Page 211 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

198 Solutions to the Exercises

q
Since we have from Rellich theorem, that u ν → u in L , ∀q ∈ [1, ∞) we obtain
the desired convergence.
Case 3: p< n. The same argument as in Case 2 leads to the result, the dif-
q
ference being that Rellich theorem gives now u ν → u in L , ∀q ∈ [1,np/ (n − p)).
Exercise 3.3.3. We here have weakened the hypothesis (H3) in the proof of the
theorem. We used this hypothesis only in the lower semicontinuity part of the
proof, so let us establish this property under the new condition. So let u ν u
¢
¡
in W 1,p (a, b); R N . Using the convexity of (u, ξ) → f (x, u, ξ) we find
Z b Z b
0 0
f (x, u ν ,u ) dx ≥ f (x, u, u ) dx
ν
a a
Z b
0
0
+ [hf u (x, u, u ); u ν − ui + hf ξ (x, u, u ); u − u i] dx .
0
0
ν
a
Since, by Rellich theorem, u ν → u in L , to pass to the limit in the second term
∞
1
of the right hand side of the inequality we need only to have f u (x, u, u ) ∈ L .
0
p
This is ascertained by the hypothesis |f u (x, u, ξ)| ≤ β (1 + |ξ| ). Similarly to
0
p
pass to the limit in the last term we need to have f ξ (x, u, u ) ∈ L , p =
0
0
p/ (p − 1); and this is precisely true because of the hypothesis |f ξ (x, u, ξ)| ≤
³ ´
p−1
β 1+ |ξ| .
7.3.3 Euler-Lagrange equations
Exercise 3.4.1. We have to prove that for u ∈ W 1,p , the following expression
is meaningful
Z
1,p
{f u (x, u, ∇u) ϕ + hf ξ (x, u, ∇u); ∇ϕi} dx =0, ∀ϕ ∈ W 0 .
Ω
Case 1: p> n. We have from Sobolev imbedding theorem (cf. Theorem
¡ ¢
1
1.42) that u, ϕ ∈ C Ω . We therefore only need to have f u (x, u, ∇u) ∈ L and
0
p
f ξ (x, u, ∇u) ∈ L ,where p = p/ (p − 1). Thisistrueifweassumethatfor
0
every R> 0,there exists β = β (R) so that ∀ (x, u, ξ) with |u| ≤ R the following
inequalities hold
³ ´
p p−1
|f u (x, u, ξ)| ≤ β (1 + |ξ| ) , |f ξ (x, u, ξ)| ≤ β 1+ |ξ| .
q
Case 2: p = n. Thistimewehave u, ϕ ∈ L , ∀q ∈ [1, ∞). We therefore have
0
p
r
to ascertain that f u ∈ L for a certain r> 1 and f ξ ∈ L .To guarantee this
claim we impose that there exist β> 0, p> s 2 ≥ 1, s 1 ≥ 1 such that
³ ´
s 1 s 2 s 1 p−1
|f u (x, u, ξ)| ≤ β (1 + |u| + |ξ| ) , |f ξ (x, u, ξ)| ≤ β 1+ |u| + |ξ| .