Philippe G.CIARLETCristinel MARDARESorin MARDARE

(Dedicated to H??m Brezis on the occasion of his 70th birthday)

1 Introduction

Throughout this paper,n designates an integer≥2.Then Endenotes the n-dimensional real Euclidean space;Mndenotes the space of all real matrices of order n;Sndenotes the subspace of all symmetric matrices in Mn;Sn>denotes the subset of all positive-definite matrices in Sn;Ondenotes the subset of all orthogonal matrices in Mn;and:={P∈On;detP=1}.The Euclidean inner product in Enis denoted by·,and the Euclidean norm in Enand the Frobenius norm in Mnare both denoted by|·|.

In all that follows,Latin indices and exponents range in the set{1,···,n}(save when otherwise indicated,as for instance when they are used for indexing sequences)and the summation convention with respect to repeated indices and exponents are used.A generic point in an open set Ω?Rnis denoted by x=(xi),and partial derivatives of the first and second order,in the classical or distributional sense,are denoted byand,respectively.

If φ :Ω → Enis a sufficiently smooth immersion from an open subset Ω ? Rninto En,then the positive-definite symmetric tensor field(gij):Ω→defined by

is a Riemannian metric in Ω,called the metric tensor field induced by the immersion φ,and the functions gij:Ω→E are called its covariant components.

The first objective of this paper is to review what can be said about the converse property:If(gij):Ω→is a sufficiently smooth field of positive-definite symmetric matrices of order n defined over an open set Ω ? Rn,then does there exist an immersion φ :Ω → Ensuch that

If the answer is yes,is such an immersion unique?

We will provide successive answers to these questions,for matrix fields(gij):Ω→with components in the spaces(in this order)

(see the first theorem in Sections 2–7).In each case,the answer relies on the following two simple,yet crucial,observations.

Assume that there exists a sufficiently smooth immersion φ :Ω → Ensuch that

By definition of an immersion,the vector fields

are thus linearly independent at each x ∈ Ω.Hence there exist functions:Ω →R such that

The first observation is that the relations ?iφ ·?jφ =gijimply(after a series of straightforward computations)that the componentsare of the form

where

The functions Γij?(resp.)are the Christoffel symbols of the first kind(resp.of the second kind)associated with matrix field(gij).

The vector fields githus satisfy a Pfa ffsystem of linear partial differential equations(the equations(1.1)above),whose coefficientsdepend only on the matrix field(gij).

The second observation is that the relationsimply that

which in turn implies that(since the vector fields g?are linearly independent)or equivalently,

The functions Rqijk(resp.)are the covariant components(resp.mixed components)of the Riemann curvature tensor field associated with the matrix field(gij).The compatibility conditions Rqijk=0 in Ω are thus necessary for the existence of the immersion φ :Ω → Ensuch that ?iφ ·?jφ =gijin Ω.

We will then show that,under the crucial assumption that the open set Ω is simply-connected(additional conditions may be imposed in some cases on the boundary of Ω),the compatibility conditions

possibly interpreted in the sense of distributions if need be,become also sufficient for the existence of immersions φ :Ω → Ensatisfying

and that such immersions φ are only defined up to composition by isometries of En(such isometries are defined in Section 2)(see Theorems 2.1,3.1,4.1,5.1,6.1 and 7.1).

The second objective of this paper is to provide a natural complement to these existence and uniqueness theorems of immersions φ :Ω → Enwith prescribed metric tensor fields(gij):Ω →,i.e.,to establish continuity theorems,showing that the immersion φ depends continuously on their metric tensor fields(gij)with respect to specific topologies(see Theorems 2.2,3.3,4.2,5.3 and 7.2).

In fact,if the metric tensors fields(gij)have components in one of the spaces C2(),C1(),W1,p(Ω),or Lq(Ω),these continuity theorems will be established as consequences of specific nonlinear Korn inequalities(see Theorems 3.2,5.2,6.2 and 8.1).More specifically,we will show that,given any immersion,k=1,2,there exist two constants C=C(φ0)>0 and δ= δ(φ0)>0 such that

(the set Isom(En)is defined in Section 2)for all immersions φ∈Ck+1(Ω;En)and?φ∈Ck+1(Ω;En)that satisfy

where

We will also show that,given any immersion φ0∈ W2,p(Ω;En),p>n,there exist two constants C=C(φ0)>0 and δ= δ(φ0)>0 such that

for all immersions φ ∈W2,p(Ω;En)andthat satisfy

Finally,we will show that,given any immersionsatisfying det?φ >0 in Ω,any 1≤q<∞and any 10 such that,for all immersionsthat satisfy detalmost everywhere in Ω,we have

(the set Isom+(En)is defined in Section 8).

2 The Classical Case:Metric Tensor Fields in C2(Ω)

We begin with the classical case,where the matrix field(gij):Ω →has components gij∈ C2(Ω).Note that this is the minimal regularity assumption that ensures that all the partial derivatives appearing in the proof of the existence and uniqueness theorem below are classical ones.

An isometry of Enis a mapping r:En→Enof class C1that preserves the Euclidean metric of En,i.e.,a mapping that satisfies

where δijdenotes the Kronecker symbol.It is well known that r is an isometry of Enif and only if there exists a vector a∈Enand an orthogonal matrix R∈Onsuch that

and that the set of all isometries of En,henceforth denoted by

is a smooth finite-dimensional manifold,since Onis a smooth manifold of dimension(see,e.g.,[1]).

Given a smooth enough mapping φ :Ω → Enand a point x ∈ Ω,the notation?φ(x)designates the n×n matrix whose j-th column vector is the vector gj(x):= ?jφ(x).

The next theorem is classical(see,e.g.,[15,4];for a detailed and self-contained proof,see[6]or[5,Theorems 8.6-1,8.7-1]).

Theorem 2.1Let Ω be a simply-connected open set in Rn,and let(gij)∈ C2(Ω;)be a matrix field whose Riemann curvature tensor field vanishes in Ω,i.e.,

where

Then there exists an immersion φ ∈ C3(Ω;En)such that

In addition,an immersion ψ ∈ C3(Ω;En)satisfies

if and only if there exists an isometry r∈Isom(En)such that

Sketch of the ProofAs already mentioned,the proof relies of the observation that,if an immersion φ ∈ C3(Ω;En)satisfies

then the n vector fields

satisfy the Pfa ffsystem of partial differential equations

where

Thus the idea of the proof of existence of a solution is to construct explicitly an immersion φ by using these equations “in reverse order”.In other words,the proof consists first in showing that there exist n vector fields gj∈ C2(Ω;En)that satisfy the Pfa ffsystem

second in showing that there exists a mapping φ ∈ C3(Ω;En)that satisfies the Poincaré system

and third in showing that the mapping φ obtained in this fashion is an immersion and satisfies the equations

(i)That the Pfa ffsystem(2.1)possesses solutions is proved as follows.Pick any point x0in Ω and any n vectors∈ Enthat satisfy

(it is easy to see that such vectors exist).

Next,given any point x ∈ Ω,let γ =(γi) ∈ C1([0,1];En)be a mapping that satisfies γ(0)=x0,γ(1)=x,and γ(t)∈ Ω for all t∈ [0,1](the image of[0,1]under the mapping γis thus a curve contained in Ω that joins the points x0and x)and,for each j,let fj∈ C1([0,1];En)be the unique solution to the system of ordinary differential equations

The assumptions that Ω is simply-connected and that the matrix field(gij)satisfies the equations together imply that the value of fjat t=1 is independent of the choice of the path γ joining x0to x(the regularity assumption gij∈ C2(Ω)is needed here),so that the n vector fields gj:Ω→Engiven by

are unambiguously defined. Then one proves that these vector fields belong to the space C2(Ω;En),that they satisfy the system(2.1),and that

(ii)That the Poincaré system(2.2)possesses solutions is proved in a similar fashion.

More specifically,let x0be any point in Ω and let f0be any vector in En.Given any point x ∈ Ω,let γ =(γi) ∈ C3([0,1];En)be any mapping that satisfies γ(0)=x0,γ(1)=x,and γ(t)∈Ω for all t∈[0,1],and let f∈C1([0,1];En)be the unique solution to the system of ordinary differential equations

The assumptions that Ω is simply-connected and the relations

(that=is a simple consequence of the definition of(see the introduction)),together imply that the value of f at t=1 is independent of the choice of the path γ joining x0to x,so that the mapping φ :Ω → Engiven by

is unambiguously defined.Then one proves that this mapping belongs to the space C3(Ω;En)and satisfies the Poincaré system(2.2).

(iii)That the mapping φ ∈ C3(Ω;En)is an immersion that satisfies ?iφ ·?jφ =gijis a simple consequence of its definition,since

which in turn implies that

(iv)Finally,in order to prove that such a mapping is unique up to isometries of En,let φ ∈ C3(Ω;En)and ψ ∈ C3(Ω;En)be two immersions that satisfy

respectively.At each point x∈Ω,letdenote the square root of the matrix and let

Then P ∈ C2(Ω;On),Q ∈ C2(Ω;On),and

Given any point x∈Ω,there exists a connected open set Ux?Ω containing x such that the restriction φ|Ux∈ C3(Ux;En)is injective,since φ ∈ C3(Ω;En)is an immersion.Let U ? Ω denote any connected and open subset such that φ|U∈ C3(U;En)is injective,let:= φ(U),and let r:= ψ ?(φ|U)?1.Then?U is a connected open subset of En,r ∈ C3(;En),and

This relation implies that

which means that r is the restriction to?U of an isometry of En.

It thus follows that,given any x∈Ω,there exists a connected open set Ux?Ω containing x and an isometry rx∈Isom(En)such that

This relation implies that rx=rywhenever Ux∩Uy?.Combined with the assumption that Ω is in particular connected(as a simply-connected set),this implies that there exists a unique isometry r∈Isom(En)such that

As a complement to Theorem 2.1,we now show that,up to an isometry of En,an immersion φ ∈ C3(Ω;En)depends continuously on its metric tensor field(gij) ∈ C2(Ω;Sn)when both spaces are equipped with their respective Fréchet topologies.

More specifically,recall that a sequenceof functions fm∈ Ck(Ω),k ∈ N,converges to f ∈ Ck(Ω)with respect to the Fréchet topology of Ck(Ω)if,for each compact subset K ? Ω,

and,if this is the case,we write

Such notions can then be clearly extended to the spaces Ck(Ω;En)and Ck(Ω;Sn),k ∈ N.

Theorem 2.2Let Ω be a connected and open subset of Rn,and let

be immersions that satisfy

where

respectively denote the covariant components of the metric tensor fields induced by the immersions φm,m ≥ 1 and φ.

Then there exist isometries rm∈Isom(En),m≥1,such that

Sketch of the ProofThe details of the proof below can be found in[7].An argument similar to that used in the proof of the uniqueness part of Theorem 2.1 shows that it suffices to prove Theorem 2.2 in the particular case where φ=id,where id denotes the identity mapping of the set Ω.So let a sequence of immersions

be given that satisfies

For each m≥1,let(gk?,m(x))denote the inverse of the matrix(),x∈Ω,let

denote the Christoffel symbols of the second kind associated with the matrix field(),and let

The definition of the Christoffel symbolsimplies on the one hand that

On the other hand,

Combined with the relations(see the proof of Theorem 2.1)

the above convergences imply that

Let x0be a point in Ω,and,for each m ≥ 1,let

wheredenotes the inverse of the square root of the matrix(gij(x0))∈.Note that the matrix Rmis orthogonal since

Let the functions rm:En→ Enand ψm:Ω → Enbe defined by

and

respectively.Then rm∈Isom(En),ψm∈C3(Ω;En)and

Therefore,

as m→∞,which in turn implies that

3 Metric Tensor Fields in C2()

A domain in Rnis a bounded and connected open subset of Rnwhose boundary is Lipschitzcontinuous,the set Ω being locally on one side of its boundary(see[19,2]).

The space Ck(),where Ωdenotes an open subset of Rnand k ∈ N,is defined as the space of all functions f ∈ Ck(Ω)that,together with all their derivatives up to order k,possess continuous extensions to the closureof Ω.In the particular case where Ωis a domain,we have

(see[20,8]).The spaceis equipped with the norm defined by

where?αis the usual multi-index notation for partial derivatives operators.

We establish in this section results similar to those in the previous section,but now for matrix fields(gij)with componentsinstead of gij∈ C2(Ω).While no boundedness or regularity assumption on the boundary of Ω was needed in the previous section,here we assume that Ω is a domain.

Theorem 3.1Let Ωbe a simply-connected domain in Rn,and letbe a matrix field whose Riemann curvature tensor field vanishes in Ω,i.e.,

where

Then there exists an immersionsuch that

In addition,an immersionsatisfies

if and only if there exists an isometry r∈Isom(En)such that

Sketch of the ProofThe details of the proof below can be found in[8].By Theorem 2.1,there exists an immersion φ ∈ C3(Ω;En)such that

It remains to prove that φ,together with its partial derivatives up to order three,possess continuous extensions to

The first step is to prove that each vector fieldtogether with its partial derivatives up to order two,possess continuous extensions to Ω.Let B ? Rnbe any open ball such that ω :=B ∩ Ω ?= ?,let x and y be any two points in the set ω,let γ =(γi)∈ C1([0,1];ω)be any mapping that satisfies γ(0)=x and γ(1)=y,and let

Since the vector fields gisatisfy the Pfa ffsystem

where

the vector fields fisatisfy the system of ordinary differential equations

Sinceand(no summation),and sinceis a compact subset of Ω,there exists a constant C1=C1(ω)such that

We then infer from the inspection of the above system of ordinary differential equations that there exists a constant C2=C2(ω)such that

The assumption that the boundary of Ω is Lipschitz continuous implies that the mapping γ can be chosen in such a way that the right-hand side of the above inequality is bounded by a constant times the diameter of the ball B.This implies that the vector fields giare uniformy continuous in Ω,hence that they can be extended by continuity up to the boundary of Ω.Since in addition the vector fields gisatisfy the Pfa ffsystem(3.1),whose coefficientsbelong tothese extensions of gibelong in fact to the space

The second step consists in showing that the mapping φ ∈ C3(Ω;En),together with its partial derivatives up to order three,possesses continuous extensions toThis is done as in the first step,but this time using the Poincaré system

satisfied by φ,instead of the Pfa ffsystem(3.1)satisfied by gi.

Finally,the uniqueness part of the theorem is a simple consequence of the uniqueness part of Theorem 2.1.

Note that,as shown in[8],the assumption that Ω is a domain in Theorem 3.1,as well as in Theorem 3.2 below,can be replaced by the weaker,but a bit too technical to be reproduced here,assumption that Ω is connected and satisfies the “geodesic property”.

The next theorem establishes a nonlinear Korn inequality in C3(),which implies in particular(see Theorem 3.3)that,up to an isometry of En,an immersion φ ∈ C3(;En)depends continuously on its metric tensor field

Theorem 3.2Let Ωbe a domain in Rn,and letbe an immersion.Then there exist two constants C=C(φ0)>0 and δ= δ(φ0)>0 such that

for all immersionsandthat satisfy

where

denote the covariant components of the metric tensor fields induced by the immersions φ0,φ,and,respectively.

Sketch of the ProofThe details of the proof below can be found in[8].

To begin with,one proves that the mappings

and

where the functionsare defined at each pointby(ak?(x)):=(aij(x))?1,are of class C∞.

Let x0∈.Using that the above mappings are in particular of class C1,one proves that there exists two constants δ= δ(φ0)>0 and D1=D1(φ0)>0 such that,for all immersionsandthat satisfies

wherethen

and

where

Next,define an isometry r∈Isom(En)by letting

where

Note that the matrices R andare orthogonal,since

Let

and define the vector fields

Then,thanks to the definition of the isometry r,

Noting that the mappingsatisfies the Poincaré system

one deduces(as in the proof of Theorem 3.1,by integrating along paths γ ∈ C1([0,1];Ω)joining x0to a generic point x∈Ω)that

where the constant D2depends only on Ω.

Since the isometries φand ψsatisfy

the vector fields giand hisatisfy the Pffaf systems(see the proof of Theorem 2.1)

respectively.Hence

which,combined with the estimate(3.3),yields(again by integrating along paths γ∈C1([0,1];Ω)joining x0to a generic point x ∈ Ω,but this time using in addition Gronwall’s inequality):

where the constant D3depends on Ωand φ0.

Finally,since the vectors(hi?gi)(x0)∈Enare the column vectors of the matrix

it follows that

where the constant D4depends only on φ0.

Combining inequalities(3.2)to(3.6)yields the announced nonlinear Korn inequality.

An immediate consequence of the nonlinear Korn inequality of Theorem 3.2 is the following convergence result,similar to that of Theorem 2.2.

Theorem 3.3Let Ω be a domain in Rn,and let

be immersions that satisfy

where

Then there exist isometries rm∈Isom(En),m≥1,such that

4 Metric Tensor Fields in C1(Ω)

Beginning with this section,we consider regularity assumptions that are weaker than those classically made,like in Section 2,for establishing the existence and uniqueness of an immersion with prescribed metric tensor field.

To begin with,we show that the theorems established in Section 2 still hold for matrix fields with coefficients gij∈ C1(Ω),instead of gij∈ C2(Ω).The notation D(Ω)designates the space of infinitely differentiable functions with compact support in Ω.

Theorem 4.1Let Ω be a simply-connected open set in Rn,and let(gij)∈ C1(Ω;)be a matrix field whose Riemann curvature tensor field vanishes in the sense of distributions,i.e.,

where

Then there exists an immersion φ ∈ C2(Ω;En)such that

In addition,an immersion ψ ∈ C2(Ω;En)satisfies

if and only if there exists an isometry r∈Isom(En)such that

Sketch of the ProofThe details of the proof sketched below can be found in[16].Like in the proof of Theorem 2.1,we first show that there exist n vector fields gj∈ C1(Ω;En)that satisfy the Pfa ffsystem

we then prove that there exists a mapping φ ∈ C2(Ω;En)that satisfies the Poincaré system

and finally,we prove that the mapping φ obtained in this fashion is an immersion and satisfies the equations

The difference,however,is that we now have to prove the existence of a solution to the Pfaffsystem(4.1)under the weaker assumption that(gij)∈ C1(Ω;).As a result,the coefficients

are only continuous in Ω,and they satisfy the relations

only in the distributional sense(while in Theorem 2.1 the relations(4.4)were satisfied in the classical sense,i.e.,pointwise).The rest of the proof,which consists in showing that the Poincaré system(4.2)has a solution φ ∈ C2(Ω;En)and that this solution satisfies the relations(4.3),uses the same arguments as those used in the proof of Theorem 2.1(save for the regularity assumptions),so we do not discuss this issue here.

We only briefly sketch the proof of the existence of a solution to the Pfa ffsystem(4.1)under the assumptions that(gij)∈ C1(Ω;)and that the functions Γij?∈ C0(Ω)satisfy(4.4)in the distributional sense.

First,one shows that,given any cube ω ? Ω whose edges are parallel to the coordinate axes of Rnand any n vectors vi∈ En,there exist n vector fields∈ C1(Ω;En)that satisfy the Pfa ffsystem

where xω=()denotes the center of the cube ω.

The vector fields gjare defined at each x∈ω by integrating the above system along a broken line joining xωto x,the edges of which are parallel to the coordinate axes of Rn.More specifically,let ε denote the half-length of the edge of the cube ω,let ω1:=(? ε,+ ε),letetc.,so that ω = ωn.Then,using relations(4.4),one shows that there exist n vector fieldssuch that

then that there exist n vector fieldssuch that

and finally,after n steps,that there exist n vector fieldssuch that

The vector fieldsthen satisfy the Pfa ffsystem(4.5).

Second,one shows that the vector fields viused in the previous step to define the vectorfieldscan be chosen in such a way that

then that

The last implication is established by showing that the Pfa ffsystem

has at most one solution(hij) ∈ C1(ω;Sn)and that both matrix fieldsand(gij)∈ C1(ω;Sn)satisfy the above Pfa ffsystem.

Third,one shows that,if ω ? Ω andare two cubes such thatifandare vector fields that satisfy respectively the Pfa ffsystems

and if there exists a pointsuch that

then

To this end,it suffices to observe that ω∩is connected,and that,for each point x∈ω∩?ω and for each mapping γ=(γi)∈C1([0,1];ω ∩)such that γ(0)=x0and γ(1)=x,the vectorfields

satisfy the same Cauchy problem.

Finally,a solution to the Pfa ffsystem(4.1)is constructed by “glueing together”the solutions of the Pfa ffsystems(4.5).This is possible thanks to the simple-connectedness of Ω and to the uniqueness result proved in the the third step above.

The proof that the immersion φ ∈ C2(Ω;En)obtained in this fashion is unique up to isometries in Enis analogous to the part(iv)in the proof of Theorem 2.1(it suffices to replace the spaces Ck(Ω;En)by the spaces Ck?1(Ω;En)at each one of their occurrences).This concludes the proof of Theorem 4.1.

The next theorem is a natural complement to Theorem 4.1,which shows that the immersion φis a continuous function of the matrix field(gij)for some ad hoc Fréchet topologies(the definition of a Fréchet topology is recalled in Section 2).

Theorem 4.2Let Ω be a connected open subset of Rn,and let

be immersions that satisfy

where

Then there exist isometries rm∈Isom(En),m≥1,such that

ProofIt suffices to replace in the proof of Theorem 2.2 the spaces Ck(Ω),k=1,2,3,by Ck?1(Ω),at each one of their occurrences.More specifically,one first shows that it suffices to consider the particular case where φ=id,so that the immersions

satisfy

Then(the notations used here are the same as in the proof of Theorem 2.2)

and

Since the vector fieldssatisfy the Pfa ffsystems

the above convergences imply that

Let x0be a point in Ω,let

let rm∈Isom(En)be defined by

and let the mapping ψm∈ C2(Ω;En)be defined by

Then

which in turn implies that

5 Metric Tensor Fields in C1(Ω)

In this section,we show that the theorems established in Section 3 still hold for matrix fields with coefficientsinstead ofAs in Theorem 3.1,the assumption that Ω is a domain can be replaced in theorem below by the weaker assumption that Ω is a connected open subset of Rnthat satisfies the “geodesic property”.

Theorem 5.1Let Ωbe a simply-connected domain in Rn,andbe a matrix field whose Riemann curvature tensor field vanishes in the sense of distributions,i.e.,

where

Then there exists an immersionsuch that

In addition,an immersionsatisfies

if and only if there exists an isometry r∈Isom(En)such that

ProofTheorem 5.1 is deduced from Theorem 4.1 in the same way that Theorem 3.1 was deduced from Theorem 2.1;in other words,Theorem 5.1 is proved by showing that any immersion φ ∈ C2(Ω;En)that satisfies

(the existence of such an immersion is guaranteed by Theorem 4.1),as well as all of its partial derivatives up to order two,possess continuous extensions toTo see that this is indeed the case,it suffices to replace in the proof of Theorem 3.1 the spaces Ck(Ω)by the spaces Ck?1(Ω)at each one of their occurrences.

The next theorem establishes a nonlinear Korn inequality in C2(),similar to the nonlinear Korn inequality inestablished in Theorem 3.2.This inequality implies in particular(see Theorem 5.3)that,up to an isometry of En,an immersiondepends continuously on its metric tensor fieldAs in Theorem 3.2,the assumption that Ω is a domain can be replaced in Theorem 5.2 by the weaker assumption that Ω is a connected open subset of Rnthat satisfies the “geodesic property”.

Theorem 5.2Let Ωbe a domain in Rn,and letbe an immersion.Then there exist two constants C=C(φ0)>0 and δ= δ(φ0)>0 such that

for all immersionsandthat satisfy

respectively,where

denote the covariant components of the metric tensor fields induced by the immersions φ0, φ,andrespectively.

ProofIt suffices to replace in the proof of Theorem 3.2 the spacesby the spacesat each one of their occurrences.

An immediate consequence of the nonlinear Korn inequality of Theorem 5.2 is the following convergence result,similar to that of Theorem 3.3.

Theorem 5.3Let Ω be a domain in Rn,and let

be immersions that satisfy

where

Then there exist isometries rm∈Isom(En),m≥1,such that

6 Metric Tensor Fields in W1,p(Ω),p>n

Given an open subset Ω of Rnand 1

where Lp(Ω)and Wm,p(Ω),m=1,2,designate the usual Lebesgue and Sobolev spaces.These spaces are respectively equipped with the norms

It is well known that,if Ω is a domain in Rnand if p>n,then the inclusion

holds and there exists a constant C0(which depends on Ω and p)such that

As is customary,a function f ∈ W1,p(Ω),i.e.,in effect an equivalence class,is identified in this case with its continuous representative.

It is also well known that,again if Ωis a domain in Rnand p>n,the space W1,p(Ω)is a Banach algebra,in the sense that the product of two functions in W1,p(Ω)is still in W1,p(Ω)and there exists a constant C1(which depends on p)such that

(particularly neat proofs of such results for domains with smooth boundaries are found in[3]).

Up to now,the metric fields(gij):Ω→(resp.(gij):→)were assumed to be at least of class C1on Ω (resp.on).We now consider the situation where the metricfields(gij):are in the space W1,p(Ω;Sn)for some p>n and Ω is a domain in Rn.It is remarkable that,under such a weak regularity assumption,an existence and uniqueness theorem analogous to the previous ones still holds as follows.

Theorem 6.1Let Ω be a simply-connected domain in Rn,let p>n,and let(gij)∈be a matrix field whose Riemann curvature tensor field vanishes in the sense of distributions,i.e.,

where

Then there exists an immersion φ ∈ W2,p(Ω;En)such that

In addition,an immersion ψ ∈ W2,p(Ω;En)satisfies

if and only if there exists an isometry r∈Isom(En)such that

Idea of the ProofFirst,notice that the assumption thatat each x∈Ω together with the inclusionimplies thatThis property,combined with the property that W1,p(Ω)is an algebra in turn implies thatHence products likeare well defined in the space(which is contained in the space L1(Ω)since p>n ≥ 2).

The existence proof follows the same pattern as before,i.e.,it consists in first finding vectorfieldsthat satisfy the Pfa ffsystem

then in finding an immersion φ ∈ W2,p(Ω;En)that satisfies the Poincaré system

Otherwise the proof is much more delicate and technical than those of the previous existence theorems.It combines a key existence theorem for Pfa ffsystems“with coefficients in Lp” (see[17,Theorem 6.8]),with an approach due to Mardare(see[18,Theorem 4.1])for solving the above Pfa ffand Poincaré systems,which relies in particular on a careful “glueing”of local solutions.

The next theorem,which is due to[11](see also[12]),establishes a nonlinear Korn inequality in W2,p(Ω),with the same function spaces for the metric tensors and for the immersions as in the above existence theorem.Note,however,that the existence of the immersions is assumed here.

Theorem 6.2Let Ωbe a domain in Rn,and let p>n.Given any ε >0,define the set

where gij:= ?iφ ·?jφ.Then there exists a constant Cε>0 such that

where

Idea of the ProofPick any point x0∈Ω.Then one shows that,given anythe matrixis orthogonal and the vector fields

satisfy the Pfa ffsystems

where

The rest of the proof relies on a series of careful estimates,which eventually allow to use a key comparison theorem between the solutions of Pfa ffsystems “with coefficients in Lp” (see[17,Theorem 4.1]).

Note that the assumption that Ω is a domain can be replaced in Theorem 6.2 by the weaker assumption that Ωis a bounded and connected open subset of Rnthat satisfies the “uniform interior cone property”(see[2]).

As in[11](see also[12]),one can then establish the following local Lipschitz-continuity result as a corollary to the nonlinear Korn inequality of Theorem 6.2.

Theorem 6.3Let Ωbe a domain in Rn,let p>n,and let φ0∈ W2,p(Ω;En)be an immersion.Then there exist constants δ= δ(φ0)>0 and C=C(φ0)>0 such that

for all φ ∈W2,p(Ω;En)and allthat satisfy

where

7 Metric Tensor Fields in (Ω),p>n

The existence and uniqueness theorems and the nonlinear Korn inequalities of Sections 3 and 5 have been extended in Section 6 from continuously-differentiable immersions in the closure of a domain to immersions in Sobolev spaces.

In this section,we show that the existence and uniqueness theorems and the continuity theorems of Sections 2 and 4 can be likewise extended from continuously-differentiable immersions in an open set to immersions that belong locally to Sobolev spaces.Note that,by contrast with Section 6,no regularity assumptions on the boundary of Ω are needed in this section.

To begin with,we show that the existence and uniqueness theorems established in Sections 2 and 6 still hold for matrix fields with coefficientsinstead of gij∈ Ck(Ω),k=1,2.Recall that a function f:Ω → E belongs to the spaceif,for each point x0in Ω,there exists an open ball B ? Ω containing x0such that f|B∈W1,p(B).

Theorem 7.1Let Ω be a simply-connected open set in Rn,let p>n,and let(gij)∈be a matrix field whose Riemann curvature tensor field vanishes in the sense of distributions,i.e.,

where

Then there exists an immersion φ ∈(Ω;En)such that

In addition,an immersion ψ ∈(Ω;En)satisfies

if and only if there exists an isometry r∈Isom(En)such that

ProofPick any point x0in Ω.Given any point x ∈ Ω,let γ ∈ C1([0,1];Ω)be any mapping that satisfies γ(0)=x0and γ(1)=x(such a mapping exists since Ωis connected).Let ε>0 denote the half-distance from the compact set γ([0,1])to the boundary of Ω,let N be the smallest integer thatand,for each m=0,1,···,N,let tm:=and let Bmdenote the open ball in Rncentered at xm:= γ(tm)with radius ε.Then x=xNand

Since Bmis a simply-connected domain in Rn,since(gij|Bm)∈W1,p(Bm;),and since the relations

imply in particular that the Riemann curvature tensor field of the matrix field(gij|Bm)vanishes in the sense of distributions on Bm,Theorem 6.1 implies that there exists an immersion φm∈W2,p(Bm;En)such that

Let the immersions

be defined for m=0 by

where

and,for m=1,2,···,N by

where

The above definition of the immersions ψm∈ W2,p(Bm;En)implies that

and that

Then the simple connectedness of Ω and the uniqueness part of Theorem 6.1 imply that the point ψN(x)∈Enonly depends on the matrix field(gij)and on x0(in particular,it is independent of the path γjoining x0to x).It follows that the mapping φ :Ω → En,where

is unambiguously defined,that

and that

The proof of the uniqueness part of the theorem is similar to the proof of the uniqueness part of Theorem 2.1,where it suffices to replace the spaces Ckbyat each one of their occurrences.

We now show that the convergence theorems established in Sections 2 and 4 still hold for immersions inp>n,instead of immersions in Ck(Ω),k=2,3.

Recall that a sequenceof functionsk=1,2,p>n,converges towith respect to the Fréchet topology ofif and only if

for each open ball B in Rnsuch that B?Ω.If this is the case,we write

Such notions can then be clearly extended to the spacesand

Theorem 7.2Let Ω be a connected open subset of Rn,let p>n,and let

be immersions that satisfy

where

Then there exist isometries rm∈Isom(En),m≥1,such that

ProofPick any point x0in Ω.For each m ≥ 1,let

let rm∈Isom(En)be defined by

and let

Note that the immersions ψmsatisfy in particular the relations

Let B be any open ball in Rnsuch that? Ω.Since the open set Ω is connected,there exists a domain ω in Rnsuch that x0∈ ω,B ? ω,and? Ω.

On the one hand,the assumptions of the theorem imply thatW2,p(ω;En),m ≥ 1,and

On the other hand,Theorem 6.3 implies that there exist constants δ= δ(φ|ω)>0 and C=C(φ|ω)>0 such that

for allthat satisfy

The two properties above imply that there exist isometries∈Isom(En)such that the immersionssatisfy

Note that the immersionssatisfy in particular the relations

The immersions ψmanddefined above are related by

whereSinceare isometries of En,there exist∈ Enand∈Onsuch that

Noting that the matrix?φ(x0)is invertible and using relations(7.1)and(7.3),we easily infer that

That

then follows by using these convergences and the convergence(7.2)in the right-hand side of the following inequality:

8 Metric Tensor Fields in Lq(Ω),1≤ q< ∞

In the previous sections,we considered immersions φ :Ω → Endefined over a connected open subset Ω of Rnand whose gradient field ?φ is at least continuous in Ω.Hence det?φ is either>0,or<0,in Ω.

In this section,we consider immersions φ :Ω → Enwhose gradient field?φ is only in Lp(Ω;Mn),which only means that det?φ ?=0 almost everywhere in Ω,so that det?φ may change sign in Ω.This is why we will assume that all the immersions φ :Ω → Enconsidered in this section preserve the orientation,in the sense that they satisfy det?φ >0 almost everywhere in Ω.This assumption naturally leads to using “proper”isometries of Enin this section(instead of isometries of Enas in the previous sections),according to the following definition.

A proper isometry of Enis an isometry of Enthat preserves the orientation of En.In other words,a proper isometry of Enis an element of the set

The next theorem establishes a nonlinear Korn inequality in W1,p(Ω)similar to those of Theorems 3.2,5.2,or 6.2,but again with respect to Lebesgue and Sobolev norms like in the nonlinear Korn inequality in W2,p(Ω)of Theorem 6.2.

Theorem 8.1Let Ωbe a domain in Rn,and let φ ∈ C1(;En)be an immersion that satisfies det?φ >0 in

Then,given any 1≤q<∞and any 10 such that,for all mappings∈W1,2q(Ω;En)that satisfy det?>0 almost everywhere in Ω,

where

denote the covariant components of the metric tensor fields induced by the immersions φ and,respectively.

Sketch of the ProofThe details of the proof below can be found in[9](see also[10]).

(i)The Poincaré-Wirtinger inequality implies that there exists a constant D such that,for all vector fields f ∈ W1,p(Ω;En),

Therefore,for each R∈and each

which next implies that

Thus it suffices to prove that there exists a constant C1=C1(p,q,φ)such that

(ii)The polar factorization theorem for invertible matrices implies that the matrix fieldscan be written as

respectively,where P(x)∈for all x∈and Q∈for almost all x∈Ω(the assumptions that det?φ>0 inand det?>0 a.e.in Ω are used here).

Since the Frobenius norm|·|is invariant under rotations,the above relations imply that,for almost all x∈Ω,

Besides,one can prove that,for each 1≤q<∞and each 1

The last two inequalities combined with step(i)show that the announced nonlinear Korn inequality will follow if one can find a constant C3=C3(p,φ)such that

(iii)The above inequality was established,first for φ=id and p=2 by Friesecke,James and M¨uller[14]under the name of“geometric rigidity lemma”,then generalized to any 1

The nonlinear Korn inequalities of Theorems 3.2,5.2,and 6.2 are established by using that the vector fields gi:= ?iφ associated with a sufficiently smooth immersion φ :Ω → Ensatisfy the Pfa ffsystem

whose coefficients are defined by

where

Note that,by contrast,the nonlinear Korn inequality of Theorem 8.1 cannot be established in the same way,since the immersions appearing there do not possess enough regularity to ensure that the above Pfa ffsystem makes sense(the above definition of the coefficientsdoes not make sense for immersions φthat are only in the space W1,p(Ω;En)).

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