Mappings of finite distortion: the degree of regularity

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Advances in Mathematics 190 (2005) 300–318

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Mappings of finite distortion: the degree of regularity Daniel Faraco,a,
Pekka Koskela,b and Xiao Zhongb b

a Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, Leipzig D-04103, Germany Department of Mathematics and Statistics, University of Jyva¨skyla¨, P.O. Box 35, FIN-40014, Finland

Received 9 May 2003; accepted 17 December 2003 Communicated by N.G. Makarov

Abstract This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let KðxÞX1 be a measurable function defined on a domain OCRn ; nX2; and such that expðbKðxÞÞAL1loc ðOÞ; b40: We show that there exist two universal constants 1;1 ðO; Rn Þ with c1 ðnÞ; c2 ðnÞ with the following property: Let f be a mapping in Wloc n jDf ðxÞj pKðxÞJðx; f Þ for a.e. xAO and such that the Jacobian determinant Jðx; f Þ is locally in L1 logc1 ðnÞb L: Then automatically Jðx; f Þ is locally in L1 logc2 ðnÞb LðOÞ: This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings. r 2004 Elsevier Inc. All rights reserved. MSC: 30C65; 26B10; 73C50

1. Introduction Let us begin with the definition. We assume that OCRn is a connected open set. We say that a mapping f : O-Rn has finite distortion if: (FD-1) f AW 1;1 ðO; Rn Þ: loc (FD-2) The Jacobian determinant Jðx; f Þ of f is locally integrable.


Corresponding author. E-mail addresses: [email protected] (D. Faraco), [email protected].fi (P. Koskela), zhong@ maths.jyu.fi (X. Zhong). 0001-8708/$ - see front matter r 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.aim.2003.12.009

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(FD-3) There is a measurable function K ¼ KðxÞX1; finite almost everywhere, such that f satisfies the distortion inequality jDf ðxÞjn pKðxÞJðx; f Þ a:e: Above, jDf ðxÞj is the operator norm of the differential matrix Df ðxÞ and Jf ðxÞ ¼ det Df ðxÞ is the pointwise Jacobian determinant of f : Observe that continuity of f is not required in the definition. However, under suitable integrability conditions on the distortion function K; or on jDf j; a mapping of finite distortion has a continuous representative [9,18,25]. This holds, in particular, when K is exponentially integrable, which is the setting of this paper. We arrive at the usual definition of a mapping of bounded distortion (a quasiregular mapping) when we require above that KALN ðOÞ: The theory of quasiregular mappings is a central topic in modern analysis with important connections to a variety of topics such as elliptic partial differential equations, complex dynamics, differential geometry and calculus of variations, and is by now well understood, see the monographs [32] by Reshetnyak, [33] by Rickman, and [20] by Iwaniec and Martin. A remarkable feature of quasiregular mappings is the self-improving regularity. In 1;n this case, condition (FD-2) assures that f AWloc ðO; Rn Þ: This is the natural regularity assumption for quasiregular mappings. Gehring [8] showed that the differential of a quasiconformal mapping (homeomorphic quasiregular mapping) of a domain OCRn is locally integrable with a power strictly greater than n; and proved the celebrated Gehring’s Lemma. This fundamental result not only extended an earlier result of Bojarski [4] but also opened up the most direct way to the analytic foundation of quasiregular mappings in Rn : A bit later, Elcrat and Meyers [6] showed that Gehring’s ideas can be further exploited to treat quasiregular mappings and partial differential systems, see also [30]. The higher integrability result admits a dual version. In two remarkable papers, Iwaniec and Martin [19] (for even dimensions) and Iwaniec [14] (for all dimensions) proved that there is an exponent qðn; KÞon such that quasiregular mappings a priori 1;q 1;n in Wloc ðOÞ with q4qðn; KÞ belong to the natural Sobolev space Wloc ðOÞ: The fundamentals of the theory of mappings of finite distortion have been recently established in a sequence of papers [3,12,16–18,23–26]. For earlier developments see e.g. [9,22,11,29]. The monograph [20] contains discussions on many basic properties of these mappings as well as connections to other topics. These recent works have established a rich theory of mappings of finite distortion under relaxed conditions on the distortion function K that do not require K to be bounded. Namely, it has been proven under the assumption of exponential integrability of K that f is continuous, sense preserving, either constant or both open and discrete, and maps sets of measure zero to sets of measure zero. The motivation for relaxing the boundedness of the distortion function partially arises from non-linear elasticity. See the paper [31] by Mu¨ller and Spector for a nice introduction to the mappings arising in that theory.

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In this paper, we consider the regularity of mappings with exponentially integrable finite distortion. It was noticed in [16,17] that if the distortion function KðxÞ satisfies expðbKðxÞÞAL1loc for some b40; the distortion inequality (FD-3) implies that the differential of our mapping of finite distortion is p-integrable for all pon and, in fact, jDf jn AL1loc : logðe þ jDf jÞ

ð1:1Þ

1;1 ðO; Rn Þ that satisfies both On the other hand, the Jacobian of each mapping f AWloc (1.1) and Jðx; f ÞX0 a.e. is locally integrable by results of Iwaniec and Sbordone [21]. Thus, the natural regularity assumption for mappings with exponentially integrable finite distortion is (1.1). First, we consider the higher regularity of mappings with exponentially integrable finite distortion. As mentioned in [20], one cannot expect quite the same sort of results as we have attained for quasiregular mappings. In this setting, we must be satisfied with only a very slight degree of improved regularity. The results in [3,16,17] showed that the scale of improved degree is logarithmic. More precisely, it was shown in [3] ðn ¼ 2Þ; [16] (n is even), [17] (all dimensions n) that (1.1) can be improved on: for each dimension nX2 and all aX0 there exists ba ðnÞX1 such that if f : O-Rn has finite distortion and the distortion function KðxÞ of f satisfies Z expðbKðxÞÞ dxoN O

for some bXba ðnÞ; then jDf ðxÞjn loga ðe þ jDf jÞAL1loc ðOÞ: However, the techniques developed in these papers do not seem to work in the general case (with arbitrary b40). Thus, this remained as a challenging and interesting open problem. As pointed out in [20], different ideas are needed to treat this situation. Our first theorem not only shows the higher integrability of the differential of a mapping with exponentially integrable distortion for any b40 but it also indicates exactly how the degree of improved regularity depends on b: Theorem 1.1. Let f be a mapping of finite distortion in OCRn ; nX2: Assume that the distortion KðxÞX1 satisfies expðbKÞAL1loc ðOÞ; for some b40: Then Jðx; f Þ loga ðe þ Jðx; f ÞÞAL1loc ðOÞ;

and

jDf jn loga1 ðe þ jDf jÞAL1loc ðOÞ;

where a ¼ c1 b and c1 ¼ c1 ðnÞ40: Moreover, for any ball B such that 2B ! O;   Z Jðx; f Þ a  Jðx; f Þ log e þ dx Jðx; f Þ2B B Z Z  pcðn; bÞ  expðbKðxÞÞ dx  Jðx; f Þ dx : 2B

2B

ð1:2Þ

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R Here and in the following, gE ¼ E g is the average of g over the set E and 2B is a ball with the same center as B and with radius twice that of B: Theorem 1.1 is sharp in the sense that the conclusion fails for a ¼ b for all b40: This is seen by considering the mapping  f ðxÞ ¼

   b=n 1 1 x ; log log e þ log e þ jxj jxj jxj

defined in the unit ball of Rn : As a consequence of Theorem 1.1, we obtain the following sharp measure distortion estimate, in which jAj denotes the Lebesgue measure of ACRn : Corollary 1.2. Let f be a continuous mapping of finite distortion in Bð0; 2Þ in Rn ; nX2: Assume that the distortion KðxÞX1 satisfies expðbKÞAL1 ðBð0; 2ÞÞ; for some b40: Then   1 j f ðEÞjpC logc1 b e þ jEj for each measurable set ECBð0; 1Þ; where c1 ¼ c1 ðnÞ and C depends only on n; b; f : The planar version of the above corollary for certain solutions of the Beltrami equation is due to David [5]. With a bit of work one can check that, at least qualitatively, Corollary 1.2 implies Theorem 1.1. Using the uniqueness results for the Beltrami equation from [5], one can thus verify a version of Theorem 1.1 in the plane 1;2 under the additional assumption that f is either homeomorphic or in Wloc ðO; R2 Þ: The sharpness of our estimate means that one can find a constant c1 for which the estimate fails. This follows from an example for distortion estimates for generalized Hausdorff measures given in [13]; our corollary also yields a sharp higherdimensional version of the dimension estimates given in [13]. Second, we study the dual version of the higher integrability Theorem 1.1. The following theorem shows that the natural regularity assumption (1.1) can be relaxed. This is the first result in this direction for mappings with exponentially integrable finite distortion. Clearly, it is an analog of Iwaniec and Martin’s [19,14] results on very weak quasiregular mappings. 1;1 Theorem 1.3. Let f AWloc ðO; Rn Þ; nX2; satisfy the distortion inequality

jDf ðxÞjn pKðxÞJðx; f Þ

a:e: in O;

where KðxÞX1 satisfies expðbKÞAL1loc ðOÞ; for some b40: There is a constant c2 ¼ c2 ðnÞ such that if jDf jn loga1 ðe þ jDf jÞAL1loc ðOÞ;

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with a ¼ c2 b; then (1.1) holds and Jðx; f ÞAL1loc ðOÞ: In particular, f is then a mapping of finite distortion. Notice that we gain more regularity for f than stated in Theorem 1.3. Indeed, invoking Theorem 1.1, it for example follows that Jðx; f Þ logc1 b ðe þ Jðx; f ÞÞAL1loc ðOÞ: x The mapping f ðxÞ ¼ logs ðe þ 1=jxjÞjxj ; defined in the unit ball of Rn ; for which Jðx; f Þ is not locally integrable when s40; shows that for each c2 o1 and any b; there are mappings for which the claim fails. Thus, Theorem 1.3 only admits improvement in finding the precise value of c2 ðnÞ: In fact, Iwaniec and Martin [19,14] not only proved the self-improved regularity of quasiregular mappings but also a so-called Caccioppoli-type inequality for the exponent n  e: From this they obtained fundamental results in the theory of removable singularities for quasiregular mappings. We also obtain a new Caccioppoli-type inequality in the settings of mappings of finite distortion. This yields the following novel result on removable sets for mappings of finite distortion. Corollary 1.4. Let b40: Let ECRn be a closed set of Ln logn1c2 b L–capacity zero, where c2 is the constant from Theorem 1.3, and let f : O\E-Rn be a bounded mapping of finite distortion. Suppose that the distortion function K satisfies Z expðbKðxÞÞ dxoN: O\E

Then f extends to a mapping of finite distortion in O with the exponentially integrable distortion function KðxÞ: It was previously only known that sets of Ln logn1 L–capacity zero are removable [16,17,27]. A weaker version of Corollary 1.4 is given in [3] in the planar case. This corollary is essentially sharp, see [3]. The proof of Theorem 1.1 is different from those in [3,16,17]. Those proofs were based on the ideas from the papers [19,14]. Our proof is inspired by the work of Gehring [8] on the reverse Ho¨lder inequality. Of course, in our case, the differential of the mapping does not satisfy the classical reverse Ho¨lder inequality but it turns out to satisfy a reverse inequality in the setting of Orlicz spaces, see (2.3). The nice paper [15] by Iwaniec largely extended the Gehring lemma to the setting of Orlicz spaces. Unfortunately, the framework of [15] does not apply to our case. In order to deal with our situation, we need to modify the original idea of Gehring. The proof of Theorem 1.1 indicates that it is possible to extract a non-homogeneous version of Gehring’s Lemma in the setting of Orlicz spaces. We believe that this more general version would be of its own interest. Concerning the proof of Theorem 1.3, it seems that the methods in [19,14] cannot be adapted to treat the case of unbounded distortion. Our approach follows the lines of [7], where the authors gave a short proof of the Caccioppoli-type inequality

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mentioned above. Their technique is inspired by a paper of Lewis [28]. The basic idea is to truncate the maximal function of the gradient along the level sets to construct a Lipschitz continuous test-function in the fashion of the pioneering work of Acerbi and Fusco [1]. In the finite distortion setting these ideas work relatively nicely as well, but a natural obstruction arises, and thus the method of [7] needs to be modified. More precisely, it is more convenient to obtain the improved regularity directly instead of using the Caccioppoli inequality and Gehring’s lemma as in [7]. Finally, we remark that it seems to be a long way to get the exact values of the constants c1 in Theorem 1.1 and c2 in Theorem 1.3. Even for quasiregular mappings, only the planar case is known [2].

2. Proofs of Theorem 1.1 and Corollary 1.2 A central ingredient in our arguments is the classical Hardy–Littlewood maximal function. Recall that, given an function gAL1 ðRn Þ; we define the Hardy–Littlewood maximal function Mg of g by Z 1 MgðxÞ ¼ sup jgðyÞj dy: r40 jBðx; rÞj Bðx;rÞ The following proposition is classical. The proof involves Vitali’s covering lemma and the Caldero´n–Zygmund decomposition, see [34, 7p and 23p]. Proposition 2.1. Let hAL1 ðRn Þ: We have for any t40 that Z Z 1 2 5n n jhðxÞj dxpjfxAR : MhðxÞ4tgjp jhðxÞj dx: 2n t jhj4t t jhj4t=2

Proof of Theorem 1.1. We rely on a result from [21] according to which Z Z Jðx; jf Þdxp  Jðx; fl Þ dx Fl

B0 \Fl

1;1 whenever fAC0N ðOÞ and f ¼ ð f1 ; f2 ; y; fn ÞAWloc ðO; Rn Þ is a Sobolev mapping with Jðx; f ÞX0 a.e. and

jDf jn AL1loc ðOÞ: log ðe þ jDf jÞ Thus Z O

fJðx; f Þ dxp

Z

j f j jDf jn1 jrfj dx: O

ð2:1Þ

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This is a reverse inequality, from which the higher-integrability result is derived. Actually, it can be proved in the same way as in the proof of Theorem 1.3 in the next section, see (3.14). Let fAC0N ðBðy; 2rÞÞ satisfy f ¼ 1 in Bðy; rÞ; 0pfp1 in Rn and jrfjp2=r; where Bðy; 2rÞ is a ball lying in O: With this choice of f; (2.1) leads to the following inequality with q ¼ n2 =ðn þ 1Þ: Z

Jðx; f Þ dxp Bðy;rÞ

2 r

2 p r

Z

j f j jDf jn1 dx

Bðy;2rÞ

Z

q

jDf j

!n1 Z q

Bðy;2rÞ

! 12 n

n2

j f j dx

:

Bðy;2rÞ

Note that this inequality remains valid if we substract from f any constant vector. In particular, it holds for f  fBðy;2rÞ replacing f : Here, fBðy;2rÞ is the average of f over Bðy; 2rÞ: Applying the Poincare´–Sobolev inequality yields Z

n2

! 12 n

j f  fBðy;2rÞ j dx

pcðnÞ

Bðy;2rÞ

!1

Z

q

q

jDf j dx

:

Bðy;2rÞ

Combining these last two inequalities, we finally obtain

1 jBðy; rÞj

Z

1 Jðx; f Þ dxpcðnÞ jBðy; 2rÞj Bðy;rÞ

Z

!n q

q

jDf j dx

ð2:3Þ

Bðy;2rÞ

whenever Bðy; 2rÞ ! O: Here jEj is the Lebesgue measure of ECRn : The above argument is standard, see Lemma 7.6.1 in [20]. Now we fix a ball B0 ¼ Bðx0 ; r0 Þ ! O: Assume that Z

Jðx; f Þ dx ¼ 1:

ð2:4Þ

B0

This assumption involves no loss of generality for us as the distortion inequality and (1.2) are homogeneous with respect to f : Let us introduce the auxiliary functions defined in Rn by h1 ðxÞ ¼ dðxÞn Jðx; f Þ; h2 ðxÞ ¼ dðxÞjDf ðxÞj; h3 ðxÞ ¼ wB0 ðxÞ;

ð2:5Þ

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where dðxÞ ¼ distðx; Rn \B0 Þ and wE is the characteristic function of the set E: We claim that 

1 jBj

1

Z

n

h1 dx B



1 pcðnÞ j2Bj

Z 2B

hq2

1 dx

q



1 þ cðnÞ j2Bj

1

Z h3 dx

n

ð2:6Þ

2B

for all balls BCRn : Indeed, we may assume that B meets B0 ; otherwise (2.6) is trivial. Our derivation of (2.6) falls naturally into two cases. Case 1: We assume that 3BCB0 : By an elementary geometric consideration we find that max dðxÞp4 min dðxÞ: xAB

xA2B

Applying (2.3) yields 

1 jBj

1

Z

n

h1 dx B



1 Z n 1 p max dðxÞ Jðx; f Þ dx B jBj B  1 Z q 1 q p cðnÞ min dðxÞ jDf j dx 2B j2Bj 2B  1 Z q 1 q p cðnÞ h dx : j2Bj 2B 2

Case 2: We assume that 3B is not contained in B0 and recall that B meets B0 : We have that 1

max dðxÞp max dðxÞpcðnÞj2B-B0 jn : xAB

xA2B

Hence we conclude that 

1 jBj

1

Z h1 dx B

n



1 Z n 1 p max dðxÞ Jðx; f Þ dx B jBj B-B0  1 Z n j2B-B0 j p cðnÞ Jðx; f Þ dx jBj B0  1 Z n 1 p cðnÞ h3 dx ; j2Bj 2B

where we used (2.4). Combining these two cases proves inequality (2.6). Since (2.6) is true for all balls BCRn ; we have the following point-wise inequality for the maximal functions. For all yARn ; 1

1

1

Mðh1 ÞðyÞn pcðnÞMðhq2 ÞðyÞq þ cðnÞMðh3 ÞðyÞn ;

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from which it follows that for l40 jfxARn : Mðh1 ÞðxÞ4ln gjp jfxARn : cðnÞMðhq2 Þ4lq gj þ jfxARn : cðnÞMðh3 ÞðxÞ4ln gj:

ð2:9Þ

We recall that h3 ðxÞ ¼ wB0 ðxÞ: So Mðh3 ÞðxÞp1 in Rn ; and then the set fxARn : cðnÞMðh3 ÞðxÞ4ln g is empty for l4l1 ¼ l1 ðnÞ: Hence jfxARn : Mðh1 ÞðxÞ4ln gjpjfxARn : cðnÞMðhq2 Þ4lq gj for all l4l1 : Now applying Proposition 2.1 yields Z h1 4l

n

h1 dxpcðnÞlnq

Z cðnÞh2 4l

hq2 dx

ð2:10Þ

for all l4l1 : We may assume that the constant cðnÞ in (2.10) is bigger than one. Let a40 be a constant, which will be chosen later, and set CðlÞ ¼

nq loga l þ loga1 l; a

where q ¼ n2 =ðn þ 1Þ as above. Notice that FðlÞ :¼

d nq a1 CðlÞ ¼ loga1 l þ loga2 l40 dl l l

for all l4l2 ¼ expððn þ 1Þ=nÞ; and that lnq FðlÞ ¼

 d  nq l loga1 l : dl

We multiply both sides of (2.10) by FðlÞ; and integrate with respect to l over ðl0 ; NÞ for l0 ¼ maxðl1 ; l2 Þ; and finally change the order of the integration to obtain that Z

Z

h1 4ln0

1 hn1

h1

Z FðlÞ dl dxpcðnÞ

l0

cðnÞh2 4l0

hq2

Z

cðnÞh2

lnq FðlÞ dl dx;

l0

that is, Z

1 ðCðhn1 Þ n

h1 4l0

 Cðl0 ÞÞh1 dxpcðnÞ

Z cðnÞh2 4l0

hn2 loga1 ðcðnÞh2 Þ dx:

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Hence, taking into account normalization (2.4), 1 a

Z

1

h1 4ln0

h1 loga hn1 dx

pcðnÞ

Z cðnÞh2 4l0

hn2 loga1 ðcðnÞh2 Þ dx þ cðn; aÞjB0 j;

ð2:11Þ

where cðnÞX1: In the remaining part of the proof, we will choose a suitable positive constant a such that the integral in the right-hand side of (2.11) can be absorbed in the left, by using the distortion inequality. Actually, this only works if we have a priori jDf jn loga1 ðe þ jDf jÞAL1loc ðOÞ: We cannot assume this. To overcome this, in the above argument we integrate with respect to l over ðl0 ; jÞ for j large, instead of over ðl0 ; NÞ: Then we proceed in the same way as above and get a similar inequality as (2.11). The proof will be then eventually concluded, by letting j-N and using the monotone convergence theorem. For simplicity, we only write down the proof for j ¼ N: We need the following elementary inequality; the proof is in the appendix. Let a; b; cðnÞX1: Then 1

ab loga1 ðcðnÞðabÞn Þp

1 CðnÞ a loga ðan Þ þ Cða; b; nÞ expðbbÞ: b

ð2:12Þ

We recall the definitions of h1 and h2 in (2.5) and notice that the distortion inequality reads as h1 ðxÞphn2 ðxÞph1 ðxÞKðxÞ: Thus, when h1 ðxÞpln0 ; we have by choosing h1 ðxÞ ¼ a and KðxÞ ¼ b in (2.12) (or directly if h1 o1) that hn2 ðxÞ loga1 ðcðnÞh2 ðxÞÞpCðn; a; bÞ expðbKðxÞÞ:

ð2:13Þ

Putting again h1 ðxÞ ¼ a; KðxÞ ¼ b in inequality (2.12) we infer from (2.11) and from (2.13) that 1 a

Z h1 4ln0

1

h1 loga hn1 dx

cðnÞ p b cðnÞ p b

Z h1 4ln0

1

h1 loga hn1 dx þ cðn; a; bÞ

Z

h1 4ln0

h1 log

a

1 hn1

Z

dx þ cðn; a; bÞ

expðbKðxÞ dx þ cðn; aÞjB0 j

B0

Z

expðbKðxÞÞ dx: B0

ð2:14Þ

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Now letting a ¼ b=ð2cðnÞÞ; (2.14) becomes Z Z 1 h1 loga hn1 dxpcðn; bÞ h1 4ln0

That is, Z

n

a

expðbKðxÞÞ dx:

B0

n

h1 loga ðe þ h1 Þ dx Z p cðn; bÞ expðbKðxÞÞ dx:

dðxÞ Jðx; f Þ log ðe þ dðxÞ Jðx; f ÞÞ dx ¼

B0

Z

B0

ð2:15Þ

B0

Noticing that in sB0 ¼ Bðx0 ; sr0 Þ with 0oso1 we have dðxÞn Xð1  sÞn rn0 Xcðn; sÞjB0 j; and taking account of the normalization (2.4), we arrive at ! Z Jðx; f Þ  Jðx; f Þ loga e þ R dx B0 Jðx; f Þ dx sB0 Z Z  ð2:16Þ pcðn; b; sÞ  expðbKðxÞÞ dx  Jðx; f Þ dx B0

B0

which proves (1.2), and the higher integrability of jDf j follows from this, the distortion inequality, inequality (2.12), and the exponential integrability of K: This concludes the proof of Theorem 1.1. & Proof of Corollary 1.2. We refer to [24] for the fact that f maps sets of measure zero to sets R of measure zero. Thus the volume of f ðEÞ can be estimated from above by E Jðx; f Þ dx: The desired volume distortion estimate thus follows from the Orlicz–Ho¨lder inequality and the higher integrability of the Jacobian given in Theorem 1.1. &

3. Proofs of Theorem 1.3 and Corollary 1.4 Central building blocks of the proof of Theorem 1.3 are the following well-known point-wise estimates for the Sobolev function, whose proofs rely on an argument due to Hedberg [10]. Lemma 3.1 (Point-wise inequalities for the Sobolev functions). Let uAW 1;q ðRn Þ; 1oqoN; and let x and y be Lebesgue points of u such that xAB0 ¼ Bðx0 ; rÞ: Then juðxÞ  uB0 jpcrMðjrujw2B0 ÞÞðx0 Þ;

ð3:1Þ

juðxÞ  uðyÞjpcjx  yjðMðjrujÞðxÞ þ MðjrujÞðyÞÞ;

ð3:2Þ

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where c ¼ cðnÞ40; wE is the characteristic function of the set E; vB0 is the average of v over B0 ¼ Bðx0 ; rÞ and Mh is the Hardy–Littlewood maximal function of h. Proof of Theorem 1.3. We start along the lines of the proof in [7]. Let jAC0N ðB0 Þ; B0 ¼ Bðx0 ; rÞ ! O; and jX0 and u ¼ f1 j and l40: Denote Fl ¼ fxABðx0 ; rÞ : MðgÞðxÞpl and x is a Lebesgue point of ug; where g ¼ jjDf j þ j f #rjj in B0 and g ¼ 0 in Rn \B0 : Then it was proved in [7] that there exists a constant c ¼ cðnÞ and a cðnÞl-Lipschitz continuous function ul such that ul ¼ u on Fl : We omit the proof of this fact, which relies on Lemma 3.1 and the Poincare inequality. Then we consider the mapping 1;q fl ¼ ðul ; jf2 ; jf3 ; y; jfn Þ: Since f AWloc ðO; Rn Þ for all qon and ul is Lipschitz we have that fl is regular enough for integration by parts and obtain that Z

Jðx; fl Þ dx ¼ 0

B0

and hence, Z

Jðx; jf Þ dxp 

Fl

Z

Jðx; fl Þ dx:

ð3:3Þ

B0 \Fl

If we expand (3.3) and use the point-wise estimates j fi rjjpCðnÞj f #rjj and jrðjfi ÞjpcðnÞg we conclude that Z

n

j Jðx; f Þ dxpcðnÞ

Z

Fl

j f #rjjg

n1

Fl

dx þ l

Z g

n1

 dx :

ð3:4Þ

B0 \Fl

Here we digress from [7]. We claim that (3.4) and Proposition 2.1 imply that Z

n

j Jðx; f Þ dxpcðnÞ

Z

gpl

j f #rjjg

gp2l

n1

dx þ cðnÞl

Z

gn1 dx:

ð3:5Þ

g4l

Indeed, by Proposition 2.1, Z g B0 \Fl

n1

Z

gn1 dx þ ln1 jfxARn : MgðxÞ4lgj

dxp g4l

p cðnÞ

Z g4l=2

gn1 dx

ð3:6Þ

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and Z

Z

jn Jðx; f Þ dx þ ln jfxARn : MgðxÞ4lgj Mgpl Z Z n j Jðx; f Þ dx þ cðnÞl gn1 dx: p

n

j Jðx; f Þ dxp

gpl

Mgpl

Combining (3.4), (3.6) and (3.7), we obtain that Z Z Z jn Jðx; f Þ dxpcðnÞ j f #rjjgn1 dx þ cðnÞl gpl

ð3:7Þ

g4l=2

gpl

gn1 dx: g4l=2

Then (3.5) follows by replacing l=2 by l: Now let a40 be a constant, which will be chosen later. Note that 1 FðlÞ :¼ ðlogð1þaÞ l  ð1 þ aÞ logð2þaÞ lÞX0 l for lXe1þa : We multiply both sides of (3.5) by FðlÞ; and integrate with respect to l over ðt; NÞ for tXl0 ¼ maxðe1þa ; e2a Þ; and finally change the order of the integration to obtain that Z Z N jn Jðx; f Þ FðlÞ dl dx B0

pcðnÞ

maxðg;tÞ

Z

j f #rjjgn1

B0

þ cðnÞ

Z g g4t

n1

Z

Z

N

FðlÞ dl dx maxðg=2;tÞ

g

lFðlÞ dl dx:

ð3:8Þ

t

Thus Z jn Jðx; f Þ 1 jn Jðx; f Þ þ dx a 2a g4t loga g got log t ! Z 1 1 n p  j Jðx; f Þ dx a loga maxðg; tÞ log1þa maxðg; tÞ B0 Z Z cðnÞ j f #rjjgn1 gn dx: dx þ cðnÞ p a 1þa a B0 log maxðg=2; tÞ g g4t log

1 2a

Z

ð3:9Þ

Observe that the first inequality holds because tXe2a : We remark here that the assumption on the regularity of f in the theorem, jDf jn AL1loc ðOÞ; log1þa ðe þ jDf jÞ implies that the integrals on the right-hand side of (3.9) are finite.

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Now we use the distortion inequality, which so far has not been used. The distortion inequality jDf jn pKðxÞJðx; f Þ and the inequality abpa logð1 þ aÞ þ eb  1

ð3:10Þ

for non-negative real numbers, imply that in the set where gðxÞXl0 we have jn jDf jn KðxÞjn Jðx; f Þ p log1þa g log1þa g     2 b 3njn Jðx; f Þ p exp KðxÞ þ : b 2 loga g Therefore, recalling the definition of g and using (3.11), Z Z Z gn jn jDf jn j f #rjjn n n dxp 2 dx þ 2 dx 1þa 1þa 1þa g g g g4t log g4t log g4t log   Z Z cðnÞ jn Jðx; f Þ cðnÞ b KðxÞ dx p dx þ exp b g4t loga g b g4t 2 Z j f #rjjn þ cðnÞ dx: 1þa g g4t log Inserting (3.12) in (3.19), and rearranging, it follows that  Z Z 1 1 cðnÞ jn Jðx; f Þ n  dx j Jðx; f Þ þ a 2a loga t got 2a b g4t log g   Z Z cðnÞ b j f #rjjn p exp KðxÞ dx þ cðnÞ dx 1þa b g4t 2 g g4t log Z cðnÞ j f #rjjgn1 dx: þ a B0 loga maxðg=2; tÞ

ð3:11Þ

ð3:12Þ

ð3:13Þ

Now let us fix a ¼ b=4cðnÞ to be the constant in the theorem. By this choice of a; the second term in the left-hand side is non-negative and therefore can be ignored. We multiply both sides of (3.13) by loga t and let t-N: We obtain by monotone convergence theorem and Lebesgue dominated convergence theorem that Z Z n j Jðx; f Þ dxpcðnÞ j f #rjjgn1 dx: ð3:14Þ B0

B0

Here we used the integrability of j f #rjjgn1 ; j f #rjjn ; and expðbKðxÞÞ to pass to the limit. Hence, (3.14) shows that Jðx; f ÞAL1loc ðOÞ; and then (1.1) follows from

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the distortion inequality using (3.10) and the integrability of expðbðKðxÞÞÞ: Thus, Theorem 1.3 is proved. & Next we shall prove the following Caccioppoli-type inequality, which is critical for the proof of Corollary 1.4: Z

jDf jn jn dxp cðn; bÞ 1þa ðe þ jDf jjÞ B0 log

Z

j f #rjjn dx aþ1n ðe þ j f #rjjÞ B0 log   Z b þ cðn; bÞ exp KðxÞ dx: 2 B0

ð3:15Þ

To this end, this time we ignore the first term in the left-hand side of (3.13), and recall the choice of a: We obtain that Z gn j f #rjjgn1 dxp cðn; bÞ dx a 1þa g g4t log B0 log maxðg=2; tÞ    Z Z j f #rjjn b þ dx þ exp KðxÞ dx : 1þa 2 g g4t log g4t

Z

ð3:16Þ

Thus, letting t ¼ l0 in (3.16) and noting that expðb2 KðxÞÞX1; results in Z gn j f #rjjgn1 dxp cðn; bÞ dx a 1þa ðe þ gÞ B0 log B0 log ðe þ gÞ   ! Z Z j f #rjjn b þ dx þ exp KðxÞ dx : 1þa 2 ðe þ gÞ B0 log B0

Z

ð3:17Þ

To estimate the first integral in the right-hand side of (3.17), we use the inequality abn1 pe

bn þ cðe; nÞan logn1 ðe þ aÞ logðe þ bÞ

for non-negative numbers a and b; and obtain that j f #rjjgn1 j f #rjj p a loga ðe þ gÞ logn ðe þ j f #rjjÞ

gn1 loga

n1 n ðe

þ gÞ gn j f #rjjn þ cðe; nÞ : p eCða; nÞ log1þa ðe þ gÞ logaþ1n ðe þ j f #rjjÞ

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By taking e ¼ cðn; bÞ=2Cða; nÞ; where cðn; bÞ is the constant in (3.17), we infer from (3.17) that Z

jDf jn jn dxp 1þa ðe þ jDf jjÞ B0 log

which proves (3.15).

Z

gn dx 1þa ðe þ gÞ B0 log Z j f #rjjn p cðn; bÞ dx aþ1n ðe þ j f #rjjÞ B0 log Z j f #rjjn þ cðn; bÞ dx aþ1 ðe þ j f #rjjÞ B0 log   Z b þ cðn; bÞ exp KðxÞ dx 2 B0 Z j f #rjjn p cðn; bÞ dx aþ1n ðe þ j f #rjjÞ B0 log   Z b þ cðn; bÞ exp KðxÞ dx; 2 B0

&

Proof of Corollary 1.4. We note that E has vanishing ðn  1Þ-dimensional Hausdorff measure. By Theorem 1.3, it thus suffices to show that jDf jn AL1loc ðOÞ; log ðe þ jDf jÞ 1þa

ð3:20Þ

where a ¼ c2 b is as in Theorem 1.3. To this end, let ZAC0N ðOÞ be an arbitrary nonnegative test function. We denote by E 0 the intersection of E with the support of Z: There exists a sequence of functions ffj gN j¼1 such that for each j we have fj AC0N ðOÞ; 0pfj p1; fj ¼ 1 on some neighborhood Uj of E 0 ; limj-N fj ðxÞ ¼ 0 for almost all xARn ; R 5. limj-N O jrfj jn logn1a ðe þ jrfj jÞ ¼ 0: 1. 2. 3. 4.

We set jj ¼ ð1  fj ÞZAC0N ðsupp Z\E 0 Þ: Let j ¼ jj in (3.15). Recall that j f j is assumed to be bounded in O and also that rjj ¼ ð1  fj ÞrZ  Zrfj : It follows from the conditions defining fj that we can

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pass to the limit (as j-N) in (3.15) to obtain that Z

jDf jn Zn dxp cðn; bÞ 1þa ðe þ jDf jn Zn Þ B0 log

Z

j f #rZjn dx aþ1n ðe þ j f #rZjÞ B0 log   Z b þ cðn; bÞ exp KðxÞ dx; 2 B0

which implies (3.20). This then proves Corollary 1.4.

&

Acknowledgments Part of this work was done when X.Z. visited the Max-Planck-Institute for Mathematics in the Sciences, Leipzig, Germany, during February of 2003. He would like to thank the institute for the hospitality. D.F. was partially supported by EU (Project #HPRN-CT-2002-00282). X.Z. was partially supported by the Academy of Finland, project 207288. The authors would also like to thank the referees for many valuable comments.

Appendix Proof of the inequality. 1

ab loga1 ðcðnÞðabÞn Þp

1 CðnÞ a loga ðan Þ þ Cða; b; nÞ expðbbÞ: b

ðA:1Þ

The case ape is easy. Suppose a4e: We have that 1

1

ab log1 ðcðnÞðabÞn Þp ab log1 ðan Þ    1 4 b log1 an p a log a þ exp b b 2    cðnÞ b a þ exp b ; p b 2

ðA:2Þ

where we used the elementary inequality abpa log a þ expð2bÞ for aX1; bX1: We also have that 1

1

loga ðcðnÞðabÞn Þp2 loga an þ cðn; aÞ loga ðcðnÞbÞ

ðA:3Þ

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which follows from ðx þ yÞa p2xa þ cðaÞya for x40; y40; a40: Combining (A.2) and (A.3) yields 1

ab loga1 ðcðnÞðabÞn Þ    1 cðnÞ b a þ exp b ð2 loga an þ cðn; aÞ loga ðcðnÞbÞÞ p b 2 1 cðnÞ ða loga an þ cðn; a; bÞ expðbbÞÞ; p b

ðA:4Þ

as desired. The last inequality holds because of the estimates 1

cðn; aÞa loga ðcðnÞbÞpa loga an þ cðn; a; bÞ expðbbÞ; and   1 1 b exp b loga an pa loga an þ cðn; a; bÞ expðbbðxÞÞ; 2 for the lower order terms which can be proved easily. &

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