aa r X i v : . [ m a t h . A P ] S e p Nonlocal H -convergence Marcus Waurick
Abstract
We introduce the concept of nonlocal H -convergence. For this we employ the theoryof abstract closed complexes of operators in Hilbert spaces. We show uniqueness ofthe nonlocal H -limit as well as a corresponding compactness result. Moreover, weprovide a characterisation of the introduced concept, which implies that local andnonlocal H -convergence coincide for multiplication operators. We provide applicationsto both nonlocal and nonperiodic fully time-dependent 3D Maxwell’s equations onrough domains. The material law for Maxwell’s equations may also rapidly oscillatebetween eddy current type approximations and their hyperbolic non-approximatedcounter parts. Applications to models in nonlocal response theory used in quantumtheory and the description of meta-materials, to fourth order elliptic problems as wellas to homogenisation problems on Riemannian manifolds are provided. Keywords: homogenisation, H -convergence, nonlocal coefficients, complexes of operators,evolutionary equations, equations of mixed type, Maxwell’s equations, plate equation, partialdifferential equations on manifoldsMSC 2010: Primary: 35B27, 74Q05 Secondary: 74Q10, 35J58, 35L04, 35M33, 35Q61 The theory of homogenisation studies the asymptotic properties of heterogeneous materialswith a macroscopic and a microscopic scale for the fictitious limit of the ratio of microscopicover macroscopic scale tending to 0. When one is to model this problem mathematically,the mentioned ratio is introduced with a parameter say ε = 1 /n , n ∈ N . For any n ∈ N oneis then given a partial differential equation, e.g., − div a n grad u n = f (1)for fixed f ∈ H − (Ω), Ω ⊆ R d open and bounded, u n ∈ H (Ω) and a n ∈ L ( L (Ω) d ) (i.e.,a bounded linear operator from L (Ω) d into L (Ω) d ) satisfying Re h a n ϕ, ϕ i > α h ϕ, ϕ i forall n ∈ N and ϕ ∈ L (Ω) d and some α >
0. Then one addresses the question, whether the(uniquely) determined sequence of solutions ( u n ) n has a (weak) limit. Assuming that u n ⇀ u weakly in H (Ω), one furthermore asks, whether there exists a ∈ L ( L (Ω) d ) (independent of f ) such that − div a grad u = f. (2)1here is a vast amount of literature concerning this or related subjects. We shall onlyrefer to the standard references [3, 18, 10, 39] for some introductory material. In almostall discussions of the subject, the attention is restricted to local coefficient sequences ( a n ) n (in this sense the approach in [14] is still considered to be local ), that is, one focusses onmultiplication operators being elements of the set M ( α, β, Ω) := { a ∈ L ∞ (Ω) d × d ; ∀ ξ ∈ C d : α k ξ k Re h a ( x ) ξ, ξ i , Re h a ( x ) − ξ, ξ i > β k ξ k for a.e. x ∈ Ω } (3)for some 0 < α < β .Particularly focussing on the model problem (1), Tartar and Murat have introduced andstudied the notion of H -convergence (see also [24]), which we call local H -convergence inorder to avoid possible misunderstandings later on. The notion reads as follows. Definition (local H -convergence, [24, Section 5], [39, Definition 6.4]) . A sequence ( a n ) n in M ( α, β, Ω) is said to be locally H -convergent to a ∈ M ( α, β, Ω), if the following conditionshold: For all f ∈ H − (Ω) = H (Ω) ∗ and ( u n ) n in H (Ω) given by (1), we obtain • ( u n ) n weakly converges in H (Ω) to some u ∈ H (Ω), • a n grad u n ⇀ a grad u , • − div a grad u = f . a is called local H -limit of ( a n ) n .Some by now standard properties of local H -convergence have been shown by their in-ventors. For instance, it is possible to associate a topology τ loc H with the above notion oflocal H -convergence (see [39, p 82]). We shall state a remarkable property of this topology: Theorem 1.1 (see e.g. [39, Theorem 6.5]) . ( M ( α, β, Ω) , τ loc H ) is a metrisable and (sequen-tially) compact Hausdorff space. As a consequence of the latter theorem, the local H -limit is unique and any sequence( a n ) n in M ( α, β, Ω) has a locally H -convergent subsequence. The arguments used to showthe latter result are based on localisation techniques. Further characterising properties forinstance as the one in [38, p 10] and concrete formulas for the limit a in case of periodiccoefficients use Tartar’s method of oscillating test functions as well as the celebrated div-curllemma (see [23]). We shall also refer to the techniques in [18] or [10], which are in turn localin nature.In recent years the interest in so-called meta-materials has emerged. Although it isgenerally rather difficult to find a precise definition for meta-materials physicists have beendealing with this notion for quite a while for coining materials with properties that are notknown for so-called ‘classical’ materials. In fact, meta-materials do not occur in nature andhave to be manufactured artificially. A subclass of these meta-materials are best described by2on-local constitutive relations, where integral operators rather than multiplication operatorsare used as coefficients, see e.g. [16, 9, 21].Other nonlocal constitutive relations can be found in nonlocal response theory related toquantum theory, see [20, Chapter 10]. Furthermore, we shall refer to the so-called McKean–Vlasov equations, see [5] and Example 2.7 below. For an account on nonlocal elasticity werefer to the recent preprint [13].Also, if the oscillations of the coefficients are ‘perpendicular’ to the differential operatorsoccurring in the differential equation nonlocal effects result after a homogenisation process.For this we refer to [37, 44, 43] as paradigmatic examples where ordinary differential equationswith infinite-dimensional state space have been considered. We also refer to [51, 45, 47] wherememory effects have been derived due to a homogenisation process.Nonlocal material models also occur, when homogenising materials with ‘soft’ and ‘stiff’components, which in turn is modelled by non-uniform coercivity estimates in the coeffi-cients with respect to n . A prominent example are equations with high-contrast or singularcoefficients, see e.g. [8, eq. (4.3)].In certain cases nonlocal homogenisation procedures have been carried out, see e.g. [16,9, 52, 42]. We shall also refer to [28, 12] for non-pde type homogenisation problems.A general theory, however, describing highly oscillatory nonlocal material models has beenmissing so far. Thus, the aim of the present article is to introduce the notion of nonlocal H -convergence . As mentioned above, the notion of nonlocal H -convergence will becomeimportant, when one analyses iterated homogenisation schemes of local models that result innonlocal limit models or, if one discusses homogenisation problems for certain meta-materialsso that nonlocal partial differential equations occur right from the start.We shall argue that local H -convergence cannot capture nonlocal coefficients. Indeed,assume in (1) we allow for general a n ∈ L ( L (Ω) d ) satisfying (suitable) uniform coercivity andboundedness conditions. In order to be consistent with local H -convergence, the nonlocal H -convergence needs to coincide, when applied to sequences in M ( α, β, Ω). So, assume that( a n ) n in L ( L (Ω) d ) locally H -converges to a , that is, apply the above definition to generaloperators in L (Ω) d . Let b ∈ L ( L (Ω) d ) with a = b on ran( ˚grad) = { q ∈ L (Ω) d ; ∃ u ∈ H (Ω) : grad u = q } . Then ( a n ) n locally H -converges to b , as well. Since ran( ˚grad) ⊥ =ker(div) = { q ∈ L (Ω) d ; div q = 0 } is infinite-dimensional as long as d >
2, we infer thatlocal H -convergence is clearly not sufficient to uniquely identify nonlocal limit operators.When introducing any notion of nonlocal H -convergence, we cannot expect propertiesof local H -convergence like independence of the attached boundary conditions ([39, Lemma10.3]) to carry over to nonlocal H -convergence. On the contrary, for the proper functionalanalytic setting the attached boundary conditions are of prime importance. We refer toExample 4.3 below for the precise argument showing that nonlocal H -convergence depends on the boundary conditions attached.However, we shall obtain a result analogous to Theorem 1.1 for the newly introducednotion (see Proposition 5.4 and Theorem 5.5), which is one of the main results of the presentexposition. Furthermore, we shall show that on M ( α, β, Ω) nonlocal H -convergence andlocal H -convergence coincide (see Theorem 5.11).3e provide an overview of the contents of this article, next.In Section 3, we will introduce nonlocal H -convergence. For the definition of nonlocal H -convergence, one observes that a certain elliptic problem with div and grad both replacedby curl with appropriate boundary conditions leads to the same homogenised limit as forthe original divergence form type equation (see (1)). Thus, quite naturally, for nonlocal H -convergence, we shall use the theory of closed complexes of operators in Hilbert spaces,which is a generalisation of the operators grad, curl, and div and will be specified in Section2. In this section, we will also recall a more detailed version of the Lax–Milgram lemma (seeTheorem 2.9 and [41]), which is crucial for our later analysis. Note that the core observationthat it is possible to formulate kernels of differential operators via the application of otherdifferential operators has been employed already in the context of Picard’s extended Maxwellsystem in order to discuss low-frequency asymptotics for the time-harmonic Maxwell’s equa-tions, see [31].The emergence of nonlocal or memory effects during the homogenisation process is rootedin the lack of continuity of the inversion mapping for linear operators in the weak operatortopology, see [43] and Proposition 2.13. It is easy to see that also multiplication is notjointly continuous in the weak operator topology either. However, a suitable combinationof projection, multiplication and inversion of the operator sequence ( a n ) n does characterise nonlocal H -convergence. This is the subject of Section 4 with its main result Theorem 4.1.The results of Section 4 will be used in order to obtain the announced variant of The-orem 1.1 in the context of nonlocal H -convergence. From the compactness statement fornonlocal H -convergence, we may then deduce Theorem 5.11 – the relationship of local andnonlocal H -convergence. This in turn yields a homogenisation result for static Maxwell typeequations under the hypothesis of H -convergence for local coefficients, see Corollary 5.14,which is interesting on its own.Using the global div-curl lemma obtained in [49], we provide a characterisation of nonlocal H -convergence in terms of (abstract) ‘div-curl quantities’ in Section 6. This characterisationis an abstract variant of [18, Lemma 4.5] and should be remindful of [38, p. 10]. Note thatthe main result of Section 6, Theorem 6.2, provides a nice way of practically computingthe nonlocal H -limit in applications. We will use Theorem 6.2 for the computation of thenonlocal H -limit for a linear variant of the McKean–Vlasov equation, see Example 6.7.The range of applicability of the main theoretical results is further touched upon in thetwo concluding Sections 7 and 8. In Section 7 we shall revisit some ideas from [47] anddiscuss a homogenisation problem for the fully time-dependent, 3D Maxwell’s equations. Infact, the main result of Section 7 generalises the main results in [1, 51] to both non-periodicand nonlocal (in both space and time) settings. We note that non-uniformly dielectric mediaas occurring for eddy current type approximations are admitted in the general homogeni-sation scheme. In fact, the underlying media may even rapidly oscillate between strictlypositive and vanishing dielectricity on different spatial domains. This oscillatory behaviourbetween hyperbolic and parabolic type problems has only recently been accessible for 1 + 1-dimensional periodic model problems, see [7, 15, 48].In Section 8 we will provide applications to a homogenisation problem of fourth order and4n adapted perspective to nonlocal homogenisation on Riemannian manifolds. The latterprovides the nonlocal counterpart of [17]. Throughout this section, we let H , H , H be Hilbert spaces. Furthermore, we let A : dom( A ) ⊆ H → H ,A : dom( A ) ⊆ H → H be densely defined and closed linear operators. Definition.
We say that ( A , A ) is a complex or sequence , if ran( A ) ⊆ ker( A ). Wecall a complex ( A , A ) closed , if both ran( A ) ⊆ H and ran( A ) ⊆ H are closed. Acomplex ( A , A ) is exact , if ran( A ) = ker( A ). A complex ( A , A ) is called compact , ifdom( A ∗ ) ∩ dom( A ) ֒ → H compactly.For short reference, we shall often address ‘exact’ for complexes, just by saying ‘( A , A )is exact’ and imply the meaning ‘( A , A ) is an exact complex’ (similarly for ‘compact’ and‘closed’).We recall some elementary properties of the theory of complexes of operators in Hilbertspaces, which we state without proof. We refer to [27, Section 2] for the proofs. Theassertions, however, follow from the closed range theorem (see e.g. [41, Corollary 2.5]) andthe orthogonal decomposition H = ker( C ) ⊕ ran( C ∗ ) for C : dom( C ) ⊆ H → H denselydefined, closed. The assertion relating compactness follows from the fact that compactoperators are compact if and only if their adjoints are. Moreover, the last statement followsfrom a contradiction argument and the fact that compact unit balls characterise finite-dimensionality. Proposition 2.1. (a) ( A , A ) is a complex if and only if ( A ∗ , A ∗ ) is a complex;(b) ( A , A ) is closed if and only if ( A ∗ , A ∗ ) is closed;(c) Assume ( A , A ) is closed. Then ( A , A ) is exact if and only if ( A ∗ , A ∗ ) is exact;(d) ( A , A ) is compact if and only if ( A ∗ , A ∗ ) is compact;(e) Let ( A , A ) be compact. Then ( A , A ) is closed and ker( A ∗ ) ∩ ker( A ) is finite-dimensional. Before we treat differential and, thus, particularly, unbounded operators, we shall statea rather trivial example of an exact complex.
Example 2.2.
Denote by H a Hilbert space and let { } = lin ∅ be the trivial Hilbert spaceconsisting of 0, only. Let ι : lin ∅ → H, H → H, ϕ ϕ . With the setting H = { } , H = H , H = H together with A = ι and A = 1, we are in the situationof the beginning of this section. Indeed, since A and A are bounded linear operators,5hey are densely defined and closed. Moreover, their ranges are closed and ran( A ) = { } =ker( A ), so that ( A , A ) is exact. A is obviously self-adjoint and A ∗ = ι ∗ is the (orthogonal)projection onto { } . By Proposition 2.1 (or direct verification), ( A ∗ , A ∗ ) is closed and exact,as well.For the time being, we focus on the 3-dimensional model case. Note that, however, thetheory carries over to the higher-dimensional setting. For this, we refer for instance to [49,Theorem 3.5] for an account on higher-dimensional situations. Other examples are treatedin Section 8. Note that exactness of the considered complexes is an incarnation of Poincar´e’slemma (see also Section 8 below). Example 2.3.
Let Ω ⊆ R open. We definegrad c : C ∞ c (Ω) ⊆ L (Ω) → L (Ω) , ϕ ( ∂ j ϕ j ) j ∈{ , , } , div c : C ∞ c (Ω) ⊆ L (Ω) → L (Ω) , ( ϕ j ) j ∈{ , , } d X j =1 ∂ j ϕ j , curl c : C ∞ c (Ω) ⊆ L (Ω) → L (Ω) , ( ϕ j ) j ∈{ , , } (cid:18) ∂ ϕ − ∂ ϕ ∂ ϕ − ∂ ϕ ∂ ϕ − ∂ ϕ (cid:19) . We set ˚grad := grad c and, similarly, ˚div , ˚curl. Furthermore, we put div := − ˚grad ∗ , grad := − ˚div ∗ , and curl := ˚curl ∗ .Before we state several examples of complexes, we shall highlight the domains of theoperators introduced and the differences between them. It is almost immediate from thedefinition of the adjoint and the distributional gradient that we havedom(grad) = H (Ω) . Since the domain of ˚grad is the closure of C ∞ c (Ω) with respect to the H (Ω)-scalar product,we obtain that dom( ˚grad) = H (Ω) , which, in turn, for Ω with Lipschitz continuous boundary (so that the boundary trace γ : H (Ω) → H / ( ∂ Ω) , u u | ∂ Ω is a well-defined, continuous operator) readsdom( ˚grad) = { u ∈ H (Ω); γ ( u ) = 0 } . Similarly, we obtaindom(curl) = { u ∈ L (Ω) ; curl u ∈ L (Ω) } =: H (curl , Ω) . Again, if we restrict ourselves to the setting of Ω with Lipschitz boundary, we may define thetangential trace by (continuous extension of) γ × : H (Ω) ⊆ H (curl , Ω) → H − / ( ∂ Ω) , u γ ( u ) × n, where n denotes the unit outward normal of ∂ Ω, which exists almost everywhere,see also [4]. In particular, we obtaindom( ˚curl) = { u ∈ H (curl , Ω); γ × ( u ) = 0 } =: H (curl , Ω) . { u ∈ L (Ω) ; div u ∈ L (Ω) } =: H (div , Ω) . Similarly, we obtain for Ω admitting a strong Lipschitz boundary ∂ Ω with unit outwardnormal n that using the normal trace operator γ n : H (Ω) ⊆ H (div , Ω) → H − / ( ∂ Ω) , q n · q again obtained by continuous extension. With this we may also writedom( ˚div) = { u ∈ H (div , Ω); γ n ( u ) = 0 } =: H (div , Ω) . (a1) If Ω is bounded in one direction, then, by Poincar`e’s inequality, ( ι , ˚grad), where ι : { } ֒ → L (Ω), is closed and exact (here H = { } , H = L (Ω) and H = L (Ω) ).Consequently, by Proposition 2.1, so is (div , ι ∗ ). In particular, this implies that divmaps onto L (Ω).(b1) If Ω is bounded with continuous boundary, by the Rellich–Kondrachov theorem, thecomplex ( ι , grad) is compact (here H = { } , H = L (Ω) and H = L (Ω) ). Thesame applies to ( ˚div , ι ∗ ), by Proposition 2.1.For the next examples we refer to [2] for the asserted compactness properties as a generalreference. We shall also refer to the references therein for a guide to the literature.(a2) If Ω is a bounded weak Lipschitz domain, that is, if Ω is a Lipschitz manifold, then( ˚grad , ˚curl) is compact (here H = L (Ω), H = H = L (Ω) ). In particular, so is(curl , div) (Weck’s selection theorem, see also [50] or [30]).(b2) If Ω is a bounded weak Lipschitz domain, then (grad , curl) is compact (here H = L (Ω), H = H = L (Ω) ). In particular, so is ( ˚curl , ˚div).We refer to [2] also for mixed boundary conditions and the respective complex and/or com-pactness properties. Example 2.4.
In the situation of the previous example, let Ω be a bounded weak Lipschitzdomain. The exactness of the considered complexes (grad , curl) and ( ˚grad , ˚curl) can beguaranteed by topological properties of the domain and its complement. In fact, using thecomplex property (ran( ˚curl) ⊆ ker( ˚div) and ran(curl) ⊆ ker(div)) we can decompose L (Ω) as follows L (Ω) = ran(grad) ⊕ ker( ˚div) = ran(grad) ⊕ (cid:0) ker( ˚div) ∩ ker(curl) (cid:1) ⊕ ran( ˚curl)and L (Ω) = ran( ˚grad) ⊕ ker(div) = ran( ˚grad) ⊕ (cid:0) ker(div) ∩ ker( ˚curl) (cid:1) ⊕ ran(curl) . Next, by Proposition 2.1, (grad , curl) is exact, if and only if ( ˚curl , ˚div) is exact, if and onlyif ker( ˚div) = ran( ˚curl), if and only if dim(ker( ˚div) ∩ ker(curl)) = { } . Thus,(grad , curl) exact ⇐⇒ H N := { q ∈ H ( ˚div , Ω) ∩ H (curl , Ω); div q = 0 , curl q = 0 } = { } . Similarly,( ˚grad , ˚curl) exact ⇐⇒ H D := { q ∈ H (div , Ω) ∩ H ( ˚curl , Ω); div q = 0 , curl q = 0 } = { } . H N describes the space of harmonic Neumann fields and H D are the harmonicDirichlet fields. Next, [30, Remark 3(a)] in conjunction with [29, Theorem 1] leads to thefollowing characterisations:(grad , curl) exact ⇐⇒ Ω simply connectedand ( ˚grad , ˚curl) exact ⇐⇒ R \ Ω connected . Next, we recall a result on the well-posedness of abstract divergence form equations.This result is the Lax–Milgram lemma with a slight twist. We shall, however, emphasise thistwist in the argument and the result. Due to the particular variational form of the consideredproblem class, one can identify elliptic problems in divergence form as the composition ofthree continuously invertible mappings . This observation is the key for the derivations tocome. For this reason we present the full proof.For the statement of the next result, we introduce for a densely defined, closed linearoperator C : dom( C ) ⊆ H → H the canonical embedding ι r ,C : ran( C ) ֒ → H . We note that ι ∗ r ,C is the orthogonal projection onto ran( C ), see [33, Lemma 3.2] for theelementary argument. Theorem 2.5 ([41, Theorem 3.1]) . Let B : dom( B ) ⊆ H → H be densely defined andclosed. Assume that B is one-to-one, ran( B ) ⊆ H closed . Let a ∈ L ( H ) be such that Re (cid:0) ι ∗ r ,B aι r ,B (cid:1) = (1 / ι ∗ r ,B ( a + a ∗ ) ι r ,B > αι ∗ r ,B ι r ,B for some α > . Then for all f ∈ dom( B ) ∗ there exists a unique u ∈ dom( B ) such that h aBu, Bv i = f ( v ) ( v ∈ dom( B )) . More precisely, we have u = B − ( ι ∗ r ,B aι r ,B ) − ( B ⋄ ) − f, where B : dom( B ) → ran( B ) , ϕ Bϕ and B ⋄ : ran( B ) → dom( B ) ∗ is given by ϕ (cid:0) dom( B ) ∋ v
7→ h ϕ, Bv i ran( B ) (cid:1) . Example 2.6.
Let Ω ⊆ R be open and bounded. Then, by Example 2.3(a1), the operator˚grad has closed range (Poincar´e‘s inequality). Moreover, since dom( ˚grad) = H (Ω), we havethat ˚grad is one-to-one. Moreover, let b ∈ L ( L (Ω) ) satisfyRe h bq, q i L (Ω) > α h q, q i L (Ω) q ∈ ran( ˚grad) and some α >
0. In this setting we may apply Theorem 2.5 to H = L (Ω), H = L (Ω) , B = ˚grad and a = b . Then, by Theorem 2.5, for all f ∈ dom( ˚grad) ∗ = H − (Ω) there exists a unique u ∈ H (Ω) such that h a ˚grad u, ˚grad v i = f ( v ) ( v ∈ H (Ω)) . Using the notation from Theorem 2.5, we realise that B : H (Ω) → ran( ˚grad) , u grad u .It is furthermore easy to see that B ⋄ = − div : ran( ˚grad) → H − (Ω).We shortly elaborate on a nonlocal differential equation of the form of the previousexample. It is a linear, static variant of the so-called McKean–Vlasov equation, see e.g. [5] Example 2.7.
Let Ω ⊆ R be open and bounded. Let k : Ω × Ω → C measurable andbounded, α >
0. Furthermore let c ∈ L ( L (Ω) ) satisfyRe h cq, q i L (Ω) > α h q, q i L (Ω) for all q ∈ ran( ˚grad). For q ∈ L (Ω) define k ∗ q := (cid:0) x Z Ω k ( x, y ) q ( y ) dy (cid:1) . Assume that k k ∗ k L ( L (Ω) ) α . Then b := c + k ∗ satisfies the conditions imposed on b inExample 2.6. Indeed, for all q ∈ ran( ˚grad) we haveRe h bq, q i = Re h cq, q i + Re h k ∗ q, q i > α h q, q i − α k q k = α h q, q i . It is worth noting that using the expressions for B and B ⋄ , we obtain as a resulting differentialequation for any f ∈ H − (Ω), where we assume that k is such that k ∗ commutes with thedistributional gradient (we refer the reader also to Remark 2.11): f = − div b ˚grad u = − div c ˚grad u − div k ∗ ˚grad u = − div c ˚grad u − div grad k ∗ u, which is of a form similar to [5] in a static, linear case.Before we turn to the proof of Theorem 2.5, we provide some particular insight for thecase a = 1. Proposition 2.8.
Let B : dom( B ) ⊆ H → H be densely defined and closed. Assume that B is one-to-one and has closed range. Then B and B ⋄ are Banach space isomorphisms. Moreprecisely, we have B ⋄ = B ′ R ran( B ) , where B ′ : ran( B ) ∗ → dom( B ) ∗ is the dual operator of B and R ran( B ) is given by R ran( B ) : ran( B ) → ran( B ) ∗ , ϕ ( v
7→ h ϕ, v i ran( B ) ) . roof. B is one-to-one and onto. Hence, an isomorphism. Thus, so is B ∗ . By unitaryequivalence using the Riesz map, we obtain that B ′ is an isomorphism, as well. Finally, R ran( B ) is the inverse of the Riesz isomorphism. Thus, we are left showing that B ⋄ = B ′ R ran( B ) holds. For this, let ϕ ∈ ran( B ). Then we have for all v ∈ dom( B ) (cid:0) B ⋄ ( ϕ ) (cid:1) ( v ) = h ϕ, Bv i ran( B ) = (cid:0) R ran( B ) ( ϕ ) (cid:1) ( Bv )= (cid:0) B ′ R ran( B ) ( ϕ ) (cid:1) ( v ) . Proof of Theorem 2.5.
We shall reformulate the left-hand side of the equation to be solved,first. For this, let π r ,B be the orthogonal projection on ran( B ). Note that π r ,B = ι r ,B ι ∗ r ,B .Let u, v ∈ dom( B ). Then we have h aBu, Bv i = h aπ r ,B Bu, π r ,B Bv i = h aι r ,B ι ∗ r ,B Bu, ι r ,B ι ∗ r ,B Bv i = h ι ∗ r ,B aι r ,B B u, B v i = (cid:0) B ⋄ ι ∗ r ,B aι r ,B B u (cid:1) ( v ) , where we used that ι ∗ r ,B B ( u ) = B ( u ) for all u ∈ dom( B ). Thus, the equation to be solvedreads B ⋄ ι ∗ r ,B aι r ,B B u = f. Under the hypotheses on a using Proposition 2.8 and Lemma 2.12(a) below, we infer boththe uniqueness and the existence result as well as the solution formula.The next result deals with the case when B is not one-to-one. Theorem 2.9 ([41, Theorem 3.1]) . Let C : dom( C ) ⊆ H → H be densely defined andclosed. Assume that ran( C ) ⊆ H closed . Let a ∈ L ( H ) be such that for some α > (cid:0) ι ∗ r ,C aι r ,C (cid:1) > αι ∗ r ,C ι r ,C . Then for all f ∈ dom( Cι r ,C ∗ ) ∗ there exists a unique u ∈ dom( Cι r ,C ∗ ) with the property h aCu, Cv i = f ( v ) ( v ∈ dom( Cι r ,C ∗ )) . More precisely, we have u = C − ( ι ∗ r ,C aι r ,C ) − ( C ⋄ ) − f, where C : dom( Cι r ,C ∗ ) → ran( C ) , ϕ Cϕ .Proof. The assertion follows from Theorem 2.5 applied to B = Cι r ,C ∗ .10 xample 2.10. Let Ω ⊆ R be open, bounded with continuous boundary. Recall thedifferential operators from Example 2.3.(a) A standard application of Theorem 2.9 would be the homogeneous Neumann bound-ary value problem, that is, h a grad u, grad ϕ i = f ( ϕ ) ( ϕ ∈ H ⊥ (Ω)) , for a ∈ L ( L (Ω) ) with Re a > α for some α > H ⊥ (Ω) := { u ∈ H (Ω); R Ω u = 0 } and f ∈ (cid:0) H ⊥ (Ω) (cid:1) ∗ . In fact, in order to solve this equa-tion for u ∈ H ⊥ (Ω) one applies Theorem 2.9 to H = L (Ω), H = L (Ω) and C = gradwith dom( C ) = H (Ω). Note that the positive definiteness condition for a is trivially satis-fied. Moreover, since Ω has continuous boundary, by Example 2.3(b1) in conjunction withProposition 2.1(e), we obtain that ran(grad) ⊆ L (Ω) is closed. It remains to show thatdom( Cι r,C ∗ ) = H ⊥ (Ω). For this we observe thatran( C ∗ ) = ran(grad ∗ ) = ran( − ˚div)= ker(grad) ⊥ = { u ∈ L (Ω); h u, i = 0 } = { u ∈ L (Ω); Z Ω u = 0 } =: L ⊥ (Ω) . Hence, dom( Cι r,C ∗ ) = H (Ω) ∩ L ⊥ (Ω) = H ⊥ (Ω). We emphasise, using the notation fromTheorem 2.9, that C : H ⊥ (Ω) → ran(grad) , u grad u is a topological isomorphism.(b) Assume in addition that Ω is simply connected, and that Ω is a bounded, weakLipschitz domain. Let again a ∈ L ( L (Ω) ) be strictly positive definite. By Example 2.3(b1),we may apply Theorem 2.9 to H = L (Ω) , H = L (Ω) and C = curl with dom( C ) = H (curl , Ω). We define H sol (curl , Ω) := { q ∈ H (curl , Ω); ˚div q = 0 } endowed with the norm from H (curl , Ω). Now, by Theorem 2.9, for all g ∈ H sol (curl , Ω) ∗ there exists a unique v ∈ H sol (curl , Ω) such that h a curl v, curl ψ i = g ( ψ ) ( ψ ∈ H sol (curl , Ω)) . Indeed, the only thing to prove is that dom( Cι r,C ∗ ) = H sol (curl , Ω). For this, we computeran( C ∗ ) = ran( ˚curl) = ker( ˚div) , where in the last equality we have used that Ω is simply connected in order that (grad , curl)and, hence, ( ˚curl , ˚div) is exact, see also Examples 2.3 and 2.4. So, dom( Cι r,C ∗ ) = H (curl , Ω) ∩ ker( ˚div) = H sol (curl , Ω). In the situation discussed here, we have with the notation fromTheorem 2.9, C : H sol (curl , Ω) → ran(curl) , q curl q , which again yields a topological iso-morphism. By construction, it follows that C ⋄ : ran(curl) → H sol (curl , Ω) ∗ is an extension of˚curl | dom( ˚curl) ∩ ran(curl) to the whole of ran(curl). Moreover, recall that also C ⋄ is a topologicalisomorphism. 11c) The example in (b) applies verbatim also to C = ˚curl. In this case, however, one hasto assume that R \ Ω is connected in order to obtaindom( Cι r,C ∗ ) = H , sol (curl , Ω) := { q ∈ dom( ˚curl); div q = 0 } . (d) The connectedness assumptions in (b) and (c) can be dispensed with to the effectthat the respective expressions of dom( Cι r,C ∗ ) are less explicit. In fact, in case of (b), onehas dom( Cι r,C ∗ ) = { q ∈ H (curl , Ω); q ∈ ran( ˚curl) } and in case of (c), one hasdom( Cι r,C ∗ ) = { q ∈ H (curl , Ω); q ∈ ran(curl) } . Remark 2.11.
We note that the variational formulation in Theorem 2.9 is (trivially) equiv-alent to h aCu, Cv i = f ( v ) ( v ∈ dom( C )) . Moreover, we see that due to the solution formula and Proposition 2.8, we obtain a thirdformulation of the latter variational equation: C ⋄ ι ∗ r ,C aι r ,C C u = f. We conclude this section with some additional elementary results needed for the analysisto come.
Lemma 2.12. (a) Let a ∈ L ( H ) with Re a > c for some c > in the sense of positivedefiniteness. Then a − ∈ L ( H ) , k a − k /c and Re (cid:0) a − (cid:1) > c/ k a k .(b) Let ( a n ) n in L ( H ) with Re a n > c for all n ∈ N and some c > . Assume that a n → a converges in the weak operator topology to some a ∈ L ( H ) . Then Re a > c .(c) Let ( a n ) n in L ( H ) bounded with Re a n > c for all n ∈ N and some c > . Assumethat a − n → b converges in the weak operator topology to some b ∈ L ( H ) . Then b − ∈ L ( H ) with k b − k sup n ∈ N k a n k /c , and Re (cid:0) b − (cid:1) > c .(d) Let ( a n ) n in L ( H ) with Re a n > α and Re (cid:0) a − n (cid:1) > /β for all n ∈ N and some α, β > . Assume that a − n → b in the weak operator topology. Then we have for a := b − that Re a > α and Re (cid:0) a − (cid:1) > /β .Proof. (a) is a straightforward consequence of the Cauchy–Schwarz inequality. (b) is easy.The assertion in (d) is a straightforward consequence of (c). Thus, we are left with showing(c). By part (a), we deduce that Re a − n > c/ sup n ∈ N k a n k . By (b), we deduce that Re b > c/ sup n ∈ N k a n k . This ensures b − ∈ L ( H ) and that k b − k sup n ∈ N k a n k /c . Finally, let ϕ ∈ H and put ψ := b − ϕ as well as ϕ n := a − n ψ . Then we computeRe h bϕ, ϕ i = Re h ψ, b − ψ i = lim n →∞ Re h ψ, a − n ψ i = lim inf n →∞ Re h a n ϕ n , ϕ n i > lim inf n →∞ c h ϕ n , ϕ n i > c h ϕ, ϕ i , ϕ n ⇀ ϕ and so k ϕ k lim inf n →∞ k ϕ n k .The following result will be of importance later on, when we compare local H -convergenceto nonlocal H -convergence. Proposition 2.13.
Let H be a Hilbert space. Then the following conditions are equivalent:(i) H is finite-dimensional.(ii) For any bounded sequence ( a n ) n in L ( H ) such that Re a n > α for all n ∈ N and some α > , a ∈ L ( H ) invertible, we have the following equivalence: ( a n ) n → a in the weak operator topology ⇐⇒ ( a − n ) n → a − in the weak operator topology . Proof.
The statement (i) implies (ii) since then the weak operator topology on L ( H ) coincideswith the norm topology.For the statement (ii) implies (i), it suffices to provide a counterexample for H = L (0 , a ∈ L ∞ (0 ,
1) with the corre-sponding multiplication operator on L (0 , L ∞ (0 , σ ( L ∞ , L ), that is, the weak*-topology on L ∞ (0 , b := (0 , / + (1 / , , where K denotes the characteristic function of a set K . Period-ically extending b to the whole of R , we put a n := ( x b ( n · x )). By [10, Theorem 2.6],we deduce that a n → ( R b ) (0 , = (0 , =: a in σ ( L ∞ , L ) as n → ∞ . On the other hand, a − n → (cid:0) R (0 ,
1) 1 b (cid:1) (0 , = (0 , = a − , which shows that (ii) is false. H -convergence for exact sequences As in the previous section, we assume that A and A are densely defined, closed linearoperators from H to H and H to H , respectively. We shall assume that ( A , A ) isclosed and exact. Note that then by Proposition 2.1 ( A ∗ , A ∗ ) is closed and exact, as well.Furthermore, we have the following orthogonal decompositions for H : H = ran( A ) ⊕ ker( A ∗ ) = ran( A ) ⊕ ran( A ∗ ) = ker( A ) ⊕ ran( A ∗ ) = ker( A ) ⊕ ker( A ∗ ) . (4)For the example cases treated in Example 2.3, the decompositions expressed in (4) areabstract variants of Helmholtz decompositions, as it will be exemplified next. Example 3.1.
Let Ω ⊆ R be a bounded weak Lipschitz domain.(a) Assume in addition that Ω is simply connected. Then by Example 2.3, we obtain that( A , A ) = (grad , curl) is exact and closed. Thus, we obtain from (4) L (Ω) = ran(grad) ⊕ ker( ˚div) = ran(grad) ⊕ ran(curl ∗ ) = ran(grad) ⊕ ran( ˚curl) . As a consequence, any q ∈ L (Ω) decomposes into q = grad ϕ + ˚curl ψ for some ϕ ∈ H ⊥ (Ω) and ψ ∈ H , sol (curl , Ω). Note that ϕ and ψ are uniquely determined and dependcontinuously on q . 13b) If the complement of Ω is connected, we infer by Example 2.3 that ( A , A ) = ( ˚grad , ˚curl)is closed and exact. As a consequence of equation (4), we obtain L (Ω) = ran( ˚grad) ⊕ ker(div) = ran( ˚grad) ⊕ ran(curl) . Thus, similar to (a), for any q ∈ L (Ω) , we find uniquely determined ϕ ∈ H (Ω) and ψ ∈ H sol (curl , Ω) such that q = ˚grad ϕ + curl ψ . The ‘potentials’ ϕ and ψ dependcontinuously on q .(c) Let Ω be simply connected with C -boundary and R \ Ω connected. Then, by (a) and(b), we have L (Ω) = ran( ˚grad) ⊕ ran(curl) = ran(grad) ⊕ ran( ˚curl) . Denoting V := ran( ˚grad) ⊥ ∩ ran(grad), we obtain L (Ω) = ran(grad) ⊕ ran( ˚curl) = ran( ˚grad) ⊕ V ⊕ ran( ˚curl) = ran( ˚grad) ⊕ ran(curl) , which implies that V ⊕ ran( ˚curl) = ran(curl). For later use, we shall describe V in moredetail as follows. Our main aim is to show that V is infinite-dimensional. For this, let q ∈ ran(grad). Then we find a uniquely determined ψ q ∈ H ⊥ (Ω) such that q = grad ψ q .In fact, this follows from Example 2.10(a) since H ⊥ (Ω) → ran(grad) , u grad u is atopological isomorphism. Hence, q ∈ V if and only if for all ϕ ∈ H (Ω) h ˚grad ϕ, grad ψ q i = 0 , which in turn is equivalent to div grad ψ q = 0 . Thus, V = { grad ψ ; ψ ∈ H ⊥ (Ω) , ∆ ψ = 0 } . Recall γ n : H (div , Ω) → H − / (Ω) to be the normal trace. Since Ω has C -boundary, wefind an infinite set W ⊆ C (Ω) with the property that for any v, w ∈ W with v = w wehave spt ∂ Ω (n · grad v ) , spt ∂ Ω (n · grad w ) = ∅ and spt ∂ Ω (n · grad v ) ∩ spt ∂ Ω (n · grad w ) = ∅ ,where spt ∂ Ω f denotes the support of a (continuous) function f on ∂ Ω. Next, usingExample 2.10(a) for any v ∈ W , we may solve for w ∈ H ⊥ (Ω) such that h grad w, grad ϕ i = h grad v, grad ϕ i ( ϕ ∈ H ⊥ (Ω)) . (5)Then grad w ∈ dom( ˚div) and so γ n (grad w ) = 0. We put u v := v − w . Then γ n (grad u v ) = γ n (grad v ) = n · grad v | ∂ Ω . Moreover, (5) (trivially) extends to all ϕ ∈ H (Ω) ⊇ H (Ω).Hence, for all ϕ ∈ H (Ω) h grad u v , grad ϕ i = h grad( v − w ) , grad ϕ i = h grad v, grad ϕ i − h grad v, grad ϕ i = 0 . Thus, grad u v ∈ V . Since lin { n · grad v ; v ∈ W } is infinite-dimensional, we infer thesame for lin { grad u v ; v ∈ W } ⊆ V . Hence, V is infinite-dimensional, which concludesthis example. 14sing the notation ι r ,C for densely defined closed linear operators C : dom( C ) ⊆ H → H from the previous section, we may define a := ι ∗ r ,A aι r ,A ∈ L (ran( A )) and a := ι ∗ r ,A ∗ aι r ,A ∗ ∈ L (ran( A ∗ )). The set of admissible (nonlocal) coefficients for which we discussthe notion of nonlocal H -convergence is described next. For α, β >
0, we define M ( α, β, ( A , A )) := { a ∈ L ( H ); Re a > α A , Re a − > (1 /β )1 A ,a continuously invertible , Re( a − ) > (1 /β )1 A ∗ , Re( a − ) − > α A ∗ } , where 1 A and 1 A ∗ are the identity operators in ran( A ) and ran( A ∗ ), respectively.We shall present a possibly nonlocal coefficient in the following example. Example 3.2.
Let Ω ⊆ R be a bounded, simply connected, weak Lipschitz domain withconnected complement, 0 < α β . Then using Example 2.3 and 2.4 both ( ˚grad , ˚curl)and (grad , curl) are exact. Moreover, according to Example 3.1(c), we have for some V thedecomposition L (Ω) = ran( ˚grad) ⊕ V ⊕ ran( ˚curl) . Denote by b ∈ L ( V ) an operator with Re b > α and Re b − > /β . Then a := π V ⊥ + ι V bι ∗ V ∈M ( α, β, ( ˚grad , ˚curl)) ∩ M ( α, β, (grad , curl)), where π V ⊥ denotes the orthogonal projectiononto V ⊥ and ι V : V ֒ → L (Ω) Indeed, for ( A , A ) = ( ˚grad , ˚curl) we have a = a − = 1 ˚grad ,( a − ) = π ran( ˚curl) + ι ∗ r , curl ι V b − ι ∗ V ι r , curl > (1 /β )1 curl , and ( a − ) − > α curl .For ( A , A ) = (grad , curl) we have ( a − ) = ( a − ) − = 1 ˚curl , a = π ran( ˚grad) + ι ∗ r , grad ι V bι ∗ V ι r , grad > α grad , and, similarly, a − > (1 /β )1 grad . We shall revisit this example in Example 4.3 below.Note that since ( A , A ) is closed and exact, both A and A satisfy the conditionsimposed on C in Theorem 2.9. Thus, the equations in the following definitions are uniquelysolvable by Theorem 2.9. We will use A : dom( A ι r ,A ∗ ) → ran( A ) , u A u and, similarly, A ∗ : dom( A ∗ ι r ,A ) → ran( A ∗ ) , v A ∗ v. Example 3.3.
We shall specify the operators A and A ∗ in two particular cases. Recallthat ι r ,A ∗ is the canonical embedding from ran( A ∗ ) = ker( A ) ⊥ into H .15a) Let Ω ⊆ R be a bounded, weak Lipschitz domain with connected complement. Then,we recall that ( A , A ) = ( ˚grad , ˚curl) with H = L (Ω), H = H = L (Ω) is closed and exact;see Examples 2.3 and 2.4. In particular, using the results from Example 2.6, we obtain that A : H (Ω) → ran( ˚grad) , u grad u and with Example 2.10(b) we get A ∗ : H sol (curl , Ω) → ran(curl) , q curl q. Recall that we also have ran( ˚curl) = ker( ˚div) ⊇ H sol (curl , Ω). Thus, A and A ∗ act the sameway as A and A ∗ , they are however arranged in the way that the domain and co-domainis restricted in order to make the calligraphic variants of the operators A and A ∗ be bothtopological isomorphisms.(b) Let Ω ⊆ R be a bounded, simply connected, weak Lipschitz domain. Then recall( A , A ) = (grad , curl) is closed and exact. In this case, we have A : H ⊥ (Ω) → ran(grad) , u grad u and A ∗ : H , sol (curl , Ω) → ran(curl) , q curl q. Definition.
Let ( A , A ) be exact and closed. Let ( a n ) n in M ( α, β, ( A , A )), a ∈ L ( H )continuously invertible. Then ( a n ) n is called nonlocally H -convergent to a with respect to ( A , A ), if the following statement holds: For all f ∈ dom( A ) ∗ and g ∈ dom( A ∗ ) ∗ let ( u n ) n in dom( A ) and ( v n ) n in dom( A ∗ ) satisfy h a n A u n , A ϕ i = f ( ϕ ) , h a − n A ∗ v n , A ∗ ψ i = g ( ψ ) ( n ∈ N ) . for all ϕ ∈ dom( A ), ψ ∈ dom( A ∗ ). Then • u n ⇀ u in dom( A ) for some u ∈ dom( A ); • v n ⇀ v in dom( A ∗ ) for some v ∈ dom( A ∗ ); • a n A u n ⇀ aA u ; a − n A ∗ v n ⇀ a − A ∗ v ; • u and v satisfy h aA u, A ϕ i = f ( ϕ ) and h a − A ∗ v, A ∗ ψ i = g ( ψ )for all ϕ ∈ dom( A ), ψ ∈ dom( A ∗ ). a is called nonlocal H -limit of ( a n ) n . If the choice of the exact complex ( A , A ) is clear fromthe context, we shall also say that ( a n ) n nonlocally H -converges to a , for short.Using the Examples 2.3 and 2.4 together with the descriptions of the Hilbert spaces inExample 2.10, we shall formulate the notion of nonlocal H -convergence in the special caseof ( ˚grad , ˚curl): 16 xample 3.4. Let Ω ⊆ R be an open, bounded, weak Lipschitz domain with connectedcomplement. Then with H = L (Ω), H = H = L (Ω) , we have that ( A , A ) = ( ˚grad , ˚curl)is closed and exact. Let α, β > a n ) n in M ( α, β, ( ˚grad , ˚curl)). Then ( a n ) n nonlocally H -converges to a ∈ M ( α, β, ( ˚grad , ˚curl)) with respect to ( ˚grad , ˚curl), if the following holds:For all f ∈ H − (Ω) and g ∈ H sol (curl , Ω) ∗ let ( u n ) n in H (Ω) and ( v n ) n in H sol (curl , Ω)satisfy h a n grad u n , grad ϕ i = f ( ϕ ) h a − n curl v n , curl ψ i = g ( ψ ) ( n ∈ N ) . for all ϕ ∈ H (Ω), ψ ∈ H sol (curl , Ω). Then • u n ⇀ u in H (Ω) for some u ∈ H (Ω); • v n ⇀ v in H (curl , Ω) for some v ∈ H (curl , Ω); • a n grad u n ⇀ a grad u ; a − n curl v n ⇀ a − curl v both convergences hold weakly in L (Ω) ; • u and v satisfy h a grad u, grad ϕ i = f ( ϕ ) and h a − curl v, curl ψ i = g ( ψ )for all ϕ ∈ H (Ω), ψ ∈ H sol (curl , Ω).
Remark 3.5.
We have formulated the notion of nonlocal H -convergence for exact and closedcomplexes only. There are two desirable steps of generalisation. A first one is to considerfinite-dimensional ‘cohomology groups’ ker( A ∗ ) ∩ ker( A ). A prime application of this arecompact complexes. Thus, the definition of nonlocal H -convergence needs to take intoaccount coefficient sequences ( a n ) n acting on or mapping into the finite-dimensional spaceker( A ∗ ) ∩ ker( A ). In applications to concrete complexes, this setting allows for compactcomplexes and in particular for more general topologies of the underlying domain Ω inExample 2.3 and Example 2.4. A second step is to consider non-closed complexes. In thelight of Example 2.3, this would pave the way to unbounded Ω.We shall analyse the relationship to local H -convergence of multiplication operators inSection 5. This requires further theoretical insight. However, before we discuss more abstracttheory for the notion just introduced, we explicitly consider the particular case of (periodic)multiplication operators, which perfectly fits into the scheme above. In the following, wewill identify a ∈ M ( α, β, Y ) (see (3) for the definition) with the corresponding multiplicationoperator acting on L ( Y ) . Example 3.6.
Let Y = [0 , and let a ∈ L ∞ ( R ) × be Y -periodic. Assume there is α, β > a ∈ M ( α, β, Y ). For n ∈ N we put a n := ( y a ( n · y )).Let A = ι and A = 1 as in Example 2.2 with H = L ( Y ) . Then it is easy to see that a n ∈ M ( α, β, ( ι , n ∈ N ) . Next, let f ∈ dom( A ) ∗ = { } ∗ = { } and g ∈ dom( A ∗ ) ∗ = L ( Y ) and let u n ∈ dom( A ) ∩ ker( A ) ⊥ = { } and v n ∈ dom( A ∗ ) ∩ ker( A ∗ ) ⊥ = L ( Y ) satisfy A ⋄ a n A u n = f, ( A ∗ ) ⋄ a − n A ∗ v n = g. u n = 0 and f = 0. Moreover, note that a n A u n = 0 for all n ∈ N . The second equation implies a − n v n = g and so v n = a n g. By [10, Theorem 2.6], we deduce that v n = a n g ⇀ ( R Y a ) g =: v . Moreover, a − n v n = g → g = (cid:16) Z Y a (cid:17) − v ( n → ∞ ) . Hence, ( a n ) n nonlocally H -converges with respect to ( ι ,
1) to R Y a .A simple modification of Example 3.6 shows that nonlocal H -convergence with respectto ( ι ,
1) is precisely convergence of ( a n ) n in the weak operator topology. Proposition 3.7.
Let H be a Hilbert space. Let ι : { } ֒ → H, , H : H → H, x x and let α, β > . Let ( a n ) n in M ( α, β, ( ι , H )) . Let a ∈ L ( H ) invertible. Then the followingconditions are equivalent:(i) a n → a in the weak operator topology as n → ∞ ,(ii) a n → a H -nonlocally with respect to ( ι , H ) as n → ∞ .Proof. The proof of (i) ⇒ (ii) follows almost literally the arguments outlined in Example 3.6.If (ii) holds, the conditions on nonlocal H -convergence imply that v n = a n g converges weaklyto v = ag for all g ∈ H . This, however, implies (i).The next example is a standard result in homogenisation, see e.g. [18, Lemma 4.5] and[10, Theorem 6.1]. Example 3.8.
Let Y = [0 , and let a ∈ L ∞ ( R ) × be Y -periodic. Assume there is α, β > a ∈ M ( α, β, Y ) with a = a ∗ . For n ∈ N we put a n := (cid:0) y a ( n · y ) (cid:1) . Notethat this particularly implies that both ( a n ) n and ( a − n ) n are bounded. Let H = L ( Y ), H = H = L ( Y ) and let A = ˚grad, A = ˚curl, and div as well as curl as in Example 2.3with Ω = ˚Y. We may also use the results of the Examples 2.6 and 2.10. Then ( A , A ) isexact and closed. (Note that exactness also follows directly with a Fourier series argument.)Moreover, it is plain that a n ∈ M ( α ′ , β ′ , ( ˚grad , ˚curl)) for some 0 < α ′ α β β ′ . Let f ∈ H − ( Y ), g ∈ H sol (curl , Y ) ∗ . Let ( u n ) n in H ( Y ) and ( v n ) n in H sol (curl , Y ) satisfy thefollowing equations − div a n ˚grad u n = f, ˚curl a − n curl v n = g, where in the latter equation, we slightly abused notation, also cf. Remark 2.11 and theconcluding comments in Example 2.10(b). By [10, Theorem 6.1], we have that ( u n ) n weaklyconverges to some u ∈ H (Ω) and there exists a constant coefficient matrix a hom with theproperty that − div a hom ˚grad u = f. Moreover, we have a n ˚grad u n ⇀ a hom ˚grad u in L ( Y ) .18ext, we set w n := a − n curl v n . Note that by the solution formula in Theorem 2.9 appliedto C = curl (see also Example 2.10(b)), the sequence ( v n ) n is bounded in H sol (curl , Y ) andby the boundedness of ( a − n ) n , the sequence ( w n ) n is bounded in L ( Y ) . Take a weaklyconvergent subsequence of ( w n ) n (in L ( Y ) ) and ( v n ) n (in H sol (curl , Y )). Denote the cor-responding limits by w and v . We do not relabel the sequences. From ˚curl w n = g anddiv a n w n = 0, it follows with [18, Lemma 4.5] that a n w n ⇀ a hom w = curl v . Hence, g = w- lim n →∞ ˚curl w n = ˚curl w = ˚curl a − curl v. Uniqueness of v follows from Theorem 2.9 and the coercivity of a hom , see [10, Section 6.3].All in all, we have shown that a n → a hom H -nonlocally with respect to ( ˚grad , ˚curl).We will show in Section 6 that a result analogous to [18, Lemma 4.5] characterises nonlocal H -convergence. Example 3.9.
Let Ω ⊆ R be an open, bounded weak Lipschitz domain with connectedcomplement. Let ( k n ) n be bounded in L ∞ (Ω × Ω) and assume that there is 0 θ < k k n ∗ k L ( L (Ω) ) θ for all n ∈ N , where we refer to Example 2.7 for a definition. Withan argument similar to that in Example 2.7 it follows thatRe(1 − k n ∗ ) > (1 − θ )1 L (Ω) ( n ∈ N ) . In particular, one has Re (cid:0) (1 − k n ∗ ) − (cid:1) > (1 /β ′ )1 L (Ω) for some β ′ >
0. Thus ( a n ) n =(1 − k n ∗ ) n is an eligible sequence in M ( α, β, ( ˚grad , ˚curl)) for some α, β >
0. So the questionis, under which circumstances can we show that1 − k n ∗ → a H -nonlocally with respect to ( ˚grad , ˚curl) as n → ∞ for some a ∈ M ( α, β, ( ˚grad , ˚curl))? Note that this question particularly deals with theoperator (1 − k n ∗ ) − = P ∞ ℓ =0 ( k n ∗ ) ℓ . For now, however, this operators is too complicated anobject to deal with. In Section 6, we shall revisit such kind of coefficients using a criterionfor nonlocal H -convergence detouring the computation of the inverse. Remark 3.10.
A quick comparison of the Examples 3.6 and 3.8 shows that the nonlocal H -limit is not independent of the underlying exact complex. In this line of reasoning it mightnot be expected that nonlocal H -convergence is independent of the considered boundaryconditions either. In fact, we will show that nonlocal H -convergence indeed depends onthe choice of boundary condition. This will be discussed in Example 4.3. We refer also toRemark 5.12 below on this matter. H -convergence As in the previous section, we shall assume throughout that A and A are densely defined,closed linear operators from H to H and H to H , respectively, with ( A , A ) closed andexact. 19ur first aim of this section is to characterise nonlocal H -convergence by means of con-vergence of operators in a block matrix representation. For this, we employ the orthogonaldecomposition mentioned in (4). For a ∈ L ( H ) we obtain a = (cid:0) ι r ,A ι r ,A ∗ (cid:1) (cid:18) ι ∗ r ,A ι ∗ r ,A ∗ (cid:19) a (cid:0) ι r ,A ι r ,A ∗ (cid:1) (cid:18) ι ∗ r ,A ι ∗ r ,A ∗ (cid:19) = (cid:0) ι r ,A ι r ,A ∗ (cid:1) (cid:18) ι ∗ r ,A aι r ,A ι ∗ r ,A aι r ,A ∗ ι ∗ r ,A ∗ aι r ,A ι ∗ r ,A ∗ aι r ,A ∗ (cid:19) (cid:18) ι ∗ r ,A ι ∗ r ,A ∗ (cid:19) =: (cid:0) ι r ,A ι r ,A ∗ (cid:1) (cid:18) a a a a (cid:19) (cid:18) ι ∗ r ,A ι ∗ r ,A ∗ (cid:19) . We shall also define the unitary operator U := (cid:0) ι r ,A ι r ,A ∗ (cid:1) ∈ L (ran( A ∗ ) ⊕ ran( A ) , H ) (6)In particluar, we then obtain a = U (cid:18) a a a a (cid:19) U ∗ With this notation at hand, we can state the first major result of this article. Keep in mindthat the block operator matrix representation rests on the generalised Helmholtz decompo-sition in equation (4).
Theorem 4.1 (Characterisation of nonlocal H -convergence) . Let α, β > , a ∈ L ( H ) , and ( a n ) n in M ( α, β, ( A , A )) . Then the following conditions are equivalent:(i) a is continuously invertible and ( a n ) n nonlocally H -converges to a ;(ii) ( a − n, ) n , (cid:0) a n, a − n, (cid:1) n , (cid:0) a − n, a n, (cid:1) n , and (cid:0) a n, − a n, a − n, a n, (cid:1) n converge in the respec-tive weak operator topologies to a − , a a − , a − a , and a − a a − a . Moreover,we have a ∈ M ( α, β, ( A , A )) . Remark 4.2.
We emphasise that due to the lack of continuity of inversion (see also Proposi-tion 2.13) and the lack of joint continuity of multiplication under the weak operator topologythe second item in Theorem 4.1 does neither imply nor is implied by convergence of any ofthe sequences ( a n, ) n , ( a n, ) n , ( a n, ) n , or ( a n, ) n under the weak operator topology. Example 4.3.
Let Ω ⊆ R be a bounded, open, simply connected C -domain with con-nected R \ Ω. We shall revisit Example 3.2 and recall that for b ∈ L ( V ) with Re b > α andRe (cid:0) b − (cid:1) > /β we have that a ( b ) = π V ⊥ + ι V bι ∗ V belongs to both M ( α, β, (grad , curl)) and M ( α, β, ( ˚grad , ˚curl)). Written as a 3-by-3 block operator matrix according to the decompo-sition L (Ω) = ran( ˚grad) ⊕ V ⊕ ran( ˚curl) ,a ( b ) maybe written as b
00 0 1 . A , A ) = ( ˚grad , ˚curl) U ∗ a ( b ) U = (cid:0) (cid:1)(cid:18) (cid:19) (cid:18) b
00 1 (cid:19) and for ( A , A ) = (grad , curl) U ∗ a ( b ) U = (cid:18) b (cid:19) (cid:18) (cid:19)(cid:0) (cid:1) . In both cases, we have a ( b ) = 0 and a ( b ) = 0.Let now ( b n ) n be a sequence in L ( V ) with Re b n > α, Re (cid:0) b − n (cid:1) > /β , and b ∈ L ( V )invertible. Then, by Theorem 4.1, we obtain that a ( b n ) → a ( b ) H -nonlocally w.r.t. ( ˚grad , ˚curl) ⇐⇒ b n → b in the weak operator topology.Again by Theorem 4.1, we see that a ( b n ) → a ( b ) H -nonlocally w.r.t. (grad , curl) ⇐⇒ b − n → b − in the weak operator topology.By Example 3.1(c), V is infinite-dimensional. Hence, by Proposition 2.13, the nonlocal H -convergence with respect to ( ˚grad , ˚curl) and with respect to (grad , curl) are not comparable .In particular, the nonlocal H -limit depends on the attached boundary conditions . Corollary 4.4.
The nonlocal H -limit is unique.Proof. Let ( a n ) n nonlocally H -converge to invertible a and b . By Theorem 4.1, we obtain a, b ∈ M ( α, β, ( A , A )). Moreover, again by Theorem 4.1, we deduce that a − = b − ; thus a = b . Hence, a a − = lim n →∞ a n, a − n, = b b − = b a − and a − a = lim n →∞ a − n, a n, = b − b = a − b . This implies a = b and a = b . Finally, a = a − a a − a + a a − a = lim n →∞ (cid:0) a n, − a n, a − n, a n, (cid:1) + a a − a = b − b b − b + a a − a = b .
21y Theorem 4.1 and the continuity of computing the adjoint in the weak operator topol-ogy as well as the fact that computing the inverse and computing the adjoint are com-mutative operations, we obtain that computing the adjoint is continuous under nonlocal H -convergence. We will provide more details of this line of reasoning in the following. Firstof all, we shall observe that the set M ( α, β, ( A , A )) is invariant under computing theadjoint. Proposition 4.5.
Let a ∈ L ( H ) , α, β > . Then a ∈ M ( α, β, ( A , A )) ⇐⇒ a ∗ ∈ M ( α, β, ( A , A )) . Proof. By a ∗∗ = a , it suffices to just prove one implication. Assume that a ∈ M ( α, β, ( A , A )).Since a is continuously invertible, so is a ∗ . We have U ∗ aU = (cid:18) a a a a (cid:19) with this we observe U ∗ a ∗ U = ( U ∗ aU ) ∗ = (cid:18) a ∗ a ∗ a ∗ a ∗ (cid:19) and, using U ∗ = U − as U is unitary, we obtain U ∗ ( a ∗ ) − U = ( U ∗ a − U ) ∗ = (cid:18) ( a − ) ∗ ( a − ) ∗ ( a − ) ∗ ( a − ) ∗ (cid:19) . Thus, Re( a ∗ ) = Re a ∗ = Re a > α, Re (cid:0) ( a ∗ ) (cid:1) − = Re( a ∗ ) − = Re(( a ) − ) ∗ = Re( a ) − > /β, and, similarly, Re (cid:0) ( a ∗ ) − (cid:1) = Re (cid:0) a − (cid:1) ∗ > /β, Re (cid:0) ( a ∗ ) − (cid:1) − = Re (cid:0)(cid:0) a − (cid:1) ∗ (cid:1) − > α, completing the proof. Corollary 4.6.
Let α, β > , a ∈ L ( H ) , ( a n ) n in M ( α, β, ( A , A )) . Then the followingconditions are equivalent:(i) a is continuously invertible and ( a n ) n nonlocally H -converges to a ;(ii) a ∗ is continuously invertible and ( a ∗ n ) n nonlocally H -converges to a ∗ ;Proof. It suffices to prove ‘(i) ⇒ (ii)’. First of all note that by Proposition 4.5, ( b n ) n := ( a ∗ n ) n is a sequence in M ( α, β, ( A , A )). Moreover, denote b := a ∗ . By Theorem 4.1, it suffices toshow that the sequences( b − n, ) n , (cid:0) b n, b − n, (cid:1) n , (cid:0) b − n, b n, (cid:1) n , and (cid:0) b n, − b n, b − n, b n, (cid:1) n b − , b b − , b − b , and b − b b − b .This, in turn, is implied by the convergence of the respective adjoints in the weak operatortopology. Using U ∗ b n U = ( U ∗ a n U ) ∗ similarly to the proof of Proposition 4.5 we have for all n ∈ N ( b − n, ) ∗ = a − n, (cid:0) b n, b − n, (cid:1) ∗ = (cid:0) b − n, (cid:1) ∗ (cid:0) b n, (cid:1) ∗ = a − n, a n, (cid:0) b − n, b n, (cid:1) ∗ = a n, a − n, (cid:0) b n, − b n, b − n, b n, (cid:1) ∗ = b ∗ n, − b ∗ n, ( b ∗ n, ) − b ∗ n, (cid:1) ∗ = a n, − a n, ( a n, ) − a n, and similarly for b and a replacing b n and a n . Since ( a n ) n nonlocally H -converges to a , byTheorem 4.1, we thus obtain that ( a ∗ n ) n nonlocally H -converges to a ∗ .The latter result implies the self-adjointness of the nonlocal H -limit given the self-adjointness of the sequence converging to it. Corollary 4.7.
Let α, β > , a ∈ L ( H ) continuously invertible, ( a n ) n in M ( α, β, ( A , A )) .Assume that ( a n ) n nonlocally H -converges to a . If a n = a ∗ n for all n ∈ N , then a = a ∗ . The proof of Theorem 4.1 needs some preparations.
Lemma 4.8.
Let a ∈ L ( H ) with a continuously invertible. Then(a) (cid:18) a a a a (cid:19) = (cid:18) a a − (cid:19) (cid:18) a a − a a − a (cid:19) (cid:18) a − a (cid:19) (b) If, in addition, a is continuously invertible, then a − a a − a is and (cid:18)(cid:0) a − (cid:1) (cid:0) a − (cid:1) (cid:0) a − (cid:1) (cid:0) a − (cid:1) (cid:19) = (cid:18) − a − a (cid:19) (cid:18) a − (cid:0) a − a a − a (cid:1) − (cid:19) (cid:18) − a a − (cid:19) = a − + a − a (cid:0) a − a a − a (cid:1) − a a − − a − a (cid:0) a − a a − a (cid:1) − − (cid:0) a − a a − a (cid:1) − a a − (cid:0) a − a a − a (cid:1) − ! In particular, we have (cid:0) a − (cid:1) − = (cid:0) a − a a − a (cid:1) and (cid:0) a − (cid:1) = − a − a (cid:0) a − (cid:1) .Proof. The first assertion follows from a direct computation. The statement in (b) is in turna straightforward consequence of the formula in (a).
Proof of Theorem 4.1.
For the proof, we refer to the solution formula for elliptic type prob-lems in Theorem 2.9. So, let n ∈ N and let f and g be as in the definition of nonlocal23 -convergence and let u n and v n be the corresponding solutions. Then we have u n = A − a − n, ( A ⋄ ) − f (7) v n = ( A ∗ ) − ( a − n ) − (( A ∗ ) ⋄ ) − g = ( A ∗ ) − (cid:0) a n, − a n, a − n, a n, (cid:1) (( A ∗ ) ⋄ ) − g (8)where the last equation follows from Lemma 4.8(b). With this, we infer a n A u n = a n U U ∗ A u n = U (cid:18) a n, a n, a n, a n, (cid:19) (cid:18) A u n (cid:19) = U (cid:18) a n, a n, a n, a n, (cid:19) (cid:18) a − n, ( A ⋄ ) − f (cid:19) = U (cid:18) ( A ⋄ ) − fa n, a − n, ( A ⋄ ) − f (cid:19) and so a n A u n = U (cid:18) ( A ⋄ ) − fa n, a − n, ( A ⋄ ) − f (cid:19) . (9)Similarly, we compute a − n A ∗ v n = a − n U U ∗ A ∗ v n = U (cid:18) ( a − n ) ( a − n ) ( a − n ) ( a − n ) (cid:19) (cid:18) a − n ) − (( A ∗ ) ⋄ ) − g (cid:19) = U (cid:18) ( a − n ) ( a − n ) − (( A ∗ ) ⋄ ) − g (( A ∗ ) ⋄ ) − g (cid:19) . Next, from Lemma 4.8(b), we deduce that( a − n ) ( a − n ) − = a − n, a n, . Thus, a − n A ∗ v n = U (cid:18) a − n, a n, (( A ∗ ) ⋄ ) − g (( A ∗ ) ⋄ ) − g (cid:19) . (10)Next, we observe that A , A ∗ , A ⋄ , ( A ∗ ) ⋄ are all isomorphisms by Proposition 2.8. Hence,the left-hand sides of (7), (8), (9), and (10) converge weakly in dom( A ), dom( A ∗ ), H , and H for all admissible f and g to the corresponding expression with a n replaced by a , if andonly if a n, , (cid:0) a n, − a n, a − n, a n, (cid:1) , a n, a − n, , and a − n, a n, converge in the respective weakoperator topologies to the corresponding expression without the additional index n . Remark 4.9.
We shall note here that the restriction to sequences ( a n ) n is not necessary. Infact, the corresponding notion of nonlocal H -convergence for nets ( a ι ) ι ∈ I ( I some directedset), is equivalent to the convergence of the corresponding operator nets in (ii) of Theorem4.1. We will exploit this fact in Section 5. 24 closer inspection of the proof of Theorem 4.1 reveals the following more detailed version. Theorem 4.10.
Let ( a n ) n in M ( α, β, ( A , A )) , a ∈ L ( H ) .Consider the following statements:(a) For all f ∈ dom( A ) ∗ let ( u n ) n in dom( A ) be such that h a n A u n , A ϕ i = f ( ϕ ) ( n ∈ N , ϕ ∈ dom( A )) . Then u n ⇀ u ∈ dom( A ) and h aA u, A ϕ i = f ( ϕ ) ( ϕ ∈ dom( A )) . (b) As (a) with the additional conclusion that a n A u n ⇀ aA u ∈ H .(c) Let a be invertible. For all g ∈ dom( A ∗ ) ∗ let ( v n ) n in dom( A ∗ ) be such that h a − n A ∗ v n , A ∗ ψ i = g ( ψ ) ( n ∈ N , ψ ∈ dom( A ∗ )) . Then v n ⇀ v ∈ dom( A ∗ ) and h a − A ∗ v, A ∗ ψ i = g ( ψ ) ( ψ ∈ dom( A ∗ )) . (d) As (c) with the additional conclusion that a − n A v n ⇀ a − A v ∈ H .(a’) Re a > α , and a − n, → a − in the weak operator topology.(b’) As in (a’) and a n, a − n, → a a − in the weak operator topology.(c’) a is continuously invertible, Re (cid:0)(cid:0) a − a a − a (cid:1) − (cid:1) > /β , and a n, − a n, a − n, a n, → a − a a − a in the weak operator topology.(d’) As in (c’) with the additional conclusion that a − ,n a ,n → a − a in the weak operatortopology.Then (a) ⇔ (a’), (b) ⇔ (b’), (c) ⇔ (c’), and (d) ⇔ (d’).Proof. Most of the things are immediate from the reformulations (7), (8), (9), and (10). Theinvertibility statements follow from Lemma 2.12.
Remark 4.11.
Let Ω ⊆ R be such that ( A , A ) = ( ˚grad , ˚curl) is compact and exact; seeagain Examples 2.3, 2.4. Let ( a n ) n in M ( α, β, Ω) and a ∈ M ( α, β, Ω).(a) If a n = a ∗ n , a = a ∗ , and let the statement (a) of Theorem 4.10 be satisfied. This isequivalent to ( a n ) n G -converging to a (as defined in [39, Definition 6.1]). Thus, weobtain the characterisation of G -convergence given in (a’) and recover the main resultin [46].(b) Condition (b) in Theorem 4.10 is equivalent to ( a n ) n H -converging to a (as definedin [39, Definition 6.4]). Hence, Theorem 4.10(b’) is an operator-theoretic descriptionof H -convergence. Note that, if in addition a n = a ∗ n and a = a ∗ and assuming Theo-rem 4.10(b), we also obtain a − n, a n, = (cid:0) a n, a − n, (cid:1) ∗ → (cid:0) a a − (cid:1) ∗ = a − a . H -convergence, a suitable char-acterisation of the convergence of a n, − a n, a − n, a n, does not follow from the refor-mulations outlined in the proof of Theorem 4.1. However, it is possible to show that a n, − a n, a − n, a n, does converge to the expected limit. In Theorem 5.11 we shallsee that local H -convergence and nonlocal H -convergence are the same concepts formultiplication operators and will, thus, show the remaining convergence result even fornon-selfadjoint sequences. In this section, we shall attach a topology to nonlocal H -convergence and show that boundedsubsets of M ( α, β, ( A , A )) are precisely the relatively compact ones under this topology.Furthermore, if H is separable, we will show that bounded subsets are metrisable, so thatthe nonlocal H -closure of bounded subsets of M ( α, β, ( A , A )) are both compact and se-quentially compact. Again let ( A , A ) be exact and closed.We recall a well-known result for the weak operator topology. Since this result is the basisfor our metrisability and compactness statement for nonlocal H -convergence, we sketch theshort proof. Theorem 5.1.
Let H , H be Hilbert spaces. Then B L ( H ,H ) := { T ∈ L ( H , H ); k T k } is compact under the weak operator topology of L ( H , H ) . If, in addition, both H and H are separable, then the weak operator topology of L ( H , H ) on B L ( H ,H ) is metrisable.Proof. Denoting the unit ball of H endowed with the weak topology by B w H , we obtain that K := Y ϕ ∈ H k ϕ k B w H is compact under the product topology by Tikhonov’s theorem and the compactness of B w H .It is elementary to show that B L ( H ,H ) ⊆ K is closed, when B L ( H ,H ) is endowed with theweak operator topology. If H is separable, then B w H is metrisable. If H is separable aswell, it is then standard to prove that K × K ∋ ( T, S ) X n ∈ N − n min { d ( T ( ϕ n ) , S ( ϕ n )) , } metrises the topology on K , where d metrises the topology on B w H and ( ϕ n ) n ∈ N is an or-thonormal basis for H .We denote by τ H the initial topology on M ( α, β, ( A , A )) such that a a − ∈ L w (ran( A )) a a a − ∈ L w (ran( A ) , ran( A ∗ )) a a − a ∈ L w (ran( A ∗ ) , ran( A )) a a − a a − a ∈ L w (ran( A ∗ ))26re continuous, where for Hilbert spaces K and K , L w ( K , K ) denotes the set of boundedlinear operators endowed with the weak operator topology. Remark 5.2.
We note that τ H is readily seen to be weaker than both the norm and thestrong operator topology on M ( α, β, ( A , A )). Examples 3.6 and 3.8 show that the weakoperator topology on M ( α, β, ( A , A )) and τ H cannot be compared in general.The following is a reformulation of Theorem 4.1. Theorem 5.3.
Let ( a n ) n in M ( α, β, ( A , A )) , a ∈ L ( H ) invertible. Then the followingconditions are equivalent:(i) ( a n ) n nonlocally H -converges to a ;(ii) a n τ H → a ∈ M ( α, β, ( A , A )) . Theorem 5.3 shows that nonlocal H -convergence (of sequences) is actually induced by thetopology τ H . Next, we show that τ H is a Hausdorff topology. Together with Theorem 5.3,this yields another proof of Corollary 4.4, the uniqueness of the nonlocal H -limit. Proposition 5.4. τ H is a Hausdorff topology.Proof. Let a, b ∈ M ( α, β, ( A , A )) with a = b . It follows that (at least) one of the equalities a − = b − a a − = b b − a − a = b − b a − a a − a = b − b b − b cannot be true. Since the weak operator topology is a Hausdorff topology, we find a suitablecontinuous semi-norm p such that one of the following 4 statements is true p (cid:0) a − (cid:1) = p (cid:0) b − (cid:1) p (cid:0) a a − (cid:1) = p (cid:0) b b − (cid:1) p (cid:0) a − a (cid:1) = p (cid:0) b − b (cid:1) p (cid:0) a − a a − a (cid:1) = p (cid:0) b − b b − b (cid:1) . This implies the assertion.The following result is the announced compactness statement.
Theorem 5.5.
The set M ( α, β, ( A , A )) := { a ∈ M ( α, β, ( A , A )); k a − a k , k a a − k β } is compact under τ H . roof. Let ( a ι ) ι be a net in M ( α, β, ( A , A )). By Theorem 5.1, we may choose a subnet( a ϕ ( ι ′ ) ) ι ′ such that a − ϕ ( ι ′ ) , → b − for some b ∈ L (ran( A )) with Re b > α and Re (cid:0) b − (cid:1) > /β . By the boundedness of ( a ι ) ι , we infer that ( a ι, ) ι and ( a ι, ) ι are bounded. Again usingTheorem 5.1, we find a subnet such that a ϕ ′ ( ι ′′ ) , a − ϕ ′ ( ι ′′ ) , b → b and b a − ϕ ′ ( ι ′′ ) , a ϕ ′ ( ι ′′ ) , → b for some b ∈ L (ran( A ∗ ) , ran( A )) and b ∈ L (ran( A ) , ran( A ∗ )). Finally, we find a subnetsuch that a ϕ ′′ ( ι ′′′ ) , − a ϕ ′′ ( ι ′′′ ) , a − ϕ ′′ ( ι ′′′ ) , a ϕ ′′ ( ι ′′′ ) , + b b − b → b for some b ∈ L (ran( A ∗ )). It is easy to see that Re b > α and Re (cid:0) b − (cid:1) > /β (see alsoLemma 2.12(d)). Similarly, using Lemma 4.8 for ( a − ι ) − = a ι, − a ι, a − ι, a ι, , it followsthat Re( b − b b − b ) > α and Re (cid:0) ( b − b b − b ) − (cid:1) > /β (see also Lemma 2.12). Next,using Lemma 4.8(a), we obtain (cid:18) b b b b (cid:19) = (cid:18) b b − (cid:19) (cid:18) b b − b b − b (cid:19) (cid:18) b − b (cid:19) . Thus, we deduce that b is continuously invertible. Hence, b ∈ M ( α, β, ( A , A )). Moreover, itis now easy to see that a ϕ ′′ ( ι ′′′ ) τ H → b . It remains to show that k b − b k β, k b b − k β . Forthis, we use lower semi-continuity of the operator norm under the weak operator topology.Thus, writing ι instead of ϕ ′′ ( ι ′′′ ) for simplicity, we obtain k b − b k = k lim ι a − ι, a ι, k lim inf ι k a − ι, a ι, k β k b b − k = k lim ι a ι, a − ι, k lim inf ι k a ι, a − ι, k β. Hence, b ∈ M ( α, β, ( A , A )). Remark 5.6.
Let
B ⊆ M ( α, β, ( A , A )) be bounded in L ( H ). Then we find α ′ , β ′ ∈ R suchthat B ⊆ M ( α ′ , β ′ , ( A , A )). Indeed, α ′ = α and β ′ = sup {k b b − k ∨ k b − b k ; b ∈ B} ∨ β are possible choices. As a consequence, we obtain with Theorem 5.5 that B is relativelycompact under τ H .Let us revisit Examples 4.3 and 3.9. Example 5.7.
We shall use the notation and operators introduced in Example 3.9. We havealready seen that there exists α, β > − k n ∗ ∈ M ( α, β, ( ˚grad , ˚curl)) for all n ∈ N .Moreover, we have assumed that (1 − k n ∗ ) n is a bounded sequence in L ( L (Ω) ). Thus, byTheorem 5.10, we find a strictly increasing sequence of positive integers κ : N → N such that(1 − k κ ( n ) ∗ ) n is nonlocally H -convergent to some a ∈ M ( α, β, ( ˚grad , ˚curl)). We emphasisethat the used compactness statement does not lead to the statement that a = (1 − k ∗ ) forsome convolution-type kernel k ∈ L ∞ (Ω × Ω). More refined arguments (or assumptions) areneeded to actually deduce that a has the desired form.28 xample 5.8. Use the assumptions and operators introduced in Example 4.3. We haveseen that for a sequence ( b n ) n in L ( V ) satisfying Re b n > α and Re (cid:0) b − n (cid:1) > /β for all n ∈ N and an invertible b ∈ L ( V ) that a ( b n ) → a ( b ) H -nonlocally w.r.t. ( ˚grad , ˚curl) ⇐⇒ b n → b in the weak operator topologyand a ( b n ) → a ( b ) H -nonlocally w.r.t. (grad , curl) ⇐⇒ b − n → b − in the weak operator topology.Thus, in these special cases, with an application of Theorem 5.10, we obtain special cases ofTheorem 5.1 for H = H = V in the separable case. Remark 5.9. (a) We have M ( α, β, ( A , A )) * { a ∈ L ( H ); Re a > α ′ , Re (cid:0) a − (cid:1) > /β ′ } forany α ′ , β ′ >
0. Indeed, a = U (cid:0) − ε ) / (cid:1) U ∗ for any ε ∈ (0 ,
1) with U as in (6) belongs to M ( α, β, ( A , A )) for some α, β > a > b = b ∗ ∈ M ( α, β, ( A , A )). Then Re b = b > α ′ for some α ′ > (cid:0) b − (cid:1) > /β ′ for some β ′ > ψ := (cid:18) − b − b (cid:19) ϕ : D (cid:18) b b b b (cid:19) ψ, ψ E = D (cid:18) b − b b − b b (cid:19) ϕ, ϕ E > α h ϕ, ϕ i > (cid:0) α/ (1 + k b − b k ) (cid:1) k ψ k . Theorem 5.10.
Assume H to be separable. Then ( M ( α, β, ( A , A )) , τ H ) is metrisableand sequentially compact.Proof. Since ran( A ) ⊆ H and ran( A ∗ ) ⊆ H both these subsets are separable. We abbre-viate M := M ( α, β, ( A , A )). We putΦ : M ∋ a a − ∈ βB L (ran( A )) Φ : M ∋ a a − a ∈ βB L (ran( A ∗ ) , ran( A )) Φ : M ∋ a a a − ∈ βB L (ran( A ) , ran( A ∗ )) Φ : M ∋ a a − a a − a ∈ βB L (ran( A ∗ )) . By Theorem 5.1 there exists metrics d , d , d , and d inducing the weak operator topologyon βB L (ran( A )) , βB L (ran( A ∗ ) , ran( A )) , βB L (ran( A ) , ran( A ∗ )) , and βB L (ran( A ∗ )) . We define d H : M × M → [0 , ∞ )( a, b ) X j,k ∈{ , } d jk (Φ jk ( a ) , Φ jk ( b )) . As in the proof of Proposition 5.4, we verify that ( M , d H ) is a metric space. Moreover, bydefinition, the identity mapping ( M , τ H ) ֒ → ( M , d H )29s continuous and onto. Since ( M , d H ) is a Hausdorff space and ( M , τ H ) is compact byTheorem 5.5, we infer that ( M , τ H ) ֒ → ( M , d H ) is a homeomorphism. Hence, ( M , τ H )is metrisable. Sequential compactness is now immediate since compact metric spaces aresequentially compact.We draw an important consequence of the compactness result, which establishes theconnection from local to nonlocal H -convergence. We recall Example 2.3(a2) and Example2.4 in order to deduce ( ˚grad , ˚curl) is compact and exact, if the underlying domain Ω is abounded weak Lipschitz domain with R \ Ω connected. In fact, due to our abstract reasoning,the assumption of ( ˚grad , ˚curl) being compact and exact is the assumption, we actually needin the next statement. Theorem 5.11.
Let Ω ⊆ R be a bounded weak Lipschitz domain with connected complement.Let ( a n ) n in M ( α, β, Ω) , a ∈ M ( α, β, Ω) . Then the following conditions are equivalent:(i) ( a n ) n locally H -converges to a , that is, for all f ∈ H − (Ω) and corresponding solutions ( u n ) n in H (Ω) of h a n grad u n , grad ϕ i = f ( ϕ ) ( ϕ ∈ H (Ω)) we have u n ⇀ u ∈ H (Ω) , a n grad u n ⇀ a grad u , where u in H (Ω) satisfies h a grad u, grad ϕ i = f ( ϕ ) ( ϕ ∈ H (Ω)) . (ii) ( a n ) n nonlocally H -converges to a with respect to ( ˚grad , ˚curl) .Proof. The implication ‘(ii) ⇒ (i)’ has been settled in Remark 4.11(b) together with Theo-rem 4.1 (see also Theorem 4.10). We shall assume (i). By Theorem 5.5, we may choose asubsequence ( a κ ( n ) ) n of ( a n ) n , which nonlocally H -converges to some b . From the implica-tion ‘(ii) ⇒ (i)’ it follows that ( a κ ( n ) ) n locally H -converges to b . Since local H -convergenceis induced by a topology, see [39, p. 82], we deduce that ( a κ ( n ) ) n locally H -converges to a .By uniqueness of the local H -limit (see again [39, p. 82]), we obtain a = b . A subsequenceprinciple concludes the proof. Remark 5.12.
Given Ω ⊆ R a simply connected bounded weak Lipschitz domain in orderthat (grad , curl) is compact and exact; see Examples 2.3 and 2.4. Let ( a n ) n and a belong to M ( α, β, Ω). By [39, Lemma 10.3] local H -convergence is independent of the attached bound-ary conditions. Thus, in particular, with an analogous proof to the one in Theorem 5.11,it is possible to show that ( a n ) n locally H -converges to a , if and only if ( a n ) n nonlocally H -converges to a with respect to (grad , curl). Remark 5.13.
Another way of stating Theorem 5.11 is the following. Let τ loc H be the(metrisable) topology induced on M ( α, β, Ω) by local H -convergence. Then( M ( α, β, Ω) , τ H ) ֒ → ( M ( α, β, Ω) , τ loc H )is a homeomorphism. Note that [39, Theorem 6.5] states that ( M ( α, β, Ω) , τ loc H ) is sequen-tially compact. Hence, so is ( M ( α, β, Ω) , τ H ).30n immediate corollary is a homogenisation result for elliptic equations involving thecurl-operator. We also refer to the explicit descriptions of the domain of the curl-operatorderived in Example 2.10(b). Corollary 5.14.
Let Ω ⊆ R be a bounded weak Lipschitz domain with connected comple-ment. Let ( a n ) n in M ( α, β, Ω) , a ∈ M ( α, β, Ω) . Assume that ( a n ) n locally H -converges to a . Then, for all g ∈ H sol (curl , Ω) ∗ and solutions ( v n ) n in H sol (curl , Ω) of h a − n curl v n , curl ψ i = g ( ψ ) ( ψ ∈ H sol (curl , Ω)) , we have v n ⇀ v ∈ H sol (curl , Ω) , a − n curl v n ⇀ a − curl v ∈ L (Ω) , where v ∈ H sol (curl , Ω) satisfies h a − curl v, curl ψ i = g ( ψ ) ( ψ ∈ H sol (curl , Ω)) . Remark 5.15. (a) In the light of Remark 4.11, we note that Corollary 5.14 particularlysettles the convergence of a n, − a n, a − n, a n, → a − a a − a as n → ∞ in the weakoperator topology.(b) As a consequence of Remark 5.12, we deduce that a similar results hold, where wereplace curl by ˚curl. Throughout this section, we shall again assume that ( A , A ) is closed and exact.In this section, we want to prove another characterisation of nonlocal H -convergence.In fact, this is the characterisation one uses in applications and can thus be viewed as themain abstract result, when characterising nonlocal H -convergence. We need variants of theoperators A ⋄ and ( A ∗ ) ⋄ that are defined on the whole of H . We put for all ϕ ∈ H A ⋄ , k ( ϕ ) = A ⋄ ( π ϕ ) and ( A ∗ ) ⋄ k ( ϕ ) = ( A ∗ ) ⋄ ( π ϕ ) , where π and π are the orthogonal projections on ran( A ) = ker( A ∗ ) ⊥ and ran( A ∗ ) =ker( A ) ⊥ . Note that this definition is consistent with A ∗ and A in the sense that we have A ⋄ , k = A ∗ on dom( A ∗ )and ( A ∗ ) ⋄ k = A on dom( A ) . Example 6.1.
Recall the setting of Example 3.3(a). We have realised that A : H (Ω) → ran( ˚grad) , u grad u . Then it is not hard to see that A ⋄ : ran( ˚grad) → H − (Ω) , q div q. On the other hand div q = 0 for all q ∈ ran( ˚grad) ⊥ = ker(div) ⊆ L (Ω) , we deduce that A ⋄ , k : L (Ω) → H − (Ω) , q div q. heorem 6.2. Let ( a n ) n in M ( α, β, ( A , A )) be bounded, a ∈ L ( H ) , H separable. Thenthe following statements are equivalent:(i) a ∈ M ( α, β, ( A , A )) , and ( a n ) n nonlocally H -converges to a ;(ii) for all ( q n ) n in H weakly convergent to some q in H and for all κ : N → N strictlymonotone we have: Given the two conditions(a) ( A ⋄ , k ( a κ ( n ) q n )) n is relatively compact in dom( A ) ∗ ,(b) (( A ∗ (cid:1) ⋄ k ( q n )) n is relatively compact in dom( A ∗ ) ∗ ,then a κ ( n ) q n ⇀ aq as n → ∞ . Remark 6.3.
In the proof of Theorem 6.2 the separability of H is used only in the im-plication ‘(ii) ⇒ (i)’, where we employ sequential compactness of M ( α, β, ( A , A )) underthe topology induced by nonlocal H -convergence. We included the separability assumptionfor convenience. For the seemingly relatively rare occasions, where non-separable Hilbertspaces are considered, we note that the corresponding reformulation of Theorem 6.2 invokes(sub)nets rather than (sub)sequences. Remark 6.4. (a) For the particular case of periodic multiplication operators in L (Ω) with a n = a ∗ n so that ( a n ) n locally H -converges to a hom , where a hom is the usual homogenisedconstant coefficient matrix, the implication ‘(i) ⇒ (ii)’ is contained in [18, Lemma 4.5].(b) In case of local H -convergence a variant of Theorem 6.2 has been stated in [38, p.10].An application of Theorem 5.11 yields another characterisation of local H -convergence.To the best of the author’s knowledge this characterisation has not been pointed out in theliterature, yet. In any case, the only important point is that ( ˚grad , ˚curl) is closed and exact;see Examples 2.3 and 2.4. Theorem 6.5.
Let Ω ⊆ R be an open, bounded weak Lipschitz domain with connectedcomplement. Let ( a n ) n in M ( α, β, Ω) , a ∈ M ( α, β, Ω) . Then the following statements areequivalent:(i) ( a n ) n locally H -converges to a ;(ii) for all ( q n ) n in L (Ω) weakly convergent to some q in L (Ω) and κ : N → N strictlymonotone we have: Given the conditions(a) (div( a κ ( n ) q n )) n is relatively compact in H − (Ω) ,(b) ( ˚curl( q n )) n is relatively compact in H sol (curl , Ω) ∗ ,then a κ ( n ) q n ⇀ aq as n → ∞ . Remark 6.6.
We note that in Theorem 6.5 (with Ω that admit a continuous extensionoperator H (Ω) → H ( R ); by Calderon’s extension theorem strong Lipschitz boundary isenough), it is possible to replace H sol (curl , Ω) ∗ by H − (Ω). We refer to [49, (the proof of)Proposition 3.10] for the details.The next example revisits the Examples 3.9 and 5.7, which is used in the already men-tioned McKean–Vlasov model ([5]) and in the so-called nonlocal response theory, see [20,Chapter 10] as well as [16, 9, 21]. Ω is assumed to be bounded and such that ( ˚grad , ˚curl)32s closed and exact, which by Examples 2.3 and 2.4 for instance corresponds to Ω being abounded Lipschitz domain with connected complement.We furthermore note that Theorem 6.5 provides the desired characterisation for nonlocal H -convergence, which avoids explicitly computing the inverses of the operators considered.In fact, this solves the problem we have encountered at the end of Example 3.9. Moreover,assuming more regularity of the integral kernels, we are also in the position to answer a partof the question raised in Example 3.9 and specified at the end of Example 5.7. Example 6.7.
Let Ω ⊆ R be bounded and such that ( ˚grad , ˚curl) is exact and closed.Let ( a n ) n be nonlocally H -converges to a with respect to ( ˚grad , ˚curl). Let k n ∗ ϕ := ( x R Ω k n ( x − y ) ϕ ( y ) dy ) for some bounded sequence ( k n ) n in W , ∞ ( R ). Assume that ( k n ) n converges in the weak*-topology to some k ∈ L ∞ ( R ). Assume further that there exists c > a n + k n ∗ = ( a n + k n ∗ ) ∗ > c Note that then we find α, β > a n + k n ∗ ∈ M ( α, β, ( ˚grad , ˚curl)) for all n ∈ N .Then ( a n + k n ∗ ) n nonlocally H -converges to a + k ∗ .In order to establish the claim, we will apply the div-curl type characterisation fromTheorem 6.2 (Theorem 6.5). So, let ( q n ) n be a weakly convergent sequence in L (Ω) withlimit q . Further, let κ : N → N be strictly monotone and assume that(a) (div( a κ ( n ) + k κ ( n ) ∗ ) q n ) n is relatively compact in H − (Ω),(b) ( ˚curl q n ) n is relatively compact in H sol (curl , Ω) ∗ .Note that since ∂ j k n ∈ L ∞ ( R ) for all j ∈ { , , } , we obtain thatdiv( k κ ( n ) ∗ q n ) = X j =1 ∂ j k κ ( n ) ∗ q n,j ∈ L (Ω) , uniformly in n . By the compactness of the embedding H (Ω) ֒ → L (Ω), we deduce that(div k κ ( n ) ∗ q n ) n is relatively compact in H − (Ω). Thus, condition (a), yields that (div( a κ ( n ) q n )) n is relatively compact in H − (Ω). Thus, by nonlocal H -convergence of ( a n ) n to a and Theo-rem 6.2, we infer that a κ ( n ) q n ⇀ aq . Thus, we are left with proving that k κ ( n ) ∗ q n ⇀ k ∗ q. For this, let ϕ ∈ L (Ω) and consider h k κ ( n ) ∗ q n , ϕ i = h q n , k κ ( n ) ∗ ϕ i . Next, we see that (div k κ ( n ) ∗ ϕ ) n is bounded in L (Ω) and so relatively compact in H − (Ω)by the boundedness of Ω. Moreover, we compute for ψ ∈ L (Ω) h k κ ( n ) ∗ ϕ, ψ i = Z Ω h k κ ( n ) ∗ ϕ ( x ) , ψ ( x ) i dx = Z Ω D Z Ω k κ ( n ) ( x − y ) ϕ ( y ) , ψ ( x ) E dx. ϕ ∈ L (Ω) . Moreover, it is easy to see that k n convergingweakly* to k implies that k n ( x − · ) converging weakly* to k ( x − · ). Thus, we infer bydominated convergence h k κ ( n ) ∗ ϕ, ψ i → h k ∗ ϕ, ψ i . By condition (b) and Theorem 6.8 below, we thus infer h k κ ( n ) ∗ q n , ϕ i = h q n , k κ ( n ) ∗ ϕ i → h q, k ∗ ϕ i = h k ∗ q, ϕ i , which shows the assertion.The proof of Theorem 6.2 needs some prerequisites. The first one is a global div-curltype result, see [49, Theorem 2.4]; see also [26] for several applications and [6, Theorem 3.1]for a Banach space setting. We shall furthermore refer to [22] and the references therein fora guide to the literature for other results and approaches to the div-curl lemma. Theorem 6.8 ([49, Theorem 2.4]) . Let ( q n ) n , ( r n ) n be weakly convergent in H . Assumethat ( A ⋄ , k q n ) n and (cid:0) ( A ∗ ) ⋄ k r n (cid:1) n are relatively compact in dom( A ) ∗ and dom( A ∗ ) ∗ , respectively.Then lim n →∞ h q n , r n i H = D w- lim n →∞ q n , w- lim n →∞ r n E H . For easy reference, we will use π and π for the orthogonal projections in H projectingon ran( A ) and ran( A ∗ ), respectively. Lemma 6.9.
Let a ∈ M ( α, β, ( A , A )) . Let v, w ∈ H . Then the following conditions areequivalent:(i) w = av ;(ii) π w = π av and π v = π a − w .Proof. Note that (i) trivially implies (ii). For the other implication, we use the block matrixrepresentation worked out in Lemma 4.8. Condition (ii) is equivalent to (cid:18) a a (cid:19) (cid:18) π vπ v (cid:19) = (cid:18) π w (cid:19) (11)and (cid:18) π v (cid:19) = (cid:18) (cid:19) (cid:18)(cid:0) a − (cid:1) (cid:0) a − (cid:1) (cid:0) a − (cid:1) (cid:0) a − (cid:1) (cid:19) (cid:18) π wπ w (cid:19) = (cid:18) − (cid:0) a − a a − a (cid:1) − a a − (cid:0) a − a a − a (cid:1) − (cid:19) (cid:18) π wπ w (cid:19) (cid:0) a − a a − a (cid:1) π v = π w − a a − π w. Next, from (11), we obtain π w = a π v + a π v . Hence, (cid:0) a − a a − a (cid:1) π v = π w − a a − (cid:0) a π v + a π v (cid:1) = π w − a π v − a a − a π v. Thus, π w = a π v + a π v. This equation together with (11) implies (i).
Lemma 6.10.
Let a ∈ L ( H ) and b ∈ M ( α, β, ( A , A )) . Then the following conditions areequivalent(i) a = b ;(ii) b − aπ = π and ab − π = π . Proof.
The implication (i) ⇒ (ii) is evidently true. Thus, we assume (ii) to hold. We aim forshowing a = b . For this, we note that b − aπ = π implies aπ = bπ . Thus, using the blockmatrix representation from Section 4, we infer (cid:18) a a a a (cid:19) (cid:18) (cid:19) = (cid:18) b b b b (cid:19) (cid:18) (cid:19) , which implies a = b and a = b . (12)Next, from ab − π = π , we obtain (cid:18) (cid:19) = (cid:18) a a a a (cid:19) (cid:18) ( b − ) ( b − ) ( b − ) ( b − ) (cid:19) (cid:18) (cid:19) = (cid:18) a a a a (cid:19) (cid:18) b − ) b − ) (cid:19) = (cid:18) a ( b − ) + a ( b − ) a ( b − ) + a ( b − ) (cid:19) Thus, using (12), we infer − b ( b − ) = a ( b − ) − b ( b − ) = a ( b − ) . Multiplying both equations by ( b − ) − from the right and using the expressions stated inLemma 4.8(b), we obtain a = − b ( b − ) ( b − ) − = − b (cid:0) − b − b ( b − ) (cid:1) ( b − ) − = b (13)35nd, similarly, a = ( b − ) − − b ( b − ) ( b − ) − = b − b b − b + b b − b = b . (14)Thus, the equations (12) together with (13) and (14) imply a = b and, hence, the assertion.We like to point out that in the implication ‘(ii) ⇒ (i)’ of Lemma 6.10, the invertibility of a is implied rather than assumed.We may now present the proof of Theorem 6.2. We note that the implication ‘(i) ⇒ (ii)’should be seen as an abstract implementation of Tartar’s method of oscillating test functions. Proof of Theorem 6.2.
We shall assume that a ∈ M ( α, β, ( A , A )) and that ( a n ) n nonlocally H -converges to a and let ( q n ) n and q be as in (ii). By Theorem 4.1, we shall assume withoutloss of generality that κ ( n ) = n since any subsequence of ( a n ) n also nonlocally H -convergesto a . By Corollary 4.6, ( a ∗ n ) n nonlocally H -converges to a ∗ . Let v ∈ dom( A ) and define v n to be the solution of h a ∗ n A v n , A ϕ i = f ( ϕ ) ( ϕ ∈ dom( A )) , where f ∈ dom( A ) ∗ is given by f ( ϕ ) = h a ∗ A v, A ϕ i ( ϕ ∈ dom( A )) . Since ( a ∗ n ) n nonlocally H -converges to a ∗ , we obtain that ( v n ) n weakly converges to some w ∈ dom( A ) satisfying h a ∗ A w, A ϕ i = h a ∗ A v, A ϕ i ( ϕ ∈ dom( A )) , which, by Theorem 2.5, leads to w = A − ( a ∗ ) − ( A ⋄ ) − f = A − ( a ∗ ) − ( A ⋄ ) − ( A ⋄ )( a ∗ ) A v = v. Moreover, by nonlocal H -convergence, we deduce a ∗ n A v n ⇀ a ∗ A v in H as n → ∞ . Wenote, in particular, that A ⋄ , k ( a ∗ n A v n ) = f and ( A ∗ ) ⋄ k A v n = 0, by the complex property. Forthe latter note that ker(( A ∗ ) ⋄ k ) = ker( A ). Without loss of generality, we may assume that( a n q n ) n weakly converges to some r ∈ H . For n ∈ N we have h a n q n , A v n i = h q n , a ∗ n A v n i . (15)Using Theorem 6.8 together with the assumptions (a) and (b) imposed on q , we infer fromequation (15) by letting n → ∞h r, A v i = h q, a ∗ A v i = h aq, A v i . Since v ∈ dom( A ) can be chosen arbitrarily, we obtain π r = π aq, (16)36here π is the orthogonal projection on ran( A ).Next, let s ∈ dom( A ∗ ). Let ( s n ) n be the sequence in dom( A ) satisfying h ( a − n ) ∗ A ∗ s n , A ∗ ψ i = h ( a − ) ∗ A ∗ s, A ∗ ψ i By the nonlocal H -convergence of ( a ∗ n ) n to a ∗ it follows (invoking Theorem 2.5 again) that s n ⇀ s ∈ dom( A ) , and ( a − n ) ∗ A ∗ s n ⇀ ( a − ) ∗ A ∗ s. Moreover, we have that ( A ∗ ) ⋄ k ( a − n ) ∗ A ∗ s n = ( A ∗ ) ⋄ k ( a − ) ∗ A ∗ s as well as A ⋄ , k A ∗ s n = 0 . Next, for n ∈ N , we have h q n , A ∗ s n i = h a n q n , ( a − n ) ∗ A ∗ s n i . By Theorem 6.8 together with the assumptions on q n , we may let n → ∞ and obtain h q, A ∗ s i = h r, ( a − ) ∗ A ∗ s i . As s ∈ dom( A ∗ ) was arbitrary, this yields π q = π a − r, (17)where π is the orthogonal projection onto ran( A ∗ ). Applying Lemma 6.9 to w = r and v = q , we obtain aq = r .We shall now assume that (ii) holds. By Theorem 5.5, we may choose a κ : N → N strictlymonotone such that of ( a κ ( n ) ) n nonlocally H -converges to some b ∈ M ( α, β, ( A , A )). Next,let f ∈ dom( A ) ∗ and g ∈ dom( A ∗ ) ∗ and let ( u n ) n as well as ( v n ) n satisfy h a κ ( n ) A u n , A ϕ i = f ( ϕ ) , h a − κ ( n ) A ∗ v n , A ∗ ψ i = g ( ψ ) , for all ϕ ∈ dom( A ) and ψ ∈ dom( A ∗ ). By nonlocal H -convergence, we obtain u n ⇀ u ∈ dom( A ) , a κ ( n ) A u n ⇀ bA uv n ⇀ v ∈ dom( A ∗ ) , a − κ ( n ) A v n ⇀ b − A v, where u and v satisfy h bA u, A ϕ i = f ( ϕ ) , h b − A ∗ v, A ∗ ψ i = g ( ψ ) , for all ϕ ∈ dom( A ) and ψ ∈ dom( A ∗ ). We observe that A ⋄ , k ( a κ ( n ) A u n ) = f, ( A ∗ ) ⋄ k A u n = 0 , A ⋄ , k A ∗ v n = 0 , ( A ∗ ) ⋄ k a − κ ( n ) A ∗ v n = g. q n = A u n or q n = a − κ ( n ) A ∗ v n , we obtain a κ ( n ) A u n ⇀ aA u, and a κ ( n ) a − κ ( n ) A ∗ v n ⇀ ab − A ∗ v. Thus, b − aA u = A u and A ∗ v = ab − A ∗ v . As f and g are arbitrary, as in the proof of‘(i) ⇒ (ii)’ we infer that u ∈ dom( A ) and v ∈ dom( A ∗ ) are arbitrary, as well. Hence, b − aπ = π and ab − π = π . By Lemma 6.10, we obtain a = b . The subsequence principle concludes the proof. In this section, we shall consider a homogenisation problem for Maxwell’s equations. Incontrast to many other discussions of homogenisation problems for the Maxwell system, weshall treat the full 3-dimensional time-dependent problem. Moreover, the setting is arrangedin a way that we may allow for the homogenisation of highly oscillatory mixed type problems,where several regions of the underlying material are considered to have no dielectricity atall. That is to say, at certain regions of the underlying domain, one may or may not use theeddy current approximation. This goes well beyond the available results in the literature.Equations having highly oscillatory change of type have also been analysed in [48, 15, 7].In these references, however, the attention is restricted to 1 + 1-dimensional model examples.For other treatments of the homogenisation of the full time-dependent 3D-Maxwell’sequations we refer to [51] and [1]. In these references, the coefficients are assumed to beperiodic. We shall furthermore refer to [35, 11], where the periodicity of the problem isexploited with the help of the Floquet–Bloch or Gelfand transformation. In particular, werefer to the seminal work [36] and the references therein.In an open set Ω ⊆ R , Maxwell’s equations are formulated as follows. Find E, H : R × Ω → R for a given J : R × Ω → R such that ∂ t εE + σE − curl H = J∂ t µB + ˚curl E = 0 , where for simplicity, we assume zero initial conditions. Moreover, ε, µ, σ ∈ L ( L (Ω) ) (di-electricity, permeability, conductivity) are given bounded linear operators with ε, µ beingselfadjoint.In the Hilbert space framework, we shall apply next, we will favour the following block-operator-matrix form (cid:16) ∂ t (cid:18) ε µ (cid:19) + (cid:18) σ
00 0 (cid:19) + (cid:18) − curl˚curl 0 (cid:19) (cid:17) (cid:18) EH (cid:19) = (cid:18) J (cid:19) . (18)Before turning to a homogenisation result for Maxwell’s equations (see in particular Exam-ple 7.12 below), we shall shortly recall the well-posedness result, which will be used in the38ollowing. For a Hilbert space H and ν > L ν ( R ; H ) := { f ∈ L ( R ; H ); Z R k f ( t ) k H exp( − νt ) dt < ∞} . We recall from [19, Corollary 2.5] that the Fourier–Laplace transformation L ν ϕ ( ξ ) := 1 √ π Z R ϕ ( t ) exp( − itξ − νt ) dt ( ϕ ∈ C c ( R ; H ))can be extended unitarily as an operator from L ν ( R ; H ) onto L ( R ; H ). Moreover, we havethat the weak derivative ∂ t realised as an operator with maximal domain in L ν ( R ; H ) enjoysthe spectral representation ∂ t = L ∗ ν ( im + ν ) L ν , where im + ν is the multiplication operator of multiplying by x ix + ν with maximaldomain. We denote for µ > H ∞ ( C Re >µ ; L ( H )) := { M : C Re >µ → L ( H ); M analytic and bounded } . For the well-posedness of Maxwell’s equations we shall employ the following theorem.
Theorem 7.1 ([32, Solution Theory]) . Let c > , µ > , M ∈ H ∞ ( C Re >µ ; L ( H )) , ν > µ .Assume that Re λM ( λ ) > c ( λ ∈ C Re >µ ) . Let A be a skew-self-adjoint operator in H . Then the operator B := ∂ t M ( ∂ t ) + A := L ∗ ν (cid:0) ( im + ν ) M ( im + ν ) + A (cid:1) L ν , where ( im + ν ) M ( im + ν ) + A (cid:1) is the (abstract) multiplication operator of multiplying by x ( ix + ν ) M ( ix + ν ) + A , is continuously invertible in L ν ( R ; H ) ; we have kB − k /c . Recall from the spectral representation for ∂ t that for ν >
0, the operator ∂ t is continu-ously invertible. Remark 7.2.
Theorem 7.1 applies to (18) with the setting H = L (Ω) ⊕ L (Ω) , M ( ∂ t ) = (cid:18) ε µ (cid:19) + ∂ − t (cid:18) σ
00 0 (cid:19) , A = (cid:18) − curl˚curl 0 (cid:19) . Note that the positive definiteness requirement translates intoRe λM ( λ ) = Re λ (cid:18)(cid:18) ε µ (cid:19) + 1 λ (cid:18) σ
00 0 (cid:19)(cid:19) = Re (cid:18) λ (cid:18) ε µ (cid:19) + (cid:18) σ
00 0 (cid:19)(cid:19) > c, for some c > λ with Re λ large enough. Thus,Re λM ( λ ) > c ⇐⇒ (cid:0) λε + Re σ > c (cid:1) ∧ (cid:0) µ > c (cid:1) . Proposition 7.3.
Let B : dom( B ) ⊆ K → K , B : dom( B ) ⊆ K → K , B : dom( B ) ⊆ K → K be densely defined and closed linear operators acting in the Hilbert spaces K , K , K , and K . Assume that ( B , B ) , ( B , B ) are compact and exact. Define ( A , A ) := (cid:16)(cid:16) B ∗ B (cid:17) , (cid:16) B − B ∗ (cid:17)(cid:17) with dom( A ) = dom( B ) ⊕ dom( B ∗ ) and dom( A ) = dom( B ∗ ) ⊕ dom( B ) with H = K ⊕ K , H = H = K ⊕ K .Then ( A , A ) is compact and exact.Proof. We frequently use Proposition 2.1 in the following. Since ( B , B ) is compact, ( B , B )is closed. As ( B , B ) is also exact, we obtain that ( B ∗ , B ∗ ) is exact, as well. Hence,ran( A ) = ran( B ∗ ) ⊕ ran( B ) = ker( B ∗ ) ⊕ ker( B ) = ker( A ) , which shows that ( A , A ) is exact. We are left with showing that ( A , A ) is compact. Forthis, we realise that dom( A ∗ ) = dom( B ) ⊕ dom( B ∗ ) . Hence,dom( A ∗ ) ∩ dom( A ) = (cid:0) dom( B ) ∩ dom( B ∗ ) (cid:1) ⊕ (cid:0) dom( B ∗ ) ∩ dom( B ) (cid:1) Since ( B , B ) is compact, so is ( B ∗ , B ∗ ). Hence, (cid:0) dom( B ) ∩ dom( B ∗ ) (cid:1) ֒ → K compactly.The compactness of ( B , B ), thus, implies that dom( A ∗ ) ∩ dom( A ) ֒ → H = K ⊕ K compactly, that is, ( A , A ) is compact. Example 7.4.
Let Ω ⊆ R be an open bounded simply connected weak Lipschitz domainwith connected complement. Then the typical situation for Maxwell’s equations for applyingProposition 7.3 is as follows: B = grad, B = curl, and B = div with dom(grad) = H (Ω),dom(curl) = H (curl , Ω) and dom(div) = H (div , Ω), K = L (Ω), K = L (Ω) , K = L (Ω) ,and K = L (Ω). The assumptions on Ω render (grad , curl) and (curl , div) exact andcompact. Indeed, the compactness of the complexes follows from Example 2.3 (a2) and(b2). Thus, Proposition 2.1(e) implies closedness of the complexes. Next, by Example 2.4,(grad , curl) is exact as Ω is simply connected. Moreover, (curl , div) is exact, if and only if( ˚grad , ˚curl) is exact, by Proposition 2.1 and the closedness of (curl , div). By Example 2.4,(curl , div) is, thus, exact since R \ Ω is connected.
Theorem 7.5 (Homogenisation theorem) . Let B : dom( B ) ⊆ K → K , B : dom( B ) ⊆ K → K , B : dom( B ) ⊆ K → K be densely defined and closed linear operators actingin the Hilbert spaces K , K , K , and K . Assume that ( B , B ) , ( B , B ) are compactand exact. Define H := K ⊕ K . Let ( M n ) n in H ∞ ( C Re >µ ; L ( H )) be bounded and let M ∈ H ∞ ( C Re >µ ; L ( H )) for some µ > .. Assume Re λM n ( λ ) > c ( λ ∈ C Re >µ ) as well as for all λ ∈ R >µ M n ( λ ) → M ( λ )40 -nonlocally with respect to (cid:16)(cid:16) B ∗ B (cid:17) , (cid:16) B − B ∗ (cid:17)(cid:17) as n → ∞ .Then (cid:16) ∂ t M n ( ∂ t ) + (cid:16) B − B ∗ (cid:17)(cid:17) − → (cid:16) ∂ t M ( ∂ t ) + (cid:16) B − B ∗ (cid:17)(cid:17) − in the weak operator topology of L ( L ν ( R ; H )) for all ν > µ .Proof. ( A , A ) := (cid:16)(cid:16) B ∗ B (cid:17) , (cid:16) B − B ∗ (cid:17)(cid:17) is compact and exact, by Proposition 7.3. Let λ ∈ R >µ . We write M n,ij ( λ ) ∈ L (ran( A j ) , ran( A i )) according to the decomposition inducedby ran( A ) ⊕ ran( A ) for all i, j ∈ { , } . Let F ∈ H . We define U n := (cid:16) λM n ( λ ) + (cid:16) B − B ∗ (cid:17)(cid:17) − F. Writing F j , U j,n for the components in ran( A j ) for j ∈ { , } , we obtain the following equiv-alent formulation for the equation defining U n : λ (cid:18) M n, ( λ ) M n, ( λ ) M n, ( λ ) M n, ( λ ) (cid:19) (cid:18) U ,n U ,n (cid:19) + (cid:18) B
00 0 (cid:19) (cid:18) U ,n U ,n (cid:19) = (cid:18) F F (cid:19) , (19)where B denotes the operator acting as (cid:16) B − B ∗ (cid:17) which is domain-wise restricted to theorthogonal complement of the null space of (cid:16) B − B ∗ (cid:17) and co-domain-wise restricted to therange of (cid:16) B − B ∗ (cid:17) . A straightforward computation shows that equation (19) equivalentlyreads as (cid:18) λM n, ( λ ) − λM n, ( λ ) M n, ( λ ) − M n, ( λ ) 0 M n, ( λ ) − M n, ( λ ) 1 (cid:19) (cid:18) U ,n U ,n (cid:19) + (cid:18) B
00 0 (cid:19) (cid:18) U ,n U ,n (cid:19) = (cid:18) F − M n, ( λ ) M n, ( λ ) − F λ M n, ( λ ) − F (cid:19) or (cid:18) U ,n U ,n (cid:19) = (cid:18) ( λ ( M n, ( λ ) − M n, ( λ ) M n, ( λ ) − M n, ( λ )) + B ) − ( F − M n, ( λ ) M n, ( λ ) − F ) − M n, ( λ ) − M n, ( λ ) U ,n + λ M n, ( λ ) − F (cid:19) . (20)By Theorem 4.1, we have that λ (cid:0) M n, ( λ ) − M n, ( λ ) M n, ( λ ) − M n, ( λ ) (cid:1) → λ (cid:0) M ( λ ) − M ( λ ) M ( λ ) − M ( λ ) (cid:1) M n, ( λ ) M n, ( λ ) − → M ( λ ) M ( λ ) − M n, ( λ ) − M n, ( λ ) → M ( λ ) − M ( λ ) M n, ( λ ) − → M ( λ ) − n → ∞ with convergence in the respective weak operator topologies. We note that, bythe identity theorem the convergence of the just mentioned operator sequences does actuallyhold for all λ ∈ C provided Re λ is large enough.Next, by the compactness of the complex ( A , A ), the operator B has compact resolvent.By Lemma 7.6 below applied to B = B , T n = λ ( M n, ( λ ) − M n, ( λ ) M n, ( λ ) − M n, ( λ )) and ϕ n = ( F − M n, ( λ ) M n, ( λ ) − F ), we deduce that ( U ,n ) n converges in norm to some U .Hence, ( U ,n ) n weakly converges to some U . Letting n → ∞ in (20) thus leads to (cid:18) U U (cid:19) = (cid:18) ( λ ( M ( λ ) − M ( λ ) M ( λ ) − M ( λ )) + B ) − ( F − M ( λ ) M ( λ ) − F ) − M ( λ ) − M ( λ ) U + λ M ( λ ) − F (cid:19) . Rearranging terms, we obtain with U = ( U , U ) (cid:16) λM ( λ ) + (cid:16) B − B ∗ (cid:17)(cid:17) U = F. This settles the proof.We complete the latter proof by stating and proving Lemma 7.6.
Lemma 7.6.
Let B : dom( B ) ⊆ H → H be skew-self-adjoint in the Hilbert space H andassume that dom( B ) ֒ → H is compact. Assume furthermore that ( T n ) n is a sequence in L ( H ) such that Re T n > c for all n ∈ N . If T n → T in the weak operator topology for some T ∈ L ( H ) as n → ∞ , then ( T n + B ) − ϕ n → ( T + B ) − ϕ in H for all ( ϕ n ) n weakly convergent to some ϕ ∈ H .Proof. Let ϕ n , ϕ be as in the statement. We define u n := ( T n + B ) − ϕ n . We obtain that ( u n ) n is bounded in dom( B ); see also [40, Lemma 2.12] for the preciseargument. Possibly choosing a subsequence (not relabelled) of ( u n ) n , we may assume that u n ⇀ u in dom( B ) for some u . In particular, we obtain that u n → u in H . Hence, in theequality ϕ n = T n u n + Bu n , we let n → ∞ and obtain ϕ = T u + Bu.
The continuous invertibility of T + B identifies u and thus the whole sequence convergesweakly in dom( B ) and strongly in H , which is the assertion. Remark 7.7.
A contradiction argument yields that the convergence implied in Lemma 7.6together with the compactness assumption is sufficient for operator norm convergence of( T n + B ) − → ( T + B ) − as n → ∞ . 42 emark 7.8. The proof of Theorem 7.5 is a variant of the rationale employed in the proof of[47, Theorem 5.5]. However, we note that the conditions in [47] are more restrictive than theones here. Indeed, in [47] only a compactness statement for ‘ G -convergence’ was obtained.Moreover, in order to prove well-posedness of the limit equation, the class of sequences M n was more restrictive in the sense that a change of type was not permitted.Since we have discussed nonlocal H -convergence with respect to operator complexes like(grad , curl) and ( ˚grad , ˚curl) only, it might be of interest to put the convergence assumed inTheorem 7.5 into perspective of nonlocal H -convergence of simpler complexes. This is donein the following result and the subsequent example. Proposition 7.9.
Let the assumptions and definitions of Proposition 7.3 be in effect, α, β > . Let ( ε n ) n in M ( α, β, ( B , B )) and ( µ n ) n in M ( α, β, ( B ∗ , B ∗ )) be bounded sequences, ε ∈ L ( K ) , µ ∈ L ( K ) be continuously invertible. Then for all n ∈ N (cid:18) ε n µ n (cid:19) ∈ M ( α, β, ( A , A )) . Moreover, the following conditions are equivalent.(i) ( ε n ) n → ε H -nonlocally w.r.t. ( B , B ) and ( µ n ) n → µ H -nonlocally w.r.t. ( B ∗ , B ∗ ) .(ii) (cid:0) (cid:0) ε n µ n (cid:1) (cid:1) n → (cid:0) ε µ (cid:1) H -nonlocally w.r.t. ( A , A ) . Example 7.10.
With the setting given in Example 7.4, we obtain for operator sequences( ε n ) n and ( µ n ) n in L ( L (Ω) ) satisfying suitable positive definiteness constraints and for ε, µ ∈ L ( L (Ω) ) that (cid:0) (cid:0) ε n µ n (cid:1) (cid:1) n → (cid:0) ε µ (cid:1) H -nonlocally w.r.t. ( A , A ) if and only if ( ε n ) n → ε H -nonlocally w.r.t. (grad , curl) and ( µ n ) n → µ H -nonlocally w.r.t. ( ˚grad , ˚curl). Proof of Proposition 7.9.
By Proposition 7.3 and (4), we have the decomposition H =ran( A ) ⊕ ran( A ∗ ). Moreover, we obtain from the exactness of ( B , B ) and ( B ∗ , B ∗ ) thedecomposition H = K ⊕ K = (cid:0) ran( B ∗ ) ⊕ ran( B ) (cid:1) ⊕ (cid:0) ran( B ) ⊕ ran( B ∗ ) (cid:1) . Next, from ran( A ) = ran( B ∗ ) ⊕ ran( B ) and ran( A ∗ ) = ran( B ) ⊕ ran( B ∗ ), we deduce usingthe operators U := (cid:0) ι r ,A ι r ,A ∗ (cid:1) U := (cid:0) ι r ,B ι r ,B ∗ (cid:1) U := (cid:0) ι r ,B ∗ ι r ,B (cid:1) , (see also (6)) that for any ε ∈ L ( K ) and µ ∈ L ( K ), we have U ∗ (cid:18) ε µ (cid:19) U = (cid:18) ε µ (cid:19) (cid:18) ε µ (cid:19)(cid:18) ε µ (cid:19) (cid:18) ε µ (cid:19) , U ∗ εU = (cid:18) ε ε ε ε (cid:19) and U ∗ µU = (cid:18) µ µ µ µ (cid:19) . These representations of ε and µ applied to ε n and µ n instead yield the first assertion ofthe present proposition; the equivalence then follows from Theorem 4.1 in a straightforwardmanner. Remark 7.11.
The main result in [1] is contained in Theorem 7.5. In fact, it suffices to takethe setting as outlined in Example 7.4; we also refer to Example 7.10. We also recall thatlocal H -convergence implies nonlocal H -convergence with respect to both ( ˚grad , ˚curl) and(grad , curl) (see also Theorem 5.11 and Remark 5.12). Moreover, we note that the coefficientstreated in [1] are arranged in a way that their Fourier–Laplace transformed images locally H -converge. We also refer to the subsequent example.A more concrete example with change of type, that is, where the underlying problemis such that the Maxwell’s equations rapidly oscillate between the parabolic eddy currentproblem and the hyperbolic full Maxwell’s equations, is considered next. Example 7.12.
Let Ω ⊆ R be such that both ( ˚grad , ˚curl) and (grad , curl) are compactand exact sequences; e.g. Ω being an open, simply connected and bounded weak Lipschitzdomain with connected R \ Ω.(a) Let ε, µ, σ ∈ L ∞ ( R ) × be [0 , -periodic with ε, µ attaining values in the (possiblycomplex) self-adjoint matrices. Define ε n ( x ) := ε ( nx ) for a.a. x ∈ R and similarly for µ n ,and σ n . Assume that there exists η > λ ∈ C Re >η we haveRe ( λε + σ ) , µ > c ( n ∈ N ) (21)for some c > ε ( x ) > c for a.a. x ∈ Ω and Re σ ( x ) > c for a.a. x ∈ Ω := R \ Ω for some c > ε on Ω and σ on Ω are allowed to vanish, while the positive definiteness condition in(21) can still be warranted. This introduces a highly oscillatory change of type. Then byTheorem 7.5 (see also Remark 7.11 and Example 7.10) (cid:18) ∂ t (cid:18) ε n µ n (cid:19) + (cid:18) σ n
00 0 (cid:19) + (cid:18) − curl˚curl 0 (cid:19)(cid:19) − → ∂ t D ε + ( · ) σ E hom ( ∂ t ) 00 D µ E hom + (cid:18) − curl˚curl 0 (cid:19) −
44n the weak operator topology of L (cid:0) L ν ( R ; L (Ω) ) (cid:1) as n → ∞ , where D µ E hom is the standardhomogenised matrix associated with µ and D ε + ( · ) σ E hom ( ∂ t ) := (cid:16) λ D ε + λ − σ E hom (cid:17) ( ∂ t ) . This is a memory term that occurs during the homogenisation process. We note here thatsuch an effect has been observed already in [18, p. 144], but also in [51, Theorem 3.2].(b) Let ε, σ as in (a) and µ n := a n + k n ∗ as in Example 6.7 (we shall also re-use the notation a and k ∗ for the operators mentioned in that example). Then the results in Example 6.7,Example 7.10, and Theorem 7.5 yield (cid:18) ∂ t (cid:18) ε n a n + k n ∗ (cid:19) + (cid:18) σ n
00 0 (cid:19) + (cid:18) − curl˚curl 0 (cid:19)(cid:19) − → ∂ t D ε + ( · ) σ E hom ( ∂ t ) 00 a + k ∗ ! + (cid:18) − curl˚curl 0 (cid:19)! − in the weak operator topology of L (cid:0) L ν ( R ; L (Ω) ) (cid:1) as n → ∞ . We emphasise that theconvolution k ∗ is computed with respect to the spatial variables. Thus, the limit model isboth nonlocal in space and time. In this section, we shall provide two more applications. In fact, since our results has beendeveloped for the abstract setting of closed complexes in Hilbert spaces and suitable operatorsas coefficients, this section may also be read as the versatility of the notion of complexes inthe analysis of partial differential equations.
Homogenisation problems for fourth order elliptic equations
In this section, we shall revisit the homogenisation problem for thin plates (see e.g. [25])In that reference, the author studied operator norm error estimates for the homogenisationproblem associated to the differential expression X i,j,s,h ∈{ , , } ∂ i ∂ j a ijsh ∂ s ∂ h , where the coefficients a ijsh are highly oscillatory. It is easy to see that the latter differentialexpression can be reformulated as div Div a Grad grad , a acts as a mappingfrom 2-tensors to 2-tensors. The variational formulation is then given by h a Grad grad u, Grad grad ϕ i = f ( ϕ )for ϕ belonging to a suitable test-function space. If we assume Grad grad to be endowedwith full homogeneous boundary conditions (i.e. the L -closure of Grad grad restricted to testfunctions compactly supported in Ω), it is possible to derive the second variational problemto be discussed for nonlocal homogenisation problems, which we will do in the following.In fact, it will turn out that the closed and exact complex involving Grad grad bases on( ˚grad , ˚curl). For more aspects of this (and an extension of this complex) we refer to thePauly–Zulehner complex ([27]; see also [34]).We introduce the following differential operators: Definition.
Let Ω ⊆ R be open and bounded. We define˚He : H (Ω) ⊆ L (Ω) → L (Ω) × ϕ ( ∂ i ∂ j ϕ ) i,j ∈{ , , } ˚Curl sym : dom( ˚curl) ∩ L (Ω) × ⊆ L (Ω) × → L (Ω) × ( ϕ ij ) i,j ∈{ , , } ˚curl( ϕ j ) j ∈{ , , } ˚curl( ϕ j ) j ∈{ , , } ˚curl( ϕ j ) j ∈{ , , } , where L (Ω) × is the set of symmetric 3-by-3 matrices with entries from L (Ω) and L (Ω) × is the set of 3-by-3 matrices with vanishing matrix trace and entries from L (Ω).For convenience of the reader, we show exactness of ( ˚He , ˚Curl sym ) with a proof inde-pendent of [27]. Recall that ( ˚grad , ˚curl) is closed and exact, for instance, if Ω is an open,bounded weak Lipschitz domain with connected complement. Theorem 8.1.
Let Ω ⊆ R open and bounded with ( ˚grad , ˚curl) exact and closed. Then ( ˚He , ˚Curl sym ) is an exact and closed complex. Lemma 8.2.
Let Ω ⊆ R open and bounded. Then the graph norm of ˚He and the H -normare equivalent on H (Ω) .Proof. Let ϕ ∈ C ∞ c (Ω). Then we have for all i, j ∈ { , , }k ϕ k L (Ω) c k ∂ j ϕ k L (Ω) c k ∂ i ∂ j ϕ k L (Ω) c k ˚He ϕ k L (Ω) c k ϕ k H (Ω) where the last and second last inequalities are trivial and the first and the second one followfrom Poincar´e’s inequality for some suitable constant c >
0. Thus, the graph norm of ˚He andthe H -norm are equivalent on C ∞ c (Ω). Since H (Ω) = C ∞ c (Ω) H we obtain the assertion.46ote that Lemma 8.2 particularly implies that ˚He is closed and that C ∞ c (Ω) is an operatorcore for ˚He. Proof of Theorem 8.1.
Let us start with the complex property. So, let ϕ ∈ C ∞ c (Ω). Then˚Curl sym ˚He ϕ = ˚curl( ˚He ϕ j ) j ∈{ , , } ˚curl( ˚He ϕ j ) j ∈{ , , } ˚curl( ˚He ϕ j ) j ∈{ , , } = ˚curl( ˚grad ∂ ϕ )˚curl( ˚grad ∂ ϕ )˚curl( ˚grad ∂ ϕ ) = 0 . By Lemma 8.2, C ∞ c (Ω) is an operator core for ˚He. Thus, ran( ˚He) ⊆ ker( ˚Curl sym ).Next, we show closedness of the complex. For this we note that H (Ω) embeds compactlyinto L (Ω). Hence, by Lemma 8.2, we deduce that ran( ˚He) is closed.Since ˚curl has closed range – by the closed graph theorem – we find c > ϕ ∈ dom( ˚curl) ∩ ker( ˚curl) ⊥ we have k ϕ k L c k ˚curl ϕ k L . It is easy to see that ker( ˚Curl sym ) = ker( ˚curl) ∩ L (Ω) × . Hence, if Φ ∈ dom( ˚Curl sym ) ∩ ker( ˚Curl sym ) ⊥ we obtain k Φ k L = X i ∈{ , , } k (Φ ij ) j ∈{ , , } k X i ∈{ , , } c k ˚curl (Φ ij ) j ∈{ , , } k = c k ˚Curl sym Φ k L . This yields closedness of the range of ˚Curl sym .We are left with showing the exactness of the complex. More precisely, it remains toprove ran( ˚He) ⊇ ker( ˚Curl sym ) . For this let Φ ∈ ker( ˚Curl sym ). Then there exists ϕ i ∈ dom( ˚grad) such that ˚grad ϕ i =(Φ ij ) j ∈{ , , } for all i ∈ { , , } since ( ˚grad , ˚curl) is exact. Since Φ is symmetric, we de-duce that ∂ j ϕ i = ∂ i ϕ j . Thus, ( ϕ i ) i ∈{ , , } ∈ ker(curl) ∩ dom( ˚grad) ⊆ ker( ˚curl). Hence, bythe exactness of ( ˚grad , ˚curl) we find ψ ∈ dom( ˚grad) such that ˚grad ψ = ( ϕ i ) i ∈{ , , } . It followsthat ψ ∈ dom( ˚He). Moreover, we obtain˚He ψ = (cid:0) ( ˚grad ∂ i ψ ) T (cid:1) i ∈{ , , } = (cid:0) ( ˚grad ϕ i ) T (cid:1) i ∈{ , , } = Φ . This shows the assertion.
Remark 8.3.
It is an easy exercise to show that ˚Curl ∗ sym = sym Curl trf , where sym M = ( M + M T ) and Curl trf is the distributional row-wise curl operator with no boundary con-ditions acting on trace-free matrices. It is remarkable that the equations for the descriptionof nonlocal H -convergence of ( a n ) n to some a with respect to the complex ( ˚He , ˚Curl sym ) areof different order. Indeed, one equation is h a n Grad grad u n , Grad grad ϕ i = f ( ϕ )47or suitable test functions ϕ and given right-hand side f . This corresponds to the 4th orderequation mentioned above. The second variational problem reads h a − n sym Curl trf v n , sym Curl trf ψ i = g ( ψ ) , which leads to a 2nd order problem, only. An H -compactness result for Riemannian manifolds The general setting developed in the previous sections allows for H -compactness results alsoon manifolds. We shall refer to the fairly recent result in [17], where the corresponding localproblem has been discussed.We refer to [50] or [30] for the precise functional analytic setting to be sketched below.Let Λ be a d -dimensional C ∞ -manifold and let Ω ⊆ Λ be an open submanifold of Λ. Forany q ∈ { , . . . , n } this induces L q (Ω), the completion of the space of compactly supported q -forms on Ω endowed with the scalar product h ω, η i = Z Ω ω ∧ ∗ η, where ∗ denotes the Hodge duality and ∧ the alternating product.Next, using the thus defined scalar product, we let d be the (distributional) exteriorderivative on L q (Ω) with values in the L q +1 (Ω). The adjoint of this operator is set to be ˚ δ .Similarly, we let ˚d be the closure of d restricted to C -forms with compact support in Ω;with adjoint δ . In order to stress the dimension of the underlying spaces, we write d q → q +1 (similarly for other operators).Assume that (cid:0) ˚d q → q +1 , ˚d q +1 → q +2 (cid:1) is exact and compact. Then the translation of ourcompactness theorem to the present setting reads as follows. Theorem 8.4.
Let ( a n ) n be a sequence in M ( α, β, (˚d q → q +1 , ˚d q +1 → q +2 )) .Then there is a convergent subsequence of ( a n ) n , which H -nonlocally converges with re-spect to (˚d q → q +1 , ˚d q +1 → q +2 ) .The nonlocal H -limit is unique. It might be worth explicitly writing down the definition of nonlocal H -convergence inthis particular setting.( a n ) n nonlocally H -converges to some a with respect to (˚d q → q +1 , ˚d q +1 → q +2 ), if and only iffor all f ∈ dom(˚d q → q +1 | ker(˚d q → q +1 ) ⊥ ) ∗ and g ∈ dom( δ q +2 → q +1 | ker( δ q +2 → q +1 ) ⊥ ) ∗ and corresponding u n and v n such that h a n ˚d u n , ˚d ϕ i = f ( ϕ ) and h a − n δv n , δϕ i = g ( ψ )for ϕ ∈ dom(˚d q → q +1 | ker(˚d q → q +1 ) ⊥ ) and ψ ∈ dom( δ q +2 → q +1 | ker( δ q +2 → q ) ⊥ ) implies u n ⇀ u and v n ⇀ v as well as a n d u n ⇀ a d u and a − n δv n ⇀ a − δv , where u and v satisfy h a d u, d ϕ i = f ( ϕ ) and h a − δv, δψ i = g ( ψ ) . Of course also the div-curl type characterisation in Theorem 6.2 carries over to the specialcase discussed here. 48
Concluding Remarks
The present article discussed a generalisation of the well-known concept of H -convergencefor nonlocal operator coefficients. This generalised concept, nonlocal H -convergence, yieldsanother characterisation of local H -convergence for local operators. In particular, this auto-matically implies convergence of a different partial differential equation (in an appropriatesense). With this observation in mind we realise that local H -convergence of multiplicationoperators readily implies a homogenisation result for Maxwell’s equations with nonperiodiccoefficients.Yet there is more, the rationale in this article provides a way of identifying the nonlocalcoefficients completely from certain solution operators of elliptic partial differential equations.Indeed, the above arguments imply that knowledge of the solution operator associated todiv a ˚grad u = f, and ˚curl a − curl v = g together with all the ‘fluxes’ a ˚grad u and a − curl v for sufficiently many f and g are needed to uniquely identify a . The above arguments alsoshow that in order to use less data than the mentioned ones, more information on a has tobe assumed a priori. Acknowledgements
The author is highly indebted to the anonymous referee for their thorough and critical review,which lead to a major improvement of the manuscript.
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InternationalJournal of Solids and Structures , 51(1):196 – 209, 2014.Marcus WaurickDepartment of Mathematics and Statistics, University of Strathclyde,Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH,ScotlandEmail: marcus.wauhugo@[email protected]@[email protected]