Page 170 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Thecaseofdimension 2 157

Corollary 6.3 The following inequality holds

1 1
Z Z
¡ 02 02 ¢ 1,2
0
u + v dx ≥ 2π uv dx, ∀u, v ∈ W per (−1, 1) .
−1 −1
Furthermore equality holds if and only if
2 2 2
(u (x) − r 1 ) +(v (x) − r 2 ) = r , ∀x ∈ [−1, 1]
3
where r 1 ,r 2 ,r 3 ∈ R are constants.

Proof. We first observe that if we replace u by u − r 1 and v by v − r 2 the
inequality remains unchanged, therefore we can assume that

Z 1 Z 1
udx = vdx =0
−1 −1
n R 1 o
1,2
and hence that u, v ∈ X = u ∈ W per (−1, 1) : −1 u (x) dx =0 .We write
the inequality in the equivalent form
Z Z Z
1 1 1
¡ 02 02 ¢ 2 ¡ 02 2 2 ¢
0
u + v − 2πuv 0 dx = (v − πu) dx + u − π u dx ≥ 0 .
−1 −1 −1
From Theorem 6.1 we deduce that the second term in the above inequality is
non negative while the first one is trivially non negative; thus the inequality is
established.
We now discuss the equality case. If equality holds we should have
Z 1
¡ 02 2 2 ¢
v = πu and u − π u dx =0
0
−1
which implies from Theorem 6.1 that

u (x)= α cos πx + β sin πx and v (x)= α sin πx − β cos πx .

Sincewecan replace u by u − r 1 and v by v − r 2 ,wehavethat
2 2 2
(u (x) − r 1 ) +(v (x) − r 2 ) = r , ∀x ∈ [−1, 1]
3
as wished.
We are now in a position to prove the isoperimetric inequality in its analytic
form; we postpone the discussion of its geometric meaning for later.