In elementary courses in mathematics, functions are often thought of as things which
have a formula associated with them and it is the formula which receives the most
attention. For example, in beginning calculus courses the derivative of a function is
defined as the limit of a difference quotient. You start with one function which tends to
be identified with a formula and, by taking a limit, you get another formula for the
derivative. A jump in abstraction occurs as soon as you encounter the derivative of a
function of n variables where the derivative is defined as a certain linear transformation
which is determined not by a formula but by what it does to vectors. When this is
understood, it reduces to the usual idea in one dimension. The idea of weak
partial derivatives goes further in the direction of defining something in terms of
what it does rather than by a formula, and extra generality is obtained when
it is used. In particular, it is possible to differentiate almost anything if the
notion of what is meant by the derivative is sufficiently weak. This has the
advantage of allowing the consideration of the weak partial derivative of a function
without having to agonize over the important question of existence but it has the
disadvantage of not being able to say much about the derivative. Nevertheless, it is
the idea of weak partial derivatives which makes it possible to use functional
analytic techniques in the study of partial differential equations and it is shown
in this chapter that the concept of weak derivative is useful for unifying the
discussion of some very important theorems. Certain things which shold be true
are.
Let Ω ⊆ ℝ^{n}. A distribution on Ω is defined to be a linear functional on C_{c}^{∞}
(Ω)
,
called the space of test functions. The space of all such linear functionals will be denoted
by D^{∗}
(Ω )
. Actually, more is sometimes done here. One imposes a topology on C_{c}^{∞}
(Ω)
making it into a topological vector space, and when this has been done, D^{′}
(Ω )
is defined
as the dual space of this topological vector space. To see this, consult the book by Yosida
[?] or the book by Rudin [?].
Example: The space L_{loc}^{1}
(Ω)
may be considered as a subset of D^{∗}
(Ω)
as
follows.
∫
f (ϕ) ≡ f (x)ϕ(x)dx
Ω
for all ϕ ∈ C_{c}^{∞}
(Ω )
. Recall that f ∈ L_{loc}^{1}
(Ω )
if fX_{K}∈ L^{1}
(Ω)
whenever K is
compact.
Example:δ_{x}∈D^{∗}
(Ω)
where δ_{x}
(ϕ)
≡ ϕ
(x)
.
It will be observed from the above two examples and a little thought that D^{∗}
(Ω )
is
truly enormous. The derivative of a distribution will be defined 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 ℝ^{n}. Then D_{xi}f makes sense and for
ϕ ∈ C_{c}^{∞}
(Ω)
∫ ∫
D f (ϕ) ≡ D f (x)ϕ (x)dx = − fD ϕdx = − f (D ϕ).
xi Ω xi Ω xi xi
This motivates the following definition.
Definition 34.2.1For T ∈D^{∗}
(Ω)
D T (ϕ) ≡ − T (D ϕ ).
xi xi
Of course one can continue taking derivatives indefinitely. Thus,
DxixjT ≡ Dxi (DxjT)
and it is clear that all mixed partial derivatives are equal because this holds for the
functions in C_{c}^{∞}
(Ω)
. In this weak sense, the derivative of almost anything exists, 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) = δ(ϕ).
0
Note that in this example, DH is not a function.
What happens when Df is a function?
Theorem 34.2.2Let Ω =
(a,b)
and suppose that f and Df are both in L^{1}
(a,b)
. Thenf is equal to a continuous function a.e., still denoted by f and
∫ x
f (x) = f (a)+ Df (t)dt.
a
In proving Theorem 34.2.2 the following lemma is useful.
Lemma 34.2.3Let T ∈D^{∗}
(a,b)
and suppose DT = 0. Then there exists a constant Csuch that
∫ b
T (ϕ) = Cϕdx.
a
Proof:T
(Dϕ)
= 0 for all ϕ ∈ C_{c}^{∞}
(a,b)
from the definition of DT = 0.
Let
∫ b
ϕ ∈ C∞ (a,b), ϕ (x)dx = 1,
0 c a 0
and let
∫ x ( ∫ b )
ψϕ (x ) = [ϕ (t)− ϕ(y)dy ϕ0(t)]dt
a a
for ϕ ∈ C_{c}^{∞}
(a,b)
. Thus ψ_{ϕ}∈ C_{c}^{∞}
(a,b)
and
( )
∫ b
Dψ ϕ = ϕ − a ϕ (y)dy ϕ0.
Therefore,
(∫ b )
ϕ = Dψ ϕ+ ϕ (y)dy ϕ0
a
and so
( )
∫ b ∫ b
T (ϕ) = T (D ψϕ) + a ϕ(y)dy T(ϕ0) = a T (ϕ0)ϕ(y)dy.
Let C = Tϕ_{0}. This proves the lemma.
Proof of Theorem 34.2.2 Since f and Df are both in L^{1}
(a,b)
,
∫ b
Df (ϕ) − Df (x)ϕ (x)dx = 0.
a
Consider
∫ (⋅)
f (⋅) − Df (t)dt
a
and let ϕ ∈ C_{c}^{∞}
(a,b)
.
( ∫ (⋅) )
D f (⋅)− Df (t)dt (ϕ)
a
∫ b ∫ b( ∫ x )
≡ − f (x)ϕ′(x)dx + Df (t)dt ϕ′(x)dx
a a a
∫ ∫
b b ′
= Df (ϕ)+ a t Df (t)ϕ (x)dxdt
∫ b
= Df (ϕ)− Df (t)ϕ (t)dt = 0.
a
By Lemma 34.2.3, there exists a constant, C, such that
( )
∫ (⋅) ∫ b
f (⋅)− Df (t)dt (ϕ) = C ϕ(x)dx
a a
for all ϕ ∈ C_{c}^{∞}
(a,b)
. Thus
∫ b ( ∫ x )
{ f (x )− Df (t)dt − C}ϕ (x )dx = 0
a a
for all ϕ ∈ C_{c}^{∞}
(a,b)
. It follows from Lemma 34.3.3 in the next section that
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^{′}
(x)
exists for a.e. x where the derivative is taken in the sense of a limit of
difference quotients and f^{′}
(x)
= Df
(x)
. This raises an interesting question. Suppose f is
continuous on [a,b] and f^{′}
(x)
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. To see an example, consider Problem 4 on Page 3129 which gives an
example of a function which is continuous on
[0,1]
, has a zero derivative for a.e. x but
climbs from 0 to 1 on [0,1]. Thus this function is not recovered from integrating its
classical derivative.
In summary, if the notion of weak derivative is used, one can at least give meaning to
the derivative of almost anything, the mixed partial derivatives are always equal, and, in
one dimension, one can recover the function from integrating its derivative. None of these
claims are true for the classical derivative. Thus weak derivatives are convenient and rule
out pathologies.