Page 182 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Chapter 7

Solutions to the Exercises

7.1 Chapter 1: Preliminaries

7.1.1 Continuous and Hölder continuous functions
Exercise 1.2.1. (i) We have

kuvk C 0,α = kuvk C 0 +[uv] C 0,α .
Since
|u(x)v(x) − u(y)v(y)|
[uv] ≤ sup
C 0,α α
|x − y|
|v(x) − v(y)| |u(x) − u(y)|
≤ kuk 0 sup α + kvk 0 sup α
C C
|x − y| |x − y|
we deduce that
kuvk C 0,α ≤ kuk C 0 kvk C 0 + kuk C 0 [v] C 0,α + kvk C 0 [u] C 0,α
≤ 2 kuk C 0,α kvk C 0,α .
k
(ii) The inclusion C k,α ⊂ C is obvious. Let us show that C k,β ⊂ C k,α .We
will prove, for the sake of simplicity, only the case k =0.Observe that
( )
½ ¾
|u(x) − u(y)| |u(x) − u(y)|
sup α ≤ sup β 6 [u] C 0,β .
|x − y|
x,y∈Ω x,y∈Ω |x − y|
0<|x−y|<1 0<|x−y|<1
Since
½ ¾
|u(x) − u(y)|
sup α ≤ sup {|u(x)| − u(y)} ≤ 2 kuk C 0
x,y∈Ω |x − y| x,y∈Ω
|x−y|≥1
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