Boundary value problems and brownian motion on fractals

Chaos,

Solitom

& Fracmls Vol. 8, No. 2, pp. 191-205, 1997 Copyright @ 1997 Elsevier Science Ltd Prixed in Great Britain. All rights reserved 0960-0779/97 $17.00 + 0.00

PII: s0960-0779(%)ooo4a-3

Boundary

Value Problems and Brownian ALF JONSSON Department

of Mathematics,

Motion

on Fractals

and HANS WALLIN

University

of Urn&

S-90187

Umea,

Sweden

Abstract-A physical state in a domain is often described by a model containing a linear partial differential equation and associated boundary conditions. The mathematical tools required to study this are well known if the boundary of the domain is smooth enough or if the boundary is smooth except for one or several corners. But in reality the boundary of the domain is usually not smooth. The typical situation is rather that the boundary is strongly broken with an intricate detailed structure and maybe that the boundary exhibits similar patterns in different scales. This means that the boundary is typically a fractal showing some kind of self-similarity: a magnification of a part of the boundary has, in some sense, the same structure as the whole boundary. A typical example of a domain in the plane having a boundary of this kind is van Koch’s snowflake domain. In the case of a fractal boundary the classical tools and theorems no longer hold. How does one provide the mathematical background in this case? This is the main topic of this survey paper. However, we also study Brownian motion on fractals. 0 1997 Elsevier Science Ltd All rights reserved

1. INTRODUCTION

Let G? be a domain in n-dimensional Euclidean space IF? and suppose that the mathematical model describing the physical problem is given by Dirichlet’s problem for Poisson’s equation: Au = f { z.l=g

in Q on X2,

(1)

where f and g are given functions and u is the unknown function. Typically, (1) is obtained as the solution to a variational problem where u is the state giving minimal energy to the physical system. This means that (1) shall be interpreted in variational form. After the basic contribution by Sobolev in the 1930s the result is described by using Sobolev spaces W:(Q), 1 c p G ~0, k non-negative integer, consisting of all functions h satisfying

Let us see what happens in a special case (see [l, 21 and the references given there for more details). Assume that f E L*(Q) and that g is, in some sense, the boundary values, called the trace, to 22 of a function u. E W:(Q). By using Riesz’s representation formula for bounded linear functionals in L2(S2), it is proved that there exists a function u E W:(Q), depending uniquely on f and uo, such that Au = f

in Q

and 191

u - z&JE tiI:(S),

A. JONSSON and H. WALLIN

192

where l@:(Q) denotes the closure of C;(Q) in w;(Q); functions in I@:(Q) are, in some generalized sense, zero on XJ. Several questions arise. (a) How do you define the trace to XJ of a function u. E W:(Q)? (b) Having answered (a), which functions g defined on as2 are traces to X2 of functions uo E w:(Q)? (c) We know that u = g on XJ in the sense that u - u. E I@:(Q). What does this mean more explicitly? (d) Does the solution u above depend uniquely on .f and g? If X2 is ‘smooth enough’ the answers to these questions have been known for a long time (see for instance [3, Ch. 21); the trace space of W:(Q) to as2 is usually denoted by Hr’*(aQ). Since these answers require a smoothness of aQ which a fractal does not have we ask: what are the answers to the questions (a)-(d) if XJ is a fractal, for instance von Koch’s snowflake curve, i.e. the boundary of the snowflake domain? The answers are given in Section 5 (Theorems 6 and 7). These results have been used by Panagiotopoulos for problems on modelling with fractals [4, 51. Regularity properties of the solution of Dirichlet’s problem (1) in variational form have been studied alot. For instance, consider the situation corresponding to g = 0 and f E L2(Q) in (1). Let u E tiT(Q) satisfy Au = f E L2(Q). This means that Au and the first order partial derivatives of u are in L*(Q) and a natural question to ask is whether this implies that all second order derivatives of u are in L*(Q). If Q is ‘smooth enough’, it is known that u E W:(Q) [3, Ch. 21 but there are simple domains with a non-convex corner such that this does not hold. The question arises: what happens if X2 is a fractal? This is discussed in Section 5.3 where some recent results by Nystrom are stated for the snowflake domain. Wavelets have in recent years become an important tool in mathematical analysis and in applications such as signal processing. In Section 7 we construct wavelets on a self-similar fractal F, and show that functions defined on F can be represented as a wavelet series

where { &, IJ$‘} is an orthonormal set of functions with respect to the invariant measure on F (see Section 7 for details). For certain fractals, such representations may be used to characterize various function spaces by means of the magnitude of the wavelet coefficients al and p’. In Section 6, we discuss atomic decomposition of functions. These are somewhat similar to wavelet decompositions but less explicit. On the other hand we can treat atomic decompositions in a much more general setting. Section 6 also contains a general trace theorem and a duality theorem. Diffusion processes on fractals have been treated a lot in recent years by physicists modelling various physical phenomena. Also, there is an extensive mathematical theory. In Section 8 we consider Brownian motion on the Sierpinski gasket, and show how function spaces can be used to clarify certain aspects of the theory for this process.

2. F’RACTALS

PRESERVING

MARKOV’S

INEQUALITY

Von Koch’s snowflake was mentioned as an example of a domain having a fractal, self-similar boundary, the snowflake curve. This curve has a very strong geometric selfsimilarity which was investigated by Hutchinson [6] inspired by Mandelbrot’s [7] description of typical objects in nature, for instance a coastline, as self-similar fractals. However, the

Boundary value problems

193

geometric self-similarity investigated by Hutchinson is too strong and gives sets which are too regular to provide good models of objects in nature. Different kinds of weaker selfsimilarity have been proposed, for instance statistical self-similarity [8, Ch. 151. The following definition determines a class of sets which give a relatively simple theory for the function spaces introduced in Section 4. It includes both non-fractal sets and fractals, and the fractals included may be self-similar in a strong sense like the Koch curve or in a much weaker sense. We denote by ??‘)kthe set of real polynomials in n real variables of total degree at most k. Definition 1.

A closed non-empty subset F of R” preserves Markov’s local inequality [9, Ch. II] if for every positive integer k there exists a constant c = c( F, n, k) such that (2)

for all P E ‘GPPkand all balls B = B(x, I) with x E F and 0 < I G 1. We call (2) Markov’s local inequality on F. If 12= 1, F = R’, the inequality (2) is true and is, in fact, the classical Markov inequality in R’ which holds with c = (deg P)* where deg P is the degree of P. It is in this case enough to prove the inequality for r = 1 since it follows for a general T by dilation. Sets preserving Markov’s local inequality were characterized geometrically in [lo, Th. 1.31. A variant of this characterization was given by Wingren [ll]: Theorem 2. F preserves Markov’s local inequality if and only if there exists a positive constant c* such that for every B = B(x, r), where x E F and 0 < r G 1, there are n + 1 affinely independent points Ui E F fl B, i = 1, . . . , n + 1, such that the n-dimensional ball inscribed in the convex hull of al, u2, . . ., a,,, has radius not less than c*r. This geometrical characterization means that sets preserving Markov’s local inequality are not too flat anywhere. For example, smooth manifolds in I?’ of dimension less than n, do not preserve Markov’s local inequality, since they are too flat, and the closure of a domain with an outgoing cusp does not preserve Markov’s local inequality, since it is too flat in the cusp. On the other hand, F preserves Markov’s local inequality if F = Et” or if F is the closure of an open set in R” with Lipschitz boundary or of an (E, +domain (see Section 5 for the definition). Also, F preserves Markov’s local inequality if F is von Koch’s snowflake curve, the ordinary Cantor set, or any other fractal which is self-similar in a strong geometric sense; see [12, Th. l] for a precise statement. By using the geometric criterion it is straightforward to construct a large class of generalized Cantor sets preserving Markov’s local inequality and having large or small Hausdorff dimension (see Section 3 for the definition). However, the dimension can never be zero. Theorem 3. Every closed non-empty set preserving Markov’s local inequality has positive Hausdorff dimension [ 131. The idea of the proof of this theorem is that, given F preserving Markov’s local inequality, the geometric criterion allows the construction of a subset of F which is a generalized Cantor set with constant ratio which implies that the subset, and hence F, has positive dimension; see [13] for details. Sets preserving Markov’s local inequality were introduced by Jonsson-Wallin in 1980; see [9]. However, based on the geometric characterization they were introduced independently by VUililii in 1986 [14] and called thick sets, and the idea of thickness has also been used by others. V&s&la et al. [15] have studied thickness in a generalized sense and invariance properties of thick sets under quasiconformal mappings.

A. JONSSON and H. WALLIN

194

3. d-SETS AND (I-MEASURES

Our function spaces on fractals F in Section 4 shall consist of P-functions on F and this means that we need a measure p on F for our integration. The obvious candidate for such a measure ~1 is the Hausdorff measure. The d-dimensional (d > 0) Hausdorff measure of E C R” is m,(E)

:= limm(,“(F) I E10

where, for a certain positive constant a(d), m&‘)(E) := a(d)inf

z(diamEi)d: {

oEi

i

> E, diamE,

G E . 1

1

The Hausdorff dimension of E, dim E, is the infimum of the set of d > 0 such that m,(E) = 0. Sets with non-integral Hausdorff dimension are often called fractals as well as sets with integral Hausdorff dimension and a fine structure. Today a fractal is often also assumed to have some kind of self-similarity (see the introduction in [S]). We want the smooth function spaces on F in Section 4 to be trace spaces to F of smooth function spaces in IV. The simplest way to guarantee this is to work with a special kind of measures p, d-measures. Definition 4. A closed non-empty subset F of R” is a d-set (0 < d G n) if there exists a Bore1 measure ~1 with support F such that, for some positive constants cl = c,(F) and ~2 =

cz(F), clrd

s

p(W,

I))

=s C2Td,

for

x E F,

O
(3)

Such a p is called a d-measure on F. If F is a d-set then mdl F defined by ?Q] F( E) = md( F fl E), for Bore1 sets E, is a d-measure on F. Also, any d-measure ~1on F is equivalent to mdl F in the sense that for some constants c3 and cq, c,~ < md] F G c4p (see [9], Ch. II). To sum up, a d-measure on a d-set F C R” is unique up to equivalence and it is given by the restriction to F of the d-dimensional Hausdorff measure in R”. The essentially unique d-measure on a d-set F is a canonical measure on F serving a similar purpose on $ as the Haar measure on a locally compact group. It is natural to extend Definition 4 and to say that an arbitrary’Bore1 set F is a d-set if (3) holds with p replaced by md] F. Examples of d-sets are F = IF’ with p equal to the n-dimensional Lebesgue measure, and geometrically self-similar sets [12, Th. 21. In particular, the Cantor set and von Koch’s snowflake curve are d-sets with d = log2/log3 and d = log4/log3, respectively. A Lipschitz domain and its closure and, more generally, an (E, 6)-domain (see Section 5) and its closure are n-sets. 4. BESOV

A natural way to define use Lipschitz conditions. that the partial derivatives or an integrated Lipschitz a nice, flexible tool we differences, and we shall

SPACES

ON FRACTALS

smoothness of order (Y, a > 0, of a function f defined in R” is to If k is the largest integer strictly smaller than (Y we can require of f up to order k satisfy a Lipschitz condition of order & - k, condition in U’(lR’), 1 < p G *. It turns out that in order to get shall define the Lipschitz condition by means of second order introduce a third index q, 1 d q 6 CQ,which in a way serves as a

195

Boundary value problems

second smoothness index. In this way we are led to the Besov spaces in R”, BP,,‘@“). We refer to [9] where several equivalent definitions are given. One way to define Besov spaces which works also when IF!?’is replaced by a fractal F is to use local or piecewise polynomial approximation. This method to describe smoothness was used in the sixties by Campanato [16] to characterize functions satisfying a Lipschitz condition, and in the seventies by Brudnyi [17]. It has been used in a systematic way by Jonsson-Wallin [9] to characterize smoothness on fractals. Let us assume that F C R” is a closed d-set preserving Markov’s local inequality. This means that F may be I? or the closure of a domain 52 with Lipschitz boundary, but the non-classical, interesting situation is when F is a fractal, for instance the boundary of a fractal domain. As a preparation we introduce some notation. Let N denote a division of R” into equally big cubes Q with side Y, half-open of the form {X = (x1, . . ., x,) E R”: ai < Xi 6 ai + r, i=l,2, . . .. n}, obtained by intersecting R” with hyperplanes orthogonal to the axes. We call such a division a net X with mesh r. Let N,, v=O, 1, . . ., be the net with mesh 2~” such that the origin is a corner of one of the cubes in the net. For Q E N,(F) := {Q E NV: Q n F # 0} and a multi-index j = basis in the (jl, . . .? in) with length (il = j, + . . . + j,, let (rrj)lj,.rlal be an orthonormal subspace G?[,] of L2@, 2Q) of polynomials of degree at most [a], and define the projection on 9,,J for every f E L’(p, 2Q): PQ

=

PQ(f)

‘=

for Y > 0,

,j~al~j~2Qf~jdh

and Pp := 0, for Y= 0. Definition 5.

Let F be a closed d-set preserving Markov’s local inequality d-measure on F. Let a> 0 and 1 s p, 4 S 03, and let f be a p-measurable defined on F. Introduce the sequence A = (A.): by

If -

and p a function

UP

P,(Pdp

== 2-‘“A,.

f belongs to the Besov space BP,‘“(F) if A E t4, and the norm of to the norm of A in lq, IIAl1q = (~~lAvlq)“q [18].

f in

Bcq(F)

is equivalent

5. DIRICHLET’S PROBLEM ON A FRACTAL DOMAIN 5.1.

The trace theorem

An open connected subset tZ of R” is an (E, @-domain, E > 0, m 2 6 > 0 [19], if whenever X, y E & and IX - y I < 6, there is a rectifiable arc y C 0 with length 1(y) joining x to y and satisfying (i) f(y) s /X - y I/E and (ii) d(z, ~2) 2 EIX - zlly - zl/lx - yl, for z E y. A Lipschitz domain $2 is an (E, S)-domain and also an n-set. If we add an ingoing cusp to Sz, then s2 is still an n-set but not an (E, @-domain. If instead we add an outgoing cusp

A. JONSSON and H. WALLIN

196

to Q, then 51 is not even an n-set. von Koch’s snowflake domain in R* is an example of a fractal domain which is an (E, @-domain. We say that f E L:,(S2) can be strictly defined at x E B U as2 if the limit f(x)

:= lim

’ J f(y)dy r+om(B(x, r) n Q) H(x,r)rm

exists. The trace, Tr f of f to X2 is defined as the function f(x) at every x E X2 where &x) exists.

flX2

(4) given by (flX2)(x)

:=

Theorem 6. Let Q C R” be an (E, @-domain such that 82 is a d-set preserving Markov’s local inequality. Let k be a positive integer, 1 < p < ~0, and /3 = k - (n - d)/p. Then the trace operator Tr: f I+ flX2 defined by means of (4) is a bounded linear surjection

Tr: W{(Q) with a bounded, 5.2

-+ Bj’P(3Q)

linear right inverse [2].

Dirichlet’s problem

We now answer the means that summarized

return to Dirichlet’s problem (1) in Section 1. By means of Theorem 6 we can questions (a)-(d) in Section 1.1. In particular, u - u. fz it:(S2) in question (c) the trace of u - u. to as2 is zero, i.e. u/aQ = uo(%2 = g. The result may be in the following theorem.

Theorem 7. Let Q be a bounded (E, @-domain in R” such that XJ f E L*(Q) and g E E$@SZ), p = 1 - (n - d)/2, the problem

is a d-set. Given

Au=finQ

24las2= g has a unique solution u E W:(QJ. Furthermore, the mapping linear mapping from L*(Q) X B28, (af2) to W:(Q) [l]. 5.3

{f,

g} ++ u is a bounded

Smoothness results

In a domain with a ‘smooth’

boundary the solution Au = f E L*(Q)

IL to the problem

(5)

1 ulaa = 0 is in W;(Q) [3, Ch. 21. This is no longer true for domains with a fractal boundary. In fact, Nystrom has proved [20, Summary, Theorem 21 that when Q is von Koch’s snowflake domain there exists an f E L*(Q) such that the solution u to (5) is not in W&Q). On the other hand, he also proved the following result [20, Summary, Theorem 31: let 8 be von Koch’s snowflake domain and f an arbitrary function in L*(Q). Then the solution u to (5) satisfies

(V*u]*d(x, as2)*dx < co, IR where d(x, XZ) is the distance from x to a&J. He also considered more general domains than von Koch’s snowflake domain and obtained more precise results on the regularity of the solution u to (5) but apparently a lot of work remains to be done to prove sharp estimates.

Boundary value problems

197

6. BESOV SPACES AND ATOMIC DECOMPOSITIONS

An atomic decomposition of a function is a representation of the function as a sum of simple functions, called atoms. Such decompositions are useful tools in the study of function spaces. Frazier and Jawerth introduced them in the study of Besov spaces on [w” in [21]. In [22] ( see also [23]) Besov spaces on closed sets were studied by means of atomic decompositions, and in [18] they were used to determine the duals of Besov spaces. We describe some of these results here in a special case, assuming that a d-set F preserving Markov’s local inequality is given, and that p is a d-measure on F, and then briefly discuss the general results. These are of interest when comparing with recent developments in the theory of function spaces on general sets. We start by defining atoms, denoting by supp a the support of a function a. Let lCpCC-2, &Z 0, and let Q be a cube with side 2~’ intersecting F, where Y is a nonnegative integer. If & > 0 we say that a function a is an ((u, p)-atom associated to Q if u E CK(Rn), where K 2 [a] + 1, and (1) suppa C 3Q, and (2) IDia( s 2-@-‘j’-d’p),

x E R”, ljl c K.

If (Y< 0 we instead require that a E P’(p) (1) supp a C

and

2Q;

(2) Ila(lLpcp)c 2-v’y; and

(3) /xYu(xjdp(x)

= 0, (yl s [-a],

if Y> 0.

The reason for having smooth atoms in the case (Y> 0, and atoms in I?‘(p) if (1:< 0, is that it was natural in the different contexts they were introduced, in [22] and [18], respectively. It may seem inappropriate to have atoms which, for a: > 0, are defined on UP when studying spaces on F, but due to a classical extension theorem of Whitney this is not of a major importance. We now define Besov spaces using atomic decompositions. Recall from Section 4 that N, is a certain net with mesh 2-“, and that X,(F) denotes those cubes from the net which intersect F. Let 1 G p, q G 00, 0 < d G n, and (Y# 0, and let F be a d-set with dmeasure p preserving Markov’s local inequality. A distribution f belongs to the Besov space BP,,q(F) if and only if it has a representation Definition 8.

f= where

up

are (a, p)-atoms

5 c SQ”Q2 v=OQ&T,,(F)

associated to Q and the numbers

(6) sp

satisfy

(,C,(Q~F,SQip)i/P)l’q <: w* The norm of f in Bz’q(F) is defined as the infimum of these sums over all possible representations. If a > 0, then this definition is equivalent to the one given in Section 4, and it is independent of K. In general the convergence in (6) must be taken in the distribution sense, by means of ( f, 4) = c >“=,c I UQf$dp, #E c,"(w"), but if a>0 one can consider f as a function in LP(p), and let the convergence in (6) be in LP(p). The given definition is non-constructive, which is a major disadvantage. In [22] it is shown that, for a > 0, the atoms can be constructed in a linear way from f. The construction uses Q~.K~QSQ

198

A. JONSSON

and H. WALLIN

approximation with orthogonal polynomials as described in Section 4 and is somewhat technical. In the next section we will, in a simple case, give a constructive decomposition in terms of wavelets. The following trace theorems are easy to prove using the definition of Besov spaces by means of atomic decompositions. In the form stated here, they were given already in [9], cf. the discussion below. The trace operator is defined as in Section 5.1 (take 52 = IV in (4), and define f(F by (flF)(x) = f(x), x E F), where a version of part (b) of the theorem was given in the case when F is the boundary of an (E, 6)-domain. Theorem 9.

Let F be a closed d-set preserving Markov’s

local inequality.

Then holds

(a) if O 0, then the trace operator f + fl F is a bounded linear surjection BP,‘“(W)

+ B$,q(F:,

with a bounded right inverse E, an extension operator; and (b) if 0 < d < n, 1 < p < to, and /I = k - (n - d)/p > 0, then the trace operator f+ a bounded linear surjection Wf#?“)

+

fl F is

B;“(F)

with a bounded right inverse E. In applications to differential equations, for example, where methods from functional analysis are used, it is often important to know the dual S’ of a space S. The following theorem describes the duals of Besov spaces, and is taken from [18]. Theorem 10. q < 00. Then

Let F be a d-set preserving (BP,‘q(F))’

Markov’s

local inequality,

LY> 0, and 1 < p,

= BP_‘:‘(F)

where p’ and q’ are the dual indices of p and q. We now comment on the assumptions on F in Definition 8 and Theorem 9 (Theorem 10 has not been proved in any greater generality). The condition that F is a d-set assures that the dimension of F is uniformly equal to d throughout the set. This makes it possible to describe the trace space BF’q( F) with the smoothness index /3 = (Y- (n - d)/p. One can replace the condition that F is a d-set with a condition that F is the support of a measure ,U which satisfies certain types of refined doubling conditions, and obtain a definition of Besov spaces and trace theorems which are analogous to the ones given above. This was shown, for spaces with low smoothness and in a setting not involving atoms, in [24] and for general spaces in [22] and [25]. This gives much more general theorems; the price one has to pay is that the smoothness of the Besov spaces cannot be described in such a transparent way as before. The condition that F preserves Markov’s local inequality is needed to give certain estimates on F for polynomials. We make two comments in this context. When working with spaces of low smoothness, e.g. with BP,‘“(F) with CY< 1, then the polynomials needed are constants so these estimates become trivial which means that the assumption becomes superfluous, For spaces with higher regularity one can also work without the assumption that F preserves Markov’s local inequality, but then one must let the elements in the Besov spaces be families of functions instead of functions. The situation is then similar to one treated in a classical extension theorem by H. Whitney, see e.g. [9]. In this context we mention two other treatments of function spaces on general sets. In [26] Besov spaces and Sobolev spaces on spaces of homogeneous type (these are metric

Boundary value problems

199

spaces equipped with a measure satisfying a doubling condition) are studied by Han and Sawyer. They use atomic decompositions in their definitions, and so their approach is related to the method in [22]. In Hajlasz and Martio [27]: Sobolev spaces on metric spaces are studied with a different method. In both these cases one works with spaces with in some sense low smoothness, so one does not expect that something corresponding to Markov’s local inequality should enter the picture.

7. WAVELETS

7.1.

ON FRACTALS

Introduction

Wavelets have become an important tool in mathematics with many applications in physics, for example in signal processing. In this section we describe a first step towards developing wavelet theory on fractals, as given in [28]. We will discuss orthogonal wavelets only. For a broad introduction to wavelets, see e.g. [29]. A wavelet & on R is constructed from a mother wavelet @ by means of dilatations and translations by means of the formula &(x) = 2k’2r#42kx -- I). The function 4 is chosen in such a way that the fucntions & become orthogonal, and with 11$112 = 1, so the factor 2k’2 guarantees that ~~~kl~~~ = 1. Thus, the functions &, - ~0 < k, I< ~0, give an orthonormal system of functions in L2(W). Also, $ is chosen so that this system is complete, and thus every function f E L2(R) has a unique representation

k=-;a

,=-cc

where the wavelet coefficients Pkl are given by Pkl

=

lm +kr(x>f(x)dx. -cc

Of course, L2(lR) is characterized by means of the wavelets coefficients: a function f belongs to L2(R) if and only if zk, /p”,, < 03 is finite, and we have /fl12 = (~k,1&)1’2. In general, it is quite difficult to find a function @ with the desired properties. There is just one elementary example, the classical Haar wavelets. These are obtained by taking $ equal to the function h which is 1 on the interval [0, l/2) (this is short for the interval from 0 to l/2 which includes 0 but excludes l/2) and -1 on [l-/2, 11. The Haar wavelets cannot be used to characterize spaces of smooth functions by means of the magnitude of the numbers l/IkrI, since they are not continuous. The situation in this respect is somewhat different on fractals. To understand this, consider the usual Cantor set C, obtained from [0, l] by taking in each step away the middle third of the remaining intervals. Then the function which is 1 on [0, l/3] fl C and -1 on [2/3, l] fl C is continuous on C, and we shall see that it can be used to generate wavelets which characterize, e.g., the space A,(C) of Lipschitz continuous functions of order a on C, if a < 1. (If a! < 1, then a function f defined on a closed set F belongs to the space A,(F) if 1 is If(x)1 c M, x E F, and If(x) - f(r)1 s Mix - Y/Y x, y E F, for some constant M .) We also construct wavelets such that (Y2 1 is covered, and treat the more general Besov spaces. 7.2

Self-similar sets

Our wavelets will be constructed on a class of self-similar sets, and we start by describing this class, following essentially [6]. We also give a simple explicit example, the Cantor set,

200

A. JONSSON

and H. WALLIN

and in the next section another one, the Sierpinski gasket, which will play a central role in that section. Let Ti: lR”+ R”, i = 1, 2, . . ., N, where n 2 1 and N 3 2, be non-constant similtudes which are also contractions, i.e. for some Ti with 0 < Ti < 1 we have (T,(x) - T,(y)1 = rilx - yl,

x, y E KY,

i = 1, 2, . . ., N.

A similtude is a composition of a dilation with scale factor Ti, a translation, a rotation and perhaps a reflection. By a theorem in [6] there exists a unique non-empty compact set K which satisfies K = UiN,lTi(K:);

(7)

K is called the invariant set (with respect to the similtudes Tj). Let s be the number such that ~~lr~ = 1. By another theorem in [6] there exists a unique positive Bore1 measure y with support K which is invariant in the sense that

P(A)= &;‘(A)) i=l

for all Bore1 sets A. We also assume that the open set condition holds. This means by definition that there exists a bounded open set V such that U z,T,(V) C: V, where the union is disjoint. Then the Hausdorff dimension of the invariant set K is the number s defined above, and 0 < m,(K) < 03. Furthermore, p is a multiple of the restriction of m, to K, and for i # i holds m,( T,(K) II Tj(K)) = 0. In fact, as observed in [12], we even have that K is a d-set with d = s, and consequently p is a d-measure on K. We write Ki,i2,,,ik for the set Ti, 0 Ti2 0 . . . 0 T,,(K). Note that Ki,iZ,,,ik is geometrically similar to K. For k fixed, the union of all the sets Ki,iZ,,,ili is K, and two different KiliZ,,,ik are essentially disjoint in the sense that they intersect on a set with m,-measure zero. It is sometimes convenient to visualize the invariant set K using the following technique. Denote for a compact set F by t(F) the set UElTi( F), let t2(F) = t(t( F)), z3( F) = z(z’( F)), and so on. If z(F) C F, then we will have that F > z(F) > s(F) > . . ., and it can be shown that K is the intersection of all the sets z”(F). As an example, let T,(x) = x/3 and T2(x) = x/3 + 2/3 and take F = [0, 11. Then t(F) consists of the two intervals [0, l/3] and [2/3, 11. t2(F) consists of the four closed intervals obtained by taking away the middle third of these two intervals and so on. It is clear that K will be the Cantor set. In this case we have, for example, that K1,2 is K intersected with the interval [2/9, 3/91. 7.3.

Wavelets

Before defining wavelets on fractals, we return to the Haar wavelets for a moment. They can also be used as a wavelet system on the interval [0, l] if we discard functions which do not have support in [0, 11, and add the function $ which is identically 1 on [0, 11. More precisely, the functions # and qkl, k 3 0, 0 6 I< 2k, form an orthonormal base in L2 ([0, 11). An important property of the Haar wavelets is that I&r dx = 0. This is useful when one wants to estimate Haar coefficients. For example, if f E h,(R) where cy< 1, then in the SUppOrt Of &[, Bkl = .hX)~kdX) b = h - f(XO))~kl(X) d x, so if x0 iS taken which is an interval of length 2-k, we get, using also that 1qklI < 2kiz, that l@k,1s M2-k”2-kp. If we want to make similar estimates for functions f of higher smoothness, we would need that the wavelets are orthogonal to polynomials of higher order. One can construct such Haar wavelets if one uses several mother functions. For example, the

Boundary value problems

201

following functions give a wavelet system where the wavelets I/,&, o = 1, 2, are orthogonal to first degree polynomials. Let G,(x) = 1 and h(x) = 2d3(x - l/2), and let r$ be given by q’(x) = (24/v7)x 5/67 if 0 6 x < l/2 and r/~‘(x) = -l/q7 if l/2 s x c 1, and I+!? by 3(x) = 2v3/7x - 6317 if 0 6 x < l/2 and q2(x) = -2d21x + lld3/7 if l/2 < x G 1. Define r& from t+!+’as before. Then the functions vi,, k 3 0, I= 0, 1, . . ., 2k - 1, o = 1, 2, and eI, 1 = 1, 2, form a complete orthonormal set in L2([0, 11). We now describe a similar construction on self-similar sets. At least on certain totally disconnected sets, it will in some sense be more effective than on [0, 11. We let K be a self-similar set as described above, and we assume furthermore that K is not a subset of some IZ - l-dimensional subspace of KY. This last assumption implies, as shown in [12], that K preserves Markov’s local inequality which is essential here. Denote by 9, the set of polynomials on R” of total degree G m. Then 9, has dimension M,, =

as a vector space, and since K preserves Markov’s local inequality it is not hard to see that it has dimension M0 also as a subspace of L2(p)), see, e.g. [30, p. 3101. Put S, = P’,, and let S1 be the space of all functions which are piecewise polynomials in the sense that the functions in S1 coincide, for i = 1, 2, . . . , N, on each Ki\(Ujzi(Ki fl Kj)) with a polynomial in 9,. Thus the functions in S1 are defined ,u-a.e. on K and we have S0 C S1. Consider S1 0 S,,, the orthogonal complement of S,, in S1, i.e. the set of functions in S1 which are orthogonal to all functions in S,. Since So has dimension M0 and S1 has dimension M,JV, the space S, 0 S,, has dimension M = M,JV - MO. Choose one orthonormal base &, &, . . ., @M0 in S, and one orthonormal base r$, 3, . . ., v, in S1 0 S,; we let the base functions be zero outside K. Next define functions Wzi *,,,i, for o = 1, 2, . . . , M, by

This means that ~~i2,,,ik is obtained by ‘compressing’ I/J~ in a linear way so that it gets support KiliP.,.ik and L2(p)-norm 1. We shall use the following shorthand notation. We denote by Ik the set of multi-indices {i = (ii, i2, . . . , ik), 1 =z i, s N, Y = 1, 2, . . . , k} and write, e.g. I&‘, i E I,, instead of W:i2,,,ik. We also put rj$ = ?+Y, o = 1, 2, . . . , M, and let I,, = (0). Then we have the following result. The functions $,, 1 G 1 s M0 and I#:, i E Ik, k 3 0, cr = 1, 2, . . ., M, form a complete orthonormal set of functions in L*(p). Consequently, any f E L2(p) has a representation Theorem 11.

k=Oa=liel,,

where cul = /f$+ dp and p’ = lf&‘dp. If f is ’ in a Besov space then we estimate the wavelet coefficients for f in Theorem 12 below. We introduce some more notation. A multiindex i = (i1, i2, . . ., ik) which is in Zk for some k, belongs to J,, Y integer, if 2~’ G diam Ki < 2-“+l (we permit here i = 0 with the interpretation that K. = K) and v. is the integer such that 2-Q s diam K < 2-“o+‘. For a given K with Hausdorff dimension s we put for a sequence {bi} where i runs over all multi-indices i E .I,,, Y2 vo,

202

A. JONSSON and H. WALLIN Il{bi}llbcq

=

i

2”P2”(liz-llP)P

c

( v= v, (

TheoremI2.

icJ,

Let 1~ p, q G a~, a > 0, f E Bft’q(K),

l/pjP

q’p

llq.

1)

and ~lt 3 [a]. Then

where c does not depend on f or the wavelet basis. A converse of this holds if the sets Ti( K) = Ki, i = 1, 2, . . . , N, are mutually disjoint, for example if K is the Cantor set. Thus in this case Besov spaces may be characterized by the magnitude of the wavelet coefficients. We remark that if p = q = 03, then BP,,q(F) coincides with the Lipschitz space A,(F), so we have in particular a result for these spaces as promised in Section 7.1. Let 1~ p, q G CQand (Y> 0, suppose that the sets K1, K,, . . ., KN are disjoint, and let f be an integrable function on K. Then

Theorem 13.

mutually

MoaIIyp + 5 llumb:.q). IlfIIB;‘(K)=sc((21 a=1 where c does not depend on

f or the wavelet basis. 8. BROWNIAN

MOTION

Diffusion processes on fractals have been studied for some time. The investigations were started in the beginning of the 1980s by physicists who were interested in transportation properties in disordered media. They use fractals as models for such media, and study, e.g. random walks on fractals, see [31]. Since then a lot of work on the subject has been done by mathematicians, who have laid the mathematical foundations both with probabilistic and analytical methods. See for example [32] and the references given there. We will here be interested in Brownian motion on the Sierpinski gasket, which has widely been used as a prototype for sets on which Brownian motion can be constructed. Our purpose is to point out how function spaces on fractals enter the picture. The results described here were given in [33]. We start by recalling how the Sierpinski gasket in R” is constructed with the methods in Section 7, and give in the same time some new notation; it does no harm to think of n as or vertices, in an n-dimensional simplex F with 2. Let ply ~2, . . ., p,,, pn+l be comers, edges of length 1 (if 12= 2 this is an equilateral triangle). Let Ti = (x - pi)/2 + pi, i = 1, 2, . . .) IZ + 1, so Ti are similtudes which leave pi fixed. The Sierpinski gasket K is the invariant set for these similtudes. The set K can also be obtained as the intersection of the sets t“(F) defined in Section 7. Note that tl(F) consists of n + 1 simplices, namely for i = 1, 2, . . ., 12+ 1, the simplices with corners pi and (p, + pi)/29 i # i. We denote the set of these simplices by .&. Analogously, $(I;) is the union of (n + 1)2 simplices, obtained by taking in the same fashion n + 1 subsimplices from each simplex in Ml. Continuing in this way we get that zk(F) is the union of (n + l)k simplices with sides of length 2-k, and we denote the set of these simplices by .I&. We also let J&J = {F}. If M is a simplex, we denote by V(M) the set of the n + 1 vertices of h4, and we put V, = UkaOUMEAkV(M). An analytic approach to the Brownian motion, and more general Markov processes, which is frequently used, is to start with Dirichlet forms, see [34]. This is because a certain class of Dirichlet forms, called regular, are equivalent to certain Markov processes. A

Boundary value problems

203

Dirichlet form (%, 9) on a real L*-space H is a closed non-negative definite symmetric bilinear form % defined in 9 x 8, where B is a dense subspace of H, having furthermore the property that it is Markovian. The last condition means that if u E B and u is the function given by u(x) = 0 if u(x) < 0, u(x) = 1 if U(X) > 1 and u(x) = u(x) otherwise, then u E 9 and %(u, u) 6 %(u, u). For example, the Dirichlet form on L*(R’) which leads to the classical Brownian motion in IF!” is given by

Here we are interested in the Dirichlet forms which have been defined in correspondance to Brownian motion on the Sierpinski gasket. The following is taken from [35], where also necessary verifications may be found. For f: V, + R’ let

~e(“‘(f~

f)

=

$

y

mM&

(

1

p m

q&cf(P)

Then it can be shown that ?~?~)(f, f) is non-decreasing and

-

f(d)*.

(9)

7

in m and we let for f: V, + R

B = {f: v* + R: %(f, f) < m}.

Each f E 9 can be uniquely extended to an element in C(K) so we have 9 C C(K) C L*(K, p)>, where ,u is the invariant measure on K. Defining @“)(f, g) as in (9) but with tf(~) - f(q))* replaced by (f(p) - f(q))(&) - g(q)) and W, d = lim,@m)tfy g> it can be shown, that (3, 9) is a regular Dirichlet form on L*( K, ,u). We consider % as a vector space equipped with the norm (]lflli,, + %(f, f))‘b. We shall identify 4. To do this we need one more function space, but we shall see that it is in general just an easily identified subspace of a Besov space. LY> 0, and let F be a d-set and ,u a Let O
If(x)

-

f(y)lP

d&)

My)

where Thenormof f in LM~7 P, q)(F) is Ilfll,,, + l14~q~

=

2-yw4.

Ilfll,,, denotes the LP@)-norm. If a < 1, then Lip(cu, p, q)(F) is the same space as BtTq( F), since it is well known that BP,“(F) can then be defined by means of first differences in this way, see [9]. If ff is non-integer and greater than 1, and k is the integer such that k < (t < k + 1, then Lida P, q)(F) is the subspace of BP,‘4(F) consisting of the functions in BP,‘“(F) whose derivatives, in a certain sense which we not make precise here, up to order k are all zero. Contrary to the situation for spaces defined on R” , this does not in general imply that the functions in Lip(cu, p, q)(F) are constants if a > 1. Note that by the discussion in Section 7, the Sierpinski gasket is a d-set with d = In (n + l)/In2, and thus Lipschitz spaces can be defined on it. The following theorem explains a connection between the Dirichlet forms defined above and these Lipschitz spaces.

204

A. JONSSON and H. WALLIN

Theorem 15.

Let K be the Sierpinski

gasket in R”, and let p = In (n + 3)/ln4.

9 = WV,

2,

Then

m)(K)

with equivalent norms. This identification allows us to use the theory of Besov spaces to clarify the structure of the space 9. For example, from Theorem 9 it follows, if p in Theorem 15 is noninteger and k < /3 < k + 1, that f E 3 if and only if f has an extension Ef which is in B$.$-d,i2(R’) and has derivatives of order not exceeding k equal to zero on K. Similarly, one can use a version of a classical imbedding theorem by Sobolev to conclude that the functions in B satisfy a Lipschitz condition of order 13- d/2. REFERENCES 1. H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Department of Mathematics, University of Umel no. 5 (1989). 2. H. Wallin, The trace to the boundary of Sobolev spaces on a snowflake, Manuscripta Math. 73, 117-125 (1991). 3. P. Grisvard, Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985). 4. P. D. Panagiotopoulos, Fractals and fractal approximation on structural mechanics, Meccanica 27, 25-33 (1992). 5. P. D. Panagiotopoulos, Fractal geometry in solids and structures, Int. .I. Solids Structures 29, 2159-2175 (1992). 6. J. E. Hutchinson, Fractals and self-similarity, Znd. Univ. Math. J. 30, 713-47 (1981). 7. B. B. Mandelbrot, The Fractal Geometry of Nature. Freeman, San Francisco (1982). 8. K. J. Falconer, The Geometry of Fractal Sets. Cambridge University Press, Cambridge (1985). 9. A. Jonsson and H. Wallin, Function Spaces on subsets of R”. .Harwood Acad. Publ. (1984). 10. A. Jonsson, P. Sjogren and H. Wallin, Hardy and Lipschitz spaces on Subsets of Iw”, Studia Math. 80, 141-166 (1984). 11. P. Wingren, Lipschitz spaces and interpolating polynomials on subsets of Euclidean space, Lecture Notes in Mathematics, Vol. 1302, pp. 424-435. Springer, Berlin (1988). 12. H. Wallin, Self-similarity, Markov’s inequality and d-sets, in Constr. theory of functions, pp. 285-297. Vama, Sofia (1993). 13. H. Wallin and P. Wingren, Dimensions and geometry of sets defined by polynomial inequalities, J. Approx. Theory 69, 231-249 (1992). 14. J. Vaislll, Bilipschitz and quasisymmetric extension properties, Ann. Acad. Sci. Fenn. Ser. A I Math. 11, 239-274 (1986). 15. J. Vlislla, M. Vuorinen and H. Wallin, Thick sets and quasisymmetric maps, Nagoya Math. J. 135, 121-148 (1994). 16. S. Campanato, Proprieta di una famiglia di spazi funzionali, Ann. SC. Norm. Sup. Pisa 18, 137-160 (1964). 17. Ju. A. Brudnyi, Piecewise Polynomial Approximation, Embedding Theorems and Rational Approximation, Lecture Notes in Mathematics, Vol. 556, pp. 73-98. Springer, Berlin (1976). 18. A. Jonsson and H. Wallin, The dual of Besov Spaces on fractals, Studia Mathematics 112, 285-300 (1995). 19. P. W. Jones, Quasiconformal mappings and extendability of functions in Sobolev spaces, Acta Math. 147, 71-88

(1981).

20. K. Nystrom, Smoothness properties of solutions or Dirichlet’s problems in domains with a fractal boundary, Department of Mathematics, University of Umel, Doctoral thesis, no. 7 (1992). 21. M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34, 777-797 (1985). 22. A. Jonsson, Besov spaces on closed sets by means of atomic decomposition, Department of Mathematics, University of Umei, no. 7 (1993). 23. A. Jonsson, Atomic decomposition of Besov spaces on closed sets, in Function Spaces, Differential Operators and Nonlinear Analysis, pp.*285-289. B. G. Teubner, Leipzig (1993). 24. A. Jonsson. Besov maces on closed subsets of Iw”, Trans. Amer. Math. Sot. 341, 355-370 (1994). 25. P. Byland, The trace of Besov spaces to sets with varying local dimension, Department of Mathematics, University of Umel, no. 9 (1992). 26. Y-S. Han and E. T. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and classical function spaces, Mem. Am. Math. Sot. 110, (1994). 27. P. Hajlasz and 0. Martio, Traces of Sobolev functions on fractal type sets and characterization of extension domains, preprint (1995). 28. A. Jonsson, Wavelets on fractals, manuscript (1995). 29. I. Daubechies, Ten Lectures on Wavelets. SIAM, Philadelphia (1992). 30. A. Jonsson. MarkovS Zneaualitv and Local Polynomial Approximation, Lecture Notes in Mathematics, Vol. 1302, pp. 303-316. Springer, Berlin (1988). . --

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31. F. Rammal and G. Toulouse, Random walks on fractal structures and percolation clusters, J. Physique Let&es 44, L13-L22 (1983). 32. M. T. Barlow, Harmonic Analysis on Fracral Spaces. Seminar BQURBAKI 44tme annee 755, 345-368, (1991-92). 33. A. Jonsson, Brownian motion on fractals and function spaces, Math. Zeit 222, 495-504 (1996). 34. M. Fukushima, Dirichlet Forms and Markov Processes. North Holland (1980). 35. M. Fukushima and T Shima, On a spectral analysis for the Sierpinski gasket, J. Potential Analysis 1, 1-35 (1992).