Page 136 - INTRODUCTION TO THE CALCULUS OF VARIATIONS

The model case: Dirichlet integral 123

Using again the properties of the operator D h we have indeed obtained that
n
n
n
∇u ∈ W 1,2 (R ; R ) and hence u ∈ W 2,2 (R ).
Step 2. (The present step, contrary to the preceding one, relies heavily on
n
the special form of the equation). Let now g ∈ W 1,2 (R ) and let us show that
n
u ∈ W 3,2 (R ). The general case g ∈ W k,2 implying that u ∈ W k+2,2 follows by
repeating the argument. The idea is simple, it consists in applying the previous
. Indeed
step to u x i = ∂u/∂x i and observing that since ∆u = g,then ∆u x i = g x i
it is elementary to see that we have, for every i =1, ..., n,
Z Z
n
(x); ∇v (x)i dx = (x) v (x) dx, ∀v ∈ W 1,2 (R ) . (4.16)
h∇u x i g x i
R n R n
n
n
To prove this, it is sufficient to establish it for v ∈ C ∞ (R ) (since C ∞ (R ) is
0 0
n
dense in W 1,2 (R )). We have, using (4.14), that
Z Z Z
­ ® ­ ®
; ∇vi dx = ∇u;(∇v) dx
h∇u x i (∇u) ; ∇v dx = −
x i x i
R n R n R n
Z Z Z
= − h∇u; ∇v x i i dx = − gv x i dx = g x i vdx .
R n R n R n
2
Since g ∈ W 1,2 , wehavethat g x i ∈ L and hence by the first step applied to
∈ W 2,2 . Since this holds for every i =1, ..., n,we have
(4.16) we get that u x i
indeed obtained that u ∈ W 3,2 . This concludes the proof of the theorem.
4.3.1 Exercises
Exercise 4.3.1 Prove Theorem 4.7 when n =1.
n
Exercise 4.3.2 Let Ω ⊂ R be an open set and let σ n−1 =meas (∂B 1 (0)) (i.e.
0
σ 1 =2π, σ 2 =4π,...). Let u ∈ C (Ω) satisfy the mean value formula, which
states that Z
1
u (x)= udσ
σ n−1 r n−1 ∂B r (x)
for every x ∈ Ω and for every r> 0 sufficiently small so that
n
B r (x)= {y ∈ R : |y − x| <r} ⊂ Ω .
Show that u ∈ C ∞ (Ω).
0
Exercise 4.3.3 We show here that if f ∈ C , then, in general, there is no
ª
©
2
2
solution u ∈ C of ∆u = f.Let Ω = x ∈ R : |x| < 1/2 and for 0 <α < 1,
define
⎧
α
⎨ x 1 x 2 |log |x|| if 0 < |x| ≤ 1/2
u (x)= u (x 1 ,x 2 )=
⎩
0 if x =0 .