SCHRIFTENREIHE DER FAKULTAT FUR MATHEMATIK
A Global div-curl-Lemmafor Mixed Boundary Conditions in Weak Lipschitz Domains
and a Corresponding Generalized A*0-A1-Lemma in Hilbert Spaces
by
Dirk Pauly
SM-UDE-812 2017
Received: July 17, 2017
A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domainsand a Corresponding Generalized A*
0-A1-Lemma in Hilbert Spaces
DIRK PAULY
Abstract. We prove global and local versions of the so called div-curl-lemma, also known as compen-
sated compactness, for mixed boundary conditions as well as bounded weak Lipschitz domains in 3D andweak Lipschitz interfaces. We will generalize our results using an abstract Hilbert space setting, which
shows corresponding results to hold in arbitrary dimensions as well as for various differential operators.
The crucial tools are Hilbert complexes and related compact embeddings.
Contents
1. Introduction 12. Definitions and Preliminaries 23. The div-rot-Lemma 44. Generalizations 54.1. Functional Analysis Toolbox 54.2. The A*
0-A1-Lemma 75. Applications 85.1. The div-rot-Lemma Revisited 85.2. Generalized Electro-Magnetics 95.3. Biharmonic Equation and General Relativity, Gravitational Waves 105.4. Linear Elasticity 11References 12
1. Introduction
We will prove a global version of the so called div-curl-lemma used for compensated compactness,stating that under certain (mixed tangential and normal) boundary conditions and (very weak) regularityassumptions on a domain Ω ⊂ R3 the following holds:
Let Ω ⊂ R3 be a bounded weak Lipschitz domain with boundary Γ and boundary parts Γt and Γn. Let
(En) and (Hn) be two sequences bounded in L2(Ω), such that (curlEn) and (divHn) are also bounded in
L2(Ω) and ν×En = 0 on Γt and ν ·Hn = 0 on Γn. Then there exist subsequences, again denoted by (En)and (Hn), such that (En), (curlEn) and (Hn), (divHn) converge weakly to E, curlE resp. H, divH in
L2(Ω) and the inner products converge as well, i.e.,∫Ω
En ·Hn →∫
Ω
E ·H.
A local version (distributional like convergence for arbitrary domains and no boundary conditionsneeded) of this div-curl-lemma is then immediately obtained.
Let Ω ⊂ R3 be an open set. Let (En) and (Hn) be two sequences bounded in L2(Ω), such that (curlEn)
and (divHn) are also bounded in L2(Ω). Then there exist subsequences, again denoted by (En) and (Hn),
Date: July 17, 2017.
1991 Mathematics Subject Classification. 35B27, 35Q61, 47B07, 46B50.Key words and phrases. div-curl-lemma, compensated compactness, mixed boundary conditions, weak Lipschitz do-
mains, Maxwell’s equations.
1
2 DIRK PAULY
such that (En), (curlEn) and (Hn), (divHn) converge weakly to E, curlE resp. H, divH in L2(Ω) and
the inner products converge in the distributional sense as well, i.e., for all ϕ ∈ C∞(Ω) it holds∫Ω
(En ·Hn)ϕ→∫
Ω
(E ·H)ϕ.
For details, see Theorem 3.1 and Corollary 3.2. We will also show a generalization to a natural Hilbertspace setting in Theorem 4.7. In Section 5 we apply this result to some more differential operators in3D and ND, appearing, e.g., in generalized electro-magnetics, for the biharmonic equation, in generalrelativity, for gravitational waves, and in the theory of linear elasticity and plasticity.
The div-curl-lemma, or compensated compactness, see the original papers by Murat [13] and Tartar[23] or [7, 22], and its variants and extensions have plenty of important applications. For an extensivediscussion and a historical overview of the div-curl-lemma see [24]. More recent discussions can befound, e.g., in [5, 25] and in the nice preprint [26]. The div-curl-lemma is widely used in the theoryof homogenization of (nonlinear) partial differential equations, see, e.g., [22]. Moreover, it is crucialin establishing compactness and regularity results for nonlinear partial differential equations such asharmonic maps, see, e.g., [9, 8, 19]. Numerical applications can be found, e.g., in [2]. It is furthera crucial tool in the homogenization of stochastic partial differential equations, especially with certainrandom coefficients, see, e.g., the survey [1] and the literature cited therein, e.g., [10].
Let us also mention that the div-curl-lemma is particularly useful to treat homogenization of problemsarising in plasticity, see, e.g., a recent preprint on this topic [21], for which the preprint [20] provides
the important key div-curl-lemma. As in [20, 21] H1(Ω)-potentials are used, these contributions are
restricted to smooth, e.g., C2 or convex, domains and to full boundary conditions. On the other hand,using Weck’s selection theorem (2.1) it is easily possible to extend these results even to bounded weakLipschitz domains of arbitrary topology and to the case of mixed boundary conditions.
Generally, for problems related to Maxwell’s equations the detour over H1(Ω) instead of using Weck’sselection theorem seems to be the wrong way to deal with such equations. Most of the arguments simplyfail, and if not, the results are usually limited to smooth domains and trivial topologies. Mixed boundaryconditions cannot be treated properly. Since the early 1970’s, see the original paper by Weck [28] for
Weck’s selection theorem, it is well known, that the H1(Ω)-detour is often not helpful and does not leadto satisfying results.
2. Definitions and Preliminaries
Let Ω ⊂ R3 be a bounded weak Lipschitz domain, see [3, Definition 2.3] for details, with boundaryΓ := ∂ Ω, which is divided into two relatively open weak Lipschitz subsets Γt and Γn := Γ \ Γt (itscomplement), see [3, Definition 2.5] for details. Note that strong Lipschitz (graph of Lipschitz functions)implies weak Lipschitz (Lipschitz manifolds) for the boundary as well as the interface. Throughout thispaper we shall assume the latter regularity on Ω and Γt.
Recently, in [3], Weck’s selection theorem, also known as the Maxwell compactness property, has beenshown to hold for such bounded weak Lipschitz domains and mixed boundary conditions. More precisely,the embedding
RΓt(Ω) ∩ DΓn(Ω) →→ L2(Ω)(2.1)
is compact, see [3, Theorem 4.7]. A short historical overview of Weck’s selection theorem is given inthe introduction of [3], see also the original paper [28] and [18, 27, 6, 29, 11, 12] for simpler proofs andgeneralizations.
Here the usual Lebesgue and Sobolev spaces are denoted by L2(Ω) and H1(Ω) as well as
R(Ω) :=E ∈ L2(Ω) : rotE ∈ L2(Ω)
, D(Ω) :=
E ∈ L2(Ω) : divE ∈ L2(Ω)
,
where we prefer to write rot instead of curl. R(Ω) and D(Ω) are also written as H(rot,Ω) or H(curl,Ω)resp. H(div,Ω) in the literature. With the help of test functions and test vector fields
C∞Γt(Ω) :=
ϕ|Ω : ϕ ∈ C∞(R3), dist(suppϕ,Γt) > 0
A Global div-curl-Lemma 3
we define the closed subspaces
H1Γt
(Ω) := C∞Γt(Ω)
H1(Ω)
, RΓt(Ω) := C∞Γt(Ω)
R(Ω)
, DΓn(Ω) := C∞Γn(Ω)
D(Ω)
(2.2)
as closures of test functions respectively vector fields. In (2.2) homogeneous scalar, tangential and normaltraces on Γt resp. Γn are generalized, respectively. To avoid case studies when using the Poincare estimate,we also define
H1∅(Ω) := H1(Ω) ∩ R⊥L2(Ω) =
u ∈ H1(Ω) :
∫Ω
u = 0.
Let us emphasize that our assumptions also allow for Rellich’s selection theorem, i.e., the embedding
H1Γt
(Ω) →→ L2(Ω)(2.3)
is compact, see, e.g., [3, Theorem 4.8]. By density we have the two rules of partial integration
∀u ∈ H1Γt
(Ω) ∀H ∈ DΓn(Ω) 〈∇u,H〉L2(Ω)
= −〈u,divH〉L2(Ω)
,(2.4)
∀E ∈ RΓt(Ω) ∀H ∈ RΓn(Ω) 〈rotE,H〉L2(Ω)
= 〈E, rotH〉L2(Ω)
.(2.5)
We emphasize that, besides Weck’s selection theorem, the resulting Maxwell estimates (Friedrichs/Poincaretype estimates), Helmholtz decompositions, closed ranges, continuous and compact inverse operators,and an adequate electro-magneto static solution theory for bounded weak Lipschitz domains and mixedboundary conditions, another important result has been shown in [3]. It holds
H1Γt
(Ω) =u ∈ H1(Ω) : 〈∇u,Φ〉
L2(Ω)= −〈u,div Φ〉
L2(Ω)for all Φ ∈ C∞Γn
(Ω),
RΓt(Ω) =E ∈ R(Ω) : 〈rotE,Φ〉
L2(Ω)= 〈E, rot Φ〉
L2(Ω)for all Φ ∈ C∞Γn
(Ω),
DΓn(Ω) =H ∈ D(Ω) : 〈divH,ϕ〉
L2(Ω)= −〈H,∇ϕ〉
L2(Ω)for all ϕ ∈ C∞Γt
(Ω),
(2.6)
i.e., strong and weak definitions of boundary conditions coincide, see [3, Theorem 4.5]. Furthermore, wedefine the closed subspaces of irrotational resp. solenoidal vector fields
R0(Ω) :=E ∈ R(Ω) : rotE = 0
, D0(Ω) :=
E ∈ D(Ω) : divE = 0
as well as
RΓt,0(Ω) := RΓt(Ω) ∩ R0(Ω), DΓn,0(Ω) := DΓn(Ω) ∩ D0(Ω).
A direct consequence of (2.1) is the compactness of the unit ball in
H(Ω) := RΓt,0(Ω) ∩ DΓn,0(Ω),
the space of so called Dirichlet-Neumann fields. Hence H(Ω) is finite dimensional. Another immediateconsequence of Weck’s selection theorem (2.1), using a standard indirect argument, is the so calledMaxwell estimate, i.e.,
∃ cm > 0 ∀E ∈ RΓt(Ω) ∩ DΓn(Ω) ∩H(Ω)⊥
L2(Ω) |E|L2(Ω)
≤ cm(| rotE|
L2(Ω)+ |divE|
L2(Ω)
)(2.7)
or, equivalently,
∀E ∈ RΓt(Ω) ∩ DΓn(Ω) |E − πE|L2(Ω)
≤ cm(| rotE|
L2(Ω)+ |divE|
L2(Ω)
),(2.8)
see [3, Theorem 5.1], where π : L2(Ω)→ H(Ω) denotes the L2(Ω)-orthonormal projector onto the Dirichlet-Neumann fields. Recent estimates for the Maxwell constant cm can be found in [14, 15, 16]. Analogously,Rellich’s selection theorem (2.3) shows the Friedrichs/Poincare estimate
∃ cf,p > 0 ∀u ∈ H1Γt
(Ω) |u|L2(Ω)
≤ cf,p| ∇u|L2(Ω),(2.9)
see [3, Theorem 4.8]. By the projection theorem, applied to the densely defined and closed (unbounded)linear operator
∇ : H1Γt
(Ω) ⊂ L2(Ω) −→ L2(Ω)
4 DIRK PAULY
with (Hilbert space) adjoint
∇∗ = −div : DΓn(Ω) ⊂ L2(Ω) −→ L2(Ω),
here we need (2.6), we get the simple Helmholtz decomposition
L2(Ω) = ∇ H1Γt
(Ω)⊕L2(Ω)
DΓn,0(Ω),(2.10)
see [3, Theorem 5.3 or (13)], which immediately implies
RΓt(Ω) = ∇ H1Γt
(Ω)⊕L2(Ω)
(RΓt(Ω) ∩ DΓn,0(Ω)
)(2.11)
as ∇ H1Γt
(Ω) ⊂ RΓt,0(Ω). Note that the decompositions (2.10) and (2.11) are orthogonal which is denoted
by ⊕L2(Ω)
. By (2.9), the range ∇ H1Γt
(Ω) is closed in L2(Ω), see also [3, Lemma 5.2]. Note that we call
(2.10) a simple Helmholtz decomposition, since the refined Helmholtz decomposition
L2(Ω) = ∇ H1Γt
(Ω)⊕L2(Ω)
H(Ω)⊕L2(Ω)
rot RΓn(Ω)
holds as well, see [3, Theorem 5.3], where also rot RΓn(Ω) is closed in L2(Ω) as a consequence of (2.7), see[3, Lemma 5.2].
3. The div-rot-Lemma
Theorem 3.1 (global div-rot-lemma). Let (En) ⊂ RΓt(Ω) and (Hn) ⊂ DΓn(Ω) be two sequences bounded
in R(Ω) resp. D(Ω). Then there exist E ∈ RΓt(Ω) and H ∈ DΓn(Ω) as well as subsequences, again denoted
by (En) and (Hn), such that (En) and (Hn) converge weakly in R(Ω) resp. D(Ω) to E resp. H togetherwith the convergence of the inner products
〈En, Hn〉L2(Ω)→ 〈E,H〉
L2(Ω).
Proof. We pick subsequences, again denoted by (En) and (Hn), such that (En) and (Hn) converge
weakly in R(Ω) resp. D(Ω) to E resp. H for some E ∈ RΓt(Ω) and H ∈ DΓn(Ω). By the simple
Helmholtz decomposition (2.11), we have the orthogonal decomposition RΓt(Ω) 3 En = ∇un + En with
some un ∈ H1Γt
(Ω) and En ∈ RΓt(Ω) ∩ DΓn,0(Ω). Then (un) is bounded in H1(Ω) by orthogonality
and the Friedrichs/Poincare estimate (2.9). (En) is bounded in R(Ω) ∩ D(Ω) by orthogonality and
rot En = rotEn, div En = 0. Hence, using Rellich’s and Weck’s selection theorems there exist u ∈ H1Γt
(Ω)
and E ∈ RΓt(Ω) ∩ DΓn,0(Ω) and we can extract two subsequences, again denoted by (un) and En, such
that un u in H1(Ω) and un → u in L2(Ω) as well as En E in R(Ω) ∩ D(Ω) and En → E in L2(Ω).
We have E = ∇u+ E, giving the simple Helmholtz decomposition for E, as, e.g., for all ϕ ∈ C∞(Ω)
〈E,ϕ〉L2(Ω)
← 〈En, ϕ〉L2(Ω)= 〈∇un, ϕ〉L2(Ω)
+ 〈En, ϕ〉L2(Ω)→ 〈∇u, ϕ〉
L2(Ω)+ 〈E, ϕ〉
L2(Ω).
Then by (2.4)
〈En, Hn〉L2(Ω)= 〈∇un, Hn〉L2(Ω)
+ 〈En, Hn〉L2(Ω)= −〈un,divHn〉L2(Ω)
+ 〈En, Hn〉L2(Ω)
→ −〈u,divH〉L2(Ω)
+ 〈E,H〉L2(Ω)
= 〈∇u,H〉L2(Ω)
+ 〈E,H〉L2(Ω)
= 〈E,H〉L2(Ω)
,
completing the proof.
Corollary 3.2 (local div-rot-lemma). Let (En) ⊂ R(Ω) and (Hn) ⊂ D(Ω) be two sequences bounded in
R(Ω) resp. D(Ω). Then there exist E ∈ R(Ω) and H ∈ D(Ω) as well as subsequences, again denoted
by (En) and (Hn), such that En E in R(Ω) and Hn H in D(Ω) together with the distributionalconvergence
∀ϕ ∈ C∞(Ω) 〈ϕEn, Hn〉L2(Ω)→ 〈ϕE,H〉
L2(Ω).
Proof. Let Γt := Γ and hence Γn = ∅. (ϕEn) is bounded in RΓ(Ω) and (Hn) is bounded in D(Ω).Theorem 3.1 shows the assertion.
A Global div-curl-Lemma 5
Remark 3.3. We note that the boundedness of (En) and (Hn) in local spaces is sufficient for Corollary3.2 to hold. Hence, no regularity or boundedness assumptions on Ω are needed, i.e., Ω can be an arbitrarydomain in R3.
4. Generalizations
The idea of the proof of Theorem 3.1 can be generalized.
4.1. Functional Analysis Toolbox. Let A :D(A) ⊂ H1 → H2 be a (possibly unbounded) closed and
densely defined linear operator on two Hilbert spaces H1 and H2 with adjoint A* :D(A*) ⊂ H2 → H1. Note
(A∗)∗ = A = A, i.e., (A,A*) is a dual pair. By the projection theorem the Helmholtz type decompositions
H1 = N(A)⊕H1 R(A*), H2 = N(A*)⊕H2 R(A)(4.1)
hold, where we introduce the notation N for the kernel (or null space) and R for the range of a linearoperator. We can define the reduced operators
A := A |R(A*)
: D(A) ⊂ R(A*)→ R(A), D(A) := D(A) ∩N(A)⊥H1 = D(A) ∩R(A*),
A* := A* |R(A)
: D(A*) ⊂ R(A)→ R(A*), D(A*) := D(A*) ∩N(A*)⊥H2 = D(A*) ∩R(A),
which are also closed and densely defined linear operators. We note that A and A* are indeed adjoint to
each other, i.e., (A,A*) is a dual pair as well. Now the inverse operators
A−1 : R(A)→ D(A), (A*)−1 : R(A*)→ D(A*)
exist and they are bijective, since A and A* are injective by definition. Furthermore, by (4.1) we havethe refined Helmholtz type decompositions
D(A) = N(A)⊕H1D(A), D(A*) = N(A*)⊕H2
D(A*)(4.2)
and thus we obtain for the ranges
R(A) = R(A), R(A*) = R(A*).(4.3)
By the closed range theorem and the closed graph theorem we get immediately the following.
Lemma 4.1. The following assertions are equivalent:
(i) ∃ cA ∈ (0,∞) ∀x ∈ D(A) |x|H1≤ cA|Ax|H2
(i∗) ∃ cA* ∈ (0,∞) ∀ y ∈ D(A*) |y|H2≤ cA* |A* y|H1
(ii) R(A) = R(A) is closed in H2.
(ii∗) R(A*) = R(A*) is closed in H1.(iii) A−1 : R(A)→ D(A) is continuous and bijective with norm bounded by (1 + c2A)1/2.
(iii∗) (A*)−1 : R(A*)→ D(A*) is continuous and bijective with norm bounded by (1 + c2A*)1/2.
In case that one of the latter assertions is true, e.g., (ii), R(A) is closed, we have
H1 = N(A)⊕H1 R(A*), H2 = N(A*)⊕H2 R(A),
D(A) = N(A)⊕H1 D(A), D(A*) = N(A*)⊕H2 D(A*),
D(A) = D(A) ∩R(A*), D(A*) = D(A*) ∩R(A),
and
A : D(A) ⊂ R(A*)→ R(A), A* : D(A*) ⊂ R(A)→ R(A*).
Remark 4.2. For the “best” constants cA, cA* the following holds: The Rayleigh quotients
1
cA:= inf
06=x∈D(A)
|Ax|H2
|x|H1
,1
cA*
:= inf0 6=y∈D(A*)
|A* y|H1
|y|H2
coincide, i.e., cA = cA* ∈ (0,∞].
6 DIRK PAULY
Lemma 4.3. The following assertions are equivalent:
(i) D(A) →→ H1 is compact.
(i∗) D(A*) →→ H2 is compact.
(ii) A−1 : R(A)→ R(A*) is compact.
(ii∗) (A*)−1 : R(A*)→ R(A) is compact.
If one of these assertions holds true, e.g., (i), D(A) →→ H1 is compact, then the assertions of Lemma 4.1and Remark 4.2 hold with cA = cA* ∈ (0,∞). Especially, the Friedrichs/Poincare type estimates hold,all ranges are closed and the inverse operators
A−1 : R(A)→ R(A*), (A*)−1 : R(A*)→ R(A)
are compact with norms∣∣A−1
∣∣R(A),R(A*)
=∣∣(A*)−1
∣∣R(A*),R(A)
= cA.
Proof. As the other assertions are easily proved or immediately clear by symmtery, we just show that (i),i.e., the compactness of
D(A) = D(A) ∩R(A*) →→ H1,
implies (i∗) as well as Lemma 4.1 (i).(i)⇒Lemma 4.1 (i): For this we use a standard indirect argument. If Lemma 4.1 (i) was wrong, there
exists a sequence (xn) ⊂ D(A) with |xn|H1= 1 and Axn → 0. As (xn) is bounded in D(A) we can
extract a subsequence, again denoted by (xn), with xn → x ∈ H1 in H1. Since A is closed, we havex ∈ D(A) and Ax = 0, hence x ∈ N(A) = 0, in contradiction to 1 = |xn|H1 → |x|H1 = 0.
(i)⇒(i∗): Let (yn) ⊂ D(A*) be a bounded sequence. Utilizing Lemma 4.1 (i) and (ii) we obtain
D(A*) = D(A*) ∩ R(A) and thus yn = Axn with (xn) ⊂ D(A), which is bounded in D(A) by Lemma4.1 (i). Hence we may extract a subsequence, again denoted by (xn), converging in H1. Therefore withxn,m := xn − xm and yn,m := yn − ym we see
|yn,m|2H2=⟨yn,m,A(xn,m)
⟩H2
=⟨
A*(yn,m), xn,m⟩H1≤ c |xn,m|H1 ,
and hence (yn) is a Cauchy sequence in H2.
Now, let A0 :D(A0) ⊂ H0 → H1 and A1 :D(A1) ⊂ H1 → H2 be (possibly unbounded) closed and denselydefined linear operators on three Hilbert spaces H0, H1, and H2 with adjoints A*
0 :D(A*0) ⊂ H1 → H0 and
A*1 :D(A*
1) ⊂ H2 → H1 as well as reduced operators A0, A*0, and A1, A*
1. Furthermore, we assume thesequence or complex property of A0 and A1, that is, A1 A0 = 0, i.e.,
R(A0) ⊂ N(A1).(4.4)
Then also A*0 A*
1 = 0, i.e., R(A*1) ⊂ N(A*
0). From the Helmholtz type decompositions (4.1) for A = A0
and A = A1 we get in particular
H1 = R(A0)⊕H1 N(A*0), H1 = R(A*
1)⊕H1 N(A1),(4.5)
and the following result for Helmholtz type decompositions:
Lemma 4.4. Let N0,1 := N(A1) ∩N(A*0). The refined Helmholtz type decompositions
N(A1) = R(A0)⊕H1N0,1, D(A1) = R(A0)⊕H1
(D(A1) ∩N(A*
0)), R(A0) = R(A0),(4.6)
N(A*0) = R(A*
1)⊕H1N0,1, D(A*
0) = R(A*1)⊕H1
(D(A*
0) ∩N(A1)), R(A*
1) = R(A*1),(4.7)
and
H1 = R(A0)⊕H1N0,1 ⊕H1
R(A*1)(4.8)
hold, which can be further refined and specialized, e.g., to
D(A1) ∩D(A*0) = D(A*
0)⊕H1 N0,1 ⊕H1 D(A1).(4.9)
Proof. By (4.5) and the complex properties we see (4.6) and (4.7), yielding directly (4.8) and (4.9).
A Global div-curl-Lemma 7
We observe
D(A1) = D(A1) ∩R(A*1) ⊂ D(A1) ∩N(A*
0) ⊂ D(A1) ∩D(A*0),
D(A*0) = D(A*
0) ∩R(A0) ⊂ D(A*0) ∩N(A1) ⊂ D(A*
0) ∩D(A1),
and using the refined Helmholtz type decompositions of Lemma 4.4 as well as the results of Lemma 4.1,Lemma 4.3, and Lemma 4.5, we immediately see:
Lemma 4.5. The following assertions are equivalent:
(i) D(A0) →→ H0, D(A1) →→ H1, and N0,1 →→ H1 are compact.
(ii) D(A1) ∩D(A*0) →→ H1 is compact.
In this case, the cohomology group N0,1 has finite dimension.
We summarize:
Theorem 4.6. Let D(A1) ∩ D(A*0) →→ H1 be compact. Then D(A0) →→ H0, D(A1) →→ H1, and
D(A*0) →→ H1, D(A*
1) →→ H2 are compact, all ranges R(A0), R(A*0), and R(A1), R(A*
1) are closed, andthe corresponding Friedrichs/Poincare type estimates hold, i.e. there exists cA0
, cA1∈ (0,∞) such that
∀ z ∈ D(A0) |z|H0 ≤ cA0 |A0 z|H1 ,(4.10)
∀x ∈ D(A*0) |x|H1
≤ cA0|A*
0 x|H0,
∀x ∈ D(A1) |x|H1≤ cA1
|A1 x|H2,
∀ y ∈ D(A*1) |y|H2 ≤ cA1 |A*
1 y|H1 .
Moreover, all refined Helmholtz type decompositions of Lemma 4.4 hold with closed ranges, especially
D(A1) = R(A0)⊕H1
(D(A1) ∩N(A*
0)).(4.11)
Proof. Apply the latter lemmas and remarks to A = A0 and A = A1.
4.2. The A*0-A1-Lemma. Let A0 and A1 be as introduced before satisfying the complex property (4.4),
i.e., A1 A0 = 0 or R(A0) ⊂ N(A1). In other words, the primal and dual sequences
D(A0) ⊂ H0A0−−−−→ D(A1) ⊂ H1
A1−−−−→ H2,
H0A*
0←−−−− D(A*0) ⊂ H1
A*1←−−−− D(A*
1) ⊂ H2
(4.12)
are Hilbert complexes of closed and densely defined linear operators. The additional assumption of closedranges R(A0) and R(A1)
(and hence also closed ranges R(A*
0) and R(A*1))
is equivalent to calling theHilbert complexes closed. The complexes are exact, if and only if N0,1 = 0.
As our main result, the following generalized global div-curl-lemma holds.
Theorem 4.7 (A*0-A1-lemma). Let D(A1) ∩ D(A*
0) →→ H1 be compact. Moreover, let (xn) ⊂ D(A1)and (yn) ⊂ D(A*
0) be two sequences bounded in D(A1) resp. D(A*0). Then there exist x ∈ D(A1) and
y ∈ D(A*0) as well as subsequences, again denoted by (xn) and (yn), such that (xn) and (yn) converge
weakly in D(A1) resp. D(A*0) to x resp. y together with the convergence of the inner products
〈xn, yn〉H1→ 〈x, y〉H1
.
Proof. Note that Theorem 4.6 can be applied. We pick subsequences, again denoted by (xn) and (yn),such that (xn) and (yn) converge weakly in D(A1) resp. D(A*
0) to x ∈ D(A1) resp. y ∈ D(A*0). By
(4.11) we get the orthogonal decomposition
D(A1) 3 xn = A0 zn + xn, zn ∈ D(A0), xn ∈ D(A1) ∩N(A*0).
(zn) is bounded in D(A0) by orthogonality and the Friedrichs/Poincare tye estimate (4.10). (xn) isbounded in D(A1) ∩D(A*
0) by orthogonality and A1 xn = A1 xn, A*0 xn = 0. Using the compact embed-
dings D(A0) →→ H0 and D(A1) ∩D(A*0) →→ H1, there exist z ∈ D(A0) and x ∈ D(A1) ∩N(A*
0) and wecan extract two subsequences, again denoted by (zn) and (xn), such that zn z in D(A0) and zn → z in
8 DIRK PAULY
H0 as well as xn x in D(A1) ∩D(A*0) and xn → x in H1. We have x = A0 z + x, giving the Helmholtz
type decomposition for x, as, e.g., for all ϕ ∈ H1
〈x, ϕ〉H1← 〈xn, ϕ〉H1
= 〈A0 zn, ϕ〉H1+ 〈xn, ϕ〉H1
→ 〈A0 z, ϕ〉H1+ 〈x, ϕ〉H1
.
Finally, we see
〈xn, yn〉H1= 〈A0 zn, yn〉H1
+ 〈xn, yn〉H1= 〈zn,A*
0 yn〉H0+ 〈xn, yn〉H1
→ 〈z,A*0 y〉H0 + 〈x, y〉H1 = 〈A0 z, y〉H1 + 〈x, y〉H1 = 〈x, y〉H1 ,
completing the proof.
Remark 4.8. By Lemma 4.5 the crucial assumption, i.e., D(A1) ∩ D(A*0) →→ H1 is compact, holds,
if and only if D(A0) →→ H0, D(A1) →→ H1 are compact and N0,1 is finite dimensional. Moreover, asBanach space adjoints we have
H0∼= H′0 →→ D(A0)′ ⇔ D(A0) →→ H0 ⇔ D(A*
0) →→ H1 ⇔ H1∼= H′1 →→ D(A*
0)′,
and
H1∼= H′1 →→ D(A1)′ ⇔ D(A1) →→ H1 ⇔ D(A*
1) →→ H2 ⇔ H2∼= H′2 →→ D(A*
1)′.
Especially, the assumption that D(A1) ∩D(A*0) →→ H1 is compact, is equivalent to dimN0,1 <∞ and
H0 →→ D(A0)′, H2 →→ D(A*1)′
are compact. Thus, we observe that Theorem 4.7 is equivalent to [26, Theorem 2.5] and that the assump-tions of Theorem 4.7 are practically equivalent to those of [26, Theorem 2.4].
5. Applications
Whenever closed Hilbert complexes like (4.12) together with the corresponding compact embeddingD(A1) ∩ D(A*
0) →→ H1 occur, we can apply the general A*0-A1-lemma, i.e., Theorem 4.7. In three
dimensions we typically have three closed and densely defined linear operators A0, A1, and A2, satisfyingthe complex properties R(A0) ⊂ N(A1) and R(A1) ⊂ N(A2), i.e.,
D(A0) ⊂ H0A0−−−−→ D(A1) ⊂ H1
A1−−−−→ D(A2) ⊂ H2A2−−−−→ H3,
H0A*
0←−−−− D(A*0) ⊂ H1
A*1←−−−− D(A*
1) ⊂ H2A*
2←−−−− D(A*2) ⊂ H3,
(5.1)
together with the crucial compact embeddings
D(A1) ∩D(A*0) →→ H1, D(A2) ∩D(A*
1) →→ H2.(5.2)
Let us recall our general assumptions on the underlying domain from Section 2: In the followingexamples we suppose that Ω ⊂ R3 is a bounded weak Lipschitz domain, see [3, Definition 2.3], withboundary Γ, which is divided into two relatively open weak Lipschitz subsets Γt and Γn := Γ \ Γt, see [3,Definition 2.5].
5.1. The div-rot-Lemma Revisited. The first and most canonical example is given by the classicaloperators from vector analysis
A0 := ∇Γt : H1Γt
(Ω) ⊂ L2(Ω) −→ L2ε(Ω); u 7→ ∇u,
A1 := µ−1rotΓt : RΓt(Ω) ⊂ L2ε(Ω) −→ L2
µ(Ω); E 7→ µ−1 rotE,
A2 := divΓtµ : µ−1DΓt(Ω) ⊂ L2µ(Ω) −→ L2(Ω); H 7→ divµH.
A0, A1, and A2 are unbounded, densely defined, and closed linear operators with adjoints
A*0 = ∇
∗Γt
= −divΓnε : ε−1DΓn(Ω) ⊂ L2ε(Ω) −→ L2(Ω); H 7→ −div εH,
A*1 = (µ−1rotΓt)
∗ = ε−1rotΓn : RΓn(Ω) ⊂ L2µ(Ω) −→ L2
ε(Ω); E 7→ ε−1 rotE,
A*2 = (divΓtµ)∗ = −∇Γn : H1
Γn(Ω) ⊂ L2(Ω) −→ L2
µ(Ω); u 7→ −∇u.
A Global div-curl-Lemma 9
Here, ε, µ : Ω → R3×3 are symmetric and uniformly positive definite L∞(Ω)-tensor fields. The complexproperties hold as
R(A0) = ∇ΓtH1Γt
(Ω) ⊂ RΓt,0(Ω) = N(A1), R(A*1) = ε−1rotΓn RΓn(Ω) ⊂ ε−1DΓn,0(Ω) = N(A*
0),
R(A1) = µ−1rotΓt RΓt(Ω) ⊂ µ−1DΓt,0(Ω) = N(A2), R(A*2) = ∇ΓnH
1Γn
(Ω) ⊂ RΓn,0(Ω) = N(A*1).
Hence, the sequences (5.1) read
H1Γt
(Ω) ⊂ L2(Ω)A0=∇Γt−−−−−→ RΓt(Ω) ⊂ L2
ε(Ω)A1=µ−1 rotΓt−−−−−−−−→ µ−1DΓt(Ω) ⊂ L2
µ(Ω)A2=divΓtµ−−−−−−−→ L2(Ω),
L2(Ω)A∗0=−divΓn ε←−−−−−−−− ε−1DΓn(Ω) ⊂ L2
ε(Ω)A∗1=ε−1 rotΓn←−−−−−−−− RΓn(Ω) ⊂ L2
µ(Ω)A∗2=−∇Γn←−−−−−−− H1
Γn(Ω) ⊂ L2(Ω).
These are the well known Hilbert complexes for electro-magnetics, which are also known as de Rhamcomplexes. Moreover, the crucial embeddings (5.2), i.e.,
D(A1) ∩D(A*0) = RΓt(Ω) ∩ ε−1DΓn(Ω) →→ L2(Ω), D(A2) ∩D(A*
1) = µ−1DΓt(Ω) ∩ RΓn(Ω) →→ L2(Ω),
are compact by Weck’s selection theorem (2.1), see [3, Theorem 4.7]. Indeed, Weck’s selection theoremsare independent of the material law tensors ε or µ. Choosing the pair (A0,A1) we get by Theorem 4.7the following:
Theorem 5.1 (global div ε-µ−1 rot-lemma). Let (En) ⊂ RΓt(Ω) and (Hn) ⊂ ε−1DΓn(Ω) be two sequences
bounded in R(Ω) resp. ε−1D(Ω). Then there exist E ∈ RΓt(Ω) and H ∈ ε−1DΓn(Ω) as well as subsequences,
again denoted by (En) and (Hn), such that (En) and (Hn) converge weakly in R(Ω) resp. ε−1D(Ω) to Eresp. H together with the convergence of the inner products
〈En, Hn〉L2ε(Ω)→ 〈E,H〉
L2ε(Ω)
.
Remark 5.2. We note:
(i) Considering (En) and (εHn) shows that Theorem 5.1 is equivalent to the global div-curl-lemmaTheorem 3.1.
(ii) Theorem 5.1 has a corresponding local version similar to the local div-curl-lemma Corollary 3.2and Remark 3.3, which holds with no regularity or boundedness assumptions on Ω.
(iii) Choosing the pair (A1,A2) we get by Theorem 4.7 a permutation of Theorem 5.1, shortly stat-
ing, that for bounded sequences (En) ⊂ µ−1DΓt(Ω) and (Hn) ⊂ RΓn(Ω) it holds (after pickingsubsequences) 〈En, Hn〉L2
µ(Ω)→ 〈E,H〉
L2µ(Ω)
.
Other examples are the following:
5.2. Generalized Electro-Magnetics. Let us allow Ω ⊂ RN or even to be a smooth Riemannianmanifold with Lipschitz boundary and interface submanifolds Γ, and Γt, Γn. Using the calculus ofalternating differential q-forms, we define the exterior derivative d and co-derivative δ = ± ∗ d ∗ asSobolev derivatives in the distributional weak sense by
Dq(Ω) :=E ∈ L2,q(Ω) : dE ∈ L2,q+1(Ω)
, ∆q+1(Ω) :=
H ∈ L2,q+1(Ω) : δ H ∈ L2,q(Ω)
.
To introduce boundary conditions we define
dq
Γt: DqΓt
(Ω) := C∞,qΓt(Ω)
Dq(Ω)
⊂ L2,q(Ω) −→ L2,q+1(Ω); E 7→ dE
as closure of the classical exterior derivative d acting on test q-forms. dq
Γtis an unbounded, densely
defined, and closed linear operator with adjoint
(dq
Γt)∗ = −δ
q+1
Γn: ∆q+1
Γn(Ω) := C∞,q+1
Γn(Ω)
∆q+1(Ω)
⊂ L2,q+1(Ω) −→ L2,q(Ω); H 7→ − δ H.
Let us introduce
A0 := dq−1
Γt, A1 := d
q
Γt, A∗0 = δ
q
Γn, A∗1 = δ
q+1
Γn.
10 DIRK PAULY
The complex properties hold as, e.g.,
R(A0) = dq−1
ΓtDq−1
Γt(Ω) ⊂ DqΓt,0
(Ω) = N(A1), R(A*1) = δ
q+1
Γn∆q+1
Γn(Ω) ⊂ ∆q
Γn,0(Ω) = N(A*
0)
by the classical properties δ δ = ± ∗ d d ∗ = 0. Hence, the sequences (5.1) read
Dq−1Γt
(Ω) ⊂ L2,q−1(Ω)A0=d
q−1Γt−−−−−−→ DqΓt
(Ω) ⊂ L2,q(Ω)A1=d
qΓt−−−−−→ L2,q+1(Ω),
L2,q−1(Ω)A∗0=δ
qΓn←−−−−− ∆q
Γn(Ω) ⊂ L2,q(Ω)
A∗1=δq+1Γn←−−−−−− ∆q+1
Γn(Ω) ⊂ L2,q+1(Ω),
which are the well known Hilbert complexes for generalized electro-magnetics, i.e., the de Rham com-plexes. Moreover, the crucial embedding (5.2), i.e.,
D(A1) ∩D(A*0) = DqΓt
(Ω) ∩ ∆qΓn
(Ω) →→ L2(Ω),
is compact by a generalization of Weck’s selection theorem (2.1), see [4] or the fundamental papers ofWeck [28] and Picard [18] for full boundary conditions. Indeed, Weck’s selection theorems are independentof the material law tensors ε or µ. Theorem 4.7 shows the following result:
Theorem 5.3 (global δ-d-lemma). Let (En) ⊂ DqΓt(Ω) and (Hn) ⊂ ∆q
Γn(Ω) be two sequences bounded
in Dq(Ω) resp. ∆q(Ω). Then there exist E ∈ DqΓt(Ω) and H ∈ ∆q
Γn(Ω) as well as subsequences, again
denoted by (En) and (Hn), such that (En) and (Hn) converge weakly in Dq(Ω) resp. ∆q(Ω) to E resp.H together with the convergence of the inner products
〈En, Hn〉L2,q(Ω)→ 〈E,H〉
L2,q(Ω).
Remark 5.4. We note:
(i) For N = 3 and q = 1 (resp. q = 2) we obtain by Theorem 5.3 again the global div-curl-lemmaTheorem 3.1.
(ii) For q = 0 (resp. q = N) as well as identifying d0
Γt= ∇Γt and ∆0
Γn(Ω) = 0 (resp. d
N
Γt= 0
and ∆NΓn
(Ω) = ∇Γn) we get by Theorem 5.3 the following trivial (by Rellich’s selection theorem)
result: For all bounded sequences (un) ⊂ H1Γt
(Ω) and (vn) ⊂ L2(Ω) there exist u ∈ H1Γt
(Ω) and
v ∈ L2(Ω) as well as subsequences, again denoted by (un) and (vn), such that (un) and (vn)
converge weakly in H1Γt
(Ω) resp. L2(Ω) to u resp. v together with the convergence of the innerproducts 〈un, vn〉L2(Ω)
→ 〈u, v〉L2(Ω)
.
(iii) Theorem 5.3 has a corresponding local version similar to the local div-curl-lemma Corollary 3.2and Remark 3.3, which holds with no regularity or boundedness assumptions on Ω.
5.3. Biharmonic Equation and General Relativity, Gravitational Waves. Let Ω ⊂ R3 enjoy ourgeneral assumptions. We introduce symmetric and deviatoric (trace-free) square integrable tensor fields
in L2(Ω;S) resp. L2(Ω;T) and as closures of the Hessian ∇∇, and Rot, Div (row-wise rot, div), appliedto test functions resp. test tensor fields, the linear operators
A0 := ∇∇Γt : H2Γt
(Ω) := C∞Γt(Ω)
H2(Ω)
⊂ L2(Ω) −→ L2(Ω; S); u 7→ ∇∇u,
A1 := RotS,Γt : RΓt(Ω; S) := C∞Γt(Ω; S)
R(Ω)
⊂ L2(Ω; S) −→ L2(Ω;T); S 7→ RotS,
A2 := DivT,Γt : DΓt(Ω;T) := C∞Γt(Ω;T)
D(Ω)
⊂ L2(Ω;T) −→ L2(Ω); T 7→ Div T.
A0, A1, and A2 are unbounded, densely defined, and closed linear operators with adjoints
A*0 = (∇∇Γt)
∗ = ˚div DivS,Γn :˚DDΓn
(Ω; S) := C∞Γn(Ω; S)
DD(Ω)
⊂ L2(Ω; S) −→ L2(Ω); S 7→ div DivS,
A*1 = Rot
∗S,Γt
= ˚sym RotT,Γn: Rsym,Γn(Ω;T) := C∞Γn
(Ω;T)Rsym(Ω)
⊂ L2(Ω;T) −→ L2(Ω; S); T 7→ sym RotT,
A*2 = Div
∗T,Γt
= − dev ∇Γn : H1Γn
(Ω) ⊂ L2(Ω) −→ L2(Ω;T); v 7→ − dev∇ v,
A Global div-curl-Lemma 11
see [17] for details. Note that u, v, and S, T are scalar, vector, and tensor (matrix) fields, respectively.The complex properties hold as
R(A0) = ∇∇ΓtH2Γt
(Ω) ⊂ RΓt,0(Ω; S) = N(A1),
R(A*1) = ˚sym RotT,Γn
Rsym,Γn(Ω;T) ⊂ ˚DDΓn,0
(Ω;S) = N(A*0),
R(A1) = RotS,Γt RΓt(Ω; S) ⊂ DΓt,0(Ω;T) = N(A2),
R(A*2) = dev ∇ΓnH
1Γn
(Ω) ⊂ Rsym,Γn,0(Ω;T) = N(A*1).
The sequences (5.1) read
H2Γt
(Ω) ⊂ L2(Ω)A0=∇∇Γt−−−−−−−−−→ RΓt (Ω; S) ⊂ L2(Ω; S)
A1=RotS,Γt−−−−−−−−−→ DΓt (Ω; T) ⊂ L2(Ω; T)A2=DivT,Γt−−−−−−−−−→ L2(Ω),
L2(Ω)A∗
0= ˚div DivS,Γn←−−−−−−−−−−−− ˚
DDΓn(Ω; S) ⊂ L2(Ω; S)
A∗1
=sym RotT,Γn←−−−−−−−−−−−−− Rsym,Γn (Ω; T) ⊂ L2(Ω; T)A∗
2=− dev ∇Γn←−−−−−−−−−−−− H1
Γn(Ω) ⊂ L2(Ω).
These are the so called Grad grad and div Div complexes, appearing, e.g., in the biharmonic problem orgeneral relativity, see [17] for details. The crucial embeddings (5.2), i.e.,
D(A1) ∩D(A*0) = RΓt(Ω;S) ∩ ˚
DDΓn(Ω;S) →→ L2(Ω;S),
D(A2) ∩D(A*1) = DΓt(Ω;T) ∩ Rsym,Γn(Ω;T) →→ L2(Ω;T),
are compact by [17, Lemma 3.22]. Choosing the pair (A0,A1) we get by Theorem 4.7 the following:
Theorem 5.5 (global div Div-Rot-S-lemma). Let (Sn) ⊂ RΓt(Ω;S) and (Tn) ⊂ ˚DDΓn
(Ω; S) be two se-
quences bounded in R(Ω) resp. DD(Ω). Then there exist S ∈ RΓt(Ω;S) and T ∈ ˚DDΓn
(Ω;S) as well as
subsequences, again denoted by (Sn) and (Tn), such that (Sn) and (Tn) converge weakly in R(Ω) resp.
DD(Ω) to S resp. T together with the convergence of the inner products
〈Sn, Tn〉L2(Ω,S)→ 〈S, T 〉
L2(Ω,S).
For the pair (A1,A2) Theorem 4.7 implies:
Theorem 5.6 (global sym Rot-Div-T-lemma). Let (Sn) ⊂ DΓt(Ω;T) and (Tn) ⊂ Rsym,Γn(Ω;T) be two
sequences bounded in D(Ω) resp. Rsym(Ω). Then there exist S ∈ DΓt(Ω;T) and T ∈ Rsym,Γn(Ω;T) as well
as subsequences, again denoted by (Sn) and (Tn), such that (Sn) and (Tn) converge weakly in D(Ω) resp.Rsym(Ω) to S resp. T together with the convergence of the inner products
〈Sn, Tn〉L2(Ω,T)→ 〈S, T 〉
L2(Ω,T).
Remark 5.7. Theorem 5.1 has a corresponding local version similar to the local div-curl-lemma Corollary3.2 and Remark 3.3, which holds with no regularity or boundedness assumptions on Ω.
5.4. Linear Elasticity. Let
A0 := sym ∇Γt : H1Γt
(Ω) ⊂ L2(Ω) −→ L2(Ω; S); v 7→ sym∇ v,
A1 := ˚Rot Rot>S,Γt:
˚RR>Γt
(Ω; S) := C∞Γt(Ω; S)
RR>(Ω)
⊂ L2(Ω; S) −→ L2(Ω; S); S 7→ Rot Rot> S,
A2 := DivS,Γt : DΓt(Ω; S) := C∞Γt(Ω; S)
D(Ω)
⊂ L2(Ω; S) −→ L2(Ω); T 7→ Div T.
A0, A1, and A2 are unbounded, densely defined, and closed linear operators with adjoints
A*0 = (sym ∇Γt)
∗ = −DivS,Γn : DΓn(Ω; S) ⊂ L2(Ω; S) −→ L2(Ω); S 7→ −DivS,
A*1 = ( ˚Rot Rot>S,Γt
)∗ = ˚Rot Rot>S,Γn:
˚RR>Γn
(Ω; S) ⊂ L2(Ω; S) −→ L2(Ω; S); T 7→ Rot Rot> T,
A*2 = Div
∗S,Γt
= − sym ∇Γn : H1Γn
(Ω) ⊂ L2(Ω) −→ L2(Ω; S); v 7→ − sym∇ v.
12 DIRK PAULY
Note that v resp. S, T are vector resp. tensor (matrix) fields. The complex properties hold as
R(A0) = sym ∇ΓtH1Γt
(Ω) ⊂ ˚RR>Γt,0(Ω;S) = N(A1),
R(A*1) = ˚Rot Rot>S,Γn
˚RR>Γn
(Ω; S) ⊂ DΓn,0(Ω; S) = N(A*0),
R(A1) = ˚Rot Rot>S,Γt
˚RR>Γt
(Ω; S) ⊂ DΓt,0(Ω;S) = N(A2),
R(A*2) = sym ∇ΓnH
1Γn
(Ω) ⊂ ˚RR>Γn,0(Ω;S) = N(A*
1).
The sequences (5.1) read
H1Γt
(Ω) ⊂ L2(Ω)A0=sym ∇Γt−−−−−−−−−→ ˚
RR>Γt(Ω; S) ⊂ L2(Ω; S)
A1= ˚Rot Rot>S,Γt−−−−−−−−−−−→ DΓt (Ω; S) ⊂ L2(Ω; S)A2=DivS,Γt−−−−−−−−→ L2(Ω),
L2(Ω)A∗0=−DivS,Γn←−−−−−−−−−− DΓn (Ω; S) ⊂ L2(Ω; S)
A∗1= ˚Rot Rot>S,Γn←−−−−−−−−−−− ˚RR>Γn
(Ω; S) ⊂ L2(Ω; S)A∗2=− sym ∇Γn←−−−−−−−−−−− H1
Γn(Ω) ⊂ L2(Ω).
These are the so called Rot Rot complexes, appearing, e.g., in linear elasticity, see [17]. The crucialembeddings (5.2), i.e.,
D(A1) ∩D(A*0) =
˚RR>Γt
(Ω;S) ∩ DΓn(Ω; S) →→ L2(Ω; S),
D(A2) ∩D(A*1) = DΓt(Ω; S) ∩ ˚
RR>Γn(Ω; S) →→ L2(Ω; S),
are compact, which can be proved by the same techniques showing [17, Lemma 3.22]. Choosing the pair(A0,A1) we get by Theorem 4.7 the following:
Theorem 5.8 (global Rot Rot>-Div-S-lemma). Let (Sn) ⊂ ˚RR>Γt
(Ω;S) and (Tn) ⊂ DΓn(Ω;S) be two
sequences bounded in RR>(Ω) resp. D(Ω). Then there exist S ∈ ˚RR>Γt
(Ω; S) and T ∈ DΓn(Ω; S) as well as
subsequences, again denoted by (Sn) and (Tn), such that (Sn) and (Tn) converge weakly in RR>(Ω) resp.
D(Ω) to S resp. T together with the convergence of the inner products
〈Sn, Tn〉L2(Ω,S)→ 〈S, T 〉
L2(Ω,S).
Remark 5.9. Let us note:
(i) Theorem 4.7 for the pair (A1,A2) implies the same result just interchanging Sn, Tn and theboundary conditions.
(ii) Theorem 5.8 has a corresponding local version similar to the local div-curl-lemma Corollary 3.2and Remark 3.3, which holds with no regularity or boundedness assumptions on Ω.
(iii) We emphasize the strong symmetry in the Rot Rot complexes of linear elasticity.
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E-mail address, Dirk Pauly: [email protected]
IN DER SCHRIFTENREIHE DER FAKULTÄT FÜR MATHEMATIK ZULETZT ERSCHIENENE BEITRÄGE: Nr. 769: Mali, O., Muzalevskiy, A., Pauly, D.: Conforming and Non-
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