Page 104 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

A general existence theorem 91

3.3.1 Exercises
Exercise 3.3.1 Prove Theorem 3.3 under the hypotheses (H1+), (H2) and (H3).

Exercise 3.3.2 Prove Theorem 3.3 if

f (x, u, ξ)= g (x, u)+ h (x, ξ)
where ¡ ¢
n
(i) h ∈ C 1 Ω × R , ξ → h (x, ξ) is convex for every x ∈ Ω,and thereexist
p> 1 and α 1 > 0, β, α 3 ∈ R such that
p
h (x, ξ) ≥ α 1 |ξ| + α 3 , ∀ (x, ξ) ∈ Ω × R n
³ ´
p−1 n
|h ξ (x, ξ)| ≤ β 1+ |ξ| , ∀ (x, ξ) ∈ Ω × R ;
¡ ¢
(ii) g ∈ C 0 Ω × R , g ≥ 0 and either of the following three cases hold
Case 1: p> n. For every R> 0,there exists γ = γ (R) such that

|g (x, u) − g (x, v)| ≤ γ |u − v|
for every x ∈ Ω and every u, v ∈ R with |u| , |v| ≤ R .
Case 2: p = n.There exist q ≥ 1 and γ> 0 so that
³ ´
q−1 q−1
|g (x, u) − g (x, v)| ≤ γ 1+ |u| + |v| |u − v|
for every x ∈ Ω and every u, v ∈ R.
Case 3: p< n.There exist q ∈ [1,np/ (n − p)) and γ> 0 so that
³ ´
q−1 q−1
|g (x, u) − g (x, v)| ≤ γ 1+ |u| + |v| |u − v|
for every x ∈ Ω and every u, v ∈ R.

N
Exercise 3.3.3 Prove Theorem 3.3 in the following framework. Let α, β ∈ R ,
N> 1 and
( )
Z b
(P) inf I (u)= f (x, u (x) ,u (x)) dx = m
0
u∈X
a
© 1,p ¡ N ¢ ª
where X = u ∈ W (a, b); R : u (a)= α, u (b)= β and
N
(i) f ∈ C 1 ¡ [a, b] × R × R N ¢ , (u, ξ) → f (x, u, ξ) is convex for every x ∈
[a, b];
(ii) there exist p> q ≥ 1 and α 1 > 0, α 2 ,α 3 ∈ R such that
N
N
p
q
f (x, u, ξ) ≥ α 1 |ξ| + α 2 |u| + α 3 , ∀ (x, u, ξ) ∈ [a, b] × R × R ;