Page 168 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Thecaseofdimension 2 155

then
1 1
Z Z
2
02
u dx ≥ π 2 u dx, ∀u ∈ X.
−1 −1
Furthermore equality holds if and only if u (x)= α cos πx + β sin πx, for any
α, β ∈ R.
Remark 6.2 (i) It will be implicitly shown below that Wirtinger inequality is
equivalent to the isoperimetric inequality.
(ii) More generally we have if
( )
Z
b
1,2
X = u ∈ W per (a, b): u (x) dx =0
a
that
2
Z b µ ¶ Z b
02 2π 2
u dx ≥ u dx, ∀u ∈ X.
a b − a a
(iii) The inequality can also be generalized (cf. Croce-Dacorogna [28]) to
à ! 1/p à ! 1/q
Z Z
b b
0 p
0 q
|u | dx ≥ α (p, q, r) |u | dx , ∀u ∈ X
a a
for some appropriate α (p, q, r) (in particular α (2, 2, 2) = 2π/ (b − a))andwhere
( )
Z b
X = u ∈ W 1,p (a, b): |u (x)| r−2 u (x) dx =0 .
per
a
(iv) We have seen in Example 2.23 a weaker form of the inequality, known
as Poincaré-Wirtinger inequality, namely
Z 1 Z 1
2
02
u dx ≥ π 2 u dx, ∀u ∈ W 0 1,2 (0, 1) .
0 0
This inequality can be inferred from the theorem by setting
u (x)= −u (−x) if x ∈ (−1, 0) .
Proof. An alternative proof, more in the spirit of Example 2.23, is proposed
in Exercise 6.2.1. The proof given here is, essentially, the classical proof of
Hurwitz. We divide the proof into two steps.
Step 1. We start by proving the theorem under the further restriction that
2
u ∈ X ∩ C [−1, 1].We express u in Fourier series
∞
X
u (x)= [a n cos nπx + b n sin nπx] .
n=1