mappings of finite distortion: monotonicity and continuity

Digital Object Identifier (DOI) 10.1007/s002220100130Invent. math. 144, 507–531 (2001)

Mappings of finite distortion:Monotonicity and continuity

Tadeusz Iwaniec1, Pekka Koskela2, Jani Onninen2,

1 Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA(e-mail: [email protected])

2 Department of Mathematics, University of Jyväskylä, P.O. Box 35, Fin-40351 Jyväskylä,Finland (e-mail: pkoskela,[email protected])

Oblatum 15-VI-2000 & 8-XI-2000Published online: 5 March 2001 – Springer-Verlag 2001

1. Introduction

We study mappings f = ( f1, ..., fn) : Ω → Rn in the Sobolev spaceW1,1

loc (Ω,Rn), where Ω is a connected, open subset of Rn with n ≥ 2.Thus, for almost every x ∈ Ω, we can speak of the linear transformationD f(x) : Rn → Rn, called differential of f at x. Its norm is defined by|D f(x)| = sup|D f(x)h| : h ∈ Sn−1. We shall often identify D f(x) withits matrix, and denote by J(x, f ) = det D f(x) the Jacobian determinant.Thus, using the language of differential forms, we can write

J(x, f )dx = d f1 ∧ ... ∧ d fn.

Most of the time the Jacobian determinant will be nonnegative and weshall refer to such mappings as orientation preserving (this need not havea topological interpretation).

Definition 1.1 A Sobolev mapping f ∈ W1,1loc (Ω,Rn) is said to have finite

distortion if there is a measurable function K = K(x) ≥ 1, finite almosteverywhere, such that

|D f(x)|n ≤ K(x)J(x, f ) a.e. (1)

Research of all authors partially supported by the Academy of Finland, project 41933.Iwaniec also by NSF grant DMS-0070807 and Onninen by the Academy of Finland, project60127. This research was done while Iwaniec was visiting at the Mathematics Department ofthe University of Jyväskylä. He wishes to thank the University for the support and hospitality.

Mathematics Subject Classification (2000): 30C65, 35J70

508 T. Iwaniec et al.

We emphasize that mappings of finite distortion are, by definition, sensepreserving. We call (1) the distortion inequality for f . It is worth recallingthat the smallest such function K , refered to as outer dilatation, is definedby

KO(x, f ) = |D f(x)|n

J(x, f )if J(x, f ) = 0

1 if J(x, f ) = 0.(2)

Note that (1) allows us to take KO(x, f ) to be any number at the pointswhere the Jacobian vanishes. Geometrically (2) means that, at the pointswhere J(x, f ) > 0, the differential takes the unit ball to an ellipsoid E andwe have KO(x, f ) = vol BO/vol E, where BO is the ball circumscribedabout E. Similarly, the ratio KI (x, f ) = vol E/vol BI , where BI is the ballinscribed in E, is known as the inner dilatation of f . Precisely, we have

KI (x, f ) = |D f(x)−1|n J(x, f ) if J(x, f ) = 0

1 if J(x, f ) = 0.(3)

There are, of course, other distortion functions of interest but we shallconfine ourselves to only these basic ones. There have been remarkable ad-vances made towards understanding and developing a theory of mappingswith finite distortion. It is the objective of the present paper to give a com-plete account of the continuity properties of mappings with finite distortion.We investigate them under minimal possible assumptions on the degree ofintegrability of the differential. However, the novelty of our results is thatno additional assumptions on the distortion function are made here.

We shall take a little time now to point out some of the advances in thecurrent literature. The origin of mappings with finite distortion can be tracedback to the work of Yu.G. Reshetnyak [28] but our account begins with thepaper of V. Goldshtein and S.K. Vodopyanov [8], in which they showedthat mappings of finite distortion in the Sobolev class W1,n

loc (Ω,Rn) areactually continuous. Most recently, F.W. Gehring and T. Iwaniec [7] verifiedthat the limit mapping f of a weakly convergent sequence of mappingsf j ∈ W1,n(Ω,Rn) with finite distortion K = K(x) also has finite distortion.While substantial progress has been made on the limit theorems, manyquestions still remain unanswered. Let us next discuss some results formappings in W1,n

loc (Ω,Rn) under suitable integrability conditions on thedistortion function. First, T. Iwaniec and V. Sveràk [19] proved that non-constant mappings in W1,2

loc (Ω,R2) with locally integrable distortion areopen and discrete. In higher dimensions, the analog of this holds whenKO ∈ L p

loc(Ω) for some p > n − 1 and f ∈ W1,nloc (Ω,Rn). It fails if

p < n − 1 (see [3]), though it remains unknown in the critical case ofp = n − 1. Let us observe that KI (x, f ) ≤ Kn−1

O (x, f ), a.e. We believethat in all dimensions the nonconstant mappings f ∈ W1,n

loc (Ω,Rn) withKI ∈ L1

loc(Ω) are open and discrete. The initial steps towards solution of

Mappings of finite distortion: Monotonicity and continuity 509

this problem were made by J. Heinonen and P. Koskela [13] while moredefinite answers were given later by J. Manfredi and E. Villamor [22], [23].

The natural Sobolev setting for mappings of finite distortion is in thespace W1,n

loc (Ω,Rn), largely due to the wish to integrate the Jacobian determi-nant by parts. However, matters are quite complicated if one does not knowa priori that the Jacobian is locally integrable or, even if so, whether it coin-cides with the so-called distributional Jacobian. The first regularity resultsbelow the natural setting were recently established by T. Iwaniec, P. Koskelaand G. Martin [15]. Assuming that J(x, f ) ∈ L1

loc(Ω) and eλK ∈ L1loc(Ω) for

some sufficiently large λ = λ(n) they proved, among other things, that infact f ∈ W1,n

loc (Ω,Rn). Also see [1] for further developments. The standingconjecture is that one can take λ = λ(n) = 1 as the critical exponent for theregularity conclusions; it is known that the Ln-integrability of the differen-tial fails for any λ < 1. The relevant examples are homeomorphic maps inW1,1

loc (Ω,Rn) and, therefore, have locally integrable Jacobian determinants.One message from the present paper is that mappings of exponentially

integrable distortion, regardless of the size of λ, always have a continuousrepresentative. More precisely, such mappings coincide almost everywherewith a continuous map. Here and in the sequel, for brevity, we simply saythat the mappings in question are continuous.

Theorem 1.2 Let f ∈ W1,1(B,Rn) satisfy the distortion inequality

|D f(x)|n ≤ K(x)J(x, f ) a.e.

in a ball B = B(0, R), where eλK is integrable for some λ > 0. If theJacobian determinant is integrable, then f is continuous and we have themodulus of continuity estimate

| f(x) − f(y)|n ≤ CK(n, λ)∫

B J(x, f ) dx

log log(

e + R|x−y|

) (4)

for all x, y ∈ B(0, R2 ).

An explicit bound for the constant here is:

CK(n, λ) ≤ C(n) −∫

BeλK(x) dx (5)

where --∫

B stands for the integral average over the ball B. Theorem 1.2is a consequence of our more general results concerning continuity ofmappings (of arbitrary finite distortion) in the Orlicz-Sobolev classesW1,Φ(Ω,Rn). For the definition of these classes see Sect. 2. The pointis that the assumption eλK ∈ L1(Ω), together with the integrability of theJacobian determinant, implies that f ∈ W1,Φ(Ω,Rn) with the Orlicz func-tion Φ(t) = tn

log(e+t) . This is exactly what we need for Theorem 1.2. In this


connection, we should point out that the Jacobian determinant of an orienta-tion preserving mapping in W1,Φ(Ω,Rn) is always locally integrable [18],and coincides with the distributional Jacobian [9]. The summability of non-negative Jacobians is the heart of our ideas. Two categories of Sobolevmappings are well suited for the question of summability of the Jacobian;the Orlicz-Sobolev spaces W1,Φ(Ω,Rn) with∫ ∞

1

Φ(t) dt

tn+1= ∞ (6)

and the so-called grand Sobolev space W1,n)(Ω,Rn). The latter consistsof mappings whose differential belongs to the space VLn(Ω) of vanishingmodulus of integrability. Without getting into technicalities, this means that

limε→0+

ε

∫Ω

|D f(x)|n−ε dx = 0. (7)

See Sect. 2 for details. Let us first consider continuity in the case of thegrand Sobolev space.

Theorem 1.3 Mappings f ∈ W1,n)(B,Rn) of finite distortion in a ballB = B(0, R) satisfy the continuity estimate

| f(x) − f(y)| ≤ C(n)R Ln

(D f ; log−1 R

|x − y|)

(8)

for all x, y ∈ B(0, R10). Here, Ln(D f ; t) stands for the modulus of integra-

bility of D f :

Ln(D f ; t) =[

t −∫

B|D f |n−t

] 1n−t

0 < t ≤ n − 1. (9)

This result is sharp in many ways. To see this, consider the mappingf(x) = x + x

|x| . An elementary computation reveals that |D f(x)| = 1 + 1|x| ,

J(x, f ) =(

1 + 1|x|

)n−1and KI (x, f ) = 1+ 1

|x| . We then see that the differ-

ential belongs to the Marcinkiewicz space weak−Ln(B) and, consequently,has bounded (though not vanishing) modulus of integrability. The reader isreferred to Sect. 3 for the discussion of this example.

Theorem 1.3 covers Orlicz-Sobolev mappings f ∈ W1,Φ(Ω,Rn) as well,but only if

Φ(t) ≥ c tn

log(e + t), (10)

confront this with Proposition 2.1.


This and other continuity results for Orlicz-Sobolev mappings with fi-nite distortion require estimates below the dimension. For an orientationpreserving mapping f = ( f1, ..., fn) in W1,n−ε(B) we have∫

B|D f |−εd f1 ∧ ... ∧ d fn ≤ C(n)ε

∫B|D f(x)|n−ε dx (11)

provided one of the coordinate functions vanishes on ∂B in the sense ofdistributions. This inequality is part of a more general spectrum of inte-gral estimates concerning wedge products of differential forms [18], [11]and [14]. The key is to average (11) with respect to ε to gain estimates thatyield continuity of mappings f ∈ W1,Φ(Ω,Rn) with finite distortion. Thiscan be done for a class of the Orlicz functions Φ satisfying (6). Notice that(6) is necessary because of the example f(x) = x + x

|x| . To illustrate ourresults let us state here some of them.

Theorem 1.4 Mappings f ∈ W1,Φ(B,Rn) with finite distortion are con-tinuous if

Φ(t) ≥ C tn

log(e + t) log log(e + t). (12)

Sharp estimates on the modulus of continuity are also available. For instance,if (12) holds as equality, then

| f(x) − f(y)| ≤ C(n)R ||D f ||Φlog log log

1n

(e + R

|x−y|) . (13)

If Φ(t) = tn logα−1(e + t) and α > 0, then

| f(x) − f(y)| ≤ C(n)R ||D f ||Φlog

αn

(e + R

|x−y|) . (14)

If Φ(t) = tn log−1(e + t), then

| f(x) − f(y)| ≤ C(n)R ||D f ||Φlog log

1n

(e + R

|x−y|) . (15)

Here x, y ∈ B(0, R2 ), where ||D f ||Φ stands for the Luxemburg Φ-norm of

the differential.A very powerful method when dealing with continuity properties of

functions is furnished by the notion of monotonicity, which goes back toH. Lebesgue [20] in 1907. Monotonicity for a continuous function u ina domain Ω ⊂ Rn simply means that

osc(u, B) ≤ osc(u, ∂B) (16)


for every ball B ⊂ Ω. Roughly speaking, u satisfies both the maximumand minimum principles in Ω. Hence the relevance for elliptic PDEs isclear. However, to effectively handle very weak solutions to the differentialinequalities such as (1) one needs to adopt a definition of the so-calledweakly monotone functions, due to J. Manfredi [21].

Definition 1.5 A real valued function u ∈ W1,1(Ω) is said to be weaklymonotone if for every ball B ⊂ Ω and all constants m ≤ M such that

ϕ = (u − M)+ − (m − u)+ ∈ W1,10 (B), (17)

we have

m ≤ u(x) ≤ M

for almost every x ∈ B.

It is worth noting that for a weakly monotone function u in the Orlicz-Sobolev class W1,Φ(Ω) we actually above have ϕ ∈ W1,Φ

0 (Ω). This latterspace stands for the completion of C∞

0 (Ω) in W1,Φ(Ω). Without saying itso every time, we are working under the integral condition (6) on the Orliczfunction Φ. It will also be required that the function τ → Φ( p

√τ) is convex

for some p > n − 1. Note that (6) is equivalent to

∫ 1

0Φ

(1

s

)dsn = ∞ (18)

where dsn = nsn−1ds, of course. To accommodate explicit bounds we definethe Φ-modulus of continuity

ω = ωΦ(t), 0 < t ≤ 1

where ω is determined uniquely from the equation

1 =∫ 1

tΦ

(ω

s

)dsn. (19)

For example, if Φ(t) = tn logα−1(e+t), α > 0, then ωΦ(t) ≈ log− αn(e + 1

t

).

Theorem 1.6 Let u ∈ W1,Φ(B) be weakly monotone in the ball B =B(0, R). Then for all Lebesgue points x, y ∈ B(0, R

2 ) it holds that

|u(x) − u(y)| ≤ C(p, n) R ||∇u||Φ ωΦ

( |x − y|R

). (20)

In particular, u has a continuous representative.


The use of weakly monotone functions is essential for our approach:Theorem 1.4 and a more general version of it will be deduced from Theo-rem 1.6 and the weak monotonicity of mappings of finite distortion in theOrlicz-Sobolev class W1,Φ(Ω,Rn). Also the proof of Theorem 1.3 is basedon similar reasoning. For a discussion of weak monotonicity of mappingsof finite distortion in Orlicz-Sobolev classes see Sect. 4 that also deals withthe question of sharpness.

There are many related works that have not been mentioned above. Firstof all the interesting paper [5] of G. David gives existence theorems for map-pings of exponentially integrable distortion in the plane. These results areobtained using the Beltrami equation. Theorem 1.2 gives new informationeven in this setting. Papers related to David’s result include [4], [26], [29]and [31]. Also see the references in these papers. In higher dimensionsreferences not discussed yet include [12] and [24].

2. Grand and Orlicz spaces

In this section we briefly review some known function spaces that will beused in the sequel. We will give sharp results on the inclusions betweenOrlicz spaces and grand Lebesgue spaces.

Let Ω be an open bounded region in Rn. The average of a functionu ∈ L1(Ω) is denoted by

uΩ = −∫

Ω

u(x)dx.

For p > 1 we shall consider functions

u ∈⋂

1≤s<p

Ls(Ω),

and define their modulus of integrability by

Lp(u; ε) =[ε −

∫Ω

|u|p−ε

] 1p−ε

, 0 < ε ≤ p − 1.

The space BL p(Ω) of functions with bounded modulus of integrability, alsoknown as the grand Lebesgue space L p)(Ω) [18], is furnished with the norm

||u||p) = sup0<ε≤p−1

Lp(u; ε).

BL p(Ω) is a Banach space. It then follows that the closure of L p(Ω) with re-spect to this norm, denoted by VL p(Ω), consists of functions with vanishingmodulus of integrability. Namely

limε→0+

Lp(u; ε) = 0

whenever u ∈ VL p(Ω).


Next recall the Marcinkiewicz space weak − L p(Ω) which consists ofall measurable functions u : Ω → Rn such that

supt>0

|x ∈ Ω : u(x) > t|t p < ∞.

We note that weak − L p(Ω) is contained in BL p(Ω) but not in VL p(Ω),see [18]. The spaces weak − L p(Ω) and VL p(Ω) are not comparable [9].

A continuous and strictly increasing function Φ : [0,∞] → [0,∞]with Φ(0) = 0 and Φ(∞) = ∞, is called an Orlicz function. The Orliczspace LΦ(Ω) is made up of all measurable functions u on Ω such that∫

Ω

Φ

( |u|k

)< ∞

for some positive k = k(u). LΦ(Ω) is equipped with the nonlinear Luxem-burg functional

||u||Φ = infk > 0 : −∫

Ω

Φ

( |u|k

)≤ 1.

If Φ is convex, then || · ||Φ defines a norm in LΦ(Ω). The Orlicz spaceLΘ(Ω) with

Θ(t) = t p logα(e + t),

also denoted by L p logαL(Ω), will play special role in our investigation.In [9] L. Greco examined relations between L p logαL(Ω) and the grandLebesgue spaces. He proved that

L p(Ω) L p log−1L(Ω) VL p(Ω) BL p(Ω) ⋂

α<−1

L p logαL(Ω).

The aim of this section is to improve on this result. The interplay betweenthe Orlicz spaces and BL p(Ω) is both satisfying and indispensable for theforthcoming results. In this regard, we give the following sharp inclusions.

Proposition 2.1 If 1 < p < ∞, then

L p log−1L(Ω) ⊂ VL p(Ω) ⊂ BL p(Ω) ⊂ LΨ(Ω),

for all Orlicz functions Ψ such that the function t → Ψ(t)t−p log t, withlarge t, decreases to zero fast enough to satisfy∫ ∞

1

Ψ(t)

t p+1dt < ∞. (21)

Convergence of this integral is necessary for the latter inclusion. There areno Orlicz spaces between L p log−1L and BL p.


Proof. We will first show that L p log−1L(Ω) is properly contained inVL p(Ω). Let h ∈ L p log−1L(Ω). Fix δ > 0 and a ∈ (1,∞) so big that

log(e + 1)

∫x∈Ω:|h(x)|>a

|h|p dx

log(e + |h|) < δ.

Then, by the elementary inequality (e + t)ε < e + tε, we obtain

ε

∫Ω

|h|p−εdx = ε

∫x∈Ω:h(x)≤a

|h|p−ε + ε

∫x∈Ω:h(x)>a

|h|p−ε

≤ εap−ε|Ω| + ε

∫x∈Ω:h(x)>a

|h|p

log(e + |h|)log(e + |h|)

|h|ε

≤ εap−ε|Ω| +∫

x∈Ω:h(x)>a

|h|p

log(e + |h|)log(e + |h|ε)

|h|ε

≤ εap−ε|Ω| + supt>1

log(e + t)

t

δ

log(e + 1)

≤ εap−ε|Ω| + δ.

Thus

0 ≤ lim supε→0

ε

∫Ω

|h|p−εdx ≤ δ.

Since δ > 0 was arbitrary, we conclude with limε→0

ε∫Ω

|h|p−εdx = 0. This

shows that h ∈ VL p(Ω), as desired. It is not difficult to see that the function

h =(|x|−n log log 1

|x|) 1

p, defined in Ω = x : |x| < 1

3 , belongs to VL p(Ω).

However, this function does not belong to L p log−1L(Ω). Consequently, thefirst inclusion in Proposition 2.1 is proper.

We shall now show that

LΦ(Ω) ⊂ BL p(Ω)

for any Orlicz function Φ(t) = H(t) t p

log t such that lim inft→∞ H(t) = 0. For this,

we must construct a function f ∈ LΦ(Ω) which is not in VL p(Ω).Since lim inf

t→∞ H(t) = 0, one can find k0 ∈ N and numbers tk > e such

that H(tk) = 2−k, for k ≥ k0. Clearly, there is a subsequence k j∞j=1 for

which∞∑j=1

log tk j

t p−1k j

≤ |Ω|.

For instance, choose tk j large enough to satisfy t1−pk j

log tk j ≤ 2− j |Ω|.


To each tk j we associate the number

aj = mink j, tk j .It is clear that there exist mutually disjoint measurable subsets E j ⊂ Ω suchthat

|E j | = aj t−pk j

log tk j .

We now define the function f by the rule

f(x) =∞∑j=1

tk j χE j (x), for all x ∈ Ω.

This yields ∫Ω

Φ(| f |)dx =∫

Ω

| f |p H(| f |) dx

log | f |

=∞∑j=1

|E j |t pk j

H(tk j )

log tk j

=∞∑j=1

aj H(tk j ) ≤∞∑j=1

k j

2k j≤ 2.

On the other hand

lim supε→0

ε

∫Ω

| f |p−εdx ≥ limj→∞

p

log tk j

∫E j

| f |p− plog tk j

= limj→∞

p|E j |log tk j

[tk j

]p− plog tk j

= limj→∞ paj e

−p = ∞.

Hence f /∈ BL p(Ω), as required.

Next suppose that an Orlicz function Ψ satisfies the conditions stated inProposition 2.1. For notational convenience we write F(t) = Ψ(t)t−p log t.Thus ∫ ∞

e

F(t) dt

t log t≤

∫ ∞

1

Ψ(t)

t p+1< ∞.

We notice, by using the Integral Test, that for sufficiently large N∞∑

k=N

F(eek) ≤

∫ ∞

0F

(eex )

dx =∫ ∞

e

F(t) dt

t log t< ∞.

Now, consider an arbitrary u ∈ BL p(Ω) and its level sets

Ωk = x ∈ Ω : eek ≤ |u(x)| < eek+1

(k = 0, 1, 2, ...).

It involves no loss of generality in assuming that ||u||p) = 1.


For every x ∈ Ωk, we have e ≤ |u(x)|e−k< ee. Therefore, for k ≥ N,

we can write ∫Ωk

Ψ(u) =∫

Ωk

u p F(u)

log u≤

∫Ωk

u p F(eek)

ek

since F(u) is decreasing for u ≥ eeN.

By the definition of the grand L p-norm we obtain∫Ωk

Ψ(u)dx ≤ ee F(eek)

e−k∫

Ωk

u p−e−k

≤ ee ||u||p−e−k

p) F(eek) = ee F

(eek)

.

Summing up we conclude with the desired estimate∫|u|≥eeN

Ψ(u) ≤ ee∫ ∞

e

F(t) dt

t log t< ∞

which shows that u ∈ LΨ(Ω).In order to see that condition (21) is necessary, we consider the function

u(x) = |x|− np in the unit ball Ω = x : |x| ≤ 1. Note that

ε −∫

Ω

|u(x)|p−ε dx = p.

Hence u belongs to BL p(Ω). Suppose now that u ∈ LΨ(Ω) for some Orliczfunction Ψ such that

∫ ∞1

Ψ(t)t p+1 dt = ∞. This means that

∫Ω

Ψ(

uk

)< ∞ for

some positive k. An elementary computation reveals that∫ ∞

1k

Ψ(t)

t p+1dt = kp

p−∫

Ω

Ψ(u

k

)dx < ∞

which gives a contradiction.We end this section with two more definitions. The completion of the

Sobolev space W1,p(Ω) under the norm

|| · ||p + ||∇ · ||p)

will be called the grand Sobolev space and denoted by W1,p)(Ω). Thus

limε→0

ε

∫Ω

|∇u|p−ε = 0

whenever u ∈ W1,p)(Ω). We do not introduce the space of functions with|∇u| in BL p(Ω) as the need will not arise. The Orlicz-Sobolev spaceW1,Φ(Ω) consists of function u ∈ W1,1(Ω) such that ∇u ∈ LΦ(Ω).


3. Example

Mappings with finite distortion in the Sobolev space W1,n(Ω,Rn) are knownto be continuous. At this point it is worthwhile to recall the familiar exampleof a map that forms a cavity at the origin [2]

f : B \ 0 → Rn : x → x + x

|x|where B denotes the unit ball in Rn. This example indicates that, in order topreserve continuity, one should not go too far from W1,n(Ω).

Clearly, the function f cannot be redefined in a set of measure zero tobecome continuous at the origin. By an easy calculation we see that

D f(x) =(

1 + 1

|x|)

I − x ⊗ x

|x|3 ,

where x ⊗ x is the n × n matrix whose i, j-entry equals xi x j . Hence

|D f(x)| = 1 + 1

|x|and

det D f(x) =(

1 + 1

|x|)n−1

.

Thus the outer dilatation function KO(x) = 1+ 1|x| belongs to weak−Ln(B).

We see at once that |D f(x)| ∈ BLn(B) and

limε→0

ε −∫B

|x|n−ε = 1.

Thus f /∈ W1,n)(B,Rn). Moreover,

−∫B

Φ

(1

|x|)

dx =∫ ∞

1

Φ(t)

tn+1dt.

Hence |D f(x)| ∈ LΦ(B,R) if and only if the defining Orlicz fuction Φsatisfies ∫ ∞

1

Φ(t)

tn+1dt < ∞. (22)

This example suggests that, in order to conclude with any sort of continuityestimate for a mapping f of finite distortion, we must assume that f ∈W1,n)(Ω,Rn) or f ∈ W1,Φ(Ω,Rn), with

∫ ∞1

Φ(t)tn+1 dt = ∞.


4. Monotonicity in W1,n(Ω,Rn)

Unlike in the classical approach, we can easily prove without gettinginto PDEs that the coordinate functions of the map f = ( f1, ..., fn) ∈W1,n(Ω,Rn) of finite distortion are weakly monotone. It is rewarding andilluminating to outline this simple proof.

Let B be a ball in Ω and suppose that for some coordinate function, saythe first one, we have

v := ( f1 − M)+ − (m − f1)+ ∈ W1,n

0 (B).

Then

∇v(x) =

0 if m ≤ f 1(x) ≤ M∇ f1(x), otherwise (say, on the set E ⊂ B).

Hence, in view of the distortion inequality (1), we can write∫Ω

|∇v(x)|nK(x)

dx ≤∫

E

|D f(x)|nK(x)

dx

≤∫

Ed f1 ∧ d f2 ∧ ... ∧ d fn

≤∫

Bdv ∧ d f2 ∧ ... ∧ d fn = 0,

by Stokes’ Theorem. Thus v ≡ 0 on B, which simply means that m ≤f1(x) ≤ M for almost every x ∈ B, as desired.

Similar considerations apply to the John Ball class of mappings:

|D f | ∈ Ln−1(Ω) and |Adj D f | ∈ Ln

n−1 (Ω) (23)

which has been studied by several people in nonlinear elasticity and calculusof variations [30], [25], [16], [13] and [12].

A slight change in the above proof leads to a result that is known tospecialists in nonlinear elasticity.

Proposition 4.1 Under the integrability hypotheses stated at (23), map-pings of finite distortion are weakly monotone.

The key here is the isoperimetric type inequality for mappings in the JohnBall class:∣∣∣∣ −

∫B

det D f(x)dx

∣∣∣∣ ≤ C(n)

n∏k=1

(−∫

∂B

∣∣d f1 ∧ ... ∧ d fk ∧ ... ∧ d fn

∣∣ nn−1

) 1n

(24)


for almost every ball B ⊂ Ω centered at the given point of Ω. This isa consequence of the estimate∣∣∣∣ −

∫B

det D f(x)dx

∣∣∣∣ ≤ C(n) −∫

∂B|Adj D f(x)| n

n−1 dx (25)

from [25] by the usual analysis of homogeneity. Here are some details. Firstwe observe the point-wise inequality

|Adj D f(x)| nn−1 ≤

n∑k=1

∣∣d f1 ∧ ... ∧ d fk ∧ ... ∧ d fn

∣∣ nn−1 . (26)

Fix any positive integer l. For k = 1, 2, ..., n we define positive numbers λkby the rule

λn

n−1k =

1l + --

∫∂B

∣∣d f1 ∧ ... ∧ d fk ∧ ... ∧ d fn

∣∣ nn−1

∏nj=1

[1l + --

∫∂B

∣∣d f1 ∧ ... ∧ d f j ∧ ... ∧ d fn

∣∣ nn−1

] 1n

.

Clearly λ1···λn = 1 and we may apply (25) to the mappings (λ1 f1, ..., λn fn)in place of ( f1, ..., fn), to obtain

∣∣∣∣ −∫

Bdet D f(x)dx

∣∣∣∣ ≤ C(n) −∫

∂B

n∑k=1

λn

1−nk

∣∣d f1 ∧ ... ∧ d fk ∧ ... ∧ d fn

∣∣ nn−1

≤ C(n)

n∑k=1

λn

1−nk

[1

l+ −

∫∂B

∣∣d f1 ∧ ... ∧ d fk ∧ ... ∧ d fn

∣∣ nn−1

]

= nC(n)

n∏j=1

[1

l+ −

∫∂B

∣∣d f1 ∧ ... ∧ d f j ∧ ... ∧ d fn

∣∣ nn−1

] 1n

.

We should remark here that the radii of the balls for which these inequalitieshold may depend on the mappings (λ1 f1, ..., λn fn). Since we are dealingonly with a countable number of such mappings, the above inequalities holdfor almost every ball B ⊂ Ω centered at the given point of Ω, and with everyl = 1, 2, ... The inequality (24) is established by letting l tend to infinity.

As before, we find that∫

B dv ∧ d f2 ∧ ... ∧ d fn = 0, by applying theisoperimetric inequality at (24) instead of Stokes’ Theorem, details beingleft to the reader.

Matters are quite different if f /∈ W1,n(Ω,Rn) because one does notknow, a priori, that the Jacobian determinant is locally integrable. First wetake on stage the mappings of finite distortion in the grand Sobolev spaceW1,n)(Ω,Rn).


5. Monotonicity in W1,n)(Ω,Rn)

Essential to our development is to establish inequality (11).

Lemma 5.1 Let f : B → Rn, be a mapping of Sobolev class W1,n−ε (B,Rn),0 ≤ ε ≤ 1, in a ball B. Then∫

B|D f(x)|−ε J(x, f ) dx ≤ C(n)ε

∫B|D f(x)|n−εdx

provided one of the coordinate functions for f = ( f1, ..., fn) belongs toW1,n−ε

0 (B).

This inequality is part of a more general spectrum of integral estimatesconcerning wedge products of differential forms [14] and [18].

Proof. We may certainly assume that 0 ≤ ε ≤ 12 , since the inequality

always holds with constant 1 in place of C(n)ε. Suppose f1 ∈ W1,n−ε0 (B).

As in [18] we begin with the Hodge decomposition

|d f1|−ε d f1 = dϕ + γ

where ϕ ∈ W1, n−ε

1−ε

0 (B) and γ ∈ Ln−ε1−ε (B). It is important to realize that γ

becomes small as ε tends to zero. Precisely, we have

||γ ||L

n−ε1−ε (B)

≤ C(n)ε ||d f1||1−εLn−ε(B)

≤ C(n)ε ||D f ||1−εLn−ε(B)

see [18].By Stokes’ Theorem we find that∫

B|d f1|−ε d f1 ∧ ... ∧ d fn =

∫Bγ ∧ d f2... ∧ d fn ≤

∫B|γ ||D f |n−1

≤ ||γ || n−ε1−ε

||D f ||n−1n−ε ≤ C(n)ε

∫B|D f(x)|n−εdx.

It remains to observe that

|D f |−ε J(x, f ) dx = |d f1|−εd f1 ∧ ... ∧ d fn

− (|d f1|−ε − |D f |−ε)

d f1 ∧ ... ∧ d fn

≤ |d f1|−εd f1 ∧ ... ∧ d fn

+( |D f |ε

|d f1|ε − 1

)|d f1| |D f |n−1−ε

≤ |d f1|−εd f1 ∧ ... ∧ d fn + ε |D f |n−ε

which follows from the elementary inequality λε − 1 ≤ ελ, for λ ≥ 1. Thelemma is proved.


Now, we are ready to prove the following extension of V. Goldstein,Yu.G. Reshetnyak and S.K. Vodop’yanov result [27] and [8]; Ln -integrabilityof the differential being relaxed.

Proposition 5.2 Coordinate functions of mappings with finite distortion inthe grand Sobolev space W1,n)(Ω,Rn) are weakly monotone.

Let us stress explicitly that this result covers mappings for which∫Ω

|D f(x)|nlog(e + |D f(x)|)dx < ∞ (27)

as it is easy to see from Proposition 2.1.

Proof. By symmetry, it suffices to prove that f1 is weakly monotone.Let B be a ball in Ω and m ≤ M be real numbers such that v =

( f1 − M)+ − (m − f1)+ ∈ W1,n−ε

0 (B) for every ε ∈ (0, 1). We have to showthat

m ≤ f1(x) ≤ M

for almost every x ∈ B.As in Sect. 4, we calculate

dv =

0 if m ≤ f 1 ≤ Md f1 otherwise (say, on a set E ⊂ B).

Applying Lemma 5.1 to the mapping (v, f2, ..., fn) : B → Rn, we obtain∫E

|D f(x)|n−ε

K(x)dx ≤

∫E

|D f(x)|−ε J( f, x) dx

≤ C(n)ε

∫B|D f(x)|n−ε dx.

Therefore, since ε∫

B |D f |n−εdx → 0, as ε → 0 and 1 ≤ K(x) < ∞ foralmost every x, by the Monotone Convergence Theorem we infer that D fvanishes almost everywhere in E. This implies that v = 0, completing theproof of Proposition 5.2.

6. Monotonicity in W1,P(Ω,Rn)

The Jacobian determinants of orientation preserving mappings inW1,P(Ω,Rn) are known to be locally integrable for a considerable classof Orlicz functions P(t) = o(tn). What we actually need, to prove mono-tonicity, is the identity ∫

Bd f1 ∧ ... ∧ d fn = 0 (28)


for f = ( f1, ..., fn) ∈ W1,P(B,Rn), with one coordinate function inW1,P

0 (B). Since we assume that J(x, f ) ≥ 0, identity (28) simply meansthat d f1 ∧ ... ∧ d fn = J(x, f ) dx ≡ 0 in B. The affirmative answer to thisquestion follows from Lemma 5.1 and it is our goal here to show how.

Consider a nonnegative decreasing function ϕ ∈ C1(0, 1] such thatlim

s→0+ ϕ(s) = ∞. Having disposed of Lemma 5.1 we can write, as a starting

point, the inequality∫B

−J(x, f )

ϕ(ε)

(∫ 1

ε

|D f(x)|−s dϕ(s)

)dx

≤ C(n)

∫B

−|D f(x)|nϕ(ε)

(∫ 1

ε

s |D f(x)|−s dϕ(s)

)dx (29)

for all 0 < ε ≤ 1. We want to pass to the limit as ε → 0. By the FatouLemma and L’Hôpital’s Rule this procedure is perfectly valid for the lefthand side, yielding the estimate∫

BJ(x, f ) dx ≤ C(n) lim

ε→0

∫B

−|D f(x)|nϕ(ε)

(∫ 1

ε

s |D f(x)|−s dϕ(s)

)dx.

(30)

The latter limit is equal to zero once we can use the Lebesgue DominatedConvergence Theorem. Indeed, by L’Hôpital’s Rule we would obtain∫

BJ(x, f ) dx ≤ C(n)

∫B

limε→0

εϕ′(ε)ϕ′(ε)

|D f(x)|n−ε dx = 0. (31)

We are, therefore, left with the task of justifying the use of the LebesgueDominated Convergence Theorem. To this effect, we only need to show thatthere is a function M ∈ L1(B) such that

− 1

ϕ(ε)

∫ 1

ε

s|D f(x)|n−s ds ≤ M(x) (32)

for sufficiently small ε > 0. This is certainly true if f ∈ W1,P(B), wherethe defining Orlicz function P satisfies

P(t) ≥ CΦ(t), C > 0, t ≥ 0 (33)

Φ(t) = sup0<ε<1

− 1

ϕ(ε)

∫ 1

ε

stn−s dϕ(s) (34)

for some ϕ as above. Clearly Φ is increasing and convex. The questionarises as to which Orlicz functions can be obtained by formula (34). Onenecessary condition on Φ is that∫ ∞

1

Φ(t) dt

tn+1= ∞. (35)


Indeed, for all 0 < ε ≤ 1 we have

− 1

ϕ(ε)

∫ 1

ε

s dϕ(s)

t1+s≤ Φ(t)

tn+1.

Fixing an arbitrary N ≥ 1 and integrating this inequality with respect to tfrom N to ∞ we arrive at

− 1

ϕ(ε)

∫ 1

ε

dϕ(s)

Ns≤

∫ ∞

N

Φ(t) dt

tn+1.

For every 0 < δ < 1 we can write

ϕ(ε) − ϕ(δ)

Nδϕ(ε)≤ − 1

ϕ(ε)

∫ δ

ε

dϕ(s)

Ns≤

∫ ∞

N

Φ(t) dt

tn+1

provided 0 < ε ≤ δ. Letting ε go to zero we obtain

1

Nδ≤

∫ ∞

N

Φ(t) dt

tn+1.

Since δ can be arbitrarily small this estimate yields

1 ≤∫ ∞

N

Φ(t) dt

tn+1

for all N ≥ 1, which is possible only when integral at (35) diverges.We should mention at this point that, modulo some minor technical

proviso, the divergence condition∫ ∞

1P(t)tn+1 dt = ∞ is sufficient to achieve

the lower bound at (33), see Appendix in [17]. The reader is also referredto some earlier and related results in [10]. We shall not enter into thesequite involved computations. Instead, we confine ourselves to the followingresult.

Theorem 6.1 The coordinate functions of a mapping f ∈ W1,P(Ω,Rn) offinite distortion, with P satisfying (33), are weakly monotone.

It is rewarding to make an explicit calculation for ϕ(s) = log es , 0 < s ≤ 1:

Φ(t) = sup0<ε<1

log−1 e

ε

∫ 1

ε

tn−s ds

≤ sup0<ε<1

tn(tε log e

ε

)log t

≤ tn

log t log log t.

Here we have assumed that t > e, to claim the inequality

tε loge

ε≥ log log t (36)


for all 0 < ε ≤ 1. Indeed, the function ε → tε log eε

has exactly onecritical point in the interval (0, 1]. This point, denoted by ε0, satisfies theequation ε0 log e

ε0= log−1 t. In particular, ε0 ≤ log−1 t and we obtain

tε log eε

≥ tε0 log eε0

≥ tε0 log(e log t) ≥ log log t. This results in the follow-ing preliminary step for the proof of Theorem 1.4.

Corollary 6.2 The coordinate functions of a mapping f ∈ W1,P(Ω,Rn) offinite distortion, with P(t) ≥ C tn

log(e+t) log log(e+t) , are weakly monotone.

7. The oscillation lemma

There is a particularly elegant geometric approach to the continuity esti-mate of monotone functions. The idea goes back to the oscillation lemmaby F.W. Gehring [6]. While many interesting implications of Gehring’slemma have been discussed in the literature, the fact that one can use itfor weakly monotone functions seems to be less familiar. It is surprisingthat the usual convolution procedure with mollifiers of Dirac distributionhas little effect on the monotonicity of functions. Consequently, we takethe time here to state and give a rigorous proof of this fact, as it might beof independent interest. Let u ∈ W1,p(B) be a Sobolev function in a ballB = B(a, R). Fix a nonnegative χ ∈ C∞

0 (B) supported in the unit ball suchthat

∫B χ(y)dy = 1. The mollifiers χ j(y) = jnχ( jy), j = 1, ..., give rise to

the sequence u j ∈ C∞(Rn) defined by

u j(x) = χ j ∗ u(x) =∫

Bu(y)χ j(x − y) dy. (37)

It is well known that u j converges to u in W1,ploc (B) and u j(x0) → u(x0),

u j(y0) → u(y0) at the Lebesgue points x0, y0 ∈ B. Here is the precisestatement concerning monotonicity.

Lemma 7.1 Let u ∈ W1,p(B) be weakly monotone in a ball B = B(a, R)and x0, y0 be Lebesgue points in B(a, r), r < R. For each δ > 0 there existsN such that

|u j(x0) − u j(y0)|p ≤ osc(u j , ∂B(a, t)) + 2δ (38)

for all j ≥ N and every t ∈ [r, R].Proof. We claim that the estimates

u j(x0) ≤ maxu j(x) : x ∈ ∂B(a, t) + δ (39)

and

u j(y0) ≥ minu j(y) : y ∈ ∂B(a, t) − δ (40)


hold for all r ≤ t ≤ R, whenever j is sufficiently large. We only needto show the first inequality. The second inequality follows by applying thefirst one to the function −u and to the point y0 in place of x0. Suppose,to the contrary, that there exists a sequence jk and radii r ≤ tk ≤ R,k = 1, 2, 3, ..., such that

u jk(x0) > maxu jk(x) : x ∈ ∂B(a, tk) + δ.

We may assume that tk converges to some number t ∈ [r, R]. Since

u jk(x) − u jk(x0) + δ < 0

on ∂B(a, tk), we have

(u jk − u jk(x0) + δ)+ ∈ W1,p0 (B(a, tk)).

Passing to the limit as k → ∞ yields

(u − u(x0) + δ)+ ∈ W1,p0 (B(a, t)).

That this function vanishes on ∂B(a, t) in the sense of distributions can beseen in various ways; details are left to the reader. As u is weakly monotoneit follows that u(x) ≤ u(x0) − δ for almost every x ∈ B(a, t). But this isimpossible since x0 is a Lebesgue point of u in B(a, r) ⊂ B(a, t).

Now inequalities (39) and (40) imply

u j(x0) − u j(y0) ≤ osc(u j, ∂B(a, t)) + 2δ.

One may interchange x0 with y0 to conclude with inequality (38), completingthe proof of Lemma (7.1).

By Fubini’s theorem we observe that the function t → ∫∂B(a,t) |∇u|p

belongs to L1loc(0, R). Consequently, its Lebesgue points form a set of full

linear measure on the interval (0, R). With these preliminaries, we can nowprove the following variant of the oscillation lemma.

Lemma 7.2 Let u ∈ W1,p(B), n − 1 < p < n, be weakly monotone ina ball B = B(a, R) and x0, y0 be Lebesgue points of u in B(a, r), r < R.Then

|u(x0) − u(y0)| ≤ C(p, n)t

(−∫

∂B(a,t)|∇u|p

) 1p

(41)

for almost every t ∈ (r, R).


Proof. We apply Sobolev’s inequality on spheres for the function u j ∈C∞(B) at (38):

|u j(x0) − u j(y0)| ≤ C(p, n)t

(−∫

∂B(a,t)|∇u j |p

) 1p

+ 2δ

for all r ≤ t ≤ R. Fix a Lebesgue point r < t0 < R of the functiont → ∫

∂B(a,t) |∇u|p. For sufficiently small ε > 0, upon integration over theinterval t0 − ε < t < t0 + ε, we obtain∫ t0+ε

t0−ε

( |u j(x0) − u j(y0)| − 2δ

C(p, n)t

)p

ωn−1tn−1 dt ≤∫

t0−ε≤|x|≤t0+ε

|∇u j |p.

Now we can pass to the limit as j → ∞. It is also legitimate to take δ = 0in this limit inequality∫ t0+ε

t0−ε

|u(x0) − u(y0)|p

C p(p, n)t pωn−1tn−1 dt ≤

∫ t0+ε

t0−ε

(∫∂B(a,t)

|∇u|p

).

Divide by 2ε and let ε go to zero to obtain

|u(x0) − u(y0)|pωn−1tn−1

C p(p, n)t p0

dt ≤∫

∂B(a,t0)|∇u|p.

This means that

|u(x0) − u(y0)| ≤ C(p, n)t0

(−∫

∂B(a,t0)|∇u|p

) 1p

as desired.

8. Modulus of continuity

Here we prove Theorem 1.6. Recall that we are working under the conditionthat the function τ → Φ( p

√τ) is convex for some p > n − 1. Thus, Φ is

also convex and Φ(s) ≥ c · sp for large s. In particular, u ∈ W1,p(B) onthe ball B = B(0, R). Because of homogeneity at (20) we may assume that||∇u||Φ = 2−n , that is

1 = −∫

B(0,R)

Φ(2n|∇u|) ≥ 2n −

∫B(0,R)

Φ(|∇u|) (42)

since Φ(Kτ) ≥ KΦ(τ) for K ≥ 1, as Φ is convex and vanishes at zero.Fix Lebesgue points x0, y0 ∈ B(0, 1

2 R) and consider the concentric ballsB(a, r) ⊂ B(a, 1

2 R) ⊂ B(0, R), where a = x0+y02 and r = |x0−y0|

2 . By theLemma 7.2 we have

|u(x0) − u(y0)|C(p, n)t

≤(

−∫

∂B(a,t)|∇u|p

) 1p


for almost all r < t < 12 R. Jensen’s inequality applied to the convex function

τ → Φ( p√

τ) yields

Φ

( |u(x0) − u(y0)|C(p, n)t

)≤ −

∫∂B(a,t)

Φ(|∇u|).

We multiply by ωn−1tn−1, and integrate over the interval (r, 12 R) to obtain

ωn−1

∫ 12 R

rΦ

( |u(x0) − u(y0)|C(p, n)t

)tn−1 dt ≤

∫B(a, 1

2 R)

Φ(|∇u|) (43)

≤∫

B(0,R)

Φ(|∇u|).

Next we make the substitution t = sR2 with 2r

R < s < 1, and arrive at∫ 1

2rR

Φ

(2|u(x0) − u(y0)|

C(p, n)sR

)dsn ≤ 2n −

∫B(0,R)

Φ(|∇u|) ≤ 1. (44)

By the definition of the Φ-modulus of continuity this means that

|u(x0) − u(y0)| ≤ C(p, n)R

2ωΦ

(2r

R

)

= 2n−1C(p, n)||∇u||Φ ωΦ

( |x0 − y0|R

)completing the proof of Theorem 1.6.

Drawing on the notation developed above, we give the following formu-las.

Examples 8.1

Φ1(t) = tn logα−1(1 + t), α > 0 ωΦ1(t) =[

logα

(e + 1

t

)]− 1n

Φ2(t) = tn

log(1 + t)ωΦ2(t) =

[log log

(e + 1

t

)]− 1n

Φ3(t) = tn

log(1 + t) log log(e + t)ωΦ3(t) =

[log log log

(e + 1

t

)]− 1n

.

Elementary computation of these formulas are left for the reader.Similar considerations for weakly monotone functions apply when the

gradient belongs to BLn(B). Recall the modulus of integrability

Ln(∇u; ε) =(

ε −∫

Ω

|∇u|n−ε

) 1n−ε

, 0 < ε ≤ n − 1.


On the other hand using (43) for the function Φ(τ) = τn−ε with 0 < ε ≤ 45 ,

we obtain

|u(x0) − u(y0)| ≤ C(n) R[1 − (

2rR

)ε] 1n−ε

(ε −

∫B|∇u|n−ε

) 1n−ε

.

We choose ε = log−1 R2r , which is legitimate if R ≥ 10r. Recall that

2r = |x0 − y0|, while the ball B(a, r) centered at a = 12 (x0 + y0) must lay

in B(0, 12 R). This certainly holds if x0, y0 ∈ B(0, 1

10 R). We then concludewith the following continuity estimate.

Proposition 8.2 Suppose u is a weakly monotone function in the grandSobolev space W1,n)(B) on a ball B = B(0, R) ⊂ Rn. Then for all Lebesguepoints x0, y0 ∈ B(a, 1

10 R) we have

|u(x0) − u(y0)| ≤ C(n) R Ln

(∇u; log−1 R

|x0 − y0|)

. (45)

Thus u has a continuous representative.

Remark. Observe that estimate (45) remains valid if |∇u| ∈ BLn(B), inwhich case we conclude that u ∈ L∞

loc(B) and obtain the uniform bound

ess osc

(u; 1

10B

)≤ C(n) R ||∇u||BLn(B). (46)

9. Final arguments

Coming to an end, we are in a position to complete the proofs of the resultsstated in Introduction.

Theorem 1.6 was completely established in Sect. 8. Theorem 1.3 followsby combining Proposition 5.2 and 8.2. Concerning Theorem 1.4, we seefrom Corollary 6.2 that the coordinate functions of f are weakly monotone.Then Theorem 1.6 provides us with the modulus of continuity estimates.According to Examples 8.1 these estimates take the form (13), (14) and(15), as desired.

It remains to complete the proof of Theorem 1.2. First we observe thatf belongs to W1,Θ(B) with Θ(t) = tn log−1(e + t), and we have

||D f ||nΘ ≤ C(n) −∫

BJ(x, f ) dx −

∫B

eλK(x) dx. (47)

Indeed,

||D f ||nΘ ≤ nn|| |D f |n||L log−1 L ≤ nn||K J(x, f )||L log−1 L

≤ Cλnn

(−∫

BJ(x, f ) dx

) (−∫

BeλK(x) dx

).


Finally, we use Theorem 1.4 and estimate (15) to conclude with the inequal-ity

| f(x) − f(y)|n ≤ C(n)Rn||D f ||nΘlog log

(e + R

|x−y|)

≤ CK (n, λ)∫

B J(x, f ) dx

log log(

e + R|x−y|

) .

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mappings of finite distortion: monotonicity and continuity

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