Surface integrals for domains with fractal boundaries ...

Surface integrals for domains with fractal boundaries and some applications to elasticity By F e o d o r M. B o r o di c hy a n d Al ek s e i Yu. Vo lo vik ovz Department of Mathematics, Glasgow Caledonian University, Cowcaddens Road, Glasgow G4 0BA, UK Received 12 November 1998; revised 10 May 1999; accepted 8 July 1999

Some possible formulations of integration of di¬erential forms over a non-smooth boundary are introduced. It is supposed that the di¬erential of a form is integrable on the whole domain. In applications to continuum mechanics, the condition of integral convergence selects the physically interesting cases, when the energy over the domain is ­ nite. Conditions providing continuity of surface integrals for domains in n with fractal boundary of non-integer Hausdor¬ dimension and non-integer box dimension for both continuous and special discontinuous di¬erential forms are described. The results obtained are applied to elastic domains in 3 with fractal boundaries. Some discontinuous di¬erential forms are considered when the forms are formed by the use of the elastic stress and displacement ­ elds. The proof of the uniqueness theorem of solutions to problems of elastostatics for bodies with fractal boundaries is given. Keywords: divergence theorem; fractal boundaries; elastic b o dies

1. Introduction The term `fractal’ is usually used to characterize some features of a physical object obeying the power law of scale where the scale varies in an interval between the upper and lower cut-o¬. However, there are purely mathematical fractals|mathematical objects of non-integer fractal dimension which have no lower cut-o¬. Such fractals should be studied using strictly mathematical approaches. Here possible formulations of integration of a di¬erential form over a non-smooth domain boundary, in particular over a fractal boundary of non-integer Hausdor¬ dimension, are discussed. There has been considerable e¬ort in the literature to formulate boundary problems of mathematical physics for domains with highly irregular boundaries (see, for example, Kats (1982, 1983, 1985, 1986, 1987, 1990, 1992 personal communication, 1993); Wallin (1991); Harrison & Norton (1991, 1992) and a series of preprints by J. Harrison that has been recently published on her Web page http://www.math. berkeley.edu/harrison). The Riemann boundary-value problem for various non-smooth Jordan curves on the complex plane was considered by Kats in a series of papers (see, for example, y Present address: Division of Applied Mathematics, Department of Mathematical Sciences, The University of Liverpool, M&O Building, Liverpool L69 3BX, UK. z Present address: Steklov Mathematical Institute, Russian Academy of Sciences, 42 Vavilova, Moscow 117966, Russia. c 2000 The Royal Society

Proc. R. Soc. Lond. A (2000) 456, 1{24

1

2

F. M. Borodich and A. Yu. Volovikov

Kats (1982, 1983, 1986, 1986, 1987, 1990, 1992 personal communication, 1993). He obtained conditions for the solvability of the problem, studied singularities of solutions and various types of solution jumps. The results obtained are also valid for fractal curves on the complex plane . Wallin (1989, 1991) considered weak or variational solutions to the Dirichlet problem for the Laplace equation in the case when the domain boundary is a fractal. The main question of his papers was how to determine the trace of Sobolev spaces to a boundary of a domain. The Ostrogradskii{Gauss theorem, or the divergence theorem, is used in various areas of physics and mechanics when we need to convert a volume integral to a surface integral or a surface integral to a line integral. For example, we encounter the need of the theorem in solid mechanics using the variational principles of total potential energy and total complementary energy (see, for example, Rabotnov 1979; Willis 1982, 1983). The Ostrogradskii{Gauss theorem in the classical form hu; ni ds =

div u dv

@V

V

is based on a supposition that a domain V in n with boundary @V is su¯ ciently smooth and we can determine the outward unit normal n to the boundary. Here u = (b1 ; : : : ; bn ) is a vector ­ eld, ds is the element of (n 1)-dimensional volume of the surface @V , dv is the element of volume of the domain and h ; i denotes the scalar product. However, if the boundary is a fractal having dimension between n 1 and n, then the unit normal cannot be de­ ned. This raises the question: how do we generalize the Ostrogradskii{Gauss theorem when @V is a fractal? This is the question of urgency because of the popularity of applications of fractal geometry to various problems for domains with rough surfaces. If we employ the usual di¬erential geometry and classical mechanics techniques of q-forms and exterior di¬erentials (see, for detail of the techniques, Arnold 1978; Abraham & Marsden 1978; Flanders 1963), then it is well known that the divergence theorem can be rewritten in the following form for smooth forms and piecewise smooth boundaries (the Newton{Leibniz{Ostrogradskii{ Gauss{Green{Stokes{Poincar´e formula) != @V

d!;

(1.1)

V

where n

( 1)j+

!=

1

bj (x) dx1 ^

^ dxj ^

^ dxn ;

j= 1

the symbol above a term means that this term is deleted. If u = (b1 ; : : : ; bn ), then hu; ni ds = @V

!

and

@V

div u dv = V

because d! = div u dx1 ^ Proc. R. Soc. Lond. A (2000)

d!; V

^ dxn :

Surface integrals for domains with fractal boundaries

3

Recently, Harrison & Norton (1991, 1992) presented an approach to the divergence theorem for domains with boundaries of non-integer box dimension. One of the methods they employed was the technique introduced by H. Whitney (1935) (see, for example, Stein 1970), of decomposition of the domain into cubes and extension of functions de­ ned on a closed set to functions de­ ned on the whole of n . These techniques were also employed by Kats (1987). However, Harrison & Norton (1992) considered not only the two-dimensional case as did Kats (1982, 1983, 1985, 1986, 1987, 1990, 1991, 1993) but also the general case of n . Formula (1.1) was often used as a de­ nition of the surface integral of a form (see discussion by Harrison & Norton (1992)), while Harrison & Norton (1992), in order to de­ ne a surface integral, used a construction which can be roughly described in the following way. They considered the Whitney extension of the form from the surface to the whole of n and proved that the integral V d! exists and does not depend on the extension under some conditions regarding box dimension of the surface and the H older’s class of !. Thus, this integral can be taken as a de­ nition of @V !. In this paper we are interested in de­ nitions of integration of di¬erential forms over a fractal closed surface which bounds a Jordan domain. We consider the case when the integral of d! over the domain is ­ nite. In applications to elasticity it will be shown that this condition of integral ­ niteness has a clear physical sense, namely this integral is elastic energy accumulated within the domain. The condition of ­ niteness allows us to consider as a fractal dimension of the surface not only the box dimension as it was considered by Kats (1983, 1987) and Harrison & Norton (1992) but also the Hausdor¬ dimension. While Harrison & Norton (1992) considered only continuous forms, we are also interested in some discontinuous forms. The cases when the integral of ! has no jump across the fractal boundary of the domain are described. Let be an open subset of n , whose boundary is a smooth Jordan hypersurface S. Let us divide by a closed Jordan hypersurface into two open subsets VI and VM . Thus, we have = VM [ [ VI or n = VM [ VI where VI and VM are internal and external subsets for , respectively. Let us suppose that ! is (n 1)-form de­ ned on n such that d! is integrable on n , i.e. the integrals d!

and

d!

VM

VI

converge. Using this supposition, the integrals ! §

and

! §

M

I

over hypersurface can be de­ ned (see de­ nitions of the integrals in x 2 a). These integrals are limits of integrals over piecewise smooth hypersurfaces tending to from inside and outside. We are interested in conditions which provide the validity of the following equation: != §

M

!: §

It is easily proved that (1.2) holds in the case when restriction of a continuous form de­ ned on the whole of Proc. R. Soc. Lond. A (2000)

(1.2)

I

is smooth and ! is the . However, equation (1.2)

4

F. M. Borodich and A. Yu. Volovikov

does not hold in the general case of fractal hypersurface even for continuous !. We will say that there is no jump of integrals if (1.2) holds and call (1.2) the property of continuity of integrals. Integrals ! §

and

! §

M

I

exist even in the case when coe¯ cients of ! are unbounded in any neighbourhood of and thus ! cannot be extended to a continuous form de­ ned on the whole of . We are interested in consideration of special discontinuous forms for which (1.2) holds under appropriate conditions. In applications to elasticity, we consider a 2form ! = A(u), which is the di¬erential of the work on displacements (see x 4). This 2-form is discontinuous even in the case when is smooth, and it is easily seen that in this case (1.2) holds. When is a fractal surface, coe¯ cients of ! = A(u) may be unbounded. If (1.2) does not hold, then it may be stated that accumulates elastic energy. The 2-form A(u) plays an important role in the proof of the uniqueness theorem of elasticity in x 4. The paper is organized as follows. In x 2 we introduce some of the possible definitions of integration of a di¬erential form over a fractal boundary of non-integer Hausdor¬ dimension and prove a theorem of continuity (theorem 2.5) of the integrals across an internal non-smooth surface for continuous forms. In x 3 we prove continuity results for some special class of bounded and unbounded discontinuous forms in n . We consider fractal hypersurfaces with non-integer Hausdor¬ dimension and also hypersurfaces having box dimension between n 1 and n. In x 4 some applications of the developed techniques to elasticity are given. De­ nitions of work and elastic energy in the case of a body with a fractal boundary are discussed. We consider here some special discontinuous di¬erential 2-form ! formed by the use of the elastic stress and displacement ­ elds and prove continuity theorems for it (theorems 4.1, 4.2). At the end of x 4 the uniqueness theorem (theorem 4.5) of solutions to problems of elastostatics for bodies with fractal boundaries is proved.

2. The surface integrals for domains with fractal boundaries Let us consider n with Cartesian coordinates x1 ; : : : ; xn and the standard orientation. Every di¬erential (n 1)-form on the space n is uniquely written in the form (Arnold 1978) n

aj (x) dx1 ^

!=

^ dxj ^

^ dxn ;

j= 1

where the aj (x) are some functions on n and ^ is the exterior product. The form is continuous (smooth) if these functions are continuous (smooth of class C 1 ). We say that ! is of class H¬ , where 0 < 1, if all coe¯ cients are H older functions of exponent , i.e. there exists constant M > 0 such that jaj (x)

aj (y)j

We write !jA = 0 where A closed if d! = 0. Proc. R. Soc. Lond. A (2000)

M jx n

yj¬

for all 1

j

n:

if !(x) = 0 for any x 2 A. We call a form !

Surface integrals for domains with fractal boundaries

5

(a) De¯nition of surface integrals of a di® erential form Let be an open subset of n , whose boundary is a smooth Jordan hypersurface S. Let us cut by a closed Jordan hypersurface . Let ! be a smooth (n 1)-form de­ ned on n = VM [ VI . We will suppose that d! is integrable on n , i.e. the integrals d!

and

d!

VM

VI

converge. This condition will always be supposed below. Let fSk g, Sk VM , be a sequence of smooth (or piecewise smooth) hypersurfaces such that (1) Sk and

bound a domain Wk with @Wk = Sk [

; and

where "k ! 0 when k ! 1.

(2) Sk lies in the "k -neighbourhood of Then we de­ ne

! = lim §

!:

1

k!

M

Sk

Here Sk is oriented as a boundary of the following domain: Vk = W k [ that @Vk = Sk . Similarly we de­ ne ! = lim §

where fTk g, Tk such that (1) Tk and

k!

I

[ VI , so

!;

1

Tk

VI , is a sequence of smooth (or piecewise smooth) hypersurfaces ~ k with @ W ~ k = Tk [ bound a domain W

~ k lies in an "k -neighbourhood of (2) W

; and

where "k ! 0 when k ! 1.

Here Tk is oriented as a boundary of its domain. The following proposition implies that these de­ nitions do not depend on the particular choice of Sk and Tk and therefore the integrals ! §

and

! §

I

M

are well de­ ned. Proposition 2.1. Denote by S 0 VM any smooth hypersurface surrounding in such a way that S 0 and bound a domain W with @W = S 0 [ . Then != §

d! + W

M

!

and

!=

S0 §

I

d!: VI

Proof . In fact, S 0 and Sk for large k bound a domain Wk0 and, by the Ostrogradskii{Gauss{Stoke s formula, ! Sk

Proc. R. Soc. Lond. A (2000)

!= S0

d! = Wk0

d! + W

d!: Wk

6

F. M. Borodich and A. Yu. Volovikov

The second equation follows from the fact that W = Wk [ Wk0 and Wk \ Wk0 = Sk . Since Wk lies in the "k -neighbourhood of , it follows from the convergence of the integral VM d! that Wk d! tends to 0 when k ! 1. Taking a limit we obtain the ­ rst formula. The second assertion is proved similarly. Thus, surface integrals in the case under consideration can be de­ ned using the Ostrogradskii{Gauss{Stoke s formula. For example, if ! is not de­ ned on S, we can de­ ne S ! by the formula != S

~ W

d! +

!; S0

~ = S [ S 0 . It is easy to check that this de­ nition does not depend on the where @ W choice of S 0 . Note 2.2. It follows directly from the de­ nition of § M ! that this integral depends only on !jVM (similarly § I ! depends only on !jVI ). In particular, if !jVM = 0, then ! = 0. § M The natural question is when do the integrals § M ! and § I ! coincide? These integrals cannot be the same and the di¬erence can occur even in the case of a continuous form (see Borodich & Volovikov 1997). (b) Some basic de¯nitions of fractal geometry The classic de­ nition says that a fractal is a set with a non-integer Hausdor¬ dimension. However, instead of the Hausdor¬ dimension, one takes often as a fractal dimension one from numerous other dimensions: box dimension, similarity dimension, etc. Let O be the totality of open balls and V be a cover of a set . Recall that for n is de­ ned as the following limit: s 0 the Hausdor® s-measure of a set (diam B)s :

mH ( ; s) = lim inf ¯ !

0+

V2 O

B;

B2 V

diam B

:

B2 V

Here V is a ­ nite or denumerable subset of O and diam B is a diameter of B. We will denote diam B simply by jBj. It was shown that the value mH ( ; s) possesses the following property: there exists the value d such that mH ( ; s) = The Hausdor® dimension of the set dimH

1; 0;

for s < d; for s > d:

is de­ ned by

= d = inffs : mH ( ; s) = 0g = supfs : mH ( ; s) = 1g:

The net s-dimension measure is de­ ned similarly to the Hausdor¬ s-measure, when open balls in the de­ nition are replaced by binary cubes. We remind that a closed binary cube is a cube [l1 2

k

; (l1 + 1)2

where l1 ; : : : ; ln are integers. Proc. R. Soc. Lond. A (2000)

k

]

[ln 2

k

; (ln + 1)2

k

];

Surface integrals for domains with fractal boundaries

7

To determine the Hausdor¬ dimension of a compact set it is enough to consider only ­ nite coverings by closed binary cubes that in addition satisfy the following condition: intersection of the interiors of any pair of cubes is empty. It is known (see Falconer 1990) that the Hausdor¬ dimension of a set does not change if we use the net measures instead of the Hausdor¬ measures. The net measures will be used below. n By de­ nition the box dimension of a bounded subset is equal to lim

"!

0

log N§ (") ; log "

where N§ (") is the least number of "-balls needed to cover Falconer 1990; Pontrjagin & Schnirelmann 1932). De¯nition 2.3. The set

(see, for example,

is d-summable if the improper integral 1

N§ (")"d

1

d"

0

converges. Note 2.4. The notion of a d-summable subset was introduced in Harrison & Norton (1992), who showed that if has box dimension less than d, then is d-summable. (c) Continuity and jumps of surface integrals for continuous forms For continuous forms we have the following theorem of continuity of integrals. Theorem 2.5. Let ! 2 H¬ ( ). If dimension of , then

!= §

n + 1, where d is the Hausdor®

> d

!: §

M

I

Proof . It is su¯ cient for any " > 0 and > 0 to ­ nd such piecewise smooth hypersurfaces S® VM and T® VI that bound a domain containing and lying in a -neighbourhood of such that ! S

! < ": T

We have n

aj (x) dx1 ^

!=

^ dxj ^

^ dxn :

j= 1

Let us consider a ­ nite covering V = fQk j k 2 Jg of by closed binary cubes and put (V ) = maxk2 J jQk j. Here J is the ­ nite set of integer numbers k of the cover V . Let us suppose that the intersections of the interiors of any two cubes are empty and that int [k2 J Qk , where int denotes the interior of a subset. Then [k2 J Qk is a Proc. R. Soc. Lond. A (2000)

8

F. M. Borodich and A. Yu. Volovikov

piecewise smooth hypersurface with boundary S® [ T® where S® It is easy to see that !=

!

@Qk

k2 J

VM

and T®

!:

S

VI . (2.1)

T

Now let Pk 2 Qk be any point and denote by !(Pk ) the form n

aj (Pk ) dx1 ^

!(Pk ) =

^ dxj ^

^ dxn :

j= 1

Since d!(Pk ) = 0, we obtain from the Ostrogradskii{Gauss{Stoke s formula that !(Pk ) = 0. Thus, @Q k

! = @Qk

!

C 0 jQk j¬ jQk jn

!(Pk )

1

= C 0 jQk j¬

+ n 1

;

@Qk

where the constant C 0 does not depend on V . Hence C0

! k2 J

@Qk

jQk j¬

+ n 1

:

k2 J

Since + n 1 > d, the last sum can be made less than " by the choice of V and in addition we can choose (V ) small enough for [ k2 J Qk to be a subset of a -neighbourhood of . n be a fractal hypersurface of Hausdor® dimension Corollary 2.6. Let d > n 1 which bounds Jordan domain VI . Let ! be a (n 1)-form of class H¬ where > d n + 1 in some neighbourhood of such that !j§ = 0. Then

!= §

! = 0: §

M

I

Proof . De­ ne forms ! 0 and ! 00 as follows: ! 0 = ! on VI and ! 0 = 0 outside VI [ ; = ! outside VI [ and ! 00 = 0 on VI . Then ! 0 and ! 00 are also of class H¬ . Now by the theorem applied to ! 00 we obtain ! 00

!00 = §

! 00 §

M

I

and it follows easily from de­ nitions of the integrals that ! 00

!= §

Thus, §

M

§

M

! = 0; similarly §

! 00 = 0:

and §

M

I

! = 0. I

Note 2.7. The condition for the H older exponent in the theorem is essential. This can be deduced from Harrison & Norton (1992) in the case n = 2. Modi­ cation of n such that ! is of class H this example gives an example of ! and d n+ 1 where d is the Hausdor¬ dimension, !jVM [ § = 0, and (see Borodich & Volovikov 1997) ! 6= §

Proc. R. Soc. Lond. A (2000)

I

! = 0: §

M

Surface integrals for domains with fractal boundaries

9

Note 2.8. In the case when is a d-summable set there is an analogue of continuity theorem (for continuous forms), which can be directly deduced from Harrison & Norton (1992). Let ! 2 H¬ ( ) and suppose that ! is smooth in n . If is d-summable and d n + 1, then != §

M

!: §

I

Note 2.9. Continuity property (1.2) holds also in the case when smooth and ! is continuous.

is piecewise

3. Discontinuous forms Above we have considered continuous forms only. However, the case of special discontinuous forms is important for applications. Below we will consider non-continuous forms and prove propositions of integral continuity across the fractal surface for forms with in­ nite (proposition 3.1) and bounded (proposition 3.2) coe¯ cients. (a) Surfaces with non-integer Hausdor® We consider the fractal surface and

dimensions

with the Hausdor¬ dimension d. Let p > 1, q > 1 1 1 + = 1: p q

Let n

aj (x) dx1 ^

=

^ dxj ^

^ dxn

j= 1 n. be a smooth (n 1)-form in n As we have mentioned, we consider forms which are smooth on n . However, they are not de­ ned on . Thus, the forms can be discontinuous on the whole domain . Let H be any hyperplane de­ ned by xj = const: We suppose that the (n 1)dimensional Lebesgue measure of H \ is equal to zero and is integrable on any \ H.

Proposition 3.1. Let u be a continuous function on , which is smooth on n and ! = u . The property of integral continuity (1:2) holds if the following conditions are satis¯ed: (1) d! is integrable on (2)

@Q

n

;

= 0 for any cube Q with faces parallel to coordinate hyperplanes;

(3) aj (x) 2 Lq (

\ H) where H is any hyperplane de¯ned as xj = const:;

Proc. R. Soc. Lond. A (2000)

10

F. M. Borodich and A. Yu. Volovikov

(4) there exists M > 0 such that, for any H and j = 1; : : : ; n, jaj (x)jq ds

M;

« \ H

(5) u 2 H¬ ( ) where

>d

1)=p and 0
d, the last sum can be made less than " by the choice of and in addition we can suppose that (V ) is small enough so that [k2 J Qk is a subset of a -neighbourhood of . V

Now we will consider bounded forms, i.e. all form coe¯ cients aj (x) are bounded. Proposition 3.2. Let u be a continuous function on which is smooth on n and ! = u . The property of integral continuity (1.2) holds if the following conditions are satis¯ed: (1) d! is integrable on (2)

@Q

n

;

= 0 for any cube Q with faces parallel to the coordinate hyperplanes;

(3) there exists D > 0 such that for any j = 1; : : : ; n we have jaj (x)j (4) u 2 H¬ ( ) where

>d

D; and

n + 1.

Proof . Using the notation of the previous proposition we have n

u

CjQk j¬

@Qk

jaj (x)j ds @ j Qk

j= 1

CjQk j¬ D

ds @Qk n 1

EjQk j¬ jQk j = EjQk j¬

+ n 1

;

(3.1)

where E is a constant. Hence ! k2 J

@Qk

jQk j¬

E

+ n 1

:

k2 J

Since + n 1 > d, the last sum can be made less than " by the choice of V and in addition we can suppose that (V ) is small enough so that [k2 J Qk is a subset of a -neighbourhood of . Note 3.3. Condition (2) in propositions 3.1 and 3.2 implies that is closed in n , i.e. d = 0. In applications to continuum mechanics, this condition is the condition of equilibrium (see Borodich & Volovikov 1997 and x 4). (b) d-summable sets We are going to discuss d-summable sets using volumes of "-neighbourhoods of . This approach is useful for further study. n. Denote by v(") the volume of a "-neighbourhood of Note that we can ­ nd positive constants C1 , C2 depending only on n, such that C1 "n N§ (") Proc. R. Soc. Lond. A (2000)

v(")

C2 "n N§ ("):

12

F. M. Borodich and A. Yu. Volovikov

Hence the convergence of 1

N§ (")"d

1

d"

0

is equivalent to the convergence of the integral 1

v(")"d

n 1

d":

0

From the integration by parts formula, it is easy to deduce that the last integral 1 converges if and only if the improper Stiltjes integral 0 "d n dv(") converges. 1 d n Thus is d-summable if and only if 0 " dv(") converges. Consider a function f de­ ned on V such that f is absolutely integrable on any · V where U · is the closure of U . Suppose jf (x)j open domain U V such that U M (d(x; ))s . We are interested in conditions which provide the convergence of the improper integral V jf j dv, dv = dx1 dxn . It is easy to see that this integral 1 converges if the following improper Stiltjes integral 0 "s dv(") converges. In fact, A

jf j dv

M

V

"s dv("); 0

where A is the diameter of . Thus we deduce that if s d n then V jf j dv converges, i.e. f 2 L1 (V ). In particular, V (d(x; ))d n dv < 1. Moreover, if is a compact subset of an open bounded set B, then is d-summable i® (d(x;

))d

n

dv < 1:

B

Thus, we have obtained the following assertion. Lemma 3.4. Let f be a function de¯ned on V and let f 2 L1 (U ) for any open domain U V such that U· V . Suppose that jf (x)j M (d(x; ))s , x 2 V . If is d-summable and s d n, then f 2 L1 (V ). Let V = VI be an open Jordan domain in

n

bounded by @V =

. Let

n

aj (x) dx1 ^

=

^ dxj ^

^ dxn

j= 1 n and let u be a continuous function on V · =V [ be an (n 1)-form on V which is smooth on V . Denote by d(x; ) the distance between x and .

,

Lemma 3.5. Suppose that V = VI is a Jordan domain in n and @V = is d-summable, n 1 < d < n. Suppose that for all 1 j n and x 2 V there exists a constant M > 0 such that jaj (x)j M (d(x; ))¬ 1 . Let u 2 H­ (V [ ) where 0 < < 1, and suppose that uj§ 0. Suppose that (i) j@u(x)=@xj j L(d(x; 1 j n, where 0 < Proc. R. Soc. Lond. A (2000)

))®

1

;

for some constants L > 0 for any x 2 V and

Surface integrals for domains with fractal boundaries (ii)

is closed in V , i.e. d = 0; and

(iii)

+

Then §

+n I

2

13

d.

u = 0.

Proof . To prove the existence of § I u we need to show that V d(u ) converges (see proposition 2.1). Since d = 0, we have d(u ) = du ^ = g(x) dx1 ^ ^ dxn . Thus jg(x)j dv:

d(u ) V

V

Since jaj (x)j

))¬

M (d(x;

1

@u(x) @xj

and

for any x 2 V and 1 j n, we have jg(x)j constant. Using lemma 3.4, we obtain that jg(x)j dv < 1

if

))¬

C(d(x;

+

2

))®

L(d(x;

d

+ ®

1

2,

where C is a

n:

V

Let us consider now the Whitney decomposition of V = VI into the union of binary k-cubes Q = [l1 2 k ; (l1 + 1)2 k ] [ln 2 k ; (ln + 1)2 k ]; where k; l1 ; : : : ; ln are integers, k 1 (see Stein 1970). For a binary k-cube Q we will also denote k by e(Q). We put W k = [e(Q) k Q. Then there exists k0 such that Wm is empty for m < k0 and Wk0 6= ;. We have Wk0 Wk Wk+ 1 V and V = [1k= k0 Wk . We have u = lim u = lim u : §

k!

I

1

1

k!

@Wk

Q Wk

@Q

Note that u = @Q

(u

u(PQ )) ;

@Q

where PQ 2 is any point such that d(Q; PQ ) = d(Q; ). It follows from the de­ nition of Whitney decomposition that d(Q; PQ ) C1 jQj where a constant C1 > 0 does not depend on Q. Since uj§ 0, we deduce that ju(x)j = ju(x) u(PQ )j C2 jQj­ for any x 2 Q. Now we have u @Wk

=

u Q Wk

jQj¬

= C3 Q Wk Q\ @Wk 6= ;

where C3 is a constant. Proc. R. Soc. Lond. A (2000)

jQj­ jQj¬

C3

@Q\ @Wk

Q Wk Q\ @Wk 6= ; + ­+ n 2

;

1

jQjn

1

14

F. M. Borodich and A. Yu. Volovikov

Harrison & Norton (1992) have proved that if is d-summable, then the sum jQjd taken over all cubes of Whitney decomposition is ­ nite (for completeness we give an alternative simple proof of that fact later), i.e. jQjd < 1; Q V

hence jQjd = 0:

lim 1

N!

Since

+

+n

2

e(Q) N

d we also have jQj¬

lim

N!

1

+ ­+ n 2

= 0:

e(Q) N

Let N (k) = Obviously limk! 1

min

Q Wk Q\ @Wk 6= ;

e(Q):

N (k) = 1. Hence jQj¬

lim

k!

1

+ ­+ n 2

=0

Q Wk Q\ @Wk 6= ;

and the proof of lemma 3.5 is complete. Now using lemma 3.4 we prove that all cubes of Whitney decomposition. We have

Q V

jQjd < 1, where the sum is taken over

jQjn = nn=2

dv: Q

Since C 0 d(x; we have jQjd

) n

jQj C 00 d(x; ) for some constants C 0 ; C 00 > 0 and any x 2 Q, C 000 (d(x; ))d n . Thus, we have

jQjd = jQjd

n

jQjn = nn=2 jQjd

n

dv

C

Q

(d(x;

))d

n

dv;

Q

where C = C 000 nn=2 . Hence using lemma 3.4, we obtain jQjd Q

C

(d(x; Q

Q

))d

n

dv = C

(d(x;

))d

n

dv < 1:

V

(c) Integral continuity for d-summable sets Now we are going to discuss the property of integral continuity (1.2) for discontinuous forms in a case when is a d-summable set. Proc. R. Soc. Lond. A (2000)

Surface integrals for domains with fractal boundaries

15

The special feature of the case of a d-summable set is that we do not need to assume the convergence of the integral V d! because it follows immediately from assumptions of propositions 3.6 and 3.9, and lemmas 3.4 and 3.5. As above we consider an (n 1)-form which is smooth on = n , where VI [ VM [ is an open domain in n . The ­ rst result is a direct consequence of lemma 3.5. Proposition 3.6. Let ! = u . Suppose is closed in

(1) (2) uj§

, i.e. d = 0 in

n

is d-summable and ;

n

0;

(3) u 2 H¬ ( ) and @u (x) @xj where x 2

n

and 0
0 such that jaj (x)j where 0 < < 1.

If + + n 2 d, then d! is integrable on continuity (1.2) holds, i.e. != §

))­

M (d(x; n

1

for any x 2

n

and the property of integral

! = 0: §

I

M

Note 3.7. In proposition 3.6 we can replace suppositions (1) and (2) by (10 ) and (20 ): (10 )

S

(20 ) uj§

= 0 for any smooth closed hypersurface S

n

; and

c where c is a constant.

In fact, it follows from (10 ) that d = 0 in to the form (u c) . Hence we have (u §

c) =

n

(u §

I

and we can apply proposition 3.2

c) = 0:

M

On the other hand it follows from (10 ) that = §

=0 §

I

M

and hence we obtain u = §

Proc. R. Soc. Lond. A (2000)

I

u = 0: §

M

16

F. M. Borodich and A. Yu. Volovikov

Our suppositions are more restrictive in the general case when u is not a constant on , namely we will consider the following system of inequalities: + 2 d n; q( 1) d n; + (n 1)=p d; 0 < < 1; 0< < 1;

(3.2)

where, as we said before, 1=p + 1=q = 1 and q > 2. Let B be a bounded open subset of n and K be a compact subset of B. For an a¯ ne plane H we de­ ne (d(x; K))d

sd;K;H;B =

d im H

ds;

B\ H

where ds is the Lebesgue measure on H. Note 3.8. The condition sd;K;H;B < 1 implies that B \ H is a d-summable set. Proposition 3.9. Let ! = u . Suppose (1) there exists D > 0 such that sd xj = c; (2)

@Q

is d-summable, and

1;H\ § ;H;«

< D for any hyperplane of the form

= 0 for any cube Q with faces parallel to coordinate hyperplanes;

(3) u 2 H¬ ( ) and @u (x) @xj where x 2

n

L(d(x;

))®

1

;

; and

(4) there exists M > 0 such that jaj (x)j where 0 < < 1.

M (d(x;

))­

1

for any x 2

n

Suppose ; ; ; q; d satisfy the system of inequalities (3:2), then d! is integrable on n and the property of integral continuity (1:2) holds, i.e. != §

I

!: §

M

Proof . As in the proof of proposition 3.1 it su¯ ces to show that (a) there exists a constant C 0 > 0 such that jaj (x)jq ds < C 0 « \ H

for any hyperplane H of the form xj = l2

k;

and

(b) for any " > 0 there exists a ­ nite covering fQ0 g of by binary cubes such that any cube has a non-empty intersection with and Q0 jQ0 jd < ". Proc. R. Soc. Lond. A (2000)

Surface integrals for domains with fractal boundaries

17

In fact, as in the proof of proposition 3.1 with the help of the assertion (a) we obtain !

jQ0 j¬

C

+ (n 1)=p

@Q0

and from (b) and the third inequality of (3:2) we deduce the needed result. Proof of (a). From the second inequality we have jaj (x)jq and since dim H = n

M q (d(x;

1 and d

))q(­

n = (d

jaj (x)jq ds

1)

M q (d(x;

1)

M q sd

(n

))d

n

;

1) we obtain DM q :

1;H\ § ;H;«

H\ «

Proof of (b). Let Q0 be a cube such that Q0 \ = 6 ;. Then for any x 2 Q0 we have 0 0 d n d(x; ) (d(x; ))d n . Integrating this inequality over jQ j and hence jQ j Q0 we obtain that jQ0 jd

C

(d(x;

))d

n

dv:

Q0

If fQ0 g is a covering of intersection with , then

by binary cubes such that any cube has a non-empty jQ0 jd

C

))d

(d(x;

n

dv:

[ Q0

Q0

Since (d(x;

))d

n

dv

«

converges, it follows that of a covering.

Q0

jQ0 jd can be made as small as we wish by the choice

In particular, consider the case following system: 2

=

in the system (3:2). Then we obtain the

s + 1; s +1 q n s+

q

1 ; q 1 ;

(3.3)

where s = d n + 1 (note that 0 < s < 1). Since q > 2, we can delete the ­ rst inequality and obtain the system s +1 q n s+ which can be easily studied. Proc. R. Soc. Lond. A (2000)

q

1 ; q 1 ;

(3.4)

18

F. M. Borodich and A. Yu. Volovikov

4. Applications to elasticity We will consider an elastic medium in a domain 2 3 with the boundary S = ST + Su . Below we will consider non-continuous forms and prove propositions of integral continuity for forms with in­ nite (proposition 2.1) and bounded (proposition 3.1) coe¯ cients. We will suppose that subscripts i and j can take the values 1; 2; 3. Under action of external forces and body forces per unit of volume Fi there arise displacements u = (u1 ; u2 ; u3 ), stresses with components of symmetric tensor of stresses ij and strains with components of symmetric tensor of in­ nitesimal strains eij : eij =

1 2

@ui @uj + : @xj @xi

The stresses satisfy the equilibrium equations @ ji + Fi = 0; @xj

(4.1)

with implied summation over the values 1; 2; 3 for the repeated subscripts. Below we will suppose that Fi = 0. The work of forces in elastic bodies on a closed contour is equal to zero. Thus, there exists a stress potential U1 (e) which is the strain energy per unit of material volume such that ij

=

@U1 (e) : @eij

(4.2)

It can be found from (4.2) that the strain energy (elastic energy) has the following form: U1 (e) =

1 e : 2 ij ij

If the body is linear elastic, then this potential is a positive-de­ nite quadratic form U1 (e) = 12 Eijkleij ekl and we obtain the `Hooke’s law’ ij = Eijklekl where Eijkl (x) are components of elastic constants at the point x. We know the displacements u on Su , i.e. we have the displacement boundary conditions ui = ui

on Su :

(4.3)

If the boundary surface is smooth, then we know the tractions Ti on ST and ij nj

= Ti

on ST ;

(4.4)

where ni are components of the unit outward normal n to the boundary ST . To close the formulation of the boundary value problem (4.1){(4.4), we have to give the boundary conditions on the fractal surface M . There are various possible ways to formulate the boundary-value problem for fractal boundaries. We will consider appropriate di¬erential forms for stress ­ elds and ­ elds of virtual displacements in Proc. R. Soc. Lond. A (2000)

Surface integrals for domains with fractal boundaries

19

order to provide us with a mathematical basis for the use of variational techniques for problems of elasticity in the case of fractal boundaries. We will assume that the work on the surface displacements is known on M . The di¬erential of the work on the displacements u is a 2-form A(u): A(u) =

ij ui

dxj ;

(4.5)

where dx1 = dx2 ^ dx3 , dx2 = dx3 ^ dx1 , dx3 = dx1 ^ dx2 , summation over repeated indices is implied, and is the so-called Hodge star operator de­ ned via the standard scalar product in 3 . In the case of a smooth boundary, the work on the surface displacements is A(u) =

ij ui

ST

dxj =

ij ui n j

ST

hT ; ui ds:

ds =

ST

ST

For example, if we consider an empty hole in an elastic body, whose points on the boundary surface do not contact with each other, then the work on the surface displacements is equal to zero: A(u) = 0: §

(4.6)

I

Thus, the functions of a sought solution u and ij satisfy the conditions of the above proposition and the conditions (4.1){(4.4) and (4.6). To use variational techniques for problems of elasticity in the case of fractal boundaries, we study conditions of continuity of surface integrals for the ­ eld of true stresses ij solving a considered boundary-value problem and a ­ eld of virtual displacements, u ~i , satisfying the displacement boundary conditions. Let us consider a 2-form ! = ij u ~i dxj . Then ! = i u ~i , where 1

=

dxj ;

1j

2

=

dxj ;

2j

3

=

dxj :

3j

From equilibrium equations we have i

=0

@Q

for any cube Q and any i. In particular, i is closed, i.e. d i = 0. In addition we suppose that d! is continuous and integrable on following integrals converge: d! < 1

, i.e. the

d! < 1:

and

VM

n

(4.7)

VI

The condition (4.7) that the integral of d! over the domain is ­ nite has a clear physical sense, namely it means that the elastic energy U« accumulated within the whole domain is ­ nite. Indeed, from (4.2) we obtain 2U«

=2

U dv = «

Proc. R. Soc. Lond. A (2000)

«

ij

@ui dv = @xj «

@(

ij ui ) dv @xj «

ui

@ ij dv = @xj

d! «

20

F. M. Borodich and A. Yu. Volovikov

due to the equilibrium equations (4.1) with Fi = 0 and since @(

d! = «

«

ij ui ) dv: @xj

Using the condition of integral convergence we obtain d! < 1:

2U« = «

Now we can apply proposition 3.1 to the forms !i = u ~i! = following.

i

and obtain the

Theorem 4.1. Let a 2-form ! be de¯ned as ! = ij u ~ i dxj . The property of integral continuity (1.2) holds if the following conditions are satis¯ed: (1) displacements u ~i are of class H¬ , where 0 < (2)

ij

2

Lq (

< 1;

H) where H is any two-dimensional plane of the form xk = c;

(3) there exists M > 0 such that for any H we have j

ij j

q

ds

M;

« \ H

(4)

>d

2=p, where d is the Hausdor® dimension of

.

Note that 2 < d < 3, hence the condition (4) implies p < 2. Therefore, q > 2. A subsequent result for bounded forms follows from proposition 3.2. We leave it to the interested reader. In the case when is a d-summable set, we obtain from proposition 3.9 the following result. Theorem 4.2. Let ! =

dxj . Suppose

~i ij u

(1) there exists D > 0 such that sd of the form xj = c;

1;H\ § ;H;«

is d-summable, 2 < d < 3, and < D for any two-dimensional plane

(2) displacements u ~i , i = 1; 2; 3, are of class H¬ and there exist L > 0 and 0 < such that @u ~i (x) L(d(x; ))® 1 @xj for any x 2

n

and 1

i

3, 1

(3) there exists M > 0 such that j where 0 < < 1.

j

3;

ij (x)j

M (d(x;

))­

1

for any x 2

n

,

Suppose ; ; ; q; d satisfy the system of inequalities (3:2) with n = 3. Then d! is integrable on n and the property of integral continuity (1.2) holds, i.e. != §

Proc. R. Soc. Lond. A (2000)

I

!: §

M

Surface integrals for domains with fractal boundaries

21

Note 4.3. The system of inequalities (3:2) in this theorem can be replaced by the system of inequalities (3:4) in the case when u ~i = ui . The following result is a direct consequence of lemma 3.5. Lemma 4.4. Let ui , i = 1; 2; 3, be of class H¬ (V· ), where 0 < < 1, and suppose that ui (x) = 0 for any x 2 and i = 1; 2; 3. Suppose further that there exist L > 0 and 0 < such that @ui (x) @xj for any x 2 V and 1 i 3, 1 If is d-summable and +

L(d(x;

j d

1

3. 1, then ij ui

§

))®

dxj = 0:

I

Proof . We put 1 = 1j dxj , 2 = 2j dxj , 3 = 3j dxj . Then ij ui dxj = ui i . Since ij is a linear combination of partial derivatives of ui , we deduce from lemma 3.5 that u1 §

1

= 0;

u2 §

I

2

= 0;

u3 §

I

3

= 0:

I

Now we are able to prove a variant of uniqueness theorem for stresses in elastic domains. ~~i , i = 1; 2; 3, be solutions of the class H¬ (V· ), where Theorem 4.5. Let u ~i and u 0 < < 1, such that ~~i j§ =u

u ~ i j§

= ui :

Suppose that there exist L > 0 and 0 < @u ~i (x) @xj for any x 2 V and 1 i 3, 1 If is d-summable and +

such that

~~i (x) @u @xj j d

L(d(x;

))®

1

3. 1, then ~ ij (x) = ~ ~ ij (x) for any x 2 V .

Proof . We will use classical arguments (see, for example, Rabotnov 1979). Let ~~i and denote by ij the corresponding stresses. Note that ui j§ ui = u ~i u 0 and ~ (x) = ~ (x) ~ (x) for any x V . 2 ij ij ij We have ij;j = 0; hence ij;j ui

dv = 0:

V

Since

ij;j

=(

ij ui );j

ij eij , ij;j ui

V

Proc. R. Soc. Lond. A (2000)

we deduce

dv =

( V

ij ui );j

dv

ij eij V

dv = 0:

22

F. M. Borodich and A. Yu. Volovikov

Using lemma 4.4, we deduce (

ij ui );j

dv =

V

ij ui §

dxj = 0:

I

Thus we obtain ij eij

dv = 0:

V

If U is a homogeneous quadratic function of eij , then by the Euler theorem on homogeneous functions we have ij eij

= eij

@U = 2U: @eij

Since U is de­ nitively positive we deduce eij (x) = 0 for any x 2 V , and hence ij (x) = 0 for all x 2 V .

5. Conclusion We have seen that modelling of a surface by a fractal makes the formulation of a boundary-value problem rather complex. There are various possible ways to formulate the boundary value problem: (1) to modify H ormander’s (1983) concept of a generalized normal (Kats 1992, personal communication); (2) to consider variational solutions to the problem and to determine the trace to the boundary (Wallin 1989, 1991); (3) to consider appropriate di¬erential forms for domains with fractal boundaries (Kats 1987, 1992 personal communication; Harrison & Norton 1991); (4) to de­ ne a generalized normal derivative using the theory of distributions or generalized functions (Kats 1992, personal communication). In this paper we followed the third way. We were interested in the case when the integral of d! over the domain is ­ nite. It has been shown that in applications to elasticity this condition has clear physical sense, namely this integral is the elastic energy accumulated within the domain and the energy is ­ nite. The condition of ­ niteness allows us to consider not only box dimension as a fractal dimension of the boundary, i.e. the case which was considered by Kats (1983, 1987) and by Harrison & Norton (1992), but also the Hausdor¬ dimension. We have described various cases when the integral of ! has no jump across the fractal boundary. We have also proved the uniqueness theorem of solutions to problems of elastostatics for bodies with fractal boundaries. The results obtained concerning the divergence theorem provide us with a mathematical basis for the use of variational techniques for problems of elasticity in the case of fractal boundaries. The authors are grateful to Professor J. R. Willis (University of Cambridge) for discussing the problem considered in this paper. F.M.B. thanks Professor B. A. Kats (Kazan, Russia) for his Proc. R. Soc. Lond. A (2000)

Surface integrals for domains with fractal boundaries

23

valuable comments concerning the problem and possible ways to its solution. A.Yu.V. thanks the Russian Foundation of Basic Research for partial support of his research by RFBR grant no. 97-01-00174. Thanks are due to the Royal Society for funding the visit of A.Yu.V. to the Department of Mathematics of Glasgow Caledonian University, during which the above work was completed.

References Abraham, R & Marsden, J. E. 1978 Foundation of mechanics. London: Benjamin/Cummings. Arnold, V. I. 1978 Mathematical methods of classical mechanics. Springer. Borodich, F. M. & Volovikov, A. Yu. 1997 Continuity of surface integrals for domains with fractal boundaries. Technical report, TR/MAT/FMB-AYV/97-80, Glasgow Caledonian University. Falconer, K. J. 1990 Fractal geometry: mathematical foundations and applications. Wiley. Flanders, H. 1963 Di® erential forms with applications to the physical sciences. Academic. Harrison, J. & Norton, A. 1991 Geometric integration on fractal curves in the plane. Indiana Univ. Math. Jl 40, 567{594. Harrison, J. & Norton, A. 1992 The Gauss{Green theorem for fractal boundaries. Duke Math. Jl 67, 575{588. H ormander, L. 1983 The analysis of linear partial di® erential operators. I. Distribution theory and Fourier analysis. Springer. Kats, B. A. 1982 The Riemann boundary value problem for a nonrecti¯able Jordan curve. Dokl. AN SSSR, 267, 789{792. (English transl. B. A. Kac Soviet Math. Dokl. 25, 695{698.) Kats, B. A. 1983 The Riemann problem on a locked Jordan curve. Izv. Vyssh. Uchebn. Zaved. Mat., no. 4, 68{80. Kats, B. A. 1985 Riemann problem for the closed Jordan curve under semi-continuous formulation. Izv. Vyssh. Uchebn. Zaved. Mat., no. 6, 14{22. Kats, B. A. 1986 Riemann boundary problem on a Jordan open curve. Izv. Vyssh. Uchebn. Zaved. Mat., no. 7, 56{60. Kats, B. A. 1987 Jump problem and the integral over a non-recti¯able curve. Izv. Vyssh. Uchebn. Zaved. Mat., no. 5, 49{57. Kats, B. A. 1990 On the solvability conditions for the Riemann boundary value problem on nonsmooth curve. Dokl. AN SSSR 314, 67{71. Kats, B. A. 1991 The Riemann boundary problem on fractal and other non-smooth curves. DrSc thesis. Physical-Technical Institute of Low Temperatures (Ukraine Academy of Science), Kharkov. Kats, B. A. 1993 To the question of integration along fractal plane contours. Siberian Math. Jl 34, 472{480. Kolmogorov, A. N. & Fomin, S. V. 1970 Introductory real analysis. Englewood Cli® s, NJ: Prentice-Hall. Kolmogorov, A. N. & Fomin, S. V. 1976 Elements of the theory of functions and functional analysis. Moscow: Nauka. Pontrjagin, L. & Schnirelmann, L. 1932 Sur une propri¶et¶e m¶ etrique de la dimension. Annals Math. 33, 156{162. Rabotnov, Y. N. 1979 Mechanics of solids. Moscow: Nauka. Stein, E. M. 1970 Singular integrals and di® erentiability properties of functions. Princeton University Press. Wallin, H. 1989 The trace to the boundary of Sobolev spaces on a snow° ake. Report S-901 87, University of Umea. Wallin, H. 1991 The trace to the boundary of Sobolev spaces on a snow° ake. Manuscripta Mathematica 73, 117{125. Proc. R. Soc. Lond. A (2000)

24

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Whitney, H. 1935 A function not constant on a connected set of critical points. Duke Math. Jl 1, 514{517. Willis, J. R. 1982 Elasticity theory of composites. In Mechanics of solids. The Rodney Hill 60th anniversary volume (ed. H. G. Hopkins & M. J. Sewell), pp. 653{686. Oxford: Pergamon. Willis, J. R. 1983 The overall elastic response of composite materials. ASME Jl Appl. Mech. 50, 1202{1209.

Proc. R. Soc. Lond. A (2000)