On the curl operator and some characterizations of matrix fields in Lipschitz domains

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Abstract

As well-known, De Rham's Theorem is a classical way to characterize vector fields as the gradient of the scalar fields, it is a tool of great importance in the theory of fluids mechanic. The first aim of this paper is to provide a useful rotational version of this theorem to establish several results on boundary value problems in the field of electromagnetism. Gurtin presented the completeness of Beltrami's representation in smooth domains for smooth symmetric matrix. The extension of Beltrami's completeness was obtained by Geymonat and Krasucki when the domain Ω is only Lipschitz and for symmetric matrix in Ls2(Ω). Our second aim is to present new extensions of Beltrami's completeness results on Lipschitz domains, firstly in the case where the data are in Ds(Ω) and secondly when the data are in Wsm,r(Ω). The third objective is to extend Saint-Venant's Theorem to the case of distributions.

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 R3 with a Lipschitz-continuous boundary. The surjectivity of the operator div:D(Ω)D0(Ω), where D0(Ω) denotes the subspace of functions in D(Ω) 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 fD(Ω) satisfyingforallφV(Ω),f,φD(Ω)D(Ω)=0, where V(Ω) denotes the subspace of vector fields in D(Ω) with divergence free, then there exists a scalar field pD(Ω) such thatf=pinΩ. 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 operatorcurl:D(Ω)V(Ω), 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 iffV(Ω)satisfiesΩfφdx=0,φKT(Ω), whereKT(Ω)={vL2(Ω),curlv=0,divv=0inΩ,vn=0onΩ} (see Section 2), then there exists φD(Ω) satisfying curlφ=f 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 1<r< and m a nonnegative integer. Then, for any fV(Ω) satisfying (2), there exists ψD(Ω) such thatcurlψ=finΩ, and there exists a constant C(r,m,Ω) such thatψWm+1,r(Ω)C(r,m,Ω)fWm,r(Ω).

ii) Let gD(Ω) and satisfiesforallφG(Ω),g,φD(Ω)D(Ω)=0, where G(Ω)={φD(Ω),curlφ=0inΩ}. Then, there exists ψD(Ω) such thatcurlψ=ginΩ.

As a close result, Borchers and Sohr in [9] have shown that if the open set Ω is of class Cm,1, where m is a nonnegative integer, then the operator curl:W0m,r(Ω)Um,r(Ω) 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 1<r< and m a nonnegative integer. Then, there exists a constant Cr,m(Ω) such that for any fUm,r(Ω), there exists ψW0m+1,r(Ω), that satisfiescurlψ=finΩ, andψWm+1,r(Ω)Cr,m(Ω)fWm,r(Ω).

ii) Let m1 be an integer and gWm,r(Ω) satisfyingφGm,r(Ω),g,φW0m,r(Ω)Wm,r(Ω)=0. Then there exists ΨWm+1,r(Ω) such thatcurlΨ=ginΩ.

The vector potential ψ obtained by Borchers and Sohr satisfies the additional property: Δm+1div ψ=0 in Ω, which needs the regularity Cm,1 of the domain Ω.

In the absence of body forces the stress equations of equilibrium take the formdivS=0inΩ,S=ST, 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:S=curl curlA for all smooth symmetric second order tensor fieldsAinΩ. Gurtin [21] gave an example of a stress field S 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 S which are self-equilibrated, i.e. for each closed regular surface C contained in Ω, the resultant force and the moment vanish. In other words, S satisfies the following condition:CSndσ=CPi×(Sn)dσ=0,for all1i3, such that Pi=εijkxkej. 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 R3 with Lipschitz-continuous boundary and S be a symmetric matrix field in Ls2(Ω) and satisfying the following conditions:divS=0inΩandΓk(Sn)eidσ=Γk(Sn)Pidσ=0,1i3,1kK.

Then, there exists a symmetric matrix field AHs2(Ω) such that curl curlA=S in Ω. Moreover, P. G. Ciarlet et al. in [12] observed that if the above symmetric matrix field S satisfies the following conditions:Sn=0onΩandSn,eiΣj=Sn,PiΣj=0,for all1i3,1jJ, where ,Σj denotes the duality pairing between H12(Σj) and H12(Σj), then AH0,s2(Ω).

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 S are in D(Ω) and to prove the above observation of P.G. Ciarlet et al. in a general case, when the components of S are in W0m,r(Ω), with m a nonnegative integer.

Theorem C

i) Let r be a real number such that 1<r<, m a nonnegative integer and S in Vs(Ω) satisfiesΣj(Sn)eidσ=Σj(Sn)Pidσ=0,forall1i3,1jJ. Then, there exists ADs(Ω) such thatcurl curlA=SinΩ, and there exists a constant C such thatAWm+2,r(Ω)CSWm,r(Ω).

ii) Let SUm,r(Ω), then there exists AW0m+2,r(Ω) such that curl curlA=S.

iii) Let SDs(Ω) and satisfiesfor allEGs(Ω),S,ED(Ω)D(Ω)=0, then, there exists ADs(Ω) such thatcurl curlA=SinΩ.

Let v be a smooth vector field defined on Ω and E=sv the corresponding strain field. It satisfies the following compatibility equations:curl curlE=0inΩ.

In 1864, A. J. C. B. de Saint-Venant announced that conversely, if Ω is a simply-connected open set of R3, then for any symmetric matrix in E=(Eij) with EijC2(Ω) which satisfies the above compatibility equations, there exists vC3(Ω) such thatsv=EinΩ. 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 EL2(Ω) satisfies the compatibility equations (6), then there exists vH1(Ω) such that (7) holds. A similar result, with EH1(Ω) and then vL2(Ω), was also obtained by Amrouche et al. [4].

In 1890, Donati proved that, if Ω is an open subset of R3 and EC2(Ω) is such thatΩE:M=0forallMDs(Ω)such thatdivM=0inΩ, then E satisfies the compatibility equations (6).

A first extension of Donati's Theorem was given by Ting [31] for symmetric matrix field EL2(Ω) when the domain Ω is bounded and Lipschitz-continuous (not necessarily simply-connected): if EL2(Ω) satisfies (8), then there exists v in H1(Ω) such that E=sv in Ω.

Another extension of Donati's Theorem was given by Moreau [26] in the case of distributions: if EDs(Ω) satisfies (8), then there exists v in D(Ω) such that E=sv in Ω. Here, Ω is an arbitrary open subset of R3.

More recently, using different proofs, some variants of Donati's Theorem have been independently obtained by Geymonat and Krasucki [16] for EWs1,p(Ω) and for ELsp(Ω) and by Amrouche et al. [4] for ELs2(Ω).

Our fourth main result is to give a general extension of Saint-Venant's Theorem when EDs(Ω).

Theorem D

Let Ω be a bounded and simply-connected open set of R3 with a Lipschitz-continuous boundary and EDs(Ω) satisfiescurl curlE=0inΩ. Then, there exists vD(Ω) such thatsv=EinΩ.

Section snippets

Notations and preliminaries

First, we recall some geometry notations. We denote by || the Euclidean norm in RN. For xΩ and d>0, we define the ball centered at x with radius d by B(x,d)={yRN,|xy|<d}. The open set Ω is starlike with respect to an open ball B(x,d), if the convex hull of the set {y}B(x,d) is contained in Ω for each yΩ. This means that it is starlike with respect to each point of this ball: for each zΩ and yB(x,d) the segment [zy] 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 R3 with a Lipschitz-continuous boundary i.e., we prove that the operatorcurl:D(Ω)V(Ω)KT(Ω) is onto. Here V(Ω)KT(Ω) denotes the space of functions vV(Ω) such that Ωvφd

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 R3 with a Lipschitz-continuous boundary and m a nonnegative integer. Then, the space V(Ω)KT(Ω) is dense in Um,r(Ω).

Proof

Step 1: we show that the linear mapping R:V(Ω)RJ defined by(R(v))j=Σjvndσ,1jJ, 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 hC1(Ω) which satisfies curlh=0 in Ω, there exists pC2(Ω) such that h=gradp. This lemma is also true in the general case where hL2(Ω) 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 hH1(Ω) 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 Ds(Ω).

Theorem 6.1

Let Ω be a bounded and connected open set of R3 with a Lipschitz-continuous boundary, r a real number such that 1<r< and m a nonnegative integer. For any matrix SV

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 R3, then any symmetric matrix field E=(Eij) with EijCN(Ω) (N2) satisfiescurl curlE=0inΩ, if and only ifΩE:Mdx=0foranyMVs(Ω). Later, Geymonat and Krasucki in [16] have proved the above equivalence when ELs2(Ω) and they have used it together with Ting's Theorem to conclude an

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