Page 38 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Sobolev spaces 25

Exercise 1.3.6 (i) Show Corollary 1.25.
(ii) Prove that if u ∈ L 1 (a, b) is such that
loc
b
Z
0
u (x) ϕ (x) dx =0, ∀ϕ ∈ C ∞ (a, b)
0
a
then u = constant, almost everywhere in (a, b).
n
1
Exercise 1.3.7 Let Ω ⊂ R be an open set and u ∈ L (Ω). Show that for every
> 0,there exists δ> 0 so that for any measurable set E ⊂ Ω
Z
meas E ≤ δ ⇒ |u (x)| dx ≤ .
E

1.4 Sobolev spaces

Before giving the definition of Sobolev spaces, we need to weaken the notion of
derivative. In doing so we want to keep the right to integrate by parts; this is
one of the reasons of the following definition.

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Definition 1.26 Let Ω ⊂ R be open and u ∈ L 1 (Ω). We say that v ∈ L 1 (Ω)
loc loc
is the weak partial derivative of u with respect to x i if
Z Z
∂ϕ
v (x) ϕ (x) dx = − u (x) (x) dx, ∀ϕ ∈ C ∞ (Ω) .
0
Ω Ω ∂x i
.
By abuse of notations we will write v = ∂u/∂x i or u x i
We will say that u is weakly differentiable if all weak partial derivatives,
,exist.
u x 1 ,..., u x n
Remark 1.27 (i) If such a weak derivative exists it is unique (a.e.), as a con-
sequence of Theorem 1.24.
(ii) All the usual rules of differentiation are easily generalized to the present
context of weak differentiability.
(iii) In a similar way we can introduce the higher derivatives.
1
(iv) If a function is C , then the usual notion of derivative and the weak one
coincide.
(v) The advantage of this notion of weak differentiability will be obvious when
defining Sobolev spaces. We can compute many more derivatives of functions
than one can usually do. However not all measurable functions can be differ-
entiated in this way. In particular a discontinuous function of R cannot be
differentiated in the weak sense (see Example 1.29).