Page 181 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

168 Isoperimetric inequality

6.3.1 Exercises
Exercise 6.3.1 Let A, B ⊂ R be compact,

a =min {a : a ∈ A} and b =max {b : b ∈ B} .

Prove that
¡ ¢
(a + B) ∪ b + A ⊂ A + B
and deduce that
M (A)+ M (B) ≤ M (A + B) .

3
Exercise 6.3.2 Denote by A the set of bounded open sets A ⊂ R whose bound-
¢
¡
2
ary ∂A is the image of a bounded smooth domain Ω ⊂ R by a C 2 Ω; R 3 map
v, v = v (x, y),with v x × v y 6=0 in Ω.Denote by L (∂A) and M (A) the area of
the boundary ∂A and the volume of A respectively.
Show that if there exists A 0 ∈ A so that
L (∂A 0 )= inf {L (∂A): M (A)= M (A 0 )}
A∈A
then ∂A 0 has constant mean curvature.