PROPERTIES OF THE ENERGY LAPLACIAN ON SIERPINSKI
GASKET TYPE FRACTALS
ANDERS ÖBERG AND KONSTANTINOS TSOUGKAS
Abstract. In this paper, we extend some results from the standard Laplacian
on the Sierpinski Gasket to the energy Laplacian. We evaluate the pointwise
formula for the energy Laplacian and then observe that we have the analogue
of Picard’s existence and uniqueness theorem also for the energy Laplacian and
we approximate functions vanishing on the boundary with functions vanishing
in a neighborhood of the boundary as in [13]. Then we generalize to the level
three Sierpinski Gasket SG3 the pointwise formula, the scaling formula for the
energy Laplacian and some of the equivalent aforementioned results.
1. Introduction.
A recent theory of analysis on self-similar fractals is currently actively being developed and the main focus is the Laplace operator in the way that Kigami defined
it, see [5], [6], [7] for details. As a prototype, many authors focus on a specific
kind of fractal called the Sierpinski Gasket. A very detailed exposition of the topics
can be found in [11], [5] and [17]. The Laplace operator is defined weakly via a
measure and depending on the measure used we have di↵erent Laplacians. The
most widely studied Laplacian is the standard Laplacian but recently attention has
been given to the so-called energy Laplacian defined by using the Kusuoka measure
as the reference measure for defining the Laplacian in a weak sense. It is less well
behaved in some sense but has some stronger properties and therefore is actively
being studied. However it lacks self-similarity and thus it is considerably more
difficult to understand its properties. The goal of this paper is to generalize some
results, such as local solvability of di↵erential equations with the Laplace operator,
scaling properties and approximations by functions vanishing on the boundary, on
some Sierpinski Gasket type fractals from the standard Laplacian to the energy
Laplacian. Before we get into further detail we give a brief overview of the theory.
Let us first define the Sierpinski Gasket. Take an equilateral triangle in R2 with
vertices {qi } and define maps Fi : R2 ! R2 with Fi = 12 (x qi ) + qi for i = 0, 1, 2.
Then, the Sierpinski Gasket is the unique compact set satisfying
SG =
2
[
Fi SG.
i=0
As a convention the Sierpinski Gasket, or more generally the fractal being studied,
is sometimes notated instead of SG as simply K. If w = (w1 , . . . , wm ) is a finite
Date: October 24, 2014.
1
2
ANDERS ÖBERG AND KONSTANTINOS TSOUGKAS
Figure 1: The SG2 and SG3 .
word, we define the mapping
Fw = Fw 1
· · · Fw m .
We call Fw K a cell of level m. The Sierpinski Gasket may be viewed as an approximation of a sequence of graphs m with vertices Vm and adjacency relations
x ⇠m y. That means, that for x, y 2 Vm we have that
x ⇠m y () x, y 2 Fw (V0 )
for some word w of length m. We will call V0 = {q0 , q1 , q2 } the boundary of the
Sierpinski Gasket, keeping in mind that it is not the actual topological boundary.
In a similar way we can define the Sierpinski Gasket of level k, with notation SGk
where instead of splitting each side of the initial triangle in two parts, we split it
into k. Then we will have k(k+1)
maps in total Fi for i = 0, ..., k(k+1)
1. Our main
2
2
focus in this paper is for k = 2, 3 where for k = 2 it is simply the usual standard
Sierpinski Gasket. They are illustrated above in Figure 1.
S1
Let V ? = m=0 Vm , if x 2 V ? V0 , then x will be called a junction point. On
these Sierpinski Gasket type fractals we can define a measure called the standard
invariant measure µ which satisfies
2
µ(Fw Fi SGk ) =
µ(Fw SGk ), i = 0, 1, ..., k 1, for any word w.
k(k + 1)
For this measure µ, we also have the self-similar identity
X
µ(A) =
µi µ(Fi 1 A).
i
A key concept in the theory of analysis on fractals plays the concept of energy. We
can define the renormalized energy of level m for two functions u and v as:
X
Em (u, v) = r m
(u(x) u(y)) (v(x) v(y))
x⇠m y
where r is the renormalization constant. The energy is E(u, v) = lim Em (u, v)
and for u = v we simply denote E(u). On a given cell we have
✓ ◆|w|
3
E(u Fw , v Fw ) =
EFw K (u, v).
5
m!1
PROPERTIES OF THE ENERGY LAPLACIAN ON SIERPINSKI GASKET TYPE FRACTALS 3
It is remarkable that every function of finite energy is continuous. In fact domE,
the space of functions of finite energy, is a dense subspace of the space of continuous
7
functions on K. For the SG2 we have that r = 35 while for the SG3 we have r = 15
.
Given the energy, we can define the Laplacian which is the main focus of our study.
Let u 2 domE. Then, u 2 dom µ and µ u = f if
Z
E(u, v) =
f vdµ for all v 2 dom0 E
K
where dom0 E denotes the subset of domE such that the functions also vanish on
the boundary. Having defined the Laplacian we are interested in the solvability of
di↵erential equations of the form
µ u = f . We have harmonic functions which
satisfy µ h = 0 for all measures µ, or equivalently defined by minimizing energy at
all levels, and the space of harmonic functions forms a three dimensional space and
each harmonic function is completely determined by its boundary values. For the
case of SG2 it is known that this is done with the harmonic extension algorithm
2
called the “ 15
5 ” rule. A similar rule exists for the SG3 that could in a similar
4
8
way be referred to as the “ 13 15
15 ” rule which goes as follows
8
4
1
2
1
h(Fw qi ) + h(Fw qj ) + h(Fw qk ) for x = Fw qi + Fw qj
15
15
3
3
3
(1.1)
1
1
h(x) = (h(Fw q1 ) + h(Fw q2 ) + h(Fw q3 )) for x = (Fw q1 + Fw q2 + Fw q3 ).
3
3
However we have solutions besides the harmonic functions. For all measures µ the
Dirichlet problem
µ u = f , u|V0 = 0
h(x) =
has a unique solution in dom
for any continuous f , given by
Z
u(x) =
G(x, y)f (y)dµ(y)
µ
K
where G(x, y) is the Green’s function. This also shows that the space dom
sufficiently rich in functions. There also exist normal derivatives
@n u(x) = lim r
m!1
m
(2u(qi )
u(Fim qi+1 )
u(Fim qi
K
is
1 ))
which help obtain the important Gauss–Green formula which is
Z
Z
X
( µ u)vdµ
( µ v)udµ =
(u@n v v@n u) for u, v 2 dom
K
µ
µ.
V0
We can also localize the definition of the normal derivative to any given junction
point. Then, since a junction point x has two addresses, we have for the same point
two normal derivatives depending on which cell we view it as. However these two
normal derivatives add up to zero, and this is called the matching condition of the
normal derivatives. This can be generalized from the SG2 to SGk where instead
of two addresses there are more depending on the cell, but they still add up to
zero. Due to the matching condition it is possible to glue together local solutions
of
µ u = f . We will use later in this paper the scaling identity of the normal
derivatives which is
(1.2)
@n u(Fw qi ) = r
|w|
@n (u Fw )(qi ).
4
ANDERS ÖBERG AND KONSTANTINOS TSOUGKAS
We also know that for any measure, there exists a pointwise formula for the
Laplacian. The graph Laplacians are defined on Vm for SGk as
X
1
(u(y) u(x)) for x 2 Vm \ V0 .
m u(x) =
(1.3)
deg(x)
y⇠m x
where deg(x) is the cardinality of the set {y ⇠m x} which due to the self-similarity
(m)
of the fractal is the same for all m. We also define the functions x which are
(m)
called piecewise harmonic splines, to be x (y) = xy for y 2 Vm . The pointwise
formula for x 2 V ? V0 becomes
✓Z
◆ 1
m
(m)
(1.4)
u(x)
=
lim
r
dµ
deg(x) m u(x)
µ
x
m!1
which translates to
3
lim 5m
µ u(x) =
2 m!1
K
m u(x)
and
µ u(x)
= 6 lim
m!1
✓
90
7
◆m
m u(x)
for the standard Laplacian for the SG2 and SG3 respectively. The Laplacian from
a self similar measure µ also satisfies the scaling property
µ (u
Fw ) =
1
(rµw )|w|
(
µ u)
Fw
where µw are the probability weights.
A problem with dom µ , where µ is the standard measure, is that it is not closed
under multiplication. If u 2 dom µ then u2 2
/ dom µ with the exception of
constant functions. This major disadvantage however can be lifted if we define
a di↵erent Laplacian with a di↵erent measure. Kusuoka defined a measure for
which the domain of its Laplacian is closed under multiplication. However the
Kusuoka measure is not self-similar and that makes it significantly more difficult
to study. There has been some recent attention towards it and it is the main
focus of this paper. Despite not having an actual self-similar identity, we have an
identity for the SG2 found in [3] that resembles a self-similar one and involves the
Radon–Nikodym derivatives. In this paper we find the analogue of this formula
for the SG3 and rediscover the pointwise formula of the energy Laplacian using
a di↵erent approach than [9]. In [14] the existence of local solutions of nonlinear
di↵erential equations of the standard Laplacian using the Picard iteration technique
was proven. In this paper we extend this result to the energy Laplacian for both
SG2 and SG3 . Moreover, on the SG2 we show that functions vanishing on the
boundary can be approximated by functions vanishing in a neighborhood of the
boundary with their Laplacians converging in Lp . This result was already known
from [13] for the standard Laplacian and here we extend it to the energy one.
2. Energy Laplacian and the Kusuoka measure.
We define first energy measures ⌫u by
⌫u (Fw K) = r
|w|
E(u Fw ).
This is called the energy measure ⌫u . Then, the Kusuoka measure is defined as
⌫ = ⌫h0 + ⌫h1 + ⌫h2
PROPERTIES OF THE ENERGY LAPLACIAN ON SIERPINSKI GASKET TYPE FRACTALS 5
where hi (qj ) = ij for i = 0, 1, 2 is the standard basis of harmonic functions. This
definition is valid for any SGk keeping in mind that the harmonic functions look
di↵erently depending on k. An equivalent definition for the Kusuoka measure would
be to define it as
⌫ 0 = ⌫h + ⌫h?
where {h, h? } is an orthonormal basis of harmonic functions modulo constants.
This definition is independent of the orthonormal basis used and in this case we
get that ⌫ 0 = 13 ⌫, or more generally ⌫ 0 = 13 E(h) if we drop the orthonormality
condition. The Kusuoka measure is singular with respect to the standard measure
and any energy measure ⌫u,v is absolutely continuous with respect to the Kusuoka
measure.
In [9] the pointwise formula was computed for the energy Laplacian on SG2 by
evaluating the factor of
Z
(m)
x d⌫
K
via the use of the carré du champs formula. Finally, it is proven for x 2 V ? V0
that
m u(x)
⌫ u(x) = 2 lim
2
2
m!1
(h
m 1 + h2 )(x)
where h1 and h2 is the orthonormal basis of harmonic functions modulo constants.
It is then used to prove that if a function belongs both in the domains of the
standard and energy Laplacian, it must necessarily be harmonic. Here, we will
rediscover the pointwise formula by evaluating it in a completely di↵erent way
which holds for all SGk . First of all, we present a result about the decay rates of
the Kusuoka measure for the case of SG2 .
Lemma 2.1. For all words w of length |w| = m we have that
✓ ◆m
3
⌫(Fw K) 6 c
.
5
Proof. We have in [3] that for any word w, i = 0, 1, 2
✓ ◆m
3
⌫(Fw Fim K) = O
.
5
Also, for any word w of length m we have according to [2] that for all harmonic
functions h
max ⌫h (Fw K) = max ⌫h (Fim K)
and thus
|w|=m
i=0,1,2
⌫(Fw K) = ⌫h1 (Fw K) + ⌫h2 (Fw K)
6 sup ⌫h1 (Fim K) + sup ⌫h2 (Fim K)
i=0,1,2
i=0,1,2
6 sup ⌫(Fim K) + sup ⌫(Fim K) 6 c
i=0,1,2
i=0,1,2
✓ ◆m
3
.
5
⇤
6
ANDERS ÖBERG AND KONSTANTINOS TSOUGKAS
Now for the evaluation of the pointwise formula. We will evaluate the pointwise
formula of the energy Laplacian for the more general Sierpinski Gasket type fractal
SGk and observe that it is exactly identical to the formula of SG2 . The reason
for this is that while for the standard Laplacian the change is visible through the
m
di↵erent convergence speed, i.e from 5m to 90
from SG2 to SG3 , in the energy
7
Laplacian the change occurs indirectly due to the di↵erent harmonic functions. For
any k in SGk we have di↵erent harmonic extension algorithms and thus the change
occurs in the m (h1 2 + h2 2 )(x) factor since it is di↵erent for di↵erent values of k.
Proposition 2.2. Let u 2 dom ⌫ . Then for all x 2 V⇤ V0 the following pointwise
formula holds with uniform limit across V⇤ V0 .
⌫ u(x)
m u(x)
.
2
2
(h
m 1 + h2 )(x)
= 2 lim
m!1
Proof. For the general case SGk and for any measure µ, it is known that we can
rewrite the pointwise formula in the form
µ u(x)
= lim
m!1
m u(x)
This is due to the fact that we have
cm (x) and we also have that
µ w(x)
with
m w(x)
µ u(x)
= lim cm (x)
m!1
µ w(x)
= 1.
= limm!1 cm (x)
µ w(x)
µ u(x)
for some
= 1.
But for the Kusuoka measure ⌫ it is known that w(x) =
result follows.
h1 (x)2 +h2 (x)2
.
2
Thus the
⇤
In [15] and [11] we have a refinement of the “ 15 25 rule” for the SG2 where it is
generalized from harmonic functions to general functions in dom µ . First we make
a correction of the formula (4.3) for [15]. The formula should read
2
(u(y0 ) + u(y1 )) +
5
2 1
6
+
(
µ u(x2 )
3 5m 5
u(x2 ) =
and so on with Rm = o(5
m
1
u(y2 )
5
2
µ u(x1 )
5
2
5
µ u(x0 ))
+ Rm
).
We remark that this formula holds for a slightly di↵erent definition of Laplacian
than the one we use here, namely with a di↵erent factor constant of 14 . That
result is obtained through a statement about the convergence of the discrete graph
Laplacians. We obtain a similar result for the energy Laplacian.
Lemma 2.3. For u 2 dom
⌫ u(x)
⌫
2 m u(x)
=
2
2
m (h1 + h2 )(x)
and x 2
/ V0 we have that
1
2rm deg(x)
2
2
m (h1 + h2 )(x)
Z
m
x (y)(
⌫ u(x)
⌫ u(y))d⌫(y).
PROPERTIES OF THE ENERGY LAPLACIAN ON SIERPINSKI GASKET TYPE FRACTALS 7
Proof. We have that
Z
(m)
⌫ u(y))d⌫(y)
x (y)( ⌫ u(x)
Z
Z
(m)
(2.1)
= ⌫ u(x)
(y)d⌫(y)
⌫ u(y)
x
1
= r m deg(x)
2
and the result follows.
m (h1
2
+ h2 2 )(x)
⌫ u(x)
(m)
x (y)d⌫(y)
r
m
deg(x)
m u(x)
⇤
Now, we will evaluate the self similar identity for the energy Laplacian in SG3 .
Of course, it is not actual self similar in the real sense of the word, but it displays
some properties similar to the one in SG2 as in [3].
First of all the energy measures ⌫hi of the harmonic functions hi will be denoted
for brevity as ⌫i . From some elementary computations it is found in [3] that
1
( ⌫0 ⌫1 + ⌫2 );
2
1
= ( ⌫0 + ⌫1 ⌫2 );
2
1
= (⌫0 ⌫1 ⌫2 ).
2
⌫h0 ,h1 =
(2.2)
⌫h0 ,h2
⌫h1 ,h2
Take the symmetric harmonic function h0 with 1 at the point q0 and zero elsewhere. Due to (1.1) we can establish the relations
8
8
8
7
h1 + h2 =
+ h0 ;
15
15
15 15
4
3
h0 + h2 ;
15
15
4
3
h0 + h1 ;
15
15
3
2
+ h0 ;
15 15
8
4
1
4
4
h0 + h2 + h1 =
+ h0 +
15
15
3
15 15
8
4
1
4
4
h0 + h1 + h2 =
+ h0 +
15
15
3
15 15
h 0 F0 = h 0 +
h 0 F1 =
(2.3)
h 0 F2 =
h 0 F3 =
h 0 F4 =
h 0 F5 =
1
h1 ;
15
1
h2 .
15
We will follow the methodology of [3] to establish a relationship of ⌫i ’s from cells
to subcells.
Theorem 2.4. Let
0
49
1 @
12
M0 =
105
12
0
4
1 @
3
M2 =
105
0
0
4
3
3
4
0
1
0
0
4
1 @
3 A , M1 =
0
105
4
3
1
0
12
4
1
@ 3
12 A , M3 =
105
49
3
12
49
12
0
12
4
1
3
0 A,
4
1
0
4 A,
12
8
ANDERS ÖBERG AND KONSTANTINOS TSOUGKAS
0
then
12
1 @
0
M4 =
105
4
0
3
4
3
1
0
4
12
1 @
0 A , M5 =
4
105
12
0
4
12
0
1
0
1
⌫0 (Fi C)
⌫0 (C)
@ ⌫1 (Fi C) A = Mi @ ⌫1 (C) A
⌫2 (Fi C)
⌫2 (C)
1
3
3 A,
4
for every cell C. Which means that,
0
1
0
1
5
⌫0
⌫0
X
@ ⌫1 A =
(2.4)
Mi @ ⌫1 A Fi 1 .
i=0
⌫2
⌫2
Proof. Let f be a continuous function on SG3 , by (2.2), (2.3),
Z
Z
15
8
7
f d⌫0 =
f F0 d⌫ 15
+ 15
h0
7 K
F0 K
✓ ◆2 Z
15 7
=
f F0 d⌫0
7 15
K
Z
49
=
f F0 d⌫0 .
105 K
On the other hand,
Z
Z
15
4
3
f d⌫0 =
f F1 d⌫ 15
h0 + 15
h2
7 K
F1 K
✓
◆
Z
Z
Z
15 16
9
24
=
f
F
d⌫
+
f
F
d⌫
+
f
F
d⌫
1
0
1
2
1
h0 ,h2
7 152 K
152 K
152 K
✓
Z
Z
Z
15 16
9
12
f F1 d⌫0 + 2
f F1 d⌫2
f F1 d⌫0
=
7 152 K
15 K
152 K
◆
Z
Z
12
12
+ 2
f F1 d⌫1
f
F
d⌫
1
2
15 K
152 K
Z
Z
Z
4
12
3
=
f F1 d⌫0 +
f F1 d⌫1
f F1 d⌫2
105 K
105 K
105 K
using (2.2) and (2.3) again.
that
Z
Z
4
f d⌫0 =
f
105 K
F K
Z 2
Z
4
f d⌫0 =
f
105
ZF3 K
ZK
12
f d⌫0 =
f
105
ZF4 K
ZK
12
f d⌫0 =
f
105
F5 K
K
In a completely similar way we continue and obtain
F2 d⌫0
3
105
Z
f
K
12
F2 d⌫1 +
105
Z
f
F2 d⌫2
K
F3 d⌫0
Z
3
F4 d⌫0
f
105 K
Z
4
F5 d⌫0 +
f
105 K
Z
4
F4 d⌫1 +
f
105 K
Z
3
F5 d⌫1
f
105 K
F4 d⌫2
F5 d⌫2
PROPERTIES OF THE ENERGY LAPLACIAN ON SIERPINSKI GASKET TYPE FRACTALS 9
Since this is true for arbitrary continuous function f , we have the following identity
49
⌫0 F0 1
105
4
12
3
+
⌫0 F1 1 +
⌫1 F 1 1
⌫2 F1 1
105
105
105
4
3
12
+
⌫0 F2 1
⌫1 F 2 1 +
⌫2 F2 1
105
105
105
4
+
⌫0 F3 1
105
12
3
4
+
⌫0 F4 1
⌫1 F 4 1 +
⌫2 F4 1
105
105
105
12
4
3
+
⌫0 F5 1 +
⌫1 F 5 1
⌫2 F5 1 .
105
105
105
By similar computations and symmetry, we get equivalent expressions for ⌫1 and
⌫2 , and thus we have the relation
0
1
0
1
5
⌫0
⌫0
X
@ ⌫1 A =
Mi @ ⌫1 A Fi 1 .
i=0
⌫2
⌫2
⌫0 =
⇤
Corollary 2.5. The Kusuoka measure satisfies the variable weight self-similar
identity
2
(2.5)
1 X
⌫=
((1 + 72Ri ) ⌫) Fi
105 i=0
5
1
1 X
+
((16
105 i=3
18Ri
and thus for any integrable f we have that
(2.6)
Z
2 Z
5 Z
1 X
1 X
f d⌫ =
(1 + 72Ri ) f Fi d⌫ +
(16
105 i=0 K
105 i=3 K
K
0
3 ) ⌫)
18Ri
Fi
3) f
1
Fi d⌫
1
⌫0
d⌫
Proof. By taking ⌫ = 1 1 1 @ ⌫1 A and Rj = d⌫j , the Radon–Nikodym
⌫2
Derivative of ⌫j in regards to ⌫ and using (2.4) we get that
0
1
5
⌫0
X
1 1 1 Mi @ ⌫1 A Fi 1
⌫=
i=0
⌫2
1
1
=
(73⌫0 + ⌫1 + ⌫2 ) F0 1 +
(⌫0 + 73⌫1 + ⌫2 ) F1 1 +
105
105
1
1
(2.7)
(⌫0 + ⌫1 + 73⌫2 ) F2 1 +
( 2⌫0 + 16⌫1 + 16⌫2 ) F3 1 +
105
105
1
1
+
(16⌫0 2⌫1 + 16⌫2 ) F4 1 +
(16⌫0 + 16⌫1 2⌫2 ) F5 1
105
105
2
5
1 X
1 X
=
((1 + 72Ri ) ⌫) Fi 1 +
((16 18Ri 3 ) ⌫) Fi 1 .
105 i=0
105 i=3
10
ANDERS ÖBERG AND KONSTANTINOS TSOUGKAS
⇤
By using the above we now obtain the “self-similar” scaling formula for the energy
Laplacian ⌫ on the SG3 .
1
Theorem 2.6. Let Qj = 225
(1 + 72Rj ) for j = 0, 1, 2 and Qj =
for j = 3, 4, 5. Then the energy Laplacian ⌫ satisfies
(2.8)
⌫ (u
Fj ) = Qj (
⌫ u)
1
225 (16
18Rj
3)
Fj
Proof. The proof follows the exact same methodology of [3]. By the definition of
the energy Laplacian and by (2.6) with f now being ( ⌫ u)v we have,
Z
E(u, v) =
( ⌫ u)vd⌫
K
2
=
1 X
105 j=0
5
+
1 X
105 j=3
Z
(1 + 72Rj )(
⌫ u)
Fj v Fj d⌫
K
Z
(16
18Rj
3 )(
⌫ u)
Fj v Fj d⌫.
K
We also know that E is self-similar and that gives us
5
15 X
E(u Fj , v Fj ).
7 j=0
E(u, v) =
Combining these two we have
0
Z
2
5
X
X
1
@ (1 + 72Rj ) +
(16
105 K j=0
j=3
5
15 X
=
7 j=0
of
Z
⌫ (u
18Rj
1
A(
3)
⌫ u)
Fj v Fj d⌫
Fj ) v Fj d⌫.
K
Since this holds for any v and thus v Fj is arbitrary, we obtain the self-similarity
⇤
⌫.
Then by iterating this formula we have the equivalent corollary of [3] for the
proper Qw as above.
Corollary 2.7. Let w = (w1 , ..., wm ) be a finite word of length m and Fw =
Fw1 .... Fwm . Define
Qw = Qwm · (Qwm
1
Fwm ) · (Qwm
2
Fw m
1
Fwm ) · · · (Qw1
Then
⌫ (u
Fw ) = Qw (
⌫ u)
Fw .
Fw 2
· · · Fw m )
PROPERTIES OF THE ENERGY LAPLACIAN ON SIERPINSKI GASKET TYPE FRACTALS11
3. Local solvability of differential equations for the energy
Laplacian.
Now, we are focusing on the SG2 and we are interested in the analogue of Picard
existence and uniqueness theorem for the local solvability of
⌫ u(x)
= F (x, u(x)).
We will follow the proof of [14] in section 2 “Local Solvability” making the appropriate changes that are required by using ⌫ instead of µ . Let F (x, u) denote a
continuous function from K ⇥R to R which satisfies also a Local Lipschitz condition
in the u-variables:
for every T > 0, there exists MT < 1 such that
|F (x, u)
F (x, u0 )| 6 MT |u
u0 |, provided |u|, |u0 | 6 T.
First we make a correction of the formula in [3]. At Theorem 2.3 the expression
Qj =
should read
1
12 d⌫j
+
15 25 d⌫
1
12 d⌫j
+
.
25 25 d⌫
Then, we have the following scaling property for the energy Laplacian.
3
⌫ (u Fj ) = Qj ( ⌫ u) Fj
5
Qj =
d⌫
j
1
for Qj = 15
+ 12
15 d⌫ . Essentially it is the same as before but it is better to illustrate
it like this with a slightly modified Qj so that we can pay attention to the factor
3
5 . And using this, on page 8 of [3] we get that
✓ ◆m
3
Qw ( ⌫ u) Fw
⌫ (u Fw ) =
5
for w = (w1 , ..., wm ) a finite word of length m and Fw = Fw1
(3.1) Qw = Qwm · (Qwm
1
Fwm ) · (Qw
2
Fw
1
Fw 2
· · · Fwm and
Fwm ) · · · (Qw1 Fw2 · · · Fwm ).
Theorem 3.1. Given F satisfying the Lipschitz condition above, for every A there
exists m such that for all choices of {aj } with |aj | 6 A, the equation
⌫ u(x) =
F (x, u(x)) on Fw K for any |w| = m and boundary conditions u(Fw qj ) = aj with
V0 = {q1 , q2 , q3 } has a unique solution.
Proof. First, we study the case in which all aj = 0. By changing the variable
x ! Fw x where w is a word of length m, then in light of (3.1) the original equation
becomes
✓ ◆m
3
(u
F
)(x)
=
Qw (x)F (Fw x, u Fw (x))
⌫
w
5
for x 2 K. Let v = u Fw . Then, the equation along with the boundary conditions
above becomes
✓ ◆m
3
Qw (x)F (Fw x, v(x))
⌫ v(x) =
5
on K with v|V0 = 0
12
ANDERS ÖBERG AND KONSTANTINOS TSOUGKAS
which we can write in an equivalent way to
✓ ◆m Z
3
v(x) =
G(x, y)Qw (y)F (Fw y, v(y))d⌫(y).
5
K
Let Gv(x) to be
Gv(x) =
✓ ◆m Z
3
G(x, y)Qw (y)F (Fw y, v(y))d⌫(y).
5
K
The space of continuous functions v is a Banach space and thus to obtain our
result it suffices to show that G satisfies the hypotheses of the contractive mapping
principle on a suitable ball (kvk1 6 T ) . Since G(x, y) is continuous and bounded
(let G0 be an upper bound), we have that
✓ ◆m
3
|Gv(x)| 6
kQw (y)k1 G0 FT
5
where FT is an upper bound for |F (x, u)| for x 2 K and |u| 6 T . First, to bound
kQw (y)k1 we have that kQw (y)k1 equals to
kQwm (y) · (Qwm
6 kQwm · Qwm
1
Fwm y) · (Qw
2
Fw
1
Fwm y) · · · (Qw1
Fw 2
· · · Qw1 k1 6 kQwm k1 · kQwm 1 k1 · · · kQw1 k1 .
1
· · · Fwm y)k1
However, we have that for any j = 0, 1, 2 that
1
12 d⌫j
1
12 d⌫j
kQj k1 = k +
k1 6
+ k
k1 .
15 15 d⌫
15 15 d⌫
Here we will notice that the Radon–Nikodym derivative is bounded by 1 since
⌫i (C) 6 ⌫(C) for any cell C. In fact there is an even stronger result in [3] where
the discontinuity of the Radon–Nikodym derivative is given along with a more
powerful bound. But in our case, it suffices to bound it by 1. Then
1
12
kQj k1 6
+
· 1 < 1.
15 15
m
And thus, we have that kQw k1 < 1. Then, |G(x)| 6 53
G0 FT so we just have to
m
3
take m large enough such that 5
G0 FT 6 T to conclude that G maps the ball
to itself. For v and v 0 in this ball, we have that
✓ ◆m
3
|Gv(x) Gv 0 (x)| 6
G0 MT kv v 0 k1
5
m
so we get the contractive mapping estimate as long as 35
G0 MT < 1 Now, to
modify the proof for the case of general {aj }. Let h(x) denote the harmonic function
which satisfies the same boundary conditions as u. Then w = u h satisfies
w|Fw V0 = 0
and solves the equation
⌫w
= F (x, h(x) + w(x))
so it is the same as in the initial special case with F being changed to
F 0 (x, u) = F (x, h(x) + u).
Note that |h(x)| 6 A, so by taking T = 2A, we have
|F 0 (x, u)| 6 FT if kuk1 6 A,
PROPERTIES OF THE ENERGY LAPLACIAN ON SIERPINSKI GASKET TYPE FRACTALS13
and we can apply the same argument as before.
⇤
Having done that then it is easy to generalize the result to the SG3 by simply
using formula (2.8). The Radon–Nikodym derivatives Rj remain bounded by 1 due
to the same reason as before. We can write then as before slightly di↵erent Qj
1
1
namely, Qj = 105
(1 + 72Rj ) for j = 0, 1, 2 and Qj = 105
(16 18Rj ) for j = 3, 4, 5
and then rewrite the scaling formula as
7
Qj ( ⌫ u) Fj .
⌫ (u Fj ) =
15
Again the only reason for writing the formula like this is for the sake of clarity to
see easily that each Qj is bounded by 1 since for j = 0, 1, 2 we have
1
1
72
1
72
(1 + 72Rj )k1 6
+k
Rj k1 6
+
<1
105
105
105
105 105
and similarly for j = 3, 4, 5 we have
1
16
18
16
18
kQj k1 = k
(16 18Rj )k1 6
+k
Rj k1 6
+
< 1.
105
105
105
105 105
7 m
Thus we obtain the contractive mapping estimate as long as ( 15
) G0 MT < 1.
It is reasonable to conjecture that this result extends also to the general case for
any SGk . Formulas similar to (2.8) would hold for the SGk and their respective
Qj quantities would be similarly bounded by 1. However, to prove it rigorously,
one would have to compute first the renormalization constant r for the energy and
the harmonic extension algorithm for the harmonic functions in SGk before the
equivalent equation (2.8) can be obtained.
kQj k1 = k
4. Approximation by functions vanishing in a neighborhood of the
boundary.
Now, we will generalize some results found in [13] in section 7 “Spline cut-o↵s” from
the standard Laplacian to the energy Laplacian for the SG2 . It is proven in [16] that
we have a “weak=strong” property for the Laplacian, as long as certain conditions
are satisfied for the test functions, namely that v(qi ) = 0 and @n v(qi ) = 0. These
conditions have been weakened in [13] to the smaller class of functions v such that
they vanish on a neighborhood of the boundary. Our goal in this section is to
obtain a similar result for the energy Laplacian on SG2 . Let H1 be the space of
multiharmonic functions such that 2⌫ u = 0.
Lemma 4.1. For any u 2 H1 we have that
!
✓ ◆2m
✓ ◆m
5
5
sup |u| +
sup |@n u| .
k ⌫ u|Fw K k1 6 c
3
3
@Fw K
@Fw K
Proof. Since u 2 H1 and thus ⌫ u is harmonic, we simply have to bound its values
on the boundary @Fw K = Fw @K. If w is the empty word, the above estimate is an
immediate consequence of the basis for H1 defined in [13] which can be equivalently
defined for the energy Laplacian as well. The general case then follows from the
m
scaling property for the energy Laplacian, where we substitute kQw k1 = 35
and
the scaling property for the normal derivatives (1.2) as well.
14
ANDERS ÖBERG AND KONSTANTINOS TSOUGKAS
⇤
Lemma 4.2. Let ⌫ be the energy Laplacian and f 2 dom
along with its normal derivatives on the boundary. Then
✓ ◆2m
3
f|@Fim K = O
5
and
@n f|@Fim K = O
⌫
such that it vanishes
✓ ◆m
3
.
5
Proof. For the first part:
First, let i = 0, 1, 2. Then,
1
1
(u(x) + u(Ri x)) + (u(x) u(Ri x))
2
2
where Ri is the reflection around the point qi . For shorter notation let’s call the
symmetric part u+ and the skew symmetric part u . Then, @n u (qi ) = 0 since
always for a skew-symmetric function the normal derivatives are zero. But then,
this implies that also @u+ (qi ) = 0 since we assumed that u has its normal derivatives
zero on the boundary. Then we have that
✓ ◆2m
3
sup |u| 6 sup |u | + sup |u+ | 6 O
5
@Fim K
@Fim K
@Fim K
u(x) =
by using theorem 3.1 and 3.6 from [12]. For the second part: We use the Gauss–
Green formula localized to Fim K with the functions u = f and v = h where h is the
harmonic function taking the values 1,-1,0 on the boundary points of Fim K (the
value 1 at the point qi ) Then, this gives us
Z
X
h ⌫ f d⌫ =
f (Fim x)@n h(Fim x) h(Fim x)@n f (Fim x).
Fim K
x2V0
By assumption, @n f (Fim qi ) = 0 so the only term in the right hand side of the form
h(Fim x)@n f (Fim x) that occurs is the single value at the vertex where h assumes
m
the value -1. The integral on the left side is O 35
since h and f are uniformly
m
bounded and the measure is O 35
and the terms of the form f (Fim x)@n h(Fim x)
m
m
are O 35
since @n h(Fim x) = O 53
and
✓ ◆2m
3
m
f (Fi x) = O
5
by the first part of this lemma.
⇤
Using these two lemmas, we are ready to prove the following theorem.
Theorem 4.3. For the energy Laplacian on SG2 , suppose that f 2 dom ⌫ and f
vanishes together with its normal derivatives on the boundary. Then there exists a
sequence of functions {fm } with each fm 2 dom ⌫ vanishing in a neighborhood of
the boundary with fm ! f uniformly, E(fm f ) ! 0 and ⌫ fm ! ⌫ f in Lp (d⌫)
for any p < 1.
PROPERTIES OF THE ENERGY LAPLACIAN ON SIERPINSKI GASKET TYPE FRACTALS15
Proof. As in the proof of theorem 7.1 on [13], we choose fm so that fm = f on
⌦m with support in ⌦m+1 . On each of the sets Fim K we take fm to be the spline
locally in S(H1 , Vm+1 ) so that fm = f and @n fm = @n f at the two boundary
points of Fim K not equal in qi and fm = 0 and @n fm = 0 in the other 4 vertices in
Vm+1 \ Fim K. Because we have matched the values of the functions and the normal
derivatives, the functions fm will be in dom ⌫ . We will show that
Z
| ⌫ fm |p d⌫ ! 0 as m ! 1 for every p < 1.
Fim K
After this, the proof is identical to the standard Laplacian case since no other
di↵erences between the standard and the energy Laplacian arise. We will use the
previous lemma. We have then that:
✓ ◆2m
✓ ◆m
5
5
k ⌫ fm |Fim K k1 6 c[
sup |f | +
sup |@f |]
3
3
@Fim K
@Fim K
and by using the lemmas above, we have that k ⌫ fm |Fim K k1 6 C and thus since
m
⌫(Fim K) = O 35
we get that
Z
| ⌫ fm |p d⌫ ! 0.
Fim K
⇤
Then, the proof for the following interesting Corollary on [13] is exactly identical
to the standard Laplacian case by using the above theorem and can be found at
[13] and we present it here too as motivation for our results.
Corollary 4.4. Let ⌫ be the energy Laplacian on SG2 . If u 2 L2 (d⌫) and
f 2 L2 (d⌫) (respectively, f is continuous), and
Z
Z
u vd⌫ =
f vd⌫
K
K
for all v 2 domC ( ⌫ ) vanishing on a neighborhood of the boundary, then u 2
domL2 ( ⌫ ) (respectively domC ( ⌫ ) and ⌫ = f .
Acknowledgements
We are grateful to Professor Robert S. Strichartz for valuable suggestions.
References
[1] Tang Donglei, Su Weiyi The Laplacian on the level 3 Sierpinski gasket via the method of
averages, Chaos Solitons Fractals, 23, Issue 4, (February 2005), 1201–1209.
[2] J. Azzam, M.A. Hall and R.S. Strichartz, Conformal energy, conformal Laplacian, and energy
measures on the Sierpinski gasket, Trans. Amer. Math. Soc. 360 (2008), 2089–2131.
[3] R. Bell, C.-W. Ho and R.S. Strichartz, Energy measures of harmonic functions on the Sierpinski gasket, to appear in the Indiana Journal of Mathematics.
[4] Marius Ionescu, Erin P.J Pearse, Luke G. Rogers, Huo-Yun. Ruan, Robert S. Strichartz
The Resolvent Kernel for PCF Self-Similar Fractals, Trans. Amer. Math. Soc. Volume 362
Number 8 (August 2010), 4451–4479.
[5] Jun Kigami, Analysis on Fractals, Cambridge Tracts in Mathematics, (2001).
[6] Jun Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc., 335
(1993), 721-755.
16
ANDERS ÖBERG AND KONSTANTINOS TSOUGKAS
[7] Jun Kigami, Harmonic analysis for resistance forms, J. Funct. Anal. 209 (2003) 399-444.
[8] S Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst.
Math. Sci, 25 (1989), 659-680.
[9] Eric D. Mbakop, Analysis on Fractals, Worcester Polytechnique Institute (2009)
[10] J. Needleman, R.S. Strichartz, A. Teplyaev, P.-L. Yung, Calculus on the Sierpinski gasket.
I. Polynomials, exponentials and power series, J. Funct. Anal. 215 (2004), no. 2, 290–340.
[11] R.S. Strichartz, Di↵erential Equations on Fractals, Princeton University Press, Princeton
2006.
[12] R.S. Strichartz and S.T. Tse, Local behavior of smooth functions for the energy Laplacian on
the Sierpinski gasket, Analysis 30 (2010), 285–299.
[13] R.S. Strichartz and Michael Usher, Splines on Fractals, Math. Proc. of the Cambridge Philos.
Soc. 129 (2000), 331–360.
[14] R.S. Strichartz, Solvability for di↵erential equations on Fractals, Journal d’Analyse Mathmatique, 96 (2005), 247–267.
[15] Oren Ben-Bassat, R.S. Strichartz, Alexander Teplyaev, What Is Not in the Domain of the
Laplacian on Sierpinski Gasket Type Fractals, J. Funct. Anal. 166 (1999), 197–217.
[16] R.S. Strichartz, Some Properties of Laplacians on Fractals, J. Func. Anal. 164,Issue 2
(1999), 191–208.
[17] Alexander Teplyaev, Energy and Laplacian on the Sierpinski gasket, Fractal Geometry and
Applications: A Jubilee of Benoit Mandelbrot, Part 1. Proceedings of Symposia in Pure Mathematics 72, Amer. Math. Soc., (2004), 131–154.
Anders Öberg, Department of Mathematics, Uppsala university, Sweden
E-mail address: anders.oberg@math.uu.se
Konstantinos Tsougkas, Department of Mathematics, Uppsala university, Sweden
E-mail address: konstantinos.tsougkas@math.uu.se