Page 52 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Sobolev spaces 39

Exercise 1.4.2 Let AC ([a, b]) be the space of absolutely continuous functions
on [a, b].This means that a function u ∈ AC ([a, b]), if for every > 0 there
exists δ> 0 so that for every disjoint union of intervals (a k ,b k ) ⊂ (a, b) the
following implication is true
X X
|b k − a k | <δ ⇒ |u (b k ) − u (a k )| < .
k k
(i) Prove that W 1,1 (a, b) ⊂ AC ([a, b]) ⊂ C ([a, b]), uptothe usualselection
of a representative.
(ii) The converse AC ([a, b]) ⊂ W 1,1 (a, b) is also true (see Section 2.2 in
Buttazzo-Giaquinta-Hildebrandt [17] or Section 9.3 in De Barra [37]).

Exercise 1.4.3 Let u ∈ W 1,p (a, b), 1 <p < ∞ and a<y < x<b.Show that
³ ´
1/p 0
u (x) − u (y)= o |x − y|
where o (t) stands for a function f = f (t) so that f (t) /t tends to 0 as t tends
to 0.

Exercise 1.4.4 (Corollary 1.45 in dimension n =1). Prove that if 1 <p < ∞,
then
p
u ν u in W 1,p (a, b) ⇒ u ν → u in L (a, b)
and even u ν → u in L ∞ (a, b).
n
Exercise 1.4.5 Show that if Ω ⊂ R is a bounded open set with Lipschitz bound-
ary, 1 <p < ∞ and if there exists a constant γ> 0 so that
ku ν k W 1,p ≤ γ
} and u ∈ W 1,p (Ω) such that
then there exist a subsequence {u ν i
u in W 1,p .
u ν i
2
Exercise 1.4.6 Let Ω =(0, 1) × (0, 1) ⊂ R and
1 ν
u ν (x, y)= √ (1 − y) sin νx .
ν
∞
Prove that u ν → 0 in L , k∇u ν k L 2 ≤ γ, for some constant γ> 0.Deduce that
a subsequence (one can even show that the whole sequence) converges weakly to
0 in W 1,2 (Ω).
1
Exercise 1.4.7 Let u ∈ W 1,p (Ω) and ϕ ∈ W 0 1,p 0 (Ω) ,where 1 p + p 0 =1 and
p> 1. Show that
Z Z
ϕdx = − uϕ dx, i =1, ..., n .
u x i
x i
Ω Ω