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Minkowski content and singular integrals
Chaos, Solitons and Fractals 17 (2003) 169–177 www.elsevier.com/locate/chaos
Minkowski content and singular integrals ubrinic Darko Z Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia Accepted 19 September 2002 Communicated by Prof. Dr. Drehc. W. Martienssen
Abstract Assume that lower and upper d-dimensional Minkowski contents of A RN are different both from 0 and 1. We show that the function dðx; AÞc is integrable in a tubular neighbourhood of A if and only if c < N d (the if part is known). von KochÕs curve and the Sierpinski gasket are shown to satisfy the Minkowski content condition. The notions of relative Minkowski content and relative box dimension are introduced in order to extend this result to singular functions generated by more general fractal sets, which may have classical upper or lower d-dimensional Minkowski content equal to 0 or 1. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The notion of Minkowski content appears naturally in the study of removable singularities of solutions of partial differential equations, see [7,13]. It is important in the study of asymptotic behavior of the eigenvalue counting function for the Dirichlet eigenvalue problem Du ¼ ku on a bounded domain X RN , where u ¼ 0 on oX, which may be fractal. See [8,10] for the detailed treatment of one-dimensional Weyl–Berry conjecture. Minkowski content is useful also in the study of singular sets of Sobolev functions, see [14]. Additional comments about Minkowski content can be found in [11, p. 80] and the references therein. In the present paper we are interested in the following problem: finding condition on a fractal set A RN , which enables to characterize Lebesgue integrability of singular function dðx; AÞc on a bounded neighbourhood of A. Here dðx; AÞ is the euclidean distance function from x to A. It is interesting that the required condition is of the same type as the one appearing in [10, Conjecture 1Õ and Theorem 2.4], namely, that the upper and lower d-dimensional Minkowski contents of A be different both from 0 and 1. See Theorems 2.3, 2.9, and 3.1 below. A sufficient condition for Lebesgue integrability of dðx; AÞc in a bounded neighbourhood of A has been stated in [12, Lemma 3.6], see also [7, the proof of Theorem 2.5, p. 42]: if the upper d-dimensional Minkowski content of A is finite, then the required sufficient integrability condition is c < N dimB A, where dimB A is the upper box dimension of A. In this paper we show that this sufficient condition is also necessary for integrability of dðx; AÞc in a bounded neighbourhood of A, assuming that both upper and lower d-dimensional Minkowski contents of A are contained in ð0; 1Þ. We extend this integrability characterization to a class of fractal sets whose lower or upper d-dimensional Minkowski contents may be 0 or 1, by introducing the notion of relative Minkowski content. The main results are stated in Theorems 2.3, 2.4 and 2.9.
ubrinic). E-mail address: [email protected] (D. Z 0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 4 4 1 - 1
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2. Relative Minkowski content and relative box dimension If A is a subset of RN and r > 0, we denote euclidean r-neighbourhood of A by Ar , that is, Ar ¼ fy 2 R : dðy; AÞ < rg. By LN we denote the standard Lebesgue measure in RN . We first introduce classical upper and lower s-dimensional Minkowski contents of set A: N
M s ðAÞ ¼ lim sup r!0
LN ðAr Þ ; rN s
Ms ðAÞ ¼ lim inf r!0
LN ðAr Þ : rN s
ð2:1Þ
The infimum of all s P 0 such that M s ðAÞ ¼ 0 is called the upper box dimension of A, and is denoted by dimB A. Here we define inf ; ¼ þ1. One can analogously define the lower box dimension dimB A. If both of them coincide, the common value is called box dimension, and is denoted by dimB A. If M s ðAÞ ¼ Ms ðAÞ for some s, then the common value is called s-dimensional Minkowski content of A, and is denoted by Ms ðAÞ. If it is contained in ð0; 1Þ we say that A is Minkowski measurable. Minkowski measurability of fractal subsets of the real line is studied in [5,8]. We shall also deal with the boundary C of a given open set X of RN . Here X may be unbounded. Let us recall the definition of lower and upper s-dimensional Minkowski content of C with respect to X: Ms ðC; XÞ ¼ lim inf r!0
LN ðCr \ XÞ ; rN s
M s ðC; XÞ ¼ lim sup r!0
LN ðCr \ XÞ : rN s
ð2:2Þ
We can define the corresponding lower and upper box dimensions dimB ðC; XÞ and dimB ðC; XÞ analogously as above. See Examples 2.1 and 2.2 below. Motivated by He and Lapidus [8], we introduce the notion of relative Minkowski content. Let X be an open subset in RN (possibily unbounded), and A any subset of RN . Let h : ð0; 1Þ ! ð0; 1Þ be a given nondecreasing function, which we call the gauge function. We define the lower h-Minkowski content of A relative to X by M ðh; A; XÞ ¼ lim inf r!0
LN ðAr \ XÞ hðrÞ rN
ð2:3Þ
and analogously the upper relative h-Minkowski content M ðh; A; XÞ. If hðrÞ ¼ rs , we obtain s-dimensional lower and upper Minkowski contents Ms ðA; XÞ and M s ðA; XÞ of A relative to X. In this case we can define lower and upper box dimensions of A relative to X in the usual way, that we denote by dimB ðA; XÞ;
dimB ðA; XÞ
ð2:4Þ Md ðA; XÞ
d
respectively. If both of them coincide, we denote the common value by dimB ðA; XÞ. Also, if ¼ M ðA; XÞ, the common value will be denoted by Md ðA; XÞ. It is easy to see that lower relative box dimension has the usual monotonicity property: if A1 A2 and X1 X2 , then dimB ðA1 ; X1 Þ 6 dimB ðA2 ; X2 Þ, and analogously for the upper relative box dimension. Also, it is clear that if A X and the distance between A and oX is positive, then we obtain classical values of Minkowski contents: Ms ðA; XÞ ¼ Ms ðAÞ and
M s ðA; XÞ ¼ M s ðAÞ:
If X is a given open set and A ¼ C its boundary, then from (2.3) we obtain generalized lower h-Minkowski contents of C, introduced in [8]. Example 2.1. It is interesting that relative box dimension can be noninteger even in the case of sets A and X having nonfractal nature. For example, if X ¼ fx ¼ ðx1 ; x2 Þ 2 R2 : 1 < x1 < 1; 0 < x2 < jx1 ja g, where a 2 ð0; 1Þ is given, and A ¼ fx 2 X : x1 ¼ 0g, then it is easy to see that Md ðA; XÞ ¼ 2=ð1 aÞ, d ¼ dimB ðA; XÞ ¼ 1 þ a. Also, it is not difficult to show that 0 < Md ðC; XÞ 6 M d ðC; XÞ < 1, d ¼ dimB ðC; XÞ ¼ 1 þ a, where C ¼ oX. Example 2.2. If X ¼ fx ¼ ðx1 ; x2 Þ 2 R2 : 1 < x1 < 0; 0 < x2 < ekx1 g, where k > 0 is given, and A ¼ fx 2 X : x2 ¼ 0g, then for hðrÞ ¼ r=ðlogð1=rÞ þ 1Þ we have Mðh; A; XÞ ¼ 2=k, dimB ðA; XÞ ¼ 1, while M1 ðA; XÞ ¼ 1. Also, 0 < M1 ðC; XÞ 6 M 1 ðC; XÞ < 1, where C ¼ oX. Our first result concerns the necessary condition for integrability of the singular function dðx; AÞc in X. It is worth noting that condition (2.5) below is the same as condition (H3) appearing in [8, Definition 2.2].
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Theorem 2.3. Let X be an open set in RN . Let h : ð0; 1Þ ! ð0; 1Þ be a nondecreasing gauge function such that hðrÞ 6 crb for all r 2 ð0; RÞ, where c, and R are positive constants independent of r, and for any q 2 ð0; 1Þ there exist positive constants m and s < N independent of r, such that hðqrÞ P mqs hðrÞ
for all r 2 ð0; RÞ:
ð2:5Þ
Assume that 0 < M ðh; A; XÞ 6 M ðh; A; XÞ < 1. If
R X
dðx; AÞc dx < 1, then c < N b.
Proof. Assume by contradiction that c P N b. Let us choose numbers c 2 ð0; M ðh; A; XÞÞ and c 2 ðM ðh; A; XÞ; 1Þ such that for all q 2 ð0; RÞ there holds cqN hðqÞ1 6 LN ðAq \ XÞ 6 cqN hðqÞ1 :
ð2:6Þ
Now we fix any q 2 ð0; 1Þ, and take any q 2 ð0; RÞ. Then LN ððAq n Aqq Þ \ XÞ ¼ LN ðAq \ XÞ LN ðAqq \ XÞ P cqN hðqÞ1 cðqqÞN hðqqÞ1 P ðc m1 qN s cÞqN hðqÞ1 P CqN b ; where we have taken q small enough, so that C :¼ c1 ðc m1 qN s cÞ > 0. Using the corresponding decomposition i AR \ X ¼ [1 i¼0 Bi , where Bi :¼ ðARi n ARiþ1 Þ \ X, Ri :¼ q R, we obtain that Z Z 1 1 1 1 X X X X N b N dðx; AÞc dx ¼ dðx; AÞc dx P Rc Rc ¼ cRN bc qðN bcÞi ¼ 1; ð2:7Þ i L ðBi Þ P i cRi AR \X
i¼0
Bi
i¼0
since N b c 6 0, and q 2 ð0; 1Þ. This contradicts
i¼0
R X
c
dðx; AÞ dx < 1.
i¼0
The following result deals with the converse of the above theorem. Its proof is similar, and hence we omit it. Theorem 2.4. Let X be an open set in RN . Let h : ð0; 1Þ ! ð0; 1Þ be a nondecreasing gauge function such that hðrÞ P cra , for all r sufficiently small, where c and a are positive constants independent of r. Assume that M ðh; A; XÞ < 1. If c < R N a, then X dðx; AÞc dx < 1. The gauge function appearing in the following corollary is useful in applications, see [8]. Corollary 2.5. Let X be an open subset of RN , and let A be a subset of RN . Assume that there exist nonnegative constants a and d such that for hðrÞ ¼
rd ; þ 1Þa
½logð1r
we have 0 < M ðh; A; XÞ 6 M ðh; A; XÞ < 1. Then d ¼ dimB ðA; XÞ, and Z dðx; AÞc dx < 1 () c < N d:
ð2:8Þ
X
Proof. To prove the sufficiency part, assume that c < N d. Taking any a 2 ðd; N cÞ it is easy to see that hðrÞ P ra provided r is small enough. R Since c < N d, we can take a slightly larger than d, in order to have c < N a. Then Theorem 2.4 implies Rthat X dðx; AÞc dx < 1. Now assume that X dðx; AÞc dx < 1. We have that ra 6 hðrÞ 6 rd for small r, with a as above. Condition (2.5) is easy to check, see [8, (H3) on p. 88]. Therefore c < N d by Theorem 2.3. It is easy to see that Corollary 2.5 can be applied to Examples 2.1 and 2.2. We now apply it to several interesting examples appearing in [8]. Example 2.6. Let C be the boundary of the open domain X in R2 discussed in [8, Example 7.5], originally defined in [3]. Here for the gauge function hðrÞ ¼ r= logðð1=rÞ þ 1Þ we have 0 < M ðh; A; XÞ 6 M ðh; A; XÞ < 1 and d ¼ dimB ðC; XÞ ¼ 1. Then by Corollary 2.5,
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Z
dðx; CÞc dx < 1
()
ð2:9Þ
c < 1:
X
Note that Md ðC; XÞ ¼ M d ðC; XÞ ¼ 1 with d ¼ 1, so that Theorems 3.4 and 3.5 below are not applicable. Example 2.7. (a) Now considerS the open set X in R2 discussed in [8, Example 7.7], originally defined in [1]. Let
aðxÞ ¼ x1 ln x and define X ¼ 1 k¼3 ðaðk þ 1Þ; aðkÞÞ ð1=2; 1=2Þ. Here we have 0 < M ðh; C; XÞ 6 M ðh; C; XÞ < 1 with hðrÞ ¼ r3=2 =½logðð1=rÞ þ 1Þ1=2 , and d ¼ dimB ðC; XÞ ¼ 3=2. By Corollary 2.5, Z dðx; CÞc dx < 1 () c < 1=2:
ð2:10Þ
X
Note that in this example we also have Md ðC; XÞ ¼ M d ðC; XÞ ¼ 1. Furthermore, it is easy to see that Md ðCÞ ¼ Md ðC; XÞ ¼ 1, and M d ðCÞ ¼ M d ðC; XÞ ¼ 1. Example 2.8. See [8, Remark 7.8(c)]. Modifying aðxÞ in the above example to aðxÞ ¼ 1=ðx ln xÞ, we have the same conclusions, including (2.10), except that here Md ðC; XÞ ¼ M d ðC; XÞ ¼ 0, so that Theorem 3.5 below is not applicable. Furthermore, Md ðCÞ ¼ Md ðC; XÞ ¼ 0, and M d ðCÞ ¼ M d ðC; XÞ ¼ 0. Now we would like to estimate growth rate of singular integrals in terms of the gauge function. Theorem 2.9. Assume that A is a closed subset of RN having Lebesgue measure zero, X an open subset of RN . Assume that c > 0 and there exists a nondecreasing gauge function h : ð0; 1Þ ! ð0; 1Þ such that Z r N c1 tN c t ! 0; t ! 0; dt < 1: ð2:11Þ hðtÞ hðtÞ 0 (a) If M ðh; A; XÞ < 1, then Z Z r dðx; AÞc dx ¼ rc LN ðAr \ XÞ þ c LN ðAt \ XÞtc1 dt: Ar \X
ð2:12Þ
0
In particular, given r0 > 0, for any r 2 ð0; r0 Þ, N c Z Z r N c1 r t dt ; þc dðx; AÞc dx 6 C hðrÞ hðtÞ 0 Ar \X
ð2:13Þ
where C ¼ supt2ð0;r0 Þ ðLN ðAt \ XÞ=tN ÞhðtÞ. (b) If 0 < M ðh; A; XÞ 6 M ðh; A; XÞ < 1, and r0 > 0 is given, then for any r 2 ð0; r0 Þ we have the following estimate: N c N c Z r N c1 Z Z r N c1 r t r t þc dt 6 þc dt ; ð2:14Þ C dðx; AÞc dx 6 C hðrÞ hðtÞ hðrÞ hðtÞ 0 0 Ar \X where C ¼ inf t2ð0;r0 Þ LN ðAt \ XÞ=tN hðtÞ. In the proof we shall use the following auxiliary result, which extends [2, Proposition 1 in Section 3.4.4]. Proposition 2.10. Let A be a closed subset of RN with Lebesgue measure equal to zero, and X an open subset of RN . Then for f ðxÞ ¼ dðx; AÞ we have jrf j ¼ 1 a.e. in RN , and for any g 2 L1 ðXÞ, Z Z 1 Z ð2:15Þ gðxÞ dx ¼ gðxÞ dH N 1 dt; 0
X
where H d dt
N 1
Z
oAt \X
is HausdorffÕs ðN 1Þ-dimensional measure. Furthermore, Z gðxÞ dx ¼ g dH N 1 for a:e: t > 0; At \X
ð2:16Þ
oAt \X
d N L ðAt \ XÞ ¼ H N 1 ðoAt \ XÞ for a:e: t > 0: dt
ð2:17Þ
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Proof. Using RademacherÕs theorem (see [2, Theorem 2 in Section 3.1.2]) and the proof from [6, Lemma 3.2.34], we obtain that Jf ¼ jrf j ¼ 1 a.e. in RN . We exploit the change of variables formula from [2, Theorem 2 in Section 3.4.3]: # Z Z "Z g dH N m dt
gðxÞJf ðxÞ dx ¼
RN
Rm
ð2:18Þ
f 1 ðtÞ
with m ¼ 1, and g 0 outside of X. This implies (2.15). It is easy to see that (2.15) ) (2.16) ) (2.17).
Proof of Theorem 2.9 (a) First note that LN ðAt \ XÞ 6 C
tN hðtÞ
for all t 2 ð0; r0 Þ, hence, by (2.11) we conclude that the set A \ X has the Lebesgue measure zero. We define gðxÞ ¼ dðx; AÞc in Ar and gðxÞ ¼ 0 otherwise, and exploit Proposition 2.10: Z Z r dðx; AÞc dx ¼ tc H N 1 ðoAt \ XÞ dt: ð2:19Þ Ar \X
0
Recalling (2.17), we integrate by parts the right hand side of (2.19), and obtain the identity (3.1) below. Here we have used that tc LN ðAt \ XÞ ¼ CCtN c =hðtÞ ! 0 as t ! 0. Estimate (2.14) follows easily from the fact that LN ðAt \ XÞ 6 CtN =hðtÞ, for t 2 ð0; r0 Þ. (b) The claim follows from (a) and from LN ðAt \ XÞ P CtN =hðtÞ, for t 2 ð0; r0 Þ.
3. Classical Minkowski content In the case of classical Minkowski content we can state stronger results than in Theorems 2.3 and 2.4. Inequality (3.2) of the following Theorem 3.1 is known, see e.g. [12, Lemma 3.6], except that our constant appearing in (3.2) is better than in [12, Lemma 3.6]. This inequality seems to have been known over a longer period, see [7, the proof of Theorem 2.5, p. 42]. Its proof was obtained using diadic decomposition of Ar . Here we provide a different proof, based on the application of Theorem 2.9, that is, generalized change of variables formula. Theorem 3.1. Assume that A is a closed subset of RN , and let r > 0. (a) Assume that d P 0 is such that M d ðAÞ < 1. If c < N d, then Z Z r dðx; AÞc dx ¼ rc LN ðAr Þ þ c LN ðAt Þtc1 dt:
ð3:1Þ
0
Ar
In particular, Z dðx; AÞc dx 6 Ar
N d LN ðAt Þ N dc sup r : N d c t2ð0;rÞ tN d
(b) Assume that there exists d ¼ dimB A, and 0 < Md ðAÞ 6 M d ðAÞ < 1. Then Z dðx; AÞc dx < 1 () c < N dimB A:
ð3:2Þ
ð3:3Þ
Ar
Furthermore, if c < N d, and r0 is a given positive number, then for all r 2 ð0; r0 Þ we have Z c1 rN dc 6 dðx; AÞc dx 6 c2 rN dc ; Ar
where c1 and c2 are two positive constants independent of r.
ð3:4Þ
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Proof of Theorem 3.1 (a) Note that dimB A < N implies that LN ðAÞ ¼ 0. The claim follows from Theorem 2.9(a), with X ¼ RN and hðrÞ ¼ rd . (b) The equivalence stated in (3.3) follows from Corollary 2.5. Estimate (3.4) is a consequence of Theorem 2.9(b), and c1 ¼
N d LN ðAt Þ ; inf t2ð0;r Þ N d c tN d 0
c2 ¼
N d LN ðAt Þ : sup N d c t2ð0;r0 Þ tN d
ð3:5Þ
The following corollary provides some examples of singular functions generated by classical fractal sets. Corollary 3.2 (a) Assume that a > 0 is given, and let A ¼ f1=na : n 2 Ng. Then Z 1 1 : dðx; AÞc dx < 1 () c < 1 aþ1 0 (b) If C is the standard Cantor set, then Z 1 log 2 : dðx; CÞc dx < 1 () c < 1 log 3 0
ð3:6Þ
ð3:7Þ
More generally, let C ðm;aÞ be generalized Cantor set defined by an integer m P 2 and a 2 ð0; 1=mÞ, see [4]. Then Z 1 log m : ð3:8Þ dðx; C ðm;aÞ Þc dx < 1 () c < 1 log 1=a 0 (c) If S is the Sierpinski gasket, then Z log 3 dðx; SÞc dx < 1 () c < 2 ; log 2 T where T is equilateral triangle in R2 containing S. (d) If K is von Koch’s curve, then for any r > 0, Z log 4 : dðx; KÞc dx < 1 () c < 2 log 3 Kr
ð3:9Þ
ð3:10Þ
Proof (a) It suffices to check that 0 < Md ðAÞ 6 M d ðAÞ < 1, see Theorem 3.1(b). We have that A is Minkowski measurable, and a=aþ1 2 Md ðAÞ ¼ ða þ 1Þ; ð3:11Þ a where d ¼ 1=ða þ 1Þ, see [10, Example 4.3]. (b) Using methods of Lapidus and Pomerence [10], or by direct computation, it follows that 1d 1 2d Md ðC ðm;aÞ Þ ¼ ; d 1d d1 1 ma mð1 aÞ
d ðm;aÞ ; Þ¼ M ðC 2ðm 1Þ m1
ð3:12Þ ð3:13Þ
where d ¼ log m= logð1=aÞ. It can be shown that the Cantor set is not Minkowski measurable, that is, the above two quantities are never equal. In the case of classical Cantor set (that is, m ¼ 2, a ¼ 1=3), we obtain Md ðCÞ
log 9 ¼ log 3=2
log 3=2 log 4
log 2= log 3 ;
M d ðCÞ ¼ 22log 2= log 3 ;
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see [10, Theorem 2.4]. For (c) and (d) see Propositions A.1 and A.2 in the Appendix A.
175
Many new examples of fractal sets satisfying conditions of Theorem 3.1 can be generated using the following proposition. It is immediate consequence of Krantz, Parks [9, Theorem 3.3.6]. Proposition 3.3. If two subsets A RN and B RM satisfy conditions 0 < Md ðAÞ 6 M d ðAÞ < 1 and 0 < Me ðBÞ 6 M e ðBÞ < 1, then the analogous condition holds for the product set A B RN þM :
ðdþeÞ 0 < Mdþe ðA BÞ < 1:
ðA BÞ 6 M
In [10, Theorem 2.4] one can see a characterization of subsets A of the real line satisfying condition 0 < Md ðAÞ 6 M d ðAÞ < 1. This condition appears in Theorem 3.1(b). Using the same proofs as above we obtain the following results involving functions that are singular on the boundary C of an open set X, where the boundary may be fractal. Theorem 3.4. Let X be an open set in RN , C ¼ oX, 0 6 d < N, and M d ðC; XÞ < 1. If c < N d, then R c dðx; CÞ dx < 1, and furthermore, X Z
dðx; CÞc dx 6
Cr \X
N d LN ðCt \ XÞ N dc sup r N d c t2ð0;rÞ tN d
for any r > 0. In particular, if R is inner radius of X, then Z N d LN ðCt \ XÞ N dc sup dðx; CÞc dx 6 R : N d c t2ð0;RÞ tN d X Theorem 3.5. Let X be an open set in RN , 0 6 d < N and 0 < Md ðC; XÞ 6 M d ðC; XÞ < 1. Then Z dðx; CÞc dx < 1 () c < N d:
ð3:14Þ
ð3:15Þ
ð3:16Þ
X
Furthermore, if c < N d, and if r0 > 0 is given, then there exist positive constants c1 and c2 independent of r 2 ð0; r0 Þ, such that Z c1 rN dc 6 dðx; CÞc dx 6 c2 rN dc : ð3:17Þ Cr \X
Appendix A. Minkowski contents of the Sierpinski gasket and von KochÕs curve The following result shows that the Sierpinski gasket has different values of lower and upper Minkowski contents, although rather close to each other. It was used in the proof of Corollary 3.2. Proposition A.1. Let S be the Sierpinski gasket. Then 0 < Md ðSÞ < M d ðSÞ < 1, where d ¼ log 3= log 2 is the box dimension of S. The values of Md ðSÞ and M d ðSÞ can be expressed explicitely, and Md ðSÞ 1:810822012, M d ðSÞ 1:814488327 (all decimal values are precise). Proof. Let T be a closed (full) equilateral triangle with unit side. In the first step we remove an open triangle with side 1=2, and in the nth step we remove 3n1 triangles with sides 1=2n (see [4, Figure 0.3]). Continuing this procedure ad infinitum we obtain the Sierpinski gasket S. Now we compute the Lebesgue measure of r-neighbourhood of S. Let r > 0 be given. The condition that a full triangle in the nth generation is not contained in the neighbourhood of its boundary is that r < hn =3, where hn is the height of the triangle. It is easy to see that the largest such n is nðrÞ ¼
1 log2 pffiffiffi 1: 2 3r
ðA:1Þ
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We define inner r-neighbourhood of any (full) triangle M as ðoMÞr \ M. Let us take any k, 1 6 k 6 nðrÞ, and consider the corresponding inner of any triangle of kth generation. An easy computation shows that its measure is pffiffir-neighbourhood ffi equal to 3r2k 3 3r2 . Hence the measure of the union of all inner r-neighbourhoods of triangles over generations k with 1 6 k 6 nðrÞ is equal to nðrÞ nðrÞ X pffiffiffi 3 3 pffiffiffi 2 nðrÞ 3 pffiffiffi 2 1 ðA:2Þ ð3r2k 3 3r2 Þ3k1 ¼ 3r 3r 3 þ 3r r : 2 2 2 2 k¼1 The remaining subset of Sr consists of the set A defined as the union of areas of all triangles fully covered with their rneighbourhoods, corresponding to generations k > nðrÞ, and the set B ¼ Tr n T . We have pffiffiffi nðrÞ 2 pffiffiffi pffiffiffi nðrÞ 3 X k1 1 3 3 3 ¼ 3 ðA:3Þ L2 ðAÞ ¼ k 2 4 4 4 4 k¼1 and clearly, L2 ðBÞ ¼ 3r þ pr2 . Hence, L2 ðSr Þ ¼ f ðrÞ þ Oðrd1 Þ; r2d
ðA:4Þ
where f ðrÞ ¼ r
d2
pffiffiffi nðrÞ ! nðrÞ 3 3 pffiffiffi 2 nðrÞ 3 3 : 3r 3r 3 þ 2 2 4 4
ðA:5Þ
pffiffiffi Let us fix any natural number n, and consider r satisfying n < log2 ð1=2 3rÞ 6 n þ 1, that is, r 2 In :¼ ½rnþ1 ; rn Þ;
1 rn :¼ pffiffiffi : 2 32n
Note that nðrÞ ¼ n for such r. It is easy to see that the function fn : In ! R, fn ðrÞ ¼ f ðrÞ, has points of minimum and maximum at r and rþ 2 In respectively, defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðd 1Þ ðd 1Þ2 þ 12 dðd 2Þ 1 pffiffiffi : r ¼ n d ; d ¼ 2 3d An easy computation shows that the values of fn ðr Þ and fn ðrþ Þ do not depend on n, as well as fn ðrnþ1 Þ ¼ fn ðrn 0Þ. Therefore, see (A.4), pffiffiffi ! 3 3 pffiffiffi 3 d 3 þ 2 3log2 d ¼ 1:810822012; M ðSÞ ¼ lim inf f ðrÞ ¼ fn ðr Þ ¼ r!0 d 2 4d pffiffiffi ! 3 3 3 pffiffiffi
d 3 þ 2 3log2 dþ ¼ 1:814488327: M ðSÞ ¼ lim sup f ðrÞ ¼ fn ðrþ Þ ¼ d 2 4d r!0 þ þ Oscillating nature of the function f ðrÞ near r ¼ 0 can nicely be seen by plotting its graph.
We do not know the precise values of lower and upper d-dimensional Minkowski content of von KochÕs curve K. However, in Proposition A.2 below we obtain that 0:9923660518 6 Md ðKÞ 6 M d ðKÞ 6 1:789812198, which enables to use Theorem 3.1 also in this case. We do not know whether von KochÕs curve is Minkowski measurable or not. Proposition A.2. Let K be von KochÕs curve, and d ¼ log 4= log 3. Then 33 11 pffiffiffi 5p 15 47 pffiffiffi p 3þ 6 Md ðKÞ 6 M d ðKÞ 6 þ 3þ : 32 288 576 32 64 64
ðA:6Þ
Sketch of the Proof. Let Cn be a decreasing sequence of compact sets in R2 defined as follows. For C1 we choose isosceles triangle with basis 1, and with both angles at the basis equal to p=6. The set C2 is obtained from C1 in two steps: (a) remove an open equilateral triangle defined by the middle third of the basis of C1 and the opposite vertex of C1 ; (b) at this moment we are left with two smaller isosceles triangles that are similar C1 , and for each of them we repeat the same
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procedure as in (a). Hence, C2 consists of four mutually congruent, isosceles triangles, that are similar to C1 . In this way we obtain a decreasing sequence of compact sets Cn . Each Cn consists of 4n mutually congruent triangles. It is easy to see that each Cn contains K, and moreover, K ¼ \1 n¼1 Cn . To estimate L2 ðKr Þ from above, let us fix r > 0. We take the largest n such that 1=3n P r, that is, nðrÞ ¼ blog3 ð1=rÞc. Since K Cn for any n, we obtain after a long, but elementary computation, that L2 ðKr Þ L2 ððCnðrÞ Þr Þ 6 ¼ gðrÞ þ Oðrd1 Þ; r2d r2d
ðA:7Þ
where f ðrÞ ¼ r
d2
! " pffiffiffi nðrÞ 4 3 5 r þ þ 8 3 3
pffiffiffi nðrÞ # pffiffiffi ! p 3 2 nðrÞ 3 3 4 r4 þ : 16 16 9 4
ðA:8Þ
pffiffiffi It is easy to see that the supremum of f ðrÞ, r > 0, is equal to ð15=32Þ þ ð47=64Þ 3 þ ðp=64Þ, and due to (A.7) it is P M d ðKÞ. To obtain an estimate for Md ðKÞ from below, we introduce the standard sequence of piecewise linear subsets Kn which approximate von KochÕs curve. First we define K1 as the set consisting of four intervals of length 1=3 each, then the set K2 consisting of 42 intervals of length 1=32 each, etc. (see [4, Figure 0.2]). Note that the set K is connected, and each of 4n subintervals constituting Kn has boundary points contained in K. We start with the following inequality: L2 ðKr Þ P L2 ððKn Þr Þ;
ðA:9Þ
which holds for all r > 0 and n. We have that L2 ððKnðrÞ Þr Þ L2 ðKr Þ P ¼ gðrÞ þ Oðrd1 Þ; 2d r r2d
ðA:10Þ
where " gðrÞ ¼ r
d2
# nðrÞ 11 4 5p 11 pffiffiffi 2 n r þ 3 r 4 : 8 3 144 72
ðA:11Þ
pffiffiffi It is easy to see that the infimum of gðrÞ, r > 0, is equal to ð33=32Þ ð11=288Þ 3 þ ð5p=576Þ, and due to (A.10) it does not exceed Md ðKÞ.
References [1] Caetano AM. Some domains where the eigenvalues of the Dirichlet Laplacian have non-power second term asymptotic estimates. J London Math Soc 1991;43(2):431–50. [2] Evans LC, Gariepy RF. Measure theory and fine properties of functions. CRC Press; 1999. [3] Evans WD, Harris DJ. Fractals, trees and the Neumann Laplacian. Math Ann 1993;296:493–527. [4] Falconer KJ. Fractal geometry. New York: John Wiley and Sons; 1990. [5] Falconer KJ. On the Minkowski measurability of fractals. Proc Amer Math Soc 1995;123(4):1115–24. [6] Federer H. Geometric measure theory. Springer-Verlag; 1969. [7] Harvey R, Polking J. Removable singularities of solutions of linear partial differential equations. Acta Math 1970;125:39–56. [8] He CQ, Lapidus ML. Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function. Mem Amer Math Soc 1997;127(608). [9] Krantz SG, Parks HR. The geometry of domains in space. Boston, MA: Birkh€auser Advanced Texts, Birkh€auser Boston, Inc; 1999. [10] Lapidus ML, Pomerance C. The Riemann Zeta-function and the one-dimensional Weyl–Berry conjecture for fractal drums. Proc London Math Soc (3) 1993;66(1):41–69. [11] Mattila P. Geometry of sets and measures in Euclidean spaces, fractals and rectifiability. Cambridge University Press; 1995. [12] Mou L. Removability of singular sets of harmonic maps. Arch Rational Mech Anal 1994;127:199–208. [13] Riihentaus J. Removable sets for subharmonic functions. Pacific J Math 2000;194:199–208. ubrinic D. Singular sets of Sobolev functions. CR Acad Sci Paris, Ser I 2002;334:539–44. [14] Z
Minkowski content and singular integrals ubrinic Darko Z Department of Applied Mathematics, Faculty of Electrical Engineering and Computing, Unska 3, 10000 Zagreb, Croatia Accepted 19 September 2002 Communicated by Prof. Dr. Drehc. W. Martienssen
Abstract Assume that lower and upper d-dimensional Minkowski contents of A RN are different both from 0 and 1. We show that the function dðx; AÞc is integrable in a tubular neighbourhood of A if and only if c < N d (the if part is known). von KochÕs curve and the Sierpinski gasket are shown to satisfy the Minkowski content condition. The notions of relative Minkowski content and relative box dimension are introduced in order to extend this result to singular functions generated by more general fractal sets, which may have classical upper or lower d-dimensional Minkowski content equal to 0 or 1. Ó 2002 Elsevier Science Ltd. All rights reserved.
1. Introduction The notion of Minkowski content appears naturally in the study of removable singularities of solutions of partial differential equations, see [7,13]. It is important in the study of asymptotic behavior of the eigenvalue counting function for the Dirichlet eigenvalue problem Du ¼ ku on a bounded domain X RN , where u ¼ 0 on oX, which may be fractal. See [8,10] for the detailed treatment of one-dimensional Weyl–Berry conjecture. Minkowski content is useful also in the study of singular sets of Sobolev functions, see [14]. Additional comments about Minkowski content can be found in [11, p. 80] and the references therein. In the present paper we are interested in the following problem: finding condition on a fractal set A RN , which enables to characterize Lebesgue integrability of singular function dðx; AÞc on a bounded neighbourhood of A. Here dðx; AÞ is the euclidean distance function from x to A. It is interesting that the required condition is of the same type as the one appearing in [10, Conjecture 1Õ and Theorem 2.4], namely, that the upper and lower d-dimensional Minkowski contents of A be different both from 0 and 1. See Theorems 2.3, 2.9, and 3.1 below. A sufficient condition for Lebesgue integrability of dðx; AÞc in a bounded neighbourhood of A has been stated in [12, Lemma 3.6], see also [7, the proof of Theorem 2.5, p. 42]: if the upper d-dimensional Minkowski content of A is finite, then the required sufficient integrability condition is c < N dimB A, where dimB A is the upper box dimension of A. In this paper we show that this sufficient condition is also necessary for integrability of dðx; AÞc in a bounded neighbourhood of A, assuming that both upper and lower d-dimensional Minkowski contents of A are contained in ð0; 1Þ. We extend this integrability characterization to a class of fractal sets whose lower or upper d-dimensional Minkowski contents may be 0 or 1, by introducing the notion of relative Minkowski content. The main results are stated in Theorems 2.3, 2.4 and 2.9.
ubrinic). E-mail address: [email protected] (D. Z 0960-0779/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 0 - 0 7 7 9 ( 0 2 ) 0 0 4 4 1 - 1
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2. Relative Minkowski content and relative box dimension If A is a subset of RN and r > 0, we denote euclidean r-neighbourhood of A by Ar , that is, Ar ¼ fy 2 R : dðy; AÞ < rg. By LN we denote the standard Lebesgue measure in RN . We first introduce classical upper and lower s-dimensional Minkowski contents of set A: N
M s ðAÞ ¼ lim sup r!0
LN ðAr Þ ; rN s
Ms ðAÞ ¼ lim inf r!0
LN ðAr Þ : rN s
ð2:1Þ
The infimum of all s P 0 such that M s ðAÞ ¼ 0 is called the upper box dimension of A, and is denoted by dimB A. Here we define inf ; ¼ þ1. One can analogously define the lower box dimension dimB A. If both of them coincide, the common value is called box dimension, and is denoted by dimB A. If M s ðAÞ ¼ Ms ðAÞ for some s, then the common value is called s-dimensional Minkowski content of A, and is denoted by Ms ðAÞ. If it is contained in ð0; 1Þ we say that A is Minkowski measurable. Minkowski measurability of fractal subsets of the real line is studied in [5,8]. We shall also deal with the boundary C of a given open set X of RN . Here X may be unbounded. Let us recall the definition of lower and upper s-dimensional Minkowski content of C with respect to X: Ms ðC; XÞ ¼ lim inf r!0
LN ðCr \ XÞ ; rN s
M s ðC; XÞ ¼ lim sup r!0
LN ðCr \ XÞ : rN s
ð2:2Þ
We can define the corresponding lower and upper box dimensions dimB ðC; XÞ and dimB ðC; XÞ analogously as above. See Examples 2.1 and 2.2 below. Motivated by He and Lapidus [8], we introduce the notion of relative Minkowski content. Let X be an open subset in RN (possibily unbounded), and A any subset of RN . Let h : ð0; 1Þ ! ð0; 1Þ be a given nondecreasing function, which we call the gauge function. We define the lower h-Minkowski content of A relative to X by M ðh; A; XÞ ¼ lim inf r!0
LN ðAr \ XÞ hðrÞ rN
ð2:3Þ
and analogously the upper relative h-Minkowski content M ðh; A; XÞ. If hðrÞ ¼ rs , we obtain s-dimensional lower and upper Minkowski contents Ms ðA; XÞ and M s ðA; XÞ of A relative to X. In this case we can define lower and upper box dimensions of A relative to X in the usual way, that we denote by dimB ðA; XÞ;
dimB ðA; XÞ
ð2:4Þ Md ðA; XÞ
d
respectively. If both of them coincide, we denote the common value by dimB ðA; XÞ. Also, if ¼ M ðA; XÞ, the common value will be denoted by Md ðA; XÞ. It is easy to see that lower relative box dimension has the usual monotonicity property: if A1 A2 and X1 X2 , then dimB ðA1 ; X1 Þ 6 dimB ðA2 ; X2 Þ, and analogously for the upper relative box dimension. Also, it is clear that if A X and the distance between A and oX is positive, then we obtain classical values of Minkowski contents: Ms ðA; XÞ ¼ Ms ðAÞ and
M s ðA; XÞ ¼ M s ðAÞ:
If X is a given open set and A ¼ C its boundary, then from (2.3) we obtain generalized lower h-Minkowski contents of C, introduced in [8]. Example 2.1. It is interesting that relative box dimension can be noninteger even in the case of sets A and X having nonfractal nature. For example, if X ¼ fx ¼ ðx1 ; x2 Þ 2 R2 : 1 < x1 < 1; 0 < x2 < jx1 ja g, where a 2 ð0; 1Þ is given, and A ¼ fx 2 X : x1 ¼ 0g, then it is easy to see that Md ðA; XÞ ¼ 2=ð1 aÞ, d ¼ dimB ðA; XÞ ¼ 1 þ a. Also, it is not difficult to show that 0 < Md ðC; XÞ 6 M d ðC; XÞ < 1, d ¼ dimB ðC; XÞ ¼ 1 þ a, where C ¼ oX. Example 2.2. If X ¼ fx ¼ ðx1 ; x2 Þ 2 R2 : 1 < x1 < 0; 0 < x2 < ekx1 g, where k > 0 is given, and A ¼ fx 2 X : x2 ¼ 0g, then for hðrÞ ¼ r=ðlogð1=rÞ þ 1Þ we have Mðh; A; XÞ ¼ 2=k, dimB ðA; XÞ ¼ 1, while M1 ðA; XÞ ¼ 1. Also, 0 < M1 ðC; XÞ 6 M 1 ðC; XÞ < 1, where C ¼ oX. Our first result concerns the necessary condition for integrability of the singular function dðx; AÞc in X. It is worth noting that condition (2.5) below is the same as condition (H3) appearing in [8, Definition 2.2].
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Theorem 2.3. Let X be an open set in RN . Let h : ð0; 1Þ ! ð0; 1Þ be a nondecreasing gauge function such that hðrÞ 6 crb for all r 2 ð0; RÞ, where c, and R are positive constants independent of r, and for any q 2 ð0; 1Þ there exist positive constants m and s < N independent of r, such that hðqrÞ P mqs hðrÞ
for all r 2 ð0; RÞ:
ð2:5Þ
Assume that 0 < M ðh; A; XÞ 6 M ðh; A; XÞ < 1. If
R X
dðx; AÞc dx < 1, then c < N b.
Proof. Assume by contradiction that c P N b. Let us choose numbers c 2 ð0; M ðh; A; XÞÞ and c 2 ðM ðh; A; XÞ; 1Þ such that for all q 2 ð0; RÞ there holds cqN hðqÞ1 6 LN ðAq \ XÞ 6 cqN hðqÞ1 :
ð2:6Þ
Now we fix any q 2 ð0; 1Þ, and take any q 2 ð0; RÞ. Then LN ððAq n Aqq Þ \ XÞ ¼ LN ðAq \ XÞ LN ðAqq \ XÞ P cqN hðqÞ1 cðqqÞN hðqqÞ1 P ðc m1 qN s cÞqN hðqÞ1 P CqN b ; where we have taken q small enough, so that C :¼ c1 ðc m1 qN s cÞ > 0. Using the corresponding decomposition i AR \ X ¼ [1 i¼0 Bi , where Bi :¼ ðARi n ARiþ1 Þ \ X, Ri :¼ q R, we obtain that Z Z 1 1 1 1 X X X X N b N dðx; AÞc dx ¼ dðx; AÞc dx P Rc Rc ¼ cRN bc qðN bcÞi ¼ 1; ð2:7Þ i L ðBi Þ P i cRi AR \X
i¼0
Bi
i¼0
since N b c 6 0, and q 2 ð0; 1Þ. This contradicts
i¼0
R X
c
dðx; AÞ dx < 1.
i¼0
The following result deals with the converse of the above theorem. Its proof is similar, and hence we omit it. Theorem 2.4. Let X be an open set in RN . Let h : ð0; 1Þ ! ð0; 1Þ be a nondecreasing gauge function such that hðrÞ P cra , for all r sufficiently small, where c and a are positive constants independent of r. Assume that M ðh; A; XÞ < 1. If c < R N a, then X dðx; AÞc dx < 1. The gauge function appearing in the following corollary is useful in applications, see [8]. Corollary 2.5. Let X be an open subset of RN , and let A be a subset of RN . Assume that there exist nonnegative constants a and d such that for hðrÞ ¼
rd ; þ 1Þa
½logð1r
we have 0 < M ðh; A; XÞ 6 M ðh; A; XÞ < 1. Then d ¼ dimB ðA; XÞ, and Z dðx; AÞc dx < 1 () c < N d:
ð2:8Þ
X
Proof. To prove the sufficiency part, assume that c < N d. Taking any a 2 ðd; N cÞ it is easy to see that hðrÞ P ra provided r is small enough. R Since c < N d, we can take a slightly larger than d, in order to have c < N a. Then Theorem 2.4 implies Rthat X dðx; AÞc dx < 1. Now assume that X dðx; AÞc dx < 1. We have that ra 6 hðrÞ 6 rd for small r, with a as above. Condition (2.5) is easy to check, see [8, (H3) on p. 88]. Therefore c < N d by Theorem 2.3. It is easy to see that Corollary 2.5 can be applied to Examples 2.1 and 2.2. We now apply it to several interesting examples appearing in [8]. Example 2.6. Let C be the boundary of the open domain X in R2 discussed in [8, Example 7.5], originally defined in [3]. Here for the gauge function hðrÞ ¼ r= logðð1=rÞ þ 1Þ we have 0 < M ðh; A; XÞ 6 M ðh; A; XÞ < 1 and d ¼ dimB ðC; XÞ ¼ 1. Then by Corollary 2.5,
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Z
dðx; CÞc dx < 1
()
ð2:9Þ
c < 1:
X
Note that Md ðC; XÞ ¼ M d ðC; XÞ ¼ 1 with d ¼ 1, so that Theorems 3.4 and 3.5 below are not applicable. Example 2.7. (a) Now considerS the open set X in R2 discussed in [8, Example 7.7], originally defined in [1]. Let
aðxÞ ¼ x1 ln x and define X ¼ 1 k¼3 ðaðk þ 1Þ; aðkÞÞ ð1=2; 1=2Þ. Here we have 0 < M ðh; C; XÞ 6 M ðh; C; XÞ < 1 with hðrÞ ¼ r3=2 =½logðð1=rÞ þ 1Þ1=2 , and d ¼ dimB ðC; XÞ ¼ 3=2. By Corollary 2.5, Z dðx; CÞc dx < 1 () c < 1=2:
ð2:10Þ
X
Note that in this example we also have Md ðC; XÞ ¼ M d ðC; XÞ ¼ 1. Furthermore, it is easy to see that Md ðCÞ ¼ Md ðC; XÞ ¼ 1, and M d ðCÞ ¼ M d ðC; XÞ ¼ 1. Example 2.8. See [8, Remark 7.8(c)]. Modifying aðxÞ in the above example to aðxÞ ¼ 1=ðx ln xÞ, we have the same conclusions, including (2.10), except that here Md ðC; XÞ ¼ M d ðC; XÞ ¼ 0, so that Theorem 3.5 below is not applicable. Furthermore, Md ðCÞ ¼ Md ðC; XÞ ¼ 0, and M d ðCÞ ¼ M d ðC; XÞ ¼ 0. Now we would like to estimate growth rate of singular integrals in terms of the gauge function. Theorem 2.9. Assume that A is a closed subset of RN having Lebesgue measure zero, X an open subset of RN . Assume that c > 0 and there exists a nondecreasing gauge function h : ð0; 1Þ ! ð0; 1Þ such that Z r N c1 tN c t ! 0; t ! 0; dt < 1: ð2:11Þ hðtÞ hðtÞ 0 (a) If M ðh; A; XÞ < 1, then Z Z r dðx; AÞc dx ¼ rc LN ðAr \ XÞ þ c LN ðAt \ XÞtc1 dt: Ar \X
ð2:12Þ
0
In particular, given r0 > 0, for any r 2 ð0; r0 Þ, N c Z Z r N c1 r t dt ; þc dðx; AÞc dx 6 C hðrÞ hðtÞ 0 Ar \X
ð2:13Þ
where C ¼ supt2ð0;r0 Þ ðLN ðAt \ XÞ=tN ÞhðtÞ. (b) If 0 < M ðh; A; XÞ 6 M ðh; A; XÞ < 1, and r0 > 0 is given, then for any r 2 ð0; r0 Þ we have the following estimate: N c N c Z r N c1 Z Z r N c1 r t r t þc dt 6 þc dt ; ð2:14Þ C dðx; AÞc dx 6 C hðrÞ hðtÞ hðrÞ hðtÞ 0 0 Ar \X where C ¼ inf t2ð0;r0 Þ LN ðAt \ XÞ=tN hðtÞ. In the proof we shall use the following auxiliary result, which extends [2, Proposition 1 in Section 3.4.4]. Proposition 2.10. Let A be a closed subset of RN with Lebesgue measure equal to zero, and X an open subset of RN . Then for f ðxÞ ¼ dðx; AÞ we have jrf j ¼ 1 a.e. in RN , and for any g 2 L1 ðXÞ, Z Z 1 Z ð2:15Þ gðxÞ dx ¼ gðxÞ dH N 1 dt; 0
X
where H d dt
N 1
Z
oAt \X
is HausdorffÕs ðN 1Þ-dimensional measure. Furthermore, Z gðxÞ dx ¼ g dH N 1 for a:e: t > 0; At \X
ð2:16Þ
oAt \X
d N L ðAt \ XÞ ¼ H N 1 ðoAt \ XÞ for a:e: t > 0: dt
ð2:17Þ
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Proof. Using RademacherÕs theorem (see [2, Theorem 2 in Section 3.1.2]) and the proof from [6, Lemma 3.2.34], we obtain that Jf ¼ jrf j ¼ 1 a.e. in RN . We exploit the change of variables formula from [2, Theorem 2 in Section 3.4.3]: # Z Z "Z g dH N m dt
gðxÞJf ðxÞ dx ¼
RN
Rm
ð2:18Þ
f 1 ðtÞ
with m ¼ 1, and g 0 outside of X. This implies (2.15). It is easy to see that (2.15) ) (2.16) ) (2.17).
Proof of Theorem 2.9 (a) First note that LN ðAt \ XÞ 6 C
tN hðtÞ
for all t 2 ð0; r0 Þ, hence, by (2.11) we conclude that the set A \ X has the Lebesgue measure zero. We define gðxÞ ¼ dðx; AÞc in Ar and gðxÞ ¼ 0 otherwise, and exploit Proposition 2.10: Z Z r dðx; AÞc dx ¼ tc H N 1 ðoAt \ XÞ dt: ð2:19Þ Ar \X
0
Recalling (2.17), we integrate by parts the right hand side of (2.19), and obtain the identity (3.1) below. Here we have used that tc LN ðAt \ XÞ ¼ CCtN c =hðtÞ ! 0 as t ! 0. Estimate (2.14) follows easily from the fact that LN ðAt \ XÞ 6 CtN =hðtÞ, for t 2 ð0; r0 Þ. (b) The claim follows from (a) and from LN ðAt \ XÞ P CtN =hðtÞ, for t 2 ð0; r0 Þ.
3. Classical Minkowski content In the case of classical Minkowski content we can state stronger results than in Theorems 2.3 and 2.4. Inequality (3.2) of the following Theorem 3.1 is known, see e.g. [12, Lemma 3.6], except that our constant appearing in (3.2) is better than in [12, Lemma 3.6]. This inequality seems to have been known over a longer period, see [7, the proof of Theorem 2.5, p. 42]. Its proof was obtained using diadic decomposition of Ar . Here we provide a different proof, based on the application of Theorem 2.9, that is, generalized change of variables formula. Theorem 3.1. Assume that A is a closed subset of RN , and let r > 0. (a) Assume that d P 0 is such that M d ðAÞ < 1. If c < N d, then Z Z r dðx; AÞc dx ¼ rc LN ðAr Þ þ c LN ðAt Þtc1 dt:
ð3:1Þ
0
Ar
In particular, Z dðx; AÞc dx 6 Ar
N d LN ðAt Þ N dc sup r : N d c t2ð0;rÞ tN d
(b) Assume that there exists d ¼ dimB A, and 0 < Md ðAÞ 6 M d ðAÞ < 1. Then Z dðx; AÞc dx < 1 () c < N dimB A:
ð3:2Þ
ð3:3Þ
Ar
Furthermore, if c < N d, and r0 is a given positive number, then for all r 2 ð0; r0 Þ we have Z c1 rN dc 6 dðx; AÞc dx 6 c2 rN dc ; Ar
where c1 and c2 are two positive constants independent of r.
ð3:4Þ
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Proof of Theorem 3.1 (a) Note that dimB A < N implies that LN ðAÞ ¼ 0. The claim follows from Theorem 2.9(a), with X ¼ RN and hðrÞ ¼ rd . (b) The equivalence stated in (3.3) follows from Corollary 2.5. Estimate (3.4) is a consequence of Theorem 2.9(b), and c1 ¼
N d LN ðAt Þ ; inf t2ð0;r Þ N d c tN d 0
c2 ¼
N d LN ðAt Þ : sup N d c t2ð0;r0 Þ tN d
ð3:5Þ
The following corollary provides some examples of singular functions generated by classical fractal sets. Corollary 3.2 (a) Assume that a > 0 is given, and let A ¼ f1=na : n 2 Ng. Then Z 1 1 : dðx; AÞc dx < 1 () c < 1 aþ1 0 (b) If C is the standard Cantor set, then Z 1 log 2 : dðx; CÞc dx < 1 () c < 1 log 3 0
ð3:6Þ
ð3:7Þ
More generally, let C ðm;aÞ be generalized Cantor set defined by an integer m P 2 and a 2 ð0; 1=mÞ, see [4]. Then Z 1 log m : ð3:8Þ dðx; C ðm;aÞ Þc dx < 1 () c < 1 log 1=a 0 (c) If S is the Sierpinski gasket, then Z log 3 dðx; SÞc dx < 1 () c < 2 ; log 2 T where T is equilateral triangle in R2 containing S. (d) If K is von Koch’s curve, then for any r > 0, Z log 4 : dðx; KÞc dx < 1 () c < 2 log 3 Kr
ð3:9Þ
ð3:10Þ
Proof (a) It suffices to check that 0 < Md ðAÞ 6 M d ðAÞ < 1, see Theorem 3.1(b). We have that A is Minkowski measurable, and a=aþ1 2 Md ðAÞ ¼ ða þ 1Þ; ð3:11Þ a where d ¼ 1=ða þ 1Þ, see [10, Example 4.3]. (b) Using methods of Lapidus and Pomerence [10], or by direct computation, it follows that 1d 1 2d Md ðC ðm;aÞ Þ ¼ ; d 1d d1 1 ma mð1 aÞ
d ðm;aÞ ; Þ¼ M ðC 2ðm 1Þ m1
ð3:12Þ ð3:13Þ
where d ¼ log m= logð1=aÞ. It can be shown that the Cantor set is not Minkowski measurable, that is, the above two quantities are never equal. In the case of classical Cantor set (that is, m ¼ 2, a ¼ 1=3), we obtain Md ðCÞ
log 9 ¼ log 3=2
log 3=2 log 4
log 2= log 3 ;
M d ðCÞ ¼ 22log 2= log 3 ;
D. Z ubrinic / Chaos, Solitons and Fractals 17 (2003) 169–177
see [10, Theorem 2.4]. For (c) and (d) see Propositions A.1 and A.2 in the Appendix A.
175
Many new examples of fractal sets satisfying conditions of Theorem 3.1 can be generated using the following proposition. It is immediate consequence of Krantz, Parks [9, Theorem 3.3.6]. Proposition 3.3. If two subsets A RN and B RM satisfy conditions 0 < Md ðAÞ 6 M d ðAÞ < 1 and 0 < Me ðBÞ 6 M e ðBÞ < 1, then the analogous condition holds for the product set A B RN þM :
ðdþeÞ 0 < Mdþe ðA BÞ < 1:
ðA BÞ 6 M
In [10, Theorem 2.4] one can see a characterization of subsets A of the real line satisfying condition 0 < Md ðAÞ 6 M d ðAÞ < 1. This condition appears in Theorem 3.1(b). Using the same proofs as above we obtain the following results involving functions that are singular on the boundary C of an open set X, where the boundary may be fractal. Theorem 3.4. Let X be an open set in RN , C ¼ oX, 0 6 d < N, and M d ðC; XÞ < 1. If c < N d, then R c dðx; CÞ dx < 1, and furthermore, X Z
dðx; CÞc dx 6
Cr \X
N d LN ðCt \ XÞ N dc sup r N d c t2ð0;rÞ tN d
for any r > 0. In particular, if R is inner radius of X, then Z N d LN ðCt \ XÞ N dc sup dðx; CÞc dx 6 R : N d c t2ð0;RÞ tN d X Theorem 3.5. Let X be an open set in RN , 0 6 d < N and 0 < Md ðC; XÞ 6 M d ðC; XÞ < 1. Then Z dðx; CÞc dx < 1 () c < N d:
ð3:14Þ
ð3:15Þ
ð3:16Þ
X
Furthermore, if c < N d, and if r0 > 0 is given, then there exist positive constants c1 and c2 independent of r 2 ð0; r0 Þ, such that Z c1 rN dc 6 dðx; CÞc dx 6 c2 rN dc : ð3:17Þ Cr \X
Appendix A. Minkowski contents of the Sierpinski gasket and von KochÕs curve The following result shows that the Sierpinski gasket has different values of lower and upper Minkowski contents, although rather close to each other. It was used in the proof of Corollary 3.2. Proposition A.1. Let S be the Sierpinski gasket. Then 0 < Md ðSÞ < M d ðSÞ < 1, where d ¼ log 3= log 2 is the box dimension of S. The values of Md ðSÞ and M d ðSÞ can be expressed explicitely, and Md ðSÞ 1:810822012, M d ðSÞ 1:814488327 (all decimal values are precise). Proof. Let T be a closed (full) equilateral triangle with unit side. In the first step we remove an open triangle with side 1=2, and in the nth step we remove 3n1 triangles with sides 1=2n (see [4, Figure 0.3]). Continuing this procedure ad infinitum we obtain the Sierpinski gasket S. Now we compute the Lebesgue measure of r-neighbourhood of S. Let r > 0 be given. The condition that a full triangle in the nth generation is not contained in the neighbourhood of its boundary is that r < hn =3, where hn is the height of the triangle. It is easy to see that the largest such n is nðrÞ ¼
1 log2 pffiffiffi 1: 2 3r
ðA:1Þ
D. Z ubrinic / Chaos, Solitons and Fractals 17 (2003) 169–177
176
We define inner r-neighbourhood of any (full) triangle M as ðoMÞr \ M. Let us take any k, 1 6 k 6 nðrÞ, and consider the corresponding inner of any triangle of kth generation. An easy computation shows that its measure is pffiffir-neighbourhood ffi equal to 3r2k 3 3r2 . Hence the measure of the union of all inner r-neighbourhoods of triangles over generations k with 1 6 k 6 nðrÞ is equal to nðrÞ nðrÞ X pffiffiffi 3 3 pffiffiffi 2 nðrÞ 3 pffiffiffi 2 1 ðA:2Þ ð3r2k 3 3r2 Þ3k1 ¼ 3r 3r 3 þ 3r r : 2 2 2 2 k¼1 The remaining subset of Sr consists of the set A defined as the union of areas of all triangles fully covered with their rneighbourhoods, corresponding to generations k > nðrÞ, and the set B ¼ Tr n T . We have pffiffiffi nðrÞ 2 pffiffiffi pffiffiffi nðrÞ 3 X k1 1 3 3 3 ¼ 3 ðA:3Þ L2 ðAÞ ¼ k 2 4 4 4 4 k¼1 and clearly, L2 ðBÞ ¼ 3r þ pr2 . Hence, L2 ðSr Þ ¼ f ðrÞ þ Oðrd1 Þ; r2d
ðA:4Þ
where f ðrÞ ¼ r
d2
pffiffiffi nðrÞ ! nðrÞ 3 3 pffiffiffi 2 nðrÞ 3 3 : 3r 3r 3 þ 2 2 4 4
ðA:5Þ
pffiffiffi Let us fix any natural number n, and consider r satisfying n < log2 ð1=2 3rÞ 6 n þ 1, that is, r 2 In :¼ ½rnþ1 ; rn Þ;
1 rn :¼ pffiffiffi : 2 32n
Note that nðrÞ ¼ n for such r. It is easy to see that the function fn : In ! R, fn ðrÞ ¼ f ðrÞ, has points of minimum and maximum at r and rþ 2 In respectively, defined by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðd 1Þ ðd 1Þ2 þ 12 dðd 2Þ 1 pffiffiffi : r ¼ n d ; d ¼ 2 3d An easy computation shows that the values of fn ðr Þ and fn ðrþ Þ do not depend on n, as well as fn ðrnþ1 Þ ¼ fn ðrn 0Þ. Therefore, see (A.4), pffiffiffi ! 3 3 pffiffiffi 3 d 3 þ 2 3log2 d ¼ 1:810822012; M ðSÞ ¼ lim inf f ðrÞ ¼ fn ðr Þ ¼ r!0 d 2 4d pffiffiffi ! 3 3 3 pffiffiffi
d 3 þ 2 3log2 dþ ¼ 1:814488327: M ðSÞ ¼ lim sup f ðrÞ ¼ fn ðrþ Þ ¼ d 2 4d r!0 þ þ Oscillating nature of the function f ðrÞ near r ¼ 0 can nicely be seen by plotting its graph.
We do not know the precise values of lower and upper d-dimensional Minkowski content of von KochÕs curve K. However, in Proposition A.2 below we obtain that 0:9923660518 6 Md ðKÞ 6 M d ðKÞ 6 1:789812198, which enables to use Theorem 3.1 also in this case. We do not know whether von KochÕs curve is Minkowski measurable or not. Proposition A.2. Let K be von KochÕs curve, and d ¼ log 4= log 3. Then 33 11 pffiffiffi 5p 15 47 pffiffiffi p 3þ 6 Md ðKÞ 6 M d ðKÞ 6 þ 3þ : 32 288 576 32 64 64
ðA:6Þ
Sketch of the Proof. Let Cn be a decreasing sequence of compact sets in R2 defined as follows. For C1 we choose isosceles triangle with basis 1, and with both angles at the basis equal to p=6. The set C2 is obtained from C1 in two steps: (a) remove an open equilateral triangle defined by the middle third of the basis of C1 and the opposite vertex of C1 ; (b) at this moment we are left with two smaller isosceles triangles that are similar C1 , and for each of them we repeat the same
D. Z ubrinic / Chaos, Solitons and Fractals 17 (2003) 169–177
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procedure as in (a). Hence, C2 consists of four mutually congruent, isosceles triangles, that are similar to C1 . In this way we obtain a decreasing sequence of compact sets Cn . Each Cn consists of 4n mutually congruent triangles. It is easy to see that each Cn contains K, and moreover, K ¼ \1 n¼1 Cn . To estimate L2 ðKr Þ from above, let us fix r > 0. We take the largest n such that 1=3n P r, that is, nðrÞ ¼ blog3 ð1=rÞc. Since K Cn for any n, we obtain after a long, but elementary computation, that L2 ðKr Þ L2 ððCnðrÞ Þr Þ 6 ¼ gðrÞ þ Oðrd1 Þ; r2d r2d
ðA:7Þ
where f ðrÞ ¼ r
d2
! " pffiffiffi nðrÞ 4 3 5 r þ þ 8 3 3
pffiffiffi nðrÞ # pffiffiffi ! p 3 2 nðrÞ 3 3 4 r4 þ : 16 16 9 4
ðA:8Þ
pffiffiffi It is easy to see that the supremum of f ðrÞ, r > 0, is equal to ð15=32Þ þ ð47=64Þ 3 þ ðp=64Þ, and due to (A.7) it is P M d ðKÞ. To obtain an estimate for Md ðKÞ from below, we introduce the standard sequence of piecewise linear subsets Kn which approximate von KochÕs curve. First we define K1 as the set consisting of four intervals of length 1=3 each, then the set K2 consisting of 42 intervals of length 1=32 each, etc. (see [4, Figure 0.2]). Note that the set K is connected, and each of 4n subintervals constituting Kn has boundary points contained in K. We start with the following inequality: L2 ðKr Þ P L2 ððKn Þr Þ;
ðA:9Þ
which holds for all r > 0 and n. We have that L2 ððKnðrÞ Þr Þ L2 ðKr Þ P ¼ gðrÞ þ Oðrd1 Þ; 2d r r2d
ðA:10Þ
where " gðrÞ ¼ r
d2
# nðrÞ 11 4 5p 11 pffiffiffi 2 n r þ 3 r 4 : 8 3 144 72
ðA:11Þ
pffiffiffi It is easy to see that the infimum of gðrÞ, r > 0, is equal to ð33=32Þ ð11=288Þ 3 þ ð5p=576Þ, and due to (A.10) it does not exceed Md ðKÞ.
References [1] Caetano AM. Some domains where the eigenvalues of the Dirichlet Laplacian have non-power second term asymptotic estimates. J London Math Soc 1991;43(2):431–50. [2] Evans LC, Gariepy RF. Measure theory and fine properties of functions. CRC Press; 1999. [3] Evans WD, Harris DJ. Fractals, trees and the Neumann Laplacian. Math Ann 1993;296:493–527. [4] Falconer KJ. Fractal geometry. New York: John Wiley and Sons; 1990. [5] Falconer KJ. On the Minkowski measurability of fractals. Proc Amer Math Soc 1995;123(4):1115–24. [6] Federer H. Geometric measure theory. Springer-Verlag; 1969. [7] Harvey R, Polking J. Removable singularities of solutions of linear partial differential equations. Acta Math 1970;125:39–56. [8] He CQ, Lapidus ML. Generalized Minkowski content, spectrum of fractal drums, fractal strings and the Riemann zeta-function. Mem Amer Math Soc 1997;127(608). [9] Krantz SG, Parks HR. The geometry of domains in space. Boston, MA: Birkh€auser Advanced Texts, Birkh€auser Boston, Inc; 1999. [10] Lapidus ML, Pomerance C. The Riemann Zeta-function and the one-dimensional Weyl–Berry conjecture for fractal drums. Proc London Math Soc (3) 1993;66(1):41–69. [11] Mattila P. Geometry of sets and measures in Euclidean spaces, fractals and rectifiability. Cambridge University Press; 1995. [12] Mou L. Removability of singular sets of harmonic maps. Arch Rational Mech Anal 1994;127:199–208. [13] Riihentaus J. Removable sets for subharmonic functions. Pacific J Math 2000;194:199–208. ubrinic D. Singular sets of Sobolev functions. CR Acad Sci Paris, Ser I 2002;334:539–44. [14] Z