Boundary Value Problems for harmonic functions on domains in Sierpinski gaskets

We study boundary value problems for harmonic functions on certain domains in the level-$l$ Sierpinski gaskets $\mathcal{SG}_l$($l\geq 2$) whose boundaries are Cantor sets. We give explicit analogues of the Poisson integral formula to recover harmonic functions from their boundary values. Three types of domains, the left half domain of $\mathcal{SG}_l$ and the upper and lower domains generated by horizontal cuts of $\mathcal{SG}_l$ are considered at present. We characterize harmonic functions of finite energy and obtain their energy estimates in terms of their boundary values. This paper settles several open problems raised in previous work.


Introduction.
A Dirichlet problem is the problem of finding a function which is harmonic in the interior of a given domain that takes continuous prescribed values on the boundary of the domain. The solvability of this problem depends on the geometry of the boundary. For a bounded domain D with sufficiently smooth boundary ∂D, the Dirichlet problem is always solvable, and the general solution is given by where G(x, y) is the Green's function for D, ∂ n G(x, s) is the normal derivative of G(x, y) along the boundary and the integration is performed on the boundary. The integral kernel ∂ n G(x, s) is called the Poisson kernel for D.
With a well developed theory of Laplacians on post-critically finite (p.c.f.) sets, originated by Kigami [4,5], it is natural to look for analogous results in the fractal context. Harmonic functions on p.c.f. self-similar sets are of finite dimension. Due to the self-similar construction of the fractal, the Dirichlet problem on the entire fractal always reduces to solving systems of linear equations and multiplying 1148 SHIPING CAO AND HUA QIU matrices. However, for the boundary value problem on bounded subsets of fractals, the knowledge remains far from clear. The study of such problem was initiated in [13] by Strichartz, where the upper domain generated by a horizontal cut of the Sierpinski gasket SG was considered. See Figure 1(a). Later it was continued in [11] and [1] to general case. In general, the boundary consists of a Cantor set together with the upper boundary vertex of SG. An explicit harmonic extension algorithm is given for solving the Dirichlet problem on such domains and the harmonic functions of finite energy are characterized in terms of their boundary values. The main tool is the Haar series expansion of the boundary values on the Cantor set with respect to the normalized Hausdorff measure by symmetry consideration. Since the only generator of the Haar basis is antisymmetric, one can localize the harmonic extension of this generator to any small scale along the boundary to get other basis harmonic functions. This observation plays a key role in their proof. However, as pointed out in [1], the results could not be extended to other fractals, even for the level-3 Sierpinski gasket SG 3 on the base of their approach. See Figure 1(b). The reason is that for SG 3 there exists a generator which is symmetric rather than antisymmetric whose harmonic extension could not be localized to small scales. On the other hand, the problem becomes much harder if we consider the domain lying below the horizontal cut instead. Except the very special case that the domains are made up of 2 m adjacent triangles of size 2 −m lying on the bottom line of SG(in this case, the boundary is a finite set), we have little knowledge. Recently, there is another natural choice of domain, namely the left half part of SG generated by a vertical cut along one of the symmetry lines of the gasket, becoming be interested. See Figure 2(a). It is the simplest example whose boundary is given as a level set of a harmonic function. In the SG setting, the boundary of the half domain consists of a countably infinite set of points, which makes it possible to study the Dirichlet problem by solving systems of countably infinite linear equations and multiplying infinite matrices. See [10] for a satisfactory discussion on this domain, including an explicit harmonic extension algorithm, the characterization of harmonic functions of finite energy, and an explicit Dirichlet to Neumann map for harmonic functions. However, if we consider the left half domain of level-l Sierpinski gasket SG l for l ≥ 3 instead, the approach in [10] is not applicable. Compared to the SG case, the essential difference is that the boundary of the left half part of SG l becomes a Cantor set together with the single left boundary vertex. See Figure  2(b).
In the following, we will use upper domain, lower domain and half domain to denote the above three types of domains respectively for simplicity. They are probably the simplest domains which should be handled in SG. In this paper, we will consider the analogues of them in level-l Sierpinski gasket. We will give explicit harmonic extension algorithms for all the three types of domains as well as the energy estimates for harmonic functions in terms of the boundary values (except the energy estimate for lower domains). This answers the questions raised in the above mentioned papers. We introduce some new techniques to overcome the difficulties we met before. In fact, for each interior point x in the domain Ω under consideration, we find certain measure µ x along the boundary ∂Ω analogous to ∂ n G(x, s)ds in Euclidean case, so that We observe that the measure µ x is closely related to the normal derivatives of some special harmonic functions along the boundary of Ω, which is crucial to our approach.
Nevertheless, these three types of domains are still the simplest domains in fractals with fractal boundary. We hope our results introduce different ideas and give insight into more general techniques for solving the Dirichlet problem and even other boundary value problems on more general fractal domains.
1.1. Preliminaries and the solvability of Dirichlet problems. Let l ≥ 2, recall that the level-l Sierpinski gasket SG l is the unique nonempty compact subset The set V 0 , which consists of the three vertices q 0 , q 1 , q 2 of the smallest triangle containing SG l , is called the boundary. For convenience, we renumber {F i } l 2 +l−2 2 i=0 so that F i (q i ) = q i for i = 0, 1, 2. SG 2 is the standard Sierpinski gasket (denoted by SG for simplicity). For SG 3 , in addition to F 0 , F 1 , F 2 , we denote by F 3 (z) = 1 3 z + 1 3 (q 1 + q 2 ), F 4 (z) = 1 3 z + 1 3 (q 0 + q 2 ) and F 5 (z) = 1 3 z + 1 3 (q 0 + q 1 ) the remaining three mappings, see Figure 3. These fractals have a well-developed theory of Laplacians, which allows us to perform analysis on them. In this paper, We will first describe the situation in more detail in the case of SG 3 for half domains and upper domains, and SG for lower domains, then extend the considerations to general SG l case.
We introduce some necessary notations. Readers can refer to textbooks [6] and [15] for precise definitions and known facts. For m ≥ 1, let W m = {0, 1, · · · , l 2 +l−2 2 } m be the collection of words with length m and W 0 = {∅}. Write W * = ∞ m=0 W m , and denote the length of w ∈ W * by |w|. For w = w 1 w 2 · · · w m ∈ W m , we define F w = F w1 • F w2 • · · · • F wm and call F w SG l a m-cell of SG l . Using the contraction and only if x = y and x, y ∈ V m belong to a same m-cell. The vertices V m , together with the edge relation ∼ m , form a graph Γ m that approximates SG l . See Figure 3 for an illustration for SG 3 .
For m ≥ 0, the natural discrete resistance form on Γ m is given by for u, v being functions defined on V m , where r = 3 5 for SG and r = 7 15 for SG 3 . For a real-valued function u defined on V * , it is easy to check that the graph energies E m (u) := E m (u, u) is an increasing sequence so that lim m→∞ E m (u) exists if we allow the value +∞. Define to be the energy of the function u and say that u ∈ domE if and only if E(u) < ∞. We regard domE ⊂ C(SG l ) since each function of finite energy admits a unique continuous extension to SG l . Moreover, domE is dense in C(SG l ). There is a natural resistance form on SG l defined as Let ν be the standard(with equal weights) self-similar probability measure on SG l . The standard Laplacian ∆ could be defined using a weak formulation. Suppose u ∈ domE and f is continuous, say u ∈ dom∆ with ∆u = f if A function h is harmonic if it minimizes the energy from each level to its next level. All the harmonic functions form a 3-dimensional space, and hence any given values on V 0 can uniquely determine a harmonic function on SG l . They are just the solutions of the equation ∆h = 0. In particular, there is an explicit extension algorithm, which determines h| V1 in terms of h| V0 and inductively h • F w | V1 in terms of h • F w | V0 for any w ∈ W * in a same manner. See Figure 4 for the exact formula for SG 3 . A harmonic function h satisfies the mean value property, that is, for each m ≥ 1, The normal derivative of a function u at a boundary point q i ∈ V 0 is defined by (cyclic notation q 3 = q 0 ) providing the limit exists. For harmonic functions, these derivatives can be evaluated without taking limit. We could localize the definition of normal derivative to any vertex in V * . Let x = F w q i be a boundary point of a m-cell F w SG l . Define ∂ w n u(x) the normal derivative of u at x with respect to the cell F w SG l to be r −m ∂ n (u • F w )(q i ). In this paper, we use the notations ∂ ← n , ∂ → n , ∂ ↑ n to represent the normal derivatives of different directions for simplicity. In particular, ∂ ↑ n u(q 0 ) = ∂ n u(q 0 ), ∂ ← n u(q 1 ) = ∂ n u(q 1 ), ∂ → n u(q 2 ) = ∂ n u(q 2 ). For u ∈ dom∆, the sum of all normal derivatives of u in different directions must vanish at each x ∈ V * \ V 0 . This is called the matching condition.
We have an analogue of Gauss-Green's formula in the fractal setting. Suppose u ∈ dom∆, then ∂ n u(q i ) exists for all q i ∈ V 0 and We also have a localized version of this formula, for any simple set A, which is defined as a finite union of cells.

SHIPING CAO AND HUA QIU
Let Ω be a half, upper or lower domain in SG l . Consider the Dirichlet problem ∆u = 0 in Ω, u| ∂Ω = f, f ∈ C(∂Ω).
Proof. First, by Lemma 8.2 in [8], if there exists a function v ∈ domE such that v| ∂Ω = f , then a solution of (1) exists, which minimizes the energy on Ω. For general case, notice that the set domE| ∂Ω := {f ∈ C(∂Ω)|∃v ∈ domE, v| ∂Ω = f } is dense in C(∂Ω), since domE is dense in C(SG l ) and ∂Ω is a closed subset of SG l . Let {f n } be a sequence of functions in domE| ∂Ω converging uniformly to f , and u n be their corresponding solutions of (1). Then {u n } also uniformly converge to a function u with u| ∂Ω = f by the maximum principle for harmonic functions. It is easy to get that u is harmonic in Ω. The uniqueness of the solution is an immediate consequence of the maximum principle.
1.2. The organization of the paper. Throughout this paper, although in different situations, we always use the same symbol Ω to denote the domain and X to denote the Cantor set boundary without causing any confusion.
In Section 2, we solve the Dirichlet problem for the half domain in SG 3 . An explicit harmonic extension algorithm is provided. Let f be the prescribed value on ∂Ω. We only need to find the explicit formula for the values of the extended harmonic function u on V 1 ∩ Ω, since if we do so, then the value of u in the 1cells contained inΩ is determined by the harmonic extension algorithm, and then the problem of finding values of u in the remaining region is essentially the same by dilation. An interesting phenomenon is that the solution could be expressed explicitly in terms of only a countable set of points which is dense in ∂Ω. We also characterize the energy estimate of solutions of finite energy in terms of their boundary values.
We consider the Dirichlet problem for the upper domain in SG 3 in Section 3. Basing on the same reason, we find the explicit formula for a finite number of crucial points, and then use dilation to continue. For the energy estimate, we still use the technique of Haar series expansion. But now we expand the boundary values with respect to a more natural probability measure rather than the normalized Hausdorff measure.
In Section 4, we deal with the lower domain in SG. Essentially the method is the same as before, but the situation is more complicated. We still obtain the explicit harmonic extension algorithm. However, it is unclear how to work out the energy estimate in term of the boundary values.
Finally, we show that our methods on the above three types of domains are still valid for general SG l and briefly state the outcomes. We present an intriguing correspondence between the normal derivatives and the boundary values of harmonic functions along boundaries on the half domain of SG, although we have no idea on how to extend it to general cases.
At the end of this section, we list some previous work on related topics. See [1,2,3,7,9,10,11,16] and the references therein. In particular, in [9], some extension problems on SG, which aim at finding a function minimizing Sobolev types of norms with certain prescribed data such as values and derivatives at a finite set, are studied. It is interesting to consider analogous problems on domains considered in this paper. We leave these as open problems for future research. The energy estimates considered in this paper characterize the restriction to the Cantor set boundary X of functions of finite energy on Ω. It is also interesting to characterize the traces on X of functions in some other Sobolev spaces, such as dom L 2 (∆ k ) defined as {u ∈ L 2 (SG l ) : ∆ j u ∈ L 2 (SG l ), ∀j ≤ k}. One may also consider how to extend a function of finite energy defined on Ω to a function of finite energy on the whole SG l , and analogous problems for other Sobolev spaces. Related problems are discussed in [10,1]. The above mentioned Sobolev spaces are easily characterized in terms of expansions in eigenfunctions of the Laplacian, see [14]. For half domains, as pointed out in [10] in the SG setting, essentially, there are no new eigenfunctions. A complete theory of the eigenspaces of the Laplacian on the upper domain in SG with X equal to the bottom line segment is given in [12].
2. Dirichlet problem on the half domain of SG 3 . In this section, we focus on solving the Dirichlet problem on the half domain of SG 3 . We will first give an extension algorithm for harmonic functions with continuous prescribed boundary values, then estimate the energies of them in terms of their boundary values. LetΩ denote the closure of Ω. It is easy to check that As shown in Section 1, to solve the Dirichlet problem (1), we only need to find the explicit algorithm for the values of the harmonic function u on V 1 ∩ Ω. For convenience, we use x ∅ , y ∅ , z ∅ to represent the three "crucial" vertices in V 1 ∩ Ω with We also denote p ∅ = F 3 q 0 .
For m ≥ 0, writeW Obviously,W m ⊂ W m andW * ⊂ W * . Denote Obviously, {x w , y w , z w } w∈W * ⊂ V * ∩ Ω and {p w } w∈W * = V * ∩ X \ {q 0 }. Now, we proceed to show how to determine the values of the harmonic function u on V 1 Ω in terms of the boundary function f . From the matching condition at To make the equations (2) enough to determine the unknowns, we need to represent the normal derivatives at x ∅ and z ∅ in terms of {u(x ∅ ), u(y ∅ ), u(z ∅ )} and the boundary data f . We will prove that there exists a signed measure on the boundary ∂Ω such that the normal derivative of u at q 1 could be evaluated as the integral of f with respect to this measure. This signed measure is determined by the normal derivative of the antisymmetric harmonic function h a along the boundary ∂Ω. See Figure 5 for the values of h a on V 1 ∩Ω. where Obviously,Ω equals the closure of m≥1 O m . See Figure 6 for O 1 and O 2 .
Applying the local Gauss-Green's formula on O m , we get It is easy to calculate the normal derivatives of h a at p w , x w , z w , So we have the estimate that 7 µ w f (p w ) by taking the limit.
For the rest part of the theorem, we introduce a sequence of harmonic functions {u n } n≥0 which are piecewise constant on X, defined as u n | Fτ X = f (p τ ), ∀τ ∈W n . The existence of such functions is ensured by Proposition 1. Moreover, it is easy to check that u n uniformly converges to u by the maximum principle for harmonic functions. Applying Gauss-Green's formula, we have For fixed n and τ ∈W n , we have that ∂ → n u n (x τ w ), ∂ ↑ n u n (z τ w ) take the same sign, and w∈W * ,|w|=m ∂ → n u n (x τ w ) + ∂ ↑ n u n (z τ w ) are uniformly bounded, as u n • 1156 SHIPING CAO AND HUA QIU F τ = c 1 + c 2 h a for some constants c 1 , c 2 . In addition, h a (x w ) and h a (z w ) converge uniformly to 0 as |w| → ∞. Thus, letting m → ∞, we get . Combining this equality with the first part of the proof, we then have Taking n → ∞, we get (3).

Remark 1.
One can regard the signed measure 3δ q1 − w∈W * 6 7 µ w δ pw as the normal derivative of h a on ∂Ω. In this opinion, Theorem 2.1 is just a result of the extended "Guass-Green's formula" acting on h a and u.
In the following, we denote µ the probability measure w∈W * 2 7 µ w δ pw on X. Thus we could write Now, we have enough information to calculate the values u(x ∅ ), u(y ∅ ), u(z ∅ ).

Theorem 2.2 (Extension algorithm).
There exists a unique solution of the Dirichlet problem (1). In addition, we have Proof. The existence and uniqueness of a solution of (1) has been shown in Propo- (2), and solving the system of linear equations, we get (5), (6) and (7).

2.2.
Energy estimate. In Theorem 2.2, we have shown that the harmonic function u could be explicitly determined by its values at only countably infinite vertices It is natural to hope that the energy estimate of u also depends on the same values as well.
Then we have the energy estimate that where C 1 , C 2 are two positive constants independent of f .
where the equality holds when u( Similarly, for any w ∈W * , we also have where c 1 , c 2 are suitable positive constants. Thus, we have Conversely, we assume without loss of generality that Q(f ) < ∞, otherwise there is nothing to prove. Consider a piecewise harmonic function v defined onΩ, which is harmonic in Figure 7 for the value of this function. It is easy to calculate the energy of v, On the other hand, E Ω (v) ≥ E Ω (u), as harmonic functions minimize the energy.
3. Dirichlet problem on upper domains of SG 3 . In this section, we deal with the Dirichlet problem on upper domains of SG 3 . Prescribe that the boundary vertices q 0 , q 1 , q 2 ∈ R 2 take the following coordinates, Then for each 0 < λ ≤ 1, define the upper domain together with the boundary See Figure 8 for an illustration. DenoteΩ λ = Ω λ ∂Ω λ the closure of Ω λ . In the following context, we write X instead of X λ when there is no confusion.
with an integer sequence 0 < m 1 < m 2 < · · · , and ι k = 1 or 2. Denote Inductively, write Set λ 0 = λ and m 0 = 0. It is easy to check the following relationship between Ω λn and Ω λn+1 , We omit the superscript λ when there is no confusion caused. See Figure 9 for an illustration.
For convenience, let It is easy to see that for λ not a triadic rational, X is homeomorphic to the space Σ λ = ∞ k=1 S ι k equipped with the product topology. Otherwise, X is a union of finite segments.
Here we give an example to help readers to get familiar with the notations.

Extension algorithm.
We still use f to denote the boundary data on ∂Ω λ and u the harmonic solution of the Dirichlet problem (1). We only need to find an explicit algorithm for the values of u at p i 's since if we do so, the problem of finding values of u in the remaining region is essentially the same after dilation. From the matching condition at each vertex p i , we have the following system of equations.
Case 1(ι 1 = 1): Case 2(ι 1 = 2): Due to the same consideration in Section 2, we need to express the normal derivatives ∂ ↑ n u(p i )'s in the above equations in terms of u(p i )'s and the boundary data f . Thus we turn to find the explicit representation of ∂ ↑ n u(q 0 ) in terms of the boundary values.

Remark 2.
In fact, we have

SHIPING CAO AND HUA QIU
for any fixed constant −∞ < c < 1. We only need small changes in the proof, and readers may refer to Theorem 4.7 for a similar discussion.
α is an increasing function of λ on ( 1 where α(1) is the root of x = Define a probability measure µ λ on X by We can easily verify that and (13). Applying the above discussion iteratively, we get Theorem 3.4. Let u be a solution of the Dirichlet problem (1). Then In addition, if u ∈ domE Ω λ , we have Proof. For n ≥ 1, let O λ,n be the simple set with boundary vertices {q 0 } {F λ w q 0 } w∈W λ n . Noticing that {O λ,n } n≥1 is expanding with n≥1 O λ,n = Ω λ , using a similar proof as that of Theorem 2.1, the theorem follows.
Proof. We only need to show that h 0 ∈ domE Ω λ . This is obvious since for any n ≥ 1, by Gauss-Green's formula, we always have By (14) and the fact that η(λ) = 2( 15 7 ) m1 1−α(λ) , the energy of h 0 is estimated by This result will be helpful in the energy estimate for general functions. Combining Theorem 3.4 with equations (10) and (11), we could calculate the values of the solution u at the "crucial" points p i 's for i ∈ W λ 1 , which are sufficient to recover u by induction.
Proof. See Proposition 1 for the existence and uniqueness of the solution. Substi- (10) or (11), after solving linear equations, we get the result.

3.2.
Haar series expansion and energy estimate. Now we consider the energy estimate for the harmonic solutions in terms of their boundary values. For a harmonic function u with boundary value u| X = f in L 2 (X, µ λ ), we will give an estimation of E Ω λ (u) in terms of the Fourier coefficients of f with respect to a Haar basis.
Before performing the energy estimate, we list two basic lemmas. Figure 11. Boundary values of h (1) , h (2) .
Proof. We only need to prove that E Ω λ (h (j) ) is bounded above and below by multiples of ( 15 In particular, we will restrict our consideration to 1 3 . So it is sufficient to assume m 1 = 1 and prove that E Ω λ (h (j) ) is bounded above and below by two positive constants.
First, we consider 2 3 < λ ≤ 1. In this case, ι 1 = 2. Let c 1 , c 2 , c 3 be some selected constants independent of λ. For each 2 3 < λ ≤ 1, write v λ the harmonic function on Ω λ which assumes 0 at q 0 and takes constant c i along X i for i = 1, 2, 3. We claim that To prove the claim, we construct another functionṽ onΩ λ b such thatṽ| by using Corollary 3.5 and the fact that η(Rλ) is decreasing on 2 3 < λ ≤ 1. In addition, we have E Ω λ b (ṽ) ≥ E Ω λ b (v λ b ), since harmonic functions minimize the energy. Combining the two inequalities, we obtain the claim. Thus On the other hand, to find an upper bound of {E Ω λ (v λ )}, consider the functionv ∈ C(A) which is harmonic in A and assumes 0 at q 0 , c i at p i for i = 1, 2, 3. It is easy to find that E Ω λ (v λ ) ≤ E A (v) by extendinḡ v to Ω λ withv| F λ i Ω Rλ = c i . The energy estimate of h (1) is a special case in the above discussion. To estimate the energy of h (2) , observe that the boundary values (h (2) | X1 , h (2) | X2 , h (2) | X3 ) vary within a compact set, denoted by C, since we always have 1 4 < µ λ 1 = µ λ 2 < 1 3 and 1166 SHIPING CAO AND HUA QIU Thus we have proved that E Ω λ (h (j) ) is bounded above and below by two positive constants, when 2 3 < λ ≤ 1. A similar discussion is valid for The next lemma shows that all the basis functions {h  Proof. We discuss in different cases. If by Theorem 3.4 and the fact that Remark 3. The proof of Lemma 3.9 also implies that E Ω λ (h (j) w , h 0 ) = 0 for each w ∈ W λ * , j ≤ ι |w|+1 . Now for the harmonic solution u of the Dirichlet problem (1), we have the following estimate in energy in terms of its boundary data f . Theorem 3.10. Let u be the harmonic function in Ω λ with boundary values u(q 0 ) = a and u| X λ = f , where Then E Ω λ (u) is bounded above and below by multiples of In particular, u has finite energy if and only if (20) is finite.
Proof. We have Since we have shown in Lemma 3.9 that the functions h 0 {h (j) w } are orthogonal in energy, Then (3.14) follows from Lemma 3.8 and Corollary 3.5.
4. Dirichlet problems on lower domains of SG. In this section, we consider the Dirichlet problem on lower domains of SG. Similar to the last section, we assume SG is contained in R 2 with boundary vertices q 0 = ( 1 where d(λ) is the smallest integer such that λ is a multiple of 2 −d(λ) and X λ = {(x, y) ∈ SG|y = 1 − λ}. We still abbreviate X − λ to X throughout this section for convenience. Figure 12 for two typical domains. For 0 ≤ λ < 1, we write λ in its binary expansion, e k (λ)2 −k , e k (λ) = 0, 1 for k ≥ 1.
We forbid infinitely consecutive 1's to make the expansion unique. Denote It is easy to check the relationship between Ω − λ and Ω − Sλ as following, if e 1 (λ) = 1.
It is natural to introduce the following sets of words Analogous to the previous two sections, we only need to find an explicit algorithm for u| Ω − λ V1 in terms of the boundary data f , since then the problem of finding values of u elsewhere in Ω − λ is essentially the same after dilation. From the matching condition at each vertex in V 1 Ω − λ , there exist the equations, and We need to express the involved normal derivatives in terms of u(F 0 q 1 ), u(F 0 q 2 ), u(F 1 q 2 ) and the boundary data f . For this purpose, we introduce two important coefficients where h λ 1 is the harmonic function with values h λ 1 (q 1 ) = 1, h λ 1 (q 2 ) = 0 and h λ 1 | X = 0. Symmetrically, we have , where h λ 2 is the harmonic function with values h λ 2 (q 1 ) = 0, h λ 2 (q 2 ) = 1 and h λ 2 | X = 0. We omit the superscript λ of h λ i when there is no confusion caused. We will discuss on how to calculate these coefficients in the second part of this section. Here we only mention the following property.
Proof. By using the maximum principle for harmonic functions, it is easy to see that η 1 is an increasing function of λ, and η 2 is a decreasing function of λ. Thus we have More precisely, we have η 1 (λ) + η 2 (λ) ≥ 3. In fact, we just need to consider the antisymmetric harmonic function h 1 − h 2 whose normal derivative at q 1 is η 1 (λ) + η 2 (λ). Using the maximum principle on the left half part of Ω − λ , one can check that η 1 + η 2 is an increasing function of λ.
By iteratively using the above matrices, we have and ∀w ∈W λ * , Proof. By using the local Gauss-Green's formula on A λ,m we get (28) holds for each m ≥ 0. Notice that for w ∈W λ * , By using Lemma 4.1 and the maximum principle for harmonic functions, we have both η 1 (S |w| λ) − η 2 (S |w| λ) ≥ 0 and h(F w q 1 ) + h(F w q 2 ) ≥ 0, in case of h(q 1 ) ≥ 0 and h(q 2 ) ≥ 0, which gives (29).
According to Lemma 4.2 and 4.3, we introduce two measures on X.
Definition 4.4. Define µ λ 1 to be the unique probability measure on X satisfying Symmetrically, define µ λ 2 by In addition, if E Ω − λ (u) < ∞, then Proof. Using the local Gauss-Green's formula on A λ,m , we have Analogous to the proof of Theorem 2.1, it is easy to see that On the other hand, by Lemma 4.1, we have Combining the above facts, we get Similarly, The rest part of the proof is similar to that of Theorem 2.1.

1172
SHIPING CAO AND HUA QIU 4.2. The calculation of η. In this section, we focus on the calculation of η 1 (λ), η 2 (λ). In particular, we will prove the following theorem. and In addition, for any fixed positive numbers c 1 > c 2 , we have Proof. By using maximum principle for harmonic functions, (a) follows easily. So we only need to prove (b). Looking at the functions h 1 + h 2 and h 1 − h 2 pictured in Figure 14, by computing their normal derivatives at q 1 , we get if e 1 (λ) = 0, if e 1 (λ) = 1.
The above equations lead to For the rest proof of Theorem 4.7(b), we need some claims and notations.
Proof of Claim 1. Let m = |w|. Noticing that h 1 (F w q i ) ≥ 0 by the maximum principle for harmonic functions, by Lemma 4.1, we have On the other hand, by Lemma 4.3, . Combining the above two inequalities, we get the desired result.
Notation. (a) For 0 ≤ λ < 1 and fixed positive numbers c 1 > c 2 , define a sequence of resistance forms E In Figure 15, we give an example of V λ m together with some conductances. We abbreviate h λ,(c1,c2) 1,m to h 1,m when there is no confusion caused. By Theorem 4.5, one can easily check that Without loss of generality, we assume that 1 − λ > 2 −j for some integer j.
Proof of Claim 2. It is easy to see the claim holds by inductively using (38) when (c 1 , For general c 1 , c 2 , the claim still holds in a completely similar way.
Proof of Claim 3. The first inequality is obtained analogously to the proof of Claim 1. The second inequality follows from Claim 2 and the fact h 1,m ∞ ≤ 1.
Noticing that j is arbitrary, we complete the proof of Theorem 4.7(b).
(b) For w ∈W λ m , the matrix M λ w is given by with j being the integer such that 0 ≤ j ≤ 2 m − 1 satisfying The proofs of Proposition 2 and 3 directly follow from elementary computations. We omit them. Proposition 3 is motivated by Theorem 5.4 of [S1], which solves the special case that λ = 1 − 2 −m .
5. Extension to SG l . In this section, we will briefly discuss how to extend previous results to level-l Sierpinski gasket SG l . 5.1. Half domains. We still use Ω to denote the half domain and X its Cantor set boundary. As shown in Section 2, to solve the Dirichlet problem on the half domain of SG l , it suffices to obtain the extension algorithm for u| V1∩Ω in terms of the boundary data f . We summarize it into following two steps.
Step 1. Find a formula for ∂ ← n u(q 1 ), in the form of Here the measure µ, as shown in Theorem 2.1 for SG 3 , is a multiple of the normal derivative of the antisymmetric harmonic function h a on X. To work out µ, we introduce some notations.
}, and prescribe that p j,∅ locates above p j+1,∅ for each See the following example for an illustration.
Example 2. In Figure 16 (a), we label the contraction mappings of SG 4 . So we haveW m = {0, 6} m . The vertices {p j,∅ } 2 j=1 are plotted in Figure 16 (b). With the above notations, introduce the pure atomic probability measure µ on X satisfying For i ∈W 1 , denote µ i = r −1 h a (F i q 1 ) and write µ w = µ w1 µ w2 · · · µ w |w| , where r is the renormalization factor of the energy on SG l . It is easy to check that ∂ → n h a (p j,w ) = µ w ∂ → n h a (p j,∅ ), so that Step 2. Solve linear equations determined by the matching conditions of normal derivatives on V 1 Ω. That is (41), the remaining problem is solving the linear equations. However, even for the values of h a , the calculation becomes much more complicated, so we could not provide a general solution of (41).
Conversely, suppose there exists such a f ∈ C(∂Ω), it must satisfies which arise from the definition of normal derivatives at p k and F k+1 0 q 1 , using the fact that −∂ ↑ n u(F k+1 0 q 1 ) = ∂ ← n u(q 1 ) + k i=0 ∂ → n u(p i ). The solution of the above equations is where f (p k ) and u(F k 0 q 1 ) converge to f (q 0 ) = −η −1 − 4 3 ∞ i=0 η i . Thus we find a unique function f which satisfies the prescribed conditions. However, it is not clear where {∂ → n u(p j,w )} w∈W * ,1≤j≤[ l 2 ] live in for general cases, even for SG 3 .
For the energy estimate part, for SG l , we have for some positive constants C 1 , C 2 , with The method is essentially the same as that for SG 3 case.

5.2.
Upper or lower domains. The Dirichlet problem on the upper and lower domains in general SG l are much more complicated.
For the upper domains, we use the infinite expansion λ = ∞ k=1 ι k · l −m k to characterize Ω λ , where {m k } k≥1 is an increasing sequence of positive integers and ι k take values from {1, 2, · · · , l − 1}. The number ι k decides the relationship between Ω R k−1 λ and Ω R k λ , where Rλ = ∞ k=2 ι k · l −(m k −m1) , and there are l − 1 choices in the SG l setting.
As for the lower domains, we refer to a different expansion with e k (λ) taking values from {0, 1, · · · , l − 1}. We forbid infinitely consecutive (l−1)'s to make the expansion unique. Similarly, different e k (λ) determines different type of relationships between Ω − S k−1 λ and Ω − S k λ , where Sλ = ∞ k=1 e k+1 (λ)l −k . The approaches in Section 3 and Section 4 to solve the Dirichlet problem on upper or lower domains still work, although the calculations involved turn to be rather complicated. We list the main steps.
Step 2. Calculate the normal derivatives of h 0 (or h 1 , h 2 ) along the Cantor set X, using the crucial coefficients η(λ) or η 1 (λ), η 2 (λ) . The normal derivatives of h 0 (or h 1 , h 2 ) hold the key to the representation of ∂ ↑ n u(q 0 ) or ∂ ← n u(q 1 ),∂ → n u(q 2 ) in terms of the boundary data f .
Step 3. Solve the linear equations determined by the matching conditions of normal derivatives on the crucial points.
Lastly, the Haar series expansion used in the energy estimate still works in general SG l cases. The key observation is that we can still use the analogue of Theorem 3.4 to show that we can decompose the harmonic solution associated with a square integrable boundary value data (with respect to a suitable choice of measure), into a summation of countably infinite, pairwise orthogonal in energy, locally supported harmonic functions with suitable piecewise constant boundary values.