Page 173 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

160 Isoperimetric inequality

6.2.1 Exercises
Exercise 6.2.1 Prove Theorem 6.1 in an analogous manner as that of Example
2.23.

Exercise 6.2.2 Let
½ Z 1 ¾
X = u ∈ W 1,2 (−1, 1) : u (x) dx =0
per
−1
and consider
1
½ Z ¾
¡ 02 2 2 ¢
(P)inf I (u)= u − π u dx : u ∈ X = m.
−1
We have seen in Theorem 6.1 that m =0 and the minimum is attained in
2
X ∩ C [−1, 1] if and only if u (x)= α cos πx + β sin πx,for any α, β ∈ R.
2
Show that these are the only minimizers in X (and not only in X∩C [−1, 1]).
2
Suggestion: Show that any minimizer of (P) is C [−1, 1].Conclude.
1
1,1
Exercise 6.2.3 Prove Step 1 of Theorem 6.4 for any u, v ∈ W per (a, b)∩C ([a, b]).
6.3 The case of dimension n
n
The above proof does not generalize to R , n ≥ 3. A completely different and
harder proof is necessary to deal with this case.
Before giving a sketch of the classical proof based on Brunn-Minkowski the-
n
orem, we want to briefly mention an alternative proof. The inequality L −
n
n ω n M n−1 ≥ 0 (L = L (∂A) and M = M (A)) is equivalent to the minimiza-
tion of L for fixed M together with showing that the minimizers are given by
spheres. We can then write the associated Euler-Lagrange equation, with a La-
grange multiplier corresponding to the constraint that M is fixed (see Exercise
6.3.2). We then obtain that for ∂A to be a minimizer it must have constant
mean curvature (we recall that a minimal surface is a surface with vanishing
mean curvature, see Chapter 5). The question is then to show that the sphere
is, among all compact surfaces with constant mean curvature, the only one to
have this property. This is the result proved by Aleksandrov, Hopf, Liebmann,
Reilly and others (see Hsiung [61], page 280, for a proof). We immediately see
that this result only partially answers the problem. Indeed we have only found
a necessary condition that the minimizer should satisfy. Moreover this method
requires a strong regularity on the minimizer.
We now turn our attention to the proof of the isoperimetric inequality. We
will need several definitions and intermediate results.