the sequence ( a n ) n . Withinour reasoning, we cannot show that the whole sequence converges, unless any power of ( a n ) n converges in the weak- ∗ topology of L ∞ . However, as we illustrated above (see e.g. Example3.11) our approach has a wide range of applications, where the method involving Young-measures might fail to work. We will finally prove our main theorems. The proof relies on elementary Hilbert space con-cepts. We emphasize that the generality of the perspective hardly allows the introduction ofYoung-measures, which have proven to be useful in particular cases (see the sections above fora detailed discussion). Before we give a detailed account of the proofs of our main theorems,we state the following auxilaury result, which we state without proof.
Lemma 6.1.
Let H be a Hilbert space, T ∈ L ( H ) . Assume that Re T ≧ c for some c > .Then k T − k ≦ c and Re T − ≧ c k T k . Proof of Theorem 2.5.
For f ∈ C ∞ ,c ( R ; H ) let u n solve ( ∂ M n + N n ) u n = f. This yields u n = M − n (cid:0) ∂ − N n M − n (cid:1) − ∂ − f = M − n ∞ X ℓ =0 (cid:0) − ∂ − N n M − n (cid:1) ℓ ∂ − f = M − n + ∞ X ℓ =1 M − n (cid:0) − ∂ − N n M − n (cid:1) ℓ ! ∂ − f. Hence, choosing an appropriate subsequence, we arrive at an expression of the form u = M hom, + ∞ X ℓ =1 M hom,ℓ ! ∂ − f. Proof of the main theorems
We remark here that due to the (standard) estimate k T k ≦ lim inf k →∞ k T k k for a sequence ( T k ) k of bounded linear operators in some Hilbert space converging to T , the series P ∞ ℓ =1 M hom,ℓ converges with respect to the operator norm if ν is chosen large enough. Using the positivedefiniteness of M n for all n ∈ N and Lemma 6.1, we deduce that Re M − n ≧ c sup n ∈ N kM n k . By kM − n k ≦ c , we conclude that Re M hom, ≧ c sup n ∈ N kM n k and Re M − hom, ≧ c sup n ∈ N kM n k . We arrive at f = ∂ ∞ X ℓ =1 M − hom, M hom,ℓ ! − M − hom, u = ∂ ∞ X k =0 − ∞ X ℓ =1 M − hom, M hom,ℓ ! k M − hom, u = ∂ ∞ X k =1 − ∞ X ℓ =1 M − hom, M hom,ℓ ! k M − hom, u = ∂ M − hom, u + ∂ ∞ X k =1 − ∞ X ℓ =1 M − hom, M hom,ℓ ! k M − hom, u. Proof of Theorem 2.7.
We observe (cid:18) ∂ M n + N n N n N n N n (cid:19) = (cid:18) N n (cid:0) N n (cid:1) − (cid:19) (cid:18) ∂ M n + N n − N n (cid:0) N n (cid:1) − N n N n (cid:19) (cid:18) (cid:0) N n (cid:1) − N n (cid:19) . Thus, with B := (cid:16) ∂ M n + N n − N n (cid:0) N n (cid:1) − N n (cid:17) − (cid:18) ∂ M n + N n N n N n N n (cid:19) − = (cid:18) − (cid:0) N n (cid:1) − N n (cid:19) (cid:16) ∂ M n + N n − N n (cid:0) N n (cid:1) − N n (cid:17) − (cid:0) N n (cid:1) − (cid:18) −N n (cid:0) N n (cid:1) − (cid:19) = (cid:18) B − (cid:0) N n (cid:1) − N n B (cid:0) N n (cid:1) − (cid:19) (cid:18) −N n (cid:0) N n (cid:1) − (cid:19) B − B N n (cid:0) N n (cid:1) − − (cid:0) N n (cid:1) − N n B (cid:0) N n (cid:1) − N n B N n (cid:0) N n (cid:1) − + (cid:0) N n (cid:1) − ! . With the Neumann series expression derived in the previous theorem, i.e., B = M − n ∂ − + ∞ X ℓ =1 M − n (cid:0) − ∂ − N n M − n (cid:1) ℓ ∂ − with N n = N n − N n (cid:0) N n (cid:1) − N n , we get that (cid:18) ∂ M n + N n N n N n N n (cid:19) − = ∞ X ℓ =0 M − n (cid:0) − ∂ − N n M − n (cid:1) ℓ ∂ − −M − n (cid:0) − ∂ − N n M − n (cid:1) ℓ ∂ − N n (cid:0) N n (cid:1) − − (cid:0) N n (cid:1) − N n M − n (cid:0) − ∂ − N n M − n (cid:1) ℓ ∂ − (cid:0) N n (cid:1) − N n M − n (cid:0) − ∂ − N n M − n (cid:1) ℓ ∂ − N n (cid:0) N n (cid:1) − ! + (cid:18) (cid:0) N n (cid:1) − (cid:19) . With Theorem 2.5, we deduce convergence of the top left corner in the latter matrix. Similarly,we deduce convergence of the other expressions. Thus, for a suitable choice of subsequences,we arrive at ∞ X ℓ =1 (cid:18) M hom,ℓ, ∂ − M hom,ℓ, M hom,ℓ, ∂ − M hom,ℓ, (cid:19) + (cid:18) M hom, , ∂ − M hom, , M hom, , ∂ − M hom, , + N hom, − , (cid:19) . We observe that ∞ X ℓ =1 (cid:18) M hom,ℓ, ∂ − M hom,ℓ, M hom,ℓ, ∂ − M hom,ℓ, (cid:19) + (cid:18) M hom, , ∂ − M hom, , M hom, , ∂ − M hom, , + N hom, − , (cid:19) = (cid:18) M (1) + (cid:18) M hom, , N hom, − , (cid:19)(cid:19) (cid:18) ∂ −
00 1 (cid:19) . Moreover, note that the operator M (1) has norm arbitrarily small if ν was chosen large enough.Hence, the operator (cid:18) M (1) + (cid:18) M hom, , N hom, − , (cid:19)(cid:19) = (cid:18) M hom, , N hom, − , (cid:19) (cid:18) M hom, , N hom, − , (cid:19) − M (1) + 1 ! is invertible. This gives (cid:18)(cid:18) M (1) + (cid:18) M hom, , N hom, − , (cid:19)(cid:19) (cid:18) ∂ −
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