Analysis

17.4 Weak Derivatives

A related concept is that of weak derivatives. Let Ω ⊆ ℝⁿ be an open set. A distribution on Ω is defined to be a linear functional on C_c^∞

, called the space of test functions. The space of all such linear functionals will be denoted by D^∗. Actually, more is sometimes done here. One imposes a topology on C_c^∞ making it into a topological vector space, and when this has been done, D^′ is defined as the dual space of this topological vector space. To see this, consult the book by Yosida [32] or the book by Rudin [29].

Example: The space L_loc¹

may be considered as a subset of D^∗ as follows.

∫ f (ϕ) ≡ f (x)ϕ (x)dx Ω

for all ϕ ∈ C_c^∞

. Recall that f ∈ L_loc¹ if fX_K ∈ L¹ whenever K is compact.

Lemma 17.2.5 makes this identification possible. Recall that this said that if f ∈ L_loc¹

and for all ϕ ∈ C_c^∞

∫ fϕdmn = 0

then f

= 0 for a.e. x.

Example: δ_x ∈D^∗

where δ_x ≡ ϕ.

It will be observed from the above two examples and a little thought that D^∗

is truly enormous. We shall define the derivative of a distribution in such a way that it agrees with the usual notion of a derivative on those distributions which are also continuously differentiable functions. With this in mind, let f be the restriction to Ω of a smooth function defined on ℝⁿ. Then D_xif makes sense and for ϕ ∈ C_c^∞

∫ ∫ D f (ϕ) ≡ D f (x)ϕ (x)dx = − fD ϕdx = − f (D ϕ). xi Ω xi Ω xi xi

Motivated by this, here is the definition of a weak derivative.

Of course one can continue taking derivatives indefinitely. Thus,

( ) DxixjT ≡ Dxi Dxj T

and it is clear that all mixed partial derivatives are equal because this holds for the functions in C_c^∞

. Thus one can differentiate virtually anything, even functions that may be discontinuous everywhere. However the notion of “derivative” is very weak, hence the name, “weak derivatives”.

Example: Let Ω = ℝ and let

{ 1 if x ≥ 0, H (x) ≡ 0 if x < 0.

Then

∫ ′ DH (ϕ) = − H (x)ϕ (x)dx = ϕ(0) = δ0(ϕ).

Note that in this example, DH is not a function.

What happens when Df is a function?

The proof of Theorem 17.4.2 depends on the following lemma.

Proof: T

= 0 for all ϕ ∈ C_c^∞ from the definition of DT = 0. Let

∞ ∫ b ϕ0 ∈ C c (a,b), a ϕ0 (x)dx = 1,

and let

( ) ∫ x ∫ b ψϕ(x) = [ϕ(t)− ϕ (y) dy ϕ0 (t)]dt a a

for ϕ ∈ C_c^∞

. Thus ψ_ϕ ∈ C_c^∞ and

( ∫ b ) D ψϕ = ϕ− ϕ(y)dy ϕ0. a

Therefore,

( ) ∫ b ϕ = D ψϕ + ϕ (y)dy ϕ0 a

and so

( ∫ b ) ∫ b T (ϕ) = T (D ψϕ)+ ϕ(y)dy T(ϕ0) = T (ϕ0)ϕ(y)dy. a a

Let C = Tϕ₀. This proves the lemma.

Proof of Theorem 17.4.2 Since f and Df are both in L¹

,

∫ b Df (ϕ)− a Df (x)ϕ (x) dx = 0.

Consider

∫ (⋅) f (⋅)− Df (t)dt a

and let ϕ ∈ C_c^∞

.

( ∫ (⋅) ) D f (⋅) − Df (t)dt (ϕ) a

∫ b ∫ b( ∫ x ) ≡ − f (x)ϕ′(x)dx+ Df (t) dt ϕ′(x)dx a a a

By Lemma 17.4.3, there exists a constant, C, such that

( ∫ (⋅) ) ∫ b f (⋅)− Df (t) dt (ϕ) = C ϕ(x)dx a a

for all ϕ ∈ C_c^∞

. Thus

∫ b ( ∫ x ) { f (x)− Df (t)dt − C}ϕ(x)dx = 0 a a

for all ϕ ∈ C_c^∞

. It follows from Lemma 17.2.5 in the next section that

∫ x f (x)− Df (t)dt− C = 0 a.e. x. a

Thus we let f

= C and write

∫ f (x) = f (a)+ xDf (t) dt.■ a

Theorem 17.4.2 says that

∫ f (x ) = f (a)+ xDf (t)dt a

whenever it makes sense to write ∫ _a^xDf

dt, if Df is interpreted as a weak derivative. Somehow, this is the way it ought to be. It follows from the fundamental theorem of calculus that f^′ exists for a.e. x in the classical sense where the derivative is taken in the sense of a limit of difference quotients and f^′ = Df. This raises an interesting question. Suppose f is continuous on [a,b] and f^′ exists in the classical sense for a.e. x. Does it follow that

∫ x f (x) = f (a)+ f ′(t)dt? a

The answer is no. You can build such an example from the Cantor function which is increasing and has a derivative a.e. which equals 0 a.e. and yet climbs from 0 to 1. Thus this function is not recovered from integrating its classical derivative. Thus, in a sense weak derivatives are more agreeable than the classical ones.