On the recovery of a manifold with boundary in R n

Mathematical Problems in Mechanics/Differential Geometry

On the recovery of a manifold with boundary in R ⁿ

Philippe G. Ciarlet

, Cristinel Mardare

aDepartment of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon, Hong Kong bLaboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, 4, place Jussieu, 75005 Paris, France

Received 1 December 2003; accepted 15 December 2003 Presented by Robert Dautray

Abstract

If the Riemann curvature tensor associated with a smooth fieldCof positive-definite symmetric matrices of ordernvanishes in a simply-connected open subsetΩ⊂Rⁿ, thenCis the metric tensor field of a manifold isometrically immersed inRⁿ.

In this Note, we first show how, under a mild smoothness assumption on the boundary ofΩ, this classical result can be extended “up to the boundary”. WhenΩis bounded, we also establish the continuity of the manifold with boundary obtained in this fashion as a function of its metric tensor field, the topologies being those of the Banach spacesC(Ω). To cite this article:

P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Résumé

Sur la reconstruction d’une variété à bord dansRⁿ. Si le tenseur de courbure de Riemann associé à un champ régulierC de matrices symétriques définies positives d’ordrens’annule sur un ouvertΩ⊂Rⁿsimplement connexe, alorsCest le champ de tenseurs métriques d’une variété plongée isométriquement dansRⁿ.

Dans cette Note, on montre d’abord, moyennant une hypothèse peu restrictive sur la régularité de la frontière deΩ, comment ce résultat classique peut être étendu “jusqu’à la frontière”. LorsqueΩest borné, on établit aussi la continuité de la variété à bord ainsi obtenue en fonction de son champ de tenseurs métriques, les topologies étant celles des espaces de BanachC(Ω).

Pour citer cet article : P.G. Ciarlet, C. Mardare, C. R. Acad. Sci. Paris, Ser. I 338 (2004).

2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

Version française abrégée

Les notations non définies ici le sont dans la version anglaise.

SoitΩun ouvert connexe et simplement connexe deRⁿ, soitx₀un point deΩ, et soitSⁿ, resp.Sⁿ_>, l’ensemble de toutes les matrices symétriques, resp. symétriques définies positives, d’ordren. SoitC=(g_ij)∈C²(Ω;Sⁿ_>)une métrique riemannienne dont le tenseur de courbure de Riemann associé s’annule dansΩ. Autrement dit,

R_·^p_{ij k}:=∂_jΓ_ik^p−∂_kΓ_ij^p+Γ_ikΓ_j ^p−Γ_ijΓ_k^p=0 dansΩ,

E-mail addresses: mapgc@cityu.edu.hk (P.G. Ciarlet), mardare@ann.jussieu.fr (C. Mardare).

1631-073X/$ – see front matter 2004 Académie des sciences. Published by Elsevier SAS. All rights reserved.

doi:10.1016/j.crma.2003.12.018

où

Γ_ij^k:=1

2g^k(∂igj +∂jgi−∂gij) et g^k

:=(gij)⁻¹.

Alors un théorème classique de géométrie différentielle (rappelé dans le Théorème 2.1) affirme qu’il existe une immersionΘ∈C³(Ω;Rⁿ)et une seule telle que

∇Θ(x)^T∇Θ(x)=C(x) pour toutx∈Ω, Θ(x0)=0 et ∇Θ(x0)=C(x0)^1/2.

Notre premier objectif est d’étendre ce résultat « jusqu’à la frontière » de l’ouvertΩ, lorsque celui-ci vérifie ce que nous appelons la « propriété géodésique » (de fait, une hypothèse peu restrictive sur la régularité de la frontière∂Ω deΩ; voir Définition 1.2). Dans ce cas, on montre en effet (Théorème 2.2) que, si les fonctions

∂^αg_ij ∈C⁰(Ω), |α|2, admettent des extensions continues sur Ω, les extensions des matrices (g_ij) restant définies positives sur l’ensemble Ω, alors les champs ∂^αΘ∈ C⁰(Ω;Rⁿ), |α|3, admettent eux aussi des extensions continues surΩ.

SoitC²(Ω;Sⁿ_>), resp.C³(Ω;Rⁿ), l’ensemble formé des champs de matrices symétriques définies positives, resp.

l’espace des champs de vecteurs, qui, avec toutes leurs dérivées partielles d’ordre2, resp. d’ordre3, admettent des extensions continues àΩ, les extensions des matrices restant définies positives sur Ω. Le Théorème 2.2 établit donc l’existence d’une application

F0:C=(gij)∈C₀²Ω;Sⁿ>

→Θ∈C³Ω;Rⁿ , où

C²₀Ω;Sⁿ>

:=

C=(g_ij)∈C²Ω;Sⁿ>

;R^p_{·ij k}=0 dansΩ .

Dans un autre travail (voir [6], et [7] pour les démonstrations détaillées), nous établissons aussi que, moyennant une hypothèse de régularité plus forte sur∂Ω, le résultat de prolongement ci-dessus combiné avec un théorème fondamental de Whitney conduit au résultat suivant de prolongement : Il existe un ouvertΩ connexe deRⁿ contenantΩet un champC∈C²(Ω;Sⁿ>)tels que CprolongeCet le tenseur de courbure de Riemann associé àCreste nul surΩ.

Notre second objectif est de montrer que, lorsque l’ouvertΩ est borné et vérifie la propriété géodésique, alors l’applicationF0est localement Lipschitz-continue, les espacesC²(Ω;Sⁿ)etC³(Ω;Rⁿ)étant munis de leurs normes usuelles (Théorème 3.1). On étend ainsi le résultat de continuité « dans le cas d’un ouvert » établi par Ciarlet et Laurent [5] pour les topologies de Fréchet des espacesC²(Ω;Sⁿ)etC³(Ω;Rⁿ).

Le présent travail trouve en partie son origine dans la théorie de l’élasticité non-linéaire tri-dimensionnelle.

Comme l’avait déjà observé Antman [1], une autre approche de cette théorie consiste en effet à considérer le champCde matrices, généralement appelées « tenseurs de Cauchy–Green » dans cette théorie, comme « l’inconnue principale », au lieu du champΘde vecteurs, appelés « déformations » dans cette théorie. Effectivement, la densité d’énergie d’un matériau hyperélastique est une fonction deC(la forme de cette dépendance a d’ailleurs joué un rôle décisif dans les travaux de Ball [2], qui ont complètement renouvelé la théorie de l’elasticité). Cependant, la partie de l’énergie totale qui prend en compte les forces appliquées, de même que les conditions aux limites, restent des fonctions deΘ. D’où le besoin, dans cette approche, d’étudier la dépendance d’un champ de déformations en fonction de son champ de tenseurs de Cauchy–Green, et cela, « jusqu’à la frontière ». Naturellement, la réalisation d’un tel « programme » devra aussi inclure la considération de normes d’espaces de Sobolev, dans l’esprit des travaux de Kohn [12], ou de ceux, plus récents, de Friesecke, James et Müller [9] ou Reshetnyak [13].

Le même genre de questions, cette fois liées à la théorie des coques non linéairement élastiques, et donc relatives à la théorie des surfaces dansR³, sont abordées dans Ciarlet [3] et Ciarlet et Mardare [8].

On trouvera les démonstrations complètes des résultats de cette Note dans [7].

1. Preliminaries

Complete proofs of Theorems 2.2 and 3.1 are found in [7].

An integern2 is chosen once and for all and Latin indices and exponents vary in the set {1,2, . . . , n}. The summation convention with respect to repeated indices and exponents is used. The notationsMⁿ, Sⁿ, and Sⁿ_>, respectively designate the sets of all square matrices, of all symmetric matrices, and of all positive-definite symmetric matrices, of ordern. The spectral norm of a matrixA∈Mⁿis denoted|A|. In any metric space, the open ball with centerx and radiusδ >0 is denotedB(x;δ).

LetΩ be an open subset ofRⁿ. IfΘ∈C¹(Ω;Rⁿ)andx∈Ω, the notation∇Θ(x)stands for the matrix inMⁿ whosei-th column is the vector∂iΘ(x)∈Rⁿ. We also define the set

C² Ω;Sⁿ>

:=

C∈C² Ω;Sⁿ

; C(x)∈Sⁿ>for allx∈Ω .

Central to this Note is the following notion of spaces of functions, vector fields, or matrix fields, “of classCup to the boundary ofΩ”.

Definition 1.1. LetΩ be an open subset ofRⁿ. For any integer1, we define the space C(Ω) as the space of all functionsf ∈C(Ω)that, together with all their partial derivatives ∂^αf, 1|α|, can be extended by continuity toΩ. Analogous definitions hold for the spacesC(Ω;Rⁿ),C(Ω;Mⁿ), and C(Ω;Sⁿ). All the continuous extensions appearing in such spaces will be identified by a bar, as exemplified in the definition of the following set:

C²Ω;Sⁿ_>

:=

C∈C²Ω;Sⁿ

; C(x)∈Sⁿ_> for allx∈ Ω .

LetΩ be a connected open subset ofRⁿ. Given two pointsx, y∈Ω, a path joiningxtoyinΩis any mapping γ∈C¹([0,1];Rⁿ)that satisfiesγ(t)∈Ω for allt∈ [0,1]andγ(0)=x andγ(1)=y. Given a pathγ joiningx toy inΩ, its length is defined byL(γ):=₁

0 |γ(t)|dt. LetΩ be a connected open subset ofRⁿ. The geodesic distance between two pointsx, y∈Ωis defined by

dΩ(x, y)=inf

L(γ); γ is a path joiningx toyinΩ .

The results of this Note are established under a specific, but mild, regularity assumption on the boundary of an open subset ofRⁿ, according to the following definition:

Definition 1.2. An open subsetΩ ofRⁿ satisfies the geodesic property if it is connected and, given any point x₀∈∂Ω and anyε >0, there existsδ=δ(x₀, ε) >0 such that

d_Ω(x, y) < ε for allx, y∈Ω∩B(x₀;δ).

Note that any connected open subset of Rⁿ with a Lipschitz-continuous boundary possesses the geodesic property.

2. Recovery of a manifold with boundary from a prescribed metric tensor field

Let a Riemannian metric(gij)∈C²(Ω;Sⁿ>)be given over an open subsetΩ ofRⁿ. The Christoffel symbols of the second kind associated with this metric are then defined by

Γ_ij^k:=1

2g^k(∂_ig_j +∂_jg_i−∂g_ij), where g^k

:=(g_ij)⁻¹, and the mixed components of its Riemann curvature tensor are defined by

R_·^p_{ij k}:=∂jΓ_ik^p−∂kΓ_ij^p+Γ_ikΓ_j ^p−Γ_ijΓ_k^p.

If this tensor vanishes inΩ andΩ is simply-connected, a classical result in differential geometry asserts that (gij)is the metric tensor field of a manifoldΘ(Ω)that is isometrically immersed inRⁿ. More precisely, we have (see, e.g., Ciarlet and Larsonneur [4, Theorem 2] for an elementary and self-contained proof):

Theorem 2.1. LetΩ be a connected and simply-connected open subset ofRⁿand let a pointx0∈Ωbe given. Let a matrix fieldC=(gij)∈C²(Ω;Sⁿ>)be given that satisfiesR^p_{·ij k}=0 inΩ. Then there exists one, and only one, immersionΘ∈C³(Ω;Rⁿ)that satisfies

∇Θ(x)^T∇Θ(x)=C(x) for allx∈Ω, Θ(x0)=0 and ∇Θ(x0)=C(x0)^1/2.

Remark 1. The additional conditions Θ(x0)=0 and ∇Θ(x0) =C(x0)^1/2 imply the uniqueness of the immersionΘ, which otherwise would be only determined up to isometries inRⁿ. In fact, any additional conditions of the formΘ(x0)=a0and∇Θ(x0)=F0, wherea0is any vector inRⁿandF0is any matrix inMⁿthat satisfies F^T₀F0=C(x0), likewise imply the uniqueness of the mappingΘ. The particular choiceF0=C(x0)^1/2made here insures that the associated mappingC(x0)∈Sⁿ_>→F0∈Mⁿ is smooth, a property that is used in the proof of Theorem 3.1. Another choice for the matrixF0that fulfills the same criterion is the upper triangular matrix that arises in the Cholesky factorization of the matrixC(x0).

The first objective of this Note is to show in the next theorem how a manifold with boundary, i.e., a subset ofRⁿ of the formΘ(Ω), can be likewise recovered from a metric tensor field that, together with some partial derivatives, can be continuously extended to the closureΩ. In other words, we wish to extend the above existence and uniqueness result “up to the boundary ofΩ”.

Theorem 2.2. Let Ω be a simply-connected open subset of Rⁿ that satisfies the geodesic property (see Definition 1.2) and let a pointx0∈Ω be given. Let there be given a matrix fieldC=(gij)∈C²(Ω;Sⁿ>)(in the sense of Definition 1.1) that satisfiesR_·^p_{ij k}=0 inΩ. Then there exists one, and only one, mappingΘ∈C³(Ω;Rⁿ) (again in the sense of Definition 1.1) that satisfies (the notations∇ΘandCrepresent the continuous extensions of the fields∇ΘandC):

∇Θ(x)^T∇Θ(x)= C(x) for allx∈ Ω, Θ(x0)=0 and ∇Θ(x0)=C(x0)^1/2.

Sketch of proof. (i) Given any mappingΘ∈C³(Ω;Rⁿ)that satisfies∇Θ(x)^T∇Θ(x)=C(x)for allx∈Ω(such mappings exist by Theorem 2.1), let

F(x):=∇Θ(x)∈Mⁿ and Γi(x):=

Γ_ij^k(x)

∈Mⁿ for allx∈Ω,

withkas the row index andj as the column index. Then the matrix fieldsF∈C²(Ω;Mⁿ)andΓi∈C¹(Ω;Mⁿ) defined in this fashion satisfy

∂_iF(x)=F(x)Γi(x) for allx∈Ω.

It is first established that the assumption(gij)∈C²(Ω;Sⁿ>)implies that the matrix fieldsΓi belong to the space C¹(Ω;Mⁿ)and that, given any compact subsetKofRⁿ, then sup_x∈K∩Ω|F(x)|<∞.

Fix a pointx₀∈∂Ωand letK₀=B(x₀;1). Then the above properties together imply that c₀:=

sup

x∈K₀∩Ω

F(x) sup

x∈K₀∩Ω

i

Γi(x)²_1/2

<+∞.

Let ε >0 be given. Because Ω satisfies the geodesic property, there exists δ(ε)∈ ]0,¹₂] such that, given any two pointsx, y∈B(x₀;δ(ε))∩Ω, there exists a pathγ =(γi)joiningx toy inΩ whose length satisfies L(γ)ε/max{c₀,2}.

Since∂iF(x)=F(x)Γi(x)for allx∈Ω, the matrix fieldY:=F◦γ ∈C¹([0,1];Mⁿ)associated with any such pathγ satisfiesY(t)=γ_i(t)Y(t)Γi(γ(t))for all 0t1. Expressing thatY(1)=Y(0)+₁

0 Y(t)dtyields, for any two pointsx, y∈B(x₀;δ(ε))∩Ω,

F(y)−F(x)=Y(1)−Y(0) sup

0t1

F

γ(t) ¹

0

γ_i(t)Γi

γ(t)dt

sup

x∈K₀∩Ω

F(x) sup

x∈K₀∩Ω

i

Γi(x)²1/2

L(γ)ε.

This inequality then implies that the fieldF ∈C²(Ω;Mⁿ)can be extended to a field that is continuous onΩ. Since ∂_iF =F Γi inΩ andΓi ∈C¹(Ω;Mⁿ), each field∂_iF ∈C¹(Ω;Mⁿ)can be extended to a field that is continuous onΩ; hence F ∈C¹(Ω;Mⁿ). Differentiating the relations ∂iF =F Γi inΩ further shows that F∈C²(Ω;Mⁿ).

Givenx0∈∂Ω, we proceed again as above, the numberδ(ε)∈ ]0,¹₂] being now chosen in such a way that L(γ)ε/max{c₁,2}, wherec₁:=^√1_n(sup_x∈K

0∩Ω|F(x)|)⁻¹<∞.

For eachx∈Ω, letg_i(x)denote thei-th column vector of the matrixF(x). The relations∂_iΘ(x)=g_i(x)for allx∈Ω then imply that the vector fieldy:=Θ◦γ ∈C¹([0,1];Rⁿ)associated with each such pathγ joiningx toy inΩ satisfiesy(t)=γ_i(t)g_i(γ(t))for all 0t1, so that, for any two pointsx, y∈B(x₀;δ(ε))∩Ω,

Θ(y)−Θ(x)=y(1)−y(0) 1

0

γ_i(t)g_i

γ(t)dt

L(γ) sup

x∈K₀∩Ω

i

g_i(x)²1/2

√

n L(γ) sup

x∈K₀∩Ω

F(x)ε.

This inequality thus shows that the field Θ ∈ C³(Ω;Rⁿ) can be extended to a field that is continuous on Ω. Differentiating the relations ∂_iΘ =g_i in Ω and noting that g_i ∈C²(Ω;Rⁿ), we finally conclude that Θ∈C³(Ω;Rⁿ). The uniqueness ofΘis then easily established. ✷

We also establish elsewhere (see [6], and [7] for complete proofs) that, under a stronger regularity assumption on

∂Ω, the above extension result combined with a fundamental theorem of Whitney leads to the following extension result: There exist a connected open subsetΩofRⁿ containingΩand a fieldC∈C²(Ω;Sⁿ_>)such thatC is an extension ofCand the Riemann curvature tensor associated withCstill vanishes inΩ.

3. Continuity of a manifold with boundary as a function of its metric tensor

LetΩ be a simply-connected open subset ofRⁿthat satisfies the geodesic property. Define the set C²₀Ω;Sⁿ>

:=

C=(gij)∈C²Ω;Sⁿ>

; R_·^p_{ij k}=0 inΩ ,

and let again a pointx₀∈Ωbe chosen once and for all. Then by Theorem 2.2, there exists a well-defined mapping F0:C₀²Ω;Sⁿ>

→C³Ω;Rⁿ

that associates with any matrix field C ∈ C₀²(Ω;Sⁿ>) the unique mapping Θ ∈ C³(Ω;Rⁿ) that satisfies

∇Θ(x)^T∇Θ(x)= C(x)for allx∈ Ω,Θ(x0)=0, and∇Θ(x0)=C(x0)^1/2.

If in addition the setΩ is bounded, the spacesC²(Ω;Sⁿ)andC³(Ω;Rⁿ)become Banach spaces and thus the setC₀²(Ω;Sⁿ_>)becomes a metric space when it is equipped with the induced topology. So natural question arises: Is the mappingF0continuous when the setC²₀(Ω;Sⁿ_>)and the spaceC³(Ω;Rⁿ)are equipped with these topologies?

The second objective of this Note is to provide the following affirmative answer to this question.

Theorem 3.1. LetΩbe a simply-connected and bounded open subset ofRⁿthat satisfies the geodesic property, let the spacesC(Ω;Mⁿ)andC(Ω;Rⁿ), 1, be equipped with their usual norms, defined by

F_,Ω= sup

x∈ Ω

|α|

∂^αF(x) for allF∈CΩ;Mⁿ ,

Θ_,Ω= sup

x∈ Ω

|α|

∂^αΘ(x) for allΘ∈CΩ;Rⁿ ,

and let the setC₀²(Ω;Sⁿ>)be equipped with the metric induced by the norm·2,Ω. Then the mapping F0:C0²

Ω;Sⁿ>

→C³Ω;Rⁿ

is continuous. It is even locally Lipschitz-continuous over the setC²₀(Ω;Sⁿ>), in the sense that, given any matrix fieldC∈C₀²(Ω;Sⁿ>), there exist constantsc(C) >0 andδ(C) >0 such that

Θ−Θ

3,ΩcCC−C_2,

Ω for allC,C∈C0²

Ω;Sⁿ>

∩BC;δC ,

whereΘ:= F0(C),Θ:= F0(C), andB(C;δ(C))denotes the open ball of centerCand radiusδ(C)in the space C²(Ω;Sⁿ).

Sketch of proof. (i) Preliminaries. The imageΘ= F0(C)∈C³(Ω;Rⁿ)of an arbitrary element C=(gij)∈ C²₀(Ω;Sⁿ>)is constructed in the following manner: First, the matrix fieldsΓi=(Γ_ij^k)∈C¹(Ω;Mⁿ)are defined in Ωby letting

Γ_ij^k=1

2g^k(∂igj +∂jgi−∂gij), where g^k

=(gij)⁻¹,

and the matrixC(x0)^1/2∈Sⁿ_>is defined as the unique square root of the matrixC(x0).

Second, the matrix fieldF ∈C²(Ω;Mⁿ)is defined as the unique one that satisfies

∂_iF(x)=F(x)Γi(x), x∈Ω, and F(x₀)=C(x0)^1/2.

Third, the vector fieldΘ∈C³(Ω;Rⁿ)is defined as the unique one that satisfies

∇Θ(x)=F(x), x∈Ω, and Θ(x0)=0.

Accordingly, the strategy consists in establishing the local Lipschitz-continuity of each one of the above factor mapping separately, according to the following steps (the proofs of which are too technical to be sketched here):

(ii) Given any matrix fieldC∈C₀²(Ω;Sⁿ>), there exist constantsc1(C) >0 andδ(C) >0 such that C(x0)^1/2−C(x0)^1/2+max

i

Γi−Γi_1,

Ωc₁CC−C

2,Ω

for all matrix fieldsC,C∈C²₀(Ω;Sⁿ>)∩B(C;δ(C)), where the matrix fieldsΓi=(Γ_ij^k)∈C¹(Ω;Mⁿ)are defined inΩ by

Γ_ij^k:=1

2g^k(∂_ig_j +∂_jg_i−∂g_ij), where(g_ij):=Cand g^k

:=(g_ij)⁻¹.

(iii) The matrix fieldsΓi,Γi∈C¹(Ω;Mⁿ)being defined in terms of the matrix fieldsC,C∈C₀²(Ω;Sⁿ>)as in (i) and (ii), let the matrix fieldsF,F∈C²(Ω;Mⁿ)satisfy

∂_iF(x)=F(x)Γi(x) for allx∈Ω and F(x₀)=C(x0)^1/2,

∂_iF(x)=F(x)Γi(x) for allx∈Ω and F(x₀)=C(x 0)^1/2.

Then, given any matrix fieldC∈C²₀(Ω;Sⁿ_>), there exists a constantc₂(C) >0 such that F−F

2,Ωc₂CC(x0)^1/2−C(x0)^1/2+max

i

Γi−Γi

1,Ω

for all matrix fieldsC,C∈C₀²(Ω;Sⁿ_>)∩B(C;δ(C)), whereδ(C) >0 is the constant found in (ii).

(iv) Let there be given matrix fieldsF,F∈C²(Ω;Mⁿ)and vector fieldsΘ,Θ∈C³(Ω;Rⁿ)that satisfy

∇Θ(x)=F(x) for allx∈Ω and Θ(x0)=0,

∇Θ(x) =F(x) for allx∈Ω and Θ(x 0)=0.

Then there exists a constantc₃>0 independent of these fields such that Θ−Θ

3,Ωc3F−F_2,

Ω. ✷

The local Lipschitz-continuity of the mappingF0established in Theorem 3.1 is to be compared with the earlier estimates of John [10,11] and Kohn [12] and the recent “theorem on geometric rigidity” of Friesecke, James and Müller [9]. Such estimates are more powerful than those found here, in the sense that they are established for Sobolev norms. However, they only imply continuity at rigid body deformations, i.e., corresponding toC=I, whereas our estimates hold “at any Cauchy–Green tensorC”. Particularly relevant here is also the continuity result for “quasi-isometric mappings” recently established by Reshetnyak [13].

Acknowledgement

The work described in this Note was supported by a grant from the Research Grants Council of the Hong Kong Special Admimistrative Region, China [Project No. 9040869, CityU 100803].

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