On Closed and Exact - and -Complexes,
Corresponding Compact Embeddings for Tensor Rotations,
and a Related Decomposition Result for Biharmonic Problems in 3D
It is shown that the first biharmonic boundary value problem on a topologically trivial domain in 3D is equivalent to three (consecutively to solve) second-order problems. This decomposition result is based on a Helmholtz-like decomposition of an involved non-standard Sobolev space of tensor fields and a proper characterization of the operator acting on this space. Similar results for biharmonic problems in 2D and their impact on the construction and analysis of finite element methods have been recently published in . The discussion of the kernel of leads to (de Rham-like) closed and exact Hilbert complexes, the -complex and its adjoint the -complex, involving spaces of trace-free and symmetric tensor fields. For these tensor fields we show Helmholtz type decompositions and, most importantly, new compact embedding results. Almost all our results hold and are formulated for general bounded strong Lipschitz domains of arbitrary topology. There is no reasonable doubt that our results extend to strong Lipschitz domains in .
Key words and phrases:biharmonic equations, Helmholtz decomposition, Hilbert complexes
1991 Mathematics Subject Classification:35G15, 58A14
Dirk Pauly]firstname.lastname@example.org Walter Zulehner]email@example.com
- 1 Introduction
- 2 Preliminaries
- 3 The - and -Complexes
- 4 Application to Biharmonic Problems
- A Proofs of Some Useful Identities
In  it was shown that the fourth-order biharmonic boundary value problem
where is a bounded and simply connected domain in with a (strong) Lipschitz boundaryiiiiii is locally a graph of a Lipschitz function. , can be decomposed into three second-order problems. The first problem is a Dirichlet-Poisson problem for an auxiliary scalar field
|the second problem is a linear elasticity Neumann problem for an auxiliary vector field|
|and, finally, the third problem is again a Dirichlet-Poisson problem for the original scalar field|
|Note that the second equation is equivalent to|
Here is a given right-hand side, , , and denote the Laplace operator, the outward normal vector to the boundary, and the derivative in this direction, respectively. In matrix notation the latter system reads as the symmetric system
with and . Throughout this paper, ‘mathrings’ indicate natural homogeneous boundary conditions for different operators. While is continuously invertible, is not on its domain of definition , but on the more regular space
which is easy to handle. We will see that the situation in is much more complicated. The differential operators , , and (for later use) denote the gradient of a scalar field, the divergence and the rotation of a vector field, respectively. The corresponding capitalized differential operators , , and denote the row-wise application of to a vector field, and to a tensor field. The prefix is used for the symmetric part of a matrix, for the skew-symmetric part we use the prefix . This decomposition is of triangular structure, i.e., the first problem is a well-posed second-order problem in , the second problem is a well-posed second-order problem in for given , and the third problem is a well-posed second-order problem in for given and . This allows to solve them consecutively analytically or numerically by means of techniques for second-order problems.
This is - in the first place - a new analytic result for fourth-order problems. But it also has interesting implications for discretization methods applied to (1.1). It allows to re-interpret known finite element methods as well as to construct new discretization methods for (1.1) by exploiting the decomposable structure of the problem. In particular, it was shown in  that the Hellan-Herrmann-Johnson mixed method (see [8, 9, 13]) for (1.1) allows a similar decomposition as the continuous problem, which leads to a new and simpler assembling procedure for the discretization matrix and to more efficient solution techniques for the discretized problem. Moreover, a novel conforming variant of the Hellan-Herrmann-Johnson mixed method was found based on the decomposition.
The main application of this paper is to derive a similar decomposition result for biharmonic problems (1.1) on bounded and topologically trivial three-dimensional domains with a (strong) Lipschitz boundary . For this we proceed as in  and reformulate (1.1) using
as a mixed problem by introducing the (negative) Hessian of the original scalar field as an auxiliary tensor field
Then the biharmonic differential equation reads
For an appropriate non-standard Sobolev space for it can be shown that the mixed problem in and is well-posed, see (4.4)-(4.5). Then the decomposition of the biharmonic problem follows from a regular decomposition of this non-standard Sobolev space, see Lemma 3.21. This part of the analysis carries over completely from the two-dimensional case to the three-dimensional case and is recalled in Section 4. To efficiently utilize this regular decomposition for the decomposition of the biharmonic problem an appropriate characterization of the kernel of the operator is required, which is well understood for the two-dimensional case, see, e.g., [3, 11, 14]. Its extension to the three-dimensional case is one of the central topics of this paper. We expect - as in the two-dimensional case - similar interesting implications for the study of appropriate discretization methods for four-order problems in the three-dimensional case.
Another application comes from the theory for general relativity and gravitational waves. There, the so called linearized Einstein-Bianchi system reads as the Maxwell’s equations
but with symmetric and deviatoric (trace-free) tensor fields and , see  for more details, especially on the modeling.
The paper is organized as follows: In Subsection 1.1 of this introduction we will present some of the main results in a non-rigorous way and the application to the three-dimensional biharmonic equation, i.e., to (1.1) for . The mathematically rigorous part, where also all precise definitions can be found, begins with preliminaries in Section 2 and introduces our general functional analytical setting. Then we will discuss the relevant unbounded linear operators, show closed and exact Hilbert complex properties, and present a suitable representation of the kernel of for the three-dimensional case in Section 3.1 for topologically trivial domains. In Section 3.2 we extend our results to (strong) Lipschitz domains with arbitrary topology based on two new and crucial compact embeddings. In the final Section 4 we give a detailed study of the application of our results to the three-dimensional biharmonic equation from Section 1.1. The proofs of some useful identities are presented in an appendix.
1.1. Some Main Results
Let be a bounded and topologically trivial strong Lipschitz domain. Based on a decomposition result of the non-standard Hilbert space
see Lemma 3.21, where denotes the symmetric -tensor fields, and a representation of the kernel of as the range of symmetric rotations of deviatoric tensor fields, i.e.,
a decomposition of the three-dimensional biharmonic problem (1.1) into three (consecutively to solve) second-order problems will be rigorously derived in Section 4. For details, see (4.11)-(4.13) and the strong equations after the corresponding proof. More precisely, the three resulting second order equations are a Dirichlet-Poisson problem for the auxiliary scalar function
a second-order Neumann type --system for the auxiliary tensor field
and, finally, a Dirichlet-Poisson problem for the original scalar function
The second system is equivalent to
In matrix notation the latter system reads as the symmetric system
with and . While is continuously invertible, is not on its domain of definition , but on the more regular space
which leads to another difficulty. This is a well known and typical situation, e.g., in the theory of static Maxwell equations, and it will turn out that it results into a symmetric saddle point system, see Theorem 4.5 and (4.25).
Hence, the kernel is an important object. In Theorem 3.12 we will show
Especially, the range is closed. The potential on the right hand side of the first line is called a regular potential and the potential on the right hand side of the second line is uniquely determined. Both potentials depend continuously on the data. Here, is the Sobolev space of deviatoric -tensor fields and
Moreover, a corresponding Poincaré type estimate
as well as a Helmholtz type decomposition
with the same properties of the respective potentials, and
where is the same as before and denotes the space of lowest order Raviart-Thomas affine linear vector fields.
Our results rely on the study of the corresponding Hilbert complex
and its dual or adjoint Hilbert complex
which will turn out to be closed (closed ranges) an exact (trivial cohomology groups). Here, the densely defined, closed, and unbounded linear operators , , and are given as closures of
with domains of definition
The adjoints are
In this contribution we will prove all important tools to handle pde-systems involving the latter operators, such as Helmholtz type decompositions, potentials, regular decompositions, regular potentials, Poincaré type estimates, closed ranges, exactness, and, most importantly, the key property, that certain canonical embeddings are compact, e.g.,
In principle, such results are known in simpler situations, e.g., in electro-magnetic theory or linear elasticity. In electro-magnetic theory (Maxwell’s equations) one has to deal with the de Rham complex (---complex), i.e., with the closed and exact Hilbert complex and its adjoint
which have a well known generalization to differential forms and exterior derivatives , and co-derivatives , as well. In linear elasticity we observe the elasticity complexes (-complexes), i.e.,
Note that these complexes admit certain symmetries, which is not the case for the --complexes. On the other hand, there is no reasonable doubt that similar results hold for the other set of boundary conditions as well, i.e., for the Hilbert complexes
We start by recalling some basic concepts and abstract results from functional analysis concerning Helmholtz decompositions, closed ranges, Friedrichs/Poincaré type estimates, and bounded or even compact inverse operators. Since we will need both the Banach space setting for bounded linear operators as well as the Hilbert space setting for (possibly unbounded) closed and densely defined linear operators, we will shortly recall these two variants.
2.1. Functional Analysis Toolbox
Let and be real Banach spaces. With we introduce the space of bounded linear operators mapping to . The dual spaces of and are denoted by and . For a given we write for its Banach space dual or adjoint operator defined by for all and all . Norms and duality in resp. are denoted by , , and .
Suppose and are Hilbert spaces. For a (possibly unbounded) densely defined linear operator we recall that its Hilbert space dual or adjoint can be defined via its Banach space adjoint and the Riesz isomorphisms of and or directly as follows: if and only if and
In this case we define . We note that has maximal domain of definition and that is characterized by
Here denotes the scalar product in a Hilbert space and is used for the domain of definition of a linear operator. Additionally, we introduce the notation for the kernel or null space and for the range of a linear operator.
Let be a (possibly unbounded) closed and densely defined linear operator on two Hilbert spaces and with adjoint . Note , i.e., is a dual pair. By the projection theorem the Helmholtz type decompositions
hold and we can define the reduced operators
which are also closed and densely defined linear operators. We note that and are indeed adjoint to each other, i.e., is a dual pair as well. Now the inverse operators
exist and they are bijective, since and are injective by definition. Furthermore, by (2.1) we have the refined Helmholtz type decompositions
|and thus we obtain for the ranges|
By the closed range theorem and the closed graph theorem we get immediately the following.
The following assertions are equivalent:
is closed in .
is closed in .
is continuous and bijective with norm bounded by .
is continuous and bijective with norm bounded by .
In case that one of the assertions of Lemma 2.1 is true, e.g., is closed, we have
For the “best” constants , we have the following lemma.
The Rayleigh quotients
coincide, i.e., , if either or exists in . Otherwise they also coincide, i.e., it holds .
From now on and throughout this paper, we always pick the best possible constants in the various Friedrichs/Poincaré type estimates.
A standard indirect argument shows the following.
Let be compact. Then the assertions of Lemma 2.1 hold. Moreover, the inverse operators
are compact with norms .
Moreover, we have
is compact, if and only if is compact.
Now, let and be (possibly unbounded) closed and densely defined linear operators on three Hilbert spaces , and with adjoints and as well as reduced operators , , and , . Furthermore, we assume the sequence or complex property of and , that is, , i.e.,
Then also , i.e., . The Helmholtz type decompositions of (2.1) for and read
|and by (2.4) we see|
yielding the refined Helmholtz type decomposition
The previous results of this section imply immediately the following.
hold. Especially, , , , and are closed, the respective inverse operators, i.e.,
are continuous, and there exist positive constants , , such that the Friedrichs/Poincaré type estimates
Note that resp. is closed, if e.g. resp. is compact. In this case, the respective inverse operators, i.e.,