Page 167 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

154 Isoperimetric inequality

weaker form (cf. Poincaré-Wirtinger inequality). This inequality reads as

1 1
Z Z
2
02
u dx ≥ π 2 u dx, ∀u ∈ X
−1 −1
n R 1 o
where X = u ∈ W 1,2 (−1, 1) : u (−1) = u (1) and udx =0 .It also states
−1
that equality holds if and only if u (x)= α cos πx + β sin πx, for any α, β ∈ R.
n
In Section 6.3 we discuss the generalization to R , n ≥ 3, of the isoperimetric
inequality. It reads as follows
n n n−1
[L (∂A)] − n ω n [M (A)] ≥ 0
n
for every bounded open set A ⊂ R with sufficiently regular boundary, ∂A;and
n
where ω n is themeasureof theunitballof R , M (A) stands for the measure
of A and L (∂A) for the (n − 1) measure of ∂A.Moreover, if A is sufficiently
regular (for example, convex), there is equality if and only if A is a ball.
The inequality in higher dimensions is considerably harder to prove; we will
discuss, briefly, in Section 6.3 the main ideas of the proof. When n =3,the
first complete proof was the one of H.A. Schwarz. Soon after there were gen-
eralizations to higher dimensions and other proofs notably by A. Aleksandrov,
Blaschke, Bonnesen, H. Hopf, Liebmann, Minkowski and E. Schmidt.
Finally numerous generalizations of this inequality have been studied in rela-
tion to problems of mathematical physics, see Bandle [9], Payne [83] and Polya-
Szegö [85] for more references.
There are several articles and books devoted to the subject, we recommend
the review article of Osserman [81] and the books by Berger [10], Blaschke [11],
Federer [45], Hardy-Littlewood-Polya [55] (for the two dimensional case) and
Webster [96]. The book of Hildebrandt-Tromba [58] also has a chapter on this
matter.

6.2 The case of dimension 2

We start with the key result for proving the isoperimetric inequality; but before
that we introduce the following notation, for any p ≥ 1,
ª
©
1,p
W per (a, b)= u ∈ W 1,p (a, b): u (a)= u (b) .
Theorem 6.1 (Wirtinger inequality). Let
½ Z 1 ¾
1,2
X = u ∈ W per (−1, 1) : u (x) dx =0
−1