Page 108 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

Euler-Lagrange equations 95

Applying Lebesgue dominated convergence theorem we deduce that (3.7) holds.
Step 3 (Derivation of (E w ) and (E)). The conclusion of the theorem follows
from the preceding step. Indeed since u is a minimizer of (P) then
1,p
I (u + ϕ) ≥ I (u) , ∀ϕ ∈ W (Ω)
0
and thus
I (u + ϕ) − I (u)
lim =0
→0
which combined with (3.7) implies (E w ).
To get (E) it remains to integrate by parts (using Exercise 1.4.7) and to find
Z
1,p
(E w ) [f u (x, u, ∇u) − div f ξ (x, u, ∇u)] ϕdx =0, ∀ϕ ∈ W (Ω) .
0
Ω
The fundamental lemma of the calculus of variations (Theorem 1.24) implies the
claim.
Step 4 (Converse). Let u be a solution of (E w ) (note that any solution of
(E) is necessarily a solution of (E w )). From the convexity of f we deduce that
1,p
for every u ∈ u 0 + W 0 (Ω) the following holds
f (x, u, ∇u) ≥ f (x, u, ∇u)+ f u (x, u, ∇u)(u − u)
+ hf ξ (x, u, ∇u);(∇u −∇u)i .
1,p
Integrating, using (E w )and the factthat u − u ∈ W (Ω) we immediately get
0
that I (u) ≥ I (u) and hence the theorem.
We now discuss some examples.
Example 3.13 In the case of Dirichlet integral we have
1 2
f (x, u, ξ)= f (ξ)= |ξ|
2
which satisfies (H3). The equation (E w )is then
Z
1,2
h∇u (x); ∇ϕ (x)i dx =0, ∀ϕ ∈ W (Ω)
0
Ω
while (E) is ∆u =0.
Example 3.14 Consider the generalization of the preceding example, where
1 p
f (x, u, ξ)= f (ξ)= |ξ| .
p
The equation (E) is known as the p-Laplace equation (so called, since when p =2
it corresponds to Laplace equation)
h i
p−2
div |∇u| ∇u =0,in Ω .