This content has been downloaded from IOPscience. Please scroll down to see the full text.

Download details:

IP Address: 147.188.128.74

This content was downloaded on 11/11/2014 at 23:08

Please note that terms and conditions apply.

The p-Laplacian on the Sierpinski gasket

View the table of contents for this issue, or go to the journal homepage for more

2004 Nonlinearity 17 595

(http://iopscience.iop.org/0951-7715/17/2/014)

Home Search Collections Journals About Contact us My IOPscience

INSTITUTE OF PHYSICS PUBLISHING NONLINEARITY

Nonlinearity 17 (2004) 595–616 PII: S0951-7715(04)61987-6

The p-Laplacian on the Sierpinski gasket

Robert S Strichartz1,3 and Carto Wong2,4

1 Mathematics Department, Malott Hall, Cornell University, Ithaca, NY 14853, USA2 Mathematics Department, Chinese University of Hong Kong, Shatin, NT, Hong Kong, People’sRepublic of China

E-mail: str@math.cornell.edu and carto@mathdb.org

Received 4 April 2003Published 9 January 2004Online at stacks.iop.org/Non/17/595 (DOI: 10.1088/0951-7715/17/2/014)

Recommended by M Viana

AbstractWe define a nonlinear p-Laplacian operator, �p, on the Sierpinski gasket, for1 < p < ∞, generalizing the linear Laplacian (p = 2) of Kigami. In thenonlinear case, the definition only gives a multivalued operator, although undermild conjectures it becomes single valued. The main result is that we can alwayssolve �pu = f with prescribed boundary values by solving an equivalentminimization problem. We use this to obtain numerical approximations to thesolution. We also study properties of p-harmonic functions.

Mathematics Subject Classification: 35J60, 28A80

1. Introduction

For functions u defined on a Riemannian manifold M , the p-Laplacian, �pu, may be definedpointwise

�pu(x) = div(|∇u|p−2∇u)(x) (1.1)

or by the weak formulation

−∫

M

(�pu)v dµ = Ep(u, v), (1.2)

where the p-energy, Ep(u, v), is given by

Ep(u, v) =∫

M

(∇u · ∇v)|∇u|p−2 dµ (see [AA]). (1.3)

3 Research supported by the National Science Foundation, Grant DMS-0140194.4 Research supported by the Mathematics Department of the Chinese University of Hong Kong, the Bankee KwanAward for Mathematics Projects and the Chung Chi Travelling Award for Mathematics.

0951-7715/04/020595+22$30.00 © 2004 IOP Publishing Ltd and LMS Publishing Ltd Printed in the UK 595

596 R S Strichartz and C Wong

Here the gradient, inner product, divergence and measure are determined by the Riemannianmetric. Note that Ep(u, v) is linear in v but not symmetric and is derivable from thediagonal form

Ep(u) =∫

M

|∇u|p dµ (1.4)

via

Ep(u, v) = 1

p

d

dtEp(u + tv)

∣∣∣∣t=0

. (1.5)

Our goal is to construct an analogous operator on the Sierpinski gasket (SG), for1 < p < ∞. When p = 2, this is the linear Laplacian of Kigami [Ki1, Ki2], which agreeswith the infinitesimal generator of a ‘Brownian motion’ process on SG [Ba] (see [S1] for a briefoverview of the subject). In [HPS], the first step in this direction was taken in the constructionof the analogue of Ep(u) for a class of fractals including SG. Here, we will restrict our attentionto SG since we are also interested in numerical approximations. We will construct Ep(u, v)

via the analogue (1.5) and use it to define �pu via the weak formulation (1.2). Because of theconvexity of Ep(u), we are able to ensure the existence of only one-sided derivatives in (1.5),so that Ep(u, v) becomes an interval-valued rather than a real-valued function, and so �pu

may not be a unique function. We say u ∈ dom(�p) and �pu = f if

−∫

SGf v dµ ∈ Ep(u, v) (1.6)

for all v in dom(Ep) vanishing on the boundary. Here µ is the standard symmetric self-similarprobability measure on SG.

To describe the results in more detail, we briefly review results and terminology from[Ki1, Ki2, HPS]. The reader should consult these references for more details. Recall that SGis the self-similar compact subset of the plane that is uniquely determined by the self-similaridentity

K =3⋃

i=1

FiK (1.7)

where {Fi} is the iterated function system (ifs) of contractive homotheties with contractionratio 1

2 and fixed points {qi}, the vertices of an equilateral triangle. We define the initial graph,�0, to be the complete graph on the vertices V0 = {q1, q2, q3}. We will regard V0 as theboundary of SG and all the approximating graphs, �m. We define �m inductively as the graphon Vm, where

Vm =3⋃

i=1

FiVm−1 (1.8)

and the edge relation x ∼m y if and only if x = Fix′, y = Fiy

′ and x ′ ∼m−1 y ′ for some i

and x ′, y ′ ∈ Vm−1. Let V∗ = ⋃Vm. Then V∗ is a dense set in SG, and continuous functions

on SG my be identified with uniformly continuous functions on V∗. Since the domain of E0 iscontained in the space of continuous functions, this means that all important notions may bedefined by the values of functions on V∗.

Let w = (w1, . . . , wm) denote a word of length |w| = m, where each wi = 1, 2 or 3,and write Fw = Fw1 ◦ · · · ◦ Fwm

. Then SG decomposes into a union of m-cells, Fw(SG), withboundary {Fwqi}. Our basic p-energy function, Ap, is a function of three real variables, outof which we construct a crude energy

E(m)p (u) =

∑|w|=m

Ap(u(Fwq1), u(Fwq2), u(Fwq3)) (1.9)

The p-Laplacian on the Sierpinski gasket 597

on level m and then a renormalized p-energy

E (m)p (u) = (rp)−mE(m)

p (u) (1.10)

for a certain factor, rp, satisfying 0 < rp < 1. When p = 2, we simply take

A2(a1, a2, a3) = |a1 − a2|2 + |a2 − a3|2 + |a3 − a1|2 (1.11)

and rp = 35 , but in general we can only say that Ap(a1, a2, a3) is bounded above and below by

a multiple of

|a1 − a2|p + |a2 − a3|p + |a3 − a1|p (1.12)

and shares some basic properties with (1.12), namely homogeneity of degree p, invarianceunder addition of constants, invariance under permutation, convexity (hence continuity) andthe Markov property. This means that Ap is determined by an even function of one variable

g(x) = Ap(−1, x, 1) (1.13)

via

Ap(a1, a2, a3) =∣∣∣∣a3 − a1

2

∣∣∣∣p

g

(2a2 − a1 − a3

|a3 − a1|)

if a1 �= a3. (1.14)

In [HPS], it is shown that there exists a choice of g(x) and rp such that given any functionu on Vm,

E (m)p (u) = min{E (m+1)

p (u) : u|Vm= u}. (1.15)

This allows us to assert that E (m)p (u) is a monotone increasing function of m, so

Ep(u) = limm→∞ E (m)

p (u) (1.16)

exists as an extended real number. We may then define dom(Ep) as the space of functionssatisfying Ep(u) < ∞. In [HPS], it is shown that dom(Ep) is a linear space and is dense inthe continuous functions. A function that achieves the minimum in (1.15) for all m � 0 iscalled p-harmonic, and one that achieves the minimum for all m � m0 is called piecewisep-harmonic of level m0. Such functions exist for any values of u on Vm0 , but uniqueness isnot proven, although it is likely to hold.

The function g is not known to be unique, although the renormalization factor rp is. Again,it seems likely that g is unique, but in any case we make one choice of g for the remainderof this paper. It would make our work a lot easier if we knew that g were differentiable, butthere does not seem to be any reason why this should be so, and it is not possible to test thisexperimentally. Note that g is determined by its values on [0, 1], but the differentiability atx = 1 requires g′(1−) = p/4.

The definition of �pu = f via (1.6) not only leaves open the possibility that there maybe more than one f but also does not assure that dom(�p) is nontrivial. It is easy to show thatp-harmonic functions belong to dom(�p) and satisfy �pu = 0. Following a similar line ofreasoning, we show that the differential equation �pu = f is equivalent to minimization ofthe functional

1

pEp(u) +

∫f u dµ (1.17)

subject to fixed boundary conditions. We are able to prove that minimizers exist, and it islikely that they are unique (this follows if Ep(u) is strictly convex). Thus dom(�p) is as largeas it should be, and the inverse operator to �p is in better shape than �p itself. We wouldvery much like to have a pointwise formula for �pu(x), analogous to Kigami’s formula in the

598 R S Strichartz and C Wong

linear case. We suggest a possible formula, but at this point it is only speculation whether ornot it is valid.

This paper is organized as follows. In section 2, we present the basic definitions of Ep(u, v)

and �pu. In section 3, we prove the equivalence of the variational problem of minimizing(1.17) and the equation �pu = f and then prove that the variational problem has solutions.In section 4 we discuss the possible pointwise formula. In section 5, we define p-normalderivatives and discuss the local behaviour of p-harmonic functions in a neighbourhood of aboundary point, but all results in this section are obtained under conjectural assumptions.

In sections 3 and 4 we supplement the theoretical development with the results of numericalapproximations. Detailed information about the programs used and more computational resultsmay be found at the website www.mathlab.cornell.edu/∼carto.

For other nonlinear differential equations on SG, see [F, S2].

2. Definitions

Our first goal is to pass from the energy Ep(u) defined on the linear space dom(Ep) to anenergy form Ep(u, v) defined for u, v ∈ dom(Ep). We want Ep(u, v) to be linear in v andsatisfy Ep(u, u) = Ep(u), and the ideal definition would simply be

Ep(u, v) = 1

p

d

dtEp(u + tv)

∣∣∣∣t=0

. (2.1)

We do not know that the derivative exists in the usual sense, but since Ep(u) is convex, we caninterpret (2.1) as an interval-valued equation. So

Ep(u, v) = [E−p (u, v), E+

p(u, v)] (2.2)

is a nonempty compact interval, and the endpoints are the one-sided derivatives. It followsfrom the theory of convex functions [Ro] that Ep(u, v) is sublinear in v, namely

Ep(u, av) = aEp(u, v) (2.3)

and

Ep(u, v1 + v2) ⊆ Ep(u, v1) + Ep(u, v2). (2.4)

Note that Ep(u + tu) = (1 + t)pEp(u) by homogeneity, and so

Ep(u, u) = [Ep(u), Ep(u)] (2.5)

reduces to a point.We can similarly define E (m)

p (u, v) as an interval by replacing Ep(u) by the approximateenergy, E (m)

p (u).

Lemma 2.1. If u is a piecewise p-harmonic function, then for sufficiently large m we have,for all m′ > m,

E (m)p (u, v) ⊆ E (m′)

p (u, v) ⊆ Ep(u, v). (2.6)

Proof. Without loss of generality we may assume that u is p-harmonic. Then E (m)p (u) =

E (m′)p (u) = Ep(u). For simplicity of notation, write E (m)

p (u, v) = [am, bm], E (m′)p (u, v) =

[am′ , bm′ ] and Ep(u, v) = [a, b]. Then

E (m)p (u + tv) =

{Ep(u) + tbm + o(t) if t > 0Ep(u) + tam + o(t) if t < 0

The p-Laplacian on the Sierpinski gasket 599

and similarly for E (m′)p (u + tv) and Ep(u + tv). Note also that E (m)

p (u + tv) � E (m′)p (u + tv) �

Ep(u + tv). It follows that

0 � E (m′)p (u + tv) − E (m)

p (u + tv) ={t (bm′ − bm) + o(t) for t > 0t (am′ − am) + o(t) for t < 0.

This implies bm′ � bm and am′ � am. A similar argument shows b � bm′ and am′ � a.QED

We can conclude from (2.6) that Ep(u, v) contains limm→∞ E (m)p (u, v) but not that they are

equal. Nevertheless, the lemma gives us a method to compute at least a portion of the intervalEp(u, v) for a dense class of functions u. Note that the computation of E (m)

p (u, v) reduces toa computation in one-dimensional calculus. If g satisfies the differentiability hypothesis, theneach E (m)

p (u, v) is a single point, and so we only have to compute it for one value of m (thelevel on which u is piecewise p-harmonic).

Definition 2.2. Let dom0(Ep) denote the subspace of dom(Ep) of functions vanishing on theboundary. For u ∈ dom(Ep) and f continuous, we say u ∈ dom(�p) and �pu = f , provided

−∫

f v dµ ∈ Ep(u, v) for all v ∈ dom0(Ep). (2.7)

More generally, we define domLq (�p) by the same condition for f only assumed to be inLq(dµ).

Note that we are not requiring �pu to be a unique function. We could conceivably have�pu = f1 and �pu = f2 with f1 �= f2 if Ep(u, v) does not always reduce to a point. Thebest we can say is that the set of all f such that �pu = f is a closed convex set. It is not clearthat dom(�p) is nontrivial, but later we will show that for every continuous f there existsu ∈ dom(�p) with �pu = f and specified boundary values. In other words, the Dirichletproblem is solvable. It seems unlikely that dom(�p) is a linear space for p �= 2, however.

Theorem 2.3. h is a p-harmonic function if and only if h ∈ dom(�p) and �ph = 0.

Proof. h is p-harmonic if and only if it minimizes p-energy among functions with the sameboundary values; in other words, Ep(u + tv) attains its minimum value at t = 0 for allv ∈ dom0(Ep). Since it is a convex function, Ep(u + tv) has a minimum at t = 0 if and onlyif 0 ∈ Ep(u, v). QED

Note that we do not rule out the possibility that �ph = f also holds for some nonzerofunction f .

3. Variational characterizations

In this section, we characterize solutions of the equation

�pu = f (3.1)

as minimizers of the functional1

pEp(u) +

∫f u dµ (3.2)

subject to fixed boundary conditions. This allows us to conclude that dom(�p) is nontrivial.It also gives us a numerical tool for approximating solutions to (3.1). We also give analogousvariational characterizations for solutions of other equations involving the p-Laplacian.

600 R S Strichartz and C Wong

Theorem 3.1. Let u ∈ dom(Ep) and f be continuous, and write bj = u(qj ). Thenu ∈ dom(�p) with �pu = f if and only if u minimizes (3.2) among all functions v ∈ dom(Ep)

satisfying v(qj ) = bj .

Proof. All such v may be written v = u+w for w ∈ dom0(Ep), and the whole line v = u+ tw

also satisfies the boundary conditions. Then

1

pEp(u + tw) +

∫f (u + tw) dµ = 1

pEp(u + tw) + t

∫f w dµ +

∫f u dµ

is a convex function of t and so has a minimum at t = 0 if and only if 0 ∈ Ep(u, w)+∫

f w dµ.The result follows from definition 2.2. QED

If we only assume f ∈ Lq , then the same result holds with u ∈ domLq (�p).In order to prove that minimizers exist, we need a kind of Holder estimate for functions

in dom(Ep).

Lemma 3.2. There exists a constant Mp such that for all u ∈ dom(Ep) we have

|u(x) − u(y)| � MpEp(u)1/p(r1/pp )m (3.3)

whenever x and y belong to the same or adjacent cells of order m.

Proof. Because rp < 1, it suffices to establish (3.3) whenever x and y are adjacent vertices inVm since we may always connect x and y via a pair of infinite chains of adjacent vertices inVk for k → ∞ and then sum the estimates. But∑

x∼my

|u(x) − u(y)|p � crmp E (m)

p (u). (3.4)

Since each summand is bounded by the sum on the left-hand side of (3.4), and E (m)p (u) � Ep(u),

we obtain (3.3). QED

Theorem 3.3. For any f ∈ Lq , q � 1, and any {bj }, there exists a function u ∈ dom(Ep)

minimizing (3.2) among all functions with boundary values {bj }. In particular, u ∈ domLq (�p)

(or dom(�p) if f is continuous) and (3.1) holds. Moreover, if Ep is strictly convex, then u isunique.

Proof. Let a be the infimum of all values of (3.2) as u varies of dom(Ep) with the givenboundary values, and let uk be a sequence of such functions such that

limk→∞

(1

pEp(uk) +

∫f uk dµ

)= a.

We claim that {uk} is uniformly bounded and uniformly equicontinuous. Since theboundary values are fixed, the uniform boundedness follows immediately from the uniformequicontinuity. Also, the uniform equicontinuity follows from lemma 3.2 if we can show that{Ep(uk)} is uniformly bounded. But this follows by a standard trick. We use∣∣∣∣

∫f u dµ

∣∣∣∣ � cε + ε‖u‖p∞

for any ε. Since

‖u‖p∞ � c1 + c2Ep(u)

by lemma 3.2 we obtain

1

pEp(u) �

(1

pEp(u) +

∫f u dµ

)+ cε + ε(c1 + c2Ep(u)).

The p-Laplacian on the Sierpinski gasket 601

By taking ε small enough that εc2 = 1/2p, we obtain uniform boundedness of {Ep(uk)} fromthe uniform boundedness of{

1

pEp(uk) +

∫f uk dµ

}.

By the Arzela–Ascoli theorem, there is a subsequence (which we may assume withoutloss of generality is the original sequence) converging uniformly to a continuous function u.We claim this is the desired minimizer. To see this, it suffices to show that

Ep(u) � lim supk→∞

Ep(uk)

since then1

pEp(u) +

∫f u dµ � lim sup

k→∞

(1

pEp(uk) +

∫f uk dµ

)= a.

Indeed, Ep(u) = supm E (m)p (u) and E (m)

p (u) = limk→∞ E (m)p (uk) because E (m)

p only depends onthe values on the finite set Vm. So

Ep(u) = supm

limk→∞

E (m)p (uk) � sup

m

lim supk→∞

Ep(uk) = lim supk→∞

Ep(uk)

since E (m)p (uk) � Ep(uk).

By theorem 3.1, u ∈ domLq (�p) (or u ∈ dom(�p) if f is continuous), and (3.1) holds.If Ep is strictly convex, then so is (3.2), and so the minimizer, and hence the solution of (3.1),is unique. QED

By similar reasoning, we can show that the equation

�pu = au + f, (3.5)

where a(x) is a non-negative function, is equivalent to the minimization of

1

pEp(u) +

∫f u dµ +

1

2

∫au2 dµ (3.6)

and minimizers exist. In this case, we have the added bonus that if a is strictly positive thenthe minimizer is unique since (3.6) is always strictly convex.

Next, we present some numerical results giving approximate solutions to (3.1) byminimizing (3.2). Of course, the idea is to replace (3.2) by

1

pE (m)

p (u) + Im(f u) (3.7)

for moderately large values of m, where Im denotes an approximation to the integral dependingonly on the values of the function on Vm. We will only work with functions f that are closeto being constant on m-cells, and we assume this is also true for the minimizer u. In that case,the difference between (3.2) and (3.7) is negligible. This means that the approximation shouldgive an accurate value for the minimum, but it is more difficult to estimate the accuracy of theminimizer.

In figures 1–3, we show the approximations to the solutions to�pu = 1 with zero boundaryconditions for the values p = 10

9 , 167 , 6. The fact that the solutions are constant along the inner

triangle may be explained by local symmetries just as in the linear case [ASST].The minimization of (3.7) is by no means easy. Here we will roughly explain

how the minimization was done. All related programs can be found at the websitewww.mathlab.cornell.edu/∼carto. The first step of minimization is that we fix the boundaryvalues, search for a function u1 defined on level m = 1 that minimizes (3.7) subject to the

602 R S Strichartz and C Wong

Figure 1. The solution to �pu = 1 with zero boundary values for p = 109 .

Figure 2. The solution to �pu = 1 with zero boundary values for p = 167 .

The p-Laplacian on the Sierpinski gasket 603

Figure 3. The solution to �pu = 1 with zero boundary values for p = 6.

boundary conditions. This can be regarded as minimizing a (convex) function Q1 of threevariables, where

Qm = 1

p(rp)−m

∑|w|=m

Ap(u(Fwq1), u(Fwq2), u(Fwq3)) +2

3m+1

∑x∈Vm\V0

f (x) u(x). (3.8)

Note that the last term on the right-hand side of (3.8) differs from Im(f u) by a constantwhenever the values of u on V0 are fixed. In that case, Qm depends only on the values of u

on Vm\V0. This is the reason why Q1 is a function of three variables. Once the minimizingfunction, u1, is found, we go ahead by finding a function u2 defined on level 2 that minimizesQ2 subject to the boundary conditions. In this step, the function u1 is used to obtain an initialguess by doing a p-harmonic extension in each of the three smaller triangles. In general, oncethe minimizing function um−1 defined on level m − 1 is found, we then view Qm as a functionof 1

2 (3m+1 − 3) variables and find a function um defined on level m that minimizes Qm. Thiscan be done by a conjugate gradient algorithm as follows:

(i) Pick a point x0 ∈ Rk , where k = 1

2 (3m+1 − 3), as our initial guess by doing a p-harmonicextension to um−1 locally (i.e. in each of the 3m−1 small triangles).

(ii) Set g0 = ∇Qm(x0) and h0 = −g0.(iii) Suppose xk , gk , hk (k � 0) are defined. Choose a point xk+1 on the half line xk + thk

(t � 0) that minimzes Qm.(iv) Set gk+1 = ∇Qm(xk+1) and hk+1 = (〈gk+1, gk+1 − gk〉/〈gk, gk〉)hk − gk+1.(v) Repeat (iii) and (iv) until for some k we have ∇Qm(xk) = 0 or ‖xk − xk−1‖ is sufficiently

small (depending on the accuracy required and the performance of the computer used).When we stop, the current xk would be our minimizer.

604 R S Strichartz and C Wong

Table 1.

m Maximum difference Lp difference

2 0.049 968 44 0.046 701 173 0.025 248 61 0.022 052 544 0.014 676 08 0.011 408 935 0.006 662 25 0.004 936 806 0.002 879 66 0.001 749 487 0.000 615 29 0.000 394 13

The same process continues until level 7 is reached. We stop at level 7 because (i) thenumber of variables involved in doing minimization increases exponentially as the levelincreases; (ii) a function defined at level 7 is just enough to produce a figure with satisfactoryresolution. Of course, one may ask why we do not do minimization directly at level 7, i.e.finding a function defined on level 7 that minimizes (3.7) subject to the boundary conditions?The answer is that if we do so we need to minimize a function with 3279 variables at one time!This terrible task is made easier in our approach since we have a better initial guess, and theresults on each level are used as the starting point for the next level. However, it is still timeconsuming.

As we mentioned before, it is difficult to estimate the accuracy of the minimizer. Wecomputed the difference between um and um−1. By the word ‘difference’, we mean

maxx∈Vm−1

|um(x) − um−1(x)|or

1

3m

∑x∈V0

|um(x) − um−1(x)|p +2

3m

∑x∈Vm−1\V0

|um(x) − um−1(x)|p

1/p

.

We call it ‘maximum difference’ or ‘Lp difference’, respectively. Table 1 shows these errorsfor p = 6, f = 1 and vanishing boundary conditions.

4. A pointwise formula?

Assuming that �pu = f uniquely determines f , it would be nice to have a pointwise formulafor the value of �pu(x) when x is a junction point, analogous to the formula

�u(x) = limm→∞

3

25m

∑y∼mx

(u(y) − u(x)) (4.1)

for the p = 2 case. In this section, we will briefly discuss some ideas of what we might expect.We assume that g satisfies the differentiability hypothesis, so that E (m)

p (u, v) is well definedas a number. We would replace (2.7) by the approximate equality

−∫

f v dµ ≈ E (m)p (u, v). (4.2)

Taking v to be the piecewise p-harmonic function on level m with v(x) = 1 and v(y) = 0 forall y ∈ Vm with y �= x, we would approximate the left-hand side of (4.2) by

−f (x)

∫v dµ = −2 · 3−m

(∫hp dµ

)f (x), (4.3)

The p-Laplacian on the Sierpinski gasket 605

where hp is the p-harmonic function with boundary values (1, 0, 0). The right-hand side of(4.2) may be computed exactly. To do this, we introduce the notation ym, zm, y ′

m, z′m for the

four neighbours of x in Vm, with (x, ym, zm) and (x, y ′m, z′

m) denoting the boundary points ofthe two level m cells containing x. Then

E (m)p (u, v) = 1

pr−mp

( ∣∣∣∣u(ym) − u(zm)

2

∣∣∣∣p−1

g′(

2u(x) − u(ym) − u(zm)

|u(ym) − u(zm)|)

+

∣∣∣∣u(y ′m) − u(z′

m)

2

∣∣∣∣p−1

g′(

2u(x) − u(y ′m) − u(z′

m)

|u(y ′m) − u(z′

m)|) )

. (4.4)

(Note that if u(ym) = u(zm) or u(y ′m) = u(z′

m), a slightly different formula must be used.)Substituting (4.3) and (4.4) into (4.2), we arrive at the approximation

�pu(x) ≈ − 1

2p∫

hp dµ

(3

rp

)m( ∣∣∣∣u(ym) − u(zm)

2

∣∣∣∣p−1

g′(

2u(x) − u(ym) − u(zm)

|u(ym) − u(zm)|)

+

∣∣∣∣u(y ′m) − u(z′

m)

2

∣∣∣∣p−1

g′(

2u(x) − u(y ′m) − u(z′

m)

|u(y ′m) − u(z′

m)|) )

. (4.5)

It is not clear how to elevate this speculation into a precise statement, such as the limitas m → ∞ of the right-hand side of (4.5) giving �pu(x) exactly. When p = 2, this isexactly (4.1). For p �= 2, we have tested (4.5) in numerical simulations.

To do this, we used the methods described in [HPS] to compute rp and the function g′.In figure 4 we display the graphs of g′ for several p values. We also need to approximatethe integral

∫hp dµ. We used ‘Simpson’s method’ from [SU] to obtain the values in table 2.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.5

1

1.5

2

2.5

x

y

Figure 4. The graphs of g′(x) on [0, 1] for several values of p. From the bottom up at x = 1,p = 10

9 , 43 , 8

5 , 169 , 20

11 , 2, 167 , 8

3 , 3, 4, 6, 10.

606 R S Strichartz and C Wong

Table 2.

p∫

hp dµ

1.1111 0.003 901 681.3333 0.162 139 301.6000 0.274 027 261.7778 0.308 095 871.8182 0.313 747 292.0000 0.333 333 332.2857 0.352 497 032.6667 0.367 312 993.0000 0.375 350 874.0000 0.387 897 056.0000 0.396 225 77

10.0000 0.399 365 55

Figure 5. The graph of hp for p = 6.

Note that the data suggest that the integral is increasing in p with

limp→∞

∫hp dµ = 0.4 and lim

p→1+

∫hp dµ = 0.

The behaviour for large p is somewhat mysterious, but the behaviour for p close to 1 maybe ‘explained’ as follows. The graph of hp is almost identically zero except in a smallneighbourhood of the boundary point where hp = 1, where the function rapidly climbs from0 to 1. Since rp is very close to 1 for such p, there is very little cost in p-energy associatedwith this rapid climb, and of course there is virtually no contribution to the p-energy fromthe region where hp is virtually constant. (In fact, all p-harmonic functions appear almostconstant for p close to 1.) It appears from our computations that limp→∞ hp exists, althoughwe cannot describe the limit function. Computations involving large p or p close to 1 becomevery inaccurate. Figure 5 shows the graph of hp for p = 6.

The p-Laplacian on the Sierpinski gasket 607

Figure 6. The pointwise formula (4.5) for u solving �pu = 1 with boundary values (0, 0, 1) forp = 20

11 (see figure 9 for the graph of u).

In figure 6 we display the result of one test of the pointwise formula (4.5), with p = 2011 . We

first computed the solution to �pu = 1 with boundary values (0, 0, 1) by minimizing (3.2) (seefigure 9). We then computed (4.5) for this function u. The answer should be the constant 1.The approximation is quite rough, with an average (L2) error 0.0586.

The next set of figures illustrates another test. In figure 7, we show the solution tothe equation �pu = u with boundary values (0, 0, 1), obtained by minimizing (3.6). Wethen computed approximations to u by two methods. In the first we computed (4.5) directlyfrom u. In the second we applied (4.5) to the solution of �pv = u (obtained by minimizing(1/p)Ep(v) +

∫u dµ with the same boundary conditions). We display only the graph of

the second approximation in figure 8, since visually the two approximations are virtuallyindistinguishable. In both cases, the approximation is rough, and yet it reveals at least somequalitative features of the solution. In the first case, the mean error is 0.0251 and the maximumerror is 0.7990. In the second case, these errors are 0.0263 and 0.8398, respectively.

In a third test, we run the operations in the reverse order. In figure 9 we display the solutionto �pu = 1 with boundary values (0, 0, 1), obtained by minimizing (3.2) with f = 1. Thiswas the function to which we applied (4.5) to obtain the function f shown in figure 6. Infigure 10, we solved �pu = f with the same boundary values by minimizing (3.2). Now theerror is much smaller, and qualitatively the two graphs are almost identical.

5. Local behaviour of p-harmonic functions

In the first part of this section, we will assume that g satisfies the differentiability conditionand also that Ep(u, v) reduces to a point for all u, v ∈ dom(Ep). We will then be able toidentify Ep(u, v) with a real-valued function, and we will by abuse of notation continue towrite Ep(u, v) for this real-valued function. Of course we have no way to verify whether ornot these assumptions are valid.

608 R S Strichartz and C Wong

–1

–0.50.5

–1

–0.5

–0.5

Figure 7. An approximation to the solution of �pu = u with boundary values (0, 0, 1) and p = 2011 ,

obtained by minimizing (3.6).

–1

–0.5

–0.4

–0.2

–0.5

–1

0.5

1

Figure 8. The pointwise formula (4.5) applied to the function v, the approximate solution to�pv = u, where u is the function in figure 7. A visually indistinguishable approximation isobtained by applying (4.5) directly to u in figure 7.

The p-Laplacian on the Sierpinski gasket 609

–0.4

–0.2

Figure 9. An approximate solution to �pu = 1 with boundary values (0, 0, 1) and p = 2011 ,

obtained by minimizing (3.2) with f = 1.

–0.4

–0.2

–0.50.5

1

Figure 10. Another approximate to the function in figure 9, obtained by minimizing (3.2) for f

equal to the function shown in figure 6.

610 R S Strichartz and C Wong

It follows from these assumptions that Ep(u, v) is linear in v. Moreover, we can strengthenlemma 2.1 to conclude that E (m)

p (u, v) = Ep(u, v) for all sufficiently large m if u is piecewisep-harmonic (all m if u is p-harmonic).

Now we want to define a p-normal derivative at each boundary point. Let ψ(m)j denote

the piecewise p-harmonic function of level m satisfying ψ(m)j (x) = 0 for all x ∈ Vm except

ψ(m)j (qj ) = 1. We define

∂pu(qj ) = limm→∞

1

pEp(u, ψ

(m)j ), (5.1)

provided the limit exists. Note that ∂pu(qj ) depends only on the values of u in an arbitrarilysmall neighbourhood of qj , is unchanged by adding a constant to u, and is homogeneous ofdegree p − 1, and so we are justified in calling it a p-normal derivative.

It is not difficult to see that if u ∈ dom(�p) (or more generally domLq (�p)), then ∂pu(qj )

exists. Indeed, since ψ(m)j − ψ

(m′)j ∈ dom0(Ep), we see that

1

pEp(u, ψ

(m)j ) +

∫(�pu)ψ

(m)j dµ (5.2)

is independent of m. Since

limm→∞

∫(�pu)ψ

(m)j dµ = 0,

it follows that the limit in (5.1) equals (5.2) for any m. Similar reasoning yields the Gauss–Green formula:

1

pEp(u, v) = −

∫(�pu)v dµ +

3∑j=1

v(qj ) ∂pu(qj ). (5.3)

We also have the self-similar identity

∂p(u ◦ Fnj )(qj ) = rn

p ∂pu(qj ). (5.4)

It is straightforward to localize the definition to define p-normal derivatives at theboundary points of any cell (the derivative depends on the cell as well as the boundarypoint). If u ∈ dom(�p), then we have the matching conditions at any junction pointthat the sum of the two normal derivatives (with respect to the two cells meeting at thejunction point) must vanish. Moreover, this leads to a gluing principle: if SG is written asa union of cells Fw(SG), disjoint except for boundary points, and if �pu = f on eachcell (meaning �p(u ◦ Fw) = 3−|w|r |w|

p f ◦ Fw) for u and f continuous functions, then�pu = f on SG if and only if the matching conditions hold at all junction points where thecells meet.

We would like to replace the right-hand side of (5.1) by

limm→∞

1

pE (m)

p (u, ψ(m)j ) (5.5)

because (in analogy with (4.4))

1

pE (m)

p (u, ψ(m)j ) = 1

pr−mp

∣∣∣∣u(ym) − u(zm)

2

∣∣∣∣p−1

g′(

2u(qj ) − u(ym) − u(zm)

|u(ym) − u(zm)|)

, (5.6)

where ym, zm are the neighbouring vertices to qj in Vm. It is not clear if (5.5) equals ∂pu(qj )

in general, but it certainly holds if u is p-harmonic. In that case, (5.5) is independent of m,

The p-Laplacian on the Sierpinski gasket 611

so we may take m = 0 in (5.6) to obtain

∂pu(qj ) = 1

p

∣∣∣∣u(qj+1) − u(qj−1)

2

∣∣∣∣ g′(

2u(qj ) − u(qj+1) − u(qj−1)

|u(qj+1) − u(qj−1)|)

(5.7)

(provided u(qj+1) �= u(qj−1)). In particular, since g′(t) = 0 if and only if t = 0, we have

∂pu(qj ) = 0 if and only if u(qj ) = u(qj+1) + u(qj−1)

2. (5.8)

We now examine more closely the local behaviour of a harmonic function in aneighbourhood of a boundary point, say q1. It is easy to localize the discussion to thebehaviour near a junction point from either side. We will assume that Ep(u) is strictly convex,so that harmonic functions are uniquely determined by their boundary values. We recallsome notation from [HPS]. We let B1(x), B2(x), B3(x) denote the values at the junctionpoints F2q3, F3q1, F1q2 of the p-harmonic function with boundary values −1, x, 1 at q1, q2,q3. Numerical evidence suggests that Bj(x) are increasing functions of x on [−1, 1], andB3(x) � B2(x) � B1(x) on [−1, 1], with equality holding only at the endpoints. Figure 11shows the graphs of these functions for p = 20

11 .For the present discussion, it is a bit more convenient to choose a different normalization,

so we define ht to be the p-harmonic function with boundary values (0, t, 1). Note that by anaffine transformation u → au + b and possibly a reflection, any p-harmonic function may betransformed into ht for some t in [−1, 1]. In particular,

ht ◦ F1 = M(t)hf (t) (5.9)

1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 10. 8

0. 6

0. 4

0. 2

0

0.2

0.4

0.6

0.8

x

y

−−−

−

−

−

−−

Figure 11. The graphs of B1(x), B2(x) and B3(x) (top to bottom) for p = 2011 .

612 R S Strichartz and C Wong

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

(a) (b)(c)

Figure 12. The graph of f (t) for different choices of p: (a) 43 , (b) 16

9 and (c) 83 . The fact that

f (0) = 12 for all p may be explained by symmetry considerations.

and it is easy to compute explicitly

M(t) =

1 + B2(2t − 1)

20 � t � 1,

1 + t

2+

(1 − t

2

)B1

(−

(1 + t

1 − t

))−1 � t � 0

(5.10)

and

f (t) =

1 + B3(2t − 1)

1 + B2(2t − 1)0 � t � 1,

1 + t − (1 − t)B1((1 + t)/(1 − t))

1 + t + (1 − t)B1(−(1 + t)/(1 − t))−1 � t � 0.

(5.11)

It is not difficult to see that the denominator in (5.11) is positive by checking it directly att = −1 and showing this is the extreme case. Since we are going to iterate (5.9) whenwe zoom in on the p-harmonic function in a neighbourhood of q1, we are interested in thedynamical system t → f (t). In figure 12, we show numerical computations of this functionfor several choices of p. It is clear that f has two fixed points, t = ±1. We will see that theseare the only fixed points, with t = −1 being repelling and t = +1 attracting all values of t in(−1, 1]. Note also that M(t) is increasing from M(−1) = B1(0) to M1(1) = (1 + B2(1))/2.Figure 13 shows M(t) for p = 20

11 .We may therefore formulate a dichotomy between p-harmonic functions with vanishing

and nonvanishing p-normal derivatives at q1. If ∂pu(q1) = 0, then u(x)−u(q1) is a multiple ofh−1, and this remains true for all zooms u(Fm

1 x)−u(q1). Thus u(x)−u(q1) is skew-symmetric

The p-Laplacian on the Sierpinski gasket 613

–1 0 0.2 0.4 0.6 0.8 10.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

–0.2–0.4–0.6–0.8

Figure 13. The graph of M(t) for p = 2011 .

with respect to the reflection preserving q1 and permuting q2 and q3, and

u(Fm1 x) − u(q1) = B1(0)m(u(x) − u(q1)). (5.12)

This gives the relatively rapid decay rate

u(x) − u(q1) = O(B1(0)m) on Fm1 (SG). (5.13)

On the other hand, if ∂pu(q1) �= 0, then u(x) − u(q1) is a multiple of ht for t �= −1, hence forlarge m we will have u(Fm

1 x) − u(q1) very close to a multiple of h1(x). This means u(x) isasymptotically symmetric about q1, with the relatively slow decay rate

u(x) − u(q1) = O

((1 + B2(1)

2

)m)on Fm

1 (SG). (5.14)

In the linear case p = 2, this dichotomy was observed in [DSV] for harmonicfunctions and extended to functions in dom(�) in [BST]. We expect that similar resultshold for functions in dom(�p) and that results in [OSY] for harmonic functions extend top-harmonic functions.

We end this section by giving a proof that f has only two fixed points. In the proof, wemake use of some reasonable assumptions. These assumptions, which will be written downin the statement of theorem 5.3, have not been proven, although they are strongly indicated bynumerical data.

Lemma 5.1. If t �= 1 is a fixed point of f , then

M(t) = (rp)1/(p−1).

Proof. Let u be the p-harmonic function with boundary values u(q1) = 0, u(q2) = 1,u(q3) = t .

614 R S Strichartz and C Wong

Note that

∂pu(q1) = limm→∞

1

pE (m)

p (u, h(m)1 )

= limm→∞

1

p2

1

(rp)m

∂

∂a1Ap(0, M(t)m, M(t)mt)

= limm→∞

sgn M(t)m

p2

( |M(t)|p−1

rp

)m∂

∂a1Ap(0, 1, t)

= ∂pu(q1) · limm→∞(sgn M(t)m)

( |M(t)|p−1

rp

)m

.

Since t �= −1, we have ∂pu(q1) �= 0, hence M(t) = (rp)1/(p−1). QED

Corollary 5.2.

B2(1) = 2(rp)1/(p−1) − 1 = 2

1 + (2g(0))1/(p−1) − 1and 21−p � rp < 1.

Theorem 5.3. f has no fixed point in [0, 1) provided that B2(x) < B2(1) for all x in [−1, 1),and it has no fixed point in (−1, 0) provided that B1(x) � B1(1) for all x in (0, 1).

Proof. If t ∈ [0, 1), then

M(t) = 1 + B2(2t − 1)

2<

1 + B2(1)

2= (rp)1/(p−1).

It follows from lemma 5.1 that there is no fixed point in [0, 1).Now, assume t ∈ (−1, 0) is a fixed point of f ; we are going to derive a contradiction.For convenience, we denote λ = (rp)1/(p−1). Note that

M(t) = 1 + t

2+

1 − t

2B1

(− 1 + t

1 − t

)and if we write x = −(1 + t)/(1 − t) (note that −1 < x < 0), then the above equality becomes

M(t) = 1 − (1 + x)/(1 − x)

2+

1 + (1 + x)/(1 − x)

2B1(x) = B1(x) − x

1 − x.

By lemma 5.1, we have M(t) = λ, i.e.

B1(x) − x

1 − x= λ

or

B1(x) = λ + (1 − λ)x. (5.15)

On the other hand,

f (t) = 1 + t − (1 − t)B1((1 + t)/(1 − t))

1 + t + (1 − t)B1(−(1 + t)/(1 − t))

= 1 − (1 + x)/(1 − x) − (1 + (1 + x)/(1 − x))B1(−x)

1 − (1 + x)/(1 − x) + (1 + (1 + x)/(1 − x))B1(x)

= x + B1(−x)

x − B1(x).

The p-Laplacian on the Sierpinski gasket 615

Since t is a fixed point, we have

x + B1(−x)

x − B1(x)= − 1 + x

1 − x

and so

B1(−x) = 1 + x

1 − x(x − B1(x)) − x.

Substitute (5.15) into the above equality. After simplification, we obtain

B1(−x) = λ − (1 − λ)x. (5.16)

However,

B1(−x) � B1(1) = λ < λ − (1 − λ)x,

which contradicts (5.16). QED

Corollary 5.4. f (t) > t for all t ∈ (−1, 1), and 1 is an attracting fixed point that attractsevery t �= −1.

Proof. Since f has no fixed point on (−1, 1), f is continuous and f (0) =(1 + B3(−1))/(1 + B2(−1)) > 0, it follows that f (t) > t for all t ∈ (−1, 1).

For any t0 ∈ (−1, 1), define tn = f (tn−1) for n � 1. Since {tn} is an increasing sequencebounded by 1, we can let

l := limn→∞ tn � 1.

Since f is continuous,

f (l) − l = limn→∞(f (tn−1) − tn) = 0.

So l is a fixed point of f . Clearly, l �= −1 and the remaining possibility is l = 1. We concludethat 1 is an attracting fixed point that attracts every t �= −1. QED

Acknowledgment

We are grateful to Alan Demlow for assistance with the numerical work.

References

[ASST] Adams B, Smith S A, Strichartz R and Teplyaev A 2003 The spectrum of the Laplacian on the pentagasketTrends in Mathematics: Fractals in Graz 2001 (Basle: Birkhauser) pp 1–24

[AA] Garzia Azorero J P and Peral Alonso I 1987 Existence and nonuniqueness for the p-Laplacian eigenvaluesCommun. Partial Diff. Eqns 12 1389–430

[Ba] Barlow M 1998 Diffusion on Fractals (Lecture Notes in Mahematics vol 1690) (Berlin: Springer)[BST] Ben-Bassat O, Strichartz R and Teplyaev A 1999 What is not in the domain of the Laplacian on Sierpinski

gasket type fractals J. Funct. Anal. 166 197–217[DSV] Dalrymple K, Strichartz R and Vinson J 1999 Fractal differential equations on the Sierpinski gasket J. Fourier

Anal. Appl. 5 203–84[F] Falconer K J 1999 Semilinear PDEs on self-similar fractals Commun. Math. Phys. 206 235–45[GRS] Gibbons M, Raj A and Strichartz R 2001 The finite element method on the Sierpinski gasket Constructive

Approx. 17 561–88[HPS] Herman P E, Peirone R and Strichartz R p-energy and p-harmonic functions on Sierpinski gasket type

fractals Potential Anal. at press[Ki1] Kigami J 1989 A harmonic calculus on the Sierpinski gasket Japan. J. Appl. Math. 6 259–90[Ki2] Kigami J 2001 Analysis on Fractals (Cambridge: Cambridge University Press)

616 R S Strichartz and C Wong

[OSY] Oberg A, Strichartz R and Yingst A 2002 Level sets of harmonic functions on the Sierpinski gasket Ark.Mat. 40 335–62

[Ro] Rockafellar R T 1970 Convex Analysis (Princeton, NJ: Princeton University Press)[S1] Strichartz R 1999 Analysis on fractals Not. Am. Math. Soc. 46 1199–208[S2] Strichartz R 2001 Harmonic mappings of the Sierpinski gasket to the circle Proc. Am. Math. Soc. 130 805–17[SU] Strichartz R and Usher M 2000 Splines on fractals Math. Proc. Camb. Phil. Soc. 129 331–60