On the curl operator and some characterizations of matrix fields in Lipschitz domains
Introduction
First, all notations and definitions not given in this introduction are mentioned in Section 2. Let Ω be a bounded and connected open set of with a Lipschitz-continuous boundary. The surjectivity of the operator , where denotes the subspace of functions in with zero average, is an important tool in the analysis of Stokes equations. This result has been shown by many authors through different techniques (see [13], [15], [25]) and it provides us with a simple proof for the following usual version of De Rham's Theorem: let satisfying where denotes the subspace of vector fields in with divergence free, then there exists a scalar field such that The first aim of this work is to present a new extension of the above theorem that we will call the rotational version of De Rham's Theorem. In the case where the open set Ω is star-shaped with respect to an open ball, Costabel et al. in [13] and Mitrea in [25] have used the properties of pseudodifferential operators to show that the operator is onto. In Section 3, we will give a new proof of this result by using the theory of singular integrals. Furthermore, we will generalize it in the case where Ω is Lipschitz but not necessarily star-shaped with respect to an open ball. More precisely, we will show that if where (see Section 2), then there exists satisfying in Ω. Next, we deduce a rotational version of De Rham's Theorem, and here we state our first main result:
Theorem A i) Let r be a real number such that and m a nonnegative integer. Then, for any satisfying (2), there exists such that and there exists a constant such that ii) Let and satisfies where . Then, there exists such that
As a close result, Borchers and Sohr in [9] have shown that if the open set Ω is of class , where m is a nonnegative integer, then the operator is onto. In Section 4, we will generalize this result in the case when the open set Ω is only Lipschitz and then we will deduce a weak rotational version of De Rham's Theorem. Our second main result is given in the following theorem:
Theorem B i) Let r be a real number such that and m a nonnegative integer. Then, there exists a constant such that for any , there exists , that satisfies and ii) Let be an integer and satisfying Then there exists such that
The vector potential ψ obtained by Borchers and Sohr satisfies the additional property: div in Ω, which needs the regularity of the domain Ω.
In the absence of body forces the stress equations of equilibrium take the form the second order symmetric tensor field being the stress in the reference configuration Ω of an elastic body. The first stress function solution of the equilibrium equation (4) was presented by Airy in [1] for the two dimensional case. The generalizations for the three dimensional case were obtained by Maxwell in [23], Morera in [27] and Beltrami in [7]. The solutions of Morera and Maxwell are special cases of the Beltrami's solution defined as follows: Gurtin [21] gave an example of a stress field satisfying (4) but which is not given by (5). So that this representation is incomplete. However the Beltrami solution is complete in the class of smooth stress fields which are self-equilibrated, i.e. for each closed regular surface contained in Ω, the resultant force and the moment vanish. In other words, satisfies the following condition: such that . For more details see [14].
An extension of this result can be found in [17] and in [18] as follows: let Ω be a bounded and connected open set of with Lipschitz-continuous boundary and be a symmetric matrix field in and satisfying the following conditions:
Then, there exists a symmetric matrix field such that in Ω. Moreover, P. G. Ciarlet et al. in [12] observed that if the above symmetric matrix field satisfies the following conditions: where denotes the duality pairing between and , then .
Our third aim in this paper is to show a new extension of the Beltrami's completeness, in the case where the components of the symmetric matrix are in and to prove the above observation of P.G. Ciarlet et al. in a general case, when the components of are in , with m a nonnegative integer.
Theorem C i) Let r be a real number such that , m a nonnegative integer and in satisfies Then, there exists such that and there exists a constant C such that ii) Let , then there exists such that . iii) Let and satisfies then, there exists such that
Let v be a smooth vector field defined on Ω and the corresponding strain field. It satisfies the following compatibility equations:
In 1864, A. J. C. B. de Saint-Venant announced that conversely, if Ω is a simply-connected open set of , then for any symmetric matrix in with which satisfies the above compatibility equations, there exists such that In fact, the first rigorous proof of sufficiency was given by Beltrami in 1886. More recently, if in addition Ω is Lipschitz, Ciarlet and Ciarlet Jr. proved that if satisfies the compatibility equations (6), then there exists such that (7) holds. A similar result, with and then , was also obtained by Amrouche et al. [4].
In 1890, Donati proved that, if Ω is an open subset of and is such that then satisfies the compatibility equations (6).
A first extension of Donati's Theorem was given by Ting [31] for symmetric matrix field when the domain Ω is bounded and Lipschitz-continuous (not necessarily simply-connected): if satisfies (8), then there exists v in such that in Ω.
Another extension of Donati's Theorem was given by Moreau [26] in the case of distributions: if satisfies (8), then there exists v in such that in Ω. Here, Ω is an arbitrary open subset of .
More recently, using different proofs, some variants of Donati's Theorem have been independently obtained by Geymonat and Krasucki [16] for and for and by Amrouche et al. [4] for .
Our fourth main result is to give a general extension of Saint-Venant's Theorem when .
Theorem D Let Ω be a bounded and simply-connected open set of with a Lipschitz-continuous boundary and satisfies Then, there exists such that
Section snippets
Notations and preliminaries
First, we recall some geometry notations. We denote by the Euclidean norm in . For and , we define the ball centered at x with radius d by . The open set Ω is starlike with respect to an open ball , if the convex hull of the set is contained in Ω for each . This means that it is starlike with respect to each point of this ball: for each and the segment is contained in Ω. Now, we can show that a bounded, starlike open set
The rotational extension of De Rham's Theorem
Mitrea [25], Costabel and Macintosch [13] have shown that if Ω is bounded and starlike with respect to an open ball, then the operator (1) is onto. In this section, we apply the singular integrals theory to give a detailed proof for this result. Then we generalize it for the case where Ω is a bounded and connected open set of with a Lipschitz-continuous boundary i.e., we prove that the operator is onto. Here denotes the space of functions such that
A weak rotational extension of De Rham's Theorem
In this section, we will use Theorem 3.4 to show another surjectivity result of the curl operator. Then, we will use this result to prove a weak rotational extension of De Rham's Theorem. First, we need the following lemma: Lemma 4.1 Let Ω be a bounded and connected open set of with a Lipschitz-continuous boundary and m a nonnegative integer. Then, the space is dense in . Proof Step 1: we show that the linear mapping defined by is onto, where J is the
A new proof of the general extension of Poincaré's Lemma
The classical Poincaré's Lemma asserts that if Ω is a simply-connected open set, then for any which satisfies in Ω, there exists such that . This lemma is also true in the general case where and Ω is a bounded and simply-connected open set with a Lipschitz-continuous boundary (see Theorem 2.9 chapter 1 in [19]). A general extension when was proved by Ciarlet and Ciarlet, Jr. (see [11]).
In this section, we study the case where h is a distribution.
Beltrami representation
In Section 3, we have shown that the operator (12) is onto. Then, we have used this surjectivity result to prove a rotational extension of De Rham's Theorem. In this section, we use the same argument to prove an extension of the Beltrami representation. First, we show a completeness of the Beltrami's representation for data in .
Theorem 6.1 Let Ω be a bounded and connected open set of with a Lipschitz-continuous boundary, r a real number such that and m a nonnegative integer. For any matrix
The general extension of Saint-Venant's Theorem
Podio-Guidugli in [28] have used a Beltrami's completeness to show the equivalence between the sufficient conditions of Donati's and Saint-Venant's Theorems: Let Ω be a smooth bounded and simply-connected open set of , then any symmetric matrix field with ( satisfies if and only if Later, Geymonat and Krasucki in [16] have proved the above equivalence when and they have used it together with Ting's Theorem to conclude an
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