Math 295a Fall Term 2004
Assignment 5
1. Let Ω ⊂Rnbe a bounded domain with smooth boundary and prove
Poincar´e’s Inequality
kukLp(Ω) ≤ck∇ukLp(Ω) , u ∈◦
W1
p(Ω) .
What does the constant cdepend on? Is the boundedness assump-
tion really necessary?
2. Let Ω ⊂Rnand p∈(1,∞) and define
Wm
p(Ω) = u∈ D0(Ω)
∂αu∈Lp(Ω) ,|α| ≤ m.
Show that Wm
p(Ω) is a Banach space if endowed with the norm k·km,p
defined by
kukm,p =X
|α|≤m
k∂αukp
Lp(Ω)1/p , u ∈Wm
p(Ω) .
Prove that W1
p(0,1) ,→BUC1−1/p([0,1]).
[Hint: Use the fact that C1([0,1]) is dense in W1
p(0,1)]
3. Prove that S(Rn) is dense in Hs(Rn) for s∈R.
4. Let Ω = B(0,1/2) and the function ube defined through
u(x, y) = loglog( 2
px2+y2),(x, y)∈Ω.
Then uis obviously not continuous in (x,y ) = (0,0). Prove that,
however, u∈H1(Ω). Let now
u(x, y) = xy log
log |(x, y)|
−log log 2,(x, y)∈Ω.
Then
u∈C1(¯
Ω) and ∂2
ju∈C(¯
Ω) , j = 1,2
or u /∈C2(¯
Ω), that is, uis a solution of the Dirichlet problem in Ω
for a continuous datum but is not twice continuously differentiable.
5. Let Ebe a Banach space and A: dom(A)⊂E−→ Ea linear,
possibly unbounded, operator on E.Ais said to be invertible if
there exists a bounded operator B∈ L(E) such that
AB = idEand BA = iddom(A).
Such an operator Acan fail to be invertible either because it has
non trivial kernel (not injective)
ker(A)6={0}
