Abstract
We revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed using an explicit description of the cohomology groups of generalised (‘harmonic’) Dirichlet and Neumann tensor fields. The main results of this contribution are the computation of the indices of Dirac type operators associated to the elasticity complex and the newly found biharmonic complex, relevant for the biharmonic equation, elasticity, and for the theory of general relativity. The differential operators are of mixed order and cannot be seen as leading order type with relatively compact perturbation. As a by-product we present a comprehensive description of the underlying generalised Dirichlet–Neumann vector and tensor fields defining the respective cohomology groups, including an explicit construction of bases in terms of topological invariants, which are of both analytical and numerical interest. Though being defined by certain projection mechanisms, we shall present a way of computing these basis functions by solving certain PDEs given in variational form. For all of this we rephrase core arguments in the work of Rainer Picard [42] applied to the de Rham complex and use them as a blueprint for the more involved cases presented here. In passing, we also provide new vector-analytical estimates of generalised Poincaré–Friedrichs type useful for elasticity or the theory of general relativity.
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1 Introduction
It is one of the greatest mathematical achievements of the twentieth century to relate the analytic notion of the Fredholm index for operators defined on Hilbert spaces to particular elliptic operators and their corresponding geometric properties of the underlying compact manifold the operators are defined on. Here, a closed, densely defined, linear operator \({\mathcal {D}}:{{\,\mathrm{dom}\,}}{\mathcal {D}}\subseteq {\mathcal {K}}_{0} \rightarrow {\mathcal {K}}_{1}\) between Hilbert spaces \({\mathcal {K}}_{0}\) and \({\mathcal {K}}_{1}\) is called a Fredholm operator, if the range, \({{\,\mathrm{ran}\,}}{\mathcal {D}}\subseteq {\mathcal {K}}_{1} \), is closed and both the kernel, \({\text {ker}}{\mathcal {D}}\), and the co-kernel, \(({{\,\mathrm{ran}\,}}{\mathcal {D}})^\bot \), are finite-dimensional. In this case, the index of \({\mathcal {D}}\), \({{\,\mathrm{ind}\,}}{\mathcal {D}}\), is given by
We refer to the concluding parts in [17, Chapter 3] for a brief round up and some standard references to the theory of unbounded Fredholm operators in Hilbert spaces. Generally spoken it is often very difficult—if not impossible—to compute either \(\dim {\text {ker}}{\mathcal {D}}\) or \(\dim ({{\,\mathrm{ran}\,}}{\mathcal {D}})^\bot \) directly. However, due to invariance under homotopies and compact perturbations, it is sometimes possible to have a better understanding of \({{\,\mathrm{ind}\,}}{\mathcal {D}}\) instead.
Indeed, one of the corner stones of results hinted at above is the celebrated Atyiah–Singer index theorem, see e.g. [25], where the (Fredholm) index for some elliptic operators defined on a manifold can be represented solely in terms of the topological properties of this manifold. The methods of proof led to the invention of K-theory, which has evolved ever since and is an active field of research. Albeit being a breakthrough in mathematics, K-theory is a rather difficult tool to work with when it comes to explicitly compute the index for particular examples. In any case there is a need to provide many examples, where it is possible to obtain an index formula, which is as explicit as possible. In fact, the Fredholm index for a perturbed Dirac operator represents physical quantities, see the concluding example in [13] and the references therein. The Witten index, a generalised version of the Fredholm index, is interesting for both physics and mathematics. Indeed, it has been shown that in particular situations the Witten index corresponds to the electromagnetic spin of a particle as well as to the spectral flow of an operator family, see the seminal paper [16].
The results in [16] are based on—among other things—a deeper understanding of the one-dimensional situation of [13], which addresses the Fredholm case. The higher-dimensional cases treated in [13] (with an index formula properly justified in [10]) were generalised in [17]. The transition from the Fredholm situation to the Witten index has been performed in [17, Chapter 14]. Again, a thorough understanding of the Fredholm case has led to further examples for the Witten index, which in turn might prove useful for both mathematics and physics.
The main contribution of the present study is to enlarge the variety of examples, where it is possible to explicitly compute the Fredholm index in terms of the topological properties of the underlying (bounded) domain \(\Omega \subseteq {\mathbb {R}}^3\) the differential (Fredholm) operators are defined on. The list of examples treated here is even more particular as it is possible to compute not only the index but also the dimension of the kernel and the co-dimension of the range in terms of topological invariants.
Moreover, this article is concerned with the explicit computation of the Fredholm index if a differential operator is ‘apparently’ of mixed order. We shall establish a collection of theorems like the following:
Theorem 1.1
Let \(\Omega \subseteq {\mathbb {R}}^3\) be open, bounded with strongFootnote 1 Lipschitz boundary. Then there exists a subspace \({\mathcal {V}}\subseteq L^{2,3\times 3}_{{\mathbb {T}}}(\Omega )\times L^{2}(\Omega )\) such that
and \({\mathcal {D}}^{*}\) are densely defined and closed Fredholm operators, where \(L^{2,3\times 3}_{{\mathbb {T}}}(\Omega )\) and \(L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\) denote the sets of trace free and symmetric \(3\times 3\) matrices with entries in \(L^{2}(\Omega )\), respectively. Moreover,
where n is the number of connected components of \(\Omega \), m is the number of connected components of its complement \({\mathbb {R}}^3\setminus {\overline{\Omega }}\), and p is the number of handles (see Sect. 10).
A closer inspection of the operator \({\mathcal {D}}\) also shows the following estimate; see also Corollary 7.7. Note that the subspace \({\mathcal {W}}\) asserted to exist in the following result—and this is the catch of the corollary—is constructed explicitly by providing a basis, see Sect. 12.2.
Corollary 1.2
There exists a finite-dimensional subspace \({\mathcal {W}}\subseteq {\mathcal {V}}\) and \(c>0\) such that for all \((T,u)\in {\mathcal {V}}\cap {\mathcal {W}}^{\bot _{L^{2}}}\) we have
In the course of the manuscript, we shall, too, describe the subspace \({\mathcal {V}}={{\,\mathrm{dom}\,}}{\mathcal {D}}\) explicitly, see Theorem 7.4 and Remark 7.5. A refined notation will indicate (full) natural boundary conditions by \(\mathring{\,\,\,\,}\) and algebraic properties of the tensor fields belonging to the domain of definition of the respective operators by \({\mathbb {S}}\) and \({\mathbb {T}}\) (symmetric and trace free), e.g., the aforementioned operators read
These operators are related to the (primal and dual) first biharmonic complex, also called \({\text {Gradgrad}}\) or \({\text {divDiv}}\) complex, i.e.,
relevant for the biharmonic equation, elasticity, and in the theory of general relativity. Here and in the following \(\iota _V\) and \(\pi _V\) denote the canonical embedding from a closed subspace V of a Hilbert space H into H and the orthogonal projection from H onto V; the space of piecewise Raviart–Thomas vector-fields, \(\mathsf {RT}_{\mathsf {pw}}\), is defined in (7.1).
We discuss another example, which is based on the second biharmonic complex where the boundary conditions are interchanged, i.e.,
leading to the operators (for the space of piecewise first order polynomials, \({\mathsf {P}}^{1}_{\mathsf {pw}}\), we refer to (8.2))
The corresponding index results can be found in Theorem 8.5 and Remark 8.6.
Finally, we address the elasticity complex, also called \({\text {CurlCurl}}\) complex, i.e., (the space of piecewise rigid motions, \(\mathsf {RM}_{\mathsf {pw}}\), is defined in (9.3))
Here, we shall discuss the Fredholm index of the operators
The solution to the corresponding index problem is provided in Theorem 9.4 and Remark 9.5. Note that in a distributional setting results concerning the computation of the dimension of the generalised Neumann fields have been obtained in [14], where a variational setting is preferred.
Here and throughout this paper, we denote by \({{\,\mathrm{grad}\,}}\), \(\text {curl}\), and \({{\,\mathrm{div}\,}}\) the classical operators from vector analysis. Moreover, \({{\,\mathrm{Grad}\,}}\) acts componentwise as \({{\,\mathrm{grad}\,}}^{\top }\) mapping vector fields to tensor fields. \({{\,\mathrm{Curl}\,}}\) and \({{\,\mathrm{Div}\,}}\) act row-wise as \(\text {curl}^{\top }\) and \({{\,\mathrm{div}\,}}\) mapping tensor fields to tensor and vector fields, respectively. \(L^{2}\)-spaces with k components (or \(k\times k\)-many components) are denoted by \(L^{2,k}\) (or \(L^{2,k\times k}\)). A similar notation is used for \(C^\infty \) and similar sets.
The approach to compute the index in situations as in Theorem 1.1 is rooted in a construction principle for Fredholm operators provided in [11]. The fundamental observation given in [11] is that it is possible to construct a Fredholm operator with the help of Hilbert complexes of densely defined and closed (possibly unbounded) linear operators, i.e,
More precisely, if \(A_{0}\), \(A_{1}\), and \(A_{2}\) are densely defined, closed linear operators defined on suitable Hilbert spaces \(H_{l}\) such that
then the block matrix operator
with its natural domain of definition is closed and densely defined. It is Fredholm, if the ranges \({{\,\mathrm{ran}\,}}A_{0}\), \({{\,\mathrm{ran}\,}}A_{1}\), and \({{\,\mathrm{ran}\,}}A_{2}\) are closed and if both kernels
and both cohomology groups
are finite-dimensional. In this case, its index is then given by
cf. Theorem 3.8. For its adjoint, which is then Fredholm as well, we simply have
In a first application of this observation presented in this article, we look at the classical de Rham complex
where again the super index \(\mathring{\;\;}\) signifies homogeneous Dirichlet boundary conditions, see Theorem 6.8 and \({\mathbb {R}}_{\mathsf {pw}}\) denotes the space of piecewise constants on \(\Omega \), see (6.3). By (1.2) in order to compute the index it is necessary to calculate the dimension of the cohomology groups, i.e., the dimension of the harmonic Dirichlet and Neumann fields
respectively. In [42], this has been done by Picard. Explicit constructions of harmonic Dirichlet and Neumann fields can also be found in [3, 15] the latter of the two references providing substantial detail to the geometrical setting. As it turns out these dimensions are related to topological properties of the underlying domain the differential operators are defined on, that is,
see Theorem 6.6. In consequence, it is possible to compute the indices for the block de Rham operators
by (1.2) in terms of m, p, and n, i.e.,
see Theorem 6.8. It is noteworthy that this index theorem provides an index theorem for the Dirac operator on open manifolds with boundary endowed with a particular boundary condition, see [45] and Sect. 6.3 below.
The proofs of the index theorems discussed here combine the structural viewpoint outlined by [11] and ideas taken from the explicit computation of the dimension of the cohomology groups in [42]. More precisely, we shall rephrase the proofs in [42] and use these reformulations as a blueprint for the proofs for other complexes. We emphasise that the construction of the generalised Neumann fields is based on subtle interactions of matrix algebra and differential operators (see Lemma 12.10) and a suitable application of so-called Poincaré maps yielding (for instance) a representation of vector fields by curve integrals over tensor fields, see e.g. Lemma 12.11. The foundation for all of this to be applicable, however, is the newly found biharmonic complex, see [37, 38], and the more familiar elasticity complex, see [39,40,41]. In [37, 38] the crucial properties and compact embedding results have been found for the biharmonic Hilbert complex underlying the computation of the index in Theorem 1.1. In [39,40,41] the corresponding results are presented for the elasticity complex. These results also stress that the mixed order differential operators discussed here cannot be viewed as a leading order term subject to a relatively compact perturbation.
As the technique computing a basis for the classical de Rham harmonic Dirichlet and Neumann fields developed in [42], we shall comment on other developments particularly concerning the computation of these basis functions in different and/or more general topological settings. Given a suitable formulation of the Hilbert complexes also in these settings, it may be possible to provide an analogous formulation of harmonic Dirichlet/Neumann tensor fields to eventually more substantially address Problem 2.2 below. A topological setting is used in [8] helping to avoid Assumption 10.3 and still being able to construct the harmonic vector fields. With a more partial differential equations oriented way a detour of Assumption 10.3 in conjunction with mixed boundary conditions is possible as demonstrated in [2, Appendix A.4]. This might provide a hint on how to address Problem 6.10. Generalising the topological setting from open subsets of \({\mathbb {R}}^3\) to situations involving a more geometric point of view, the authors of [9, 12, 18, 24] discussed properties of the de Rham complex in general and the corresponding harmonic Dirchlet and Neumann fields in particular.
Before we come to a more in depth description of the structure of the paper, we emphasise the importance of a deeper understanding of Hilbert complexes for index theory and other areas of partial differential equations.
Next to [11] (and others), the notion of Hilbert complexes in connection with (Fredholm) index theory has also been addressed in [46] and references therein. The work in [46] is particularly interesting as the authors address manifolds with boundary. The focus is on characterising the Fredholm property (i.e., the finiteness of the cohomology groups) for certain complexes with boundary in terms of the principal symbol. Here, the Fredholm property of the Hilbert complex (i.e., with the terminology of [46], that a Hilbert complex is a Fredholm complex) follows from suitable compactness criteria (see e.g. [38, 55]) and all the dimensions of the cohomology groups and not just the index of the complex are addressed here explicitly.
An understanding of Hilbert complexes in connection with partial differential equations involving the classical vector analytic operators \({{\,\mathrm{div}\,}}\), \(\text {curl}\), and \({{\,\mathrm{grad}\,}}\) led to [43], where the kernel of the classical Maxwell operator is written by means of other differential operators. The resulting Picard’s extended Maxwell system is useful for numerical studies [49] as well as for the study of the low frequency asymptotics of Maxwell’s equations [43]. More involved low frequency asymptotics for Maxwell’s equations can be found in the series [29,30,31,32,33] based on the series [56,57,58,59] for the reduced scalar and elasticity wave equations. The connections of the extended Maxwell system and the Dirac operator are drawn in [45] and shortly commented in this manuscript below.
Rather recently, the notion of Hilbert complexes (reusing the idea of writing the kernel of differential operators by means of other operators) has found applications in the context of homogenisation theory of partial differential equations. More precisely, it was possible to derive a certain operator-theoretic version of the so-called \({{\,\mathrm{div}\,}}\)-\(\text {curl}\) lemma (see [26, 48]), which implied a whole family of \({{\,\mathrm{div}\,}}\)-\(\text {curl}\) lemma-type results, see [35, 50].
Furthermore, the abstract \({{\,\mathrm{div}\,}}\)-\(\text {curl}\) result together with theory from Hilbert complexes are used to define and study the notion of nonlocal H-convergence, [51]. The applications presented in [51, 52] as well as in [28] use the assumption of exactness of the Hilbert complex, that is, triviality of certain cohomology groups. It is one corollary of the present study to describe the topological properties of the domains the differential operators are defined on to yield exact Hilbert complexes making the theory of nonlocal H-convergence applicable, see also [51, Section 8]. This then results in new homogenisation and compactness theorems for nonlocal homogenisation problems. We postpone further results in this direction to future studies.
We shortly outline the plan of the paper next. The main results, that is, the dimensions of the cohomology groups and the indices of the operators involved, are summarised in Sect. 2. In Sect. 3, we briefly recall the notion of Hilbert complexes of densely defined and closed linear operators. Also, we provide a small introduction to the construction principle for Fredholm operators provided in [11]. As we slightly deviate from the approach presented there we recall some of the proofs for convenience of the reader. As an addendum to Sect. 3, we provide an abstract set of Poincaré–Friedrichs inequalities in Sect. 4 and outline an abstract perspective to variable coefficents in Sect. 5. In order to have a first non-trivial yet rather elementary example at hand, we present the so-called Picard’s extended Maxwell system in Sect. 6. This sets the stage for the index theorem for the Dirac operator provided in Sect. 6.3. In Sect. 7, we recall the first biharmonic complex and provide a more explicit formulation of Theorem 1.1 (see Theorem 7.4). Similar results will be presented in Sect. 8 for the second biharmonic complex and in Sect. 9 for the elasticity complex. Section 10 is concerned with the topological setting introduced in [42] forming our main assumption on \(\Omega \). The Sects. 11 and 12 address the computation of bases and hence the dimensions of the generalised Dirichlet and Neumann vector and tensor fields for the different complexes, respectively, and thus concluding the proofs of our main results. In passing, we also provide partial differential equations whose unique solutions will correspond to the basis functions under consideration. This is particularly important for numerically computing these basis functions. Amongst these PDEs we recover the one in [14], when the generalised Neumann fields for the elasticity complex are concerned (see in particular Remark 12.37; the regularity assumptions on \(\Omega \) are the same here). In Sect. 13 we provide a small conclusion.
Note that unlike to many research topics in the analysis of partial differential equations (and other topics), we shall use \(\Omega \) being ‘open’ and a ‘domain’ as synonymous terms. In particular, we shall not imply \(\Omega \) to satisfy any connectivity properties, when calling \(\Omega \) a domain.
2 A brief overview of the main results
We employ the notations and assumptions of Sect. 1. In particular, we shall assume that \(\Omega \subseteq {\mathbb {R}}^3\) is a bounded, strong Lipschitz domain. The number of connected components of \(\Omega \) is n, the number of connected components of \({\mathbb {R}}^3{\setminus }{\overline{\Omega }}\) is m, and p denotes the number of handles (see Assumption 10.3 below). We introduce the cohomology groups
i.e., the Dirichlet and Neumann fields
We will compute the dimensions of the kernels \(N_{0}\), \(N_{2,*}\), i.e.,
and the dimensions of the cohomology groups \(K_{1}\), \(K_{2}\), i.e.,
and the indices \({{\,\mathrm{ind}\,}}{\mathcal {D}}\), \({{\,\mathrm{ind}\,}}{\mathcal {D}}^{*}\) of the involved Fredholm operators, i.e.,
Remark 2.1
We observe that in all of our examples, where generally the operators \(A_{j}\) carry the boundary condition and the adjoints \(A_{j}^{*}\) do not have any boundary condition, the dimensions of the first and second cohomology groups \(K_{1}\) and \(K_{2}\) (Dirichlet fields and Neumann fields) are given by
respectively. The indices of \({\mathcal {D}}\) (see (1.1)) and \({\mathcal {D}}^{*}\) are
Remark 2.1 leads to the following problem that seems to be open:
Problem 2.2
Is it possible to find differential operators on \(\Omega \subseteq {\mathbb {R}}^3\) (bounded, strong Lipschitz domain) of the form (1.1) as discussed in Remark 2.1 that violate the general index formula for \({\mathcal {D}}\) in (2.1)?
3 The construction principle and the index theorem
In this section, we provide the basic construction principle, which is the basis for the operators in question. The theory in more general terms has been developed already in [11]. Here, we rephrase the situation with a slightly more particular viewpoint. For the convenience of the reader, we carry out the necessary proofs here.
Throughout this section, we let \(H_{0}, H_{1}, H_{2}, H_{3}\) be Hilbert spaces, and
be densely defined and closed linear operators.
Definition 3.1
Let \(A_{0},A_{1},A_{2}\) be defined as above.
-
We call a pair \((A_{0},A_{1})\) a complex (Hilbert complex), if \({{\,\mathrm{ran}\,}}A_{0}\subseteq {\text {ker}}A_{1}\). In this situation we also write
$$\begin{aligned} H_{0}\xrightarrow {A_{0}} H_{1}&\xrightarrow {A_{1}} H_{2}. \end{aligned}$$ -
We say a complex \((A_{0},A_{1})\) is closed, if \({{\,\mathrm{ran}\,}}A_{0}\) and \({{\,\mathrm{ran}\,}}A_{1}\) are closed.
-
A complex \((A_{0},A_{1})\) is said to be compact, if the embedding \({{\,\mathrm{dom}\,}}A_{1}\cap {{\,\mathrm{dom}\,}}A_{0}^{*}\hookrightarrow H_{1}\) is compact.
-
The triple \((A_{0},A_{1},A_{2})\) is called a (closed/compact) complex, if both \((A_{0},A_{1})\) and \((A_{1},A_{2})\) are (closed/compact) complexes.
-
We say that a complex \((A_{0},A_{1},A_{2})\) is maximal compact, if \((A_{0},A_{1},A_{2})\) is a compact complex and both embeddings \({{\,\mathrm{dom}\,}}A_{0}\hookrightarrow H_{0}\) and \({{\,\mathrm{dom}\,}}A_{2}^{*}\hookrightarrow H_{3}\) are compact as well.
Remark 3.2
The ‘FA-ToolBox’ (‘Functional-Analysis-Tool Box’) from [34,35,36, 38,39,40,41] shows that
As a consequence, we obtain \((A_{0},A_{1},A_{2})\) is a (closed/compact/maximal compact) complex if and only if \((A_{2}^{*},A_{1}^{*},A_{0}^{*})\) is a (closed/compact/maximal compact) complex.
Throughout this section, we assume that \((A_{0},A_{1},A_{2})\) is a complex, i.e.,
We define the operator
In block operator matrix notation, we have
From the introduction, we recall the notation
and
The aim of this section is to provide a proof of Theorem 3.8 below. As a standard tool for this and related results, we recall the standard orthogonal decompositions
Using (3.2), by the complex property we get
and hence we obtain the following (abstract) Helmholtz type decomposition
We gather some elementary facts about \({\mathcal {D}}\).
Proposition 3.3
\({\mathcal {D}}\) is a densely defined and closed linear operator.
Proof
For the closedness of \({\mathcal {D}}\), we let \(\big ((x_{k},y_{k})\big )_{k}\) be a sequence in \({{\,\mathrm{dom}\,}}{\mathcal {D}}\) with \(\big ((x_{k},y_{k})\big )_{k}\) converging to some (x, y) in \(H_{2}\times H_{0}\) and \(({\mathcal {D}}(x_{k},y_{k}))_{k}\) converging to (w, z) in \(H_{3}\times H_{1}\). One readily sees using the closedness of \(A_{2}\) that \(x\in {{\,\mathrm{dom}\,}}A_{2}\) and \(A_{2}x=w\). Next, we observe that \({{\,\mathrm{ran}\,}}A_{0}\subseteq {\text {ker}}A_{1}\,\bot _{H_{1}}\,{{\,\mathrm{ran}\,}}A_{1}^{*}\). Hence, \((A_{1}^{*}x_{k})_{k}\) and \((A_{0} y_{k})\) are both convergent to some \(z_{1}\in H_{1}\) and \(z_{2}\in H_{1}\), respectively. By the closedness of both \(A_{1}^{*}\) and \(A_{0}\), we thus deduce that \(x\in {{\,\mathrm{dom}\,}}A_{1}^{*}\) and \(y\in {{\,\mathrm{dom}\,}}A_{0}\) with \(z_{1}=A_{1}^{*}x\) and \(z_{2}=A_{0}y\) as well as \(z=z_{1}+z_{2}=A_{1}^{*}x+A_{0}y\).
For \({\mathcal {D}}\) being densely defined, we see that by assumption, \({{\,\mathrm{dom}\,}}A_{0}\) is dense in \(H_{0}\). Hence, it suffices to show that \({{\,\mathrm{dom}\,}}A_{2}\cap {{\,\mathrm{dom}\,}}A_{1}^{*}\) is dense in \(H_{2}\). Thus, as \({{\,\mathrm{dom}\,}}A_{2}\) and \({{\,\mathrm{dom}\,}}A_{1}^{*}\) are dense in \(H_{2}\), we deduce by (3.3) that \({{\,\mathrm{dom}\,}}A_{2} \cap \overline{{{\,\mathrm{ran}\,}}A_{2}^{*}}\) and \({{\,\mathrm{dom}\,}}A_{1}^{*} \cap \overline{{{\,\mathrm{ran}\,}}A_{1}}\) are dense in \(\overline{{{\,\mathrm{ran}\,}}A_{2}^{*}}\) and \(\overline{{{\,\mathrm{ran}\,}}A_{1}}\), respectively. Thus, the decomposition in (3.5) implies that \({{\,\mathrm{dom}\,}}A_{2}\cap {{\,\mathrm{dom}\,}}A_{1}^{*}\) is dense in \(H_{2}\), which yields the assertion. \(\square \)
Theorem 3.4
\({\mathcal {D}}^{*}=\begin{pmatrix}A_{2}^{*}&{}\quad A_{1}\\ 0&{}\quad A_{0}^{*}\end{pmatrix}\). More precisely,
Proof
Note that
holds by definition since for all \((x,y)\in {{\,\mathrm{dom}\,}}{\mathcal {D}}=({{\,\mathrm{dom}\,}}A_{2}\cap {{\,\mathrm{dom}\,}}A_{1}^{*})\times {{\,\mathrm{dom}\,}}A_{0}\) and for all \((w,z)\in {{\,\mathrm{dom}\,}}A_{2}^{*}\times ({{\,\mathrm{dom}\,}}A_{1}\cap {{\,\mathrm{dom}\,}}A_{0}^{*})\)
Let \((w,z)\in {{\,\mathrm{dom}\,}}{\mathcal {D}}^{*}\) and set \((u,v):={\mathcal {D}}^{*}(w,z)\). For \(y\in {{\,\mathrm{dom}\,}}A_{0}\) we have \((0,y)\in {{\,\mathrm{dom}\,}}{\mathcal {D}}\). We obtain
Hence, \(z\in {{\,\mathrm{dom}\,}}A_{0}^{*}\) and \(A_{0}^{*}z = v\).
For all \(x\in {{\,\mathrm{dom}\,}}A_{2}\cap {{\,\mathrm{dom}\,}}A_{1}^{*}\) we see \((x,0)\in {{\,\mathrm{dom}\,}}{\mathcal {D}}\) and deduce that
Let \(\pi _{2}\) denote the orthonormal projector onto \(\overline{{{\,\mathrm{ran}\,}}A_{2}^{*}}\) in (3.3). Then for \({\widetilde{x}}\in {{\,\mathrm{dom}\,}}A_{2}\) we have
and by (3.6)
Thus \(w\in {{\,\mathrm{dom}\,}}A_{2}^{*}\) and \(A_{2}^{*}w=\pi _{2}u\). Analogously, let \(\pi _{1}\) denote the orthonormal projector onto \(\overline{{{\,\mathrm{ran}\,}}A_{1}}\) in (3.3). Then for \({\widetilde{x}}\in {{\,\mathrm{dom}\,}}A_{1}^{*}\) we have
and by (3.6)
Thus \(z\in {{\,\mathrm{dom}\,}}A_{1}\) and \(A_{1}z=\pi _{1}u\). Therefore, \((w,z)\in {{\,\mathrm{dom}\,}}A_{2}^{*}\times ({{\,\mathrm{dom}\,}}A_{1}\cap {{\,\mathrm{dom}\,}}A_{0}^{*})\). Moreover, using the orthonormal projector \(\pi _{0}\) onto \(K_{2}\) in (3.5) we see for \(x\in K_{2}\) by (3.6)
yielding \(\pi _{0}u=0\). Finally, by (3.5) we arrive at
completing the proof. \(\square \)
Lemma 3.5
With the settings (3.1) and (3.2), the kernels of \({\mathcal {D}}\) and \({\mathcal {D}}^{*}\) read
Proof
For \((x,y)\in {\text {ker}}{\mathcal {D}}\) we have \(A_{2}x=0\) and \(A_{1}^{*}x+A_{0}y=0\). By orthogonality and the complex property, i.e., \({{\,\mathrm{ran}\,}}A_{0}\subseteq {\text {ker}}A_{1}\,\bot _{H_{1}}\,{{\,\mathrm{ran}\,}}A_{1}^{*}\), we see \(A_{1}^{*}x=A_{0}y=0\). The assertion about \({\text {ker}}{\mathcal {D}}^{*}\) (use Theorem 3.4 and Remark 3.2) follows analogously. \(\square \)
With Lemma 3.5 at hand, the following result is immediate.
Corollary 3.6
The closures of the ranges of \({\mathcal {D}}\) and \({\mathcal {D}}^{*}\) are given by
Lemma 3.7
Let \((A_{0},A_{1},A_{2})\) be a maximal compact Hilbert complex. Then the embedding \({{\,\mathrm{dom}\,}}{\mathcal {D}}\hookrightarrow H_{2}\times H_{0}\) is compact, and so is the embedding \({{\,\mathrm{dom}\,}}{\mathcal {D}}^{*}\hookrightarrow H_{3}\times H_{1}\).
Proof
Let \(\big ((x_{k},y_{k})\big )_{k}\) be a \(({{\,\mathrm{dom}\,}}{\mathcal {D}})\)-bounded sequence in \({{\,\mathrm{dom}\,}}{\mathcal {D}}\). Then, as in the proof of Lemma 3.5, by orthogonality and the complex property \((x_{k})_{k}\) is a \(({{\,\mathrm{dom}\,}}A_{2}\cap {{\,\mathrm{dom}\,}}A_{1}^{*})\)-bounded sequence in \({{\,\mathrm{dom}\,}}A_{2}\cap {{\,\mathrm{dom}\,}}A_{1}^{*}\) and \((y_{k})_{k}\) is a \(({{\,\mathrm{dom}\,}}A_{0})\)-bounded sequence in \({{\,\mathrm{dom}\,}}A_{0}\). Since \((A_{0},A_{1},A_{2})\) is maximal compact, we can extract converging subsequences of \((x_{k})_{k}\) and \((y_{k})_{k}\). Analogously, using Theorem 3.4 and Remark 3.2, we see that also \({{\,\mathrm{dom}\,}}{\mathcal {D}}^{*}\hookrightarrow H_{3}\times H_{1}\) is compact, finishing the proof. \(\square \)
We now recall the abstract index theorem taken from [11] formulated for the present situation.
Theorem 3.8
Let \((A_{0},A_{1},A_{2})\) be a maximal compact Hilbert complex. Then \({\mathcal {D}}\) and \({\mathcal {D}}^{*}\) are Fredholm operators with indices
Proof
Utilising the ‘FA-ToolBox’ from, e.g., [34,35,36, 38,39,40,41], and Lemma 3.7 we observe that both ranges \({{\,\mathrm{ran}\,}}{\mathcal {D}}\) and \({{\,\mathrm{ran}\,}}{\mathcal {D}}^{*}\) are closed and that both kernels \({\text {ker}}{\mathcal {D}}\) and \({\text {ker}}{\mathcal {D}}^{*}\) are finite-dimensional. Therefore, both \({\mathcal {D}}\) and \({\mathcal {D}}^{*}\) are Fredholm operators. The index \({{\,\mathrm{ind}\,}}{\mathcal {D}}=\dim {\text {ker}}{\mathcal {D}}-\dim {\text {ker}}{\mathcal {D}}^{*}\) is then easily computed with the help of Lemma 3.5. \(\square \)
4 Abstract Poincaré–Friedrichs type inequalities
Let us mention some additional features of the ‘FA-ToolBox’ from [34,35,36, 38,39,40,41]. Lemma 3.7 and Theorem 3.8 imply some additional results for the reduced operators
Corollary 4.1
Let \((A_{0},A_{1},A_{2})\) be a maximal compact Hilbert complex. Then the inverse operators \({\mathcal {D}}_{\mathsf {red}}^{-1}:{{\,\mathrm{ran}\,}}{\mathcal {D}}\rightarrow {{\,\mathrm{ran}\,}}{\mathcal {D}}^{*}\) and \(({\mathcal {D}}^{*}_{\mathsf {red}})^{-1}:{{\,\mathrm{ran}\,}}{\mathcal {D}}^{*}\rightarrow {{\,\mathrm{ran}\,}}{\mathcal {D}}\) are compact. Moreover, \({\mathcal {D}}_{\mathsf {red}}^{-1}:{{\,\mathrm{ran}\,}}{\mathcal {D}}\rightarrow {{\,\mathrm{dom}\,}}{\mathcal {D}}_{\mathsf {red}}\) and \(({\mathcal {D}}^{*}_{\mathsf {red}})^{-1}:{{\,\mathrm{ran}\,}}{\mathcal {D}}^{*}\rightarrow {{\,\mathrm{dom}\,}}{\mathcal {D}}^{*}_{\mathsf {red}}\) are continuous and, equivalently, the Friedrichs–Poincaré type estimates
hold for all \((x,y)\in {{\,\mathrm{dom}\,}}{\mathcal {D}}_{\mathsf {red}}\) and for all \((w,z)\in {{\,\mathrm{dom}\,}}{\mathcal {D}}^{*}_{\mathsf {red}}\) with the same optimal constant \(c_{{\mathcal {D}}}>0\).
The latter estimates are additive combinations of the corresponding estimates for \(A_{0}\) and \((A_{2},A_{1}^{*})\) as well as \(A_{2}^{*}\) and \((A_{1},A_{0}^{*})\), respectively.
Remark 4.2
The compactness assumptions (maximal compact) are not needed to render \({\mathcal {D}}\) and \({\mathcal {D}}^{*}\) Fredholm operators. It suffices to assume that \((A_{0},A_{1},A_{2})\) is a closed Hilbert complex with finite-dimensional kernels \(N_{0}\) and \(N_{2,*}\) and finite-dimensional cohomology groups \(K_{1}\) and \(K_{2}\). In this case, the latter Friedrichs–Poincaré type estimates still hold and \({\mathcal {D}}_{\mathsf {red}}^{-1}\) and \(({\mathcal {D}}_{\mathsf {red}}^{*})^{-1}\) are still continuous.
Remark 4.3
There are simple relations between the primal, dual, and adjoint complexes, when \({\mathcal {D}}\) is considered. More precisely, let us denote the latter primal operators \({\mathcal {D}}\) and \({\mathcal {D}}^{*}\) of the primal complex \((A_{0},A_{1},A_{2})\) by
and the dual operators corresponding to the dual complex \((A_{2}^{*},A_{1}^{*},A_{0}^{*})\) by
By Remark 3.2\((A_{0},A_{1},A_{2})\) is a maximal compact complex, if and only if \((A_{2}^{*},A_{1}^{*},A_{0}^{*})\) is a maximal compact complex. Note that we may weaken the assumptions along the lines sketched in Remark 4.2. Theorem 3.8 shows that \({\mathcal {D}}^{p}\), \(({\mathcal {D}}^{p})^{*}\), \({\mathcal {D}}^{d}\), \(({\mathcal {D}}^{d})^{*}\) are Fredholm operators with indices
Next we observe
Hence
Note that basically \({\mathcal {D}}^{d}\) and \(({\mathcal {D}}^{p})^{*}\) as well as \({\mathcal {D}}^{p}\) and \(({\mathcal {D}}^{d})^{*}\) are the ‘same’ operators.
5 The case of variable coefficients
Note that the Hilbert space adjoints \(A_{l}^{*}\) depend on the particular choice of the inner products (metrics) of the underlying Hilbert spaces \(H_{l}\). A typical example is simply given by ‘weighted’ inner products induced by ‘weights’ \(\lambda _{l}\), \(l\in \{0,1,2,3\}\), i.e., symmetric and positive topological isomorphisms (symmetric and positive bijective bounded linear operators) \(\lambda _{l}:H_{l}\rightarrow H_{l}\) inducing inner products
where \({\widetilde{H}}_{l}:=H_{l}\) (as linear space) equipped with the inner product \(\langle \,\cdot \,,\,\cdot \,\rangle _{{\widetilde{H}}_{l}}\). A sufficiently general situation is defined by \(\lambda _{0}:={{\,\mathrm{Id}\,}}\), \(\lambda _{3}:={{\,\mathrm{Id}\,}}\), and \(\lambda _{1},\lambda _{2}\) being symmetric and positive topological isomorphisms, as well as \({\widetilde{H}}_{l}:=\big (H_{l},\langle \lambda _{l}\,\cdot \,,\,\cdot \,\rangle _{H_{l}}\big )\), \(l\in \{0,1,2,3\}\). Then the modified operatorsFootnote 2
form again a primal and dual Hilbert complex, i.e.,
and we can define
The closedness of the operators \({\widetilde{A}}_{l}\) and the complex properties are easily checked. Moreover, it is not hard to see that the closedness of \(({\widetilde{A}}_{0},{\widetilde{A}}_{1},{\widetilde{A}}_{2})\) is implied by the closedness of \((A_{0},A_{1},A_{2})\). Remark 3.2, Proposition 3.3, Theorem 3.4, Lemma 3.5, and Corollary 3.6 can be applied to \(({\widetilde{A}}_{0},{\widetilde{A}}_{1},{\widetilde{A}}_{2})\) as well. In particular,
It is possible to relate the statements in Lemma 3.7 and Theorem 3.8 to the corresponding ones of the original complex \((A_{0},A_{1},A_{2})\). This will be done next.
Lemma 5.1
The compactness properties and the dimensions of the kernels and cohomology groups of the latter complexes are independent of the weights \(\lambda _{l}\). More precisely,
- (i):
-
\({\widetilde{N}}_{0}=N_{0}\) and \({\widetilde{N}}_{2,*}=N_{2,*}\), as \({{\,\mathrm{dom}\,}}{\widetilde{A}}_{0}={{\,\mathrm{dom}\,}}A_{0}\) and \({{\,\mathrm{dom}\,}}{\widetilde{A}}_{2,*}={{\,\mathrm{dom}\,}}A_{2,*}\),
- (ii\(_{{\mathbf {1}}}\)):
-
\(\dim \big ({\text {ker}}A_{1}\cap (\lambda _{1}^{-1}{\text {ker}}A_{0}^{*})\big ) =\dim {\widetilde{K}}_{1} =\dim K_{1} =\dim ({\text {ker}}A_{1}\cap {\text {ker}}A_{0}^{*})\),
- (ii\(_{{\mathbf {2}}}\)):
-
\(\dim \big ({\text {ker}}A_{2}\cap (\lambda _{2}^{-1}{\text {ker}}A_{1}^{*})\big ) =\dim {\widetilde{K}}_{2} =\dim K_{2} =\dim ({\text {ker}}A_{2}\cap {\text {ker}}A_{1}^{*})\),
- (iii\(_{{\mathbf {1}}}\)):
-
\({{\,\mathrm{dom}\,}}{\widetilde{A}}_{1}\cap {{\,\mathrm{dom}\,}}{\widetilde{A}}_{0}^{*} ={{\,\mathrm{dom}\,}}A_{1}\cap (\lambda _{1}^{-1}{{\,\mathrm{dom}\,}}A_{0}^{*}) \hookrightarrow {\widetilde{H}}_{1}\) compactly
\(\Leftrightarrow \;{{\,\mathrm{dom}\,}}A_{1}\cap {{\,\mathrm{dom}\,}}A_{0}^{*} \hookrightarrow H_{1}\) compactly,
- (iii\(_{{\mathbf {2}}}\)):
-
\({{\,\mathrm{dom}\,}}{\widetilde{A}}_{2}\cap {{\,\mathrm{dom}\,}}{\widetilde{A}}_{1}^{*} ={{\,\mathrm{dom}\,}}A_{2}\cap (\lambda _{2}^{-1}{{\,\mathrm{dom}\,}}A_{1}^{*}) \hookrightarrow {\widetilde{H}}_{2}\) compactly
\(\Leftrightarrow \;{{\,\mathrm{dom}\,}}A_{2}\cap {{\,\mathrm{dom}\,}}A_{1}^{*} \hookrightarrow H_{2}\) compactly.
Proof
For the proof we follow in close lines the ideas of [6, Theorem 6.1], where [6] is the extended version of [7]. (i) is trivial and it is sufficient to show only (ii\(_{1}\)) and (iii\(_{1}\)).
For (ii\(_{1}\)), let \(\mu \) be another weight having the same properties as \(\lambda _{1}\). Similar to (3.3), (3.5) we have by orthogonality in \({\widetilde{H}}_{1}\) and by the complex property
and we note that \({\widetilde{H}}_{1}=H_{1}\) and \({\text {ker}}{\widetilde{A}}_{1}={\text {ker}}A_{1}\) as sets. We denote the \({\widetilde{H}}_{1}\)-orthonormal projector along \(\overline{{{\,\mathrm{ran}\,}}A_{0}}\) onto \(\lambda _{1}^{-1}{\text {ker}}A_{0}^{*}\) by \(\pi \). Then, by (5.1), we deduce
We consider the linear mapping
Then \({\widehat{\pi }}\) is injective. Indeed, let \(y\in {\text {ker}}A_{1}\cap (\mu ^{-1}{\text {ker}}A_{0}^{*})\) with \({\widehat{\pi }}y=\pi y=0\). Then \(y\in \overline{{{\,\mathrm{ran}\,}}A_{0}}\) and \(\mu y\in {\text {ker}}A_{0}^{*}\). Since \(\overline{{{\,\mathrm{ran}\,}}A_{0}}\ \bot _{H_{1}}{\text {ker}}A_{0}^{*}\), using that \(\mu \ge \mu _{0}\) in the sense of positive definiteness for some \(d>0\), we infer \(\mu _{0}|y|_{H_{1}}^{2}\le \langle \mu y,y\rangle _{H_{1}}=0\). Thus
The other inequality \(\ge \) is deduced by symmetry (in \(\mu \) and \(\lambda _{1}\)) and hence equality holds.
For (iii\(_{1}\)), we use a similar decomposition strategy. Let \(\mu \) be as before and let
be compact. Moreover, let us consider a bounded sequence
i.e., \((y_{k})_{k}\), \((A_{1}y_{k})_{k}\), \((A_{0}^{*}\,\mu \,y_{k})_{k}\) are bounded. Similar to (5.1) we get
and \({{\,\mathrm{dom}\,}}{\widetilde{A}}_{1}={{\,\mathrm{dom}\,}}A_{1}\) and \({{\,\mathrm{dom}\,}}{\widetilde{A}}_{0}^{*}=\lambda _{1}^{-1}{{\,\mathrm{dom}\,}}A_{0}^{*}\) as sets. Now, we apply these decompositions to \((y_{k})_{k}\). First, we \({\widetilde{H}}_{1}\)-orthogonally decompose \(y_{k}\in {{\,\mathrm{dom}\,}}A_{1}\) into
with
Therefore \((v_{k})_{k}\) is bounded in \({{\,\mathrm{dom}\,}}A_{1}\cap (\lambda _{1}^{-1}{\text {ker}}A_{0}^{*})\) and by (5.2) we can extract an \(H_{1}\)-converging subsequence, again denoted by \((v_{k})_{k}\). Second, we \({\widetilde{H}}_{1}\)-orthogonally decompose \(\lambda _{1}^{-1}\mu y_{k}\in \lambda _{1}^{-1}{{\,\mathrm{dom}\,}}A_{0}^{*}\) into
with
Hence \((w_{k})_{k}\) is bounded in \({\text {ker}}A_{1}\cap (\lambda _{1}^{-1}{{\,\mathrm{dom}\,}}A_{0}^{*})\) and by (5.2) we can extract an \(H_{1}\)-converging subsequence, again denoted by \((w_{k})_{k}\). Finally, again by \(H_{1}\)-orthogonality, i.e., \(u_{k}\in \overline{{{\,\mathrm{ran}\,}}A_{0}}\,\bot _{H_{1}}{\text {ker}}A_{0}^{*}\ni \lambda _{1}z_{k}\),
for some \(c>0\) independently of k, l, which shows that \((y_{k})_{k}\) is an \(H_{1}\)-Cauchy sequence in \(H_{1}\). Thus \({{\,\mathrm{dom}\,}}A_{1}\cap (\mu ^{-1}{{\,\mathrm{dom}\,}}A_{0}^{*})\hookrightarrow H_{1}\) is compact. \(\square \)
Now we can formulate the counterparts of Lemma 3.7 and Theorem 3.8. The proofs follow immediately by Lemma 5.1.
Lemma 5.2
Maximal compactness does not depend on the weights \(\lambda _{l}\). More precisely: \((A_{0},A_{1},A_{2})\) is a maximal compact Hilbert complex, if and only if the Hilbert complex \(({\widetilde{A}}_{0},{\widetilde{A}}_{1},{\widetilde{A}}_{2})\) is maximal compact. In either case, \({{\,\mathrm{dom}\,}}{\widetilde{{\mathcal {D}}}}\hookrightarrow {\widetilde{H}}_{2}\times {\widetilde{H}}_{0}\) and \({{\,\mathrm{dom}\,}}{\widetilde{{\mathcal {D}}}}^{*}\hookrightarrow {\widetilde{H}}_{3}\times {\widetilde{H}}_{1}\) are compact.
Theorem 5.3
The Fredholm indices do not depend on the weights \(\lambda _{l}\). More precisely: Let \((A_{0},A_{1},A_{2})\) be a maximal compact Hilbert complex. Then \({\mathcal {D}}\), \({\widetilde{{\mathcal {D}}}}\), \({\mathcal {D}}^{*}\), and \({\widetilde{{\mathcal {D}}}}^{*}\) are Fredholm operators with indices
6 The De Rham complex and its indices
As a first application of our abstract findings, in this section, we specialise to a particular choice of the operators \(A_{0}\), \(A_{1}\), \(A_{2}\). Also, we will show that the assumptions of Theorem 3.8 are satisfied for this particular choice of operators. We will, thus, obtain an index formula. The computations of the dimensions of the occurring cohomology groups date back to [42].
Definition 6.1
Let \(\Omega \subseteq {\mathbb {R}}^3\) be an open set. We put
and further define the densely defined and closed linear operators
In terms of classical definitions and notions, we record the following equalities (that are easily seen):
6.1 Picard’s extended Maxwell system
We want to apply the index theorem in the following situation of the classical de Rham complex:
We note
The complex properties, i.e., \(A_{1}A_{0}\subseteq 0\) and \(A_{2}A_{1}\subseteq 0\), are based on Schwarz’s lemma ensuring that \(\text {curl}_{c}{{\,\mathrm{grad}\,}}_{c}=0\) and \({{\,\mathrm{div}\,}}_{c}\text {curl}_{c}=0\).
Proposition 6.2
Let \(\Omega \subseteq {\mathbb {R}}^3\) be open. Then
and by Remark 3.2 the same holds for the adjoints (operators without homogeneous boundary conditions).
Proof
See, e.g., [47, Proposition 6.1.5]. \(\square \)
Theorem 6.3
(Picard–Weber–Weck selection theorem, [44, 53, 55]) Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded weakFootnote 3 Lipschitz domain. Then
are both compactly embedded into \(H_{1}=H_{2}=L^{2,3}(\Omega )\).
Remark 6.4
Proposition 6.2 in conjunction with Theorem 6.3 and Rellich’s selection theorems show that \((\mathring{{{\,\mathrm{grad}\,}}},\mathring{\text {curl}},\mathring{{{\,\mathrm{div}\,}}})\) is a maximal compact complex. By Remark 3.2 so is the dual complex \((-{{\,\mathrm{grad}\,}},\text {curl},-{{\,\mathrm{div}\,}})\).
Note that
where we recall from the introduction the classical harmonic Dirichlet and Neumann fields \({\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )\) and \({\mathcal {H}}_{N}^{\mathsf {Rhm}}(\Omega )\), respectively.
Definition 6.5
Let \(\Omega \subseteq {\mathbb {R}}^3\) be bounded and open. Then we denote by
-
n the number of connected components of \(\Omega \),
-
m the number of connected components of the complement \({\mathbb {R}}^3{\setminus }{\overline{\Omega }}\),
-
p the number of handles of \(\Omega \), see Assumption 10.3.
For p to be well-defined we suppose Assumption 10.3 to hold.
The dimensions of the cohomology groups are given as follows.
Theorem 6.6
([42, Theorem 1]) Let \(\Omega \subseteq {\mathbb {R}}^3\) be open and bounded with continuous boundary. Moreover, suppose Assumption 10.3. Then
In comparison to [42, Theorem 1] a modified proof of Theorem 6.6 is provided in the Sects. 11.1 and 12.1. Note that in [42] unbounded domains where considered as well, which necessitates a slightly different rationale.
Remark 6.7
Note that for \(\Omega \) to have a continuous boundaryFootnote 4 is equivalent for it to have the segment property, see, e.g., [4, Remark 7.8 (a)].
Let us introduce the space of piecewise constants by
where
Theorem 6.8
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded weak Lipschitz domain. Then \({\mathcal {D}}^{\mathsf {Rhm}}\) is a Fredholm operator with index
If additionally \(\Gamma =\partial \Omega \) is continuous and Assumption 10.3 holds, then
Proof
We recall Remark 6.4 and apply Theorem 3.8 together with (6.2), the observations
and Theorem 6.6. \(\square \)
Remark 6.9
By Theorem 3.8 the adjoint of the de Rham operator \(({\mathcal {D}}^{\mathsf {Rhm}})^{*}\) is Fredholm as well with index \({{\,\mathrm{ind}\,}}({\mathcal {D}}^{\mathsf {Rhm}})^{*}=-{{\,\mathrm{ind}\,}}{\mathcal {D}}^{\mathsf {Rhm}}\). Moreover, Picard’s extended Maxwell system is given by
with \(({\mathcal {M}}^{\mathsf {Rhm}})^{*}=-{\mathcal {M}}^{\mathsf {Rhm}}\) and \({{\,\mathrm{ind}\,}}{\mathcal {M}}^{\mathsf {Rhm}}=\dim {\text {ker}}{\mathcal {M}}^{\mathsf {Rhm}}-\dim {\text {ker}}({\mathcal {M}}^{\mathsf {Rhm}})^{*}=0\). Moreover, \(\dim {\text {ker}}{\mathcal {M}}^{\mathsf {Rhm}}=n+m+p-1\) as
6.2 Variable coefficients and Poincaré–Friedrichs type inequalities
The construction of a maximal compact Hilbert complex is also possible for mixed boundary conditions as well as for inhomogeneous and anisotropic media, such as constitutive material laws, see, e.g., [5, 35, 36]. For mixed boundary conditions we note the following:
Problem 6.10
In order to provide a greater variety of index theorems, it would be interesting to compute the dimensions of the harmonic Dirichlet and Neumann fields also in the situation of mixed boundary conditions. This would be particularly interesting for the more involved situations described below. At least for the authors of this article it is completely beyond their expertise in geometry and topology and it appears to be an open problem as to which index formulas could be expected in terms of subcohomologies and related concepts. In the de Rham situation discussed in this section, note that Fredholmness is guaranteed by the compactness result in [5] in conjunction with Theorem 3.8 for a suitably large class of underlying sets and boundaries.
For inhomogeneous and anisotropic media (constitutive material laws) we have:
Remark 6.11
As mentioned before, a maximal compact Hilbert complex can also be constructed for inhomogeneous and anisotropic media. These may be considered as weights \(\lambda _{l}\) as presented in Theorem 5.3. For Maxwell’s equations a typical situation is given by the choices \(\lambda _{0}:={{\,\mathrm{Id}\,}}\), \(\lambda _{3}:={{\,\mathrm{Id}\,}}\), and \(\lambda _{1}:=\varepsilon ,\lambda _{2}:=\mu :\Omega \rightarrow {\mathbb {R}}^{3\times 3}\) being symmetric and uniformly positive definite \(L^{\infty }(\Omega )\)-matrix (tensor) fields. Let us introduce the Hilbert spaces \(L^{2,3}_{\varepsilon }(\Omega ):={\widetilde{H}}_{1}:=\big (L^{2,3}(\Omega ),\langle \varepsilon \,\cdot \,,\,\cdot \,\rangle _{L^{2,3}(\Omega )}\big )\) and similarly \(L^{2,3}_{\mu }(\Omega ):={\widetilde{H}}_{2}\) as well as \({\widetilde{H}}_{0}={\widetilde{H}}_{3}=H_{0}=H_{3}=L^{2}(\Omega )\). We look at
i.e., the de Rham complex, cf. (6.1),
Lemmas 5.1, 5.2, and Theorem 5.3 show that the compactness properties, the dimensions of the kernels and cohomology groups, the maximal compactness, and the Fredholm indices of the de Rham complex do not depend on the material weights \(\varepsilon \) and \(\mu \). More precisely,
-
\(\dim \big ({\text {ker}}(\mathring{\text {curl}})\cap \big (\varepsilon ^{-1}{\text {ker}}({{\,\mathrm{div}\,}})\big )\big ) \!=\!\dim \big ({\text {ker}}(\mathring{\text {curl}})\cap {\text {ker}}({{\,\mathrm{div}\,}})\big ) \!=\!\dim {\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )=m-1\),
-
\(\dim \big (\big (\mu ^{-1}{\text {ker}}(\mathring{{{\,\mathrm{div}\,}}})\big )\cap {\text {ker}}(\text {curl})\big ) =\dim \big ({\text {ker}}(\mathring{{{\,\mathrm{div}\,}}})\cap {\text {ker}}(\text {curl})\big ) =\dim {\mathcal {H}}_{N}^{\mathsf {Rhm}}(\Omega )=p\),
-
\({{\,\mathrm{dom}\,}}(\mathring{\text {curl}})\cap \big (\varepsilon ^{-1}{{\,\mathrm{dom}\,}}({{\,\mathrm{div}\,}})\big ) \hookrightarrow L^{2,3}_{\varepsilon }(\Omega )\) compactly
\(\Leftrightarrow \;{{\,\mathrm{dom}\,}}(\mathring{\text {curl}})\cap {{\,\mathrm{dom}\,}}({{\,\mathrm{div}\,}}) \hookrightarrow L^{2,3}(\Omega )\) compactly,
-
\(\big (\mu ^{-1}{{\,\mathrm{dom}\,}}(\mathring{{{\,\mathrm{div}\,}}})\big )\cap {{\,\mathrm{dom}\,}}(\text {curl}) \hookrightarrow L^{2,3}_{\mu }(\Omega )\) compactly
\(\Leftrightarrow \;{{\,\mathrm{dom}\,}}(\mathring{{{\,\mathrm{div}\,}}})\cap {{\,\mathrm{dom}\,}}(\text {curl}) \hookrightarrow L^{2,3}(\Omega )\) compactly,
-
\((\mathring{{{\,\mathrm{grad}\,}}},\mu ^{-1}\mathring{\text {curl}},\mathring{{{\,\mathrm{div}\,}}}\mu )\) is maximal compact iff \((\mathring{{{\,\mathrm{grad}\,}}},\mathring{\text {curl}},\mathring{{{\,\mathrm{div}\,}}})\) is maximal compact,
-
\(-{{\,\mathrm{ind}\,}}({\widetilde{{\mathcal {D}}}}^{\mathsf {Rhm}})^{*}={{\,\mathrm{ind}\,}}{\widetilde{{\mathcal {D}}}}^{\mathsf {Rhm}} ={{\,\mathrm{ind}\,}}{\mathcal {D}}^{\mathsf {Rhm}}=p-m-n+1\).
At this point, see Lemma 3.5, Corollary 3.6, and (6.5), we note that the kernels and ranges are given by
Finally, Corollary 4.1 yields additional results for the corresponding reduced operators
Corollary 6.12
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded weak Lipschitz domain with continuous boundary. Then
are compact. Furthermore,
are continuous and, equivalently, the Friedrichs–Poincaré type estimate
holds for all (E, u) in
or (u, E) in
with some optimal constant \(c_{{\mathcal {D}}^{\mathsf {Rhm}}}>0\).
Note that the latter estimate is an additive combination of the well-known Friedrichs–Poincaré estimates for \({{\,\mathrm{grad}\,}}\) and the well-known Maxwell estimates for \((\text {curl},{{\,\mathrm{div}\,}})\).
6.3 The Dirac operator
In this section, we flag up a relationship of the Dirac operator and Picard’s extended Maxwell system. Let the assumptions of Theorem 6.8 be satisfied. The extended Maxwell operator is an operator that is surprisingly close to the Dirac operator, see [45]. We shall carry out this construction in the following. Recall from Remark 6.9 that Picard’s extended Maxwell system is given by the operator
Next, we introduce the Dirac operator. For this, we define the Pauli matrices
Setting
we define the Dirac operator
We have not specified the domain of definition of \({\mathcal {Q}}\), yet. For now, we only record that \(C^{\infty ,2}_{c}(\Omega )\subseteq {{\,\mathrm{dom}\,}}{\mathcal {Q}}\), and the domain of definition of \({\mathcal {Q}}\) corresponding to \({\mathcal {M}}\) is provided below, see also Proposition 6.13. We introduce the unitary operators from \(L^{2,4}(\Omega )\) into itself
Then the operators \({\mathcal {L}}\) (Dirac operator) and \({\mathcal {M}}\) (Picard’s extended Maxwell operator) are unitarily equivalent. More precisely, we have with V from Proposition 6.13
and, consequently, \({\mathcal {Q}}\) with domain \({{\,\mathrm{dom}\,}}(V^{*}U^{*}{\mathcal {D}}WV)={{\,\mathrm{dom}\,}}({\mathcal {D}}WV)\) is a Fredholm operator. Moreover, we have \({{\,\mathrm{ind}\,}}{\mathcal {L}}=0\) and
We conclude this section by stating the missing proposition used above. The proofs of which are straightforward and will therefore be omitted. In a similar fashion, they can be found in [45]. For the next result we use \(L^{2}_{{\mathbb {R}}}(\Omega )\) and \(L^{2}_{{\mathbb {C}}}(\Omega )\) to denote the Hilbert space \(L^{2}(\Omega )\) with the reals and the complex numbers as respective underlying field.
Proposition 6.13
(Realification of \({\mathcal {L}}\)) It holds:
- (i):
-
\(V:L^{2}_{{\mathbb {C}}}(\Omega )\rightarrow L^{2,2}_{{\mathbb {R}}}(\Omega )\) with \(Vf:=(\mathfrak {R}f,\mathfrak {I}f)\) is unitary.
- (ii):
-
\(V i V^{*} =\begin{pmatrix} 0 &{}\quad 1\\ -1 &{}\quad 0\end{pmatrix}\).
- (iii):
-
\({\widetilde{{\mathcal {Q}}}}:= V{\mathcal {Q}}V^{*} =\partial _{1} \begin{pmatrix} 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad 1\\ 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\end{pmatrix} +\partial _{2}\begin{pmatrix} 0 &{}\quad 0 &{}\quad 0 &{}\quad -1\\ 0 &{}\quad 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ -1 &{}\quad 0 &{}\quad 0 &{}\quad 0\end{pmatrix} +\partial _{3}\begin{pmatrix} 1 &{}\quad 0 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad -1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad 0 &{}\quad -1\end{pmatrix}\) with \({{\,\mathrm{dom}\,}}{\widetilde{{\mathcal {Q}}}} = V{{\,\mathrm{dom}\,}}{\mathcal {Q}}\,V^{*}\).
7 The first biharmonic complex and its indices
In this section, we focus on our first main result and properly define the operators involved in the formulation of Theorem 1.1. Thus, we introduce the first biharmonic complex (see [37, 38]) constructed for biharmonic problems and general relativity, but also relevant in problems for elasticity. It will be interesting to see that the differential operator is apparently of mixed order rather than just of first order. It is worth noting that the apparently leading order term is not dominating the lower order differential operators.
Definition 7.1
Let \(\Omega \subseteq {\mathbb {R}}^3\) be an open set. We put
and further define the densely defined and closed linear operators
We shall apply the index theorem in the following situation of the first biharmonic complex:
Introducing the space of piecewise Raviart–Thomas fields by
for \(\mathsf {cc}(\Omega )\) see (6.4), we can write the first biharmonic complex as
The foundation of the index theorem to hold is the following compactness result established by Pauly and Zulehner. Note that it holds \({{\,\mathrm{dom}\,}}(\mathring{{\text {Gradgrad}}})=H^{2}_{0}(\Omega )\) and \({{\,\mathrm{dom}\,}}({{\,\mathrm{devGrad}\,}})=H^{1,3}(\Omega )\).
Theorem 7.2
([38, Lemma 3.22, Theorem 3.23]) Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then \((\mathring{{\text {Gradgrad}}},\mathring{{{\,\mathrm{Curl}\,}}}_{{\mathbb {S}}},\mathring{{{\,\mathrm{Div}\,}}}_{{\mathbb {T}}})\) is a maximal compact Hilbert complex.
We observe and define
The dimensions of the cohomology groups are given as follows.
Theorem 7.3
Let \(\Omega \subseteq {\mathbb {R}}^3\) be open and bounded with continuous boundary. Moreover, suppose Assumption 10.3. Then
Proof
We postpone the proof to Sects. 11.2 and 12.2. \(\square \)
The proper formulation of the first main result, Theorem 1.1, reads as follows.
Theorem 7.4
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then \({\mathcal {D}}^{\mathsf {bih},1}\) is a Fredholm operator with index
If additionally Assumption 10.3 holds, then
Proof
Using Theorem 7.2, we apply Theorem 3.8 together with (7.3) and the observations
see [38, Lemma 3.2, Lemma 3.3], and Theorem 7.3. \(\square \)
Remark 7.5
By Theorem 3.8 the adjoint \(({\mathcal {D}}^{\mathsf {bih},1})^{*}\) is Fredholm as well with index simply given by \({{\,\mathrm{ind}\,}}({\mathcal {D}}^{\mathsf {bih},1})^{*}=-{{\,\mathrm{ind}\,}}{\mathcal {D}}^{\mathsf {bih},1}\). Similar to Remark 6.9 we define the extended first biharmonic operator
with \(({\mathcal {M}}^{\mathsf {bih},1})^{*}=-{\mathcal {M}}^{\mathsf {bih},1}\) and \({{\,\mathrm{ind}\,}}{\mathcal {M}}^{\mathsf {bih},1}=0\). Moreover, \(\dim {\text {ker}}{\mathcal {M}}^{\mathsf {bih},1}=4(n+m+p-1)\) as \({\text {ker}}{\mathcal {M}}^{\mathsf {bih},1} =\mathsf {RT}_{\mathsf {pw}}\times {\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {bih},1}(\Omega )\times {\mathcal {H}}_{N,{\mathbb {T}}}^{\mathsf {bih},1}(\Omega )\times \{0\}\).
Variable Coefficients and Poincaré–Friedrichs Type Inequalities. Inhomogeneous and anisotropic media may also be considered for the first biharmonic complex, cf. Remark 6.11.
Remark 7.6
Let \(\lambda _{0}:={{\,\mathrm{Id}\,}}\), \(\lambda _{3}:={{\,\mathrm{Id}\,}}\), \(\lambda _{1}:=\varepsilon :\Omega \rightarrow {\mathbb {R}}^{3\times 3\times 3\times 3}\), and \(\lambda _{2}:=\mu :\Omega \rightarrow {\mathbb {R}}^{3\times 3\times 3\times 3}\) be \(L^{\infty }(\Omega )\)-tensor fields such that the induced respective operators on \(L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\) and \(L^{2,3\times 3}_{{\mathbb {T}}}(\Omega )\) are symmetric and strictly positive definite. Moreover, let us introduce
and similarly \(L^{2,3\times 3}_{{\mathbb {T}},\mu }(\Omega ):={\widetilde{H}}_{2}\) as well as \({\widetilde{H}}_{0}=H_{0}=L^{2}(\Omega )\), \({\widetilde{H}}_{3}=H_{3}=L^{2,3}(\Omega )\). We look at
i.e., the first biharmonic complex, cf. (7.2),
Lemma 5.1, Lemma 5.2, and Theorem 5.3 show that the compactness properties, the dimensions of the kernels and cohomology groups, the maximal compactness, and the Fredholm indices of the first biharmonic complex do not depend on the material weights \(\varepsilon \) and \(\mu \). More precisely,
Note that the kernels and ranges are given by
see Lemma 3.5, Corollary 3.6, and (7.4). Corollary 4.1 shows additional results for the corresponding reduced operators
Corollary 7.7
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then
are compact. Furthermore,
are continuous and, equivalently, the Friedrichs–Poincaré type estimates
hold for all (T, u) in
for all (v, S) in
with some optimal constant \(c_{{\mathcal {D}}^{\mathsf {bih},1}}>0\).
8 The second biharmonic complex and its indices
The second major application of the abstract findings in the Sects. 3, 4, and 5 is concerned with the second biharmonic complex. The needed operators are provided next. It is worth recalling the definitions of the operators \({{\,\mathrm{devGrad}\,}}\), \({{\,\mathrm{symCurl}\,}}_{{\mathbb {T}}}\), and \({\text {divDiv}}_{{\mathbb {S}}}\) from Definition 7.1.
Definition 8.1
Let \(\Omega \subseteq {\mathbb {R}}^3\) be an open set. We put
and further define the densely defined and closed linear operators
We shall apply the index theorem in the following situation of the second biharmonic complex:
where we used the space of piecewise first order polynomials (for \(\mathsf {cc}(\Omega )\) see (6.4))
Note that \({{\,\mathrm{dom}\,}}(\mathring{{{\,\mathrm{devGrad}\,}}})=H^{1,3}_{0}(\Omega )\) by [38, Lemma 3.2].
Lemma 8.2
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then the regularity \({{\,\mathrm{dom}\,}}({\text {Gradgrad}})=H^{2}(\Omega )\) holds and there exists \(c>0\) such that for all \(u\in H^{2}(\Omega )\)
Proof
Let \(u\in {{\,\mathrm{dom}\,}}({\text {Gradgrad}})\). Then \({{\,\mathrm{grad}\,}}u\in H^{-1,3}(\Omega )\) and \({{\,\mathrm{Grad}\,}}{{\,\mathrm{grad}\,}}u\in L^{2,3\times 3}(\Omega )\). Nečas’ regularity yields \({{\,\mathrm{grad}\,}}u\in L^{2,3}(\Omega )\) and thus \(u\in H^{1}(\Omega )\) and \({{\,\mathrm{grad}\,}}u\in H^{1,3}(\Omega )\). Hence \(u\in H^{2}(\Omega )\) and by Nečas’ inequality (see [27]) we have
showing the desired estimate. \(\square \)
Theorem 8.3
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then the second biharmonic complex \((\mathring{{{\,\mathrm{devGrad}\,}}},\mathring{{{\,\mathrm{symCurl}\,}}}_{{\mathbb {T}}},\mathring{{\text {divDiv}}}_{{\mathbb {S}}})\) is a maximal compact Hilbert complex.
Proof
The assertions can be shown by using the ‘FA-ToolBox’ from [34,35,36, 38,39,40,41]. In particular, the crucial compact embeddings can be shown by the same techniques used in the proof of [40, Theorem 4.7]. \(\square \)
We observe and define
Theorem 8.4
Let \(\Omega \subseteq {\mathbb {R}}^3\) be open and bounded with continuous boundary. Moreover, suppose Assumption 10.3. Then
Proof
We postpone the proof to Sects. 11.3 and 12.3. \(\square \)
Theorem 8.5
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then \({\mathcal {D}}^{\mathsf {bih},2}\) is a Fredholm operator with index
If additionally Assumption 10.3 holds, then
Proof
Using Theorem 8.3 apply Theorem 3.8 together with (8.3), the observations
by using [38, Lemma 3.2 (i)], and Theorem 8.4. \(\square \)
Remark 8.6
By Theorem 3.8 the adjoint \(({\mathcal {D}}^{\mathsf {bih},2})^{*}\) is Fredholm as well with index simply given by \({{\,\mathrm{ind}\,}}({\mathcal {D}}^{\mathsf {bih},2})^{*}=-{{\,\mathrm{ind}\,}}{\mathcal {D}}^{\mathsf {bih},2}\). Similar to Remark 6.9 and Remark 7.5 we define the extended second biharmonic operator
with \(({\mathcal {M}}^{\mathsf {bih},2})^{*}=-{\mathcal {M}}^{\mathsf {bih},2}\) and \({{\,\mathrm{ind}\,}}{\mathcal {M}}^{\mathsf {bih},2}=0\). Moreover, \(\dim {\text {ker}}{\mathcal {M}}^{\mathsf {bih},2}=4(n+m+p-1)\) as \({\text {ker}}{\mathcal {M}}^{\mathsf {bih},2} ={\mathsf {P}}_{\mathsf {pw}}^{1}\times {\mathcal {H}}_{D,{\mathbb {T}}}^{\mathsf {bih},2}(\Omega )\times {\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {bih},2}(\Omega )\times \{0\}\).
Variable Coefficients and Poincaré–Friedrichs Type Inequalities. Inhomogeneous and anisotropic media may also be considered for the second biharmonic complex, cf. Remark 6.11 and Remark 7.6.
Remark 8.7
Recall the notations from Remark 7.6 and set \(\lambda _{0}:={{\,\mathrm{Id}\,}}\), \(\lambda _{3}:={{\,\mathrm{Id}\,}}\), \(\lambda _{1}:=\varepsilon \), \(\lambda _{2}:=\mu \), and \({\widetilde{H}}_{1}:=L^{2,3\times 3}_{{\mathbb {T}},\varepsilon }(\Omega )\), \({\widetilde{H}}_{2}:=L^{2,3\times 3}_{{\mathbb {S}},\mu }(\Omega )\), \({\widetilde{H}}_{0}=H_{0}=L^{2,3}(\Omega )\), \({\widetilde{H}}_{3}=H_{3}=L^{2}(\Omega )\). We look at
i.e., the second biharmonic complex, cf. (8.1),
Lemmas 5.1, 5.2, and Theorem 5.3 show that the compactness properties, the dimensions of the kernels and cohomology groups, the maximal compactness, and the Fredholm indices of the second biharmonic complex do not depend on the material weights \(\varepsilon \) and \(\mu \). More precisely,
Note that the kernels and ranges are given by
see Lemma 3.5, Corollary 3.6, and (8.4). Corollary 4.1 shows additional results for the corresponding reduced operators
Corollary 8.8
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then
are compact. Furthermore,
are continuous and, equivalently, the Friedrichs-Poincaré type estimates
hold for all (S, v) in
for all (u, T) in
with some optimal constant \(c_{{\mathcal {D}}^{\mathsf {bih},2}}>0\).
9 The elasticity complex and its indices
This section is devoted to adapt our main results Theorems 1.1, 7.4, and 8.5, to the elasticity complex, see [39,40,41] for details. Its elasticity differential operator is of mixed order as well, this time in the center of the complex. As before for the biharmonic operators, the leading order term is not dominating the lower order differential operators.
Definition 9.1
Let \(\Omega \subseteq {\mathbb {R}}^3\) be an open set. We put
and further define the densely defined and closed linear operators
We want to apply the index theorem in the following situation of the elasticity complex:
The foundation of the index theorem to follow is the following compactness result established in [39,40,41]. Note that we have by Korn’s inequalities \({{\,\mathrm{dom}\,}}(\mathring{{{\,\mathrm{symGrad}\,}}})=H^{1,3}_{0}(\Omega )\) and \({{\,\mathrm{dom}\,}}({{\,\mathrm{symGrad}\,}})=H^{1,3}(\Omega )\).
Theorem 9.2
([40, Theorem 4.7]) Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then \((\mathring{{{\,\mathrm{symGrad}\,}}},\mathring{{\text {CurlCurl}}}{}^{\top }_{{\mathbb {S}}},\mathring{{{\,\mathrm{Div}\,}}}_{{\mathbb {S}}})\) is a maximal compact Hilbert complex.
We observe and define
The dimensions of the cohomology groups are given as follows.
Theorem 9.3
Let \(\Omega \subseteq {\mathbb {R}}^3\) be open and bounded with continuous boundary. Moreover, suppose Assumption 10.3. Then
Proof
We postpone the proof to the Sects. 11.4 and 12.4. \(\square \)
Let us introduce the space of piecewise rigid motions by (for \(\mathsf {cc}(\Omega )\) see (6.4))
Theorem 9.4
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then \({\mathcal {D}}^{\mathsf {ela}}\) is a Fredholm operator with index
If additionally Assumption 10.3 holds, then
Proof
Using Theorem 9.2 apply Theorem 3.8 together with (9.2), the observations
see [39, Lemma 3.2], and Theorem 9.3. \(\square \)
Remark 9.5
By Theorem 3.8 the adjoint \(({\mathcal {D}}^{\mathsf {ela}})^{*}\) is Fredholm as well with index simply given by \({{\,\mathrm{ind}\,}}({\mathcal {D}}^{\mathsf {ela}})^{*}=-{{\,\mathrm{ind}\,}}{\mathcal {D}}^{\mathsf {ela}}\). Similar to Remark 6.9, Remark 7.5, and Remark 8.6 we define the extended elasticity operator
with \(({\mathcal {M}}^{\mathsf {ela}})^{*}=-{\mathcal {M}}^{\mathsf {ela}}\) and \({{\,\mathrm{ind}\,}}{\mathcal {M}}^{\mathsf {ela}}=0\). Moreover, \(\dim {\text {ker}}{\mathcal {M}}^{\mathsf {ela}}=6(n+m+p-1)\) as \({\text {ker}}{\mathcal {M}}^{\mathsf {ela}} =\mathsf {RM}_{\mathsf {pw}}\times {\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )\times {\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )\times \{0\}\).
Variable coefficients and Poincaré–Friedrichs type inequalities. Inhomogeneous and anisotropic media may also be considered for the elasticity complex, cf. Remarks 6.11, 7.6, and 8.7.
Remark 9.6
Recall the notations from Remark 7.6 and Remark 8.7 and set \(\lambda _{0}:={{\,\mathrm{Id}\,}}\), \(\lambda _{3}:={{\,\mathrm{Id}\,}}\), \(\lambda _{1}:=\varepsilon \), \(\lambda _{2}:=\mu \), and \({\widetilde{H}}_{3}={\widetilde{H}}_{0}=H_{3}=H_{0}=L^{2,3}(\Omega )\), \({\widetilde{H}}_{1}:=L^{2,3\times 3}_{{\mathbb {S}},\varepsilon }(\Omega )\), \({\widetilde{H}}_{2}:=L^{2,3\times 3}_{{\mathbb {S}},\mu }(\Omega )\). We look at
i.e., the elasticity complex, cf. (9.1),
Lemmas 5.1, 5.2, and Theorem 5.3 show that the compactness properties, the dimensions of the kernels and cohomology groups, the maximal compactness, and the Fredholm indices of the elasticity complex do not depend on the material weights \(\varepsilon \) and \(\mu \). More precisely,
Note that the kernels and ranges are given by
see Lemma 3.5, Corollary 3.6, and (9.4). Corollary 4.1 shows additional results for the corresponding reduced operators
Corollary 9.7
Let \(\Omega \subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Then
are compact. Furthermore,
are continuous and, equivalently, the Friedrichs–Poincaré type estimate
holds for all (S, v) in
or (v, S) in
with some optimal constant \(c_{{\mathcal {D}}^{\mathsf {ela}}}>0\).
10 The main topological assumptions
In Theorems 6.6, 7.3, 8.4, and 9.3 we have seen that the dimensions of the harmonic Dirichlet and Neumann fields are given by the topological invariants of the open and bounded set \(\Omega \) and its complement
i.e., by
-
n, the number of connected components \(\Omega _{k}\) of \(\Omega \), i.e., \(\Omega =\dot{\bigcup }_{k=1}^{n}\,\Omega _{k}\),
-
m, the number of connected components \(\Xi _{\ell }\) of \(\Xi \), i.e., \(\Xi =\dot{\bigcup }_{\ell =0}^{m-1}\,\Xi _{\ell }\),
-
p, the number of handles of \(\Omega \), see Assumption 10.3.
Note that \(\mathsf {cc}(\Omega )=\{\Omega _{1},\dots ,\Omega _{n}\}\) and \(\mathsf {cc}(\Xi )=\{\Xi _{0},\dots ,\Xi _{m-1}\}\). We have claimed
The concluding sections of this manuscript are devoted to provide the corresponding proofs in detail. For the de Rham complex we follow in close lines the arguments of Picard in [42] introducing some simplifications for bounded domains and trivial material tensors \(\varepsilon \) and \(\mu \). These ideas will be adapted and modified for the proofs of the corresponding results of the other Hilbert complexes.
Assumption 10.1
\(\Omega \subseteq {\mathbb {R}}^{3}\) is open and bounded with segment property, i.e., \(\Omega \) has a continuous boundary \(\Gamma :=\partial \Omega \), see Remark 6.7.
Assumption 10.2
\(\Omega \subseteq {\mathbb {R}}^{3}\) is open, bounded, and \(\Gamma \) is strong Lipschitz.
In view of Assumption 10.1 and Assumption 10.2 we note:
-
Assumption 10.1 guarantees that \(m,n\in {\mathbb {N}}\) are well-defined. In particular, we have \(\mathrm {int}\,\Xi _{\ell }\ne \emptyset \) for all \(\ell \in \{0,\dots ,m-1\}\).
-
Assumption 10.2 simplifies some arguments, in particular, all ranges in the crucial Helmholtz type decompositions used in our proofs are closed, cf. Remarks 12.4, 12.16, 12.26, and 12.36. We emphasise that all our results presented in the following still hold with Assumption 10.2 replaced by the weaker Assumption 10.1. In this case, however, the computation (and verification of the existence of) the Fredholm index in the sections above is more involved. In fact, it is not clear if the mentioned ranges are closed and in some of our arguments we need to use some additional density and approximation arguments.
-
Our results concerning the bases and dimensions of the generalised Dirichlet and Neumann fields extend naturally to exterior domains, i.e., domains with bounded complement \(\Xi \). For simplicity and to avoid even longer and more technical proofs we restrict ourselves to the case of bounded domains \(\Omega \) here.
The key topological assumptions to be satisfied by \(\Omega \) to compute a basis for the Neumann fields and for p to be well-defined, is described in detail next. For this, we recall the construction in [42].
Assumption 10.3
([42, Section 1]) Let \(\Omega \subseteq {\mathbb {R}}^3\) be open and bounded. There are \(p\in {\mathbb {N}}_{0}\) piecewise \(C^1\)-curves \(\zeta _{j}\) and p \(C^{2}\)-surfaces \(F_{j}\), \(j\in \{1,\ldots , p\}\), with the following properties:
- (A1):
-
The curves \(\zeta _{j}\), \(j\in \{1,\ldots , p\}\), are pairwise disjoint and given any closed piecewise \(C^1\)-curve \(\zeta \) in \(\Omega \) there exists uniquely determined \(\alpha _{j}\in {\mathbb {Z}}\), \(j\in \{1,\ldots , p\}\), such that for all \(\Phi \in {\text {ker}}(\text {curl})\) being continuously differentiable we have
$$\begin{aligned} \int _{\zeta }\langle \Phi ,{{\,\mathrm{d}\,}}\lambda \rangle = \sum _{j=1}^{p}\alpha _{j}\int _{\zeta _{j}}\langle \Phi ,{{\,\mathrm{d}\,}}\lambda \rangle . \end{aligned}$$ - (A2):
-
\( F_{j}\), \(j\in \{1,\ldots , p\}\), are pairwise disjoint and \( F_{j}\cap \zeta _{k}\) is a singleton, if \(j=k\), and empty, if \(j\ne k\).
- (A3):
-
If \(\Omega _{c}\in \mathsf {cc}(\Omega )\), i.e., \(\Omega _{c}\) is a connected component of \(\Omega \), then \(\Omega _{c}{\setminus }\bigcup _{j=1}^{p} F_{j}\) is simply connected.
p is called the topological genus of \(\Omega \) and the curves \(\zeta _{j}\) are said to represent a basis of the respective homology group of \(\Omega \).
It is worth mentioning the following local regularity results for the Dirichlet and Neumann fields (see Lemma 12.2 below), which are crucial for the construction of the Neumann fields,
In particular, all Dirichlet and Neumann fields of the respective cohomology groups are continuous and square integrable.
11 The construction of the Dirichlet fields
Let us denote the unbounded connected component of \(\Xi \) by \(\Xi _{0}\) and its boundary by \(\Gamma _{0}:=\partial \Xi _{0}\). The remaining connected components of \(\Xi \) are \(\Xi _{1},\dots ,\Xi _{m-1}\) with boundaries \(\Gamma _{\ell }:=\partial \Xi _{\ell }\). Note that none of \(\Gamma _{0},\dots ,\Gamma _{m-1}\) need to be connected. Furthermore, let us introduce an open (and bounded) ball \(B\supset {\overline{\Omega }}\) and set \({\widetilde{\Xi }}_{0}:=B\cap \Xi _{0}\). Then the connected components of \(B{\setminus }{\overline{\Omega }}\) are \({\widetilde{\Xi }}_{0}\) and \(\Xi _{1},\dots ,\Xi _{m-1}\). Moreover, let
with disjoint supports such that \(\xi _{\ell }=0\) in a neighbourhood of \(\Xi _{0}\) and in a neighbourhood of \(\Xi _{k}\) for all \(k\in \{1,\dots ,m-1\}\), \(k\ne \ell \), as well as \(\xi _{\ell }=1\) in a neighbourhood of \(\Xi _{\ell }\). In particular, \(\xi _{\ell }=0\) in a neighbourhood of \(\Gamma _{0}\) and in a neighbourhood of \(\Gamma _{k}\) for all \(k\in \{1,\dots ,m-1\}\), \(k\ne \ell \), and \(\xi _{\ell }=1\) in a neighbourhood of \(\Gamma _{\ell }\). Theses indicator type functions \(\xi _{\ell }\) will be used to construct a basis for the respective Dirichlet fields.
11.1 Dirichlet vector fields of the classical de Rham complex
In this section, we rephrase the core arguments of [42] in the simplified setting of bounded domains and trivial materials \(\varepsilon \) and \(\mu \). In order to highlight the apparent similarities and to motivate our rationale carried out for more involved situations later on, we shall present the construction for Dirichlet fields (and similarly for Neumann fields) in a seemingly great detail.
For the de Rham complex, see also (3.3) and (3.4), we have the orthogonal decompositions
Remark 11.1
We have \({{\,\mathrm{dom}\,}}(\mathring{{{\,\mathrm{grad}\,}}},\Omega )=H^{1}_{0}(\Omega )\). Moreover, the range in (11.2) is closed due to the Friedrichs estimate
which follows from Assumption 10.1. We recall that for the Friedrichs estimate to hold it suffices to assume that \(\Omega \) is open and bounded only.
Let us denote by \(\pi :L^{2,3}(\Omega )\rightarrow {\text {ker}}({{\,\mathrm{div}\,}},\Omega )\) the orthogonal projector along \({{\,\mathrm{ran}\,}}(\mathring{{{\,\mathrm{grad}\,}}},\Omega )\) onto \({\text {ker}}({{\,\mathrm{div}\,}},\Omega )\), which is well-defined according to (11.2). Moreover, we observe by (11.2) that \(\pi ({\text {ker}}(\mathring{\text {curl}},\Omega ))={\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )\). Recall \(\xi _{\ell }\) from (11.1). Then for \(\ell \in \{1,\dots ,m-1\}\)
Again relying on (11.2) (and Remark 11.1) for all \(\ell \in \{1,\ldots ,m-1\}\), we find uniquely determined \(\psi _{\ell }\in H^{1}_{0}(\Omega )\) such that
We will show that
defines a basis of \({\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )\). The first step for showing this statement is the next lemma.
Lemma 11.2
Let Assumption 10.1 be satisfied. Then \({\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )={{\,\mathrm{lin}\,}}{\mathcal {B}}^{\mathsf {Rhm}}_{D}\).
Proof
Let \(H\in {\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )={\text {ker}}(\mathring{\text {curl}},\Omega )\cap {\text {ker}}({{\,\mathrm{div}\,}},\Omega )\). In particular, by the homogeneous boundary condition its extension by zero, \({\widetilde{H}}\), to B belongs to \({\text {ker}}(\mathring{\text {curl}},B)\). As B is topologically trivial (and smooth and bounded), there exists (a unique) \(u\in H^{1}_{0}(B)\) such that \({{\,\mathrm{grad}\,}}u={\widetilde{H}}\) in B, see, e.g., [38, Lemma 2.24]. As \({{\,\mathrm{grad}\,}}u={\widetilde{H}}=0\) in \(B{\setminus }{\overline{\Omega }}\), u must be constant in each connected component \({\widetilde{\Xi }}_{0},\Xi _{1},\dots ,\Xi _{m-1}\) of \(B{\setminus }{\overline{\Omega }}\). Due to the homogenous boundary condition at \(\partial \!B\), u vanishes in \({\widetilde{\Xi }}_{0}\). Therefore, \(H={{\,\mathrm{grad}\,}}u\) in \(\Omega \) and \(u\in H^{1}_{0}(B)\) such that \(u|_{{\widetilde{\Xi }}_{0}}=0\) and \(u|_{\Xi _{\ell }}=:\alpha _{\ell }\in {\mathbb {R}}\) for all \(\ell \in \{1,\dots ,m-1\}\). Let us consider
with \(u_{\ell }\) from (11.3). The extension by zero of \(\psi _{\ell }\), \({\widetilde{\psi }}_{\ell }\), to the whole of B belongs to \(H^{1}_{0}(B)\). Hence as an element of \(H^{1}(B)\) we see that
vanishes in \(\Xi _{\ell }\) for all \(\ell \in \{0,\ldots ,m-1\}\). Thus \({\widehat{u}}={\widehat{u}}_{B}|_{\Omega }\in H^{1}_{0}(\Omega )\) by Assumption 10.1, and we compute
finishing the proof. \(\square \)
Before we show linear independence of the set \({\mathcal {B}}^{\mathsf {Rhm}}_{D}\), we highlight the possibility of determining the functions constructed here by solving certain PDEs. This can be used for numerically determining a basis for \({\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )\).
Remark 11.3
(Characterisation by PDEs)
- (i):
-
It is not difficult to see that \(\psi _{\ell }\in H^{1}_{0}(\Omega )\) as in (11.3) can be found as the solution of the standard variational formulation
$$\begin{aligned} \forall \,\phi \in H^{1}_{0}(\Omega )\qquad \langle {{\,\mathrm {grad}\,}}\psi _{\ell },{{\,\mathrm {grad}\,}}\phi \rangle _{L^{2,3}(\Omega )} =\langle {{\,\mathrm {grad}\,}}\xi _{\ell },{{\,\mathrm {grad}\,}}\phi \rangle _{L^{2,3}(\Omega )}, \end{aligned}$$i.e., \(\psi _{\ell }=\Delta _{D}^{-1}\Delta \xi _{\ell }\), where \(\Delta _{D}={{\,\mathrm{div}\,}}\mathring{{{\,\mathrm{grad}\,}}}\) denotes the Laplacian with standard homogeneous Dirichlet boundary conditions on \(\Omega \).
- (ii):
-
As a consequence of (i) and (11.3), we obtain \(u_{\ell } =\xi _{\ell }-\psi _{\ell } =(1-\Delta _{D}^{-1}\Delta )\xi _{\ell }\in H^{1}(\Omega )\) and
$$\begin{aligned} {{\,\mathrm{grad}\,}}u_{\ell }&=(1-{{\,\mathrm{grad}\,}}\Delta _{D}^{-1}{{\,\mathrm{div}\,}}){{\,\mathrm{grad}\,}}\xi _{\ell }. \end{aligned}$$Let us also mention that for \(\ell \in \{1,\ldots ,m-1\}\), \(u_{\ell }\) solves in classical terms the Dirichlet Laplace problem
$$\begin{aligned} \begin{aligned} -\Delta u_{\ell }=-{{\,\mathrm{div}\,}}{{\,\mathrm{grad}\,}}u_{\ell }&=0&\text {in }\Omega ,\\ u_{\ell }&=1&\text {on }\Gamma _{\ell },\\ u_{\ell }&=0&\text {on }\Gamma _{k},\, k\in \{0,\dots ,m-1\}{\setminus }\{\ell \}, \end{aligned} \end{aligned}$$(11.5)which is uniquely solvable. In particular, for all \(\ell \in \{1,\dots ,m-1\}\), \(u_{\ell }=0\) on \(\Gamma _{0}\).
- (iii):
-
u (representing \(H={{\,\mathrm{grad}\,}}u\)) constructed in the proof of Lemma 11.2 solves the linear Dirichlet Laplace problem
$$\begin{aligned} -\Delta u=-{{\,\mathrm{div}\,}}{{\,\mathrm{grad}\,}}u=-{{\,\mathrm{div}\,}}H&=0&\text {in }\Omega ,\\ u&=0&\text {on }\Gamma _{0},\\ u&=\alpha _{\ell }\in {\mathbb {R}}&\text {on }\Gamma _{\ell },\,\ell \in \{1,\dots ,m-1\}, \end{aligned}$$which is uniquely solvable as long as the constants, \(\alpha _{\ell }\), are prescribed.
Lemma 11.4
Let Assumption 10.1 be satisfied. Then \({\mathcal {B}}^{\mathsf {Rhm}}_{D}\) is linearly independent.
Proof
Let \(\alpha _{\ell }\in {\mathbb {R}}\) for all \(\ell \in \{1,\ldots ,m-1\}\) such that
Then \({{\,\mathrm{grad}\,}}u=0\) in \(\Omega \), i.e., u is constant in each connected component of \(\Omega \). We show \(u=0\). Since \(\psi _{\ell }\in H_{0}^1(\Omega )\) and \(\xi _{\ell }\in H_{0}^1(B)\) we can extend \(u_{\ell }=\xi _{\ell }-\psi _{\ell }\) from (11.3) to B by setting
Note \({\widetilde{u}}_{\ell }\in H_{0}^1(B)\). Moreover, for all \(\ell \in \{1,\ldots ,m-1\}\), we have \({\widetilde{u}}_{\ell }=\xi _{\ell }=1\) in \(\Xi _{\ell }\) and \({\widetilde{u}}_{\ell }=\xi _{\ell }=0\) in \({\widetilde{\Xi }}_{0}\cup \bigcup _{k\in \{1,\ldots ,m-1\}{\setminus }\{\ell \}}\Xi _{k}\). Then
with \({\widetilde{u}}=0\) in \({\widetilde{\Xi }}_{0}\) and \({{\,\mathrm{grad}\,}}{\widetilde{u}}=0\) in \(B{\setminus }{\overline{\Omega }}\) as well as \({{\,\mathrm{grad}\,}}{\widetilde{u}}={{\,\mathrm{grad}\,}}u=0\) in \(\Omega \) by assumption. Hence, \({{\,\mathrm{grad}\,}}{\widetilde{u}}=0\) in B, showing \({\widetilde{u}}=0\) in B. In particular, \(u=0\) in \(\Omega \), and \(\alpha _{\ell }={\widetilde{u}}|_{\Xi _{\ell }}=0\) for all \(\ell \in \{1,\dots ,m-1\}\), finishing the proof. \(\square \)
Theorem 11.5
Let Assumption 10.1 be satisfied. Then \(\dim {\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )=m-1\) and a basis of \({\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )\) is given by (11.4).
Proof
Use Lemmas 11.2 and 11.4. \(\square \)
11.2 Dirichlet tensor fields of the first biharmonic complex
For the first biharmonic complex, see also (3.3), (3.5), and (11.2), we have the orthogonal decompositions
Remark 11.6
By [38, Lemma 3.3] we have \({{\,\mathrm{dom}\,}}(\mathring{{\text {Gradgrad}}},\Omega )=H^{2}_{0}(\Omega )\). Moreover, the range in (11.6) is closed by the Friedrichs type estimate
which holds by Assumption 10.1. Similar to Remark 11.1 it suffices to have \(\Omega \) to be open and bounded.
We define \(\pi :L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\rightarrow {\text {ker}}({\text {divDiv}}_{{\mathbb {S}}},\Omega )\) to be the projector onto \({\text {ker}}({\text {divDiv}}_{{\mathbb {S}}},\Omega )\) along \({{\,\mathrm{ran}\,}}(\mathring{{\text {Gradgrad}}},\Omega )\). By (11.6) we obtain \(\pi \big ({\text {ker}}(\mathring{{{\,\mathrm{Curl}\,}}}_{{\mathbb {S}}},\Omega )]\big )={\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {bih},1}(\Omega )\). We recall the functions \(\xi _{\ell }\) from (11.1). In contrast to the derivation for the de Rham complex, here the second order nature of \({\text {Gradgrad}}\) necessitates the introduction of polynomials \({\widehat{p}}_{j}\) given by
for \(j\in \{1,2,3\}\). We define \(\xi _{\ell ,j}:=\xi _{\ell }{\widehat{p}}_{j}\) for all \(\ell \in \{1,\dots ,m-1\}\) and \(j\in \{0,\dots ,3\}\). In particular, for all \(j\in \{0,\dots ,3\}\) and \(\ell \in \{1,\dots ,m-1\}\) we have \(\xi _{\ell ,j}=0\) in a neighbourhood of \(\Xi _{k}\) for all \(k\in \{0,\dots ,m-1\}{\setminus }\{\ell \}\) and \(\xi _{\ell ,j}={\widehat{p}}_{j}\) in a neighbourhood of \(\Xi _{\ell }\). Then
By (11.6) (and the Friedrichs type estimate for \(\mathring{{\text {Gradgrad}}}\), see Remark 11.6) there exists a unique \(\psi _{\ell ,j}\in H^{2}_{0}(\Omega )\) such that
where
We shall show that
defines a basis of \({\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {bih},1}(\Omega )\). In order to show that the linear hull of \({\mathcal {B}}^{\mathsf {bih},1}_{D}\) generates \({\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {bih},1}(\Omega )\), we cite the following prerequisite.
Lemma 11.7
([38, Theorem 3.10 (i) and Remark 3.11 (i)]) Let \(D\subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Assume D is topologically trivial, i.e., D is simply connected and \({\mathbb {R}}^3{\setminus } D\) is connected. Then
Lemma 11.8
Let Assumption 10.1 be satisfied. Then \({\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {bih},1}(\Omega )={{\,\mathrm{lin}\,}}{\mathcal {B}}^{\mathsf {bih},1}_{D}\).
Proof
We follow in close lines the arguments used in the proof of Lemma 11.2. For this, let \(S\in {\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {bih},1}(\Omega )={\text {ker}}(\mathring{{{\,\mathrm{Curl}\,}}}_{{\mathbb {S}}},\Omega )\cap {\text {ker}}({\text {divDiv}}_{{\mathbb {S}}},\Omega )\). In particular, by the homogeneous boundary condition its extension by zero, \({\widetilde{S}}\), to B belongs to \({\text {ker}}(\mathring{{{\,\mathrm{Curl}\,}}}_{{\mathbb {S}}},B)\). As B is topologically trivial (and smooth and bounded), there exists (a unique) \(u\in H^{2}_{0}(B)\) such that \({\text {Gradgrad}}u={\widetilde{S}}\) in B, see Lemma 11.7 and (11.6) applied to \(\Omega =D=B\). Since \({\text {Gradgrad}}u={\widetilde{S}}=0\) in \(B{\setminus }{\overline{\Omega }}\), u must belong to \({\mathsf {P}}^{1}\), the polynomials of order at most 1, in each connected component \({\widetilde{\Xi }}_{0},\Xi _{1},\dots ,\Xi _{m-1}\) of \(B{\setminus }{\overline{\Omega }}\). Due to the homogenous boundary condition at \(\partial \!B\), u vanishes in \({\widetilde{\Xi }}_{0}\). Therefore, \(S={\text {Gradgrad}}u\) in \(\Omega \) and \(u\in H^{2}_{0}(B)\) is such that \(u|_{{\widetilde{\Xi }}_{0}}=0\) and \(u|_{\Xi _{\ell }}=:p_{\ell }=:\sum _{j=0}^{3}\alpha _{\ell ,j}{\widehat{p}}_{j}\in {\mathsf {P}}^{1}\), for some unique \(\alpha _{\ell ,j}\in {\mathbb {R}}\), for all \(\ell \in \{1,\dots ,m-1\}\) and \(j\in \{0,\ldots ,3\}\). Let us consider
with \(u_{\ell ,j}\) from (11.7). The extension \({\widetilde{\psi }}_{\ell ,j}\) of \(\psi _{\ell ,j}\) by zero to B belongs to \(H^{2}_{0}(B)\). Hence as an element of \(H^{2}(B)\) we see that
vanishes in all \(\Xi _{\ell }\). Thus \({\widehat{u}}={\widehat{u}}_{B}|_{\Omega }\in H^{2}_{0}(\Omega )\) by Assumption 10.1, and we compute
finishing the proof. \(\square \)
Similar to the case of the de Rham complex, we have a look at a possible numerical implementation for the computation of the basis functions. Naturally, the PDEs in question differ from one another quite substantially.
Remark 11.9
(Characterisation by PDEs)
- (i):
-
The functions \(\psi _{\ell ,j}\in H^{2}_{0}(\Omega )\) introduced just above (11.7) can be characterised as solutions by the standard variational formulation
$$\begin{aligned} \forall \,\phi \in H^{1}_{0}(\Omega )\qquad \langle {{\,\mathrm {grad}\,}}\psi _{\ell },{{\,\mathrm {grad}\,}}\phi \rangle _{L^{2,3}(\Omega )} =\langle {{\,\mathrm {grad}\,}}\xi _{\ell },{{\,\mathrm {grad}\,}}\phi \rangle _{L^{2,3}(\Omega )}, \end{aligned}$$i.e., \(\psi _{\ell ,j}=({\Delta }_{DD}^{2})^{-1}\Delta ^{2}\xi _{\ell ,j}\), where \(\Delta ^{2}_{DD}={\text {divDiv}}_{{\mathbb {S}}}\mathring{{\text {Gradgrad}}}\) is the bi-Laplacian with both the functions as well as the derivatives satisfying homogeneous Dirichlet boundary conditions.
- (ii):
-
With the statement in (i) together with (11.7), we deduce for all \(\ell \in \{1,\ldots ,m-1\}\) and \(j\in \{0,\ldots ,3\}\)
$$\begin{aligned} u_{\ell ,j} =\xi _{\ell ,j}-\psi _{\ell ,j} =\big (1-({\Delta }_{DD}^{2})^{-1}\Delta ^{2}\big )\xi _{\ell ,j}\in H^{2}(\Omega ). \end{aligned}$$Hence,
$$\begin{aligned} {\text {Gradgrad}}u_{\ell ,j}&=\big (1-{\text {Gradgrad}}({\Delta }_{DD}^{2})^{-1}{\text {divDiv}}_{{\mathbb {S}}}\big ){\text {Gradgrad}}\xi _{\ell ,j}. \end{aligned}$$For all \(\ell \in \{1,\ldots ,m-1\}\) and \(j\in \{0,\ldots ,3\}\), \(u_{\ell ,j}\) solve in classical terms the biharmonic Dirichlet problem
$$\begin{aligned} \begin{aligned} \Delta ^{2}u_{\ell ,j}={\text {divDiv}}_{{\mathbb {S}}}{\text {Gradgrad}}u_{\ell ,j}&=0&\text {in }\Omega ,\\ u_{\ell ,j}={\widehat{p}}_{j},\quad {{\,\mathrm{grad}\,}}u_{\ell ,j}={{\,\mathrm{grad}\,}}{\widehat{p}}_{j}&=e^{j}&\text {on }\Gamma _{\ell },\\ u_{\ell ,j}=0,\quad {{\,\mathrm{grad}\,}}u_{\ell ,j}&=0&\text {on }\Gamma _{k},\,k\in \{0,\dots ,m-1\}{\setminus }\{\ell \}, \end{aligned} \end{aligned}$$(11.9)which is uniquely solvable. In particular, we have for all \(\ell \in \{1,\dots ,m-1\}\) and all \(j\in \{0,\dots ,3\}\) that \(u_{\ell ,j}=0\) and \({{\,\mathrm{grad}\,}}u_{\ell ,j}=0\) on \(\Gamma _{0}\), where we denote by \(e^{j}\), \(j\in \{1,2,3\}\), the Euclidean unit vectors in \({\mathbb {R}}^{3}\) and set \(e^{0}:=0\in {\mathbb {R}}^{3}\).
- (iii):
-
In classical terms, u (representing \(S={\text {Gradgrad}}u\)) derived in the proof of Lemma 11.8 solves the linear biharmonic Dirichlet problem
$$\begin{aligned} \Delta ^{2}u={\text {divDiv}}_{{\mathbb {S}}}{\text {Gradgrad}}u={\text {divDiv}}_{{\mathbb {S}}}S&=0&\text {in }\Omega ,\\ u=0,\quad {{\,\mathrm{grad}\,}}u&=0&\text {on }\Gamma _{0},\\ u=p_{\ell }\in {\mathsf {P}}^{1},\quad {{\,\mathrm{grad}\,}}u&={{\,\mathrm{grad}\,}}p_{\ell }\in {\mathbb {R}}^3&\text {on }\Gamma _{\ell },\,\ell \in \{1,\dots ,m-1\}, \end{aligned}$$which is uniquely solvable as long as the polynomials, \(p_{\ell }\) in \({\mathsf {P}}^{1}\), are prescribed.
Lemma 11.10
Let Assumption 10.1 be satisfied. Then \({\mathcal {B}}^{\mathsf {bih},1}_{D}\) is linearly independent.
Proof
For \(\ell \in \{1,\ldots ,m-1\}\) and \(j\in \{0,\ldots ,3\}\) we take \(\alpha _{\ell ,j}\in {\mathbb {R}}\) such that
Then \({\text {Gradgrad}}u=0\) in \(\Omega \), i.e., u belongs to \({\mathsf {P}}^{1}_{\mathsf {pw}}\), see (8.2). We will show \(u=0\). For this we extend \(u_{\ell ,j}=\xi _{\ell ,j}-\psi _{\ell ,j}\) (see (11.7)) to B via (note that \(\xi _{\ell ,j}\in H_{0}^{2}(B)\) and \(\psi _{\ell ,j}\in H_{0}^{2}(\Omega )\))
Note that \({\widetilde{u}}_{\ell ,j}\in H_{0}^{2}(B)\). For all \(\ell \in \{1,\ldots ,m-1\}\), \(j\in \{0,\ldots ,3\}\) we have \({\widetilde{u}}_{\ell ,j}=\xi _{\ell ,j}={\widehat{p}}_{j}\) in \(\Xi _{\ell }\) and \({\widetilde{u}}_{\ell ,j}=\xi _{\ell ,j}=0\) in \({\widetilde{\Xi }}_{0}\cup \bigcup _{k\in \{1,\ldots ,m-1\}{\setminus }\{\ell \}}\Xi _{k}\). Then
with \({\widetilde{u}}=0\) in \({\widetilde{\Xi }}_{0}\) and \({\text {Gradgrad}}{\widetilde{u}}=0\) in \(B{\setminus }{\overline{\Omega }}\) as well as \({\text {Gradgrad}}{\widetilde{u}}={\text {Gradgrad}}u=0\) in \(\Omega \) by assumption. Hence, \({\text {Gradgrad}}{\widetilde{u}}=0\) in B, showing \({\widetilde{u}}=0\) in B. In particular, \(u=0\) in \(\Omega \), and \(\sum _{j=0}^{3}\alpha _{\ell ,j}{\widehat{p}}_{j}={\widetilde{u}}|_{\Xi _{\ell }}=0\) for all \(\ell \in \{1,\dots ,m-1\}\). We conclude \(\alpha _{\ell ,j}=0\) for all \(j\in \{0,\dots ,3\}\) and all \(\ell \in \{1,\ldots ,m-1\}\), finishing the proof. \(\square \)
Theorem 11.11
Let Assumption 10.1 be satisfied. Then \(\dim {\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {bih},1}(\Omega )=4(m-1)\) and a basis of \({\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {bih},1}(\Omega )\) is given by (11.8).
Proof
Use Lemmas 11.8 and 11.10. \(\square \)
11.3 Dirichlet tensor fields of the second biharmonic complex
The rationale to derive a set of basis functions for the second biharmonic complex is somewhat similar to the first one. For the second biharmonic complex, similar to (3.3), (3.5), and (11.2), (11.6), we have the orthogonal decompositions
Remark 11.12
[38, Lemma 3.2] yields \({{\,\mathrm{dom}\,}}(\mathring{{{\,\mathrm{devGrad}\,}}},\Omega )=H^{1,3}_{0}(\Omega )\). Moreover, the range in (11.10) is closed by the Friedrichs type estimate
which holds by Assumption 10.1. Again, \(\Omega \) being open and bounded would be sufficient already. Indeed, the estimate mentioned here is based on the Friedrichs estimate provided in Remark 11.1 and the following observations similar to the proof of Korn’s inequality, cf. Remark 11.18: From \({{\,\mathrm{dom}\,}}(\mathring{{{\,\mathrm{devGrad}\,}}},\Omega )=H^{1,3}_{0}(\Omega )\) it suffices to show (11.11) for smooth vector fields v with compact support in \(\Omega \). It is elementary to see that for matrices T in \({\mathbb {R}}^{3\times 3}\) and the Frobenius norm \(|T|_{{\mathbb {R}}^{3\times 3}}\) we have \(|T|_{{\mathbb {R}}^{3\times 3}}^{2}=|{{\,\mathrm{dev}\,}}T|_{{\mathbb {R}}^{3\times 3}}^{2}+\frac{1}{3}|{{\,\mathrm{tr}\,}}T|_{{\mathbb {R}}}^{2}\), where \({{\,\mathrm{dev}\,}}T=T-\frac{1}{3}({{\,\mathrm{tr}\,}}{T}){{\,\mathrm{Id}\,}}\) is the deviatoric (‘trace-free’) part of T and \({{\,\mathrm{tr}\,}}T\) is the trace of T. Integration by parts shows \(|{{\,\mathrm{Grad}\,}}v|_{L^{2,3\times 3}(\Omega )}^{2}=|\text {curl}v|_{L^{2,3}(\Omega )}^{2}+|{{\,\mathrm{div}\,}}v|_{L^{2}(\Omega )}^{2}\ge |{{\,\mathrm{div}\,}}v|_{L^{2}(\Omega )}^{2}\) for all \(v\in C_{c}^{\infty ,3}(\Omega )\). Thus, from \({{\,\mathrm{tr}\,}}{{\,\mathrm{Grad}\,}}v={{\,\mathrm{div}\,}}v\) we infer
Hence, \(2|{{\,\mathrm{Grad}\,}}v|_{L^{2,3\times 3}(\Omega )}^{2}\le 3|{{\,\mathrm{devGrad}\,}}v|_{L^{2,3\times 3}(\Omega )}^{2}\), and inequality (11.11) follows from Remark 11.1.
Using (11.10), we define the orthogonal projector \(\pi :L^{2,3\times 3}_{{\mathbb {T}}}(\Omega )\rightarrow {\text {ker}}({{\,\mathrm{Div}\,}}_{{\mathbb {T}}},\Omega )\) along \({{\,\mathrm{ran}\,}}(\mathring{{{\,\mathrm{devGrad}\,}}},\Omega )\) and we have \(\pi ({\text {ker}}(\mathring{{{\,\mathrm{symCurl}\,}}}_{{\mathbb {T}}},\Omega ))={\mathcal {H}}_{D,{\mathbb {T}}}^{\mathsf {bih},2}(\Omega )\). Recalling \(\xi _{\ell }\in C^{\infty }_{c}({\mathbb {R}}^{3})\) from (11.1) and introducing the Raviart–Thomas fields \({\widehat{r}}_{j}\) given by
for \(j\in \{1,2,3\}\), we define \(\xi _{\ell ,j}:=\xi _{\ell }{\widehat{r}}_{j}\) for all \(\ell \in \{1,\dots ,m-1\}\) and all \(j\in \{0,\dots ,3\}\). It is easy to see that
Due to Remark 11.12 in conjunction with (11.10), we find unique \(\psi _{\ell ,j}\in H^{1,3}_{0}(\Omega )\) such that
with
We shall show that
defines a basis of \({\mathcal {H}}_{D,{\mathbb {T}}}^{\mathsf {bih},2}(\Omega )\).
Lemma 11.13
([38, Theorem 3.10 (iv) and Remark 3.11 (i)]) Let \(D\subseteq {\mathbb {R}}^3\) be a bounded strong Lipschitz domain. Assume that D is topologically trivial. Then
Lemma 11.14
Let Assumption 10.1 be satisfied. Then \({\mathcal {H}}_{D,{\mathbb {T}}}^{\mathsf {bih},2}(\Omega )={{\,\mathrm{lin}\,}}{\mathcal {B}}^{\mathsf {bih},2}_{D}\).
Proof
Let \(T\in {\mathcal {H}}_{D,{\mathbb {T}}}^{\mathsf {bih},2}(\Omega )={\text {ker}}(\mathring{{{\,\mathrm{symCurl}\,}}}_{{\mathbb {T}}},\Omega )\cap {\text {ker}}({{\,\mathrm{Div}\,}}_{{\mathbb {T}}},\Omega )\) and let \({\widetilde{T}}\) be the extension of T by zero onto B. Then \({\widetilde{T}}\in {\text {ker}}(\mathring{{{\,\mathrm{symCurl}\,}}}_{{\mathbb {T}}},B)\). As B is topologically trivial (and smooth and bounded), by Lemma 11.13 there exists (a unique vector field) \(v\in H^{1,3}_{0}(B)\) such that \({{\,\mathrm{devGrad}\,}}v={\widetilde{T}}\) in B. Since \({{\,\mathrm{devGrad}\,}}v={\widetilde{T}}=0\) in \(B{\setminus }{\overline{\Omega }}\), v is a Raviart–Thomas vector field, \(v\in \mathsf {RT}\), in each connected component \({\widetilde{\Xi }}_{0},\Xi _{1},\dots ,\Xi _{m-1}\) of \(B{\setminus }{\overline{\Omega }}\). Due to the boundary condition of \(v\in H^{1,3}_{0}(B)\), v vanishes in \({\widetilde{\Xi }}_{0}\). Therefore, \(T={{\,\mathrm{devGrad}\,}}v\) in \(\Omega \) and \(v\in H^{1,3}_{0}(B)\) is such that \(v|_{{\widetilde{\Xi }}_{0}}=0\) and \(v|_{\Xi _{\ell }}=:r_{\ell }=:\sum _{j=0}^{3}\alpha _{\ell ,j}{\widehat{r}}_{j}\in \mathsf {RT}\), for some \(\alpha _{\ell ,j}\in {\mathbb {R}}\), for all \(\ell \in \{1,\dots ,m-1\}\) and \(j\in \{0,\ldots ,3\}\). Define
with \(v_{\ell ,j}\) from (11.12). Since \({\widetilde{\psi }}_{\ell ,j}\in H^{1,3}_{0}(B)\), where \({\widetilde{\psi }}_{\ell ,j}\) is the extension of \(\psi _{\ell ,j}\) by zero to B, as an element of \(H^{1,3}(B)\) we see that
vanishes in all \(\Xi _{\ell }\). Thus \({\widehat{v}}={\widehat{v}}_{B}|_{\Omega }\in H^{1,3}_{0}(\Omega )\) by Assumption 10.1, and
yields the assertion. \(\square \)
Remark 11.15
(Characterisation by PDEs)
- (i):
-
Denoting \(\Delta _{{\mathbb {T}},D}:={{\,\mathrm{Div}\,}}_{{\mathbb {T}}}\mathring{{{\,\mathrm{devGrad}\,}}}\) the ‘deviatoric’ Laplacian with homogeneous Dirichlet boundary conditions, we see that \(\psi _{\ell ,j}={\Delta }_{{\mathbb {T}},D}^{-1}\Delta _{{\mathbb {T}}}\xi _{\ell ,j}\) with \(\Delta _{{\mathbb {T}}}:={{\,\mathrm{Div}\,}}_{{\mathbb {T}}}{{\,\mathrm{devGrad}\,}}\), which corresponds to the variational formulation
$$\begin{aligned}&\forall \,\phi \in H^{1,3}_{0}(\Omega )\quad \langle {{\,\mathrm{devGrad}\,}}\psi _{\ell ,j},{{\,\mathrm{devGrad}\,}}\phi \rangle _{L^{2,3\times 3}_{{\mathbb {T}}}(\Omega )}\\&\quad \qquad \qquad \qquad \qquad =\langle {{\,\mathrm{devGrad}\,}}\xi _{\ell ,j},{{\,\mathrm{devGrad}\,}}\phi \rangle _{L^{2,3\times 3}_{{\mathbb {T}}}(\Omega )}. \end{aligned}$$ - (ii):
-
For all \(\ell \in \{1,\dots ,m-1\}\) and all \(j\in \{0,\dots ,3\}\) we have
$$\begin{aligned} v_{\ell ,j} =\xi _{\ell ,j}-\psi _{\ell ,j} =(1-\Delta _{{\mathbb {T}},D}^{-1}\Delta _{{\mathbb {T}}})\xi _{\ell ,j}\in H^{1,3}(\Omega ) \end{aligned}$$and deduce
$$\begin{aligned} {{\,\mathrm{devGrad}\,}}v_{\ell ,j}&=(1-{{\,\mathrm{devGrad}\,}}\Delta _{{\mathbb {T}},D}^{-1}{{\,\mathrm{Div}\,}}_{{\mathbb {T}}}){{\,\mathrm{devGrad}\,}}\xi _{\ell ,j}. \end{aligned}$$In classical terms, this reads
$$\begin{aligned} \begin{aligned} -\Delta _{{\mathbb {T}}}v_{\ell ,j}&=0&\text {in }\Omega ,\\ v_{\ell ,j}&={\widehat{r}}_{j}&\text {on }\Gamma _{\ell },\\ v_{\ell ,j}&=0&\text {on }\Gamma _{k},\,k\in \{0,\dots ,m-1\}{\setminus }\{\ell \}, \end{aligned} \end{aligned}$$(11.14)which is uniquely solvable.
- (iii):
-
In classical terms, v (representing \(T={{\,\mathrm{devGrad}\,}}v\)) from the proof of Lemma 11.14 solves the linear elasticity type Dirichlet problem
$$\begin{aligned} -\Delta _{{\mathbb {T}}}v=-{{\,\mathrm{Div}\,}}_{{\mathbb {T}}}{{\,\mathrm{devGrad}\,}}v=-{{\,\mathrm{Div}\,}}_{{\mathbb {T}}}T&=0&\text {in }\Omega ,\\ v&=0&\text {on }\Gamma _{0},\\ v&=r_{\ell }\in \mathsf {RT}&\text {on }\Gamma _{\ell },\,\ell \in \{1,\dots ,m-1\}, \end{aligned}$$which is uniquely solvable given the knowledge of \(r_{\ell }\) in \(\mathsf {RT}\).
Lemma 11.16
Let Assumption 10.1 be satisfied. Then \({\mathcal {B}}^{\mathsf {bih},2}_{D}\) is linearly independent.
Proof
Let \(\alpha _{\ell ,j}\in {\mathbb {R}}\) with \(\ell \in \{1,\ldots ,m-1\}\) and \(j\in \{0,\ldots ,3\}\) be such that
Then \({{\,\mathrm{devGrad}\,}}v=0\) in \(\Omega \), i.e., \(v\in \mathsf {RT}\) in each connected component of \(\Omega \). We show \(v=0\). Recalling \(v_{\ell ,j}=\xi _{\ell ,j}-\psi _{\ell ,j}\) in \(\Omega \) from (11.7) and using \(\xi _{\ell ,j} \in H_{0}^1(B)\) and \(\psi _{\ell ,j}\in H_{0}^1(\Omega )\), we define
Note that \({\widetilde{v}}_{\ell ,j}\in H^{1,3}_{0}(B)\). For all \(\ell \in \{1,\dots ,m-1\}\) and \(j\in \{0,\dots ,3\}\), we obtain \({\widetilde{v}}_{\ell ,j}=\xi _{\ell ,j}={\widehat{r}}_{j}\) in \(\Xi _{\ell }\) and \({\widetilde{v}}_{\ell ,j}=\xi _{\ell ,j}=0\) in \({\widetilde{\Xi }}_{0}\cup \bigcup _{k\in \{1,\ldots ,m-1\}{\setminus }\{\ell \}}\Xi _{k}\). Then
with \({\widetilde{v}}=0\) in \({\widetilde{\Xi }}_{0}\) and \({{\,\mathrm{devGrad}\,}}{\widetilde{v}}=0\) in \(B{\setminus }{\overline{\Omega }}\) as well as \({{\,\mathrm{devGrad}\,}}{\widetilde{v}}={{\,\mathrm{devGrad}\,}}v=0\) in \(\Omega \) by assumption. Hence, \({{\,\mathrm{devGrad}\,}}{\widetilde{v}}=0\) in B, showing \({\widetilde{v}}=0\) in B. In particular, \(v=0\) in \(\Omega \), and \(\sum _{j=0}^{3}\alpha _{\ell ,j}{\widehat{r}}_{j}={\widetilde{v}}|_{\Xi _{\ell }}=0\) for all \(\ell \in \{1,\dots ,m-1\}\). We conclude \(\alpha _{\ell ,j}=0\) for all \(\ell \in \{1,\dots ,m-1\}\) and \(j\in \{0,\dots ,3\}\). \(\square \)
Theorem 11.17
Let Assumption 10.1 be satisfied. Then \(\dim {\mathcal {H}}_{D,{\mathbb {T}}}^{\mathsf {bih},2}(\Omega )=4(m-1)\) and a basis of \({\mathcal {H}}_{D,{\mathbb {T}}}^{\mathsf {bih},2}(\Omega )\) is given by (11.13).
Proof
Use Lemmas 11.14 and 11.16. \(\square \)
11.4 Dirichlet tensor fields of the elasticity complex
For the elasticity complex, similar to (3.3), (3.5), and (11.2), (11.6), (11.10), we have the orthogonal decompositions
Remark 11.18
[39, Lemma 3.2] implies \({{\,\mathrm{dom}\,}}(\mathring{{{\,\mathrm{symGrad}\,}}},\Omega )=H^{1,3}_{0}(\Omega )\). Moreover, the range in (11.15) is closed by the Friedrichs type estimate (and follows from the standard first Korn’s inequality and Remark 11.1)
which holds by Assumption 10.1. Again, \(\Omega \) open and bounded is sufficient for (11.16). Indeed, Korn’s first inequality is easy to see as follows: For a tensor \(T\in {\mathbb {R}}^{3\times 3}\) we have \(|T|^{2}_{{\mathbb {R}}^{3\times 3}}=|{{\,\mathrm{sym}\,}}T|^{2}_{{\mathbb {R}}^{3\times 3}}+|{{\,\mathrm{skw}\,}}T|^{2}_{{\mathbb {R}}^{3\times 3}}\). Hence,
By \(|{{\,\mathrm{Grad}\,}}v|_{L^{2,3\times 3}(\Omega )}^{2}=|\text {curl}v|_{L^{2,3}(\Omega )}^{2}+|{{\,\mathrm{div}\,}}v|_{L^{2}(\Omega )}^{2}\ge |\text {curl}v|_{L^{2,3}(\Omega )}^{2}\) for all \(v\in H^{1,3}_{0}(\Omega )\), we get Korn’s first inequality \(|{{\,\mathrm{Grad}\,}}v|_{L^{2,3\times 3}(\Omega )}^{2}\le 2|{{\,\mathrm{symGrad}\,}}v|_{L^{2,3\times 3}(\Omega )}^{2}\).
The orthogonal projector from \(L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\) onto \({\text {ker}}({{\,\mathrm{Div}\,}}_{{\mathbb {S}}},\Omega )\) along \({{\,\mathrm{ran}\,}}(\mathring{{{\,\mathrm{symGrad}\,}}},\Omega )\) is denoted by \(\pi \). From (11.15), we deduce \(\pi ({\text {ker}}(\mathring{{\text {CurlCurl}}}{}^{\top }_{{\mathbb {S}}},\Omega ))={\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )\). Recall the functions \(\xi _{\ell }\in C^{\infty }_{c}({\mathbb {R}}^{3})\) from (11.1) and introduce rigid motions \({\widehat{r}}_{j}\) given by
for \(j\in \{1,2,3\}\). We define \(\xi _{\ell ,j}:=\xi _{\ell }{\widehat{r}}_{j}\) for all \(\ell \in \{1,\dots ,m-1\}\) and for all \(j\in \{1,\dots ,6\}\). Then
We find unique \(\psi _{\ell ,j}\in H^{1,3}_{0}(\Omega )\) such that
with
We shall show that
defines a basis of \({\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )\).
Lemma 11.19
([39, Theorem 3.5]) Let \(D\subseteq {\mathbb {R}}^3\) a bounded, topologically trivial, strong Lipschitz domain. Then
Proof
The result follows by [38, Corollary 2.29] for \(m=1\) in conjunction with the formulas in [38, Appendix]; see [39, Theorem 3.5] and [40] for the details. \(\square \)
Lemma 11.20
Let Assumption 10.1 be satisfied. Then \({\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )={{\,\mathrm{lin}\,}}{\mathcal {B}}^{\mathsf {ela}}_{D}\).
Proof
We follow in close lines the arguments used in the proofs of Lemmas 11.2, 11.8, and 11.14. Let \(S\in {\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )={\text {ker}}(\mathring{{\text {CurlCurl}}}{}^{\top }_{{\mathbb {S}}},\Omega )\cap {\text {ker}}({{\,\mathrm{Div}\,}}_{{\mathbb {S}}},\Omega )\). and \({\widetilde{S}}\) its extension to B by zero. Then \({\widetilde{S}}\in {\text {ker}}(\mathring{{\text {CurlCurl}}}{}^{\top }_{{\mathbb {S}}},B)\). By Lemma 11.19, as B is topologically trivial (and smooth and bounded), there exists (a unique) \(v\in H^{1,3}_{0}(B)\) such that \({{\,\mathrm{symGrad}\,}}v={\widetilde{S}}\) in B. Since \({{\,\mathrm{symGrad}\,}}v={\widetilde{S}}=0\) in \(B{\setminus }{\overline{\Omega }}\), v is a rigid motion, i.e., \(v\in \mathsf {RM}\), in each connected component \({\widetilde{\Xi }}_{0},\Xi _{1},\dots ,\Xi _{m-1}\) of \(B{\setminus }{\overline{\Omega }}\). Since \(v\in H^{1,3}_{0}(B)\), v vanishes in \({\widetilde{\Xi }}_{0}\). Thus, \(S={{\,\mathrm{symGrad}\,}}v\) in \(\Omega \) with some \(v\in H^{1,3}_{0}(B)\) and we have \(v|_{\Xi _{\ell }}=:r_{\ell }=:\sum _{j=1}^{6}\alpha _{\ell ,j}{\widehat{r}}_{j}\in \mathsf {RM}\) for \(\alpha _{\ell ,j}\in {\mathbb {R}}\) and all \(\ell \in \{1,\dots ,m-1\}\), \(j\in \{1,\ldots ,6\}\). Let
with \(v_{\ell ,j}\) from (11.17). With \({\widetilde{\psi }}_{\ell ,j} \in H^{1,3}_{0}(B)\), the extension of \(\psi _{\ell ,j}\) by zero, we see, as an element of \(H^{1,3}(B)\), that
vanishes in \(\Xi _{\ell }\) for all \(\ell \). Thus \({\widehat{v}}={\widehat{v}}_{B}|_{\Omega }\in H^{1,3}_{0}(\Omega )\) by Assumption 10.1, and we conclude
finishing the proof. \(\square \)
For numerical purposes, we again highlight the partial differential equations satisfied by the functions constructed here.
Remark 11.21
(Characterisation by PDEs)
- (i):
-
For all \(\ell \in \{1,\ldots ,m-1\}\), \(j\in \{1,\ldots ,6\}\), the vector field \(\psi _{\ell ,j}\in H^{1,3}_{0}(\Omega )\) can be found with the help of the standard variational formulation
$$\begin{aligned}&\forall \,\phi \in H^{1,3}_{0}(\Omega )\quad \langle {{\,\mathrm{symGrad}\,}}\psi _{\ell ,j},{{\,\mathrm{symGrad}\,}}\phi \rangle _{L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )}\\&\quad \qquad \qquad \qquad \qquad =\langle {{\,\mathrm{symGrad}\,}}\xi _{\ell ,j},{{\,\mathrm{symGrad}\,}}\phi \rangle _{L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )}, \end{aligned}$$i.e., \(\psi _{\ell ,j}={\Delta }_{{\mathbb {S}},D}^{-1}\Delta _{{\mathbb {S}}}\xi _{\ell ,j}\), where \(\Delta _{{\mathbb {S}},D}={{\,\mathrm{Div}\,}}_{{\mathbb {S}}}\mathring{{{\,\mathrm{symGrad}\,}}}\) and \(\Delta _{{\mathbb {S}}}={{\,\mathrm{Div}\,}}_{{\mathbb {S}}}{{\,\mathrm{symGrad}\,}}\) are the ‘symmetric’ Laplacians with homogeneous Dirichlet boundary conditions and no boundary conditions, respectively.
- (ii):
-
For all \(\ell \in \{1,\ldots ,m-1\}\), \(j\in \{1,\ldots ,6\}\) we have
$$\begin{aligned} v_{\ell ,j} =\xi _{\ell ,j}-\psi _{\ell ,j} =(1-\Delta _{{\mathbb {S}},D}^{-1}\Delta _{{\mathbb {S}}})\xi _{\ell ,j}\in H^{1,3}(\Omega ) \end{aligned}$$and thus
$$\begin{aligned} {{\,\mathrm{symGrad}\,}}v_{\ell ,j}&=(1-{{\,\mathrm{symGrad}\,}}\Delta _{{\mathbb {S}},D}^{-1}{{\,\mathrm{Div}\,}}_{{\mathbb {S}}}){{\,\mathrm{symGrad}\,}}\xi _{\ell ,j}. \end{aligned}$$In classical terms, \(v_{\ell ,j}\) solves the linear elasticity Dirichlet problem
$$\begin{aligned} \begin{aligned} -\Delta _{{\mathbb {S}}}v_{\ell ,j}&=0&\text {in }\Omega ,\\ v_{\ell ,j}&={\widehat{r}}_{j}&\text {on }\Gamma _{\ell },\\ v_{\ell ,j}&=0&\text {on }\Gamma _{k},\,k\in \{0,\dots ,m-1\}{\setminus }\{\ell \}. \end{aligned} \end{aligned}$$(11.19)which is uniquely solvable.
- (iii):
-
In classical terms, v (representing \(S={{\,\mathrm{symGrad}\,}}v\)) from the proof of Lemma 11.20 solves the linear elasticity Dirichlet problem
$$\begin{aligned} -\Delta _{{\mathbb {S}}}v=-{{\,\mathrm{Div}\,}}_{{\mathbb {S}}}S&=0&\text {in }\Omega ,\\ v&=0&\text {on }\Gamma _{0},\\ v&=r_{\ell }\in \mathsf {RM}&\text {on }\Gamma _{\ell },\,\ell \in \{1,\dots ,m-1\}, \end{aligned}$$which is uniquely solvable as long as the rigid motions \(r_{\ell }\) in \(\mathsf {RM}\) are prescribed.
Lemma 11.22
Let Assumption 10.1 be satisfied. Then \({\mathcal {B}}^{\mathsf {ela}}_{D}\) is linearly independent.
Proof
Let \(\alpha _{\ell ,j}\in {\mathbb {R}}\) for all \(\ell \in \{1,\dots ,m-1\}\), \(j\in \{1,\ldots ,6\}\) such that
Then \({{\,\mathrm{symGrad}\,}}v=0\) in \(\Omega \), i.e., \(v\in \mathsf {RM}\) in each connected component of \(\Omega \). We show \(v=0\). Recall \(v_{\ell ,j}=\xi _{\ell ,j}-\psi _{\ell ,j}\) in \(\Omega \). Using \(\psi _{\ell ,j}\in H_{0}^1(\Omega )\) and \(\xi _{\ell ,j}\in H_{0}^1(B)\) we extend \(v_{\ell ,j}\) to B via
Note that \({\widetilde{v}}_{\ell ,j}\in H_{0}^1(B)\). Then, for all \(\ell \in \{1,\dots ,m-1\}\) and all \(j\in \{1,\dots ,6\}\) we have \({\widetilde{v}}_{\ell ,j}=\xi _{\ell ,j}=0\) in \({\widetilde{\Xi }}_{0}\cup \bigcup _{k\in \{1,\ldots ,m-1\}{\setminus }\{\ell \}}\Xi _{k}\) and \({\widetilde{v}}_{\ell ,j}=\xi _{\ell ,j}={\widehat{r}}_{j}\) in \(\Xi _{\ell }\). Thus
with \({\widetilde{v}}=0\) in \({\widetilde{\Xi }}_{0}\) and \({{\,\mathrm{symGrad}\,}}{\widetilde{v}}=0\) in \(B{\setminus }{\overline{\Omega }}\) as well as \({{\,\mathrm{symGrad}\,}}{\widetilde{v}}={{\,\mathrm{symGrad}\,}}v=0\) in \(\Omega \) by assumption. Hence, \({{\,\mathrm{symGrad}\,}}{\widetilde{v}}=0\) in B, showing \({\widetilde{v}}=0\) in B. In particular, \(v=0\) in \(\Omega \), and \(\sum _{j=1}^{6}\alpha _{\ell ,j}{\widehat{r}}_{j}={\widetilde{v}}|_{\Xi _{\ell }}=0\) for all \(\ell \in \{1,\dots ,m-1\}\). We conclude \(\alpha _{\ell ,j}=0\) for all \(\ell \in \{1,\dots ,m-1\}\) and all \(j\in \{1,\dots ,6\}\), finishing the proof. \(\square \)
Theorem 11.23
Let Assumption 10.1 be satisfied. Then \(\dim {\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )=6(m-1)\) and a basis of \({\mathcal {H}}_{D,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )\) is given by (11.18).
Proof
12 The construction of the Neumann fields
The construction of the Neumann fields is more involved than the one for the generalised harmonic Dirichlet fields. We start off with some general definitions and remarks all the basis constructions have in common.
Since \(\Omega \) consists of the connected components \(\Omega _{k}\), i.e., \(\mathsf {cc}(\Omega )=\{\Omega _{1},\ldots ,\Omega _{n}\}\), we have by Assumption 10.3 (A3) for all \(k\in \{1,\dots ,n\}\) that \(\Omega _{k}{\setminus }\bigcup _{j=1}^{p} F_{j}\) is simply connected. We define
For \(j\in \{1,\ldots ,p\}\), let \({\widehat{F}}_{j}\subseteq {\widetilde{F}}_{j}\) be two stacked, open, and simply connected neighbourhoods of \(F_{j}\), i.e,
let
and let \(\theta _{j}\in C^{\infty }(\Omega _{F})\) be a bounded (together with all derivatives) function with the following properties:
-
\(F_{j}\subseteq \Upsilon _{j}\subseteq {\widetilde{\Upsilon }}_{j}\).
-
\(\Upsilon _{j}\) and \({\widetilde{\Upsilon }}_{j}\) are (nonempty, open, and) simply connected.
-
\({\widetilde{F}}_{j}\) are pairwise disjoint.
-
\(\Upsilon _{j}{\setminus } F_{j}=\Upsilon _{j,0}\dot{\cup }\Upsilon _{j,1}\) and \({\widetilde{\Upsilon }}_{j}{\setminus }F_{j}={\widetilde{\Upsilon }}_{j,0}\dot{\cup }{\widetilde{\Upsilon }}_{j,1}\) with \(\Upsilon _{j,0}\subseteq {\widetilde{\Upsilon }}_{j,0}\) and \(\Upsilon _{j,1}\subseteq {\widetilde{\Upsilon }}_{j,1}\) (which are all nonempty, open, and simply connected).
-
\({\overline{\Upsilon }}_{j,0}\cap {\overline{\Upsilon }}_{j,1}={\overline{F}}_{j}\).
-
\({{\,\mathrm{supp}\,}}\theta _{j}\subseteq \overline{{\widetilde{\Upsilon }}_{j,1}}\).
-
\(\theta _{j}|_{\Upsilon _{j,0}}=0\) and \(\theta _{j}|_{\Upsilon _{j,1}}=1\).
Additionally, for all \(l\in \{1,\dots ,p\}\) we pick curves
-
\(\zeta _{x_{l,0},x_{l,1}}\subseteq \zeta _{l}\) with fixed starting points \(x_{l,0}\in \Upsilon _{l,0}\) and fixed endpoints \(x_{l,1}\in \Upsilon _{l,1}\).
Remark 12.1
Roughly speaking, \({\widetilde{\Upsilon }}_{j}{\setminus } F_{j}\) consists of exactly two open, nonempty, and simply connected components \({\widetilde{\Upsilon }}_{j,0}\) and \({\widetilde{\Upsilon }}_{j,1}\), on which subsets \(\Upsilon _{j,0}\) (one side) and \(\Upsilon _{j,1}\) (other side) the indicator function \(\theta _{j}\) is 0 and 1, respectively. Note that \(\Upsilon _{j,0}\) and \(\Upsilon _{j,1}\) touch each other at the whole surface \(F_{j}\), i.e., \({\overline{\Upsilon }}_{j,0}\cap {\overline{\Upsilon }}_{j,1}={\overline{F}}_{j}\). Moreover, \({\widetilde{\Upsilon }}_{j}\) are pairwise disjoint and \(\theta _{j}\) is supported in \(\overline{{\widetilde{\Upsilon }}_{j,1}}\). As a consequence, \(\theta _{j}\) cannot be continuously extended to \(\Omega \). On the other hand, \({{\,\mathrm{grad}\,}}\theta _{j}=0\) in \(\Upsilon _{j}{\setminus }F_{j}\), and hence \({{\,\mathrm{grad}\,}}\theta _{j}\) can be continuously extended to \(\Upsilon _{j}\) and thus to \(\Omega \). Note that for all \(l,j\in \{1,\ldots ,p\}\) it holds \(\theta _{j}(x_{l,0})=0\) and \(\theta _{j}(x_{l,1})=\delta _{l,j}\).
For the construction of bases and to compute the dimensions of the Neumann fields it is crucial that these fields are sufficiently regular, e.g., continuous in \(\Omega \). We even have the following local regularity results.
Lemma 12.2
(local regularity of the cohomology groups) Let \(\Omega \subseteq {\mathbb {R}}^{3}\) be open. Then
Proof
Vector fields in \({\mathcal {H}}_{D}^{\mathsf {Rhm}}(\Omega )\cup {\mathcal {H}}_{N}^{\mathsf {Rhm}}(\Omega )\) are harmonic and thus belong to \(C^{\infty ,3}(\Omega )\).
Let
Then S can be represented locally, e.g., in any topologically trivial and smooth subdomain \(D\subseteq \Omega \), by \(S={\text {Gradgrad}}u\) with some \(u\in H^{2}(D)\), see Lemma 11.7. Therefore, \({\text {divDiv}}_{{\mathbb {S}}}{\text {Gradgrad}}u=0\) in D. Local regularity for the biharmonic equation shows \(u\in C^{\infty }(D)\) and hence \(S={\text {Gradgrad}}u\in C^{\infty ,3\times 3}(D)\), i.e., \(S\in C^{\infty ,3\times 3}(\Omega )\).
Next, let
Then, for any topologically trivial and smooth subdomain \(D\subseteq \Omega \) we find \(v\in H^{1,3}(D)\) such that \(T={{\,\mathrm{devGrad}\,}}v\), see Lemma 11.13. Thus \({{\,\mathrm{Div}\,}}_{{\mathbb {T}}}{{\,\mathrm{devGrad}\,}}v=0\) in D. Local elliptic regularity shows \(v\in C^{\infty ,3}(D)\) and hence \(T={{\,\mathrm{devGrad}\,}}v\in C^{\infty ,3\times 3}(D)\), i.e., \(T\in C^{\infty ,3\times 3}(\Omega )\).
Finally, let
For \(D\subseteq \Omega \) smooth, bounded, and topologically trivial, we find \(v\in H^{1,3}(D)\) representing \(S={{\,\mathrm{symGrad}\,}}v\), see Lemma 11.19. Thus, \({{\,\mathrm{Div}\,}}_{{\mathbb {S}}}{{\,\mathrm{symGrad}\,}}v=0\) in D. Local elliptic regularity shows \(v\in C^{\infty ,3}(D)\) and thus \(S={{\,\mathrm{symGrad}\,}}v\in C^{\infty ,3\times 3}(D)\), i.e., \(S\in C^{\infty ,3\times 3}(\Omega )\). \(\square \)
12.1 Neumann vector fields of the classical de Rham complex
Similar to our reasoning for the generalised harmonic Dirichlet fields, we start off with the arguably easiest case of the de Rham complex. Since we rely on the rephrasing of Picard’s ideas in the forthcoming sections, we carry out the full construction of the harmonic Neumann fields. Note that we heavily use the functions and sets introduced at the beginning of Sect. 12, cf. Remark 12.1. Let \(j\in \{1,\ldots ,p\}\). By definition \(\theta _{j}=0\) outside of \({\widetilde{\Upsilon }}_{j,1}\) and \(\theta _{j}\) is constant on each connected component \(\Upsilon _{j,0}\) and \(\Upsilon _{j,1}\) of \(\Upsilon _{j}{\setminus }F_{j}\). Hence \({{\,\mathrm{grad}\,}}\theta _{j}=0\) in \(\Upsilon _{j}{\setminus }F_{j}\) and—due to the support condition—also \(\theta _{j}=0\) in \(\dot{\bigcup }_{l\in \{1,\ldots ,p\}{\setminus }\{j\}}\Upsilon _{l}{\setminus }F_{l}\). Thus, \({{\,\mathrm{grad}\,}}\theta _{j}\) can be continuously extended by zero to \(\Theta _{j}\in C^{\infty ,3}(\Omega )\cap L^{2,3}(\Omega )\) with \(\Theta _{j}=0\) in \(\dot{\bigcup }_{l\in \{1,\ldots ,p\}}\Upsilon _{l}\).
Lemma 12.3
Let Assumption 10.3 be satisfied. Then \(\Theta _{j}\in {\text {ker}}(\text {curl},\Omega )\).
Proof
Let \(\Phi \in C^{\infty ,3}_{c}(\Omega )\). As \({{\,\mathrm{supp}\,}}\Theta _{j}\subseteq {\widetilde{\Upsilon }}_{j}{\setminus }\Upsilon _{j}\) we can pick another cut-off function \(\varphi \in C^{\infty }_{c}(\Omega _{F})\) with \(\varphi |_{{{\,\mathrm{supp}\,}}\Theta _{j}\cap {{\,\mathrm{supp}\,}}\Phi }=1\). Then
as \(\varphi \Phi \in C^{\infty ,3}_{c}(\Omega _{F})\). \(\square \)
Let \(l,j\in \{1,\ldots ,p\}\). We recall from the latter proof and from Remark 12.1 that \({{\,\mathrm{supp}\,}}\Theta _{j}\subseteq {\widetilde{\Upsilon }}_{j}{\setminus }\Upsilon _{j}\) and thus
where we recall \(\zeta _{x_{l,0},x_{l,1}}\subseteq \zeta _{l}\) with chosen starting points \(x_{l,0}\in \Upsilon _{l,0}\) and respective endpoints \(x_{l,1}\in \Upsilon _{l,1}\). Hence we define functionals \(\beta _{l}\) in the way that
Let Assumption 10.1 be satisfied. For the de Rham complex, similar to (3.3), (3.5), and (11.2), we have the orthogonal decompositions
Remark 12.4
By definition \({{\,\mathrm{dom}\,}}({{\,\mathrm{grad}\,}},\Omega )=H^{1}(\Omega )\), and the range in (12.2) is closed by the Poincaré estimate
which is implied by a contradiction argument using Rellich’s selection theorem as Assumption 10.1 holds.
Similar to the case of harmonic Dirichlet fields, we denote in (12.2) the orthogonal projector along \({{\,\mathrm{ran}\,}}({{\,\mathrm{grad}\,}},\Omega )\) onto \({\text {ker}}(\mathring{{{\,\mathrm{div}\,}}},\Omega )\) by \(\pi \). By Lemma 12.3 for all \(j\in \{1,\ldots ,p\}\) there exists some \(\psi _{j}\in H^{1}(\Omega )\) (unique up to \({\mathbb {R}}_{\mathsf {pw}}\)) such that
By Lemma 12.2 we have \({\mathcal {H}}_{N}^{\mathsf {Rhm}}(\Omega )\subseteq C^{\infty ,3}(\Omega )\). Therefore, \(\Theta _{j}\in C^{\infty ,3}(\Omega )\) and we obtain \({{\,\mathrm{grad}\,}}\psi _{j}\in C^{\infty ,3}(\Omega )\). Hence, \(\psi _{j}\in H^{1}(\Omega )\cap C^{\infty }(\Omega )\) and the following path integrals are well-defined and can be computed by (12.1), i.e., for all \(l,j\in \{1,\dots ,p\}\)
We will show that
defines a basis of \({\mathcal {H}}_{N}^{\mathsf {Rhm}}(\Omega )\).
Also for the harmonic Neumann fields, we provide a possible variational formulation for obtaining \(\psi _{j}\) constructed here:
Remark 12.5
(Characterisation by PDEs) Let \(j\in \{1,\ldots ,p\}\). Then \(\psi _{j}\in H^{1}(\Omega )\cap {\mathbb {R}}_{\mathsf {pw}}^{\bot _{L^{2}(\Omega )}}\) satisfies
i.e., \(\psi _{j}=\Delta _{N}^{-1}\big ({{\,\mathrm{div}\,}}\Theta _{j},\nu \cdot \Theta _{j}|_{\Gamma }\big )\), where \(\Delta _{N}\subseteq {{\,\mathrm{div}\,}}{{\,\mathrm{grad}\,}}\) is the Laplacian with inhomogeneous Neumann boundary conditions restricted to a subset of \(H^{1}(\Omega )\cap {\mathbb {R}}_{\mathsf {pw}}^{\bot _{L^{2}(\Omega )}}\). Therefore,
In classical terms, \(\psi _{j}\) solves the Neumann Laplace problem
which is uniquely solvable.
Lemma 12.6
Let Assumptions 10.1 and 10.3 be satisfied. Then \({\mathcal {H}}_{N}^{\mathsf {Rhm}}(\Omega )={{\,\mathrm{lin}\,}}{\mathcal {B}}^{\mathsf {Rhm}}_{N}\).
Proof
Let \(H\in {\mathcal {H}}_{N}^{\mathsf {Rhm}}(\Omega )={\text {ker}}(\mathring{{{\,\mathrm{div}\,}}},\Omega )\cap {\text {ker}}(\text {curl},\Omega )\subseteq C^{\infty ,3}(\Omega )\) (see Lemma 12.2), and define the numbers
We shall show that
The aim is to prove that there exists \(u\in H^{1}(\Omega )\) such that \({{\,\mathrm{grad}\,}}u={\widehat{H}}\), since then
Using (12.3), we obtain
Note that \({\widehat{H}}\in {\text {ker}}(\text {curl},\Omega )\cap C^{\infty ,3}(\Omega )\). Hence, by Assumption 10.3 (A.1) we have for any closed piecewise \(C^1\)-curve \(\zeta \) in \(\Omega \)
Recall the connected components \(\Omega _{1},\dots ,\Omega _{n}\) of \(\Omega \). For \(1\le k\le n\) let \(\Omega _{k}\) and some \(x_{0}\in \Omega _{k}\) be fixed. By (12.6) and the fundamental theorem of calculus the function \(u:\Omega \rightarrow {\mathbb {R}}\) given by
where \(\zeta (x_{0},x)\) is any piecewise \(C^1\)-curve connecting \(x_{0}\) with x, is well defined, i.e., independent of the choice of the respective curve \(\zeta (x_{0},x)\), and belongs to \(C^{\infty }(\Omega _{k})\) with \({{\,\mathrm{grad}\,}}u={\widehat{H}}\in L^{2,3}(\Omega _{k})\). ThusFootnote 5\(u\in L^{2}(\Omega _{k})\), see, e.g., [22, Theorem 2.6 (1)] or [1, Theorem 3.2 (2)], and hence \(u\in H^{1}(\Omega _{k})\), showing \(u\in H^{1}(\Omega )\). \(\square \)
Remark 12.7
Note that in the latter proof the existence of \(u\in H^{1}(\Omega _{k})\) with \({{\,\mathrm{grad}\,}}u={\widehat{H}}\) in \(\Omega _{k}\) is well-known, if the connected component \(\Omega _{k}\) of \(\Omega \) is even simply connected. Indeed, in this case \({\text {ker}}(\text {curl},\Omega _{k})={{\,\mathrm{ran}\,}}({{\,\mathrm{grad}\,}},\Omega _{k})\).
Lemma 12.8
Let Assumptions 10.1 and 10.3 be satisfied. Then \({\mathcal {B}}^{\mathsf {Rhm}}_{N}\) is linearly independent.
Proof
Let \(\sum _{j=1}^{p}\gamma _{j}\pi \Theta _{j}=0\) for some \(\gamma _{j}\in {\mathbb {R}}\). Then (12.3) implies
for all \(l\in \{1,\ldots ,p\}\). \(\square \)
Theorem 12.9
Let Assumptions 10.1 and 10.3 be satisfied. Then \(\dim {\mathcal {H}}_{N}^{\mathsf {Rhm}}(\Omega )=p\) and a basis of \({\mathcal {H}}_{N}^{\mathsf {Rhm}}(\Omega )\) is given by (12.4).
Proof
Use Lemmas 12.6 and 12.8. \(\square \)
12.2 Neumann tensor fields of the first biharmonic complex
The main difference of the constructions to come to the one in the previous section is the introduction of \(\beta \): a suitable collection of functionals that very easily allows for testing of linear independence and for a straightforward application of Assumption 10.3 (A1). As a preparation for this, we need the next results. The first one—also important for the sections to come—is rather combinatorical and analyses the interplay between vector analysis and matrix calculus; the second and third one deal with so-called Poincaré maps, which form the foundation of the construction of the desired set of functionals. Note that for the subsequent sections Lemma 12.10 is of independent interest. For this, we introduce for \(v\in {\mathbb {R}}^3\) the skew-symmetric matrix
and the corresponding isometric mapping \({{\,\mathrm{spn}\,}}:{\mathbb {R}}^{3}\rightarrow {\mathbb {R}}^{3\times 3}_{{{\,\mathrm{skw}\,}}}\).
Lemma 12.10
Let \(u\in C^{\infty }_{c}({\mathbb {R}}^{3})\), \(v,w\in C^{\infty ,3}_{c}({\mathbb {R}}^{3})\), and \(S\in C^{\infty ,3\times 3}_{c}({\mathbb {R}}^{3})\). Then:
-
\(({{\,\mathrm{spn}\,}}v)\,w=v\times w=-({{\,\mathrm{spn}\,}}w)\,v\) and \(({{\,\mathrm{spn}\,}}v)({{\,\mathrm{spn}\,}}^{-1}S)=-Sv\), if \({{\,\mathrm{sym}\,}}S=0\)
-
\({{\,\mathrm{sym}\,}}{{\,\mathrm{spn}\,}}v=0\) and \({{\,\mathrm{dev}\,}}(u{{\,\mathrm{Id}\,}})=0\)
-
\({{\,\mathrm{tr}\,}}{{\,\mathrm{Grad}\,}}v={{\,\mathrm{div}\,}}v\) and \(2{{\,\mathrm {skw}\,}}{{\,\mathrm {Grad}\,}}v={{\,\mathrm {spn}\,}}\mathrm {curl}\,v\)
-
\({{\,\mathrm{Div}\,}}(u{{\,\mathrm{Id}\,}})={{\,\mathrm{grad}\,}}u\) and \({{\,\mathrm{Curl}\,}}(u{{\,\mathrm{Id}\,}})=-{{\,\mathrm{spn}\,}}{{\,\mathrm{grad}\,}}u\),
in particular, \(\mathrm {curl}{{\,\mathrm {Div}\,}}(u{{\,\mathrm {Id}\,}})=0\) and \(\mathrm {curl}{{\,\mathrm {spn}\,}}^{-1}{{\,\mathrm {Curl}\,}}(u{{\,\mathrm {Id}\,}})=0\)
and \({{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}(u{{\,\mathrm{Id}\,}})=0\)
-
\({{\,\mathrm {Div}\,}}{{\,\mathrm {spn}\,}}v=-\mathrm {curl}\,v\) and \({{\,\mathrm {Div}\,}}{{\,\mathrm {skw}\,}}S=-\mathrm {curl}{{\,\mathrm {spn}\,}}^{-1}{{\,\mathrm {skw}\,}}S\),
in particular, \({{\,\mathrm{div}\,}}{{\,\mathrm{Div}\,}}{{\,\mathrm{skw}\,}}S=0\)
-
\({{\,\mathrm{Curl}\,}}{{\,\mathrm{spn}\,}}v=({{\,\mathrm{div}\,}}v){{\,\mathrm{Id}\,}}-({{\,\mathrm{Grad}\,}}v)^{\top }\)
and \({{\,\mathrm{Curl}\,}}{{\,\mathrm{skw}\,}}S=({{\,\mathrm{div}\,}}{{\,\mathrm{spn}\,}}^{-1}{{\,\mathrm{skw}\,}}S){{\,\mathrm{Id}\,}}-({{\,\mathrm{Grad}\,}}{{\,\mathrm{spn}\,}}^{-1}{{\,\mathrm{skw}\,}}S)^{\top }\)
-
\({{\,\mathrm{dev}\,}}{{\,\mathrm{Curl}\,}}{{\,\mathrm{spn}\,}}v=-({{\,\mathrm{dev}\,}}{{\,\mathrm{Grad}\,}}v)^{\top }\)
-
\(-2{{\,\mathrm {Curl}\,}}{{\,\mathrm {sym}\,}}{{\,\mathrm {Grad}\,}}v=2{{\,\mathrm {Curl}\,}}{{\,\mathrm {skw}\,}}{{\,\mathrm {Grad}\,}}v=-({{\,\mathrm {Grad}\,}}\mathrm {curl}\,v)^{\top }\)
-
\(2{{\,\mathrm{spn}\,}}^{-1}{{\,\mathrm{skw}\,}}{{\,\mathrm{Curl}\,}}S={{\,\mathrm{Div}\,}}S^{\top }-{{\,\mathrm{grad}\,}}{{\,\mathrm{tr}\,}}S={{\,\mathrm{Div}\,}}\big (S-({{\,\mathrm{tr}\,}}S){{\,\mathrm{Id}\,}}\big )^{\top }\),
in particular, \(\mathrm {curl}{{\,\mathrm {Div}\,}}S^{\top }=2\mathrm {curl}{{\,\mathrm {spn}\,}}^{-1}{{\,\mathrm {skw}\,}}{{\,\mathrm {Curl}\,}}S\)
and \(2{{\,\mathrm{skw}\,}}{{\,\mathrm{Curl}\,}}S={{\,\mathrm{spn}\,}}{{\,\mathrm{Div}\,}}S^{\top }\), if \({{\,\mathrm{tr}\,}}S=0\)
-
\({{\,\mathrm{tr}\,}}{{\,\mathrm{Curl}\,}}S=2{{\,\mathrm{div}\,}}{{\,\mathrm{spn}\,}}^{-1}{{\,\mathrm{skw}\,}}S\), in particular, \({{\,\mathrm{tr}\,}}{{\,\mathrm{Curl}\,}}S=0\), if \({{\,\mathrm{skw}\,}}S=0\),
and \({{\,\mathrm{tr}\,}}{{\,\mathrm{Curl}\,}}{{\,\mathrm{sym}\,}}S=0\) and \({{\,\mathrm{tr}\,}}{{\,\mathrm{Curl}\,}}{{\,\mathrm{skw}\,}}S={{\,\mathrm{tr}\,}}{{\,\mathrm{Curl}\,}}S\)
-
\(2({{\,\mathrm{Grad}\,}}{{\,\mathrm{spn}\,}}^{-1}{{\,\mathrm{skw}\,}}S)^{\top }=({{\,\mathrm{tr}\,}}{{\,\mathrm{Curl}\,}}{{\,\mathrm{skw}\,}}S){{\,\mathrm{Id}\,}}-2{{\,\mathrm{Curl}\,}}{{\,\mathrm{skw}\,}}S\)
-
\(3{{\,\mathrm{Div}\,}}({{\,\mathrm{dev}\,}}{{\,\mathrm{Grad}\,}}v)^{\top }=2{{\,\mathrm{grad}\,}}{{\,\mathrm{div}\,}}v\)
-
\(2{{\,\mathrm {Curl}\,}}{{\,\mathrm {sym}\,}}{{\,\mathrm {Grad}\,}}v=-2{{\,\mathrm {Curl}\,}}{{\,\mathrm {skw}\,}}{{\,\mathrm {Grad}\,}}v=-{{\,\mathrm {Curl}\,}}{{\,\mathrm {spn}\,}}\mathrm {curl}\,v=({{\,\mathrm {Grad}\,}}\mathrm {curl}\,v)^{\top }\)
-
\(2{{\,\mathrm {Div}\,}}{{\,\mathrm {sym}\,}}{{\,\mathrm {Curl}\,}}S=-2{{\,\mathrm {Div}\,}}{{\,\mathrm {skw}\,}}{{\,\mathrm {Curl}\,}}S=\mathrm {curl}{{\,\mathrm {Div}\,}}S^{\top }\)
-
\({{\,\mathrm{Curl}\,}}({{\,\mathrm{Curl}\,}}{{\,\mathrm{sym}\,}}S)^{\top }={{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}({{\,\mathrm{Curl}\,}}S)^{\top }\)
-
\({{\,\mathrm{Curl}\,}}({{\,\mathrm{Curl}\,}}{{\,\mathrm{skw}\,}}S)^{\top }={{\,\mathrm{skw}\,}}{{\,\mathrm{Curl}\,}}({{\,\mathrm{Curl}\,}}S)^{\top }\)
All formulas extend to distributions as well.
Proof
Almost all formulas can be found in [38, Lemma 3.9] and [38, Lemma A.1]. It is elementary to show that \({{\,\mathrm{skw}\,}}T=0\) implies \({{\,\mathrm{skw}\,}}{{\,\mathrm{Curl}\,}}({{\,\mathrm{Curl}\,}}T)^{\top }=0\), and that \({{\,\mathrm{sym}\,}}T=0\) implies \({{\,\mathrm{sym}\,}}{{\,\mathrm{Curl}\,}}({{\,\mathrm{Curl}\,}}T)^{\top }=0\). Note that the needed (straightforward-to-prove) formulas for this are provided in [39, Appendix B]. Hence \({{\,\mathrm{sym}\,}}\) commutes with \({{\,\mathrm{Curl}\,}}{{\,\mathrm{Curl}\,}}^{\top }\) as
and so does \({{\,\mathrm{skw}\,}}\). \(\square \)
In Lemma 12.11 below for a tensor field T the operation \(T{{\,\mathrm{d}\,}}\lambda :=\big (\langle \mathrm {row}_{\ell }T,{{\,\mathrm{d}\,}}\lambda \rangle \big )_{\ell =1,2,3}\) has to be understood row-wise, i.e., the transpose of the \(\ell \)th row of T is denoted by \(\mathrm {row}_{\ell }T\), yielding the vector object \(T{{\,\mathrm{d}\,}}\lambda \). More precisely,
with some parametrisation \(\varphi \in C^{1,3}_{\mathsf {pw}}\big ([0,1]\big )\) of a piecewise \(C^1\)-curve \(\zeta _{x_{0},x}\) connecting \(x_{0}\in \Omega \) and \(x\in \Omega \). Furthermore, we define
The first statement concerned with describing vector fields and their divergence by means of curve integrals over tensor fields reads as follows.
Lemma 12.11
Let \(x,x_{0}\in \Omega \) and let \(\zeta _{x_{0},x}\subseteq \Omega \) be a piecewise \(C^{1}\)-curve connecting \(x_{0}\) and x.
- (i):
-
Let \(v\in C^{\infty ,3}(\Omega )\). Then v and its divergence \({{\,\mathrm{div}\,}}v\) can be represented by
$$\begin{aligned}&v(x)-v(x_{0})-\frac{1}{3}\big ({{\,\mathrm{div}\,}}v(x_{0})\big )(x-x_{0})\\&\quad =\int _{\zeta _{x_{0},x}}{{\,\mathrm{devGrad}\,}}v{{\,\mathrm{d}\,}}\lambda +\frac{1}{2}\int _{\zeta _{x_{0},x}} \Big (\int _{\zeta _{x_{0},y}}\big \langle {{\,\mathrm{Div}\,}}({{\,\mathrm{devGrad}\,}}v)^{\top },{{\,\mathrm{d}\,}}\lambda \big \rangle \Big ){{\,\mathrm{Id}\,}}{{\,\mathrm{d}\,}}\lambda _{y} \end{aligned}$$and
$$\begin{aligned} {{\,\mathrm{div}\,}}v(x)-{{\,\mathrm{div}\,}}v(x_{0}) =\frac{3}{2}\int _{\zeta _{x_{0},x}}\big \langle {{\,\mathrm{Div}\,}}({{\,\mathrm{devGrad}\,}}v)^{\top },{{\,\mathrm{d}\,}}\lambda \big \rangle . \end{aligned}$$ - (ii):
-
Let \(T\in C^{\infty ,3\times 3}(\Omega )\). Then
$$\begin{aligned}&\int _{\zeta _{x_{0},x}} \Big (\int _{\zeta _{x_{0},y}}\langle {{\,\mathrm{Div}\,}}T^{\top },{{\,\mathrm{d}\,}}\lambda \rangle \Big ){{\,\mathrm{Id}\,}}{{\,\mathrm{d}\,}}\lambda _{y} \\&\quad =\int _{\zeta _{x_{0},x}} (x-y)\big \langle ({{\,\mathrm{Div}\,}}T^{\top })(y),{{\,\mathrm{d}\,}}\lambda _{y}\big \rangle . \end{aligned}$$
Proof
For (i), let
and observe \(3{{\,\mathrm{Div}\,}}T^{\top }=2{{\,\mathrm{grad}\,}}{{\,\mathrm{div}\,}}v\) by Lemma 12.10. Thus
Therefore,
Moreover,Footnote 6\(\displaystyle \int _{\zeta _{x_{0},x}}{{\,\mathrm{Id}\,}}{{\,\mathrm{d}\,}}\lambda _{y} =\int _{\zeta _{x_{0},x}}{{\,\mathrm{Grad}\,}}y{{\,\mathrm{d}\,}}\lambda _{y} =x-x_{0}\). For (ii) we compute
with \(\varphi \) parametrising \(\zeta _{x_{0},x}\) as above. \(\square \)
Proposition 12.12
Let \(x_{0}\in \Omega _{0}\in \mathsf {cc}(\Omega )\) and let \(S,T\in C^{\infty ,3\times 3}(\Omega _{0})\).
- (a):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega _{0}\) closed, piecewise \(C^1\)-curves
$$\begin{aligned} \int _{\zeta }\big \langle {{\,\mathrm{Div}\,}}T^{\top },{{\,\mathrm{d}\,}}\lambda \big \rangle =0. \end{aligned}$$ - (ii):
-
For all \(\zeta _{x_{0},x},{\widetilde{\zeta }}_{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curves connecting \(x_{0}\) with x
$$\begin{aligned} \int _{\zeta _{x_{0},x}}\big \langle {{\,\mathrm{Div}\,}}T^{\top },{{\,\mathrm{d}\,}}\lambda \big \rangle =\int _{{\widetilde{\zeta }}_{x_{0},x}}\big \langle {{\,\mathrm{Div}\,}}T^{\top },{{\,\mathrm{d}\,}}\lambda \big \rangle . \end{aligned}$$ - (iii):
-
There exists \(u\in C^{\infty }(\Omega _{0})\) such that \({{\,\mathrm{grad}\,}}u={{\,\mathrm{Div}\,}}T^{\top }\).
In the case one of the above conditions is true the function
for some \(\zeta _{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curve connecting \(x_{0}\) with x is a (well-defined) possible choice for u in (iii).
- (b):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega _{0}\) closed, piecewise \(C^1\) curves
$$\begin{aligned} \int _{\zeta }S{{\,\mathrm{d}\,}}\lambda =0. \end{aligned}$$ - (ii):
-
For all \(\zeta _{x_{0},x},{\widetilde{\zeta }}_{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\) curves connecting \(x_{0}\) with x
$$\begin{aligned} \int _{\zeta _{x_{0},x}}S{{\,\mathrm{d}\,}}\lambda =\int _{{\tilde{\zeta }}_{x_{0},x}}S{{\,\mathrm{d}\,}}\lambda . \end{aligned}$$ - (iii):
-
There exists \(v\in C^{\infty ,3}(\Omega _{0})\) such that \({{\,\mathrm{Grad}\,}}v=S\).
In the case one of the above conditions is true the vector field
for some \(\zeta _{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curve connecting \(x_{0}\) with x is a (well-defined) possible choice for v in (iii).
- (c):
-
Let \(S:=T+\frac{1}{2}\,u{{\,\mathrm{Id}\,}}\) with \(u\in C^{\infty }(\Omega _{0})\) and \({{\,\mathrm{grad}\,}}u={{\,\mathrm{Div}\,}}T^{\top }\) as in (a), (iii). Moreover, let \(v\in C^{\infty ,3}(\Omega _{0})\) such that \({{\,\mathrm{Grad}\,}}v=S\) as in (b), (iii). Then
- (i):
-
\({{\,\mathrm{tr}\,}}T=0\),
- (ii):
-
\({{\,\mathrm{devGrad}\,}}v =T\)
are equivalent. In either case, we have
Proof
(a): The conditions (i) and (ii) are clearly equivalent. Assuming (ii), we obtain that the choice of u in (12.7) is well-defined. By the fundamental theorem of calculus it follows that this u satisfies the equation in (iii) and, consequently, \(u\in C^{\infty }(\Omega _{0})\). If, on the other hand, (iii) is true, then again using the fundamental theorem of calculus, we obtain (ii).
(b): The proof of the equivalence follows almost exactly the same way as for (a).
(c): We compute \({{\,\mathrm{devGrad}\,}}v={{\,\mathrm{dev}\,}}S={{\,\mathrm{dev}\,}}(T+\frac{1}{2}u{{\,\mathrm{Id}\,}})={{\,\mathrm{dev}\,}}T\). Hence (i) and (ii) are equivalent. Finally, if (i) or (ii) is true, then by the complex property
and we conclude \({{\,\mathrm{grad}\,}}u={{\,\mathrm{Div}\,}}T^{\top }={{\,\mathrm{Div}\,}}({{\,\mathrm{devGrad}\,}}v)^{\top }=\frac{2}{3}{{\,\mathrm{grad}\,}}{{\,\mathrm{div}\,}}v\) by Lemma 12.10. \(\square \)
Remark 12.13
Related to Proposition 12.12 we note with Lemma 12.10 the following:
- (i):
-
For \(T\in C^{\infty ,3\times 3}(\Omega )\) we have
$$\begin{aligned} \text {curl}{{\,\mathrm{Div}\,}}T^{\top } =2{{\,\mathrm{Div}\,}}{{\,\mathrm{symCurl}\,}}T. \end{aligned}$$ - (ii):
-
For \(T\in C^{\infty ,3\times 3}_{{\mathbb {T}}}(\Omega )\) and \(S:=T+\frac{1}{2}\,u{{\,\mathrm{Id}\,}}\) with \({{\,\mathrm{grad}\,}}u={{\,\mathrm{Div}\,}}T^{\top }\) it holds
$$\begin{aligned} {{\,\mathrm{Curl}\,}}S ={{\,\mathrm{Curl}\,}}T-\frac{1}{2}{{\,\mathrm{spn}\,}}{{\,\mathrm{grad}\,}}u ={{\,\mathrm{Curl}\,}}T-{{\,\mathrm{skw}\,}}{{\,\mathrm{Curl}\,}}T ={{\,\mathrm{symCurl}\,}}T. \end{aligned}$$ - (iii):
-
If \(\Omega _{0}\) is simply connected, Proposition 12.12 (a), (iii) and (b), (iii) are equivalent to \(\text {curl}{{\,\mathrm{Div}\,}}T^{\top }=0\) and \({{\,\mathrm{Curl}\,}}S=0\), respectively.
Arguing for each connected component separately (and using formulas (12.7) and (12.8) on every connected component), we obtain the following more condensed version of Proposition 12.12.
Corollary 12.14
Let \(S,T\in C^{\infty ,3\times 3}(\Omega )\).
- (a):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega \) closed, piecewise \(C^1\)-curves \(\int _{\zeta }\big \langle {{\,\mathrm{Div}\,}}T^{\top },{{\,\mathrm{d}\,}}\lambda \big \rangle =0\).
- (ii):
-
There exists \(u\in C^{\infty }(\Omega )\) such that \({{\,\mathrm{grad}\,}}u={{\,\mathrm{Div}\,}}T^{\top }\).
- (b):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega \) closed, piecewise \(C^1\)-curves \(\int _{\zeta }S{{\,\mathrm{d}\,}}\lambda =0\).
- (ii):
-
There exists \(v\in C^{\infty ,3}(\Omega )\) such that \({{\,\mathrm{Grad}\,}}v=S\).
- (c):
-
Let \(S=T+\frac{1}{2}\,u{{\,\mathrm{Id}\,}}\) with \(u\in C^{\infty }(\Omega )\) and \({{\,\mathrm{grad}\,}}u={{\,\mathrm{Div}\,}}T^{\top }\) as in (a), (ii). Moreover, let \(v\in C^{\infty ,3}(\Omega )\) with \({{\,\mathrm{Grad}\,}}v=S\) as in (b), (ii). Then \({{\,\mathrm{tr}\,}}T=0\) in \(\Omega \) if and only if \({{\,\mathrm{devGrad}\,}}v=T\) in \(\Omega \).
The construction of the harmonic Neumann tensor fields for the first biharmonic complex forms a nontrivial adaptation of the rationale developed in the previous section for the de Rham complex. We shortly recall that for \(j\in \{1,\ldots ,p\}\), by construction, \(\theta _{j}=0\) outside of \({\widetilde{\Upsilon }}_{j,1}\) and that \(\theta _{j}\) is constant on each connected component \(\Upsilon _{j,0}\) and \(\Upsilon _{j,1}\) of \(\Upsilon _{j}{\setminus } F_{j}\). Let \({\widehat{r}}_{k}\) be the Raviart–Thomas fields from Sect. 11.3 given by \({\widehat{r}}_{0}(x):=x\) and \({\widehat{r}}_{k}(x):=e^{k}\) for \(k\in \{1,2,3\}\). We define the vector fields \(\theta _{j,k}:=\theta _{j}{\widehat{r}}_{k}\) and note \({{\,\mathrm{devGrad}\,}}\theta _{j,k}=0\) in \(\dot{\bigcup }_{l\in \{1,\ldots ,p\}}\Upsilon _{l}{\setminus } F_{l}\) for all \(j\in \{1,\ldots ,p\}\) and all \(k\in \{1,2,3\}\). Thus \({{\,\mathrm{devGrad}\,}}\theta _{j,k}\) can be continuously extended by zero to \(\Theta _{j,k}\in C^{\infty ,3\times 3}(\Omega )\cap L^{2,3\times 3}_{{\mathbb {T}}}(\Omega )\) with \(\Theta _{j,k}=0\) in all the neighbourhoods \(\Upsilon _{l}\) of all the surfaces \(F_{l}\), \(l\in \{1,\ldots ,p\}\).
Lemma 12.15
Let Assumption 10.3 be satisfied. Then \(\Theta _{j,k}\in {\text {ker}}({{\,\mathrm{symCurl}\,}}_{{\mathbb {T}}},\Omega )\).
Proof
Let \(\Phi \in C^{\infty ,3\times 3}_{c,{\mathbb {S}}}(\Omega )\). As \({{\,\mathrm{supp}\,}}\Theta _{j,k}\subseteq {\widetilde{\Upsilon }}_{j}{\setminus }\Upsilon _{j}\) we can pick another cut-off function \(\varphi \in C^{\infty }_{c}(\Omega _{F})\) with \(\varphi |_{{{\,\mathrm{supp}\,}}\Theta _{j,k}\cap {{\,\mathrm{supp}\,}}\Phi }=1\). Then
as \(\varphi \Phi \in C^{\infty ,3\times 3}_{c}(\Omega _{F})\), where in the second to last equality sign, we used that the \({{\,\mathrm{Curl}\,}}\) applied to a symmetric tensor fields is trace-free, i.e., deviatoric, see Lemma 12.10. \(\square \)
Next, we note that for \(l,j\in \{1,\dots ,p\}\) and \(k\in \{0,\dots ,3\}\) and for the curves \(\zeta _{x_{l,0},x_{l,1}}\subseteq \zeta _{l}\) with the chosen starting points \(x_{l,0}\in \Upsilon _{l,0}\) and respective endpoints \(x_{l,1}\in \Upsilon _{l,1}\) we can compute by Lemma 12.11
and
Thus, for \(l\in \{1,\dots ,p\}\) and \(\ell \in \{0,\dots ,3\}\) we have functionals \(\beta _{l,\ell }\), given by
for \(l,j\in \{1,\dots ,p\}\) and \(k\in \{0,\dots ,3\}\), as well as
for \(l,j\in \{1,\dots ,p\}\) and \(\ell \in \{1,2,3\}\) and \(k\in \{0,\dots ,3\}\). Therefore, we have
Let Assumption 10.2 be satisfied. For the first biharmonic complex, similar to (3.3), (3.5), we have the orthogonal decompositions
Remark 12.16
By Assumption 10.2, [38, Lemma 3.2] yields \({{\,\mathrm{dom}\,}}({{\,\mathrm{devGrad}\,}},\Omega )=H^{1,3}(\Omega )\). As a consequence using Rellich’s selection theorem, the range in (12.11) is closed and the Poincaré type estimate
holds, see also [38, Lemma 3.2].
Let \(\pi :L^{2,3\times 3}_{{\mathbb {T}}}(\Omega )\rightarrow {\text {ker}}(\mathring{{{\,\mathrm{Div}\,}}}_{{\mathbb {T}}},\Omega )\) denote the orthogonal projector onto \({\text {ker}}(\mathring{{{\,\mathrm{Div}\,}}}_{{\mathbb {T}}},\Omega )\) along \({{\,\mathrm{ran}\,}}({{\,\mathrm{devGrad}\,}},\Omega )\), see (12.11). We have \(\pi \big ({\text {ker}}({{\,\mathrm{symCurl}\,}}_{{\mathbb {T}}},\Omega )\big )={\mathcal {H}}_{N,{\mathbb {T}}}^{\mathsf {bih},1}(\Omega )\). By Lemma 12.15 there exists some \(\psi _{j,k}\in H^{1,3}(\Omega )\) such that
As \({\mathcal {H}}_{N,{\mathbb {T}}}^{\mathsf {bih},1}(\Omega )\subseteq C^{\infty ,3\times 3}(\Omega )\), cf. (10.1), we conclude by \(\pi \Theta _{j,k},\Theta _{j,k}\in C^{\infty ,3\times 3}(\Omega )\) that also \({{\,\mathrm{devGrad}\,}}\psi _{j,k}\in C^{\infty ,3\times 3}(\Omega )\) and hence \(\psi _{j,k}\in C^{\infty ,3}(\Omega )\). Thus all path integrals over the closed curves \(\zeta _{l}\) are well-defined. Furthermore, we observe by Lemma 12.11
and
Therefore, \(\beta _{l,\ell }({{\,\mathrm{devGrad}\,}}\psi _{j,k})=0\) and by (12.10) we have
for all \(l,j\in \{1,\dots ,p\}\) and all \(\ell ,k\in \{0,1,2,3\}\). We shall show that
defines a basis of \({\mathcal {H}}_{N,{\mathbb {T}}}^{\mathsf {bih},1}(\Omega )\).
Remark 12.17
(Characterisation by PDEs) Note that \(\psi _{j,k}\in H^{1,3}(\Omega )\cap \mathsf {RT}_{\mathsf {pw}}^{\bot _{L^{2,3}(\Omega )}}\) can be found by the variational formulation
i.e., \(\psi _{j,k}=\Delta _{{\mathbb {T}},N}^{-1}\big ({{\,\mathrm{Div}\,}}_{{\mathbb {T}}}\Theta _{j,k},\Theta _{j,k}\nu |_{\Gamma }\big )\), where \(\Delta _{{\mathbb {T}},N}\subseteq {{\,\mathrm{Div}\,}}_{{\mathbb {T}}}{{\,\mathrm{devGrad}\,}}\) denotes the ‘deviatoric’ Laplacian with inhomogeneous Neumann boundary conditions restricted to a subset of \(H^{1,3}(\Omega )\cap \mathsf {RT}_{\mathsf {pw}}^{\bot _{L^{2,3}(\Omega )}}\). Therefore,
In classical terms, \(\psi _{j,k}\) solves the Neumann elasticity type problem
which is uniquely solvable.
Lemma 12.18
Let Assumption 10.2 as well as Assumption 10.3 be satisfied. Then we have \({\mathcal {H}}_{N,{\mathbb {T}}}^{\mathsf {bih},1}(\Omega )={{\,\mathrm{lin}\,}}{\mathcal {B}}^{\mathsf {bih},1}_{N}\).
Proof
Let \(H\in {\mathcal {H}}_{N,{\mathbb {T}}}^{\mathsf {bih},1}(\Omega )={\text {ker}}(\mathring{{{\,\mathrm{Div}\,}}}_{{\mathbb {T}}},\Omega )\cap {\text {ker}}({{\,\mathrm{symCurl}\,}}_{{\mathbb {T}}},\Omega )\subseteq C^{\infty ,3\times 3}_{{\mathbb {T}}}(\Omega )\), see Lemma 12.2. With the above introduced functionals \(\beta _{l,0}\) and \(b_{l}\), \(l\in \{1,\dots ,p\}\), we recall
and define for \(l\in \{1,\dots ,p\}\) and \(\ell \in \{1,2,3\}\) the numbers
We shall show that
Similar to the proof of Lemma 12.6, the aim is to prove the existence of \(v\in H^{1,3}(\Omega )\) such that \({{\,\mathrm{devGrad}\,}}v={\widehat{H}}\), since then
For finding v, we will apply Corollary 12.14 and Remark 12.13 to \(T={\widehat{H}}\). By (12.12) we observe for all \(l\in \{1,\ldots ,p\}\)
Note that \({{\,\mathrm{Div}\,}}{\widehat{H}}^{\top }\in {\text {ker}}(\text {curl},\Omega )\cap C^{\infty ,3}(\Omega )\) by Remark 12.13 (i) as \({\widehat{H}}\) belongs to \({\text {ker}}({{\,\mathrm{symCurl}\,}}_{{\mathbb {T}}},\Omega )\cap C^{\infty ,3\times 3}(\Omega )\). Thus, by Assumption 10.3 (A.1) for any closed piecewise \(C^1\)-curve \(\zeta \) in \(\Omega \)
Let \(u\in C^{\infty }(\Omega )\) be as in Corollary 12.14 (a), (ii), i.e., \({{\,\mathrm{grad}\,}}u={{\,\mathrm{Div}\,}}{\widehat{H}}^{\top }\), and define \(S:\Omega \rightarrow {\mathbb {R}}^{3\times 3}\) by
Our next aim is to show condition (b), (ii) of Corollary 12.14. For this, let \(l\in \{1,\ldots ,p\}\). Note that \(\zeta _{x_{l,0},x_{l,1}}\subseteq \zeta _{l}\subseteq \Omega _{0}\) for some \(\Omega _{0}\in \mathsf {cc}(\Omega )\). Then we have with \(c:=u(x_{l,1})\in {\mathbb {R}}\) for all \(x\in \zeta _{l}\)
where \(\zeta _{x_{l,1},x}\) denotes the path from \(x_{l,1}\) to x along \(\zeta _{l}\). Moreover,
Next, we consider the closed curve \(\zeta _{l}\) as the closed curve \(\zeta _{x_{l,1},x_{l,1}}\) with circulation 1 along \(\zeta _{l}\). Then, using Lemma 12.11 and the definition of \(b_{l}\), we compute
Hence, for \(\ell \in \{1,2,3\}\) we get by (12.12) recalling (12.15)
Therefore, \(\int _{\zeta _{l}}S{{\,\mathrm{d}\,}}\lambda =0\) for all \(l\in \{1,\ldots ,p\}\). Note that \(S\in {\text {ker}}({{\,\mathrm{Curl}\,}},\Omega )\cap C^{\infty ,3\times 3}(\Omega )\) by Remark 12.13 (ii) as \({\widehat{H}}\in {\text {ker}}({{\,\mathrm{symCurl}\,}}_{{\mathbb {T}}},\Omega )\cap C^{\infty ,3\times 3}_{{\mathbb {T}}}(\Omega )\). Thus, by Assumption 10.3 (A.1) for any closed piecewise \(C^1\)-curve \(\zeta \) in \(\Omega \)
Hence, Corollary 12.14 (b) and (c) (note \({{\,\mathrm{tr}\,}}{\widehat{H}}=0\)) imply the existence of a smooth vector field \(v:\Omega \rightarrow {\mathbb {R}}^{3}\) such that \({{\,\mathrm{devGrad}\,}}v={\widehat{H}}\). Finally, similar to the end of the proof of Lemma 12.6, elliptic regularity and, e.g., [22, Theorem 2.6 (1)] or [1, Theorem 3.2 (2)], show that \(v\in C^{\infty ,3}(\Omega _{0})\) and \({{\,\mathrm{devGrad}\,}}v\in L^{2,3\times 3}_{{\mathbb {T}}}(\Omega _{0})\) imply \(v\in H^{1,3}(\Omega _{0})\) for all \(\Omega _{0}\in \mathsf {cc}(\Omega )\) and thus \(v\in H^{1,3}(\Omega )\), completing the proof. (Let us note that \(v\in H^{1,3}(\Omega )\) implies also \(S\in L^{2,3\times 3}(\Omega )\) and hence \(u\in L^{2}(\Omega )\).) \(\square \)
Lemma 12.19
Let Assumptions 10.2 and 10.3 be satisfied. Then \({\mathcal {B}}^{\mathsf {bih},1}_{N}\) is linearly independent.
Proof
Let \(\gamma _{j,k}\in {\mathbb {R}}\), \(j\in \{1,\ldots ,p\}\), \(k\in \{0,\ldots ,3\}\), be such that \(\sum _{j=1}^{p}\sum _{k=0}^{3}\gamma _{j,k}\pi \Theta _{j,k}=0\). Then (12.12) implies for \(l\in \{1,\dots ,p\}\)
finishing the proof. \(\square \)
Theorem 12.20
Let Assumptions 10.2 and 10.3 be satisfied. Then \(\dim {\mathcal {H}}_{N,{\mathbb {T}}}^{\mathsf {bih},1}(\Omega )=4p\) and a basis of \({\mathcal {H}}_{N,{\mathbb {T}}}^{\mathsf {bih},1}(\Omega )\) is given by (12.13).
Proof
Use Lemmas 12.18 and 12.19. \(\square \)
12.3 Neumann tensor fields of the second biharmonic complex
The rationale for the second biharmonic complex in comparison to the first one has to be changed appropriately. For this we also use Lemma 12.10, the Poincaré maps, however, differ from one another.
In Lemma 12.21 below for a tensor field S and a parametrisation \(\varphi \in C^{1,3}_{\mathsf {pw}}\big ([0,1]\big )\) of a curve \(\zeta \) we define
Lemma 12.21
Let \(x,x_{0}\in \Omega \) and let \(\zeta _{x_{0},x}\subseteq \Omega \) be a piecewise \(C^{1}\)-curve connecting \(x_{0}\) and x.
- (i):
-
Let \(u\in C^{\infty }(\Omega )\). Then u and its gradient \({{\,\mathrm{grad}\,}}u\) can be represented by
$$\begin{aligned} u(x)-u(x_{0})-\big \langle {{\,\mathrm{grad}\,}}u(x_{0}),x-x_{0}\big \rangle&=\int _{\zeta _{x_{0},x}}\big \langle \int _{\zeta _{x_{0},y}}{\text {Gradgrad}}u{{\,\mathrm{d}\,}}\lambda ,{{\,\mathrm{d}\,}}\lambda _{y}\big \rangle \end{aligned}$$and
$$\begin{aligned} {{\,\mathrm{grad}\,}}u(x)-{{\,\mathrm{grad}\,}}u(x_{0})&=\int _{\zeta _{x_{0},x}}{\text {Gradgrad}}u{{\,\mathrm{d}\,}}\lambda . \end{aligned}$$ - (ii):
-
For all \(S\in C^{\infty ,3\times 3}(\Omega )\)
$$\begin{aligned} \int _{\zeta _{x_{0},x}}\big \langle \int _{\zeta _{x_{0},y}}S{{\,\mathrm{d}\,}}\lambda ,{{\,\mathrm{d}\,}}\lambda _{y}\big \rangle =\int _{\zeta _{x_{0},x}}\big \langle x-y,S(y){{\,\mathrm{d}\,}}\lambda _{y}\big \rangle . \end{aligned}$$
Proof
For (i),
i.e.,
Therefore,
Using \(\varphi \in C^{1,3}_{\mathsf {pw}}\big ([0,1]\big )\) as a parametrisation of \(\zeta _{x_{0},x}\), we conclude the proof of (i) by
For (ii), we compute
again with \(\varphi \) parametrising \(\zeta _{x_{0},x}\). \(\square \)
Proposition 12.22
Let \(x_{0}\in \Omega _{0}\in \mathsf {cc}(\Omega )\) and let \(w\in C^{\infty ,3}(\Omega _{0})\) and \(S\in C^{\infty ,3\times 3}(\Omega _{0})\).
- (a):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega _{0}\) closed, piecewise \(C^1\)-curves
$$\begin{aligned} \int _{\zeta }S{{\,\mathrm{d}\,}}\lambda =0. \end{aligned}$$ - (ii):
-
For all \(\zeta _{x_{0},x},{\widetilde{\zeta }}_{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curves connecting \(x_{0}\) with x
$$\begin{aligned} \int _{\zeta _{x_{0},x}}S{{\,\mathrm{d}\,}}\lambda =\int _{{\widetilde{\zeta }}_{x_{0},x}}S {{\,\mathrm{d}\,}}\lambda . \end{aligned}$$ - (iii):
-
There exists \(v\in C^{\infty ,3}(\Omega _{0})\) such that \({{\,\mathrm{Grad}\,}}v=S\).
In the case one of the above conditions is true the vector field
$$\begin{aligned} x\mapsto v(x)=\int _{\zeta _{x_{0},x}}S{{\,\mathrm{d}\,}}\lambda \end{aligned}$$(12.18)for some \(\zeta _{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curve connecting \(x_{0}\) with x is a (well-defined) possible choice for v in (iii).
- (b):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega _{0}\) closed, piecewise \(C^1\) curves
$$\begin{aligned} \int _{\zeta }\langle w,{{\,\mathrm{d}\,}}\lambda \rangle =0. \end{aligned}$$ - (ii):
-
For all \(\zeta _{x_{0},x},{\widetilde{\zeta }}_{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\) curves connecting \(x_{0}\) with x
$$\begin{aligned} \int _{\zeta _{x_{0},x}}\langle w,{{\,\mathrm{d}\,}}\lambda \rangle =\int _{{\widetilde{\zeta }}_{x_{0},x}}\langle w,{{\,\mathrm{d}\,}}\lambda \rangle . \end{aligned}$$ - (iii):
-
There exists \(u\in C^{\infty }(\Omega _{0})\) such that \({{\,\mathrm{grad}\,}}u=w\).
In the case one of the above conditions is true the function
$$\begin{aligned} x\mapsto u(x)=\int _{\zeta _{x_{0},x}}\big \langle w,{{\,\mathrm{d}\,}}\lambda \big \rangle \end{aligned}$$(12.19)for some \(\zeta _{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curve connecting \(x_{0}\) with x is a (well-defined) possible choice for u in (iii).
- (c):
-
Let \(v\in C^{\infty ,3}(\Omega _{0})\) with \({{\,\mathrm{Grad}\,}}v=S\) as in (a), (iii) and let \(u\in C^{\infty }(\Omega _{0})\) with \({{\,\mathrm{grad}\,}}u=v\) as in (b), (iii). Then \({\text {Gradgrad}}u=S\), \({{\,\mathrm{skw}\,}}S=0\), and \({{\,\mathrm{Curl}\,}}_{{\mathbb {S}}}S=0\).
Proof
The statements in (a) and (b) are straightforward consequences of the fundamental theorem of calculus and follow essentially the same lines as (a) and (b) of Proposition 12.12. Schwarz’s Lemma and the complex property show (c). \(\square \)
Remark 12.23
Related to Proposition 12.22 we note with Lemma 12.10 the following:
- (i):
-
For \(v\in C^{\infty ,3}(\Omega )\) we have \(\text {curl}v=2{{\,\mathrm{spn}\,}}^{-1}{{\,\mathrm{skw}\,}}{{\,\mathrm{Grad}\,}}v\).
- (ii):
-
If \(\Omega _{0}\) is simply connected, Proposition 12.22 (a), (iii) and (b), (iii) are equivalent to \({{\,\mathrm{Curl}\,}}S=0\) and \(\text {curl}w=0\), respectively.
Similar to the first biharmonic complex, there exists an analogous version of Proposition 12.22 irrespective of the components. We only formulate the following slightly weaker statement, which is an easy consequence of Proposition 12.22.
Corollary 12.24
Let \(w\in C^{\infty ,3}(\Omega )\) and \(S\in C^{\infty ,3\times 3}(\Omega )\).
- (a):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega \) closed, piecewise \(C^1\)-curves \(\int _{\zeta }S{{\,\mathrm{d}\,}}\lambda =0\).
- (ii):
-
There exists \(v\in C^{\infty ,3}(\Omega )\) such that \({{\,\mathrm{Grad}\,}}v=S\).
- (b):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega \) closed, piecewise \(C^1\)-curves \(\int _{\zeta }\langle w,{{\,\mathrm{d}\,}}\lambda \rangle =0\).
- (ii):
-
There exists \(u\in C^{\infty }(\Omega )\) such that \({{\,\mathrm{grad}\,}}u=w\).
- (c):
-
Let \(v\in C^{\infty ,3}(\Omega )\) with \({{\,\mathrm{Grad}\,}}v=S\) as in (a), (ii) and let \(u\in C^{\infty }(\Omega )\) with \({{\,\mathrm{grad}\,}}u=v\) as in (b), (ii). Then \({\text {Gradgrad}}u=S\), \({{\,\mathrm{skw}\,}}S=0\), and \({{\,\mathrm{Curl}\,}}_{{\mathbb {S}}}S=0\).
Next, we turn to the actual construction of the Neumann fields for the second biharmonic complex. Let \(j\in \{1,\ldots ,p\}\). For this, recall from the beginning of Sect. 12 that \(\theta _{j}\) is constant on each connected component \(\Upsilon _{j,0}\) and \(\Upsilon _{j,1}\) of \(\Upsilon _{j}{\setminus } F_{j}\) and vanishes outside of \({\widetilde{\Upsilon }}_{j,1}\). Moreover, let \({\widehat{p}}_{k}\) be the polynomials from Sect. 11.2 given by \({\widehat{p}}_{0}(x):=1\) and \({\widehat{p}}_{k}(x):=x_{k}\) for \(k\in \{1,2,3\}\). We define the functions \(\theta _{j,k}:=\theta _{j}{\widehat{p}}_{k}\) and note \({\text {Gradgrad}}\theta _{j,k}=0\) in \(\dot{\bigcup }_{l\in \{1,\ldots ,p\}}\Upsilon _{l}{\setminus } F_{l}\) for all \(k\in \{1,2,3\}\). Thus \({\text {Gradgrad}}\theta _{j,k}\) can be continuously extended by zero to \(\Theta _{j,k}\in C^{\infty ,3\times 3}(\Omega )\cap L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\) with \(\Theta _{j,k}=0\) in all the neighbourhoods \(\Upsilon _{l}\) of all the surfaces \(F_{l}\), \(l\in \{1,\ldots ,p\}\).
Lemma 12.25
Let Assumption 10.3 be satisfied. Then \(\Theta _{j,k}\in {\text {ker}}({{\,\mathrm{Curl}\,}}_{{\mathbb {S}}},\Omega )\).
Proof
Let \(\Phi \in C^{\infty ,3\times 3}_{c,{\mathbb {T}}}(\Omega )\). As \({{\,\mathrm{supp}\,}}\Theta _{j,k}\subseteq {\widetilde{\Upsilon }}_{j}{\setminus }\Upsilon _{j}\) we can pick another cut-off function \(\varphi \in C^{\infty }_{c}(\Omega _{F})\) with \(\varphi |_{{{\,\mathrm{supp}\,}}\Theta _{j,k}\cap {{\,\mathrm{supp}\,}}\Phi }=1\). Then
as \(\varphi \Phi ,{{\,\mathrm{Curl}\,}}(\varphi \Phi )\in C^{\infty ,3\times 3}_{c}(\Omega _{F})\). \(\square \)
Similar to the first biharmonic complex, we introduce a set of functionals.
For \(l,j\in \{1,\dots ,p\}\) and \(k\in \{0,\dots ,3\}\) and for the curves \(\zeta _{x_{l,0},x_{l,1}}\subseteq \zeta _{l}\) with the chosen starting points \(x_{l,0}\in \Upsilon _{l,0}\) and respective endpoints \(x_{l,1}\in \Upsilon _{l,1}\) we can compute by Lemma 12.21
and
Thus, for \(l\in \{1,\dots ,p\}\) and \(\ell \in \{0,\dots ,3\}\) we have functionals \(\beta _{l,\ell }\) given by
for \(l,j\in \{1,\dots ,p\}\) and \(\ell \in \{1,2,3\}\) and \(k\in \{0,1,2,3\}\), as well as
for \(l,j\in \{1,\dots ,p\}\) and \(k\in \{0,1,2,3\}\). Therefore, we have
Let Assumption 10.2 be satisfied. For the second biharmonic complex, similar to (3.3), (3.5), we have the orthogonal decompositions
Remark 12.26
Lemma 8.2 shows \({{\,\mathrm{dom}\,}}({\text {Gradgrad}},\Omega )=H^{2}(\Omega )\). Thus, employing a contradiction argument together with Rellich’s selection theorem, we obtain the Poincaré type estimate
as Assumption 10.2 holds. Thus, the range in (12.21) is closed.
Let \(\pi :L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\rightarrow {\text {ker}}(\mathring{{\text {divDiv}}}_{{\mathbb {S}}},\Omega )\) denote the orthogonal projector onto \({\text {ker}}(\mathring{{\text {divDiv}}}_{{\mathbb {S}}},\Omega )\) along \({{\,\mathrm{ran}\,}}({\text {Gradgrad}},\Omega )\), see (12.21). In particular, \(\pi \big ({\text {ker}}({{\,\mathrm{Curl}\,}}_{{\mathbb {S}}},\Omega )\big )={\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {bih},2}(\Omega )\). By Lemma 12.25 there exists some \(\psi _{j,k}\in H^{2}(\Omega )\) such that
As \({\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {bih},2}(\Omega )\subseteq C^{\infty ,3\times 3}(\Omega )\), see Lemma 12.2, we conclude by \(\pi \Theta _{j,k},\Theta _{j,k}\in C^{\infty ,3\times 3}(\Omega )\) that also \({\text {Gradgrad}}\psi _{j,k}\in C^{\infty ,3\times 3}(\Omega )\) and hence \(\psi _{j,k}\in C^{\infty }(\Omega )\). Hence all path integrals over the closed curves \(\zeta _{l}\) are well-defined. Furthermore, we observe by Lemma 12.21
and
Therefore, \(\beta _{l,\ell }({\text {Gradgrad}}\psi _{j,k})=0\) and by (12.20) we get
for all \(l,j\in \{1,\dots ,p\}\) and all \(\ell ,k\in \{0,1,2,3\}\). We shall show that
defines a basis of \({\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {bih},2}(\Omega )\).
Remark 12.27
(Characterisation by PDEs) Let \(j\in \{1,\dots ,p\}\) and \(k\in \{0,\ldots ,3\}\). Then \(\psi _{j,k}\in H^{2}(\Omega )\cap ({\mathsf {P}}^{1}_{\mathsf {pw}})^{\bot _{L^{2}(\Omega )}}\) can be found by the variational formulation
i.e., \(\psi _{j,k}=(\Delta _{NN}^{2})^{-1}\big ({\text {divDiv}}_{{\mathbb {S}}}\Theta _{j,k},\Theta _{j,k}\nu |_{\Gamma },\nu \cdot {{\,\mathrm{Div}\,}}\Theta _{j,k}|_{\Gamma }\big )\), where \(\Delta _{NN}^{2}\subseteq {\text {divDiv}}_{{\mathbb {S}}}{\text {Gradgrad}}\) is the bi-Laplacian with inhomogeneous Neumann type boundary conditions restricted to a subset of \(H^{2}(\Omega )\cap ({\mathsf {P}}^{1}_{\mathsf {pw}})^{\bot _{L^{2}(\Omega )}}\). Therefore,
In classical terms, \(\psi _{j,k}\) solves the biharmonic Neumann problem
which is uniquely solvable.
Lemma 12.28
Let Assumptions 10.2 and 10.3 be satisfied. Then \({\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {bih},2}(\Omega )={{\,\mathrm{lin}\,}}{\mathcal {B}}^{\mathsf {bih},2}_{N}\).
Proof
Let \(H\in {\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {bih},2}(\Omega )={\text {ker}}(\mathring{{\text {divDiv}}}_{{\mathbb {S}}},\Omega )\cap {\text {ker}}({{\,\mathrm{Curl}\,}}_{{\mathbb {S}}},\Omega )\subseteq C^{\infty ,3\times 3}_{{\mathbb {S}}}(\Omega )\), see Lemma 12.2. With the above introduced functions \(\beta _{l,0}\) and \(b_{l}\), \(l\in \{1,\dots ,p\}\), we recall
and define for \(l\in \{1,\dots ,p\}\) the numbers
We shall show that
Similar to the proof of Lemma 12.6, the aim is to prove that there exists \(u\in H^{2}(\Omega )\) such that \({\text {Gradgrad}}u={\widehat{H}}\), since then
For this, we shall apply Corollary 12.24 and Remark 12.23 to \(S={\widehat{H}}\). By (12.22) we observe for \(\ell \in \{1,2,3\}\) and \(l\in \{1,\ldots ,p\}\)
Note that \({\widehat{H}}\in {\text {ker}}({{\,\mathrm{Curl}\,}},\Omega )\cap C^{\infty ,3\times 3}(\Omega )\). Thus by Assumption 10.3 (A.1) for any closed piecewise \(C^1\)-curve \(\zeta \) in \(\Omega \)
By Corollary 12.24 (a), we find \(v\in C^{\infty ,3}(\Omega )\) such that \({{\,\mathrm{Grad}\,}}v={\widehat{H}}\). Next, let \(l\in \{1,\ldots ,p\}\). Then, with \(\zeta _{x_{l,0},x_{l,1}}\subseteq \zeta _{l}\subseteq \Omega _{0}\) for some \(\Omega _{0}\in \mathsf {cc}(\Omega )\), we obtain with \(c:=v(x_{l,1})\in {\mathbb {R}}^{3}\) for all \(x\in \zeta _{l}\)
and
We consider the closed curve \(\zeta _{l}\) as the closed curve \(\zeta _{x_{l,1},x_{l,1}}\) with circulation 1 along \(\zeta _{l}\). By Lemma 12.21, the definition of \(\beta _{l,0}\), and (12.22) we have
Note that \(v\in {\text {ker}}(\text {curl},\Omega )\cap C^{\infty ,3}(\Omega )\) by Remark 12.23 (i) as \({{\,\mathrm{Grad}\,}}v={\widehat{H}}\in L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\). Therefore, by Assumption 10.3 (A.1) for any closed piecewise \(C^1\)-curve \(\zeta \) in \(\Omega \)
Hence, by Corollary 12.24 (b), we find \(u\in C^{\infty }(\Omega )\) with \({{\,\mathrm{grad}\,}}u=v\) and thus
Similar to the end of the proof of Lemma 12.6, elliptic regularity and, e.g., [22, Theorem 2.6 (1)] or [1, Theorem 3.2 (2)] show that \(v\in C^{\infty ,3}(\Omega )\) together with \({{\,\mathrm{Grad}\,}}v\in L^{2,3\times 3}(\Omega )\) imply \(v\in H^{1,3}(\Omega )\). Then, analogously, \(u\in C^{\infty }(\Omega )\) and \({{\,\mathrm{grad}\,}}u=v\in L^{2,3}(\Omega )\) imply \(u\in H^{1}(\Omega )\) and hence \(u\in H^{2}(\Omega )\), completing the proof. \(\square \)
Lemma 12.29
Let Assumptions 10.2 and 10.3 be satisfied. Then \({\mathcal {B}}^{\mathsf {bih},2}_{N}\) is linearly independent.
Proof
Let \(\gamma _{j,k}\in {\mathbb {R}}\), \(j\in \{1,\ldots ,p\}\), and \(k\in \{0,\ldots ,3\}\) be such that \(\displaystyle \sum _{j=1}^{p}\sum _{k=0}^{3}\gamma _{j,k}\pi \Theta _{j,k}=0\). Then (12.22) implies for \(l\in \{1,\dots ,p\}\)
finishing the proof. \(\square \)
Theorem 12.30
Let Assumptions 10.2 and 10.3 be satisfied. Then \(\dim {\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {bih},2}(\Omega )=4p\) and a basis of \({\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {bih},2}(\Omega )\) is given by (12.23).
Proof
Use Lemmas 12.28 and 12.29. \(\square \)
12.4 Neumann tensor fields of the elasticity complex
The concluding example for our general construction principle is the elasticity complex. Again, we require some preparations regarding integration along curves and the operators involved in the elasticity complex. We need the formulas providing the interaction of matrix and vector analytic operations as outlined in Lemma 12.10 (in particular, note we defined \({{\,\mathrm{spn}\,}}\) there.)
In Lemma 12.31 below for a tensor field S and a parametrisation \(\varphi \in C^{1,3}_{\mathsf {pw}}\big ([0,1]\big )\) of a curve \(\zeta \) in \(\Omega \) we define
Lemma 12.31
Let \(x,x_{0}\in \Omega \) and let \(\zeta _{x_{0},x}\subseteq \Omega \) be a piecewise \(C^{1}\)-curve connecting \(x_{0}\) to x.
- (i):
-
Let \(v\in C^{\infty ,3}(\Omega )\). Then v and its rotation \(\text {curl}v\) can be represented by
$$\begin{aligned}&v(x)-v(x_{0})-\frac{1}{2}\big (\mathrm {curl}\,v(x_{0})\big )\times (x-x_{0})\\&\quad =\int _{\zeta _{x_{0},x}}{{\,\mathrm {symGrad}\,}}v{{\,\mathrm {d}\,}}\lambda +\int _{\zeta _{x_{0},x}}\int _{\zeta _{x_{0},y}} {{\,\mathrm {spn}\,}}\big (({{\,\mathrm {Curl}\,}}{{\,\mathrm {symGrad}\,}}v)^{\top }{{\,\mathrm {d}\,}}\lambda \big ){{\,\mathrm {d}\,}}\lambda _{y} \end{aligned}$$and
$$\begin{aligned} \mathrm {curl}\,v(x)-\mathrm {curl}\,v(x_{0}) =2\int _{\zeta _{x_{0},x}}({{\,\mathrm {Curl}\,}}{{\,\mathrm {symGrad}\,}}v)^{\top }{{\,\mathrm {d}\,}}\lambda . \end{aligned}$$ - (ii):
-
Let \(S\in C^{\infty ,3\times 3}(\Omega )\). Then
$$\begin{aligned}&\int _{\zeta _{x_{0},x}}\int _{\zeta _{x_{0},y}} {{\,\mathrm{spn}\,}}\big (({{\,\mathrm{Curl}\,}}S)^{\top }{{\,\mathrm{d}\,}}\lambda \big ){{\,\mathrm{d}\,}}\lambda _{y} \\&\quad =\int _{\zeta _{x_{0},x}} {{\,\mathrm{spn}\,}}\big (({{\,\mathrm{Curl}\,}}S)^{\top }(y){{\,\mathrm{d}\,}}\lambda _{y}\big )(x-y). \end{aligned}$$
Proof
For (i), let
and observe \(2{{\,\mathrm{Curl}\,}}S=-2{{\,\mathrm{Curl}\,}}{{\,\mathrm{skw}\,}}{{\,\mathrm{Grad}\,}}v=({{\,\mathrm{Grad}\,}}\text {curl}v)^{\top }\) by Lemma 12.10. Thus
Therefore, by Lemma 12.10
Moreover, with \(\varphi \in C^{1,3}_{\mathsf {pw}}\big ([0,1]\big )\) parametrising \(\zeta _{x_{0},x}\)Footnote 7
For (ii), we compute
with \(\varphi \) from above. \(\square \)
Proposition 12.32
Let \(x_{0}\in \Omega _{0}\in \mathsf {cc}(\Omega )\) and let \(S,T\in C^{\infty ,3\times 3}(\Omega _{0})\).
- (a):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega _{0}\) closed, piecewise \(C^1\)-curves
$$\begin{aligned} \int _{\zeta }({{\,\mathrm{Curl}\,}}S)^{\top }{{\,\mathrm{d}\,}}\lambda =0. \end{aligned}$$ - (ii):
-
For all \(\zeta _{x_{0},x},{\widetilde{\zeta }}_{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curves connecting \(x_{0}\) with x
$$\begin{aligned} \int _{\zeta _{x_{0},x}}({{\,\mathrm{Curl}\,}}S)^{\top }{{\,\mathrm{d}\,}}\lambda =\int _{{\widetilde{\zeta }}_{x_{0},x}}({{\,\mathrm{Curl}\,}}S)^{\top }{{\,\mathrm{d}\,}}\lambda . \end{aligned}$$ - (iii):
-
There exists \(w\in C^{\infty ,3}(\Omega _{0})\) such that \({{\,\mathrm{Grad}\,}}w=({{\,\mathrm{Curl}\,}}S)^{\top }\).
In either of the above cases,
for some \(\zeta _{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curve connecting \(x_{0}\) with x is a (well-defined) possible choice for w in (iii).
- (b):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega _{0}\) closed, piecewise \(C^1\)-curves
$$\begin{aligned} \int _{\zeta }T{{\,\mathrm{d}\,}}\lambda =0. \end{aligned}$$ - (ii):
-
For all \(\zeta _{x_{0},x},{\widetilde{\zeta }}_{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curves connecting \(x_{0}\) with x
$$\begin{aligned} \int _{\zeta _{x_{0},x}}T{{\,\mathrm{d}\,}}\lambda =\int _{{\widetilde{\zeta }}_{x_{0},x}}T{{\,\mathrm{d}\,}}\lambda . \end{aligned}$$ - (iii):
-
There exists \(v\in C^{\infty ,3}(\Omega _{0})\) such that \({{\,\mathrm{Grad}\,}}v=T\).
In either of the above cases,
for some \(\zeta _{x_{0},x}\subseteq \Omega _{0}\) piecewise \(C^1\)-curve connecting \(x_{0}\) with x is a (well-defined) possible choice for v in (iii).
- (c):
-
Let \(T:=S+{{\,\mathrm{spn}\,}}w\) with \(w\in C^{\infty ,3}(\Omega _{0})\) and \({{\,\mathrm{Grad}\,}}w=({{\,\mathrm{Curl}\,}}S)^{\top }\) as in (a), (iii). Moreover, let \(v\in C^{\infty ,3}(\Omega _{0})\) with \({{\,\mathrm{Grad}\,}}v=T\) as in (b), (iii). Then
- (i):
-
\({{\,\mathrm{skw}\,}}S=0\),
- (ii):
-
\({{\,\mathrm{symGrad}\,}}v=S\)
are equivalent. In either case, we have
Proof
The proofs of (a) and (b) are easy (fundamental theorem of calculus) and follow in a similar way to Propositions 12.22. For (c), we compute \({{\,\mathrm{symGrad}\,}}v={{\,\mathrm{sym}\,}}T={{\,\mathrm{sym}\,}}S\). Hence (i) and (ii) are equivalent. Finally, if (i) or (ii) is true, then by the complex property
and we conclude \({{\,\mathrm{Grad}\,}}w=({{\,\mathrm{Curl}\,}}{{\,\mathrm{sym}\,}}{{\,\mathrm{Grad}\,}}v)^{\top }=\frac{1}{2}{{\,\mathrm{Grad}\,}}\text {curl}v\) by Lemma 12.10. \(\square \)
The respective result for the whole of \(\Omega \) reads as follows.
Corollary 12.33
Let \(S,T\in C^{\infty ,3\times 3}(\Omega )\).
- (a):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega \) closed, piecewise \(C^1\)-curves \(\int _{\zeta }({{\,\mathrm{Curl}\,}}S)^{\top }{{\,\mathrm{d}\,}}\lambda =0\).
- (ii):
-
There exists \(w\in C^{\infty ,3}(\Omega )\) such that \({{\,\mathrm{Grad}\,}}w=({{\,\mathrm{Curl}\,}}S)^{\top }\).
- (b):
-
The following conditions are equivalent:
- (i):
-
For all \(\zeta \subseteq \Omega \) closed, piecewise \(C^1\)-curves \(\int _{\zeta }T{{\,\mathrm{d}\,}}\lambda =0\).
- (ii):
-
There exists \(v\in C^{\infty ,3}(\Omega )\) such that \({{\,\mathrm{Grad}\,}}v=T\).
- (c):
-
Let \(T=S+{{\,\mathrm{spn}\,}}w\) with \(w\in C^{\infty ,3}(\Omega )\) and \({{\,\mathrm{Grad}\,}}w=({{\,\mathrm{Curl}\,}}S)^{\top }\) as in (a), (ii). Moreover, let \(v\in C^{\infty ,3}(\Omega )\) with \({{\,\mathrm{Grad}\,}}v=T\) as in (b), (ii). Then \({{\,\mathrm{skw}\,}}S=0\) in \(\Omega \) if and only if \({{\,\mathrm{symGrad}\,}}v=S\) in \(\Omega \).
Remark 12.34
Related to Proposition 12.32 and Corollary 12.33 we note with Lemma 12.10 the following:
- (i):
-
For \(S\in C^{\infty ,3\times 3}_{{\mathbb {S}}}(\Omega )\) and \(T:=S+{{\,\mathrm{spn}\,}}w\) with \({{\,\mathrm{Grad}\,}}w=({{\,\mathrm{Curl}\,}}S)^{\top }\) it holds
$$\begin{aligned} {{\,\mathrm{Curl}\,}}T= & {} {{\,\mathrm{Curl}\,}}S+({{\,\mathrm{div}\,}}w){{\,\mathrm{Id}\,}}-(({{\,\mathrm{Curl}\,}}S)^{\top })^{\top } \\= & {} {{\,\mathrm{tr}\,}}{{\,\mathrm{Grad}\,}}w ={{\,\mathrm{tr}\,}}{{\,\mathrm{Curl}\,}}S ={{\,\mathrm{tr}\,}}{{\,\mathrm{Curl}\,}}{{\,\mathrm{skw}\,}}S =0. \end{aligned}$$ - (ii):
-
If \(\Omega _{0}\) is simply connected, Proposition 12.32 (a), (iii) and (b), (iii) are equivalent to \({{\,\mathrm{Curl}\,}}({{\,\mathrm{Curl}\,}}S)^{\top }=0\) and \({{\,\mathrm{Curl}\,}}T=0\), respectively.
Next, we provide the construction of the basis tensor fields for the Neumann fields for the elasticity complex. Let \(j\in \{1,\ldots ,p\}\). From the beginning of Sect. 12 recall that \(\theta _{j}\) is constant on each connected component \(\Upsilon _{j,0}\) and \(\Upsilon _{j,1}\) of \(\Upsilon _{j}{\setminus }F_{j}\) and vanishes outside of \({\widetilde{\Upsilon }}_{j,1}\). Let \({\widehat{r}}_{k}\) be the rigid motions (Nedelec fields) from Sect. 11.4 given by \({\widehat{r}}_{m}(x):=e^{m}\times x={{\,\mathrm{spn}\,}}(e^{m})\,x\) and \({\widehat{r}}_{m+3}(x):=e^{m}\) for \(m\in \{1,2,3\}\). We define the vector fields \(\theta _{j,k}:=\theta _{j}{\widehat{r}}_{k}\) and note \({{\,\mathrm{symGrad}\,}}\theta _{j,k}=0\) in \(\dot{\bigcup }_{l\in \{1,\ldots ,p\}}\Upsilon _{l}{\setminus } F_{l}\) for all \(k\in \{1,\ldots ,6\}\). Thus \({{\,\mathrm{symGrad}\,}}\theta _{j,k}\) can be continuously extended by zero to \(\Theta _{j,k}\in C^{\infty ,3\times 3}(\Omega )\cap L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\) with \(\Theta _{j,k}=0\) in all the neighbourhoods \(\Upsilon _{l}\) of all surfaces \(F_{l}\), \(l\in \{1,\ldots ,p\}, k\in \{1,\ldots ,6\}\).
Lemma 12.35
Let Assumption 10.3 be satisfied. Then \(\Theta _{j,k}\in {\text {ker}}({\text {CurlCurl}}^{\top }_{{\mathbb {S}}},\Omega )\).
Proof
Let \(\Phi \in C^{\infty ,3\times 3}_{c,{\mathbb {S}}}(\Omega )\). As \({{\,\mathrm{supp}\,}}\Theta _{j,k}\subseteq {\widetilde{\Upsilon }}_{j}{\setminus }\Upsilon _{j}\) we can pick another cut-off function \(\varphi \in C^{\infty }_{c}(\Omega _{F})\) with \(\varphi |_{{{\,\mathrm{supp}\,}}\Theta _{j,k}\cap {{\,\mathrm{supp}\,}}\Phi }=1\). Then
as \(\varphi \Phi ,{\text {CurlCurl}}^{\top }_{{\mathbb {S}}}(\varphi \Phi )\in C^{\infty ,3\times 3}_{c,{\mathbb {S}}}(\Omega _{F})\) by Lemma 12.10. \(\square \)
Next, we construct functionals similar to the previous sections. Here, however, due to the complex structure, we need six times as many (instead of four) as for the de Rham complex. For starters, note that for \(l,j\in \{1,\dots ,p\}\) and \(k\in \{1,\dots ,6\}\) and for the curves \(\zeta _{x_{l,0},x_{l,1}}\subseteq \zeta _{l}\) with the chosen starting points \(x_{l,0}\in \Upsilon _{l,0}\) and respective endpoints \(x_{l,1}\in \Upsilon _{l,1}\) we can computeFootnote 8 by Lemma 12.31
and
Thus, for \(l\in \{1,\dots ,p\}\) and \(\ell \in \{1,\dots ,6\}\) we have functionals \(\beta _{l,\ell }\) given by
Then for \(l,j\in \{1,\dots ,p\}\) and for \(\ell \in \{1,2,3\}\)
i.e.,
and for \(\ell \in \{4,5,6\}\)
i.e.,
where we define
In particular, \((x_{l,1})_{\widehat{\ell -3,k}}=0\) if \(\ell -3=k\) or \(\ell \in \{1,2,3\}\) or \(k\in \{4,5,6\}\). Therefore, we have for \(l,j\in \{1,\dots ,p\}\) and \(k,\ell \in \{1,\dots ,6\}\)
Let Assumption 10.2 be satisfied. For the elasticity complex, similar to (3.3), (3.5), see also (12.2), (12.11), (12.21), we have the orthogonal decompositions
Remark 12.36
[39, Lemma 3.2] yields \({{\,\mathrm{dom}\,}}({{\,\mathrm{symGrad}\,}},\Omega )=H^{1,3}(\Omega )\). Thus, as before, a combination of Rellich’s selection theorem and a contradiction argument implies the Poincaré type estimate
Thus, the range in (12.31) is closed.
Let \(\pi :L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\rightarrow {\text {ker}}(\mathring{{{\,\mathrm{Div}\,}}}_{{\mathbb {S}}},\Omega )\) denote the orthogonal projector along \({{\,\mathrm{ran}\,}}({{\,\mathrm{symGrad}\,}},\Omega )\) according to (12.31). We infer \(\pi \big ({\text {ker}}({\text {CurlCurl}}^{\top }_{{\mathbb {S}}},\Omega )\big )={\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )\). Thus, using Lemma 12.35, for all \(j\in \{1,\ldots ,p\}\) and \(k\in \{1,\ldots ,6\}\) we find \(\psi _{j,k}\in H^{1,3}(\Omega )\) such that
Next, Lemma 12.2 implies \({\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )\subseteq C^{\infty ,3\times 3}(\Omega )\). Thus, \(\pi \Theta _{j,k},\Theta _{j,k}\in C^{\infty ,3\times 3}(\Omega )\) implies \({{\,\mathrm{symGrad}\,}}\psi _{j,k}\in C^{\infty ,3\times 3}(\Omega )\) and hence \(\psi _{j,k}\in C^{\infty ,3}(\Omega )\). Therefore, all path integrals over the closed curves \(\zeta _{l}\) are well-defined. Furthermore, we observe by Lemma 12.31
and
Hence, \(\beta _{l,\ell }({{\,\mathrm{symGrad}\,}}\psi _{j,k})=0\) and by (12.30)
for all \(l,j\in \{1,\dots ,p\}\) and all \(\ell ,k\in \{1,\dots ,6\}\). We shall show that
defines a basis of \({\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )\).
The tensor fields \(\Theta _{j,k}\) being constructed explicitly, we provide a way of finding \(\psi _{j,k}\) by means of solutions of PDEs.
Remark 12.37
(Characterisation by PDEs) Let \(j\in \{1,\dots ,p\}\) and \(k\in \{1,\dots ,6\}\). Then \(\psi _{j,k}\in H^{1,3}(\Omega )\cap \mathsf {RM}_{\mathsf {pw}}^{\bot _{L^{2,3}(\Omega )}}\) can be found by solving the following elasticity PDE formulated in the standard variational formulation
i.e., \(\psi _{j,k}=\Delta _{{\mathbb {S}},N}^{-1}\big ({{\,\mathrm{Div}\,}}_{{\mathbb {S}}}\Theta _{j,k},\Theta _{j,k}\nu |_{\Gamma }\big )\), where \(\Delta _{{\mathbb {S}},N}\subseteq {{\,\mathrm{Div}\,}}_{{\mathbb {S}}}{{\,\mathrm{symGrad}\,}}\) is the ‘symmetric’ Laplacian with inhomogeneous Neumann boundary conditions restricted to a subset of \(H^{1,3}(\Omega )\cap \mathsf {RM}_{\mathsf {pw}}^{\bot _{L^{2,3}(\Omega )}}\). Therefore,
In classical terms, \(\psi _{j,k}\) solves the Neumann elasticity problem
which is uniquely solvable.
Lemma 12.38
Let Assumptions 10.2 and 10.3 be satisfied. Then \({\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )={{\,\mathrm{lin}\,}}{\mathcal {B}}^{\mathsf {ela}}_{N}\).
Proof
Let \(H\in {\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )={\text {ker}}(\mathring{{{\,\mathrm{Div}\,}}}_{{\mathbb {S}}},\Omega )\cap {\text {ker}}({\text {CurlCurl}}^{\top }_{{\mathbb {S}}},\Omega )\subseteq C^{\infty ,3\times 3}_{{\mathbb {S}}}(\Omega )\), see Lemma 12.2. With the above introduced functionals \(a_{l}\) and \(b_{l}\), \(l\in \{1,\dots ,p\}\), we recall
and define for \(l\in \{1,\dots ,p\}\) the numbers
where we recall \((x_{l,1})_{\widehat{\ell -3,k}}=(\delta _{\ell ,4}+\delta _{\ell ,5}+\delta _{\ell ,6}) (\delta _{1,k}+\delta _{2,k}+\delta _{3,k})(1-\delta _{\ell -3,k})(x_{l,1})_{\widehat{\ell -3,k}}\) by definition, see (12.30). We shall show that
Similar to the proof of Lemma 12.6 (or 12.18, 12.28) the aim is to prove the existence of \(v\in H^{1,3}(\Omega )\) such that \({{\,\mathrm{symGrad}\,}}v={\widehat{H}}\), since then
For the construction of v, we apply Corollary 12.33 and Remark 12.34 to \(S={\widehat{H}}\). In order to show condition Corollary 12.33 (a), (i), we observe for \(l\in \{1,\dots ,p\}\) and \(\ell \in \{1,2,3\}\) by (12.32)
Note that \({{\,\mathrm{Curl}\,}}({{\,\mathrm{Curl}\,}}{\widehat{H}})^{\top }={\text {CurlCurl}}^{\top }_{{\mathbb {S}}}{\widehat{H}}=0\), i.e., \(({{\,\mathrm{Curl}\,}}{\widehat{H}})^{\top }\in {\text {ker}}({{\,\mathrm{Curl}\,}})\cap C^{\infty ,3\times 3}(\Omega )\). Thus, by Assumption 10.3 (A.1) for any closed piecewise \(C^1\)-curve \(\zeta \) in \(\Omega \)
Hence, by Corollary 12.33 (a), (ii), there exists \(w\in C^{\infty ,3}(\Omega )\) such that
We define \(T:={\widehat{H}}+{{\,\mathrm{spn}\,}}w\). Let \(l\in \{1,\ldots ,p\}\). Then \(\zeta _{x_{l,0},x_{l,1}}\subseteq \zeta _{l}\subseteq \Omega _{0}\) for some \(\Omega _{0}\in \mathsf {cc}(\Omega )\). With \(c:=w(x_{l,1})\in {\mathbb {R}}^{3}\) we compute for all \(x\in \zeta _{l}\)
and
Again, we consider the curve \(\zeta _{l}\) as the closed curve \(\zeta _{x_{l,1},x_{l,1}}\) with circulation 1 along \(\zeta _{l}\). By Lemma 12.31 and by the definition of \(b_{l}\) we have for \(l\in \{1,\dots ,p\}\)
Hence, for \(\ell \in \{4,5,6\}\) we get by (12.32)
Therefore, \(\int _{\zeta _{l}}T{{\,\mathrm{d}\,}}\lambda =0\) for all \(l\in \{1,\dots ,p\}\). Note that \(T\in {\text {ker}}({{\,\mathrm{Curl}\,}})\cap C^{\infty ,3\times 3}(\Omega )\) by Remark 12.34 as \(S={\widehat{H}}\in C^{\infty ,3\times 3}_{{\mathbb {S}}}(\Omega )\) and \(T=S+{{\,\mathrm{spn}\,}}w\) with \({{\,\mathrm{Grad}\,}}w=({{\,\mathrm{Curl}\,}}{\widehat{H}})^{\top }\). Thus, by Assumption 10.3 (A.1) for any closed piecewise \(C^1\)-curve \(\zeta \) in \(\Omega \)
Hence, by the symmetry of \({\widehat{H}}\) and Corollary 12.33 (b), (c), there exists \(v\in C^{\infty ,3}(\Omega )\) such that \({{\,\mathrm{Grad}\,}}v=T\) as well as \({{\,\mathrm{symGrad}\,}}v={\widehat{H}}\). Similar to the end of the proof of Lemma 12.6, elliptic regularity and, e.g., [22, Theorem 2.6 (1)] or [1, Theorem 3.2 (2)] show that \(v\in C^{\infty ,3}(\Omega )\) with \({{\,\mathrm{symGrad}\,}}v\in L^{2,3\times 3}_{{\mathbb {S}}}(\Omega )\) implies \(v\in H^{1,3}(\Omega )\), completing the proof. (Let us note that \(v\in H^{1,3}(\Omega )\) implies also \(T\in L^{2,3\times 3}(\Omega )\) and hence \(w\in L^{2,3}(\Omega )\).) \(\square \)
Lemma 12.39
Let Assumptions 10.2 and 10.3 be satisfied. Then \({\mathcal {B}}^{\mathsf {ela}}_{N}\) is linearly independent.
Proof
Let \(\gamma _{j,k}\in {\mathbb {R}}\), \(j\in \{1,\ldots ,p\}\), \(k\in \{1,\ldots ,6\}\), be such that \(\displaystyle \sum _{j=1}^{p}\sum _{k=1}^{6}\gamma _{j,k}\pi \Theta _{j,k}=0\). Then (12.32) implies for \(l\in \{1,\dots ,p\}\)
finishing the proof. \(\square \)
Theorem 12.40
Let Assumptions 10.2 and 10.3 be satisfied. Then \(\dim {\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )=6p\) and a basis of \({\mathcal {H}}_{N,{\mathbb {S}}}^{\mathsf {ela}}(\Omega )\) is given by (12.33).
Proof
13 Conclusion
The index theorems presented rest on the abstract construction principle provided in [11] and the results on the newly found biharmonic complex from [37, 38] and the elasticity complex from [39,40,41]. With this insight it is possible to construct basis fields for the generalised harmonic Dirichlet and Neumann tensor fields. The number of basis fields of the considered cohomology groups provides additional topological invariants. The construction of the generalised Dirichlet fields being somewhat similar to the de Rham complex, the same for the generalised Neumann fields requires some more machinery particularly relying on the introduction of Poincaré maps defining the functionals.
Furthermore, we have outlined numerical strategies to compute the generalised Neumann and Dirichlet fields in practice. In passing we have also provided a set of Friedrichs–Poincaré type estimates and included variable coefficients relevant for numerical studies.
All these constructions heavily rely on the choice of boundary conditions and we emphasise that the considered mixed order operators cannot be viewed as leading order plus relatively compact perturbation, when it comes to computation of the Fredholm index. In particular, techniques from pseudo-differential calculus successfully applied to obtain index formulas for operators defined on non-compact manifolds or compact manifolds without boundary, see e.g. [19, 20], are likely to be very difficult to be applicable in the present situation. It would be interesting to see, whether the operators considered above defined on an unbounded domain enjoy similar index formulas (maybe a comparable Witten index of some sort) even though the operator itself might not be of Fredholm type anymore.
Notes
The boundary of a strong Lipschitz domain is locally a graph of some Lipschitz function.
E.g., we compute \({\widetilde{A}}_{0}^{*}\). Let \(y\in {{\,\mathrm{dom}\,}}{\widetilde{A}}_{0}^{*}\). Then for \(x\in {{\,\mathrm{dom}\,}}{\widetilde{A}}_{0}={{\,\mathrm{dom}\,}}A_{0}\)
$$\begin{aligned} \langle x,{\widetilde{A}}_{0}^{*}y\rangle _{H_{0}} =\langle x,{\widetilde{A}}_{0}^{*}y\rangle _{{\widetilde{H}}_{0}} =\langle {\widetilde{A}}_{0}x,y\rangle _{{\widetilde{H}}_{1}} =\langle A_{0}x,\lambda _{1}y\rangle _{H_{1}}, \end{aligned}$$showing that \(\lambda _{1}y\in {{\,\mathrm{dom}\,}}A_{0}^{*}\) and \(A_{0}^{*}\lambda _{1}y={\widetilde{A}}_{0}^{*}y\).
The boundary of a weak Lipschitz domain is a 2-dimensional submanifold of the 3-dimensional Lipschitz manifold \({\overline{\Omega }}\) with boundary.
A boundary being locally representable as the graph of a continuous function.
Alternatively, note \(\displaystyle \int _{\zeta _{x_{0}x}}{{\,\mathrm{Id}\,}}{{\,\mathrm{d}\,}}\lambda _{y} =\int _{0}^{1}{{\,\mathrm{Id}\,}}\varphi '(s){{\,\mathrm{d}\,}}s =\int _{0}^{1}\varphi '(s){{\,\mathrm{d}\,}}s=x-x_{0}\) with the parametrisation \(\varphi \) of \(\zeta _{x_{0},x}\) from above.
Alternatively, we can compute with \({{\,\mathrm{Id}\,}}={{\,\mathrm{Grad}\,}}y\)
$$\begin{aligned}&\int _{\zeta _{x_{0},x}}\underbrace{{{\,\mathrm{spn}\,}}\text {curl}v(x_{0})}_{=({{\,\mathrm{spn}\,}}\text {curl}v(x_{0})){{\,\mathrm{Id}\,}}}{{\,\mathrm{d}\,}}\lambda _{y} \\&\quad ={{\,\mathrm{spn}\,}}\text {curl}v(x_{0})\int _{\zeta _{x_{0},x}}{{\,\mathrm{Grad}\,}}y{{\,\mathrm{d}\,}}\lambda _{y} =\big ({{\,\mathrm{spn}\,}}\text {curl}v(x_{0})\big )(x-x_{0}). \end{aligned}$$Note that \(\text {curl}{\widehat{r}}_{k}=2e^{k}\) for \(k\in \{1,2,3\}\), since, e.g.,
$$\begin{aligned} \text {curl}{\widehat{r}}_{1}(x)&=\text {curl}\,(e^{1}\times x) =\text {curl}\,(x_{2}\,e^{1}\times e^{2}+x_{3}\,e^{1}\times e^{3}) \\&=\text {curl}\,(x_{2}e^{3}-x_{3}e^{2})\\&={{\,\mathrm{grad}\,}}\,(x_{2})\times e^{3}-{{\,\mathrm{grad}\,}}\,(x_{3})\times e^{2} =e^{2}\times e^{3}-e^{3}\times e^{2} =2e^{1}. \end{aligned}$$
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Acknowledgements
We cordially thank Rainer Picard, TU Dresden, and Walter Zulehner, JKU Linz, for many inspiring and fruitful discussions. We particularly thank Walter Zulehner for his kind help related to the Poincaré maps necessary for the construction of Neumann fields. We also thank the anonymous referee for their thorough review helping to improve the quality of the manuscript.
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Pauly, D., Waurick, M. The index of some mixed order Dirac type operators and generalised Dirichlet–Neumann tensor fields. Math. Z. 301, 1739–1819 (2022). https://doi.org/10.1007/s00209-021-02947-9
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DOI: https://doi.org/10.1007/s00209-021-02947-9
Keywords
- Dirac operator
- Picard’s extended Maxwell system
- Fredholm index
- Cohomology
- Hilbert complex
- Elasticity complex
- biharmonic complex
- Harmonic Dirichlet and Neumann tensors