Page 138 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Some general results 125

The proof of such a theorem relies on the De Giorgi-Nash-Moser theory. In
the course of the proof, one transforms the nonlinear Euler-Lagrange equation
into an elliptic linear equation with bounded measurable coefficients. Therefore
to obtain the desired regularity, one needs to know the regularity of solutions of
such equations and this is precisely the famous theorem that is stated below (see
Giaquinta [47] Theorem 2.1 of Chapter II). It was first established by De Giorgi,
then simplified by Moser and also proved, independently but at the same time,
by Nash.
Theorem 4.13 Let Ω ⊂ R n be a bounded open set and v ∈ W 1,2 (Ω) be a
solution of
n Z
X h i 1,2
(x) ϕ (x) dx =0, ∀ϕ ∈ W (Ω)
a ij (x) v x i 0
x j
i,j=1
Ω
where a ij ∈ L ∞ (Ω) and, denoting by γ> 0 aconstant,
n
X 2 n
a ij (x) λ i λ j ≥ γ |λ| ,a.e. in Ω and ∀λ ∈ R .
i,j=1
Then there exists 0 <α < 1 so that v ∈ C 0,α (D), for every D ⊂ D ⊂ Ω.

Remark 4.14 It is interesting to try to understand, formally, the relationship
between the last two theorems, for example in the case where f = f (x, u, ξ)=
f (ξ).The coefficients a ij (x) and the function v in Theorem 4.13 are, respec-
∈ W 1,2 is
(∇u (x)) and u x i in Theorem 4.11. The fact that v = u x i
tively, f ξ i ξ j
proved by the method of difference quotients presented in Theorem 4.9.
The two preceding theorems do not generalize to the vectorial case u : Ω ⊂
n
N
R −→ R ,with n, N > 1. In this case only partial regularity can, in general, be
proved. We give here an example of such a phenomenon due to Giusti-Miranda
(see Giaquinta [47] Example 3.2 of Chapter II).
n
Example 4.15 Let n,an integer, be sufficiently large, Ω ⊂ R be the unit ball
and u 0 (x)= x.Let
⎡ ⎤ 2
à !
n
n ³ ´ i j
X j 2 X 4 u u j
f (x, u, ξ)= f (u, ξ)= ξ + ⎣ δ ij + ξ ⎦
i 2 i
n − 2 1+ |u|
i,j=1 i,j=1
j j
where ξ stands for ∂u /∂x i and δ ij is the Kronecker symbol (i.e., δ ij =0 if
i
i 6= j and δ ij =1 if i = j). Then u (x)= x/ |x| is the unique minimizer of
½ Z ¾
1,2 n
(P) inf I (u)= f (u (x) , ∇u (x)) dx : u ∈ u 0 + W (Ω; R ) .
0
Ω