Geometric rigidity of conformal matrices

Geometric rigidity of conformal matrices

DANIELFARACO ANDXIAOZHONG

Abstract. We provide a geometric rigidity estimate `a la Friesecke-James-M¨uller for conformal matrices. Namely, we replace SO(n)by an arbitrary compact set of conformal matrices, bounded away from 0 and invariant under SO(n), and rigid motions by M¨obius transformations.

Mathematics Subject Classification (2000):30C65 (primary); 49J45 (secondary).

1.

Introduction

This paper is concerned with the so-called geometric rigidity estimates for confor- mal matrices. Recently, Friesecke, James and M¨uller developed a successful new approach to the classical problem of dimension-reduction in nonlinear elasticity [7, 8]. A fundamental ingredient was the following rigidity estimate for the group SO(n)= {A∈ M^n×n: A^tA= I,detA=1}of special orthogonal matrices in

R

^n. Theorem 1.1. Let⊂

R

ⁿbe a bounded Lipschitz domain and n≥2. There exists a constant C₁ = C₁()with the property that for eachv ∈ W^1,2(,

R

ⁿ), there exists R ∈SO(n)such that

Dv−RL²()≤C₁dist_SO(n)(Dv)L²(). (1.1) Theorem 1.1 has been used in a number of related problems concerning dimension- reduction, e.g. [3, 6, 18] and [17]. In all the applications it is crucial that the dependence between the left- and right-hand side is linear and that vis any gen- eral Sobolev mapping (the classical result of John [15] gives anL²−L^∞estimate valid for locally bi-Lipschitz maps). Theorem 1.1 makes quantitative the following classical result of Reshetnyak [19] for sequences.

Theorem 1.2. Let{vj} ∈W^1,2()be a weakly convergent sequence in W^1,2. Then there exists R∈SO(n)such that

jlim→∞dist_SO(n)(Dvj)_L²₍₎ =0⇒ lim

j→∞Dvj −R_L²₍₎=0. (1.2) Pervenuto alla Redazione il 22 aprile 2005 e in forma definitiva il 14 settembre 2005.

A natural question raised by Theorem 1.1 is to determine when a qualitative rigidity theorem like Theorem 1.2 can in fact be made quantitative in the sense of Theo- rem 1.1. That is, let E ⊂ M^n×n be such that for a weakly convergent sequence {vj} ∈W^1,2()it holds that

jlim→∞dist_E(Dvj)_L²₍₎ =0⇒ lim

j→∞Dvj −Dϕ_L²₍₎=0 (1.3) whereϕ∈W^1,2(,

R

ⁿ)is such thatDϕ∈Ea.e.

Then, we would like to know if there exists a constantC()such that for every v∈W^1,2(,

R

ⁿ)there existsϕwithDϕ∈Eand

Dv−Dϕ_L²₍₎≤C()dist_E(Dv)_L²₍₎. (1.4) In particular, since Reshetnyak proved results related to Theorem 1.2 for subsets of the set of conformal matrices

CO₊(n)= {A∈M^n×n : A=ρR, whereρ∈

R

₊^andR∈SO(n)}, in [8] it was posed the question whether his results can made quantitative in the sense of Theorem 1.1.

In this work we show that ifEis a compact subset of CO₊(n), invariant under SO(n)and with 0 ∈/ E, a quantitative rigidity estimate holds (see Theorem 1.4).

Before stating the result, we need to recall that for E

^CO+(n)the solutions of the differential inclusion

Dϕ∈E, ϕ ∈W^1,2() (1.5)

are described by the Liouville Theorem.

Theorem 1.3 (Liouville Theorem). Let⊂

R

ⁿ,n≥3and letϕ∈W^1,n(,

R

ⁿ) be such that

Dϕ(x)∈CO₊(n)a.e. x ∈. (1.6) Then

ϕ(x)=b+ Ax,orϕ(x)=b+A R x−a

|x−a|² (1.7)

where b∈

R

ⁿ,a∈

R

ⁿ\, A∈CO₊(n)and R =diag(1, . . . ,−1).

The Liouville Theorem has a long history. Liouville established it forC³map- pings in [16], Gehring for homeomorphisms inW^1,n(,

R

ⁿ)in [9] and Reshetnyak removed the injectivity assumption in [19]. In fact,W^1,nis not the borderline case.

Iwaniec [13] proved that there exists a critical threshold p_n < n such that Liou- ville theorem holds for mappings inW^1,p(,

R

ⁿ),p≥ p_n. Moreover, Iwaniec and

Martin [14] proved that ifn =2m andmis an integer then p_n =m and the result is optimal. Fornodd p_nis conjectured to be alsoⁿ₂.

Geometrically, the Liouville theorem relates CO₊(n)with the special M¨obius group

M

n. The M¨obius groupM¨ob(n)is the group generated by reflections on spheres and hyperplanes. Then

M

nconsists of M¨obius transformations preserving orientation (see [1] for an introduction of the geometry of the M¨obius group and its discrete subgroups). It turns out that any M¨obius transform can be represented as the composition of an affine mapping and an inversion with respect to a sphere.

Analytically, this yields formula (1.7).

We are interested in compact subsets of CO₊(n)invariant under SO(n). For technical reasons we also assume that these sets have finitely many connected com- ponents. Let us introduce the notation

E^m,M =(CO₊(n)∩B(0,M))\B(0,m),

mSO(n)= {m R: R∈SO(n)}. (1.8) Then our conditions onEimply that it can be represented by,

E= ∪ⁿ_i=¹₁E^mi,Mi ∪ⁿ_i=²₁m_iSO(n). (1.9) In addition, since the sets E are compact the solutions to (1.5) are in particular M¨obius transforms. We denote them by

M

_n^E(). An interesting new feature re- spect to the other nonlinear sets for which quantitative rigidity estimates are avail- able (e.g. [2]) is that

M

_n^E() contains non affine solutions. We are now in the position to state the rigidity estimate:

Theorem 1.4. Let E ⊂ CO₊(n)be compact, finitely connected, with 0 ∈/ E and such that

SO(n)E= E.

Let

⊂

R

ⁿ,n≥3andbe a bounded domain. Then,

i) There exists a constant C₂ =C₂(E, , )such that for anyv∈W^1,2(,

R

ⁿ) there existsϕ∈

M

n^E(,

R

ⁿ)such that

|Dϕ−Dv|²≤C₂

dist²_E(Dv). (1.10) ii) Letbe Lipschitz. Then, a constant C₃=C₃(E, )such that

|Dϕ−Dv|²≤C₃

dist²_E(Dv) exists if and only if E = ∪ⁿ_i=³₁m_iSO(n).

We do not know if we could put E = CO₊(n)in Theorem 1.4. However if we restrict our attention to mappings of finite distortion with suitable integrability conditions on the distortion function a rigidity estimate holds, as it will be shown elsewhere. The proof relies heavily on Theorem 1.4 and a recent result of Koskela, Hencl, and Zhong [12] about the doubling properties of the Jacobian of this kind of mappings. The casen = 2 is very different. It holds that CO₊(2)is a linear subspace and by Weyl‘s lemmaW^1,1solutions to the differential inclusion

Dϕ∈CO₊(2)

are holomorphic functions. Then the rigidity estimate for E = CO₊(2)holds (for every exponent 1 < p ≤ ∞) by the boundedness of the Ahlfors-Beurling trans- form (for a proof see [5]). However, since holomorphic function are not as rigid as M¨obious transforms, it remains open what is the situation for the corresponding set E^M,min two dimensions.

Our approach to Theorem 1.4 is essentially a combination of ideas of [8] with those developed by Reshetnyak in his study of the stability of Liouville theorem respect to different parameters and classes of functions (See [21], in particular The- orems 3.2 and 5.2). The paper is organized as follows. In Section 2 we state some basic facts about M¨obius mappings and solutions to elliptic equations. Section 3 is devoted to prove that arbitrary M¨obious transforms are approximated in the sense of Theorem 1.4 by mappings in

M

^E_n(). In this section, we provide an example showing that, if

M

^En()contains non affine mappings, a global estimate like in Theorem 1.4 (ii) does not hold even in this simplified setting.

Section 4 constitutes the crux of the paper. We prove Theorem 1.4 fora ball.

The proof is based on the fact that E is related to elliptic equations globally and locally. Globally, there exists a smooth uniformly convex mapping F :

R

ⁿ →

R

such that for everyA∈E

F(A_i)=det(A), (1.11)

where A_i is any row of the matrix A. The existence of such a mappingFenable us to write

v=w+z, where each of the coordinatesz_i satisfies the equation

div(D F(Dz_i))=0, and

|∇w|²≤C

dist²_E(Dv). Hence, it suffices to prove Theorem 1.4 for such a mapping z. Next, the regularity theory of elliptic equations implies that z_i en- joys a priori estimates. In particular, the modulus of continuity of Dzis uniformly bounded. This allows the use of a compactness argument to deduce that it is essen- tially enough to prove Theorem 1.4 for mappingszsuch that

dist(Dz,I)L^∞() <<1 (1.12)

and the modulus of continuity of Dz is uniformly bounded. In this situation we can use the local equation which is given by the tangent plane to CO₊(n). We proceed by adapting the ideas of Reshetnyak [21] to our situation. Beside a Korn type inequality for the tangent plane to CO₊(n), a degree argument involving the exponential map of

M

n is needed to choose the M¨obius mapping closest toz. In this way, one obtains a mapping ϕ ∈

M

n()satisfying (1.10). However, these arguments do not imply thatϕ ∈

M

n^E(). The desired mappingϕis obtained by applying section 3 toϕ. Let us remark that this is the only moment along the whole proof where the SO(n)invariance ofEis used. The last section is devoted to prove Theorem 1.4 for an arbitrary

.

ACKNOWLEDGEMENT. We thank Stefan M¨uller for bringing the problem to our at- tention and for many interesting discussions and suggestions. Part of this research took part when D. F. was visiting Wuhan Institute of Physics and Mathematics, and when X. Z. was visiting the Max-Planck Institute for Mathematics in the Sciences.

We would like thank the hospitality and research environment in both institutions.

This work was partially supported by the EU Research Training NetworksHYper- bolic and Kinetic Equations, contract HPRN-CT-2002-00282 and Phase Transi- tions in crystalline solids, contract FMRX-CT 98-0229.

2.

Notation and preliminaries

We will denote byC₁,C₂, . . . ,C_nconstants which will be used during the whole paper, whereas c₁,c₂, . . . ,c_n will be used for different constants within the same proof.

Concerning sets in

R

ⁿ, we will use B for balls and B(a,r)where we want to specify the center a and the radiusr. For balls centered at the origin we use B_r = B(0,r). Given a ball B(a,r),h B = B(a,hr). For a measurable setL,|L| denotes its Lebesgue measure. Let K ⊂

R

^{n ands} ∈

R

^{. Then K}s = {y ∈

R

^{n :} dist(y,K)≤s}.

Let A=(a_{i j})∈ M^n×n then|A|stands for the operator norm. Given a closed set E ∈ M^n×n dist(A) =inf_B∈E|A−B|. Let us remark the choice of the opera- tor norm in our definition of the distance is motivated to simplify the constants in section 3 but of course any other norm would do.

Let E be the set in Theorem 1.4. Since 0 ∈/ E andE is compact there exists 0<m <M <∞such that

E⊂ E^m,M (2.1)

where E^m,Mwas introduced in (1.9).

We will used the notation E for an auxiliary set different E but satisfying the assumptions of Theorem 1.4. In particular it will satisfy (2.1) for some other numbersmandM.

Given a closed set⊂

R

^n,

M

n()are M¨obius transforms which are finite in ,

M

n(1, 2)stands for M¨obius transform mapping1 onto2. Let us recall the notation

M

n^E()= {ϕ∈W^1,2(): Dϕ(x)∈Ea.e.x ∈}.

We discuss now several basic properties of

M

n()and more precisely of

M

_n^E(B). In particular we show in the first lemma that mappingsϕ∈

M

n^E(B)can be handled in an uniform way. The main reason is that for ϕ ∈

M

^E_n(B) ϕ⁻¹(∞), the center of the sphere associated toϕ, is bounded away fromBindependently ofϕ. For the notation, recall that forϕ∈

M

n,ϕ(B)is a ball.

Lemma 2.1. Letϕ ∈

M

_n^E(B)and let m,M be the constants in (2.1). Then the following three properties hold:

(1) mⁿ≤ ^|ϕ(_|B^B_|^)| ≤Mⁿ.

(2) ϕ∈

M

n(h₀B)where h₀=h₀(^M_m) >1.

(3) Let s <1. Then there exists a number h₁ = h₁(s,E) < 1such thatϕ(s B)⊂ h₁ϕ(B)and sϕ(B)⊂ϕ(h₁B).

Proof. Since forA∈CO₊(n),|A|ⁿ =det(A), it follows that

|ϕ(B)| =

B

J_ϕ=

B

|Dϕ|ⁿ (1) follows from (2.1).

We prove (2) forϕ ∈

M

^E_n(B)not affine. It follows from (1.7) that for such ϕ there exists r ∈

R

^{and a} ∈

R

ⁿ. such that |Dϕ(x)| = r²|x − a|⁻². Thus, max_B|Dϕ| = r²dist_B(a)⁻²and min_B|Dϕ| = r²(dist_B(a)+diam(B))⁻². Since ϕ∈

M

n^E(B)we have that

r²dist_B(a)⁻²

r²(dist_B(a)+diam(B))⁻² ≤ M m, i.e.

dist_B(a)≥ diam(B) M

m −1 . Henceh₀=1+2 ¹

M m−1.

For (3), we firstly observe there is no loss of generality in assuming thatB = B₁andϕ(B₁)= B₁. The general case follows by considering similaritiesT_B,T_ϕ(B) such thatT_B(B₁)=BandT_ϕ(B)(B₁)=ϕ(B). Then the mappingϕ˜ =T_ϕ(⁻¹_B)◦ϕ◦T_B

satisfies thatϕ(˜ B₁)= B₁andm²≤ |Dϕ| ≤˜ M². It is easy to check that if the thesis holds forϕ˜it also holds forϕ.

Therefore there is no loss of generality assuming ϕ ∈

M

^E_n(B₁,B₁). Since

1

M ≤ |Dϕ⁻¹| ≤ _m¹, (2) implies thatϕ⁻¹ ∈

M

n(h₀B₁)for the sameh₀. Thus, ϕ(∞)=b, with|b| ≥h₀.

On the other hand,ϕ(B₁)= B₁implies thatϕfixesSⁿ⁻¹. Thus, it conjugates with the mapping R = _|x^x_|2 (see [1, Theorem 3.2.4]), i.e. ϕ(x) = R◦ϕ◦ R(x). Putting x = 0 yields ϕ(0) = R ◦ϕ◦ R(0) = R(b) = _|b^b_|2. Recall now that

M

n(B₁,B₁)is the group of isometries of(B₁,d), whereddenotes the hyperbolic metric of B₁(see [1, Chapter 3]). Thus, by triangle inequality,

d(ϕ(x),0)≤d(ϕ(x), ϕ(0))+d(ϕ(0),0)=d(x,0)+d b

|b|²,0

. (2.2) The hyperbolic distance is related with the Euclidean distance by the formula d(x,0) = log(¹₁^+|_−|^x_x^|_|). Therefore we deduce from (2.2) that ifx ∈ s B₁ andϕ ∈

M

_n^E(B₁,B₁),|ϕ(x)|satisfies the inequality

|ϕ(x)| +1

|ϕ(x)| −1 ≤ (s+1)(h₀+1) (s−1)(h₀−1). Henceh₁in the claim (3) of the lemma is implicitly defined by

h₁+1

h₁−1 ≤ (s+1)(h₀+1) (s−1)(h₀−1).

The assertion sϕ(B) ⊂ ϕ(h₁B). is equivalent to ϕ⁻¹(s B) ⊂ h₁B and hence it follows from the above reasoning.

The following proposition relies on the fact that

M

n is a finite dimensional manifold and

M

n(h B)is a compact manifold. Thus all metrics are equivalent.

Proposition 2.2 ( [21, Chapter 4. Lemma 2.5 ] ). Let B = B(a,r) and ϕ, ψ ∈

M

n(h B)with h > 1. Then there exists a constant C₄ = C₄(h)such that for any x,y∈ B we have that

|Dϕ(x)−Dψ(x)| ≤ C4

|B|

B|Dϕ−Dψ| (2.3)

and,

|Dϕ(x)−Dϕ(y)| ≤ 1 rC₄max

B |Dϕ||x−y|. (2.4)

The next lemma states that derivatives of M¨obius transforms are like constants in the following sense: If they are sufficiently close in a ball, they are also close in appropriate dilations of this ball.

Lemma 2.3 ([21, Chapter 4. Lemma 4.1]). Letϕ∈

M

nsuch that the inequality

|ϕ(x)−x| ≤r

holds for all x ∈ B(a,r), where > 0. Let h > 0. Then there exist constants α=α(h)and C₅=C₅(h)such that if < αthen the inequality

|Dϕ(x)−I| ≤C₅ holds for all x ∈h B.

We conclude the section by introducing the elliptic equations needed in Sec- tion 4 and DeGiorgi-Nash’s Theorem on the regularity of the solutions.

Definition 2.4. LetF :

R

ⁿ →

R

be a convex function such that|F(A)| ≤ C(1+

|A|^p). Then a mapping z ∈ W^1,p(,

R

ⁿ);z = (z₁, . . . ,z_n) is said to be F- harmonic inif each of the coordinatesz_i satisfies that

div(D F(Dz_i))=0 in. Equivalently,z_i minimizes

F(Dv)respect to its own boundary values.

Proposition 2.5 ([4, 10]). Let z be an F -harmonic mapping in B. Let F :

R

ⁿ →

R

be a C^∞ uniformly convex and such that|D F(A)| ≤ C_F|A|, |D²F| ≤ C_F for C_F >0. Let0≤h <1. Then, z∈C^∞(h B)and there exists a number0< α <1, α=α(h,F)and a constant C₆=C₆(h,F)such that

[Dz]_Cα(h B)≤C₆

B

|Dz|^{2 1}₂

.

3.

M

_n^E

as a subset of M

n

In this section we prove Theorem 1.4 forv∈

M

n()\

M

n^E(). Firstly, we reduce the situation to the case whereEis connected in Proposition 3.1. The essential point in the proof is Proposition 2.2 stating that ifϕ∈

M

_n^EthenDϕis Lipschitz and the Lipschitz constant depends only onE. Thus, there is no loss of generality assuming E = E^m.M. We treat this case in Proposition 3.2. The essential observation for the proof of Proposition 3.2 is that ifϕ /∈

M

n^E(), then there exists a ballB˜ ⊂\ with | ˜B|¹ⁿ ≈ dist(, ∂) such that for x ∈ ˜B, Dϕ(x) /∈ E. This, and that for

ϕ ∈

M

^E_n the oscillation of |Dϕ| is uniformly controlled, provides us a constant C =C(,E)such that

min

dist²_MSO(n)(Dϕ),

dist²_mSO(n)(Dϕ)

≤C

B˜

dist²_E(Dϕ).

Therefore, we can conclude by means of Theorem 1.1.

If one tries to follow a similar scheme for proving an estimate up to the bound- ary one faces the situation of the Example 3.3 presented at the end of the section.

Proposition 3.1. Let E, , as in Theorem1.4and let{E_i}_i^I₌₁be the connected components of E. Then, there exists C₇ = C₇(E, , )such that for everyϕ ∈

M

n()

mini

dist²_Ei(Dϕ)

≤C₇

dist²_∪iEi(Dϕ). (3.1) Proof. By induction, it is enough to prove the theorem forEhaving two connected componentsE =E₁∪E₂. Letm₁=min_E1|A| ≤max_E1|A| =m₂<min_E2|A| = M₁ ≤ max_E2|A| = M₂ andρ = M₁ −m₂ = dist(E₁,E₂) (Here we use the invariance of E under SO(n)). We firstly observe that there exist constantsc₁ = c₁(E),c₂=c₂(E)such that if either|A| ≥2M₂or|A| ≤_2m¹₁ then

dist_Ei(A)≤c₁dist_E(A), (3.2) and dist_E(A)≥c₂.

Let^E = {x ∈ : _2m¹

1 ≤ |Dϕ(x)| ≤ 2M₂}. Suppose that|^E| ≤ ¹₂||. Then we have that 2|\^E| ≥ ||and therefore

dist²_E

1(Dϕ)≤c₁

\^Edist²_E(Dϕ)d x +4M₂²||

≤c₁

\^Edist²_E(Dϕ)d x +8M₂²

c₂ c₂|\^E|

≤c₃

\^Edist²_E(Dϕ), and (3.1) holds. Hence, we can assume that|^E| ≥ ¹₂||.

Now, if

dist_E1(Dϕ(x))≥ ρ 2, for everyx ∈E it would follow that

dist_E2(Dϕ(x))≤c₄dist_E(Dϕ),

withc₄= ⁴_ρM₂.Together with (3.2) we would have that dist_E2(Dϕ(x))≤max{c₁,c₄}dist_E(Dϕ) for allx∈and (3.1) would be trivial.

Since the same argument works if we exchange E₁and E₂ we are left to the case where there exist two pointsx₁,x₂∈^E such that

dist_E(Dϕ(x₁))=dist_E1(Dϕ(x₁))≤ ρ 2, dist_E(Dϕ(x₂))=dist_E2(Dϕ(x₂))≤ρ

2.

Then, sinceis connected we can assume to be connected. Thus, there exists x₀ ∈ with|Dϕ(x₀)| = ^M+₂^m and consequentlydi st_Ei(Dϕ(x₀)) = ^ρ₂ for i = 1,2. Since

, there is no loss of generality assuming the existence ofr₁ = r₁(, ) such that B(x₀,r₁) ⊂ . Furthermore, since 2|^E| ≥ || and is bounded there existsr₂=r₂(,n)such thatB(x₀,r₂)⊂^E.

Now it follows from Proposition 2.2 that Dϕ_|B(x₀,r₂) is Lipschitz with a con- stant L(E,r₂). Thus, there is new r₃ = r₃(, ,E) such that on B(x₀,r₃), dist_E(Dϕ(x))≥ ^ρ₄. But now this implies that

dist²_E(Dϕ)=

\^Edist²_E(Dϕ)+

^Edist²_E(Dϕ)

≥

\^Edist²_E(Dϕ)+

B(x₀,r₃)dist²_E(Dϕ)

≥

\^Edist²_E(Dϕ)d x +c₅ρ²r₃ⁿ≥c₆

dist²_E1(Dϕ), wherec₆=max{^c_M^5ρ2^2r3n

2||,c²₁}. The proof is concluded.

Proposition 3.2. Let, and E as in Theorem1.4. Let E ⊂ E. Then, there exists a constant C₈=C₈(, ,E)such that for everyϕ∈

M

n^E()there exists ϕ ∈

M

_n^E()satisfying

|Dϕ−Dϕ|²≤C₈

dist²_E(Dϕ). (3.3) Proof. By Theorem 1.1 and Proposition 3.1 it suffices to prove the thesis for

E= E^m,M 0<m <M <∞.

Letϕ∈

M

_n^E()and suppose thatϕ_|∈/

M

_n^E(). Then either max|Dϕ|>

M or min|Dϕ|<m. We consider the two possibilities separately.

Casemax|Dϕ|>M.

Letd =dist(, ∂). We will deduce (3.3) from the next two claims:

There existsc₁=c₁(, ,E)such that (max

_d 2

|Dϕ| −M)²≤c₁

dist²_E(Dϕ). (3.4) There existsc₂=c₂(, )such that if min|Dϕ|< Mwe have that

(max

_d 2

|Dϕ| −M)≥c₂(M−min

|Dϕ|). (3.5)

Let us supposed we have proved (3.4) and (3.5) and deduce the thesis from them.

Indeed, (3.4) and (3.5) imply that ifx ∈and|Dϕ(x)| ≤M, then it holds that dist²_MSO(n)(Dϕ(x))≤c₃

dist²_E(Dϕ), with c₃ = _c^c12

2

. On the other hand if |Dϕ(x)| ≥ M, then dist²_MSO(n)(Dϕ(x)) = dist²_E(Dϕ(x)). Hence,

dist²_MSO(n)(Dϕ)≤c₃

dist²_E(Dϕ) (3.6) and thus (3.3) would follow from Theorem 1.1 withϕ= AxwhereA∈ MSO(n). To prove the claims (3.4) and (3.5) we need further notation. Since ϕ ∈

M

n^E() we have that|Dϕ(x)| = r²|x −a|⁻² where r ∈

R

^anda ∈

R

ⁿ \. Let x₀ ∈ be such that|Dϕ(x₀)| = max|Dϕ| > M and let L(t) :

R

→

R

ⁿ be defined by L(t)= x₀+t_|^a_a⁻₋^x_x⁰

0|. By triangle inequality ifx ∈ B(L(^d₄),^d₄)then

|Dϕ(x)| ≥M. Thus for such anx

dist_E(Dϕ(x))= |Dϕ(x)| −M=dist_MSO(n)(Dϕ(x)). (3.7) Therefore we can apply Theorem 1.1 to ϕ in B(L(^d₄),^d₄). Together with (3.7) it yields an R_ϕ ∈SO(n)such that

B(L(^d₄),^d₄)|Dϕ−M R_ϕ|²≤c₄

B(L(^d₄),^d₄)dist²_E(Dϕ). (3.8) Sinceϕ ∈

M

n(), ϕ ∈

M

n(B(L(^d₄,4^d₄)). Thus, we can apply Proposition 2.2 to obtain

Dϕ−M R_ϕL∞(B(L(^d₄),^d₄))≤ C₄(4)

|B(L(^d₄),^d₄)|

B(L(^d₄),^d₄)|Dϕ−M R_ϕ|. (3.9)

Combining (3.8), (3.9) and triangle inequality yields Dϕ

L

d

2 − |M R_ϕ| 2

≤c₁

dist²_E(Dϕ), (3.10) withc₁= _|B^c₍⁴_L^C₍⁴d⁽⁴⁾

4),^d₄)|. We have obtained the desired (3.4).

For (3.5), we observe that the function f(t) = |Dϕ(L(t))|is convex and in- creasing fort <d. Let|Dϕ(x₁)| =min|Dϕ(x)|. Then|Dϕ(x₁)| = f(t₁)where t₁ = |x₁−a| − |a−x₀|. Since f(0) > M, t₁ < 0 and by triangle inequality

−t₁≤diam(). Now since f is convex it holds that f

d 2

− f(0)≥ d

−2t₁(f(0)− f(t₁)). (3.11) Replacing f by its values, we see that (3.11) is indeed (3.5) withc₂ = _2diam^d₍). Therefore the proof is concluded in the case max|Dϕ|> M.

Case min|Dϕ|<m.

By considering the inverse mapping ϕ⁻¹ we will reduce the situation to the case max|Dϕ| ≥ M. Sinceϕ ∈

M

n^E() we can apply Proposition 2.1 to find a constants =s(E, , )such that dist(ϕ(), ∂ϕ()) ≥s. Hence we can argue as in the previous case withϕ⁻¹ in the place ofϕ and _m¹ in the place of M. We obtain that if inf|Dϕ(x)| < m, there exists c₅ = c₅(E,s), R ∈ SO(n) and

B⊂ϕ()\ϕ()with|B| ≥c₆sⁿ such

ϕ()

1

mR−Dϕ⁻¹²≤c₅

B

dist²₁

mSO(n)(Dϕ⁻¹) (3.12) and for allx ∈B|Dϕ⁻¹(x)| ≥ _m¹. Now recall that ifϕ∈

M

_n^E()mⁿ ≤J_ϕ ≤Mⁿ and that forρ ∈

R

^dist_ρSO(n)(Dϕ(x))= |ρ− |Dϕ(x)||. Therefore we can change variables in (3.12) to obtain that

|m R⁻¹−Dϕ|²≤c₇

ϕ⁻¹(B)dist²_mSO(n)(Dϕ) (3.13) with c₇ = c₇(E,E, , ).We also used that|A− B| = |A||B||A⁻¹− B⁻¹|. Finally forx∈ϕ⁻¹(B)|Dϕ(x)|<mand hence dist²_mSO(n)(Dϕ(x))=dist²_E(Dϕ(x)). Thus, the desired estimate

|m R⁻¹−Dϕ|²≤c₇

ϕ⁻¹(B)dist²_E(Dϕ) (3.14) follows and the proof is concluded.

In the following exampleE= E^m,M.

Example 3.3. There exists a sequenceϕj ∈

M

n(B1)such that

jlim→∞

inf_ψ∈ME n(B1)

B₁|Dϕj −Dψ|²

B₁dist²_E(Dϕj) = ∞.

Proof. Let ϕ = r_|x^x_−λ^−λ_e^e1

1|² with λ and r chosen so that |Dϕ(e₁)| = M and

|Dϕ(−e₁)| = m where e₁ = (1,0. . . ,0). Let us consider the sequence ϕj = r ^x−(λ−

1 j)e₁

|x−(λ−¹_j)e₁|² and sett_j = |Dϕj(e₁)| −M. On one hand, we have that

B₁

dist²_E(Dϕj)≤t²_j|{x ∈B₁: dist²_E(Dϕj) >0}|

=t²_j|{x ∈B₁:|Dϕj| ≥M}|.

(3.15)

On the other hand, letψ ∈

M

_n^E(B₁)andϕj defined as above with ¹_j ≤ ^λ−₂¹. Then ϕj andψ ∈

M

n(h B)for someh>1,hnot depending of j. Thus, Proposition 2.2 yields a constantc₁=C₄(h)such that

t²_j =(|Dϕj(e₁)| −M)² ≤ |Dϕj(e₁)−Dψ(e₁)|²

≤c₁

B₁

|Dϕj−Dψ|². (3.16)

If we put (3.15) and (3.16) together, we obtain that inf_ψ∈ME

n(B₁)

B₁|Dϕj−Dψ|²

B₁dist²_E(Dϕj) ≥ c₁

|x ∈B₁ :|Dϕj| ≥M|, which proves the claim.

4.

Proof of Theorem 1.4 in a ball

The starting point for the proof of Theorem 1.1 in [7] is that thanks to the structure of the set SO(n)it suffices to prove Theorem 1.1 for mappings whose components are uniformly Lipschitz harmonic functions. We start along the same lines than in [7] since Eenjoys a nice structure as well.

4.1. Reduction to Lipschitz mappings

Proposition 4.1. Let E ⊂ E^m,M. There exists constants C₉ = C₉(n), C₁₀ = C₁₀(M)such that ifv∈W^1,2(B)there existsvM ∈W^1,∞()such that

• DvML^∞(B) ≤C₉M and

•

B|Dv−DvM|²≤C₁₀

Bdist²_E(Dv).

Proof. The appendix A.1 in [7] yieldsvMsatisfying

• DvML^∞(B) ≤c₁M

•

B|Dv−DvM|²≤c₂

{x∈B:|Dv|≥2M}|Dv|². Since if|A| ≥2M,|A| ≤2dist_E(A)we are done.

4.2. Reduction to F-harmonic mappings

As SO(n)is related to harmonic functions so is CO₊(n)ton-harmonic functions.

However since then-harmonic equation is degenerate the arguments below applied to n-harmonic functions would yield a wrong exponent in (4.7). Instead we take advantage of E being a compact subset of CO₊(n)bounded away from 0. This allows us to modify| · |ⁿnear 0 and∞constructing a functionFsuch thatF(A)=

|A|ⁿ for matrices A ∈ E but with the right growth at 0 and ∞. In this way, the set Eis related to minimizers of a variational problem

F(Dv), whereFhas the appropriate growth.

Throughout the section then-tuple ofn-vectors(A₁,A₂, . . . ,A_n)stands for the matrix with rows(A₁,A₂, . . . ,A_n)and the cofactor matrix ofA,(Cof(A))i j =

∂i jdet(A).

Lemma 4.2. There exists a functionψ ∈C^∞(

R

,

R

)such that the function F(A):

R

ⁿ →

R

defined by F(A) = ψ(|A|) belongs to C^∞(

R

ⁿ,

R

) and satisfies that

D F(A)≤C_F(1+ |A|)and_C¹

F ≤ D²F(A)≤C_F with C_F =C_F(E). Moreover, for every A ∈ E it holds that(D F(A₁),D F(A_i),D F(A_n)) = Cof(A). Equiva- lentlyCof(A)= DF where˜ F˜(A)= ⁿ_i=1F(A_i).

Proof. LetF(A)=ψ(|A|). Then a direct calculation gives that D F(A)=ψ(|A|) A

|A|, D²F(A)=ψ(|A|) A

|A| ⊗ A

|A|

+ ψ(|A|)

|A|

I − A

|A|⊗ A

|A|

.

(4.1)

If we put

ψ1(x)=n(n−1) _x

0

_y

0

zⁿ⁻²χ₍0,M)(z)+(1−χ₍0,M)(z))Mⁿ⁻²d zd y

,

ψ1(x)=xⁿχ0,M+((1−χ₍0,M)(x))ax²+bx+cwitha,b≥0 and it satisfies the claims of the Lemma 4.2 except in a neighborhood of the origin. Then we consider ψ2(x) = max{(^m₂¹)ⁿ⁻²x², ψ1(x)}. This new function satisfies the claims but it is not smooth. We repair this by replacingψ2 by a smooth approximation of it in a neighborhood of^m₂¹. Letrandbe small numbers, andϕ∈C₀^∞(^m₂¹−2r,^m₂¹+2r) ϕ = 1 on(^m₂¹ −r,^m₂¹ +r)and let ρ be an approximation of the identity. Then setting

ψ_,r =ϕ(ρ∗ψ2)+(1−ϕ)ψ2

forandr small enough it is not hard to check that F(A)=ψ_,r(|A|)satisfies all the conditions concerning regularity in the statement of the lemma. By (4.1) the bounds for the derivatives of F depend on the bounds for the derivatives ofϕ_,r, which in turn depend onm and M. Hence the constantC_F depends on the setE.

Concerning regularity sinceψ ∈C^∞and is quadratic near 0 the smoothness of F follows.

Finally, if A = (A₁,A₂, . . . ,A_n) ∈ CO₊(n), det(A)= |A|ⁿ = |A_i|ⁿ where

|A_i| is the Euclidean norm of the vector A_i. Since for A ∈ E F(A_i) = |A_i|ⁿ, it holds that F˜(A) =ndet(A)as well. If we differentiate both sides of this equality we obtain that for A ∈ E DF˜(A)= (D F(A₁),D F(A_i),D F(A_n)) = Cof(A)as claimed.

Seth(A)= DF˜(A)−Cof(A). Lemma 4.3. Letv∈W^1,2(B,

R

ⁿ). Then

v=w+z,

where z is an F -harmonic mapping in B andw∈W₀^1,2(B,

R

ⁿ)is such that

B|Dw|²≤C_F

B|h(Dv)|².

Proof. To obtain the decomposition we solve the following system Div(DF˜(Dz))=0 in B

z=v on∂B. (4.2)

Here for a matrix valued function A(x)the operator Div(A(x))yields ann-vector with div(A_i(x))as components. Thus, given that

DF˜(Dz)=(D F(Dz₁),D F(Dz₂), . . . ,D F(Dz_n)) we can express

Div(DF˜(Dz)=(div(Dz₁(x)),·,div(D_n(x)))