Page 26 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Continuous and Hölder continuous functions 13

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(i) If u : R → R, u = u (x 1 , ..., x n ), we will denote partial derivatives by
either of the following ways
∂u
=
D j u = u x j
∂x j
µ ¶
∂u ∂u
∇u =grad u = , ..., =(u x 1 , ..., u x n ) .
∂x 1 ∂x n
(ii) We now introduce the notations for the higher derivatives. Let k ≥ 1 be
an integer; an element of
⎧ ⎫
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⎨ ⎬
X
A k = a =(a 1 , ..., a n ) ,a j ≥ 0 an integer and a j = k ,
⎩ ⎭
j=1
will be called a multi-index of order k. We will also write, sometimes, for such
elements
n
X
|a| = a j = k.
j=1
Let a ∈ A k , we will write

∂ |a| u
a
a 1
D u = D ...D u = .
a n
1 n a 1 a n
∂x ...∂x n
1
k
k
a
We will also let ∇ u =(D u) .In other words, ∇ u contains all the partial
a∈A k
0
1
derivatives of order k of the function u (for example ∇ u = u, ∇ u = ∇u).
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Definition 1.4 Let Ω ⊂ R be an open set and k ≥ 0 be an integer.
a
(i) The set of functions u : Ω → R which have all partial derivatives, D u,
k
a ∈ A m , 0 ≤ m ≤ k, continuous will be denoted by C (Ω).
Ω is the set of C (Ω) functions whose derivatives up to the order
(ii) C k ¡ ¢ k
k can be extended continuously to Ω. It is equipped with the following norm
a
kuk C k =max sup |D u (x)| .
0≤|a|≤k
x∈Ω
k
k
(iii) C (Ω)= C (Ω) ∩ C 0 (Ω).
0
∞ T ¡ ¢ ∞ T ¡ ¢
k
(iv) C ∞ (Ω)= C (Ω), C ∞ Ω = C k Ω .
k=0 k=0