Page 172 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Thecaseofdimension 2 159

We, however, know from Corollary 6.3 that
Z 1 Z 1
¡ 02 02 ¢ 1,2
0
ϕ + ψ dx ≥ 2π ϕψ dx, ∀ϕ, ψ ∈ W per (−1, 1)
−1 −1
which implies the claim
2 1,1 1
[L (u, v)] − 4πM (u, v) ≥ 0, ∀u, v ∈ W (a, b) ∩ C ([a, b]) .
per
The uniqueness in the equality case follows also from the corresponding one in
Corollary 6.3.
1
Step 2. Wenow removethehypothesis u, v ∈ W 1,1 (a, b) ∩ C ([a, b]).As
per
1
before, given u, v ∈ W 1,1 (a, b),we can find u ν ,v ν ∈ W 1,1 (a, b) ∩ C ([a, b]) so
per per
that
u ν ,v ν → u, v in W 1,1 (a, b) ∩ L ∞ (a, b) .
Therefore, for every > 0,wecan find ν sufficiently large so that
2 2
[L (u, v)] ≥ [L (u ν ,v ν )] − and M (u ν ,v ν ) ≥ M (u, v) −
and hence, combining these inequalities with Step 1, we get
2 2
[L (u, v)] −4πM (u, v) ≥ [L (u ν ,v ν )] −4πM (u ν ,v ν )−(1 + 4π) ≥− (1 + 4π) .
Since is arbitrary, we have indeed obtained the inequality.
We now briefly discuss the geometrical meaning of the inequality obtained
in Theorem 6.4. Any bounded open set A, whose boundary ∂A is a closed curve
1,1
which possesses a parametrization u, v ∈ W per (a, b) so that its length and area
are given by
Z
b p
02
02
L (∂A)= L (u, v)= u + v dx
a
Z b Z b
1
0
0
0
M (A)= M (u, v)= (uv − vu ) dx = uv dx
2 a a
will therefore satisfy the isoperimetric inequality
2
[L (∂A)] − 4πM (A) ≥ 0 .
This is, of course, the case for any simple closed smooth curve, whose interior is
A.
One should also note that very wild sets A can be allowed. Indeed sets A
that can be approximated by sets A ν that satisfy the isoperimetric inequality
and which are so that
L (∂A ν ) → L (∂A) and M (A ν ) → M (A) , as ν →∞
also verify the inequality.