www.elsevier.com/locate/anihpc
Invertibility of Sobolev mappings under minimal hypotheses
Leonid V. Kovalev
a,1, Jani Onninen
a,2, Kai Rajala
b,∗,3aDepartment of Mathematics, Syracuse University, Syracuse, NY 13244, USA
bDepartment of Mathematics and Statistics, University of Jyväskylä, PO Box 35 (MaD), FI-40014, University of Jyväskylä, Finland Received 15 May 2009; received in revised form 14 September 2009; accepted 14 September 2009
Available online 1 October 2009
Abstract
We prove a version of the Inverse Function Theorem for continuous weakly differentiable mappings. Namely, a nonconstant W1,nmapping is a local homeomorphism if it has integrable inner distortion function and satisfies a certain differential inclusion.
The integrability assumption is shown to be optimal.
©2009 Elsevier Masson SAS. All rights reserved.
MSC:primary 30C65; secondary 26B10, 26B25
Keywords:Local homeomorphism; Differential inclusion; Finite distortion
1. Introduction
Throughout this paper Ω is a bounded domain in Rn. The classical Inverse Function Theorem states that if f:Ω→Rnis continuously differentiable and the differential matrixDf (x)is invertible at some pointx, thenf is a homeomorphism in a neighborhood ofx. We are interested in a version of the Inverse Function Theorem for con- tinuous weakly differentiable mappings. In this context the invertibility of the differential matrix is not sufficient. As an example, consider the winding mappingf:R3→R3written in cylindrical coordinates asf (r, θ, z)=(r,2θ, z).
Althoughf is Lipschitz and its Jacobian determinantJ (x, f )equals 2 for a.e. x∈Rn, this mapping is not a local homeomorphism.
Let us introduce the following subset ofn×nmatrices.
M(δ)=
A∈Rn×n: Aξ, ξδ|Aξ||ξ|for allξ∈Rn ,
where−1δ1. Note thatδ= −1 imposes no condition on the matrix. When−1< δ <0, the setM(δ) is not convex and the differential inclusion
* Corresponding author.
E-mail addresses:lvkovale@syr.edu (L.V. Kovalev), jkonnine@syr.edu (J. Onninen), kirajala@maths.jyu.fi (K. Rajala).
1 Supported by the NSF grant DMS-0913474.
2 Supported by the NSF grant DMS-0701059.
3 Supported by the Academy of Finland.
0294-1449/$ – see front matter ©2009 Elsevier Masson SAS. All rights reserved.
doi:10.1016/j.anihpc.2009.09.010
Df (x)∈M(δ) for a.e.x∈Ω, (1.1) cannot be integrated to yield a pointwise inequality forf.
The winding mapping does not satisfy (1.1) for anyδ >−1. Even so, this differential inclusion does not by itself guarantee thatf is locally invertible, e.g., f (x1, x2)=(x1,0). There are also such examples with strictly positive Jacobian [14, Example 18]. To quantify the invertibility of a matrix A∈Rn×n, we introduce the inner distortion KI(A)∈ [1,∞].
KI(A)=
⎧⎪
⎨
⎪⎩
An
(detA)n−1, detA >0,
1, A=0,
∞, otherwise.
(1.2)
HereAstands for the cofactor matrix ofAand·is the operator norm. To shorten the notation we writeKI(x, f )= KI(Df (x))and
KΩ[f] := 1
|Ω|
Ω
KI(x, f )dx,
where|Ω| is the Lebesgue measure ofΩ. Iff ∈W1,n(Ω,Rn)andKI(x, f ) <∞ a.e., thenf has a logarithmic modulus of continuity [4,9]; that is,
f (a)−f (b) nC(n)
2BDfn
log(e+2 diamB|a−b| ), a, b∈B, 2BΩ.
In this paper we always takef to be its continuous representative.
If moreoverKΩ[f]<∞andf is invertible, then the inverseh:=f−1is aW1,n-mapping and
Ω
KI(x, f )dx=
f (Ω)
Dhn,
see [1, Theorem 9.1]. ThusKΩ[f]controls the modulus of continuity off−1, should it exist. Our main result ad- dresses its existence.
Theorem 1.1.Suppose thatf ∈Wloc1,n(Ω,Rn)is a nonconstant mapping such thatKΩ[f]<∞. If there existsδ >−1 such thatDf (x)∈M(δ)for almost everyx∈Ω, thenf is a local homeomorphism.
This theorem is already known in the planar casen=2 [14, Theorem 4]. The assumptionKΩ[f]<∞cannot be replaced by
ΩKIq(x, f )dx <∞for anyq <1, see [14, Example 18] or [2, Example 1].
Our proof of Theorem 1.1 is based on two results of independent interest. The first step toward proving that a mapping is a local homeomorphism is to show that it is discrete and open; that is, preimages of points are discrete sets and images of open sets are open.
Theorem 1.2.Letf:Ω→Rnbe a mapping inWloc1,n(Ω,Rn)such thatJ (x, f ) >0a.e. If(Df )−1∈L∞(Ω,Rn×n), thenf is discrete and open.
The challenging Iwaniec–Šverák conjecture asserts even more: a nonconstant mapping f ∈Wloc1,n(Ω,Rn)with KΩ[f]<∞ is discrete and open. So far this conjecture was proved only forn=2 in [10]. Partial results in this direction were recently obtained in [6–8,15,19,20].
Another crucial ingredient of our proof of Theorem 1.1 is an estimate for the multiplicity N (y, f, A) :=
#(f−1(y)∩A) of a local homeomorphismf in terms of the integral ofKI(·, f )in dimensionsn3. This result (Theorem 5.1) continues the line of development that began in 1967 with the celebrated Global Homeomorphism Theorem of Zorich [24].
The proof of Theorem 1.1 proceeds as follows. The differential inclusion (1.1) allows us to approximate f by mappings fλ(x):=f (x)+λx to which Theorem 1.2 can be applied. The results of [14] yield that fλ is a local
homeomorphism. By virtue of Theorem 5.1 the mappingsfλ have uniformly bounded multiplicity, which leads to a bound for the essential multiplicity off. This additional information suffices to show thatf is discrete and open, see Proposition 2.2 below. Sincef is a limit of local homeomorphismsfλ, the conclusion follows.
Different approaches to the invertibility of Sobolev mappings were pursued in [2,3,5,16,18,22], see also references therein.
2. Background
In this section we collect necessary notation and preliminaries. An open ball with centeraand radiusris denoted byB(a, r):= {x∈Rn:|x−a|< r}. Its boundary is the sphereS(a, r). Ifλ >0 andB=B(a, r), thenλB=B(a, λr) andλS=S(a, λr). In addition,B=B(0,1),Br=B(0, r),S=S(0,1)andSr=S(0, r).
LetHdstand for thed-dimensional Hausdorff measure which agrees with the Lebesgue measure whendcoincides with the space dimension. The Hausdorff distance dH(E, F )between nonempty bounded setsEandF is defined as the infimum of numbers >0 such that the-neighborhood ofEcontainsF and vice versa.
Given a continuous mappingf:Ω →Rn and a setE⊂Ω, we denote by N (y, f, E)the cardinality (possibly infinite) of the setf−1(y)∩E. Ify∈Rn\f (∂Ω), the local degree off aty with respect to a domainG⊂Ω is denoted by deg(y, f, G). We writef:A−→hom Bto indicate thatf is a homeomorphism fromAontoB.
LetΓ be a family of paths (parametrized curves) inRn,n2. The image ofγ∈Γ is denoted by|γ|. We letΥΓ be the set of all Borel functionsρ:Rn→ [0,∞]such that
γ
ρds1
for every locally rectifiable pathγ∈Γ. The functions inΥΓ are called admissible forΓ. For a given weightω:Rn→ [0,∞]we define
MωΓ = inf
ρ∈ΥΓ
ρ(x)nω(x)dx,
and callMωΓ the weighted conformal modulus ofΓ. Here it suffices to haveωdefined on a Borel set containing
γ∈Γ|γ|. Whenω≡1 we obtain the conformal modulusMΓ. We will also use the spherical modulus with respect to a sphereS,
MSΓ = inf
ρ∈ΥΓ
S
ρ(y)ndHn−1(y).
The reader may wish to consult the monographs [21,23] for basic properties of moduli of path families. The following generalization of the Poletsky inequality relates the moduli ofΓ and of its image underf, denoted byf Γ.
Proposition 2.1.(See [12].) Suppose thatf ∈W1,n(Ω,Rn)is a discrete and open mapping with KΩ[f]<∞. If Γ is a family of paths contained inΩ, then
Mf Γ MKI(·,f )Γ. (2.1)
We will use the following result, which establishes the Iwaniec–Šverák conjecture under an additional assumption on the multiplicity off.
Proposition 2.2.Suppose thatf ∈Wloc1,n(Ω,Rn)is a nonconstant mapping withKΩ[f]<∞. LetB be a ball such that2BΩ. If
ess lim sup
r→0
r1−n
S(a,r)
N (y, f, B)dHn−1(y) <∞ (2.2)
for everya∈Rn, thenf is discrete and open inB.
This proposition is a consequence of [20, Theorem 2.2]. Although [20, Theorem 2.2] requires that ess sup
0<t <1
∂(t B)
Df (x)
|f (x)−a|n−1dHn−1(x) <∞, this condition is only used to obtain (2.2).
3. Preliminary results
For the sake of brevity, the connected component of a set A that contains a pointx ∈A will be called thex- component ofA.
Proposition 3.1.Suppose thatf ∈Wloc1,n(Ω,Rn)is a mapping such thatKΩ[f]<∞. Letx∈Ω andy=f (x). If thex-component off−1(y)is{x}, thenf is discrete and open in some neighborhood ofx.
Proof. Pickr >0 such thatB(x, r)Ω and letUj be thex-component of(f−1B(y,1/j ))∩B(x, r),j =1,2, . . .. Since the setsUj⊂Rnare nested, compact, and connected, their intersectionEis also connected. On the other hand, x∈E⊂f−1(y), henceE= {x}. It follows that diam(Uj)→0 asj→ ∞. Let us fixj such thatUjΩ. Note that Uj coincides with thex-component off−1B(y,1/j ).
We claim that f is quasilight in Uj; that is, the connected components off−1(w)∩Uj are compact for all w∈Rn. If not, then there existsz∈Uj such that thez-component off−1(f (z))intersects∂Ujat some pointb. Since f (b)=f (z)∈B(y,1/j ), there existst >0 such thatf B(b, t )⊂B(y,1/j ). This contradicts the definition ofUj. Therefore,f is quasilight inUj. By [19, Theorem 1.1]f is discrete and open inUj. 2
For the convenience of the reader we state two preliminary results from [21, III.3].
Lemma 3.2. (See [21, III.3.1].) Letf:Ω →Rn be a local homeomorphism and letQbe a simply connected and locally pathwise connected set inRn. SupposeP is a component off−1Qsuch thatP ⊂Ω. Thenf:P −→hom Q. If in additionQis relatively locally connected, thenf:P −→hom Q.
Lemma 3.3.(See [21, III.3.3].) Letf:Ω→Rn be a local homeomorphism and letA, B⊂Ω be two sets such that f is homeomorphic inAand inB. IfA∩B= ∅and iff A∩f Bis connected, thenf is homeomorphic inA∪B.
Given a sphereS=S(a, r)⊂Rn, and a pointp∈S, letCS(p, φ)be the open spherical cap ofSwith centerpand opening angleφ∈(0, π],
CS(p, φ)=
y∈S: y−a, p−a> r2cosφ .
For instanceCS(p, π/2)is a hemisphere andCS(p, π )is a punctured sphere. For anyφ∈(0, π]the capCS(p, φ) contains the pointp.
The following topological lemma forms the main step of the proof of Zorich Global Homeomorphism Theorem, see [21, III.3].
Lemma 3.4.Letf:Ω→Rnbe a local homeomorphism,Ω⊂Rn,n3. Suppose we have the following:
(i) GΩ such thatf:G−→hom GwhereGis convex;
(ii) G⊂DΩand there isa∈∂G∩∂D;
(iii) a ballB⊂Rnthat containsa=f (a)and such thatS=∂BmeetsGat some pointb.
Let b=f−1(b)∩Gand denote byCS∗(b, φ)the component off−1CS(b, φ)containingb. Then there exists0<
φ0< πsuch thatCS∗(b, φ0)⊂Dand the closure ofCS∗(b, φ0)meets∂D.
Proof. Letφ0be the supremum of allφsuch thatCS∗(b, φ)⊂D. First we observe thatφ0>0. Indeed, sincef is a local homeomorphism, there exists a neighborhoodV ⊂Dofbsuch thatf:V −→hom f (V ). Ifφis sufficiently small, thenCS(b, φ)f (V ), henceCS∗(b, φ)V ⊂D. It remains to show thatφ0< π.
Suppose to the contrary that φ0=π. Since CS∗(b, π )⊂D, it follows from Lemma 3.2 thatf:C∗S(b, π )−→hom CS(b, π )=S(here the assumptionn3 is used). SinceS∗:=C∗S(b, π )is homeomorphic toS, it separatesRninto two components. LetUbe the bounded component ofRn\S∗. Then the boundary off (U )is contained inSwhich impliesf (U )=B. Moreover,f:U−→hom Bby Lemma 3.2. Sinceb∈U∩Gand sincef (U )∩f (G)=B∩Gis convex (hence connected), Lemma 3.3 yields thatf is homeomorphic inU∪G.
This leads to a contradiction. SinceU∪G⊂Dit follows thatalies on the boundary ofU∪G. On the other hand, f (a)=a∈f (U )is an interior point off (U∪G). 2
We shall use a geometric lemma which is essentially contained in [13].
Lemma 3.5.Suppose we are given a ballB(y0, r)⊂Rn, a pointy1∈S(y0, r)and a connected setEthat containsy0 and some pointy2∈S(y0, r). Then there existq∈B(y0, r)and0< σ <2rsuch that for everyσ < t <4σ/3,
(i) y1∈B(q, t );
(ii) S(q, t )∩E= ∅;
(iii) S(q, t )⊂B(y0,2r)\B(y0, r/10).
Proof. Letα be the angle at the point(y0+y1)/2 formed by the line segments fromy0 to(y0+y1)/2 and from (y0+y1)/2 toy2. There are two cases possible.
Case 1.0α < π/2, or, equivalently,|y1−y2|> r. In this case we choose q =(y0+y1)/2 and σ =3r/5. For σ < t <4σ/3 we haveB(y0, r/10)⊂B(q, t )andy1∈B(q, t ). At the same time,y2∈/B(q, t )because
|y2−q|>
√3 2 r= 5
2√ 3σ >4
3σ.
Thus, all conditions (i)–(iii) are satisfied.
Case 2.π/2απ, or, equivalently,|y1−y2|r. This time we chooseq =(y1+y2)/2 andσ = |y1−y2|/2.
Since|y0−q|(√
3/2)r, it follows thatB(q, t )∩B(y0, r/10)= ∅provided that t <
√ 3 2 − 1
10
r.
This is indeed the case, because 4
3σ2 3r <
√ 3 2 − 1
10
r.
All conditions (i)–(iii) are met. 2 4. Proof of Theorem 1.2
Let(Df )−1∞=L. First we observe that the inner distortion off is locally integrable because
KI(x, f )=Df (x)−1nJ (x, f )LnDfn for a.e.x∈Ω. (4.1) We may assume that B4=B(0,4)Ω. It suffices to show that f is discrete and open inB. We will do this by proving that (2.2) holds. Without loss of generality, a in (2.2) equals 0. Fix 1< t <2 and 3< T <4 so that Hn−1(fSt) <∞andHn−1(fST) <∞. By the area formula we have
Rn
N (y, f,BT)dy=
BT
J (x, f )dx <∞. Therefore, for almost every 0< R <∞we have
SR
N (y, f,BT)dHn−1(y) <∞ and Hn−1
f (ST)∩SR
=0. (4.2)
We fix suchR <1/(2L)so that (4.2) holds, and let M:=R1−n
SR
N (y, f,BT)dHn−1(y).
Our goal is to prove that r1−n
Sr
N (y, f,B)dHn−1(y)M for a.e. 0< r < R. (4.3)
Letr < Rbe such thatHn−1(f (St)∩Sr)=0, and denote byE⊂Sthe set of unit vectorsvfor which
deg(Rv, f,BT) <deg(rv, f,Bt). (4.4)
LetIv:[r, R] →Rnbe the parametrized line segmentIv(s)=sv. By Proposition 3.1, eitherf−1(sv)has a nontrivial component for somersR, orf is discrete and open in a neighborhood off−1(Iv[r, R]), denoted byUv. By using the co-area formula as in [20, Lemma 2.4], we see that the former possibility only occurs forv∈F1where Hn−1(F1)=0. The mappingf is discrete and open in the open setU:=
{Uv: v∈E\F1}. It follows from (4.4) and basic properties of path lifting [21, Section II.3] that for eachv∈E\F1the segmentIvhas a maximalf-liftingIv∗ starting atBtand leavingBT.
Denote
f(x):=lim inf
z→x
|f (z)−f (x)|
|z−x| .
By our assumption on(Df )−1there exists a Borel null setF⊂Ωsuch thatf(x)1/Lforx∈Ω\F. LetF2be the set ofv∈E\F1such that eitherIv∗is unrectifiable orH1(|Iv∗|∩F ) >0. Since the measure ofFis zero, it follows that the family of curvesΓF := {Iv∗: v∈F2}has zero weighted modulus for any locally integrable weight. In particular, MKIΓF =0. SinceΓF ⊂U we can apply (2.1) and obtainM{Iv: v∈F2} =0, which impliesHn−1(F2)=0.
Forv∈E\(F1∪F2)we have H1
Iv∗
LH1(Iv) < LR <1
2, (4.5)
which contradicts the fact thatIv∗begins atBt and leavesBT. ThusE⊂F1∪F2. As a consequence,Hn−1(E)=0, which means deg(rv, f,Bt)deg(Rv, f,BT)forHn−1-a.e.v∈S. Since deg(y, f,Bt)=N (y, f,Bt)for a.e.y∈Rn [8, Proposition 2], inequality (4.3) follows. This completes the proof of Theorem 1.2 via Proposition 2.2.
5. Multiplicity of local homeomorphisms
In 1967 Zorich [24] proved that a local homeomorphismf:Rn→Rn,n3, withKI(·, f )∈L∞(Rn)must be a global homeomorphism. Martio, Rickman and Väisälä [16] gave a local version of this result. Namely, iff: 2B→Rn, n3, is a local homeomorphism with bounded distortionKI, then its radius of injectivity in B is bounded from below by a constant depending only onnand ess supKI. As a consequence, the multiplicityN (y, f, B)is bounded byC(n,ess supKI)for ally∈Rn.
The boundedness ofKI can be replaced by the condition exp
λKI1/(n−1)
∈L1(2B),
but this cannot be relaxed any further [13,17]. Surprisingly, the multiplicity bound remains true under a much weaker condition, namelyKI ∈L1. Example 7.2 below shows that KIq ∈L1 withq <1 does not suffice. The mappings fj(z)=ej zshow that all results discussed here fail whenn=2.
Theorem 5.1.Letf ∈Wloc1,n(Ω,Rn),n3, be a local homeomorphism such thatKΩ[f]<∞. IfB is a ball such that4BΩ, thenN (y, f, B)C(n,K4B[f])for ally∈Rn.
Proof. We may assume thatBis the unit ballB. Letx1, . . . , xm∈f−1(y)∩B. Moreover, letrjbe the largest radiusr so that thexj-componentU (xj, r)of f−1B(y, r)satisfies U (xj, r)⊂B3. By Lemma 3.2f is a homeomorphism fromU (xj, rj)ontoB(y, rj). We denote bysj the largest radiusssuch thatB(xj, s)⊂U (xj, rj). Thenf B(xj, sj) intersects bothy andS(y, rj). We notice that sincexj∈Band since the ballsB(xj, sj)are pairwise disjoint, there exist at mostN (n)indicesj for whichsj1. Thus we may assume thatB(xj, sj)⊂B2for every 1jm.
We now fix 1jmand a pointaj∈U (xj, rj)∩S3. We apply Lemma 3.5 withB(y0, r)=B(y, rj),y1=f (aj) andE=f (B(xj, sj)), obtaining a pointqj and a numberσj>0. For σj< t <4σj/3 choosewt ∈B(xj, sj)such thatf (wt)∈S(qj, t ). We apply Lemma 3.4 withG=U (xj, rj),D=B3,a=aj,B=B(qj, t )andb=f (wt). As a result we obtain 0< φt < π such that the spherical cap Ct :=CS(qj,t )(f (wt), φt)satisfiesCt∗⊂B3andC∗t ∩S3
contains some pointct. Consequently, for every pathγjoiningf (wt)andf (ct)inCt, the maximalf-liftingγ∗ofγ starting atwt starts fromB2and leavesB3. Following [23, 10.2], we will choose a particular familyΓt of such paths.
Let us say that a circular arc isshortif it is contained in a half-circle. The familyΓt will consist of all short circular arcs that connectf (wt)tof (ct)within Ct. More precisely, lethbe a Möbius transformation that mapsf (wt)to infinity andS(qj, t )\ {f (wt)}toRn−1. Observe thath(Ct)is the complement of a ball inRn−1. The convexity of Rn−1\h(Ct)implies that there exists an(n−2)-hemisphereV such thath(f (ct))+sv∈h(Ct)for everys >0 and v∈V.
Introduce a family of curvesIv:[0,∞)→Ct, defined by Iv(s)=h−1
h f (ct)
+s−1v ,
and denote byIv∗the maximalf-lifting ofIvstarting atwt. Now let 0< (v) <∞be the smallest number such that Iv∗((v))∈S3. Let
Γt=
Iv∗|[0,(v)]: v∈Vt .
We writef Γt for the image ofΓtunderf.
There is a lower bound for the spherical modulus off Γt, namely [23, Theorem 10.2]
MS(f Γt)C(n)
t . (5.1)
Let
Γj= {γ: γ∈f Γtfor someσj< t <4σj/3},
and letΓj∗be the family of the corresponding liftsγ∗starting atwt. Then integrating (5.1) we obtain
MΓj
4σj/3 σj
C(n)
t dtC(n). (5.2)
As observed earlier, everyγ∈Γj∗starts atB2and leavesB3. We denote byEj the smallest closed subset ofB3\B2
that contains|γ| ∩(B3\B2)for allγ∈Γj∗. Note that Ej⊂f−1
B(y,2rj)\B(y, rj/10)
(5.3) by part (iii) of Lemma 3.5. Since the characteristic functionχEj is an admissible function forΓj∗, we have
MKIΓj∗
Ej
KI(x, f )dx. (5.4)
The generalized Poletsky inequalityMΓjMKIΓj∗[14, Theorem 4.1], together with (5.2) and (5.4) yield mC(n)
m j=1
MΓj m j=1
Ej
KI(x)dx
sup
x∈B3\B2
m j=1
χEj(x)
×
3B
KI(x, f )dx. (5.5)
Claim 1.There existsM=M(n,K4B[f])such that m
j=1
χEj(x)M for everyx∈B3\B2. (5.6)
By virtue of (5.5), Theorem 5.1 follows from Claim 1. In the rest of this section we prove (5.6).
Letx ∈B3\B2be a point covered byM of the setsEj. After relabeling we havex∈Ej for 1j M, and r1r2· · ·rM. Since disjoint sets have disjoint preimages, (5.3) impliesrM20r1.
Chooseτ >0 such thatB(x, τ )⊂B3andf is injective inB(x, τ ). For 1j Mthere existsγj∗∈Γj∗which meetsB(x, τ ). Letwj be the starting point ofγj∗, and letγj be the subcurve ofγj∗that begins atwj and ends once it meetsB(x, τ ).
Claim 2.For1jMthere is a curveτj that joinsytof (wj)withinB(y, rj)in such a way that the union of|τj| and |f ◦γj| can be mapped onto a line segment by anL-biLipschitz mappingg:Rn→Rn. HereLis a universal constant.
Proof. Note that the imagef ◦γj is a short circular arc contained in the sphereS(q, t )of Lemma 3.5. Part (iii) of Lemma 3.5 implies
dist
y,|f◦γj| dist
y, S(q, t )
1
10rj 1
40diam|f◦γj|. (5.7)
There are two cases. Ify∈B(q, t ), thenτj is the line segment connectingytof (wj). By virtue of (5.7), the distance fromy toS(q, t )is comparable tot. Therefore, the angle betweenτj and the sphereS(q, t )is bounded from below by a universal constant, and the claim follows.
Suppose that y /∈B(q, t ). Letρj := |f (wj)−y|. Note thatrj/10ρjrj. Letp be the point of the sphere S(y, ρj)that is farthest fromq, namely
p=y−ρj q−y
|q−y|.
We chooseτj as the union of the line segment connectingytopand the geodesic arc onS(y, ρj)fromptof (wj).
Once again, the angle betweenτj and the sphereS(q, t )is bounded from below by a universal constant. 2
Letηj, 1jM, be the curve obtained by concatenating−(f◦γj)with−τj, where−indicates the reversal of orientation. Note thatηj begins inf B(x, τ ), proceeds along a circular arc tof (wj), and ends aty. Itsf-liftingη∗j starting inB(x, τ )is contained inB3and ends atxj.
Claim 3.There exists=(n, M)such that→0asM→ ∞, and
1i<jminM dH
|ηi|,|ηj|
r1/L. (5.8)
Proof. We begin our proof of Claim 3 by observing that |ηj| ⊂B(y,2rM)⊂B(y,40r1). For >0 let Z = {z1, . . . , zN}be an(r1/L)-net inB(y,40r1), whereN=N (, n). The set of all nonempty subsets ofZis an(r1/L)- net in the set of all nonempty closed subsets ofB(y,40r1)equipped with the Hausdorff metric. IfM >2N, then by
the pigeonhole principle there existi < jsuch that|ηi|and|ηj|are within the distance(r1/L)from the same subset ofZ. Claim 3 follows. 2
Fixi,j, andas in Claim 3, and letg:Rn→Rnbe theL-biLipschitz mapping from Claim 2. By replacingf withg◦f, which has a comparable distortion functionKI, we may assume that|ηj|is a line segment. Forδ >0 we denote byW (δ)the openδ-neighborhood of|ηj|. LetW∗(δ)be thexj-component off−1W (δ).
Claim 4.Ifδ > r1, thenW∗(δ)∩S4= ∅.
Proof. Sinceδ > r1, we have|ηi| ⊂W (δ). Suppose to the contrary thatW∗(δ)⊂B4. ThenW∗(δ)Ω, which by Lemma 3.2 implies thatf:W∗(δ)→W (δ)is a homeomorphism. This contradicts the fact that thef-liftings of ηi andηj starting inB(x, τ )end at different points, namelyxi andxj. 2
Letδ0 be the supremum of all numbersδ such that W∗(δ)⊂B4. Since f is a local homeomorphism,δ0>0.
By Lemma 3.2,f:W∗(δ)−→hom W (δ)for every 0< δ < δ0. By Claim 4 we haveδ0r1.
Choose a pointa∈∂W∗(δ0)∩S4. Leta=f (a). Sincea∈∂W (δ0), there existsp∈ |ηj|such that|a−p| =δ0. Forδ0< t <12diam|ηj|choosebt∈ |ηj| ∩S(p, t ). We apply Lemma 3.4 withG=W∗(δ0),D=B4,a=a,B= B(p, t ) and b=bt. As a result we obtain 0< φt < π such that the spherical cap Ct :=CS(p,t )(bt, φt)satisfies Ct∗⊂B4andC∗t ∩S4contains some pointct. Consequently, for every pathγjoiningbtandf (ct)inCt, the maximal f-liftingγ∗ofγ starting atf−1(bt)∩ |η∗j|starts fromB3and leavesB4. LetΓ be the family of all such pathsγ and letΓ∗be the family of the liftsγ∗. From [23, Theorem 10.2] we have
MΓ C(n)
diam(η j)/2 r1
dt
t C(n)logdiam(ηj) 2r1
.
By (5.7) we have diam|ηj|cr1with a universal constantc >0. Therefore, MΓ C(n)log1
. (5.9)
On the other hand, since the characteristic functionχB4\B3 is an admissible function forΓj, we obtain MKIΓ∗
B4\B3
KI(x, f )dx.
Combining this with (5.9) and using the Poletsky inequality again, we have C(n,K4B[f]), hence M C(n,K4B[f]). This gives (5.6). The proof of Theorem 5.1 is complete. 2
6. Proof of Theorem 1.1
Denotefλ(x)=f (x)+λx,λ >0. Thenfλ∈Wloc1,n(Ω,Rn). Moreover, by [14, Lemma 10], KI
x, fλ
C(δ, n)KI(x, f ) and Dfλ−1
(x)C(δ, λ) (6.1)
for almost every x ∈Ω. Thus fλ is discrete and open for every λ >0 by Theorem 1.2. Furthermore, by [14, Lemma 13] fλ is a local homeomorphism. (Although [14, Lemma 13] imposes a stronger condition on the dis- tortion off, this condition is only used to ensure thatf is discrete and open.) Sincefλ→f locally uniformly, the following proposition implies thatf is a local homeomorphism, completing the proof of Theorem 1.1.
Proposition 6.1.Suppose that a mappingf ∈Wloc1,n(Ω,Rn)withKΩ[f]<∞ can be uniformly approximated by local homeomorphismsfj∈Wloc1,n(Ω,Rn)such thatsupjKΩ[fj]<∞. Thenf is a local homeomorphism.
Proof. By [14, Proposition 7] it suffices to show thatf is discrete and open. If n=2, this is due to Iwaniec and Šverák [10]. Thus we assume thatn3. LetB=B(x0, R)be a ball such that 8BΩ. We will show that
N (y, f, B)C for a.e.y∈Rn, (6.2)
whereC <∞does not depend ony. Proposition 2.2 will then imply thatf is discrete and open inB. Applying Theorem 5.1 tofj, we obtain
N (y, fj,2B)C for everyy∈Rn, whereCdepends only on supjKΩ[fj]andn.
We fix R < t <2R so thatHn−1(f S(x0, t )) <∞, and a pointy∈f B\f S(x0, t ). Letd=dist(y, f S(x0, t )).
Sincefj →f locally uniformly, there existsj0such that|fj(x)−f (x)|< d/2 for allj j0and allx∈S(x0, t ).
Consequently, the restrictions offj andf toS(x0, t )are homotopic via the straight-line homotopy that takes values inRn\ {y}. It follows that
deg
y, f, B(x0, t )
=deg
y, fj, B(x0, t )
N (y, fj,2B)C
for alljj0. SinceN (y, f, B)N (y, f, B(x0, t ))=deg(y, f, B(x0, t ))for almost everyy∈Rn, we conclude that (6.2) indeed holds. The proof is complete. 2
7. Concluding remarks
Corollary 7.1. Suppose that f ∈Wloc1,n(Rn,Rn)is a nonconstant mapping such thatKI(·, f )∈L1loc(Rn). If there existsδ >−1such thatDf (x)∈M(δ)for almost everyx∈Rn, thenf is a homeomorphism.
Proof. As in the proof of Theorem 1.1 we have thatfλ(x)=f (x)+λx is a local homeomorphism for allλ >0.
Since(Dfλ)−1∈L∞(Rn), it follows from [14, Lemma 12] that lim inf
x→a
|fλ(x)−fλ(a)|
|x−a| λ
2 >0 (7.1)
for alla∈Rn. By a theorem of John [11, p. 87],fλis a homeomorphism. Sincef is discrete and open by Theorem 1.1, we can apply [14, Proposition 7] and conclude thatf is a homeomorphism. 2
Sharpness of Theorem 5.1 is demonstrated by the following example which combines the ideas from [2] and [13].
Example 7.2.For anyq <1 there exists a sequence of mappingsfj∈W1,3(B,R3)such that sup
j
B
KIq(x, fj)dx <∞ and N
0, fj, B(0,1/4)
→ ∞.
Proof. By a version of Zorich’s construction (see [9,21]) there exists a mapping φ∈ W1,3(R3,R3) such that KI(·, φ)∈L∞(R3),φ is a local homeomorphism outside of R×(2Z+1)2, and φ is 4-periodic in the last two variables. Therefore, it suffices for us to construct biLipschitz homeomorphismsfj:B→R3such that
(i) supj
BKIq(x, fj)dx <∞;
(ii) fj(B)⊂D×R(hereD⊂R2is the unit disc);
(iii) fj(B1/4)contains a line segment{0} × [−L, L] ⊂R2×RwhereL→ ∞asj→ ∞.
The compositionsφ◦fjwill be mappings with large multiplicity.
Fory∈R3lets(y)=
y12+y22. Forα >2 we define a mappingx=g(y)by xi=s(y)α−1yi, i=1,2;
x3=s(y)y3.
Sinces(x)=s(y)α, the inverse mappingy=f (x)outside the set{s(x)=0}is given by yi=s(x)1/α−1xi, i=1,2;
y3=s(x)−1/αx3, s(x)=0.
Let Ω= {x∈R3: s(x) <1, |x3|<1}and Ω=f (Ω). We restrict our attention to y∈Ω, where in particular s(y) <1. Elementary computations show that
Dg(y)Cmax
s(y),|y3| and J (y, g)Cs(y)2(α−1)+1.
Therefore, Dg(y)3
J (y, g) Cs(y)2(1−α)−1max
s(y)3,|y3|3
. (7.2)
Since
Dg(y)3
J (y, g) =KI(x, f ),
inequality (7.2) can be used to estimateKI(x, f )as follows.
KI(x, f )Cs(x)(2(1−α)−1)/αmax
s(x)3/α, s(x)−3/α|x3|3
Cs(x)−(2α+2)/α, where at the last step we used|x3|<1. We achieve
ΩKI(x, f )qdx <∞by choosingαlarge enough so that 2α+2
α q <2.
The mappingf constructed thus far is not inW1,3, and is not even continuous. However, this can be corrected by replacings(y)withsj(y)=
y12+y22+1/j2. The mappingx=gj(y)given by xi=sj(y)α−1yi, i=1,2;
x3=sj(y)y3,
is biLipschitz; we denote the inverse byfj. The computation ofDgjandJ (·, gj)goes through exactly as before and shows that the integral ofKIq(·, fj)is bounded independently ofj. Sincegj(0,0, y3)=(0,0, y3/j ), we have fj(0,0, x3)=(0,0, j x3). Thus, this mappingfj fulfills the requirements (i)–(iii). 2
Acknowledgements
The research was partially carried out during Rajala’s visit to Syracuse University. He wishes to thank the De- partment of Mathematics for its hospitality. Part of this research was done when Kovalev and Onninen were visiting Princeton University. Also, the authors are thankful to the referee for careful reading of the manuscript and several valuable comments.
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