Page 171 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

158 Isoperimetric inequality

1,1
Theorem 6.4 (Isoperimetric inequality). Let for u, v ∈ W per (a, b)
Z
b p
L (u, v)= u + v dx
02
02
a
Z b Z b
1
M (u, v)= (uv − vu ) dx = uv dx .
0
0
0
2 a a
Then
2 1,1
[L (u, v)] − 4πM (u, v) ≥ 0, ∀u, v ∈ W per (a, b) .
1,1
1
Moreover, among all u, v ∈ W per (a, b) ∩ C ([a, b]), equality holds if and only if
2 2 2
(u (x) − r 1 ) +(v (x) − r 2 ) = r , ∀x ∈ [a, b]
3
where r 1 ,r 2 ,r 3 ∈ R are constants.
Remark 6.5 The uniqueness holds under fewer regularity hypotheses that we
do not discuss here. We, however, point out that the very same proof for the
1,1
uniqueness is valid for u, v ∈ W per (a, b) ∩ C 1 piec ([a, b]).
Proof. We divide the proof into two steps.
Step 1.We first prove the theorem under the further restriction that u, v ∈
1
W 1,1 (a, b) ∩ C ([a, b]). We will also assume that
per
02
02
u (x)+ v (x) > 0, ∀x ∈ [a, b] .
This hypothesis is unnecessary and can be removed, see Exercise 6.2.3.
We start by reparametrizing the curve by a multiple of its arc length, namely
⎧ Z
x √
⎪ 2 02 02
⎪ y = η (x)= −1+ u + v dx
⎨
L(u,v)
a
⎪
⎪ ¡ ¢ ¡ ¢
⎩ −1 −1
ϕ (y)= u η (y) and ψ (y)= v η (y) .
1
It is easy to see that ϕ, ψ ∈ W 1,2 (−1, 1) ∩ C ([−1, 1]) and
per
q L (u, v)
02
02
ϕ (y)+ ψ (y)= , ∀y ∈ [−1, 1] .
2
We therefore have
Z µ Z 1 ¶1/2
1 q £ ¤
02
02
02
02
L (u, v)= ϕ (y)+ ψ (y) dy = 2 ϕ (y)+ ψ (y) dy
−1 −1
Z 1
0
M (u, v)= ϕ (y) ψ (y) dy .
−1