A Functional Analytic Perspective to the div-curl Lemma
aa r X i v : . [ m a t h . F A ] A p r A Functional Analytic Perspective to the div-curlLemma
Marcus Waurick
Abstract
We present an abstract functional analytic formulation of the celebrated div-curllemma found by F. Murat and L. Tartar. The viewpoint in this note relies on sequencesfor operators in Hilbert spaces. Hence, we draw the functional analytic relation of thediv-curl lemma to differential forms and other sequences such as the Grad grad-sequencediscovered recently by D. Pauly and W. Zulehner in connection with the biharmonicoperator.
Keywords: div-curl-lemma, compensated compactness, de Rham complexMSC 2010: 35A15 (35A23 46E35)
In the year 1978 a groundbreaking result in the theory of homogenisation has been found byFrancois Murat and Luc Tartar, the celebrated div-curl lemma ([10] or [18]):
Theorem 1.1.
Let Ω ⊆ R d open, ( u n ) n , ( v n ) n in L (Ω) d weakly convergent. Assume that (div u n ) n = ( d X j =1 ∂ j u n ) n , (curl u n ) n = (cid:0) ( ∂ j u ( k ) n − ∂ k u ( j ) n ) j,k (cid:1) n are relatively compact in H − (Ω) and H − (Ω) d × d , respectively.Then ( h u n , v n i C d ) n converges in D ′ (Ω) and we have lim n →∞ h u n , v n i C d = h lim n →∞ u n , lim n →∞ v n i C d . Ever since people were trying to generalise the latter theorem in several directions. Forthis we refer to [1], [9], [5], and [8] just to name a few. It has been observed that thelatter theorem has some relationship to the de Rham cohomology, see [18]. We shall alsorefer to [21], where the Helmholtz decomposition has been used for the proof of the div-curl lemma for the case of 3 space dimensions. We will meet the abstract counter part ofthe Helmholtz projection in our abstract approach to the div-curl lemma. In any case, the1equence property of the differential operators involved plays a crucial role in the derivationof the div-curl lemma. Note that, however, there are results that try to weaken this aspect,as well, see [4]. In this note, in operator theoretic terms, we shall further emphasise theintimate relation of the sequence property of operators from vector analysis and the div-curl lemma. In particular, we will provide a purely functional analytic proof of the div-curllemma. More precisely, we relate the so-called “global” form ([17]) of the div-curl lemma tofunctional analytic realisations of certain operators from vector analysis, that is, to compactsequences of operators in Hilbert spaces. Moreover, having provided this perspective, wewill also obtain new variants of the div-curl lemma, where we apply our abstract findings tothe Pauly–Zulehner Grad grad-sequence, see [11] and [15]. With these new results, we havepaved the way to obtain homogenisation results for the biharmonic operator with variablecoefficients, which, however, will be postponed to future research.The next section contains the functional analytic prerequisites and our main result itself– the operator-theoretic version of the div-curl lemma. The subsequent section is devotedto the proof of the div-curl lemma with the help of the results obtained in Section 2. In theconcluding section, we will apply the general result to several examples. div - curl Lemma
We start out with the definition of a (short) sequence of operators acting in Hilbert spaces.Note that in other sources sequences are also called “complexes”. We use the usual notationof domain, range, and kernel of a linear operator A , that is, dom( A ), ran( A ), and ker( A ).Occasionally, we will write dom( A ) to denote the domain of A endowed with the graph norm. Definition.
Let H j be Hilbert spaces, j ∈ { , , } . Let A : dom( A ) ⊆ H → H , and A : dom( A ) ⊆ H → H densely defined and closed. The pair ( A , A ) is called a (short)sequence , if ran( A ) ⊆ ker( A ). We say that the sequence ( A , A ) is closed , if both ran( A ) ⊆ H and ran( A ) ⊆ H are closed. The sequence ( A , A ) is called compact , if dom( A ) ∩ dom( A ∗ ) ֒ → H is compact.We recall some well-known results for sequences of operators in Hilbert spaces, we referto [11] and the references therein for the respective proofs. Theorem 2.1.
Let ( A , A ) be a sequence. Then the following statements hold:(a) ( A ∗ , A ∗ ) is a sequence;(b) ( A , A ) is closed if and only if ( A ∗ , A ∗ ) is closed.(c) ( A , A ) is compact if and only if ( A ∗ , A ∗ ) is compact;(d) if ( A , A ) is compact, then ( A , A ) is closed.(e) ( A , A ) is compact if and only if both dom( A ) ∩ ker( A ) ⊥ ֒ → ker( A ) ⊥ and dom( A ∗ ) ∩ ker( A ∗ ) ⊥ ֒ → ker( A ∗ ) ⊥ are compact and ker( A ∗ ) ∩ ker( A ) is finite-dimensional. Next, we need to introduce some notation.
Definition.
Let H , H be Hilbert spaces, A : dom( A ) ⊆ H → H . Then we define thecanonical embeddings 2a) ι ran( A ) : ran( A ) ֒ → H ;(b) ι ker( A ) : ker( A ) ֒ → H ;(c) π ran( A ) := ι ran( A ) ι ∗ ran( A ) ;(d) π ker( A ) := ι ker( A ) ι ∗ ker( A ) .If a densely defined closed linear operator has closed range, it is possible to continuouslyinvert this operator in an appropriate sense. For convenience of the reader and since theoperator to be defined in the next theorem plays an important role in the following, weprovide the results with the respective proofs. Note that the results are known, as well, seefor instance again [11]. Theorem 2.2.
Let H , H Hilbert spaces, A : dom( A ) ⊆ H → H densely defined andclosed. Assume that ran( A ) ⊆ H is closed. Then the following statements hold:(a) B := ι ∗ ran( A ) Aι ran( A ∗ ) is continuously invertible;(b) B ∗ = ι ∗ ran( A ∗ ) A ∗ ι ran( A ) ;(c) the operator c A ∗ : H → dom( B ) ∗ , ϕ ( v
7→ h ϕ, Av i H ) is continuous; and c B ∗ := c A ∗ | ran( A ) is an isomorphism that extends B ∗ .Proof. We prove (a). Note that by the closed range theorem, we have ran( A ∗ ) ⊆ H is closed.Moreover, since ker( A ) ⊥ = ran( A ∗ ), we have that B is injective and since ι ∗ ran( A ) projects ontoran( A ), we obtain that B is also onto. Next, as A is closed, we infer that B is closed. Thus, B is continuously invertible by the closed graph theorem.For the proof of (b), we observe that B ∗ is continuously invertible, as well. Moreover,it is easy to see that B ∗ = A ∗ on dom( A ∗ ) ∩ ker( A ∗ ) ⊥ , see also [19, Lemma 2.4]. Thus, theassertion follows.In order to prove (c), we note that c A ∗ is continuous. Next, it is easy to see that c B ∗ extends B ∗ . We show that c B ∗ is onto. For this, let ψ ∈ dom( B ) ∗ . Then there exists w ∈ dom( B )such that h w, v i H + h Bw, Bv i H = ψ ( v ) ( v ∈ dom( B )) . Define ϕ := ( B − ) ∗ w + Bw ∈ ran( A ). Then we compute for all v ∈ dom( B )( c B ∗ ϕ )( v ) = h ϕ, Bv i H = h ( B − ) ∗ w + Bw, Bv i H = h w, B − Bv i H + h Bw, Bv i H = ψ ( v ) . Hence, c B ∗ ϕ = ψ . We are left with showing that c B ∗ is injective. Let c B ∗ ϕ = 0. Then, for all v ∈ dom( B ) we have 0 = h ϕ, Bv i H . Hence, ϕ ∈ dom( B ∗ ) and B ∗ ϕ = 0. Thus, ϕ = 0, as B ∗ is one-to-one. Hence, c B ∗ is one-to-one. 3 emark 2.3. In the situation of the previous theorem, we remark here a small pecularityin statement (c): One could also define˜ A ∗ : H → dom( A ) ∗ , ϕ ( v
7→ h ϕ, Av i H )to obtain an extension of A ∗ . In the following, we will restrict our attention to the consid-eration of c A ∗ . The reason for this is the following fact:dom( A ) ∗ ⊇ ran( ˜ A ∗ ) ∼ = ran( c A ∗ ) ⊆ dom( B ) ∗ , where the identification is given by c A ∗ ϕ ( ˜ A ∗ ϕ ) | dom( B ) ( ϕ ∈ H ) . Indeed, let ϕ ∈ H . Thensup v ∈ dom( A ) , k v k dom( A ) | ( ˜ A ∗ ϕ )( v ) | = sup v ∈ dom( A ) , k v k dom( A ) |h ϕ, Av i H | = sup v ∈ dom( A ) ∩ ker( A ) ⊥ , k v k dom( A ) |h ϕ, Av i H | = sup v ∈ dom( B ) , k v k dom( B ) |h ϕ, Av i H | = sup v ∈ dom( B ) , k v k dom( B ) | ( c A ∗ ϕ )( v ) | . The latter remark justifies the formulation in the div-curl lemma, which we state next.
Theorem 2.4.
Let ( A , A ) be a closed sequence. Let ( u n ) n , ( v n ) n in H be weakly convergent.Assume ( c A ∗ u n ) n , ( c A v n ) n to be relatively compact in dom( A ) ∗ and dom( A ∗ ) ∗ , respectively. Further, assume that ker( A ∗ ) ∩ ker( A ) is finite dimensional.Then lim n →∞ h u n , v n i H = h lim n →∞ u n , lim n →∞ v n i H . We emphasise that in this abstract version of the div-curl lemma no compactness condi-tion on the operators A and A is needed.On the other hand, it is possible to formulate a statement of similar type without theusage of (abstract) distribution spaces. For this, however, we have to assume that ( A , A ) isa compact sequence. The author is indebted to Dirk Pauly for a discussion on this theorem.It is noteworthy that the proof for both Theorem 2.4 and 2.5 follows a commonly knownstandard strategy to prove the so-called ‘Maxwell compactness property’, see [20, 13, 2].4 heorem 2.5. Let ( A , A ) be a compact sequence. Let ( u n ) n , ( v n ) n be weakly convergentsequences in dom( A ∗ ) and dom( A ) , respectively.Then lim n →∞ h u n , v n i H = h lim n →∞ u n , lim n →∞ v n i H . In order to prove Theorem 2.4 and 2.5 we formulate a corollary of Theorem 2.2 first.
Corollary 2.6.
Let H , H be Hilbert spaces, A : dom( A ) ⊆ H → H densely defined andclosed. Assume that ran( A ) ⊆ H is closed. Let B be as in Theorem 2.1. For ( ϕ n ) n in H the following statements are equivalent:(i) ( c A ∗ ϕ n ) n is relatively compact in dom( B ) ∗ ;(ii) ( π ran( A ) ϕ n ) n is relatively compact in H .If ( ϕ n ) n weakly converges to ϕ in H , then either of the above conditions imply π ran( A ) ϕ n → π ran( A ) ϕ in H .Proof. From ran( A ) = ker( A ∗ ) ⊥ and ker( c A ∗ ) = ker( A ∗ ), we deduce that c A ∗ ϕ = c A ∗ π ran( A ) ϕ for all ϕ ∈ H . Next, c A ∗ π ran( A ) ϕ = c B ∗ ι ∗ ran( A ) ϕ for all ϕ ∈ H . Thus, as c B ∗ is an isomorphismby Theorem 2.2, we obtain that (i) is equivalent to ( ι ∗ ran( A ) ϕ n ) n being relatively compact inran( A ). The latter in turn is equivalent to (ii), since ( ι ∗ ran( A ) ϕ n ) n being relatively compact is(trivially) equivalent to the same property of ( ι ran( A ) ι ∗ ran( A ) ϕ n ) n = ( π ran( A ) ϕ n ) n .The last assertion follows from the fact that π ran( A ) is (weakly) continuous. Indeed, weakconvergence of ( ϕ n ) n to ϕ implies weak convergence of ( π ran( A ) ϕ n ) n to π ran( A ) ϕ . This togetherwith relative compactness implies π ran( A ) ϕ n → π ran( A ) ϕ with the help of a subsequence argu-ment. Corollary 2.7.
Let H , H be Hilbert spaces, A : dom( A ) ⊆ H → H densely defined andclosed. Assume dom( A ) ∩ ker( A ) ⊥ H ֒ → H compact. Let ( ϕ n ) n weakly converging to ϕ in dom( A ∗ ) . Then lim n →∞ π ran( A ) ϕ n = π ran( A ) ϕ in H .Proof. We note that – by a well-known contradiction argument – dom( A ) ∩ ker( A ) ⊥ H ֒ → H compact implies the Poincar´e type inequality ∃ c > ∀ ϕ ∈ dom( A ) ∩ ker( A ) ⊥ : k ϕ k H c k Aϕ k H . The latter together with the closedness of A implies the closedness of ran( A ) ⊆ H . Thus,Theorem 2.2 is applicable. Let B as in Theorem 2.2.We observe that the assertion is equivalent to lim n →∞ ι ∗ ran( A ) ϕ n = ι ∗ ran( A ) ϕ in ran( A ). Wecompute with the help Theorem 2.2 for n ∈ N ι ∗ ran( A ) ϕ n = ( B ∗ ) − B ∗ ι ∗ ran( A ) ϕ n = ( B ∗ ) − ι ∗ ran( A ∗ ) A ∗ ι ran( A ) ι ∗ ran( A ) ϕ n = ( B ∗ ) − ι ∗ ran( A ∗ ) A ∗ π ran( A ) ϕ n = ( B ∗ ) − ι ∗ ran( A ∗ ) A ∗ ϕ n .
5y hypothesis, A ∗ ϕ n ⇀ A ∗ ϕ in H and so ι ∗ ran( A ∗ ) A ∗ ϕ n ⇀ ι ∗ ran( A ∗ ) A ∗ ϕ in ran( A ∗ ) as n → ∞ since ι ∗ ran( A ∗ ) is (weakly) continuous. Next B − is compact by assumption and thus so is( B ∗ ) − . Therefore ( B ∗ ) − ι ∗ ran( A ∗ ) A ∗ ϕ n → ( B ∗ ) − ι ∗ ran( A ∗ ) A ∗ ϕ in ι ran( A ) . The assertion followsfrom ( B ∗ ) − ι ∗ ran( A ∗ ) A ∗ ϕ = ι ∗ ran( A ) ϕ . Proof of Theorem 2.4 and Theorem 2.5.
By the sequence property, we deduce that π ran( A ) π ker( A ) and π ran( A ∗ ) π ker( A ∗ ) . By Corollary 2.6 (Theorem 2.4) or Corollary 2.7 (Theo-rem 2.5), we deduce that π ran( A ) u n → π ran( A ) u and π ran( A ∗ ) v n → π ran( A ∗ ) v in H . Fromker( A ) ∩ ker( A ∗ ) being finite-dimensional (cf. Theorem 2.1), we obtain π ker( A ) ∩ ker( A ∗ ) u n → π ker( A ) ∩ ker( A ∗ ) u as π ker( A ) ∩ ker( A ∗ ) is compact. Thus, we obtain for n ∈ N h u n , v n i H = h ( π ran( A ) + π ker( A ∗ ) ∩ ker( A ) + π ker( A ∗ ) ∩ ran( A ∗ ) ) u n , ( π ran( A ∗ ) + π ker( A ) ) v n i H = h u n , π ran( A ∗ ) v n i H + h ( π ran( A ) + π ker( A ∗ ) ∩ ker( A ) + π ker( A ∗ ) ∩ ran( A ∗ ) ) u n , π ker( A ) v n i H = h u n , π ran( A ∗ ) v n i H + h π ran( A ) u n , π ker( A ) v n i H + h π ker( A ∗ ) ∩ ker( A ) u n , π ker( A ) v n i H → h lim n →∞ u n , lim n →∞ v n i H . A closer look at the proof of our main result reveals the following converse of Theorem2.4:
Theorem 2.8.
Let ( A , A ) be a closed sequence. Assume that for all weakly convergentsequences ( u n ) n , ( v n ) n in dom( A ∗ ) and dom( A ) , respectively, we obtain lim n →∞ h u n , v n i H = h lim n →∞ u n , lim n →∞ v n i H . Then ker( A ∗ ) ∩ ker( A ) is finite-dimensional. For the proof of the latter, we need the next proposition:
Proposition 2.9.
Let H be a Hilbert space. Then the following statements are equivalent:(a) H is infinite-dimensional;(b) there exists ( u n ) n weakly convergent to such that c := lim n →∞ h u n , u n i exists with c = 0 .Proof. Let H be infinite-dimensional. Without loss of generality, we may assume that H = L (0 , π ). Then u n := sin( n · ) → n → ∞ and Z π (sin( nx )) dx → π Z π (sin( x )) dx > . If H is finite-dimensional, then weak convergence and strong convergence coincide, and thedesired sequence cannot exist. 6 roof of Theorem 2.8. Suppose that ker( A ∗ ) ∩ ker( A ) is infinite-dimensional. Choose ( u n ) n in ker( A ∗ ) ∩ ker( A ) as in Proposition 2.9. Then, clearly, ( u n ) n is weakly convergent indom( A ∗ ) and dom( A ). Hence,0 = h lim n →∞ u n , lim n →∞ u n i H = lim n →∞ h u n , u n i H = c = 0 . We will need the next abstract results for the proof of the div-curl lemma in the nextsection. Note that this is only needed for the formulation of the div-curl lemma where thedivergence and the curl operators are considered to map into H − . For this, we need somenotation. Let A ∈ L ( H , H ). The dual operator A ′ ∈ L ( H ∗ , H ∗ ) is given by( A ′ ϕ )( ψ ) := ϕ ( Aψ ) . We also define A ⋄ : H → H ∗ via A ⋄ := A ′ R H , where R H : H → H ∗ denotes the Rieszisomorphism. Proposition 2.10.
Let H , H , D Hilbert spaces, A : dom( A ) ⊆ H → H densely definedand closed. Assume D ֒ → dom( A ) continuously and ran( A | D ) = ran( A ) ⊆ H closed. Define A : D → H , ϕ Aϕ . Then c A ∗ = A ⋄ , that is, for every v ∈ H we have A ⋄ v can be uniquelyextended to an element of dom( A ) ∗ , the extension is given by c A ∗ v , where c A ∗ is given inTheorem 2.2.Proof. Let v ∈ H . Then for all ϕ ∈ D we have (cid:16)c A ∗ v (cid:17) ( ϕ ) = h v, Aϕ i H = h v, A ϕ i H = R H v ( A ϕ ) = ( A ′ R H v )( ϕ ) = ( A ⋄ v )( ϕ ) . Since A is continuous, it is densely defined and closed, hence B := ι ∗ ran( A ) A ι ran( A ∗ ) is aHilbert space isomorphism from D ∩ ker( A ) ⊥ D to ran( A ) = ran( A ), by Theorem 2.2. Notethat AB − = id ran( A ) = id ran( A ) . For ψ ∈ dom( A ) and v ∈ H , we define( A ⋄ v ) e ( ψ ) := ( A ⋄ v ) ( B − Aψ ) . Next, if ψ ∈ dom( A ), then with the above computations, we obtain( A ⋄ v ) e ( ψ ) = ( A ⋄ v ) ( B − Aψ ) = h v, AB − Aψ i H = h v, Aψ i H = (cid:16)c A ∗ v (cid:17) ( ψ ) . Thus, ( A ⋄ v ) e indeed extends A ⋄ v and coincides with c A ∗ v . We infer also the continuityproperty for A ⋄ v . The uniqueness property follows from ran( A ) = ran( A ).From Proposition 2.10 it follows that ran( c A ∗ ) = ran( A ⋄ ). This is the actual fact used inthe following. Lemma 2.11 ([11, Lemma 2.14]) . Let H , H , H Hilbert spaces, A ∈ L ( H , H ) onto. Then ran( A ⋄ ) ⊆ H ∗ is closed and ( A ⋄ ) − ∈ L (ran( A ⋄ ) , H ) . roof. By the Riesz representation theorem A ⋄ and A ′ are unitarily equivalent. Thus, itsuffices to prove the assertions for A ′ instead of A ⋄ . By the closed range theorem, ran( A ′ )is closed, since ran( A ) = H is. Next, A is onto, hence A ′ ∈ L ( H ∗ , H ∗ ) is one-to-one, and,thus, by the closed graph theorem, we obtain that ( A ′ ) − maps continuously from ran( A ′ )into H ∗ . Corollary 2.12.
Let H , H be Hilbert spaces, A : dom( A ) ⊆ H → H densely defined andclosed, C : dom( C ) ⊆ H → H densely defined, closed. Assume that ran( A ) ⊆ H is closed, dom( C ) ֒ → dom( A ) continuous.If ran( A ) = { Aϕ ; ϕ ∈ dom( C ) } , (1) then ran( c A ∗ ) = dom( B ) ∗ ⊆ dom( C ) ∗ is closed, where B is given in Theorem 2.2.Proof. Since dom( C ) ֒ → dom( A ) continuously, we obtain that A : dom( C ) → ran( A ) = ran( B ) , ϕ Aϕ is continuous. Moreover, by (1), we infer that A is onto. Hence, by Lemma 2.11, we obtainthat ran( A ⋄ ) ⊆ dom( C ) ∗ is closed. Thus, we are left with showing that ran( A ⋄ ) = dom( B ) ∗ .By Proposition 2.10, we realise that ran( A ⋄ ) = ran( c A ∗ ) = ran( c B ∗ ). By Theorem 2.2, we getthat c B ∗ maps onto dom( B ) ∗ . Remark 2.13.
Corollary 2.12 particularly applies to A = C . div - curl lemma Before we formulate Theorem 3.2, the classical div-curl lemma, we need to introduce somedifferential operators from vector calculus.
Definition.
Let Ω ⊆ R d open. We definegrad c : C ∞ c (Ω) ⊆ L (Ω) → L (Ω) d , ϕ ( ∂ j ϕ j ) j ∈{ ,...,d } div c : C ∞ c (Ω) ⊆ L (Ω) d → L (Ω) , ( ϕ j ) j ∈{ ,...,d } d X j =1 ∂ j ϕ j Grad c : C ∞ c (Ω) d ⊆ L (Ω) d → L (Ω) d × d , ( ϕ j ) j ∈{ ,...,d } ( ∂ k ϕ j ) j,k ∈{ ,...,d } Div c : C ∞ c (Ω) d × d ⊆ L (Ω) d × d → L (Ω) d , ( ϕ j,k ) j,k ∈{ ,...,d } ( d X k =1 ∂ k ϕ j,k ) j ∈{ ,...,d } Curl c : C ∞ c (Ω) d ⊆ L (Ω) d → L (Ω) d × d , ( ϕ j ) j ∈{ ,...,d } ( ∂ k ϕ j − ∂ j ϕ k ) j,k ∈{ ,...,d } = Grad ϕ − (Grad ϕ ) T . Moreover, we set ˚grad := grad c and, similarly, ˚div , ˚Div , ˚Curl , ˚Grad. Furthermore, we putdiv := − ˚grad ∗ , Div := − ˚Grad ∗ , grad := − ˚div ∗ , Grad := − ˚Div ∗ and Curl := (2 ˚Div skew) ∗ ,where skew A := ( A − A T ) denotes the skew symmetric part of a matrix A .8 emark 3.1. It is an elementary computation to establish that the operators just introducedwith ˚ are restrictions of the ones without.As usual, we define, H − (Ω) := dom( ˚grad) ∗ . We may now formulate the classical div-curllemma. We slightly rephrase the lemma, though. Theorem 3.2 (div-curl lemma – global version) . Let ( u n ) n , ( v n ) n in L ( B (0 , d weaklyconvergent, with [ n ∈ N (spt u n ∪ spt v n ) ⊆ B (0 , δ ) = { x ∈ R d ; k x k δ } for some δ < . Assume (div u n ) n , (Curl u n ) n are relatively compact in H − ( B (0 , and H − ( B (0 , d × d , resp.Then lim n →∞ h u n , v n i L = h lim n →∞ u n , lim n →∞ v n i L . We recall here that in [17], Theorem 3.2 is called “global div-curl lemma”. We providethe connection to the classical, the “local” version of it, in the following remark.
Remark 3.3 (div-curl lemma – local version) . We observe that the assertions in Theorem1.1 and in Theorem 3.2 are equivalent. For this, observe that Theorem 1.1 implies Theorem3.2. Indeed, for Ω = B (0 , ϕ ∈ C ∞ c ( B (0 , ϕ = 1 on the compact set S n ∈ N (spt u n ∪ spt v n ).Then, by Theorem 1.1 and putting u := lim n →∞ u n and v := lim n →∞ v n , we obtain h u n , v n i L = Z Ω ϕ h u n , v n i → Z Ω ϕ h u, v i = h u, v i . On the other hand, let the assumptions of Theorem 1.1 be satisfied. With the help ofTheorem 3.2, we have to prove that for all ϕ ∈ C ∞ c (Ω) we get Z Ω ϕ h u n , v n i → Z Ω ϕ h u, v i . (2)To do so, we let ψ ∈ C ∞ c (Ω) be such that ψ = 1 on spt ϕ . Then there exists R > ψ ⊆ B (0 , R ). By rescaling the arguments, the statement in (2) follows from Theorem3.2, once we proved that(div( ψu n )) n = ( ψ div( u n ) + grad( ψ ) u n ) n , (Curl( ψv n )) n = (2 skew((grad ψ ) v Tn ) + ψ Curl v n ) n is relatively compact in H − ( B (0 , R + 1)) and H − ( B (0 , R + 1)) d × d . This, however, followsfrom the hypothesis and the compactness of the embedding L ( B (0 , ֒ → H − ( B (0 , H = L ( B (0 , ,H = L ( B (0 , d ,A := ˚grad ,A := ˚Curl . ( ∗ ) Proposition 3.4.
With the setting in ( ∗ ) , ( A , A ) is a sequence.Proof. By Schwarz’s lemma, it follows for all ϕ ∈ C ∞ c ( B (0 , ϕ = ˚Curl( ∂ j ϕ ) j ∈{ ,...,d } = ( ∂ k ∂ j ϕ − ∂ j ∂ k ϕ ) j,k ∈{ ,...,d } = 0 . Thus, ˚Curl ˚grad ⊆ Theorem 3.5.
With the setting in ( ∗ ) , ( A , A ) is compact. For the proof of Theorem 3.5, we could use compactness embedding theorems such asWeck’s selection theorem ([20]) or Picard’s selection theorem ([13]). However, due to thesimple geometric setting discussed here, it suffices to walk along the classical path of showingcompactness by proving Gaffney’s inequality and then using Rellich’s selection theorem. Weemphasise, however, that meanwhile there have been developed sophisticated tools detouringGaffney’s inequality, to obtain compactness results for very irregular Ω, which do not satisfyGaffney’s inequality. For convenience of the reader, we shall provide a proof of Theorem 3.5using the following regularity result for the Laplace operator, see [7, Teorema 10 and 14] orsince we use the respective result for a d -dimensional ball, only, see [6, Inequality (3,1,1,2)].For this, we denote the Dirichlet Laplace operator by ∆ := div ˚grad. Theorem 3.6.
Let Ω ⊆ R d open, bounded and convex. Then for all u ∈ dom(∆) , we have u ∈ dom(Grad ˚grad) and k Grad ˚grad u k L (Ω) d × d k ∆ u k L (Ω) . Based on the latter estimate, we shall prove Friedrich’s inequality. For the proof of which,we will follow the exposition of [16]. Since the exposition in [16] is restricted to 2 or 3 spatialdimensions, only, we provide a proof for the “multi- d ”-case in the following. Theorem 3.7 ([16, Theorem 2.2]) . Let Ω ⊆ R d open, bounded, convex. Then dom( ˚Curl) ∩ dom(div) ֒ → dom(Grad) . Moreover, we have k Grad u k L (Ω) d k ˚Curl u k L (Ω) d × d + k div u k L (Ω) for all u ∈ dom( ˚Curl) ∩ dom(div) . emma 3.8 ([16, Lemma 2.1]) . Let Ω ⊆ R d open, bounded. Denote V := { ϕ ; ∃ ψ ∈ C ∞ c (Ω) d : ϕ = ψ + ˚grad( − ∆ + 1) − div ψ } . Then V is dense in dom( ˚Curl) ∩ dom(div) .Proof. First of all note that V ⊆ X := dom( ˚Curl) ∩ dom(div). Indeed, for ϕ = ψ + ˚grad( − ∆ +1) − div ψ for some ψ ∈ C ∞ c (Ω), we get ˚Curl ϕ = ˚Curl ψ ∈ L (Ω) d × d , by Proposition 3.4.Moreover, div ϕ = ( − ∆ + 1) − div ψ ∈ L (Ω). Thus, V ⊆ X . Next, we show the densityproperty. For this, we endow X with the scalar product h u, v i X := h ˚Curl u, ˚Curl v i + h div u, div v i + h u, v i . Let u ∈ V ⊥ X ⊆ X . We need to show that u = 0. For all ψ ∈ C ∞ c (Ω) and w := ( − ∆+1) − div ψ we have0 = h u, ψ + ˚grad w i X = h ˚Curl u, ˚Curl ψ i + h div u, div ψ i + h div u, div ˚grad w i + h u, ψ i + h u, ˚grad w i = h ˚Curl u, ˚Curl ψ i + h div u, div ψ i + h div u, ∆ w i + h u, ψ i − h div u, w i = h ˚Curl u, ˚Curl ψ i + h u, ψ i . Thus, ( ˚Curl ∗ ˚Curl +1) u = 0, which yields u = 0.Before we come to the proof of Theorem 3.7, we mention an elementary formula to beused in the forthcoming proof: For all ψ ∈ C ∞ c (Ω) d we have − ∆ I d × d ψ = − Div Grad ψ = − Div Curl ψ − grad div ψ. Proof of Theorem 3.7.
By Lemma 3.8 it suffices to show the inequality for u ∈ V . For this,let ψ ∈ C ∞ c (Ω) d and put u := ψ + ˚grad w with w := ( − ∆ + 1) − div ψ . We compute k Grad u k = k Grad( ψ + ˚grad w ) k = h Grad ψ, Grad ψ i + 2 Re h Grad ψ, Grad ˚grad w i + k Grad ˚grad w k . We aim to discuss every term in the latter expression separately. We have h Grad ψ, Grad ψ i = −h Div Grad ψ, ψ i = −h Div Curl ψ, ψ i − h grad div ψ, ψ i = −h Div skew Curl ψ, ψ i + h div ψ, div ψ i = 12 h Curl ψ, Curl ψ i + h div ψ, div ψ i . h Grad ψ, Grad ˚grad w i = −h Div Grad ψ, ˚grad w i = −h Div Curl ψ, ˚grad w i − h grad div ψ, ˚grad w i = h div Div Curl ψ, w i − h grad div ψ, ˚grad w i = −h grad div ψ, ˚grad w i . By Theorem 3.6, we estimate k Grad ˚grad w k k ∆ w k = k w − div ψ k = k w k − h w, div ψ i + k div ψ k . Note that since div ψ ∈ C ∞ c (Ω), we obtain from w = ( − ∆ + 1) − div ψ that h ˚grad w, ˚grad div ψ i + h w, div ψ i = h div ψ, div ψ i . Thus, all together, k Grad u k h Curl ψ, Curl ψ i + h div ψ, div ψ i − h grad div ψ, ˚grad w i + k w k − h w, div ψ i + k div ψ k = 12 h Curl ψ, Curl ψ i + h div ψ, div ψ i + 2 Re h w, div ψ i − h div ψ, div ψ i + k w k − h w, div ψ i + k div ψ k = 12 h Curl ψ, Curl ψ i + k w k = 12 k Curl u k + k div u k . Proof of Theorem 3.5.
By Theorem 3.7 as B (0 ,
1) is convex, we obtain thatdom( A ) ∩ dom( A ∗ ) = dom( ˚Curl) ∩ dom(div) ֒ → dom(Grad) . On the other hand dom(Grad) ֒ → L ( B (0 , d is compact by Rellich’s selection theorem.This yields the assertion. Lemma 3.9.
Assume the setting in ( ∗ ) . Then ker(div) ∩ ker( ˚Curl) = { } .Proof. The assertion follows from the connectedness of B (0 , Proposition 3.10.
Assume the setting in ( ∗ ) . Then ran( d ˚Curl) ⊆ H − (Ω) d × d is closed. roof. In this proof, we need to consider the differential operators on various domains. Toclarify this in the notation, we attach the underlying domain as an index to the differentialoperators in question, that is, grad = grad Ω and when the domains are considered we writedom(grad) = dom(grad , Ω) and similarly for ran and ker. We apply Corollary 2.12 to A =˚Curl B (0 , , C = ˚Grad B (0 , . Note that ran( A ) is closed by Theorem 3.5 and Theorem 2.1.Thus, we are left with showing thatran( ˚Curl , B (0 , { ˚Curl B (0 , ϕ ; ϕ ∈ dom( ˚Grad , B (0 , } . From Proposition 3.4 and by Theorem 3.7, we inferran( ˚Curl B (0 , ) = { ˚Curl B (0 , ϕ ; ϕ ∈ ker(div , B (0 , ∩ dom( ˚Curl , B (0 , } = { ˚Curl B (0 , ϕ ; ϕ ∈ dom(Grad , B (0 , ∩ dom( ˚Curl , B (0 , } . So, let ψ = Curl B (0 , ϕ for some ϕ ∈ dom( ˚Curl , B (0 , ∩ dom(Grad , B (0 , ϕ and ψ by zero to B (0 , ϕ e and ψ e . Note that ϕ e ∈ dom( ˚Curl , B (0 , B (0 , ϕ e = ψ e . By the above applied to Ω = B (0 , ϕ r ∈ dom( ˚Curl , B (0 , ∩ dom(Grad , B (0 , B (0 , ϕ r = ˚Curl B (0 , ϕ e = ψ e . Thus, ϕ r − ϕ e ∈ ker( ˚Curl , B (0 , , B (0 , , by Lemma 3.9. Thus, we find u ∈ dom( ˚grad , B (0 , B (0 , u = ϕ r − ϕ e . On B (0 , \ B (0 ,
1) we have 0 = ϕ e = ϕ r − grad B (0 , \ B (0 , u. Therefore, grad B (0 , \ B (0 , u = ϕ r on B (0 , \ B (0 , u ∈ dom(Grad grad , B (0 , \ B (0 , H ( B (0 , \ B (0 , . By Calderon’s extension theorem, there exists u e ∈ dom(Grad grad , B (0 , H ( B (0 , u e = u on B (0 , \ B (0 , . Next, we observe that ϕ r , := ϕ r − grad B (0 , u e ∈ dom(Grad , B (0 , u − u e ∈ dom(grad , B (0 , ϕ r = ϕ r , − grad B (0 , ( u − u e ) . Moreover, on B (0 , \ B (0 , ϕ r , = 0 as well as u − u e = 0. Thus, ϕ r , ∈ dom( ˚Grad , B (0 , u − u e ∈ dom( ˚grad , B (0 , ψ = Curl B (0 , ϕ = Curl B (0 , ϕ r = Curl B (0 , ( ϕ r , − ˚grad B (0 , ( u − u e )) = ˚Curl B (0 , ϕ r , . Therefore,ran( ˚Curl , B (0 , { ˚Curl B (0 , ϕ ; ϕ ∈ dom( ˚Grad , B (0 , ∩ dom( ˚Curl , B (0 , } = { ˚Curl B (0 , ϕ ; ϕ ∈ dom( ˚Grad , B (0 , } . emma 3.11. Let Ω ⊆ R d open, bounded, ϕ ∈ L (Ω) d with spt ϕ ⊆ Ω . Then dom( ˚Div skew) ∗ ∋ Curl ϕ = ˚Curl ϕ ∈ dom(Div skew) ∗ Proof.
We have dom(Div skew) ∗ ֒ → dom(( ˚Div skew) ∗ . Let η ∈ C ∞ c (Ω) with the property η = 1 on spt ϕ . Then for all ψ ∈ dom(Div skew) we have ηψ ∈ dom( ˚Div skew) and so, h ˚Curl ϕ, ψ i = h ϕ, ψ i = h ϕ, ηψ i = h ϕ, ηψ i = h Curl ϕ, ηψ i . Thus, there is κ > ψ ∈ dom(Div skew) | ( ˚Curl ϕ )( ψ ) | = | (Curl( ϕ )( ψ )) | = | (Curl( ϕ )( ηψ )) | κ k ψ k dom(Div skew) . This yields the assertion.Finally, we can prove the div-curl lemma with operator-theoretic methods. We shall alsoformulate a simpler version of the div-curl lemma, which needs less technical preparations.In fact, the simpler version only uses Theorem 2.5 and Theorem 3.5.
Proof of Theorem 3.2.
We apply Theorem 2.4 with the setting in ( ∗ ). For this, by Lemma3.11, we note that Curl v n = ˚Curl v n = d ˚Curl v n . With Theorem 2.4 at hand, we need toestablish that ( d ˚Curl v n ) n is relatively compact in dom( ˚Curl ∗ ) ∗ . By Corollary 2.12 appliedto C = A = ˚Curl ∗ , the latter is the same as showing that ( d ˚Curl v n ) n is relatively compactin ran( d ˚Curl). On the other hand, by Proposition 3.10, ran( d ˚Curl) is closed in H − (Ω) d × d .Thus, since ( d ˚Curl v n ) n is relatively compact in H − (Ω) d × d , we get that ( d ˚Curl v n ) n is relativelycompact in dom( ˚Curl ∗ ) ∗ . This yields the assertion.Theorem 2.5 with the setting in ( ∗ ) reads as follows. Note that the assertion follows fromTheorem 3.5. Theorem 3.12.
Let ( u n ) n in dom(div) and ( v n ) n in dom( ˚Curl) be weakly convergent se-quences. Then lim n →∞ h u n , v n i L (Ω) d = h lim n →∞ u n , lim n →∞ v n i L (Ω) d . It is well-known that the sequence property and the compactness of the sequence istrue also for submanifolds of R d and the covariant derivative on tensor fields of appropriatedimension and its adjoint. We conclude this exposition with a less known sequence. ThePauly–Zulehner Grad grad-complex, see [11].14 n Example – the Pauly–Zulehner- Grad grad -complex
In the whole section, we let Ω ⊆ R to be a bounded Lipschitz domain. We will denote bycurl the usual 3-dimensional curl operator that maps vector fields to vector fields. Somedefinitions are in order Definition.
We define ◦ grad r grad : ˚ H (Ω) ⊆ L (Ω) → L (Ω) , ϕ grad r grad ϕ. ˚curl r , sym : dom( ˚curl r ) ∩ L (Ω) ⊆ L (Ω) → L (Ω) , ϕ ˚curl r ϕ ˚div r , dev : dom( ˚div r ) ∩ L (Ω) ⊆ L (Ω) → L (Ω) d , ϕ Div ϕ div div r , sym : dom(div Div sym ) ⊆ L (Ω) → L (Ω) , ϕ div Div ϕ, sym curl r , dev : dom(curl r ) ∩ L (Ω) ⊆ L (Ω) → L (Ω) , ϕ sym curl r ϕ, dev grad r : H (Ω) ⊆ L (Ω) → L (Ω) , ϕ dev grad r ϕ. The subscript r refers to row-wise application of the vector-analytic operators, where it isattached. Moreover, as before, we have attached a “˚” above the differential operators inquestion, if we consider the completion of smooth tensor fields with compact support withappropriate norm. The operators dev and sym are the projections on the deviatoric and symmetric parts of 3 × A ∈ C × , we putdev A := A −
13 tr( A ) I × , sym A = 12 ( A + A T ) . Moreover, we define L (Ω) := dev [ L (Ω) × ] as well as L (Ω) := sym [ L (Ω) × ].Next, we gather some of the main results of Pauly–Zulehner: Theorem 3.13 ([11, Lemma 3.5, Remark 3.8, and Lemma 3.21]) . The pairs ◦ grad r grad , ˚curl r , sym ! , (cid:16) ˚curl r , sym , ˚div r , dev (cid:17) , ( − dev grad r , sym curl r , dev ) , (cid:18) sym curl r , dev , div div r , sym (cid:19) are compact sequences. Moreover, we have ◦ grad r grad ∗ = div div r , sym , ˚curl ∗ r , sym = sym curl r , dev , ˚div ∗ r , dev = − dev grad r . We have now several theorems being consequences of our general observation in Theorem2.4. We will formulate the versions for Theorem 2.4 only. The analogues to Theorem 2.5 arestraightforwardly written down, which we will omit here.15 heorem 3.14. (a) Let ( u n ) n , ( v n ) n be weakly convergent sequences in L (Ω) . Assumethat (div div r , sym u n ) n , ( ˚curl r , sym v n ) n are relatively compact in dom( ◦ grad r grad) ∗ and dom(sym curl r ) ∗ . Then lim n →∞ h u n , v n i = h lim n →∞ u n , lim n →∞ v n i . (b) Let ( u n ) n , ( v n ) n be weakly convergent sequences in L (Ω) . Assume that (sym curl r , dev u n ) n , ( ˚div r , dev v n ) n are relatively compact in dom( ˚curl r , sym ) ∗ and dom(dev grad r ) ∗ . Then lim n →∞ h u n , v n i = h lim n →∞ u n , lim n →∞ v n i . Acknowledgements
This work was carried out with financial support of the EPSRC grant EP/L018802/2: Math-ematical foundations of metamaterials: homogenisation, dissipation and operator theory. Agreat deal of this research has been obtained during a research visit of the author at theRICAM for the special semester 2016 on Computational Methods in Science and Engineer-ing organised by Ulrich Langer and Dirk Pauly, et al. The wonderful atmosphere and thehospitality extended to the author are gratefully acknowledged.
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