User manual | Properties of the energy Laplacian on the Sierpinski Gasket Konstantinos Tsougkas

```U.U.D.M. Project Report 2014:4
Properties of the energy Laplacian on the
Konstantinos Tsougkas
Examensarbete i matematik, 30 hp
Handledare och examinator: Anders Öberg
Januari 2014
Department of Mathematics
Uppsala University
Properties of the energy Laplacian
Konstantinos Tsougkas
January 27, 2014
Abstract
In this paper we study the recent topic of analysis on fractals,
taking as our primary focus the Sierpinski Gasket. We examine how a
Laplacian is created on it in regards to different measures and we study
its properties. Then we extend some results of the standard Laplacian
to the Kusuoka one.
1
Acknowledgements
I would like to express my gratitude to my advisor Anders Öberg for his
valuable guidance every step along the way of this work. He introduced me
to this fascinating topic and then had the patience to help me wherever I
needed and provide me constant encouragement, comments and suggestions.
I also thank immensely my family whose constant support and love gave
me motivation. Without them this work would not have been possible and
thus, this is dedicated to them.
Contents
1 Introduction
4
2 Energy measures and the Kusuoka measure
16
3 Local behavior of functions in dom∆µ
20
4 Local Solvability of Differential equations for the energy
Laplacian
28
5 Approximation by functions vanishing in a neighborhood of
the boundary
32
3
1
Introduction
Let us define the Sierpinski gasket, SG. 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 Sieprinski Gasket is the unique compact set satisfying
SG =
2
[
Fi SG.
i=0
Also, as a convention we may refer to the Sierpinski Gasket as SG or K. If
w = (w1 , . . . , wm ) is a finite word, we can also define the mapping
Fw = Fw1 ◦ · · · ◦ Fwm .
We call Fw K a cell of level m. The Sierpinski Gasket may be viewed as an
approximation of a sequence of graphs Γm with vertices Vm and adjacency
relations x ∼m y. That means, that for x, y ∈ Vm we have that
x ∼m y ⇐⇒ x, y ∈ Fw (V0 )
for some word w of length m.
S∞We want the vertices to be nested V0 ⊆ V1 ⊆
V2 . . . with the union V∗ = m=0 Vm , a dense set of the fractal. In the case
of the Sierpinski Gasket, we take V0 = {qi }2i=0 as the vertices of an original
equilateral triangle and define
Vm =
2
[
Fi (Vm−1 ).
i=0
4
We call V0 the boundary of the Sierpinski Gasket and every point in V∗ a
junction point. We will be concerned with functions u : K → R.
Now, we create a measure on the Sierpinski Gasket. In this thesis we
will focus on two measures, namely the standard measure and the Kusuoka
measure. The standard measure is a special case of a self-similar measure
created in the following way:
Assign probability weights µi with
2
X
µi = 1 ,with each µi > 0 and then set
i=0
µ(Fw K) =
2
Y
µwi for |w| = m.
i=0
Then, for the standard measure we just set all µi = 1/3. On the Sierpinski
Gasket the standard invariant measure µ satisfies
1
µ(Fw Fi SG) = µ(Fw SG),
3
i = 0, 1, 2, for any word w.
We also have the self-similar identity
X
µ(A) =
µi µ(Fi−1 A).
i
Now, after having defined a measure we can define integrals and create
and enrich our theory. This is standard as in usual calculus, and since we
have uniform continuity due to the compact set K it suffices to define it as
Z
X
f dµ = lim
f (xw )µ(Fw K).
K
m→∞
|w|=m
A key concept in the theory of analysis on fractals plays the concept of
energy. On each graph G we construct an energy E(u, v) for two functions
u and v with:
X
EG (u, v) =
(u(x) − u(y)) (v(x) − v(y))
x∼y
where the sum extends over all the edges of the graph. For u = v we simply
denote EG (u). This is a bilinear form. If we have a graph G and G0 with
vertices V and V 0 respectively, and V ⊂ V 0 and we also have a function u
5
defined on V , then we can extend it on V 0 in many possible ways. However,
there is at least one way to do it so that it minimizes EG0 (u). Such an energyminimizing extension will be called a harmonic extension and write it ũ. In
the Sierpinski Gasket, it turns out that EG0 (ũ) = 53 EG (u). This leads us to
define the so called renormalized graph energies, Em (u) = (r−m Em (u)). In
our case, it turns out r = 35 . There exists a simple extension algorithm in
order to obtain a harmonic extension of a function defined on Vm to Vm+1 .
This algorithm is called the “ 15 − 25 rule” and it goes as follows:
If on a given cell Vm we have that the function takes boundary values a, b, c
and the junctions points of the next level sub-cell, directly opposite of these
points respectively take values x, y, z then we have that
1
2
x = (b + c) + a
5
5
2
1
y = (a + c) + b
5
5
2
1
z = (a + b) + c
5
5
This equations are obtained easily by using calculus. By computing the
value of the energy, and taking the derivatives of x, y, z equal to zero to
minimize it, and then solving accordingly. If ũ is the harmonic extension
of u then it holds that Em+1 (ũ) = Em (u) and in general we have that
Em (u) ≤ Em+1 (u). We define a harmonic function h to be one that minimizes
Em (h) at all levels for the given boundary values on V0 . This can be done
by following the “ 15 − 52 ” rule and thus by this rule, it is obvious that a
harmonic function is defined completely by its boundary values. Thus, we
have a space of harmonic functions called H0 which is three-dimensional
with a basis {h0 , h1 , h2 } with hi (qj ) = δi,j . Thus we have that for harmonic
extension functions at each point x ∈ Vm+1 \ Vm , ũ(x) is the average of
the values at the four neighboring points in Vm+1 . In a similar fashion as
standard analysis, we have a maximum principle for harmonic functions too,
and thus they take their maximum value at the boundary. We also have the
following key lemma.
Lemma 1.1. Let u, v be defined on Vm , let ũ be the harmonic extension of
u and let v 0 be any extension of v to Vm+1 . Then
Em+1 (ũ, v 0 ) = Em (u, v).
6
We define then the energy of u as
E(u) = lim Em (u) or similarly
m→∞
E(u, v) = lim Em (u, v)
m→∞
This energy plays a central role in our theory and we call functions u such
that E(u) < ∞ as functions with finite energy and we denote u ∈ domE.
A very important property of functions of finite energy is that they are
continuous. In fact, they are Hölder continuous. Moreover, domE is dense
in C(K, R) and also domE forms an algebra. This energy E(u, v) is also
a bilinear form and forms an inner product on the space of domE modulo
constants. In fact, domE/constants forms a Hilbert space with that inner
product.
We are now ready to define the main object of our study, the Laplacian.
Definition 1.2. Let u ∈ domE. Then, u ∈ dom∆µ and ∆µ u = f if
Z
E(u, v) = −
f vdµ for all v ∈ dom0 E
K
where dom0 E denotes the functions of finite energy that vanish on the boundary.
If we mean the standard Laplacian, that is the Laplacian with the standard
measure, then we can simply write ∆u instead of ∆µ u without any confusion.
We have that dom∆µ is a real vector space. However just by the definition
it is not so clear that there are any nontrivial functions in dom∆µ . But a
bit later on we will see a theorem which shows that it is a very rich space
because for every continuous function f there exists u ∈ dom∆µ such that
∆µ u = f . Initially, a very important fact is that the space contains harmonic
functions and they have Laplacian zero, which also is an equivalent way of
defining harmonic functions. This fact holds true for all measures µ.
Theorem 1.3. If h ∈ H0 , we have that h ∈ dom∆µ and ∆µ h = 0. Conversely, if u ∈ dom∆µ and ∆µ u = 0 then u is harmonic.
Proof. We know that harmonic functions h have the property that Em (h, v)
is independent of m, so E(h, v) = E0 (h, v) = 0 since v vanishes on the
boundary. Thus ∆µ h = 0.
For the opposite direction we will use some very specific functions v. Let
(m)
m and a point x ∈ Vm ∈
/ V0 . Define ψx as the piecewise harmonic function
7
(m)
at level m that satisfies ψx (y) = δxy for all x, y ∈ Vm . Then we have that
ψxm ∈ dom0 E since x ∈
/ V0 . Then, by using the fact that ∆µ u = 0 we get
(m)
that E(u, ψx ) = 0. But then by reversing the roles of u and v we have that
(m)
(m)
(m)
E(u, ψx ) = Em (u, ψx ). But then, this condition that Em (u, ψx ) = 0
means that
X
(u(x) − u(y)) = 0
y∼m x
and that u|Vm is harmonic. But this is true for all m, and thus u is harmonic.
A very big drawback of dom∆µ is that it is not closed under multiplication.
If u ∈ dom∆µ then u2 ∈
/ dom∆µ . This will be explored later, and this
fact is completely dependent on the measure µ. This major disadvantage
however can be lifted if we create a different Laplacian with a different
measure. Namely, Kusuoka created a measure called the Kusuoka measure
and with that measure we have that the domain of its Laplacian is closed
under multiplication. The major scope of this thesis is investigating the
properties and the differences between the two Laplacians, with the Kusuoka
and the standard measure.
The definition we used above for the Laplacian is called a weak definition and there is an equivalent pointwise formula. First, we define a graph
Laplacian
∆m u(x) =
X
(u(y) − u(x)) for all x ∈ Vm V0 .
y∼m x
Then our pointwise formula would be the following:
Z
−1
−m
(m)
∆µ u(x) = lim r
ψx dµ
∆m (x).
m→∞
K
In the case of the standard Laplacian, we simply get
∆u(x) =
3
lim 5m ∆m u(x)
2 m→∞
The reason for this, is that we already have that r =
need to estimate
Z
ψx(m) dµ.
K
8
3
5
and thus we only
(m)
To do this, let x ∈ K. Then the function ψx has support in the two m-cells
meeting at x. If Fw K is one of these cells with vertices x, y, z then we get
that
ψx(m) + ψy(m) + ψz(m) = 1
and this holds as an identity. Thus
m
Z
1
(m)
(m)
(m)
(ψx + ψy + ψz )dµ = µ(Fw K) =
.
3
Fw K
But due to symmetry all the summands have the same integral, so
Z
ψx(m) dµ =
Fw K
1
3m+1
.
Along with the other m-cell, we get finally that
Z
ψx(m) dµ =
Fw K
2
.
3m+1
We can also create normal derivatives at the boundary points {qi }.
Definition 1.4. Let x a boundary point and u a continuous function on K.
We say ∂n u(x) exists if the right hand side limit exists and
X
∂n u(x) = lim r−m
(u(x) − u(y)).
m→∞
y∼m x
This is called the normal derivative at the point x.
For the case of the Sierpinski Gasket this can be viewed as
m
5
(2u(qi ) − u(Fim qi+1 ) − u(Fim qi−1 )) .
∂n u(x) = lim
m→∞ 3
We can also localize the definition of the normal derivative. If we have a cell
Fw K and x = Fw qi be a boundary point of that cell, then we can say that
∂n u(x) with respect to the cell Fw K is the same formula with the y in the
formula lie inside Fw K. We can also view this as a scaling property for the
normal derivative, namely
∂n u(Fw qi ) = r−|w| ∂n (u ◦ Fw )(qi ).
9
However at this point it is important to note that a junction point has two
addresses Fw qi and Fw0 qi0 and thus there is another local derivative on that
point x with respect to the other cell. Thus a normal derivative is not only
with respect to any junction point but also viewed according to the cell of
that junction point.
However we have an important connection to these two derivatives, and
that is that they sum to zero. This is called the matching condition for
normal derivatives at x.
Proposition 1.5. Suppose u ∈ dom∆µ . Then at each junction point x =
Fw qi = Fw0 qi0 , the local derivatives exist and
∂n u(Fw qi ) + ∂n u(Fw0 qi0 ) = 0
Proof. The existence follows from the scaling property. Now, we have that
X
∂n u(Fw qi ) + ∂n u(Fw0 qi0 ) = lim r−m
(u(x) − u(y))
m→∞
y∼m x
since the neighbors y of x lie in either Fw K or Fw0 K. However since we have
that u ∈ dom∆µ we know that
Z
−1
lim r−m
ψx(m) dµ
∆m u(x)
m→∞
exists and since
is also zero.
K
(m)
R
K
ψx dµ → 0 we see that the above right hand side limit
In a similar fashion, we also define the tangential derivatives:
Definition 1.6. Let qi be a boundary point. The tangential derivative at
the point qi is the limit
∂T u(qi ) = lim 5m (u(Fim (qi+1 ) − Fim (qi−1 ))
m→∞
if the limit exists.
Note, that the tangential derivative may not always exist. However, later
on, we will have a theorem confirming its existence under suitable conditions.
One of the most important equations which is a very powerful tool for our
theory is that of the Gauss-Green formula.
10
Theorem 1.7. Suppose u and v are in dom∆µ . Then
Z
Z
X
(∆µ v)udµ =
(u∂n v − v∂n u).
(∆µ u)vdµ −
K
K
V0
If we choose v = 1 in the above formula, then we get that
Z
X
(∆µ u)dµ =
∂n u.
K
V0
Another version of the formula is also
Z
X
(∆µ u)vdµ +
E(u, v) = −
v∂n u.
K
V0
We also note a consequence of the formula. If u ∈ dom∆µ and ∂n u(x) = 0
for every x ∈ V? then u is constant.R To see this, by
Pchoosing v = 1 in the
Gauss-Green formula we have that Fw K ∆µ udµ = v0 ∂n u = 0 for all cells
Fw K which imples ∆µ = 0 and thus u is harmonic. But we know how to
compute the normal derivatives of harmonic functions, and we don’t get all
zeros unless the function is constant.
A big scope of the theory is to provide solutions to the “differential equation” ∆µ u = f for a continuous f . To derive a solution we create a special
function called Green’s function.
Definition 1.8. Green’s function is defined by
G(x, y) = lim GM (x, y)
M →∞
with GM (x, y) defined by
GM (x, y) =
M
X
m=0
(m+1)
X
z 0 ∈V
g(z, z 0 )ψz(m+1) (x)ψz 0
(y)
m+1 \Vm
where g(z, z 0 ) = 0 when z and z 0 are not in the same cell of level m + 1,
g(z, z) =
and
9 3 m
50 5
for z ∈ Vm+1 \ Vm
3 3 m
g(z, z ) =
50 5
0
for z 6= z 0 , z and z 0 in the same level m + 1 cell.
11
Then, now that we have defined Green’s function, we arrive at this very important result that gives us existence of a huge class of functions in dom∆µ .
Using the Green’s function is also the main tool we use for solving differential
equations on fractals.
Theorem 1.9. On the Sierpinski Gasket, the Dirichlet problem
−∆µ u = f , u|V0 = 0
has a unique solution in dom∆µ for any continuous f , given by
Z
u(x) =
G(x, y)f (y)dµ(y)
K
where G(x, y) is the Green’s function. If we don’t have Dirichlet boundary
conditions then the solution is given by
Z
u(x) =
G(x, y)f (y)dµ(y) + h(x)
K
where h(x) is a harmonic function with the same boundary values as u.
We have already seen that a function h such that ∆µ h = 0 is called
harmonic and the space of those functions is H0 . We can also define spaces
of multiharmonic functions in the same way by defining
Hk = {u | ∆k+1
µ u = 0}
for k = 0, 1, 2, ...
Then this space has dimension 3k + 3 and we are interested in creating a
basis. There have been many different construction of basis but one way
of doing it that is particularly interesting is creating “monomials” that are
j
the analogue of the functions { xj! } on the real line. These monomials are
also centred around a specific boundary point with the ulterior motive of
creating Taylor series.
Definition 1.10. We define the monomials {Pki ∈ Hk } to have the k-jet
consisting of all 0’s except for one 1:
∆j Pki (q0 ) = δjk δi1
∂n ∆j Pki (q0 ) = δjk δi2
∂T ∆j Pki (q0 ) = δjk δi3
for j 6 k.
12
From that we can observe that this means ∆Pki = P(k−1)i and thus we
can recursively find Pki by
Z
Pki (x) = −
G(x, y)P(k−1)i (y)dµ(y) + h(x)
K
for a harmonic function h(x) defined by the j = 0 case.
Definition 1.11. Define as follows
αj = Pj1 (q1 ), βj = Pj2 (q1 ), γj = Pj3 (q1 )nj = ∂n Pj1 (q1 ), tj = ∂T Pj2 (q1 ).
Note that by symmetry we have Pj1 (q2 ) = αj , Pj2 (q2 ) = βj and Pj3 (q2 ) =
−γj , so that all values of monomials at boundary points are expressible in
terms of α’s, β’s and γ’s.
Then, in order to obtain some decay rates for the monomials, we use the
following relations
Lemma 1.12. The following recursion relations hold:
j−1
αj =
4 X
αj−` α`
j
5 −5
for
j≥2
`=1
γj =
j−1
X
4
5j+1 − 5
αj−` γ`
for
j≥1
`=0
j−1
1 X 2 j−`
2
4
βj = j
5 αj−` β` − αj−` 5` β` + αj−` β`
5 −1
5
3
5
for
j ≥ 1,
`=0
with initial data α0 = 1, α1 = 1/6, β0 = −1/2, γ0 = 1/2. In particular,
γj = 3αj+1 .
Lemma 1.13. There exists a constant c such that
0 < aj < c(j!)−log5/log2
for all j.
Then by using the above two lemmas we get the following main theorem.
13
Theorem 1.14. (i) For any r < ∞ there exists cr such that
kPj1 k∞ 6 cr r−j
or more precisely
1
kPj1 k∞ = −∞.
j→∞ j
(ii) There exists c such that
lim
kPj2 k∞ 6 cλ2 −j
and
lim −λ2 j Pj2 = φ
j→∞
where φ is a λ2 -Neumann eigenfunction of ∆ which is R0 -symmetric and
vanishes ok F0 K, the limit existing uniformly and in energy.
The proofs of the above lemmas and theorem are quite lengthy and detailed
and can be found at [3]. Having found these rates for the monomials, a
theory of Taylor series can be created. It should also be noted, that the
polynomials, i.e the sum of the monomials Pij , or equivalently solutions of
∆nµ P = 0, behave differently from the standard polynomials we are used to
in analysis, and many key results that are true in standard analysis, do not
hold here. A key example is the Stone-Weierstrass theorem. Polynomials
cannot uniformly approximate all continuous functions.
To see this, first we must note a very interesting and curious fact about
the Laplacian on SG. As usual in analysis we define eigenfunctions and
eigenfunction in the natural way as solutions of
∆u = λu
and the Dirichlet boundary conditions if the boundary vanishes, while the
Neumann boundary conditions if the normal derivatives vanish at the boundary. A striking difference however between standard analysis and analysis on
fractals is that in standard analysis we cannot have joint Dirichlet-Neumann
eigenfunctions. However, surprisingly, in this theory of Laplacian on fractals
we can have certain eigenfunctions also seen as “localized eigenfunctions”
that satisfy both Dirichlet and Neumann conditions.
Then, let u be a joint Dirichlet-Neumann eigenfunction. Thus if P (x) is a
polynomial we have that
Z
P (x)u(x)dµ = 0
K
14
since by the definition of the monomials we have that ∆Pki = P(k−1)i and
thus we can use repeatedly the Gauss-Green formula to reduce the order of
the polynomial since due to the joint D − N conditions we have that all the
terms u(qi )=0 and ∂n u(qi ) = 0.
15
2
Energy measures and the Kusuoka measure
While with the standard measure we know many results concerning the
behavior of functions on the Sierpinski gasket, we see that the standard
measure has also many drawbacks. Namely, the domain of its Laplacian
is not an algebra. To overcome this we create a different measure called
the Kusuoka measure. Many results that are known about the standard
measure still remain open if we use the Kusuoka measure instead. To define
the Kusuoka measure we need to define first the energy measures. Define a
measure νu by
νu (Fw K) = r−|w| E(u ◦ Fw ).
This is called the energy measure νu . For energy measures and u, v ∈ domE
we have the important carré du champs formula
Z
1
1
1
f dνu,v = E(f u, v) + E(u, f v) − E(f, uv).
2
2
2
K
Definition 2.1. Let {h1 , h2 } be an orthonormal basis for the space of harmonic functions modulo constants with respect to the energy inner product.
Then the Kusuoka measure is defined as
ν = νh1 + νh2
Proposition 2.2. The Kusuoka measure is independent on the choice of
the orthonormal basis.
Proof. Let {h, h0 } be an orthonormal basis for the Kusuoka measure and
let {h1 , h2 } be a different one such that ν 0 = νh1 + νh2 . Then there exist
a, b, c, d such that h1 = ah + bh0 and h2 = ch + dh0 . Because of the change
of basis we have that the matrix
a b
M=
c d
is a rotation matrix and thus we have the properties
a2 + c2 = 1, b2 + d2 = 1, ab + cd = 0.
Then, we see that
16
ν 0 (Fw K) = r−|w| (E(h1 ◦ Fw ) + E(h2 ◦ Fw )
= r−|w| (a2 E(h ◦ Fw ) + b2 E(h0 ◦ Fw )
+c2 E(h ◦ Fw ) + d2 E(h0 ◦ Fw )
+abE(h ◦ Fw , h0 ◦ Fw ) + cdE(h ◦ Fw , h0 ◦ Fw ))
= νh (Fw K) + νh0 (Fw K)
We can take as basis h1 = √12 (0, 1, 1) and h2 = √16 (0, 1, −1) to be the
orthogonal harmonic functions defining the Kusuoka measure.
An equivalent definition for the Kusuoka measure would be to define it as
ν 0 = νh0 + νh1 + νh2
but in this case we get that ν 0 = 3ν with respect to the previous definition.
Now, with this new measure, by exactly the same definition as before we
create a different Laplacian, the Kusuoka Laplacian ∆ν which has different
properties. A very important result is that every energy measure is absolutely continuous with respect to the Kusuoka measure. In fact the Kusuoka
measure is also singular with respect to the standard measure.
Now, in [7], a pointwise formula is obtained for the Kusuoka Laplacian.
Proposition 2.3. Let u ∈ dom∆ν . Then for all x ∈ V∗ V0 the following
pointwise formula holds with uniform limit across V∗ V0
∆ν u(x) = 2 lim
m→∞
∆m u(x)
∆m (h1 2 + h2 2 )(x)
Proof. First of all, we have already mentioned before that the pointwise
formula for any measure is
Z
−1
−m
(m)
∆µ u(x) = lim r
ψx dµ
∆µ u(x).
m→∞
K
R
−1
(m)
To compute the K ψx dµ
where µ is now the Kusuoka measure, we
use the carré du champs formula and thus we have
Z
1
1
ψx(m) dν = E(ψx(m) h1 , h1 )+E(ψx(m) h2 , h2 )− E(ψx(m) , h1 2 )− E(ψx(m) , h2 2 ).
2
2
K
17
But we have that
E(ψx(m) h1 , h1 ) = E0 (ψx(m) h1 , h1 ) = 0
E(ψx(m) h2 , h2 ) = E0 (ψx(m) h2 , h2 ) = 0.
And thus we have that
Z
1
ψx(m) dν = − E(ψx(m) , h1 2 + h2 2 ) = r−m ∆m (h1 2 + h2 2 ).
2
K
Concluding, we get that
∆ν u(x) = lim r
m→∞
−m
Z
ψx(m) dµ
−1
∆m u(x) = 2 lim
m→∞
K
∆m u(x)
∆m (h1 2 + h2 2 )(x)
It would be interesting to try to see how similar dom∆ and dom∆ν are.
The following theorem shows us that they are quite different and in fact
they only coincide on the space of harmonic functions.
Theorem 2.4. dom∆ ∩ dom∆ν = H0
Similarly as before with the standard measure, we can create polynomials
Pij that are the basis for multiharmonic functions for the Kusuoka measure.
However, while the decay rates kPij k∞ are known for the polynomials of
the standard measure, it remains an open problem to find estimates for the
Kusuoka polynomials. In fact, numerical estimation shows that the decay
rates are different. As for a self-similar identity, the Kusuoka measure is
not self-similar in the way the standard measure is. However, we have the
following result.
Proposition 2.5. The Kusuoka measure satisfies the variable self-similar
identity
2 X
1
12
+ Ri ν ◦ Fi −1
15 15
i=0
where Ri =
dνi
dν
the Radon-Nikodym derivative of νi .
18
Using this relation a scaling identity for the Kusuoka Laplacian can be
derived. We will introduce the scaling identity in section 4, where we will
also make extensive use of it. Perhaps the single most important advantage
of the Kusuoka Laplacian over the standard one is that the domain of the
Kusuoka Laplacian forms an algebra. This is a key fact that is not true for
the standard Laplacian. A proof will be given for that in the next section by
evaluating the properties of the decay rates of functions. Thus, for example
if u ∈ dom∆ then u2 ∈
/ dom∆. However if u ∈ dom∆ν then u2 ∈ dom∆ν
and
dνu
∆ν u2 = 2u∆ν u + 2
.
dν
This advantage makes it clear that the Kusuoka Laplacian is one that is
worth studying and perhaps despite its apparent disadvantages due to the
lack of self-similarity, in some sense is better behaved than the standard one.
However, we have also have the following theorem which shows us that the
Theorem 2.6. Let h be harmonic function with νh = aν0 + bν1 + cν2 and
h0 be a harmonic function orthonormal to h under the energy inner product.
Then, if C is a cell on K:
i) inf
x∈C
ii) sup
x∈C
dνh
=0
dν
dνh
2
= (a + b + c)
dν
3
However we have some form of limited continuity, if we restrict the derivative to the set of vertices V? then it is continuous on the edges of every
triangle.
19
3
Local behavior of functions in dom∆µ
Now, in this section, we would like to turn our attention to some results concerning the local behavior of functions and namely their rate of convergence
to junction points. Let qi any boundary point and define
εim (u) = sup |u(x) − u(qi )|.
Fim K
We would like to get some results about how fast it is decaying to zero.
We will split our analysis into two parts. First with the standard measure
and next with the Kusuoka measure.
For the standard measure, we have the following:
Lemma 3.1. Let u ∈ dom∆ and consider any (m-1)-cell with boundary
vertices y0 , y1 , y2 and let x0 , x1 , x2 ∈ Vm Vm−1 be the vertex in that cell
with xj opposite yj . Then
2
1
2 1 6
2
2
u(x2 ) = (u(y0 )+u(y1 ))+ u(y2 )+ m ( ∆u(x2 )+ ∆u(x1 )+ ∆u(x0 ))+Rm
5
5
35 5
5
5
and so on with Rm = o(5−m )
Theorem 3.2. Let u ∈ dom∆. If ∂n u(q0 ) 6= 0 then
m
m
3
3
6 εm 6 c2
c1
5
5
while if ∂u(q0 ) = 0 then
εm 6 cm5−m
Proof. Without loss of generality assume that u(q0 ) = 0. First, we will prove
estimates for u(F0m q1 ) and u(F0m q2 ) and then show that these estimates
transfer to the entire cell. Formula (2.4.9) in [4] gives us that
m
m
1
3
m
m
2u(q0 ) − u(F0 q1 ) − u(F0 q2 ) =
∂n u(q0 ) + O
.
5
5
Now, in our case this gives
u(F0m q1 )
+
u(F0m q2 )
m
m
3
1
∂n u(q0 ) + O
.
=−
5
5
20
Now, by using the lemma above, and subtracting at the points F0m (q1 ) and
F0m (q2 ) we obtain that
1
u(F0m q1 ) − u(F0m q2 ) = (u(F0m−1 q1 ) − u(F0m−1 q2 )) + O(5−m ).
5
This is a recursion relation for the difference, and it is easy to see that this
implies
|u(F0m q1 ) − u(F0m q2 )| 6 cm5−m .
Then, if ∂n u(q0 ) 6= 0 we get that
m
m
3
3
m
6 |u(F0 qj )| 6 c2
c1
5
5
while if ∂n u(q0 ) = 0 we get that |u(F0m qj )| 6 cm5−m
Now, to see that these estimates transfer to the entire cell. We will use a
generic argument here. We recall first the scaling identity for the Laplacian
∆(u ◦ Fw ) = 5−m f ◦ Fw .
Then, if u ∈ dom∆ and ∆u = f and Fw K is an m-cell, |w| = m, and we
also have that
|u(Fw qi )| 6 a for i = 0, 1, 2
then we can write u ◦ Fw = h + g where h is the harmonic function taking
the same boundary values as u ◦ Fw and
Z
−m
G(x, y)f (Fw y)dµ(y).
g(x) = −5
Then by the maximum principle for harmonic functions we have that |h| 6 a
and we also have that
|g(x)| 6 c0 kf k∞ 5−m
R
where c0 = sup G(x, y)dµ(y) So we obtain
y∈K
|u| 6 a + c0 kf k∞ 5−m
on Fw K.
We know that the normal derivatives always exist. However while it is
not true in general for the tangential ones, we have the following sufficient
condition.
21
Theorem 3.3. Assume u ∈ dom∆ and ∆u satisfies a Hölder condition of
some order. Then ∂T u(q0 ) exists.
Using the above decay rates, we can also prove the very important result which is the main weakness of the standard Laplacian, and that is the
domain of the Laplacian does not form an algebra.
Corollary 3.4. Let u be any nonconstant function in dom∆. Then u2 is
not in dom∆.
Proof. Let x0 be a junction point such that ∂n u(x0 ) 6= 0. We can always
find such a point since we assumed u is non constant. Then we can write
u(x0 ) = y. Thus u = (u − y) + y and thus u2 = (u − y)2 + 2y(u − y) + y 2 .
Obviously 2y(u − y) + y 2 ∈ dom∆ and thus we must show that (u − y)2 ∈
/
dom∆. By localizing the decay rates we have that ∂n (u − y)(x0 ) 6= 0 and
thus by squaring the decay rates we have
2m
2m
2 3
2 3
(c1 )
6 ε̃m 6 (c2 )
5
5
where
ε̃m = sup |u − y|
in the m-cells containing x0 .
Then by assuming that (u − y)2 is in dom∆ we have a contradiction since
these decay
rates are impossible to be also compatible with the decay rates
3 m
of Θ 5 .
Now, we would like to prove similar results for the Kusuoka measure. We
would like to have an estimate of how the Kusuoka measure changes as we
are zooming in on individual cells. A simple first result is to study how it
changes by zooming in one direction.
Lemma 3.5. For the Kusuoka measure we have that:
m m
3
1
m
+
ν(F0 K) =
5
15
Proof. It is easy to see by the definition of the energy measures that
m
3
νh1 (F0m K) =
5
and that
νh2 (F0m K)
=
22
1
15
m
.
Then, since ν = νh1 + νh2 we get the result.
However this result has been significantly strengthened in [2] to have that
for an arbitrary junction point, and thus we have the following lemma.
Lemma 3.6. For the sequence {Fw Fim K} which converges to the point
Fw (qi ) we have that
m
3
m
ν(Fw Fi K) = Θ
5
To obtain some results about the decay rates, we will need some results
about the Green’s function. The following theorem gives us the necessary
tools. The proof of the theorem can be found in [5]
R
Theorem 3.7. If φ(x) = K |G(x, y) − G(Rx, y)|dν(y) where R is the reflection about q0 . Then,
2m
3
m
φ(F0 q1 ) = Θ
.
5
If ξ(x) = supy∈K |G(x, y)|, then
ξ(F0m q1 ) = Θ
m
3
5
Now we are ready to prove important results for the decay rates of functions.
Theorem 3.8. If u is skew-symmetric, u ∈ dom∆ν , then
2m
3
εm (u) = O
5
Proof. Write
Z
u(x) =
G(x, y)f (y)dν(y) + h1 (x) = ũ(x) + h1 (x)
K
where f = ∆ν (u) is a continuous function and h1 is a harmonic function
taking the same boundary values as u. Note that u is skew-symmetric
implies h1 is skew-symmetric and hence
m
1
.
h1 = Θ
5
23
On the other hand,
Z
ũ(x) = | G(x, y)f (y)dν(y)|
K
Z
1
= | (G(x, y) − G(Rx, y)f (y)dν(y)|
2 K
Z
1
|(G(x, y) − G(Rx, y)|dν(y).
6 kf k∞
2
K
2m
By the theorem above we have sup∂(F0m K) ũ = O 35
. To extend the
m
m
estimate to the entire cell, we fix F0 K and write ũ ◦ F0 = g̃ + h2 , where h2
is the harmonic function taking the same boundary values as ũ ◦ F0m . Then,
we have
m m Z
m Z
3
3
1
m
m
g̃(x) = −
G(x, y)f (F0 y)dνh (y) +
G(x, y)f (F0 y)dνh0 (y)
5
5
15
K
K
where h and h0 are the harmonic functions used in the definition of the
Kusuoka measure. Since f is continuous, we can regard the integrals as
O(1) so,
2m
3
g̃(x) = O
.
5
m
2m
Together with the fact that h2 = Θ 51
= O 35
we are done.
Theorem 3.9. Let u ∈ dom∆ν , u(q0 ) = 0, then
m
3
εm (u) = O
5
Proof. Similar as before, we can extend the results to the entire cell, so we
will estimate only on the boundary. We have that
Z
u(x) =
G(x, y)∆ν u(y)dν(y) + h1 (x) = ũ(x) + h1 (x)
K
and h1 is harmonic taking the same boundary values as u. Then we have
that since h1 (q0 ) = 0 it must be a linear combination of the orthonormal
m
basis of harmonic functions for the Kusuoka measure,
so h1 = O 35 . Also,
m
and thus we are done.
by the theorem above we have that |ũ| = O 35
24
In the same way that we have a sufficient condition for the existence of the
tangential derivative for the standard Laplacian, we have a slightly different
one also for the Kusuoka one.
Proposition
3.10. If u is skew-symmetric and u ∈ dom∆ν 2 then εm (u) =
1 m
O 5
and ∂T u(q0 ) exists.
Using the following lemma, we can also obtain results about more general
functions.
Lemma 3.11. If u is symmetric, u ∈ dom∆ν and u(q0 ) = 0 then
m
m Z
3
3
2u(q0 ) = −
∂n u(q0 ) +
ψq(m)
∆ν udν
0
5
5
F0 K
Theorem 3.12. Let u be symmetric and u(q0 ) = 0 and u ∈ dom∆ν k+1 for
k = 1, 2, 3... with ∂n ∆ν j u(q0 ) = 0 and ∆ν j+1 u(q0 ) = 0 for j < k. Then
(2k+1)m
3
εm (u) = O
5
Proof. The estimate will be done similarly as before only on the boundary.
The proof is one of induction. For k = 1 we have that ∂n u(q0 ) = 0 and
∆ν u ∈ dom|∆ν and that ∆ν u(q0 ) = 0. Then by the above lemma we have
that
m
1
3
m
|u(F0 q1 )| 6 εm (∆ν u)
ν(F0m K).
2
5
By using then the decay rates above to estimate εm (∆ν u) we are done. The
induction is based on the above lemma.
Now, we will generalize the results in [5] to any random junction point.
We will see that the convergence bounds obtained for the boundary points
are exactly the same for all junction points of the Sierpinski Gasket.
Lemma 3.13. Let y be a junction point such that y = Fw (qi ). For any u
in the domain of the Laplacian, we have that
sup |u(x) − u(y)| = sup |u ◦ Fw (x) − u ◦ Fw (qi )|
Fw Fim K
Fim K
for any word w and m ∈ N.
25
Proof. Let w be a word and u be in the domain of the Laplacian. Then we
know that u is continuous. We have that Fi are continuous functions, then
Fw are also continuous, and since K is a compact set, then u ◦ Fw has a
maximum and minimum value on u ◦ Fw (K) which is also compact. Then,
we have that there exists a y1 ∈ Fw Fim K such that:
sup |u(x) − u(y)| = |u(y1 ) − u(y)|.
Fw Fim K
But then, since y1 ∈ Fw Fim K it must be that there exists a y2 ∈ K such
that y1 = Fw Fim (y2 ) So,
|u(y1 ) − u(y)| = |u(Fw Fim (y2 )) − u(y)| = |(u ◦ Fw )(Fim (y2 )) − u(y)|
6 sup |(u ◦ Fw )(K) − u(y)|.
Fim K
Using the exact same argument, we obtain the other inequality as well.
Lemma 3.14. Define εim (u) = supFim K |u(x) − u(qi )|, for i=0,1,2. Then,
m
εim (u) 6 O 35
Proof. Let R be the clockwise rotation around the center point of the Sierpinski gasket by 90 degrees. Namely, if (x0 , y0 ) is the center of the Sierpinski
gasket, then R(x, y) = (x0 + (y − y0 ), y0 − (x − x0 )). Then, obviously R is
a continuous function, and (R ◦ F1 )(K) = F0 K. Now, let u ∈ dom∆. We
have that
sup |u(x)| = sup |u ◦ R−1 ◦ R(x)| = sup |(u ◦ R−1 ) ◦ R(x)|.
F1m K
F1m K
F1m K
But if we call g = u ◦ R−1 then we have that g is continuous, F1m K is
compact, and thus we have
sup |g(R(x))|
F1m K
is actually attained by say, y1 ∈ F1m K and thus
sup |g(R(x))| = |g(R(y1 ))|.
F1m K
26
But then, R(y1 ) ∈ F0m K so
sup |g(R(x))| = |g(R(y1 ))| 6 sup |g(x)|.
F1m K
F0m K
But since g ∈ dom∆ we have that supF0m K |g(x)| 6 O
m
3
.
sup |u(x)| 6 O
5
F1m K
3 m
.
5
So,
In an exact similar way, but instead with rotation anticlockwise, we obtain
that
m
3
sup |u(x)| 6 O
5
F2m K
Proposition 3.15. The rate of convergence of a function in thedomain of
m
the Laplacian to an arbitrary junction point, is bounded by O 35
Proof. Define
εim (u) = sup |u(x) − u(qi )|.
Fim K
Now, let y be any junction point that is not in V0 . Then, it is known that
y has two addresses, namely y = Fw (qi ) and y = Fw0 (qi0 ) with w 6= w0 and
i 6= i0 . Then, let {Am }m be a sequence of cells with {Am }m → y. Define,
ε̃m (u) = supAm |u(x) − u(y)|. Then, it is clear that
ε̃m (u) 6
sup |u(x) − u(y)| +
sup
|u(x) − u(y)|
Fw0 Fim
0 K
Fw Fim K
(Since, it is true that {Fw Fim K} → y and {Fw0 Fim
0 K} → y). Then, using
the Lemma above, we have that
ε̃m (u) 6 sup |(u ◦ Fw (x)) − u ◦ Fw (qi )| + sup |(u ◦ Fw0 (x)) − u ◦ Fw0 (qi0 )|.
Fim K
Fim
0 K
But, if we call now u ◦ Fw (x) = g(x) and u ◦ Fw0 (x) = h(x) then it is clear
that both g and h belong in the domain of the Laplacian. Thus,
ε̃m (u) 6
εim (g)
+
0
εim (h)
27
m
3
6O
.
5
4
Local Solvability of Differential equations for the
energy Laplacian
In this section we are interested in studying the solvability of differential
equations of the form
∆µ u = f.
We have already seen that due to the existence of the Green’s function
we have solutions. However what is of particular interest is that we have
solvability for other certain subsets of K.
Theorem 4.1. Let Ω be an open subset of K not containing any points of
V0 . Then the equation
−∆µ u = f on Ω
has a solution for any continuous f on
Ω.
This result is independent of the measure µ and the proof can be found
in [8]. Of course the solution is not unique since we can always add an
harmonic function on Ω.
Now, we are interested in the analogue of Picard existence and uniqueness
theorem for the local solvability of
−∆ν u(x) = F (x, u(x)).
We already have results for the standard Laplacian in [8]. We are interested
in extending them for the Kusuoka Laplacian as well. We will follow the
proof of [8] in section 2 “Local Solvability” making the appropriate changes
that are required by using ∆ν instead of ∆.
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 < ∞ such that
|F (x, u) − F (x, u0 )| 6 MT |u − u0 |, provided |u|, |u0 | 6 T.
Theorem 4.2. 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
−∆ν 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.
28
First of all, before we begin the proof, we note that there is a typo in [2].
At Theorem 2.3 the expression
Qj =
1
12 dνj
+
15 25 dν
Qj =
12 dνj
1
+
.
25 25 dν
should be replaced with
Then, we have the following scaling property for the energy Laplacian.
3
∆ν (u ◦ Fj ) = Qj (∆ν u) ◦ Fj
5
for Qj =
1
15
+
12 dνj
15 dν .
And using this, on page 8 of [2] we get that
m
3
∆ν (u ◦ Fw ) =
Qw (∆ν u) ◦ Fw
5
for w = (w1 , ..., wm ) a finite word of length m and Fw = Fw1 ◦ Fw2 ◦ · · · ◦ Fwm
and
Qw = Qwm · (Qwm−1 ◦ Fwm ) · (Qw−2 ◦ Fw−1 ◦ Fwm ) · · · (Qw1 ◦ Fw2 ◦ · · · ◦ Fwm ).
In the following, we will use the estimate from [2]
ν(Fim K)
m
3
=Θ
5
from which it is obvious that
ν(Fim K) = O
m
3
.
5
Now we are ready to prove the theorem.
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, the original equation becomes
−∆ν (u ◦ Fw )(x) =
m
3
Qw (x)F (Fw x, u ◦ Fw (x))
5
29
for x ∈ K. Let v = u ◦ Fw . Then, the equation along with the boundary
conditions above becomes
m
3
−∆ν v(x) =
Qw (x)F (Fw x, v(x)) on K
5
v|V0 = 0
which we can write in an equivalent way to
v(x) =
m Z
3
G(x, y)Qw (y)F (Fw y, v(y))dν(y).
5
K
Let Gv(x) to be
m Z
3
Gv(x) =
G(x, y)Qw (y)F (Fw y, v(y))dν(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 (kvk∞ 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)k∞ G0 FT
5
where FT is an upper bound for |F (x, u)| for x ∈ K and |u| 6 T First, to
bound kQw (y)k∞ we have that
kQw (y)k∞ = kQwm (y)·(Qwm−1 ◦Fwm y)·(Qw−2 ◦Fw−1 ◦Fwm y) · · · (Qw1 ◦Fw2 ◦· · ·◦Fwm y)k∞
6 kQwm · Qwm−1 · · · Qw1 k∞ 6 kQwm k∞ · kQwm−1 k∞ · · · kQw1 k∞ .
However, we have that for any j = 0, 1, 2 that
kQj k∞ = k
12 dνj
1
12 dνj
1
+
k∞ 6
+ k
k∞ .
15 15 dν
15 15 dν
Here, we will use the key result from [2], Theorem 3.5 page 10. This
theorem proves not only the discontinuity of the Radon-Nikodym derivative
but also gives us bounds on its values with respect to the energy measure
taken. In our case we see that the Radon-Nikodym derivative is bounded
by 23 . Then
30
kQj k∞ 6
1
12 2
3
+
· = .
15 15 3
5
2m
G0 FT so we
And thus, we have that kQw k∞ 6 ( 35 )m . Then, |G(x)| 6 35
3 2m
G0 FT 6 T to conclude
just have to take m large enough such that 5
0
that G maps the ball to itself. For v and v in this ball, we have that
2m
3
|Gv(x) − Gv (x)| 6
G0 MT kv − v 0 k∞
5
2m
so we get the contractive mapping estimate as long as 35
G0 MT < 1.
0
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)|Fw V0 = 0 and solves the equation
−∆ν 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 kuk∞ 6 A,
and we can apply the same argument as before.
31
5
Approximation by functions vanishing in a neighborhood of the boundary
Now, we will generalize some results found in [6] in section 7 “Spline cut-offs”
from the standard Laplacian to the Kusuoka Laplacian. It is proven in [10]
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 [6] 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 Kusuoka Laplacian
as well.
Lemma 5.1. For any u ∈ H1 we have that
k∆ν u|Fw K k∞
!
2m
m
5
5
6c
sup |u| +
sup |∂n u| .
3
3
∂Fw K
∂Fw K
Proof. Since ∆ν u is harmonic, we simply have to bound its values on ∂Fw K =
Fw ∂K. Now, for w equal to the empty word, the above estimate is an immediate consequence of the basis for H1 defined in [6]. The general case then
follows from the scaling
m property for the Kusuoka Laplacian, where we substitute kQw k∞ = 53
and the scaling property for the normal derivatives
as well.
Lemma 5.2. Let ∆ν be the Kusuoka Laplacian and f ∈ dom∆ν such that
it vanishes along with its normal derivatives on the boundary. Then
f|∂Fim K
2m
3
=O
5
and
∂n f
|∂Fim K
m
3
=O
.
5
Proof. For the first part:
First, let i = 0, 1, 2. Then,
1
1
u(x) = (u(x) + u(Ri x)) + (u(x) − u(Ri x))
2
2
32
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
by using theorem 3.1 and 3.6 from [5].
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
−
h∆f dν =
Fim K
X
f (Fim x)∂n h(Fim x) − h(Fim x)∂n f (Fim x).
x∈V0
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
mvertex
where h assumes the value -1. The integral on the left side is O 53
since
3 m
h and f are uniformly bounded and the
measure
is
O
and
the
terms
of
5
3 m
5 m
m
m
m
the form f (Fi x)∂n h(Fi x) are O 5
since ∂n h(Fi x) = O 3
and
f (Fim x)
2m
3
=O
5
by the first part of this lemma.
Using these two lemmas, we are ready to prove the following theorem.
Theorem 5.3. For the Kusuoka Laplacian on SG, suppose that f ∈ dom∆ν
and f vanishes together with its normal derivatives on the boundary. Then
there exists a sequence of functions {fm } with each fm ∈ domE vanishing in
a neighborhood of the boundary with fm → f uniformly, E(fm − f ) → 0 and
∆ν fm → ∆ν f in Lp (dν) for any p < ∞.
33
Proof. As in the proof of theorem 7.1 on [6], we choose fm so that fm = f
on Ωm with support in Ωm+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∆ν .
We will show that
Z
Fim K
|∆ν fm |p dν → 0 as m → ∞ for every p < ∞.
After this, the proof is identical to the standard Laplacian case since no
other differences between the standard and the energy Laplacian arise. We
will use the previous lemma. We have then that:
k∆ν fm |Fim K k∞
2m
m
5
5
6 c[
sup |f | +
sup |∂n f |]
3
3
∂Fim K
∂Fim K
and by using the lemmas above, we have that k∆ν fm |Fim K k∞ 6 C and thus
m
we get that
since ν(Fim K) = O 35
Z
|∆ν fm |p dν → 0
Fim K
Then, the proof for the following interesting Corollary on [6] is exactly
identical to the standard Laplacian case by using the above theorem and
can be found at [6].
Corollary 5.4. Let ∆ be the Kusuoka Laplacian on SG. If u ∈ L2 (dν) and
f ∈ L2 (dν) (respectively, f is continuous), and
Z
Z
u∆vdν =
f vdν
K
K
for all v ∈ domC (∆ν ) vanishing on a neighborhood of the boundary, then
u ∈ domL2 (∆ν ) (respectively domC (∆ν ) and ∆ν = f .
34
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