Page 209 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

196 Solutions to the Exercises

7.3 Chapter 3: Direct methods

7.3.1 The model case: Dirichlet integral

Exercise 3.2.1. The proof is almost completely identical to that of Theorem
3.1; only the first step is slightly different. So let {u ν } be a minimizing sequence
n o
I (u ν ) → m =inf I (u): u ∈ W 1,2 (Ω) .
0
Since I (0) < +∞,wehavethat m< +∞. Consequently wehavefrom Hölder
inequality that
Z Z
1 2
m +1 ≥ I (u ν )= |∇u ν | dx − g (x) u ν (x) dx
2
Ω Ω
Z
1 2 1 2
≥ |∇u ν | − kgk L 2 ku ν k L 2 = k∇u ν k L 2 − kgk L 2 ku ν k L 2 .
Ω 2 2
Using Poincaré inequality (cf. Theorem 1.47) we can find constants (independent
of ν) γ > 0,k =1, ..., 5,sothat
k
2 2
m +1 ≥ γ ku ν k W 1,2 − γ
3
2
1 W 1,2 − γ ku ν k W 1,2 ≥ γ ku ν k 4
and hence, as wished,
ku ν k W 1,2 ≤ γ .
5
7.3.2 A general existence theorem
Exercise 3.3.1. As in the preceding exercise it is the compactness proof in
Theorem 3.3 that has to be modified, the remaining part of the proof is essentially
unchanged. Let therefore {u ν } be a minimizing sequence, i.e. I (u ν ) → m.We
have from (H2) that for ν sufficiently large

p q
m +1 ≥ I (u ν ) ≥ α 1 k∇u ν k L p − |α 2 |ku ν k L q − |α 3 | meas Ω .
From now on we will denote by γ > 0 constants that are independent of ν.
k
Since by Hölder inequality we have

Z µZ ¶ q/p µZ ¶ (p−q)/p
q q p (p−q)/p q
ku ν k L q = |u ν | ≤ |u ν | dx =(meas Ω) ku ν k L p
Ω Ω Ω
we deduce that we can find constants γ and γ such that
1 2
p q
m +1 ≥ α 1 k∇u ν k L p − γ ku ν k L p − γ 2
1
p q
≥ α 1 k∇u ν k L p − γ ku ν k W 1,p − γ .
2
1