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84 Polynomial Approximation of Differential Equations
5.4 Weak derivatives
We recall that the space Ck ( Ī), where k ≥ 1 is an integer, is given by the functions
whose derivatives, of order less or equal to k, exist and are continuous in Ī. Following
section 5.3, when I is bounded, a norm for Ck ( Ī) is obtained by setting
(5.4.1) u C k (Ī) := u C 0 (Ī) +
k
d
mu dxm
m=1
C 0 (Ī)
, u ∈ C k ( Ī), k ≥ 1.
With this norm C k ( Ī), k ≥ 1, is a complete space. In particular, when u ∈ Ck ( Ī),
∀k ∈ N, we say that u ∈ C∞ ( Ī). We are mainly concerned with a particular subspace
of C∞ ( Ī). This is denoted by C∞ 0 (I) and consists of all the functions φ ∈ C∞ ( Ī) such
that there exists a closed and bounded interval strictly included in I, outside of which φ
vanishes. Therefore, since I is open, any function in C ∞ 0 (I) is zero in a neighborhood of
the possible boundary points of I. Therefore, all the derivatives of a function in C ∞ 0 (I)
still belong to C ∞ 0 (I).
Up to this point, we only considered subspaces of the set of continuous functions.
This is against the policy of expanding our functional spaces. We now give meaning
to the differential calculus of measurable functions by introducing the concept of weak
derivative. A Lebesgue integrable function u is differentiable in a weak sense, if there
exists another integrable function v such that
(5.4.2)
uφ ′
dx = −
I
vφ dx, ∀φ ∈ C
I
∞ 0 (I).
The function v is the weak derivative of u which is indicated using classical notation.
This definition is not in conflict with the usual one. Actually, we claim that, if u ∈ C 1 ( Ī),
then v ∈ C0 ( Ī) is exactly the standard derivative of u. To prove this statement, we just
integrate the left-hand side of (5.4.2) by parts. Recalling the vanishing conditions on φ,
we obtain as requested
(5.4.3)
(u
I
′ − v)φ dx = 0, ∀φ ∈ C ∞ 0 (I) =⇒ v ≡ u ′ .
The implication in (5.4.3) can be proven with a little knowledge of basic calculus.