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2.1. Variational Formulation 53

For instance, there exists Cantor’s function with the following properties:

f :[0, 1] → R, f ∈ C([0, 1]), f =0, f is not constant, f (x)exists with

f (x) = 0 for all x ∈ [0, 1].
x
Here the fundamental theorem of calculus, f(x)= f (s) ds+f(0), and

0
thus the principle of integration by parts, are no longer valid.
Consequently, additional conditions for ∂ i u are necessary.
To prepare an adequate definition of the space V, we extend the definition
of derivatives by means of their action on averaging procedures. In order
to do this, we introduce the multi-index notation.
A vector α =(α 1 ,...,α d ) of nonnegative integers α i ∈{0, 1, 2,...} is
d
called a multi-index.The number |α| := α i denotes the order (or
i=1
length)of α.
d
For x ∈ R let
α
x := x α 1 ··· x α d . (2.15)
1 d
A shorthand notation for the differential operations can be adopted by this:
For an appropriately differentiable function u let
α
∂ u := ∂ α 1 ··· ∂ α d u. (2.16)
1 d
We can obtain this definition from (2.15) by replacing x by the symbolic
vector
T
∇ := (∂ 1 ,...,∂ d )
of the first partial derivatives.
For example, if d =2 and α =(1, 2), then |α| =3 and
3
∂ u
2
α
∂ u = ∂ 1 ∂ u = .
2 2
∂x 1 ∂x
2
k
Now let α be a multi-index of length k and let u ∈ C (Ω). We then
∞
obtain for arbitrary test functions ϕ ∈ C (Ω) by integration by parts
0

α
α
∂ uϕ dx =(−1) k u∂ ϕdx.
Ω Ω
β
The boundary integrals vanish because ∂ ϕ =0on ∂Ω for all multi-indices
β.
Therefore, we make the following definition:
2
Definition 2.9 v ∈ L (Ω) is called the weak (or generalized) derivative
α
2
∂ u of u ∈ L (Ω) for the multi-index α if for all ϕ ∈ C (Ω),
∞
0

α
vϕ dx =(−1) |α| u∂ ϕdx.
Ω Ω