For each question you may use the results of previous questions even if you have
not answered them.
Exercise 1. State the Lax-Milgram lemma (or theorem).
Exercise 2. For any R > 0 let BR be the ball of radius R in R3 , centered at 0:
BR = {(x1 , x2 , x3 ) ∈ R3 : x21 + x22 + x23 < R2 }.
(0.1)
Fix a function c ∈ L∞ (R3 ) and a function f ∈ L2 (R3 ).
For each R > 0 we consider the following problem, which we call P(R):
Find uR : BR → R such that:
−∆uR (x) + c(x)uR (x) = f (x) for x ∈ BR ,
uR (x) = 0 for x ∈ ∂BR .
1) What is the variational formulation of problem P(R)?
2) Show that there exists C > 0 such that for all R > 0 we have:
Z
Z
1
2
2
∀u ∈ H0 (BR )
|u| ≤ CR
| grad u|2 .
BR
(0.2)
(0.3)
(0.4)
BR
3) Deduce that for small enough R, the variational formulation of problem P(R)
has one and only one solution uR ∈ H10 (BR ).
Exercise 3. Define:
R+ = {x ∈ R : x > 0}.
(0.5)
Fix p ∈ R with p > 1.
Choose f ∈ C01 (R+ ). Define g : R+ → R by, for t > 0:
g(t) =
1
t
Z
t
f (s) d s.
(0.6)
0
1) Show that there exist C > 0 and T > 0 such that for t > T :
|g(t)| ≤
C
.
t
(0.7)
Deduce that g ∈ Lp (R+ ).
2) Show that, for t > 0:
f (t) = tg 0 (t) + g(t).
Deduce that:
Z
∞
f |g|
p
p−1
(0.8)
Z
∞
sign g = (p − 1)
0
|g|p ,
(0.9)
0
where, for a ∈ R we define:

 1
0
sign a =

−1
3) Deduce:
kgkLp (R+ ) ≤
if a > 0,
if a = 0,
if a < 0.
p
kf kLp (R+ ) .
p−1
1
(0.10)
(0.11)
Exercise 4. As in Exercise 3 we define:
R+ = {x ∈ R : x > 0},
(0.12)
and fix p ∈ R with p > 1.
For any function u : R+ → R, we define the function Au : R+ → R as follows.
For t > 0:
u(t)
(Au)(t) =
.
(0.13)
t
1) Use the results of Exercise 3 to show that A defines a continuous linear
operator from W01,p (R+ ) to Lp (R+ ).
2) Given u ∈ C01 (R+ ) and λ > 0, define uλ : R+ → R by, for t > 0 :
uλ (t) = u(t/λ).
(0.14)
Express kuλ kLp (R+ ) , k(uλ )0 kLp (R+ ) and kA(uλ )kLp (R+ ) in terms of kukLp (R+ ) , ku0 kLp (R+ ) ,
kAukLp (R+ ) and λ.
3) Show that the operator A : W01,p (R+ ) → Lp (R+ ) is not compact.
2