Mean value properties of harmonic functions on Sierpinski gasket type fractals

Transcription

1 Mean value properties of harmonic functions on Sierpinski gasket type fractals arxiv: v3 [math.ca] 6 May 203 HUA QIU AND ROBERT S. STRICHARTZ Abstract. In this paper, we establish an analogue of the classical mean value property for both the harmonic functions and some general functions in the domain of the Laplacian on the Sierpinski gasket. Furthermore, we extend the result to some other p.c.f. fractals with Dihedral-3 symmetry. Keywords. Sierpinski gasket, Laplacian, harmonic function, mean value property, analysis on fractals. Mathematics Subject Classification (2000) 28A80 Introduction It is well known that harmonic functions (i.e., solutions of the Laplace equation u = 0, where = d 2 i= ) possess the mean value property: Namely, if u is harmonic on a x 2 i domain Ω R d, then for every closed ball B r (x) Ω of a center x Ω and radius r > 0 the average of u over B r (x) equals to the value of x, i.e., u(y)dy = u(x), B r (x) B r(x) where B r (x) is the volume of the ball B r (x). There is a similar statement for mean values on spheres. More generally, if u is not assumed harmonic but u is a continuous function, then lim r 0 for the appropriate dimensional constant c n. ( ) u(y)dy u(x) = c r 2 n u(x) (.) B r (x) B r(x) The research of the first author was supported by the National Science Foundation of China, Grant 09008, and the Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions. The research of the second author was supported in part by the National Science Foundation, Grant DMS

2 What are the fractal analogs of these results? The analytic theory on p.c.f. fractals was developed by Kigami [3, 4, 5] following the work of several probabilists who constructed stochastic processes analogous to Brownian motion, thus obtaining a Laplacian indirectly as the generator of the process. See the book of Barlow [] for an account of this development. Since analysis on fractals has been made possible by the analytic definition of Laplacian, it is natural to explore the properties of these fractal Laplacians that are natural analogs of results that are known for the usual Laplacian. As for the fractal analog of the mean value property, we won t state the nature of the sets on which we do the averaging here, but will say that if K is a fractal set and x K, we investigate whether there is a sequence of sets B k (x) containing x with k B k(x) = {x} such that u(y)dy = u(x) µ(b k (x)) B k (x) for every harmonic function u. Moreover, for general u not assumed harmonic, is there a formula analogous to (.)? In the present paper, we will mainly deal with the Sierpinski gasket SG. This set is a key example of fractals on which a well established theory of Laplacian exists [3 7]. Since the mean value property plays a very important role in the usual theory of harmonic functions, it is of independent interest to understand the similar property of harmonic functions on the Sierpinski gasket. We will prove that for each point x SG \V 0, (V 0 is the boundary of SG.) there is a sequence of mean value neighborhoods B k (x) depending only on the location of x in SG. {B k (x)} forms a system of neighborhoods of the point x satisfying k B k(x) = {x}. On such sequences, we get the fractal analogs of the mean value properties of both the harmonic functions and the general functions which belong to the domain of the fractal Laplacian satisfying some natural continuity assumption. We also investigate the extent to which our method can be applicable to other p.c.f. selfsimilar sets, but it seems that it strongly depends on the symmetric properties of both the geometric structure and the harmonic structure of the fractals. The paper is organized as follows: In Section 2 we briefly introduce some key notions from analysis on the Sierpinski gasket. In Section 3 and Section 4, we prove the mean value property for harmonic functions and general functions on SG respectively. Section 5 contains a further extension of the mean value property to p.c.f. self-similar fractals with Dihedral-3 symmetry. An interesting open question is to what extent the results of Section 4 can be extended to this class of fractals. See [2] for a related result concerning solutions of divergence form elliptic operators. 2

3 2 Analysis on the Sierpinski gasket For the convenience of the reader, we collect some key facts from analysis on SG that we need to state and prove our results. These come from Kigami s theory of analysis on fractals, and may be found in [3, 4, 5]. An elementary exposition may be found in [6, 7]. Recall that SG is the attractor of the i.f.s (iterated function system) in the plane consisting of three homotheties {F 0,F,F 2 } with contraction ratio /2 and fixed points equal to the three vertices {q 0,q,q 2 } of an equilateral triangle. Then SG is the unique nonempty compact set satisfying 2 SG = F i (SG). (2.) i=0 We refer to the sets F i (SG) as cells of level one, and by iterating (2.) we obtain the splitting of SG into cells of higher level. For a word w = (w,w 2,,w m ) of length m, the set F w (SG) = F w F w2 F wm (SG) with w i {0,,2}, is called an m-cell. The fractal SG can be realized as the limit of a sequence of graphs Γ 0,Γ, with vertices V 0 V. The initial graph Γ 0 is just the complete graph on V 0 = {q 0,q,q 2 }, which is considered the boundary of SG. See Fig. 2.. Note that SG is connected, but just barely: there is a dense set of points J, called junction points, defined by the condition that x J if and only if U \{x} is disconnected for all sufficiently small neighborhoods U of x. It is easy to see that J consists of all images of {q 0,q,q 2 } under iterates of the i.f.s. The vertices {q 0,q,q 2 } are not junction points. All other points in SG will be called generic points. In the SG case, J = V \V 0, where V = m V m. However, it is not true for general p.c.f. self-similar sets. In all that follows, we assume that SG is equipped with Γ 0 Γ Γ 2 Fig. 2.. The first 3 graphs, Γ 0,Γ,Γ 2 in the approximation to the Sierpinski gasket. the self-similar probability measure µ that assigns the measure 3 m to each m-cell. 3

4 We define the unrenormalized energy of a function u on Γ m by E m (u) = (u(x) u(y)) 2. x my The energy renormalization factor is r = 3 5, so the renormalized graph energy on Γ m is E m (u) = r m E m (u), and we can define the fractal energy E(u) = lim m E m (u). We define dome as the space of continuous functions with finite energy. Then E extends by polarization to a bilinear form E(u,v) which serves as an inner product in this space. The standard Laplacian may then be defined using the weak formulation: u dom with u = f if f is continuous, u dome, and E(u,v) = fvdµ for all v dom 0 E, where dom 0 E = {v E : v V0 = 0}. There is also a pointwise formula (which is proven to be equivalent in [7]) which, for points in V \V 0 computes u(x) = 3 2 lim m 5m m u(x), where m is a discrete Laplacian associated to the graph Γ m, defined by for x not on the boundary. m u(x) = (u(y) u(x)) y mx It is not necessary to invoke the measure to define harmonic functions, although it is true that these are just the solutions of h = 0. The more direct definition is that h(x) = 4 y mx forevery nonboundarypoint andevery m. This canbeviewed asamean valuepropertyof h at the junction points. The space of harmonic functions is 3-dimensional and the values at the 3 boundary points may be freely assigned. Moreover, there is a simple efficient algorithm, the 2 rule, for computing the values of a harmonic function exactly at 5 5 all vertex points in terms of the boundary values. The harmonic functions satisfy the maximum principle, i.e., the maximum and minimum are attained on the boundary and 4 h(y)

5 only on the boundary if the function is not constant. We call a continuous function h a piecewise harmonic spline of level m if h F w is harmonic for all w = m. The Laplacian satisfies the scaling property and by iteration for F w = F w F w2 F wm. (u F i ) = 5 ( u) F i (u F w ) = 5 m( u) F w Although there is no satisfactory analogue of gradient, there is a normal derivative n u(q i ) defined at boundary points by n u(q i ) = lim m y mq i r m (u(q i ) u(y)), the limit existing for all u dom. The definition may be localized to boundary points of cells: for each point x V m \V 0, there are two cells containing x as a boundary point, hence two normal derivatives at x. For u dom, the normal derivatives at x satisfy the matching condition that their sum is zero. The matching conditions allow us to glue together local solutions to u = f. As is shown in [3, 4, 7], the Dirichlet problem for the Laplacian can be solved by integrating against an explicitly given Green s function. Recall that the Green s function G(x,y) is a uniform limit of G M (x,y) as M goes to the infinity, with G M defined by M G M (x,y) = g(z,z )ψ z (m+) (x)ψ (m+) z (y) m=0 z,z V m+ \V m and { g(z,z) = 9 50 rm for z V m+ \V m, g(z,z ) = 3 50 rm for z,z V m+ \V m with z,z F w (SG) for w = m, and z z, where ψ m z (x) denotes a piecewise harmonic spline of level m satisfying ψ (m) z (x) = δ z (x) for x V m. 3 Mean value property of harmonic functions on SG Lemma 3.. (a) Let C be any cell with boundary points p 0,p,p 2, and h any harmonic function. Then hdµ = µ(c) C 3 (h(p 0)+h(p )+h(p 2 )). 5

6 (b) Let p be any junction point, and C, C 2 the two m-cells containing p. Then hdµ = h(p). µ(c C 2 ) C C 2 Proof. The space of harmonic functions on C is three-dimensional. A simple basis {h 0,h,h 2 } is obtained by taking h j (p j ) = and h j (p k ) = 0 for k j. Noticing that h 0 + h + h 2 is identically on C, by symmetry, h C idµ = µ(c) for each i. Hence 3 (a) follows. (b) follows by combining (a) for C = C and C = C 2 with the mean value property of h at p. Note that (b) gives a trivial solution to the problem of finding mean value neighborhoods for junction points. p C 0 p 0 D w (2) C w C 2 p2 C D w D () w D w Fig. 3.. C w and its three neighboring cells. The right part of the figure refers to the proof of Lemma 4.. Given a point x in SG \V 0, consider any cell F w (SG)(denote it by C w ) containing the point x, with boundary points F w q i = p i. Choose the cell C w small enough, such that it does not intersect V 0. Then it must have three neighboring cells C 0, C and C 2 of the same level with C i intersecting C w at p i. Denote by D w the union of C w and its three neighbors. See Fig. 3.. In this section, we will describe a method to find a subset B of D w, containing C w, such that for any harmonic function h, the mean value of h over B is equal to its value at x, i.e., M B (h) = h(x) where M B (h) is defined by M B (h) = hdµ. µ(b) Then we will call the set B a k level mean value neighborhood of x associated to C w where k is the length of w. 6 B

7 Let h be a harmonic function on SG. The harmonic extension algorithm implies that there exist coefficients {a i (x)} depending only on the relative position of x and C w such that h(x) = a i (x)h(p i ). i Moreover, since constants are harmonic we must have a i (x) = i and by the maximum principle all a i (x) 0. Let W denote the triangle in R 3 with boundary points (,0,0),(0,,0) and (0,0,) and π W the plane in R 3 containing W. So {(a 0 (x),a (x),a 2 (x))} W for any x C w. However, not every point in W occurs in this way. On the other hand, given a set B such that C w B D w, by linearity we have M B (h) = i a i h(p i ) (3.) for some coefficients (a 0,a,a 2 ) depending only on the relative geometry of B and C w. Again we must have a i = by considering h. So (a 0,a,a 2 ) π W. (Later we will show that (a 0,a,a 2 ) does not have to belong to W for some sets B.) Thus we have a map, denoted by T from the collection of B s to π W. If we can show that the image of the map T covers the triangle W for some reasonable class of sets B, then we can get a set B over which the mean value property holds for all harmonic functions. Moreover, if we can prove T is one-to-one, then we get a mean value neighborhood B of x associated to C w, that is unique within the collection of sets we are considering. The above is the basic idea of our method. Hence, the remaining task in this section is to find a suitable class B of sets B such that there is a map T from B to π W, such that T (B) covers the triangle W. Comparing with the usual mean value neighborhoods (they are just balls in the Euclidean case), it is reasonable to require B to be as simple as possible. They should be connected, possess some symmetry properties, depend only on the relative geometry of x and C w, and be independent of the level of C w and the location of C w. In the following, we use ρ to denote the distance from p 0 to the line containing p and p 2, namely, ρ is the length of the height of the minimal equilateral triangle containing C w. Call ρ the size of C w. 7

8 Definition 3.. Let c 0,c,c 2 be three real numbers satisfying 0 c i, denote by B(c 0,c,c 2 ) the set B(c 0,c,c 2 ) = C w E 0 E E 2, where each E i is a sub-triangle domain in C i obtained by cutting C i symmetrically with a line at the distance c i ρ away from the vertex p i. Remark. See Fig. 3.2 for a sketch of B(c 0,c,c 2 ). For example, B(0,0,0) = C w and B(,,) = D w. Denote by B = {B(c 0,c,c 2 ) : 0 c i } the natural 3-parameter family of all such sets. Each member of B contains C w and is contained in D w. Denote by σ : B Λ the natural one-to-oneprojectionwith σ(b(c 0,c,c 2 )) = (c 0,c,c 2 ), whereλ = {(c 0,c,c 2 ) : 0 c i }. p 0 E 0 C w p p 2 E 2 E B(c 0,c,c 2 ) Fig The relative geometry of B(c 0,c,c 2 ) and C w. For each vector (c 0,c,c 2 ) Λ, there is a unique vector (a 0,a,a 2 ) π W corresponding to the set B(c 0,c,c 2 ), satisfying (3.) where B is replaced by B(c 0,c,c 2 ). This defines a map T from Λ to π W. Then T described above from B to π W is exactly T = T σ. The following lemma shows that the value T(c 0,c,c 2 ) is independent of the particular choice of C w, which benefits from the symmetric properties of the set B(c 0,c,c 2 ). Lemma 3.2. T(c 0,c,c 2 ) is independent of the particular choice of C w. 8

9 Proof. Let h be a harmonic function. First we consider the integral E i hdµ. Denote by {s i,t i,p i } the boundary points of C i. By linearity, µ(c w) E i hdµ can be expressed as a non-negative linear combination of {h(s i ),h(t i ),h(p i )}, which by symmetry must have the form E i hdµ = (m i h(p i )+n i (h(s i )+h(t i )))µ(c w ), (3.2) for some appropriate non-negative coefficients m i,n i. Notice that in (3.2), the coefficients m i,n i are independent of the location of C i in SG. Actually, they only depend on the relative position of E i in C i, i.e., m i,n i depend only on c i. Using the mean value property at p i, namely 4h(p i ) = h(p i )+h(p i+ )+h(s i )+h(t i ), we obtain E i hdµ = (m i h(p i )+n i (4h(p i ) h(p i ) h(p i+ )))µ(c w ) = ((4n i +m i )h(p i ) n i (h(p i )+h(p i+ )))µ(c w ). Notice thattheratio ofµ(e i )to µ(c i ) also depends onlyonc i. Combined withlemma 3.(a), we see that (a 0,a,a 2 ) = T(c 0,c,c 2 ) is independent of the particular choice of C w, depending only on (c 0,c,c 2 ). WewillshowtheimageofthemapT coversthetrianglew. Moreprecisely, T(c 0,c,c 2 ) will fill out a set W which is a bit larger than W. Denote by P 0 = (,0,0), P = (0,,0) and P 2 = (0,0,) the three boundary points of the triangle W in R 3 and by O the center point of W. O Γ y P 2 Γ Q 0 Fig a /6 region of W surrounded by OQ 0, OP 2 and P 2 Q 0. Lemma 3.3. T(0,0,) = P 2 and T(0,,) = Q 0 where Q 0 = {, 5, 5 } is a point in π W located outside of W. 9

10 Proof. From Definition 3., B(0,0,) = C w C 2. Hence by Lemma 3.(b), for any harmonicfunctionh, wehave M B(0,0,) (h) = h(p 2 ). Thisimplies T(0,0,) = P 2. Similarly, B(0,,) = C w C C 2, then for any harmonic function h, still using Lemma 3., we get M B(0,,) (h) = which gives T(0,,) = Q 0. ( ) hdµ+ hdµ hdµ 3µ(C) C w C C w C 2 C w = 9 h(p 0)+ 5 9 h(p )+ 5 9 h(p 2), Lemma 3.4. T({(0,c,) : 0 c }) is a continuous curve lying outside of W, joining P 2 and Q 0. (See Fig. 3.3.) Proof. From Lemma 3.3, by varying c continuously between 0 and we trace a continuous curve P 2 Q 0 joining P 2 and Q 0. So we only need to prove the curve P 2 Q 0 lies outside of W. To prove this, we consider the set B = B(0,c,) for 0 c. In this case B = C w E C 2. Given a harmonic function h, by the proof of Lemma 3.2, we have E hdµ = ((4n +m )h(p ) n (h(p 0 )+h(p 2 )))µ(c w ), for some appropriate non-negative coefficients m,n depending only on c. On the other hand, we have by Lemma 3.(b). Hence B C w C 2 hdµ = 2h(p 2 )µ(c w ), hdµ = hdµ+ hdµ E C w C 2 = ( n h(p 0 )+(4n +m )h(p )+(2 n )h(p 2 ))µ(c w ). The coefficient of h(p 0 ) is always less than 0. Moreover, it equals to 0 if and only if E = (c=0). Hence T(0,c,) will always lie on the outside of the triangle W as c varies between 0 and. Now we come to the main result of this section. 0

11 Theorem 3.. The map T from B to π W fills out a region W which contains the triangle W. Proof. WeonlyneedtoprovethemapT fromb toπ W fillsouta/6regionsurrounded by the line segments OQ 0, OP 2 and the curve P 2 Q 0 as shown in Fig Then we will get the desired result by exploiting the symmetry. Consider a subfamily B = {B(0,0,c) : 0 c } of B. If we restrict the map T to B, by varying c continuously between 0 and we trace a curve (it is a line segment, which follows from the symmetry of E 2 ) in W joining the center O and the vertex point P 2. Consider another subfamily B 2 = {B(0,c,c) : 0 c } of B. If we restrict the map T to B 2, by varying c continuously between 0 and we trace a curve (it is also a line segment, which follows from the symmetric effect of E and E 2 ) in W joining the center O and the point Q 0 across the boundary line P P 2 with Q 0 located outside of W, where Q 0 is the point defined in Lemma 3.3. Fix a number 0 y. Consider a subfamily C y = {B(0,c,y) : 0 c y} of B. If we restrict the map T to C y, by varying c continuously between 0 and y we trace a curve Γ y joining the two points T(0,0,y) and T(0,y,y). The first endpoint T(0,0,y) lies on the line segment OP 2 and the second endpoint T(0,y,y) lies on the line segment OQ 0. (See Fig for Γ y.) When y = 0, the curve Γ 0 draws back to the single center point O. When y =, by Lemma 3.4, the curve Γ is a continuous curve located outside of the triangle W. Moreover, P 2 is the only common points of Γ and W. Hence if we vary y continuously between 0 and, we can fill out the /6 region surrounded by the line segments OQ 0, OP 2 and the curve P 2 Q 0. Remark. In the proof of the above theorem, we actually only consider those sets B in B which are contained in the union of C w and subsets of only two neighbors. See Fig Of course, the map T restricted to this subfamily is one-to-one, which can be easily seen from the proof. Hence instead of B, the map T is one-to-one from B onto W, where B = {B(0,c,c 2 ) : 0 c i } {B(c 0,0,c 2 ) : 0 c i } {B(c 0,c,0) : 0 c i }. Based on the discussion in the beginning of this section, we then have Theorem 3.2. For each point x SG \ V 0, there exists a system of mean value neighborhoods B k (x) with k B k(x) = {x}. Proof. Let k 0 be the smallest value of k such that there exists a k level cell C w containing x but not intersecting V 0. (k 0 depends on the location of x in SG.) Then

12 Fig The 3 shapes of B B associated to C w shown in Fig. 3.. by using Theorem 3. we can find a sequence of words w (k) of length k (k k 0 ) and a sequence of mean value neighborhoods B k (x) associated to C w (k). Obviously, {B k (x)} k k0 will form a system of neighborhoods of the point x satisfying k k 0 B k (x) = {x}. 4 Mean value property of general functions on SG In this section, we extend the mean value property to more general functions on SG. Given a point x in SG \ V 0 and a cell C w = F w (SG) containing x, for each mean value neighborhood B of x associated to C w, we assign a constant c B to B. We want M B (u) u(x) c B u(x) foruindom. Moreprecisely, let{b k (x)} k k0 bethesystemofmeanvalueneighborhoods of the point x; we want lim k c Bk (x) ( MBk (x) u(x) ) = u(x) (4.) for appropriate functions in the domain of, which is the desired fractal analog of (.). For this purpose, let v be a function on SG satisfying v. For each point x in SG \V 0, and each mean value neighborhood B of x, define c B by c B = M B (v) v(x). Note that the result is independent of which v, because any two such functions differ by a harmonic function and the equality M B (h) h(x) = 0 always holds for any harmonic function h. So we can choose v(x) = G(x, y)dµ(y), 2

13 which vanishes on the boundary of SG. Here G is Green s function. Then We will prove that c B is controlled by the size of C w. More precisely, we will prove: Theorem 4.. Let x SG \ V 0 and B be a k level mean value neighborhood of x. c 0 5 k c B c 5 k for some constant c 0,c which are independent of x. To prove Theorem 4., we need the explicit expression for the function v. Recall from Section 2 that v(x) is the uniform limit of v M (x) for v M (x) = G M (x,y)dµ(y). Interchanging the integral and summation, M v M (x) = m=0 g(z,z ) z,z V m+ \V m ψ (m+) z (y)dµ(y)ψ z (m+) (x). Notice that for each z V m+ \ V m, ψ z (m+) is a piecewise harmonic spline of level (m+) satisfying ψ z (m+) (y) = δ z (y) for y V m+. More precisely, ψ z (m+) is supported in the two (m+)-cells meeting at z. If F τ (SG) is one of these cells with vertices z,z and z 2, then ψ (m+) z +ψ (m+) z (ψ z (m+) F τ(sg) +ψ (m+) z 2 +ψ (m+) z restricted to F τ (SG) is identically. Thus +ψ (m+) z 2 )dµ = µ(f τ (SG)) = 3 m+. By symmetry all three summands have the same integral, so F ψ(m+) τ(sg) z dµ =. 3 m+2 Together with the contribution from the other (m+)-cell we find for each z V m+ \V m, ψ (m+) z (y)dµ(y) = 2 3m+2. (4.2) Hence v M (x) = 2 9 M m=0 g(z,z )ψ 3 m z (m+) (x). z,z V m+ \V m Substituting the exact value of g(z,z )(see Section 2 and details in [7] page 50) into it, we 3

14 get v M (x) = 2 9 = 2 9 = 5 M m=0 M m=0 M m=0 3 m 3 m w =mz,z F w(v 0 )\F w(v ) w =mz F w(v 0 )\F w(v ) 5 mφ m(x) g(z,z )ψ (m+) z (x) ( 9 50 rm rm )ψ (m+) z (x) for Thus φ m (x) = v(x) = 5 ψ z (m+) z V m+ \V m m=0 (x). 5 mφ m(x). Remark. The function v is invariant under Dihedral-3 symmetry. This is a direct corollary of the fact that each φ m (x) is invariant under D 3 symmetry. Due to the above remark, we may assume that D w associated to C w has a fixed shape as shown in Fig. 3. without loss of generality. We now show that although c B depends on the relative position of x in C w, it does not depend on the location of x or C w in SG. Lemma 4.. Let x,x be two distinct points in SG \ V 0. Let C w and C w be two k and k level neighboring cells of x and x respectively. Denote by B and B two mean value neighborhoods of x and x respectively. If B and B have the same shapes (the same relative locations associated to C w and C w respectively), then c B = 5 k k c B. In particular, if B and B have the same levels and same shapes, then c B = c B. Proof. D w can be decomposed into a union of a k level cell D w () and a (k ) level cell D w (2) as shown in Fig. 3.. Denote by q the junction point connecting D w () and D w (2). Similarly, D w can also be written as a union of a k cell D () w and a (k ) cell D (2) w with a junction point q connecting them. Let τ be the linear function mapping D w onto D w. Suppose D w () = F α (SG) and D w (2) = F β (SG) where α and β are the corresponding words of D w () and D w (2) respectively. Similarly, denote by α and β the corresponding words of D () w and D (2) w. Hence we can 4

15 write τ as τ(z) = F α Fα (z) if z D() w, and τ(z) = F β F β (z) if z D(2) w. In particular, τ(q) = q and τ(x) = x. Consider the function (v F α 5 k k v F α ) defined on SG. Noting that α = k and α = k, using the scaling property of (see details in [7] page 33), we have (v F α 5 k k v F α ) = r α 3 α v F α 5 k k r α 3 α v F α = 0, which shows that the difference between v F α and 5 k k v F α is a harmonic function. Hence the difference between v and 5 k k v τ on D () w is harmonic. A similar discussion will show that the difference between v and 5 k k v τ on D (2) w is also harmonic. Since the matching condition on normal derivatives of (v 5 k k v τ) at q holds obviously, we have proved that (v 5 k k v τ) = 0 on D w, i.e., the function (v 5 k k v τ) is harmonic on D w. By the definition c B = M B (v) v(x) and c B = M B (v) v(x ). Notice that for the second equality, by changing variables we can write c B = M B (v τ) v τ(x). Hence c B 5 k k c B = M B (v 5 k k v τ) (v 5 k k v τ)(x) = 0, since (v 5 k k v τ) is a harmonic function on D w. Proof of Theorem 4.. Estimate of c B from above. From Lemma 4., since c B depends only on the relative geometry of B and C w and the size of C w, but not on the location of C w, we may assume that D w is contained in a (k 2) level cell C in SG without loss of generality. By the definition of c B, we may write ( ) c B = M B (v) v(x) = lim v M dµ v M (x). M µ(b) B Substituting the exact formula of v M into it, we get for c B = 5 m=0 φ m = 5 m (M B(φ m ) φ m (x)), ψ z (m+). z V m+ \V m Noticethateachφ m isapiecewiseharmonicsplineoflevel m+. Sowhenm+ k 2, φ m is harmonic in the cell C, which yields that M B (φ m ) φ m (x) = 0. So the first k 2 5

16 terms in the infinite series of v will contribute 0 to c B. Hence c B = 5 5 (M B(φ m m ) φ m (x)). It is easy to see that this implies c B 5 m=k 2 m=k 2 Then by the maximum principle, we finally get c B 5 Estimate of c B from below. φ 5 m m (y) φ m (x) dµ(y). µ(b) B m=k 2 p 0 5 m = k. o C w p p 2 Fig. 4.. a /3 region of C w. x Without loss of generality, we assume that x is located in the /3 region of C w as shown in Fig. 4., i.e., x is contained in the triangle T p,p 2,o, where o is the geometric center of C w. Then by the proof of Theorem 3., B is a subset of the union of C w and two of its neighbors C and C 2. Hence we can write B = C w E E 2, where E i = B C i. Claim. Let B = F 0 (SG) Ẽ Ẽ2, where Ẽi is a triangle obtained by cutting F i (SG) symmetrically with a line below the top vertex F i q 0.(see Fig. 4.2.) If B and B have the same shapes, then c B = 5 k c B. This is a direct corollary of Lemma 4.. We only need to prove that c B for B defined in Claim has a positive lower bound. For simplicity of notation, in all that follows, we write B instead of B. In other words, 6

17 q 0 F 0 (SG) Ẽ Ẽ 2 q q 2 Fig a sketch of B. we only need to consider B whose associate cell C w is F 0 (SG). In this setting, p i = F 0 q i, C = F (SG) and C 2 = F 2 (SG). We write v = 5ṽ where ṽ is the non-negative function defined by For each M 0, denote by ṽ = ṽ M = m=0 M m=0 5 mφ m. 5 mφ m the partial sum of the first M + terms of ṽ. Then ṽ M converges to ṽ uniformly as M. We have the following three claims on ṽ. Claim 2. 0 ṽ on SG and ṽ takes constant along the maximal inner upsidedown triangle contained in SG. Proof. Considerthepartialsumfunctionṽ M. Obviously, ṽ M isa(m+)-levelpiecewise harmonic function on SG. For convenience, denote by the maximal inner upside-down triangle contained in SG. We divide the vertices V M+ into three parts, V M+, V V M+, where V M+ consists of those vertices lying along, V M+ at distance 2 (M+) from, and V M+ M+ and consists of those vertices consists of the remain vertices. Then by using the rule, an inductive argument shows that ṽ M on V M+, ṽ M 5 M on V and ṽ M on V 5 M M+. Since ṽ is the uniform limit of ṽ M and V M+ goes to the infinity, we then have 0 ṽ on SG and ṽ on. M+, goes to as M Claim 3. For each x contained in the triangle T p,p 2,o, ṽ(x) Proof. For τ = (0,,),(0,,2),(0,2,) and (0,2,2), by using the harmonic extension 7

18 algorithm, namely, the 2 rule, we get that ṽ(f τ q 0 ) = ṽ 2 (F τ q 0 ) = 5 mφ m(f τ q 0 ) = = 25 25, m=0 where 4 5, 3 5 and are the values of φ 0, φ and φ 2 at F τ q 0 respectively. Also, for those τ, by Claim 2, we have ṽ(f τ q ) = ṽ 2 (F τ q ) = ṽ(f τ q 2 ) = ṽ 2 (F τ q 2 ) =. Notice that for each point x in the triangle T p,p 2,o, x is contained in one of the four 3- level cells F 0 (SG), F 02 (SG), F 02 (SG) and F 022 (SG). Since ṽ 2 is harmonic in each such cell, by using the maximal principle, we get that ṽ 2 (x) Hence ṽ(x) 24 since each term in the infinite series of ṽ is non-negative. 25 Claim 4. M B (ṽ) 7. 8 Proof. First of all we prove that ṽ(y)dµ(y) = 5 8. F 0 (SG) We need to compute φ F 0 (SG) m(y)dµ(y) for each non-negative integer m. For each m 0, φ m (y)dµ(y) = 2 3 3m+ 3 = 2 m+2 9, F 0 (SG) by using (4.2) and the fact that φ m = z V m+ \V m ψ z (m+). Hence F 0 (SG)ṽ(y)dµ(y) = 2 9 m=0 5 m = 5 8. By our assumption, the mean value neighborhood B can be written as B = F 0 (SG) E E 2, where E i = B C i. Hence we have ( ) M B (ṽ) = ṽ(y)dµ(y) + ṽ(y)dµ(y)+ ṽ(y)dµ(y) µ(b) F 0 (SG) E E 2 ( ) ṽ(y)dµ(y) + dµ(y)+ dµ(y) µ(b) F 0 (SG) E E 2 = 5/8+µ(E )+µ(e 2 ) /3+µ(E )+µ(e 2 ), 8

19 wheretheinequality followsfromclaim2. Since0 µ(e )+µ(e 2 ) 2 3, 5 in x 0, 5/8+µ(E )+µ(e 2 ) /3+µ(E )+µ(e 2 ) 5/8+2/3 /3+2/3 = 7 8. Hence we always have M B (ṽ) 7 8. Now we turn to estimate c B. Obviously, c B = M B (v) v(x) = 5 (M B(ṽ) ṽ(x)). 8 +x 3 +x isincreasing By Claim 3 and Claim 4, we notice that M B (ṽ) ṽ(x) = Hence c B > 0. On the other hand, given a point x and C w = F w (SG) a k level neighborhood of x, for any u dom, we write u = h (k) +( u(x))v +R (k) on C w, where h (k) is a harmonic function defined by h (k) +( u(x))v Cw = u Cw. It is not hard to prove the following estimate: Lemma 4.2. Let u dom with g = u satisfying the following Hölder condition g(y) g(x) cγ k, (0 < γ < ) for all y C w. Then the remainder satisfies R (k) = O (( γ ) 5 )k on C w (hence also on B k (x)). Proof. It is easy to check that R (k) (y) = u(y) u(x) and R (k) (y) vanishes on the boundary of C w. Hence R (k) is given by the integral of u(y) u(x) on C w against a scaled Green s function. Noticing that the scaling factor is ( 5 )k and we then get the desired result. u(y) u(x) cγ k, 9

20 This looks like a Taylor expansion remainder estimate of u at x. See more details on this topic in [8]. Remark. If we require u dom 2, then the remainder R (k) satisfies ( R (k) = O ( 3 5 ) 5 )k on C w (hence also on B k (x)). The reason is that in this case u satisfies the Hölder condition that u(y) u(x) c( 3 5 )k for all y C w, because 2 u is assumed continuous, see [8], Theorem 8.4. Using the above lemma and Theorem 4., we then have the following main result of this section. Theorem 4.2. Let u dom with g = u satisfying the Hölder condition g(y) g(x) cγ k for some γ with 0 < γ <, for all x,y belonging to the same k level cell. Then lim k c Bk (x) ( MBk (x)(u) u(x) ) = u(x). Proof. Using Taylor expansion of u and noticing that M Bk (x)(h (k) ) h (k) (x) = 0, M Bk (x)(v) v(x) = c Bk (x), we have ( MBk (x)(u) u(x) ) u(x) = ( MBk (x)(r (k) ) R (k) (x) ) c Bk (x) = c Bk (x) c Bk (x) O (( γ ) 5 )k = O(γ k ). Hence letting k, we get the desired result. 5 p.c.f. fractals with Dihedral-3 symmetry The results for SG should extend to other p.c.f. fractals which possess symmetric properties of both the geometric structure and the harmonic structure. We assume that a regular harmonic structure is given on a p.c.f. self-similar fractal K. The reader is referred to [4, 7] for exact definitions and any unexplained notations. We assume now that V 0 = 3 and all structures possess full D 3 symmetry. This means there exists a group G of homeomorphisms of K isomorphic to D 3 that acts as permutations on V 0, and G preserves the harmonic structures and the self-similar measure. Assume that the fractal K is the invariant set of a finite iterated function system of contractive similarities. We denote these maps {F i } i=,,n with N 3. Let r i denote 20

21 the i-th resistance renormalization factor and µ i denote the i-th weight of the self-similar measure µ on K. In general, it is not necessary that all r i s and all µ i s be the same, but here we must have r 0 = r = = r N and µ 0 = µ = = µ N from the above Dihedral-3 symmetry assumption. We denote V 0 = {q 0,q,q 2 } the set of boundary points. Examples. (i) The Sierpinski gasket SG. In this case all r i = 3/5 and all µ i = /3. (ii) The hexagasket, or fractal Star of David, can be generated by 6 maps with simultaneously rotate and contract by a factor of /3 in the plane. Thus V 0 consists of 3 points of an equilateral triangle, and V consists of the vertices of the Star of David, as shown in Fig. 5.. Although the same geometric fractal can be constructed by using contractions which do not rotate, this gives rise to a different self-similar structure (in particular with V 0 = 6). Our choice of self-similar structure destroys the D 6 symmetry of the geometric fractal, but it has the advantage of easier computation. In this case, all r i = 3/7 and all µ i = /6. Note that in this example there exist points in V that are not junction points. (iii) The level 3 Sierpinski gasket SG 3, obtained by taking 6 contractions of ratio /3 as Γ 0 Γ Fig. 5.. The first 2 graphs, Γ 0,Γ in the approximation to the hexagasket. shown in Fig Here we have all r i = 7/5 and µ i = /6. Note that all seven vertices in V \ V 0 are junction points, but the one in the middle intersects three -cells. In a similar manner we could define SG n for any value of n 2. We prove that there are results analogous to Theorem 3., which yield the existence of mean value neighborhoods associated to K. Given a point x in K\V 0, consider any cell F w K = C w with boundary points p 0,p,p 2 containing the point x. Without losing of generality, we may require that the cell C w does not intersect V 0. For each i, denote by C i,,,c i,li the neighboring cells of C w of the same size, intersecting C w at p i, where l i is the number of such cells. It is possible 2

22 Fig The graph of the V vertices of the level 3 Sierpinski gasket. that l i = 0 for some i since p i may be a non-junction point. If this is true, the matching condition says that the normal derivative of any harmonic function h must be zero at this point, which yields that the value of h at this point is the mean value of the values of h at the other two boundary points of C w. In other words, the restriction of all global harmonic functions in C w is two dimensional. Denote by D w the union of C w and all its neighboring cells, i.e., D w = C w i,j Two cells C w and C w are said to have the same neighborhood type if they have the same relative geometry with respect to D w and D w respectively. It is obvious that there only exist finitely many distinct types. For example, for SG, all cells have exactly only one neighborhood type. For SG 3, the number of the finite types is 3. For SG n (n 4), the number of the finite types becomes 4. For the hexagasket gasket, the number of the finite types is 2. C i,j. Let h be a harmonic function on K. Given a set B containing C w, define M B (h) = hdµ µ(b) the mean value of h over B. We are interested in an identity B M B (h) = i a i h(p i ) (5.) for some coefficients (a 0,a,a 2 ) satisfying a i =. Notice that this is true for SG. In that setting, a harmonic function is uniquely determined by its values on the boundary of 22

23 any given cell C w because the harmonic extension matrix associated with C w is invertible. However, in the general case, the harmonic extension matrices may not be invertible. So we can not prove (5.) for every set B simply by linearity. However, it will suffice to show that the equality (5.) holds for certain specified sets B. Consider a set B which is a subset of D w, containing C w. Then B must be made up of four parts, i.e., B = C w E 0 E E 2 where E i = B C i with C i = l i j= C i,j. It is possible that C i may be empty since p i may be a nonjunction point. We can also subdivide each E i into l i small pieces, i.e., E i = j E i,j for E i,j = E i C i,j. For each i, we require that E i,,,e i,li be of the same size and shape. Moreover, in analogy with the SG case, we require that each E i,j to be a symmetric (under the reflection symmetry that fixes p i ) cutoff sub-triangle of C i,j, containing p i as one of its vertex points. This means that there is a straight line L i,j, symmetric under the reflection symmetry fixing p i, cutting C i,j into two parts, and E i,j is the one containing p i. For each E i,j, define the distance between p i and the line L i,j the size of E i,j. Of course, for each fixed i, E i,,,e i,li have the same sizes. We call the common value the size of E i. Suppose the size of every C i,j is ρ. (Of course, they are all equal.) Then for each i, the size of E i is c i ρ where the coefficient 0 c i. Hence we can write the set B = B(c 0,c,c 2 ). (If p i is a nonjunction point, then c i should always be 0.) For example, suppose that the boundary points of C w consist of junction points, then B(0,0,0) = C w and B(,,) = D w. Denote by B = {B(c 0,c,c 2 ) : 0 c i } the family of all such sets. Then we can show that the formula (5.) holds for each B B. Proposition 5.. Let B B, then for any harmonic function h, we have (5.) for some coefficients (a 0,a,a 2 ) independent of h. Moreover, i a i =. Proof. Each B B can be written as B = C w E 0 E E 2. Given a harmonic function h on K, for fixed i, we first consider the integral E i hdµ. Obviously, E i hdµ = j E i,j hdµ. For each j l i, denote by {z i,j,w i,j,p i } the boundary points of C i,j. Since each E i,j is contained in C i,j, µ(c w) E i,j hdµ can be expressed as a linear combination of h(p i ),h(z i,j ) and h(w i,j ) with non-negative coefficients independent of the harmonic function h. Since 23

24 the set E i,j is symmetric under the reflection symmetry fixing p i, the two coefficients with respect to h(z i,j ) and h(w i,j ) must be equal. In other words, we can write E i,j hdµ = (m i,j h(p i )+n i,j h(z i,j )+n i,j h(w i,j ))µ(c w ) for m i,j,n i,j 0. Moreover, since for each fixed i, E i,j are in the same relative position associated to C i,j for different j s, E i,j hdµ can be expressed as a linear combination of h(p i ),h(z i,j ),h(w i,j ) with the same coefficients for different j s. Hence we can write ( ) hdµ = m i h(p i )+n i (h(z i,j )+h(w i,j )) µ(c w ), E i for suitable coefficients m i,n i 0. The mean value property at the point p i says that j (h(z i,j )+h(w i,j )) = (2l i +2)h(p i ) (h(p i )+h(p i+ )). j Combining the above two equalities, we get E i hdµ = ((m i +2l i n i +2n i )h(p i ) n i h(p i ) n i h(p i+ ))µ(c w ). On the other hand, by the linearities and symmetries of both the harmonic structure and the self-similar measure, hdµ = µ(c w) C w 3 (h(p 0 )+h(p )+h(p 2 )). Since the ratio of µ(e i,j ) to µ(c w ) depends only on c i, we have proved that M B (h) can be viewed as a linear combination of the values of h on the boundary points of C w, i.e., M B (h) = i a i h(p i ), wherethecombinationcoefficientsareindependentofh. Moreover, wemusthave a i = by considering h. Remark. This means that M B (h) is a weighted average of the values h(p 0 ),h(p ) and h(p 2 ). Moreover, if one of the boundary points, for example p 2, is a nonjunction point, then by the fact that h(p 2 ) = 2 (h(p 0)+h(p )), we have M B (h) = a 0 h(p 0 )+a h(p )+ 2 a 2(h(p 0 )+h(p )) = ã 0 h(p 0 )+ã h(p ) 24

25 for ã 0 = a a 2 and ã = a + 2 a 2. We also have ã 0 +ã =. Hence in this case, we can also view M B (h) as a weighted average of the values of h(p 0 ) and h(p ). Remark 2. The proof of Proposition 5. shows that (a 0,a,a 2 ) depends only on the neighborhood type of C w and the relative position of B associated to C w, and does not depend on the particular choice of C w. In other words, if we consider a cell C w with a given neighborhood type, then for each set B B with the expression B = B(c 0,c,c 2 ), the coefficients (a 0,a,a 2 ) of B depend only on (c 0,c,c 2 ). The following is the main result in this section. Theorem 5.. Given a point x K\V 0, let C w be a cell containingx, not intersecting V 0, and let D w be the union of C w and its neighboring cells of the same size. Then there exists a mean value neighborhood B of x satisfying C w B D w. Moreover, for each point x K \ V 0, there exists a system of mean value neighborhoods B k (x) with k B k(x) = {x}. Proof. We need to classify the distinct neighborhood types into three cases according to the number of nonjunction points in the set of boundary points of C w. Case. All boundary points of C w are junction points. This case is similar to what we have described in the SG setting. Let W denote the triangle in R 3 with boundary points P 0 = (,0,0),P = (0,,0) and P 2 = (0,0,) and π W the plane containing W. Notice that from Proposition 5., (a 0,a,a 2 ) π W for each B. We use T to denote the map from B to π W. From Remark 2 of Proposition 5., the map T is uniquely determined by the neighborhood type of C w. Let B be a subfamily contained in B defined by B = {B(0,c,c 2 ) : 0 c i } {B(c 0,0,c 2 ) : 0 c i } {B(c 0,c,0) : 0 c i }, i.e., those elements B in B which have the decomposition form B = C w E E 2 or B = C w E 0 E 2, or B = C w E 0 E. Then we have Claim. The map T from B to π W fills out a region W which contains the triangle W. Moreover, T is one-to-one from B onto W. Proof. The proof is similar to the SG case. The only difference is the line segments OQ 0 and OP 2 described in the proof of Theorem 3. may become continuous curves ÔQ 0 and ÔP 2 in the general setting. Case 2. There is one nonjunction point (for example, p 2 ) among the boundary points of C w. In this case, there is no neighboring cell intersecting C w at the point p 2. Hence E 2 will always be empty. So B = {B(c 0,c,0) : 0 c i } for this case. 25

26 As shown in Remark of Proposition 5., for any harmonic function h on K, B B, M B (h) is a weighted average of h(p 0 ) and h(p ), i.e., M B (h) = a 0 h(p 0 )+a h(p ) with a 0,a independent of h, satisfying a 0 + a =. Let I denote the line segment in R 2 with endpoints P 0 = (,0),P = (0,) and ρ I the line containing I. Notice that from Remark of Proposition 5., (a 0,a ) ρ I for each B. We still use T to denote the map from B to ρ I. From Remark 2 of Proposition 5., the map T is uniquely determined by the neighborhood type of C w. We may write T(B(c 0,c,0)) = (a 0,a ) for each set B(c 0,c,0). We will show the image of the map T covers the line segment I. Similar to Case, let B be a subfamily contained in B defined by B = {B(c 0,0,0) : 0 c 0 } {B(0,c,0) : 0 c }, i.e., those elements B in B which have the decomposition form B = C w E 0 or B = C w E. Then we have Claim 2. The map T from B to ρ I fills out the line segment I. Moreover, T is a one-to-one map on B. Proof. The proof is similar to Case. Denote by O = (, ) the midpoint of I. We 2 2 only prove the map T from B to ρ I fills out half of the line segment I. Then we will get the desired result by symmetry. Let h be a harmonic function on K. We consider T ({(B(c,0,0)) : 0 c }). When c = 0, B(0,0,0) = C w and M Cw (h) = 3 (h(p 0)+h(p )+h(p 2 )). Combining this with the fact that we get h(p 2 ) = 2 (h(p 0)+h(p )), M Cw (h) = 2 (h(p 0)+h(p )). Hence T(B(0,0,0)) is the midpoint O of I. When c =, B(,0,0) = C w C 0, and an easy calculation gives that M Cw C 0 = h(p 0 ). Hence T (B(,0,0)) is the endpoint P 0. So if we vary c continuously between 0 and, we can fill out the line segment joining O and P 0, which is half of I. Case 3. There are two nonjunction points (for example, p and p 2 ) among the boundary points of C w. In this case, let h be any harmonic function on K. By the matching condition on both points p and p 2, h must be constant on the whole cell C w. Hence for every point x C w, we could view C w itself as the mean value neighborhood of x. 26

27 Hence the proof of Theorem 5. is completed by using a same argument as that of Theorem 3.2. We should mention here that the result can also be extended to some other p.c.f. fractals including the 3-dimensional Sierpinski gasket. However, it seems that some strong symmetric conditions of both the geometric and the harmonic structures should be required. Acknowledgements. This work was done while the first author was visiting the Department of Mathematics, Cornell University. He expresses his sincere gratitude to the department for their hospitality. We would also like to thank the anonymous referees for several important suggestions which led to the improvement of the manuscript. References [] M. T. Barlow, Diffusion on fractals. In Lectures Notes in Mathematics, vol Springer, Berlin, 998. [2] I. Blank and Z. Hao, The mean value theorem for divergence form elliptic operators, preprint. [3] J. Kigami, A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math., 6 (989), [4] J. Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335 (993), [5] J. Kigami, Analysis on Fractals, Cambrideg University Press, New York, 200. [6] R. S. Strichartz, Analysis on fractals, Not. Am. Math. Soc., 46 (999), [7] R. S. Strichartz, Differential equations on fractals: a tutorial. Princeton University Press, Princeton, NJ, [8] R. S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Func. Anal., 74 (2000), (Hua Qiu) DEPARTMENT OF MATHEMATICS, NANJING UNIVERISITY, NAN- JING, 20093, CHINA address: huatony@gmail.com (Robert S. Strichartz) DEPARTMENT OF MATHEMATICS, MALOTT HALL, COR- NELL UNIVERSITY, ITHACA, NY address: str@math.cornell.edu 27