Nonlinear Korn Inequalities on a Hyp ersurface∗

1 Introduction

The notation and terminology used in this paper is described in the next section.

In a recent article[6],the authors and P.G.Ciarlet established(in particular)the following nonlinear Korn inequalities on a surface in R3:Let 12 be a domain,and letbe an immersion such that the vector fieldis also of class C1 overThen,for eachε>0,there exists a constant Cεsuch that

and

for all immersionssuch thatand

whereandrespectively,denote the covariant components of the first fundamental form,the covariant components of the second fundamental form,and the principal radii of curvature,of the surfaceat the pointThey also showed that,if in addition

on some relatively open subsetof the boundary ofω,then

for some constant Cε,γ0(depending on ε and γ0 in particular).

Since is an immersion of class C1,there existsε= ε(θ)>0 such that the restriction,still denotedΘfor convenience,of the mappingΘto the closure of set

The objective of this paper is to generalize the above inequalities to hyper surfaces(submanifolds of co-dimension 1 in R n+1,n>2)and to weaken the assumptions on the immersion,in particular by eliminating the restrictions in terms of the parameterεon its principal radii of curvature and first fundamental form.This will be done at the expense of replacing in the right-hand side of the above inequalities the matrix fieldby the matrix field and the matrix field.

More specifically,we establish the following nonlinear Korn inequalities on a hypersurface in R n+1(cf.Theorems 3.1–3.2 and 4.1–4.2):Let 1n be a domain,and let θ ∈ C1(ω;R n+1)be an immersion whose unit normal vector fieldis of class C1 over ω.Then there exists a constant C such that

and

for all immersionssuch that

Furthermore,if in addition

on some relatively open subsetof the boundary ofω,then

for some(other)constant Cγ0(depending in particular on γ0).

Finally,we also show that,ifθ and γ0 are such that θ(γ0)is not contained in any affine subspace of dimension 6(n−1)of R n+1,then the assumption above can be dropped and the last inequality still holds,possibly with a different constant Cγ0.

It is worth noticing that some of the results established in this paper,like Lemma 3.3,Theorem 3.2,Lemma 4.1,and Lemma 4.3,generalize to immersionsprevious results due to Ciarlet and Mardare[8],like Lemma 3,Theorem 2,and Lemma 4 in ibid.,about immersionsTo see this,it suffices to particularize the immersions considered in this paper to immersions of the formand to notice that in this case

so that

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Some of the results of this paper were announced in[13].

2 Preliminaries

In this article,all vector spaces are over R.Scalars and scalar functions are denoted by normal letters,while vectors,matrices,vector fields and matrix fields are denoted by boldface letters.

For each positive integers k and l,the notationsandrespectively,designate the space of real matrices with l rows and k columns,the space of all real square matrices of order l,the space of all antisymmetric matrices of order l,the space of all symmetric matrices of order l,the set of all positive-definite symmetric matrices of order l,and the set of all real proper orthogonal matrices of order l.

The identity matrix in M l is denoted I.

The Euclidean norm of a vector v=(v i)∈R l and the Frobenius norm of a matrix F=(F ij)∈ M l×k are denoted by

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Note that the above norms are invariant under rotations,in the sense that

The notation Isom+(R l)designates the set of all proper isometries of R l,i.e.,

A domain U in R n is a bounded,connected,open subset of R n with a Lipschitz-continuous boundary,the set U being locally on the same side of its boundary(see,e.g.,[2]or[14]).

Let U be an open subset of R k and let 1 6 p<∞.Given a smooth enough vector field v=(v i):U → R l,we let denote the gradient matrix of the vector v at each point x=(x j)∈U,i.e.,

where i denotes the row index.

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The usual Lebesgue and Sobolev spaces are respectively denoted by L p(U)and W 1,p(U).

The space of vector fields with components v i∈L p(U)is denoted by L p(U;R l)and the corresponding norm is defined by

The space of vector fields with components v i∈W 1,p(U)is denoted by W 1,p(U;R l)and the corresponding norm is defined by

The space of matrix fields with components F ij∈ L p(U)is denoted by L p(U;M l×k)and the corresponding norm is defined by

The notationdesignates the space of all vector fields v∈C1(U;R l)that,together with their gradientspossess continuous extensions to the closure of U.If U is a domain,one can show,by using Whitney’s extension theorem(cf.[15]),that any vector fieldpossesses an extension to the space C1(R n;R n)(cf.[7]).

In all that follows,n designates an integer>2,Latin indices and exponents range in the set{1,2,···,n+1}save when they are used for indexing sequences,Greek indices and exponents range in the set{1,2,···,n},and the summation convention for repeated indices or exponents is used in conjunction with these rules.

Given an open subsetω of R n,we letwhere(yα)denotes a generic point in ω.A mapping is an immersion if the vectors∂αθ(y)are linearly independent at each point y∈ ω.A mapping ,is an immersion if the vectors∂αθ(y)are linearly independent at almost all point y∈ω.

Given an open subsetΩ of R n+1,we let,where(x i)denotes a generic point inΩ.A mapping is an immersion if the vectorsare linearly independent at each point

Thenotation[v 1v 2 ···v n]designatesthematrix whose i-th column vector is v i,i=1,2,···,n.Given a matrix A,its component at i-th row and α-th column is denotedLikewise,given a vector v,its i-th component is denoted(v)i.

Given any n vectors,the exterior product

For notational brevity,we shall drop the explicit dependence on the exponent p in the various constants found in the nonlinear Korn inequalities appearing below,as well as in their proofs.

where V is a(n+1)×(n+1)square matrix whose last n column vectorsare v 1,···,v n(in this order)and Cof V designates the cofactor matrix of V(Cof V:=(det V)V−T if the matrix V is invertible).

We conclude this section by enunciating two lemmas about the geometry of hypersurfaces in R n+1,n≥2,which are straightforward generalizations of similar lemmas given in[5]in the particular case n=2 and p=2.These lemmas show that some classical definitions and properties pertaining to hypersurfaces in R n+1 still hold under less stringent regularity assumptions than the usual ones(these definitions and properties are traditionally given and established under the assumptions that the immersions denoted θ in Lemmas 2.1–2.2 below belong to the space

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Note that the functionsappearing in these lemmas,respectively,denote the covariant and contravariant components of the first fundamental form and the covariant and mixed components of the second fundamental form.For such classical notions of the differential geometry of hyper surfaces,see[1,3–4,11–12].

The notations,and(g ij)respectively designate matrices in M n and M n+1 with components,and g ij,the index or exponent denoted here α,or i,designating the row index.

Lemma 2.1 Letω be a domain in R n and letbe an immersion such thatwhere

Then the functions

whereTo further estimate the integrand appearing in the right-hand side,we note that,given any vectors uα ∈ R n+1 and vα ∈ R n+1,α =1,···,n,and any x3∈ R,we have

Lemma 2.2 Let 1n and let there be given an immersion θ ∈ W 1,p(ω;R n+1)such that a n+1∈ W 1,p(ω;R n+1),where

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Then the functions

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belong to the spaceBesides,

Furthermore,define the mapping by

Then Θ ∈ W 1,p(ω×]− δ,δ[;R n+1)for any δ>0.

3 Nonlinear Korn Inequalities on a Hyp ersurface Without Boundary Conditions

In this section,we establish two nonlinear Korn inequalities on a hypersurface“without boundary conditions” (cf.Theorems 3.1 and 3.2);by “without boundary conditions”,we mean that the values of the two immersions and on the boundary ofω are arbitrary.

In order to find a lower bound of the left-hand side of the above inequality in terms of L p(ω)-norms ofand∇θ,we proceed as in the proof of Theorem 4.2 in Ciarlet,Malin,and Mardare[6];we deduce in this fashion that there exists a constant c2(θ)such that

is the vector whose components are defined by

Lemma 3.1 Let 1n,and letbe an immersion.Then there exists a constantsuch that,for all

We now generalize the above geometric rigidity lemma to hypersurfaces in R n+1,instead of open subsetsΩ of R n.The definition of the vector field a n+1(θ)is given in Lemmas 2.1–2.2.

Lemma 3.2 Letω be a domain in R n,letbe an immersion such thatand let 12(θ)such that

for all immersionssuch that

Proof Given a mappingθthat satisfies all the assumptions of Lemma 3.2,define the mappingΘ∈C1(ω×R;R n+1)by

Likewise,given any mappingthat satisfies the assumptions of Lemma 3.2,define the mapping

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isan immersion of class C1.Sincetherestriction,still denotedfor convenience,of themappingto the setΩbelongsto the space W 1,p(Ω;R n+1)by Lemma 2.2,all theassumptions of Lemma 3.1 are satisfied.Hence there exists a constant c1(θ)such that

for any mappingsatisfying the assumptions of Lemma 3.2.

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In what follows,our purposeisto find estimates of both the left-hand side and the right-hand side of the above inequality in terms of∇θand.

The point of departure of the proofs of these nonlinear Korn inequalities on a hypersurface is the following generalization,established in[8,Lemma 2],of a geometric rigidity lemma,due to Friesecke,James and Müller[10]for p=2,and later extended to p>1 by Conti[9](in[9–10],the mapping Θ was the identity mapping).

where ε= ε(θ)is the constant defined above.

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The next step is to find an upper bound of the L p(Ω)-norm of the

Hence there exists a constant c4(θ)such that

To this end,we first deduce that

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To estimate the integrand appearing in the right-hand side,we note that,given any vectors v i∈ R n+1,i=1,···,n+1,we have

by Jensen’s inequality applied to the convex function,whereand

by applying recursively the inequality for all a≥0 and t≥0,whereCombining these inequalities with the previous one,we next deduce that

wherebelong to the spaceBesides,

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(again by Jensen’s inequality).Using this inequality to estimate the right-hand side of the previous one,we finally deduce that

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Consequently,the announced inequality follows with

We are now in a position to establish our first nonlinear Korn inequality on a hypersurface without boundary conditions.

Theorem 3.1 Letω be a domain in R n,letbe an immersion such thatand let 13(θ)such that

for all immersionsthat satisfy

Proof The proof consists in finding an upper bound of the right-hand side of the inequality of Lemma 3.2 featuring the fundamental forms ofθandinstead ofandSo,webegin by expressingeach termin terms of

Letθandsatisfy the assumptions of the theorem.Then

which in turn imply that

The matrix polar decomposition theorem shows that

where

Hence

Next,combining the Weingarten equations

with the relations

(deduced from the above expressions of andimplies that

Then we infer from the above relations that

Consequently,we get

Then the announced inequality is obtained withby using the aboveine quality in the right-hand side of the inequality of Lemma 3.2.

In the remaining of this section,we establish our second nonlinear Korn inequality on a hypersurface without boundary conditions.It shows that the infimum in the left-hand side of the nonlinear Korn inequality of Theorem 3.1 can be dropped if a weaker norm ofand(e a n+1−a n+1)is added to its right-hand side.

The key for proving such a nonlinear Korn inequality is the following lemma.

Lemma 3.3 Letω be a domain in R n,let 1and let U⊂M n+1 be any non-empty set.Then there exists a constant C4(θ)such that,for all

Proof Assume on the contrary that,for each k ∈ N∗,there exists a vector fieldsuch that

Let

Then,for each k>1,

Since the space W 1,p(ω;R n+1)(1p(ω;R n+1)is compact,it follows that there exists a subsequence(l)of(k)such that,as

Then the above inequality implies that u=0 inωand that there exists a sequence(R l)⊂U such that,as l→∞,

The above relation combined with the weak convergence inimplies that

Since the sequence(F l)belongs to a subspace of L p(ω;M(n+1)×n)of finite dimension,namely

it follows that

Hence

Since in addition init follows that

This contradicts that.Therefore,the announced inequality holds.

We arenow in a position to establish our second nonlinear Korn inequality on a hypersurface without boundary conditions.

Theorem 3.2 Letω be a domain in R n,letbe an immersion such thatand let 1such that

for all immersionsthat satisfy

Proof The proof isan immediateconse quence of Theorem 3.1 combined with the inequality of Lemma 3.3 with(applied twice,to estimate bothand

4 Nonlinear Korn Inequality on a Hypersurface with Boundary Conditions

In this section,we establish a nonlinear Korn inequality on a hypersurface for mappings subjected to specific boundary conditions(cf.Theorems 4.1–4.2 below).We consider two types of boundary conditions,one corresponding to the case where the hypersurface is kept fixed on a portion of its boundary,and one corresponding to the case where both the hypersurface and its positively-oriented unit normal vector field are kept fixed on a portion of the boundary of the hypersurface.

We begin by showing that it is possible to drop the termin the right-hand side of the inequality of Lemma 3.3 by imposing a boundary condition on a portion γ0 of∂ω.Note however the stronger assumptions of Lemma 4.1 compared with those of Lemma 3.3:The extra regularity forθand the particular choice

Lemma 4.1 Letω be a domain in R n,let 1be an immersion,and letγ0 be a relatively open subset of∂ω such thatθ(γ0)is not contained in any affine subspace of R n+1 of dimension 6(n − 1).Then there exists a constant C6(γ0,θ)such that

for allthat satisfies

Proof Assume on the contrary that for all k∈N∗,there existssatisfying on γ0 such that

It follows that and there existssuch that

Let

Then,for all k∈N∗,

Consequently,there existand a subsequence(l)of(k)such that,as l→∞,

As in the proof of Lemma 3.3,the last two convergences together imply that

Since then inwe deduce that

We now provethat u=0,which will end the proof since it contradictsthat

Letη∈[0,∞]and letbe such that,for some subsequence(m)of(l),

We distinguish three cases:η=∞,0<η<∞andη=0.

First,ifη= ∞,then inHence in L p(ω;M(n+1)×n).But u=0 on γ0 since u l=0 on γ0 and in W 1,p(ω;R n+1).Therefore u=0.

Second,if 0< η < ∞,then inHence there exists a∈R n+1 such thatSinceonγ0 for all k∈N∗,and since inwe have that u=0 on γ0.Therefore.Consequently,

This means that the isometry r: for allsatisfies r(x)=x for all x ∈ θ(γ0).Since the set of all fixed points of an isometry of R n+1 is either R n+1(if the isometry is the identity mapping)or an affine subspace of R n+1 of dimension 6(n−1)(otherwise),the assumption on θ(γ0)of Lemma 4.1 implies that r(x)=x for all x ∈ R n+1.Therefore,ηa=0 and R=I.Sinceη>0,this next implies that u=0.

Finally,assume thatη=0.Then the convergence

in L p(ω;M(n+1)×n)implies that each component of the matrixvanishes almost everywhere in ω.Hence the matrixcoincides with the zero matrix M(n+1)×n for almost all y∈ω.This means that

for some negligible subset N ofω.Since the set S contains(at least)n linearly independent vectors(recall thatθis an immersion by assumption),one has R=I.

Furthermore,there exists a subsequence ofstill indexed by m for simplicity,that convergesalmost every where inω.Sinceθisan immersion,the reexist n linearly independent vectors v 1,v 2,···,v n(for example,∂1θ(y),···,∂nθ(y)at some point y ∈ ω)such that,for allα ∈ {1,···,n},as

It follows that,for each v ∈ E:=span{v 1,···,v n},

Hence,given any orthonormal basis{e1,···,e n}in E,for allα ∈ {1,2,···,n},

In order to prove that the sequenceconverges in M n+1,it suffices to prove that the sequencewhereis the unique unit vector such that{e1,···,e n+1}is a positive basis in R n+1,converges in R n+1.Let

Then,for allα ∈ {1,2,···,n},there exists fα ∈ R n+1 such that

Sinceone hasConsequently,

Therefore,since aswe have

The above convergences show that there exists a matrix A∈M n+1 such that,on the one hand,

On the other hand,the relation implies that,for all m,

Letting m→∞in the above equality(recall that as m→∞)we getwhich means that A is antisymmetric.

To summarize,we proved that,if asthen,even if it means extracting a subsequence,

Hencealmost everywhere inω,which in turn implies that there exists a∈R n+1 such that

Besides,the trace of u on γ0 vanishes(since inas and u l:=vanishes onγ0).Therefore the infinitesimal isometryµ: for all x∈ R n+1 vanishes on the setθ(γ0).In other words,

Then the assumption on θ(γ0)of Lemma 4.1 implies that u=0,since the set Kerµ is either R n+1(if a=0 and A=0)or an affine subspace of R n+1 of dimension 6(n−1)(otherwise).To prove the last assertion,one notices on the one hand that either Keror Kerµ=x0+Ker A,where x0 isa particular solution of A x=a;but on theother hand Ker A⊕Im A=R n+1,so dim(Ker A)=(n+1)−Im A,and dim(Im A)is either 0(if A=0)or dim(Im A)>2 if

In order to prove our first nonlinear Korn inequality of this section(cf.Theorem 4.1 below),where and a n+1 do not necessarily coincide onγ0,we need to supplement the estimate ofprovided by Lemma 4.1 by an estimate ofThis is the object of the following lemma.

Lemma 4.2 Letω,p,θ and γ0 satisfy the assumptions of Lemma 4.1.Assume in addition thatbelongs toThen there exists a constantsuch that

for all immersionsthat satisfyandonγ0.

Proof Assumethat thisassertion isfalse.Then for all k∈N∗,thereexistandsuch thatonγ0 and

Let

Then for all k∈N∗and,by using the same arguments as in the proof of Lemma 4.1,there exists a subsequence(l)of(k)such that,for someand

On the other hand,Lemma 4.1 shows that,for some constantwhich in turn implies that

Hence as l→ ∞,so thatTherefore,as l→ ∞,

Sinceθis an immersion,the above relation implies(as in the proof of Lemma 4.1)that,as l→∞,

Therefore,Hence

a contradiction.

We are now in a position to establish our first nonlinear Korn inequality on a hypersurface with boundary conditions:It is similar to the one established in Theorem 3.2,but without the termsandin its right-hand side.

Theorem 4.1 Letω be a domain in R n,let 1be an immersion such that,and letγ0 be a relatively open subset of∂ω such thatθ(γ0)is not contained in any affine subspace of R n+1 of dimension 6(n − 1).Then there exists a constant C8(γ0,θ)such that

for all immersionsthat satisfyandonγ0.

Proof This inequality is an immediate consequence of Theorem 3.1 and Lemma 4.2.

In the remaining of this section,we show that the Korn inequality on a hypersurface of Theorem 4.1 holds different set of assumptions.Specifically,it shows that the assumption onθ(γ0)can be dropped provided the mappingssatisfy the additional boundary condition

To do so,we first need to prove the next lemma,which removesthe restriction onγ0 imposed in Lemma 4.2 at the expense of adding boundary conditions for the normal vector fields a n+1 and to the hypersurfacesθ(ω)and

Lemma 4.3 Letω be a domain in R n,let 10 be any non-empty relatively open subset of∂ω and letbe an immersion that satisfies a n+1=a n+1(θ)∈Then there exists a constant C9(γ0,θ)such that

for all immersionsthat satisfyon γ0,and

Proof Assume that such a constant does not exist.Then for all k ∈ N∗,there existssatisfyingonγ0 andonγ0,such that

Let

The previous inequality shows that,for each k∈ N∗,µk>0 and there existssuch that

Let

Clearly,

Since the spaceis reflexive(recall that 1is compact,the above inequality implies that there existand a subsequence(l)of(k)such that,as l→∞,

Since in addition,as k→∞,

the sequencesandconverge strongly in L p(ω;M(n+1)×n)(cf.the proof of Lemma 3.3).Hence

In what follows,we will show that the last two convergences imply that

which will yield a contradiction with

To this end,we distinguish three cases:µ=∞,0<µ<∞,µ=0.

First,assume thatµ=∞.Then in M n+1,which combined with the last two convergences above implies that

Since in addition on also vanishes a.e.in ω.

Second,assume that 0<µ<∞.Then the convergences

imply that

Hence there exists a∈R n+1 such that

Since onγ0 and onγ0,the above relations imply that there exists y∗ ∈ γ0 such that

where τα(y∗),α =1,2,···,n − 1,denote a(n − 1)-tuple of linearly independent vectors in R n+1 that are orthogonal to a n+1(y∗).

More specifically,sinceω is a domain in R n,∂ω is locally the graph of a Lipschitz function.In particular,there exist 1 6 j 6 n and a Lipschitz functionψj:U→R,where U is an open set of R n−1,and an open ball V in R n,such that

It is well-known thatψj is differentiable at almost all points y′∈ U.Define the mapping ψ:U→R n by letting

Letdenote any point whereψj is differentiable and letThen the(n−1)vectorsare well defined,and they are linearly independent.

Since

we have in particular that

where

Note that the vectorsτ1(y∗),···,τn−1(y∗)are linearly independent since θ is an immersion at y∗,so the vectors ∂1θ(y∗),···,∂nθ(y∗)are linearly independent.Indeed,since

the rank of the matrix in the left-hand side is(n−1).

Note also that the vectors τα(y∗)are orthogonal to a n+1(y∗)since

and ∂βθ(y∗)·a n+1(y∗)=0 for allβ ∈ {1,···,n}by the definition of a n+1(y∗).

So we just proved that

Since dim E=n and,it follows that R=I,because,if{e1,···,e n+1}is an orthogonal basis in R n+1 such that E=span{e1,···,e n}and det[e1 ···e n+1]>0,then,for allα =1,···,n,

so that.Therefore,either Re n+1=e n+1,or Re n+1= −e n+1.But,Re n+1= −e n+1 implies that

a contradiction.Consequently,R=I and so

Besides, on γ0,so that a=0.

Third,assume thatµ=0.We then infer from the convergence

which implies in particular that

that the limit satisfies R=I.Indeed,if on the contrarythen

Furthermore,let y∗ ∈ ω and let.Then ε>0(since rankand dim Ker(I−R)6 n−1).Sinceand as l→ ∞,there exists δ>0 and l0∈N∗such that

for all y ∈ B(y∗,δ)and all l>l0.Therefore,for all l>l0,

which implies thata contradiction.

Next,using that the column vector fields ofare linearly independent at each point ofwe deduce(as in the proof of Lemma 3.3)from the convergencethat there exists an antisymmetric matrix A ∈ A n+1 and a subsequence(m)of(l)such that

Hence

which in turn implies that there exists a∈R n+1 such that

Besides,since

we have

Therefore,using that onγ0 and onγ0,the same argument as that used in the case 0< µ< ∞ shows that there exists y∗ ∈ γ0 such that

for some vectorsτα(y∗)∈ R n+1,α =1,2,···,n−1,that are linearly independent and orthogonal to a n+1(y∗).Thus the antisymmetric matrix A ∈ A n+1 satisfies

where E denotes the subspace of dimension n of R n+1 spanned by the vectors τ1(y∗), ···,Hence A=0.To see this,letThenand

Hence,Av=0 for all v∈R n+1,which means that A=0.

Finally,since for almost all y ∈ ω and since on γ0,the vector a vanishes.Therefore,

We are now in a position to establish our second nonlinear Korn inequality on a hypersurface with boundary conditions.

Theorem 4.2 Letω be a domain in R n,let 10 be any non-empty relatively open subset of∂ω,and letbe an immersion such thatThen there exists a constant C10(γ0,θ)such that

for all immersionsthat satisfyon and

Proof It suffices to estimate the left-hand side in Theorem 3.1 by applying Lemma 4.3.

Acknowledgement The authors would like to thank Professor Philippe G.Ciarlet for stimulating discussions that initiated this research and for his hospitality at the City University of Hong Kong.M.Malin is also very grateful to the City University of Hong Kong for its support during the preparation of this manuscript.

References

[1]Abraham,R.,Marsden,J.E.and Ratiu,T.,Manifolds,Tensor Analysis,and Applications,Springer-Verlag,New York,1988.

[2]Adams,R.A.,Sobolev Spaces,Academic Press,New York,1975.

[3]Ciarlet,P.G.,An Introduction to Differential Geometry,with Applications to Elasticity,Springer-Verlag,Heidelberg,2005.

[4]Ciarlet,P.G.,Linear and Nonlinear Functional Analysis with Applications,SIAM,Philadelphia,2013.

[5]Ciarlet,P.G.,Gratie,L.and Mardare,C.,A nonlinear Korn inequality on a surface,J.Math.Pures Appl.,85,2006,2–16.

[6]Ciarlet,P.G.,Malin M.and Mardare,C.,New nonlinear estimates for surfaces in terms of their fundamental forms,C.R.Acad.Sci.Paris,Ser.I,355,2017,226–231.

[7]Ciarlet,P.G.and Mardare,C.,Recovery of a manifold with boundary and its continuity as a function of its metric tensor,J.Math.Pures Appl.,83,2004,811–843.

[8]Ciarlet,P.G.and Mardare,C.,Nonlinear Korn inequalities,J.Math.Pures Appl.,104,2015,1119–1134.

[9]Conti,S.,Low-Energy Deformations of Thin Elastic Plates:Isometric Embeddings and Branching Patterns,Habilitationsschrift,Universität Leipzig,2004.

[10]Friesecke,G.,James,R.D.and Müller,S.,A theorem on geometric rigidity and the derivation of nonlinear plate theory from three dimensional elasticity,Comm.Pure Appl.Math.,55,2002,1461–1506.

[11]Klingenberg,W.,A Course in Differential Geometry,Springer-Verlag,Berlin,1978.

[12]Kühnel,W.,Differential Geometry:Curves-Surfaces-Manifolds,American Mathematical Society,Providence,2002.

[13]Malin,M.and Mardare,C.,Nonlinear estimates for hypersurfaces in terms of their fundamental forms,C.R.Acad.Sci.Paris,Ser.I,355,2017,1196–1200.

[14]Nečas,J.,Les Méthodes Directes en Théorie des Equations Elliptiques,Masson,Paris,1967.

[15]Whitney,H.,Analytic extensions of differentiable functions defined in closed sets,Trans.Amer.Math.Soc.,36,1934,63–89.