Page 20 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Presentation of the content of the monograph 7

Furthermore, equality holds if and only if A is a disk (i.e., ∂A is a circle).
We can rewrite it into our formalism (here n =1 and N =2) by parametriz-
ing the curve
∂A = {u (x)= (u 1 (x) ,u 2 (x)) : x ∈ [a, b]}
and setting
Z
b q
02
02
L (∂A)= L (u)= u + u dx
1 2
a
Z b Z b
1
M (A)= M (u)= (u 1 u − u 2 u ) dx = u 1 u dx .
0
0
0
2
1
2
2 a a
The problem is then to show that
√
(P) inf {L (u): M (u)= 1; u (a)= u (b)} =2 π.
n
The problem can then be generalized to open sets A ⊂ R with sufficiently
regular boundary, ∂A,and it readsas
n n n−1
[L (∂A)] − n ω n [M (A)] ≥ 0
n
where ω n is the measure of the unit ball of 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.
0.3 Presentation of the content of the mono-
graph
To deal with problems of the type considered in the previous section, there are,
roughly speaking, two ways of proceeding: the classical and the direct meth-
ods. Before describing a little more precisely these two methods, it might be
N
enlightening to first discuss minimization problems in R .
N
Let X ⊂ R , F : X → R and
(P) inf {F (x): x ∈ X} .
The first method consists, if F is continuously differentiable, in finding solu-
tions x ∈ X of
0
F (x)= 0,x ∈ X.
Then, by analyzing the behavior of the higher derivatives of F, we determine if x
is a minimum (global or local), a maximum (global or local) or just a stationary
point.