O UTLINE :
k,p
• definition of H (Ω), where Ω is a domain in Rn
• density properties of C ∞ functions in H k,p
• theorems:
– interpolation
– embedding
– compactness
• introduction to manifolds and extension of previous results
• apply to DP for general elliptic operators (not just ∆) on general manifolds
N OTATION :
α1
αn
∂
• ∇α u = ∂x
. . . ∂x∂n
u
1
Here
α
is
a
multi-index
with
α = (α1 , . . . , αn ) ∈ Nn
Pn
• |α| = 1 αi the length of the multi-index.
P
1/2
α 2
• k∇k uk =
|α|=k |∇ u|
Sobolev spaces of functions f : Rn → R
Let Ω ⊆ Rn be a domain. Unless otherwise
noted, we consider Ω to be open and connected.
Definition: The H k,p norm of u ∈ C k Ω is:
kukH k,p (Ω)
k
X
≡
k∇l ukLp (Ω)
l=1
where kukLp (Ω)
R
≡ ( Ω |u|p )1/p
Definition: H k,p (Ω) ≡ C ∞ Ω
k·kH k,p
, the Sobolev space H k,p (Ω)
k.kH k,p
Definition: H0k,p (Ω) ≡ Cc∞ Ω
k,p
Definition: Hloc
(Ω) ≡ {u : Ω → R | u|Ω1 ∈ H k,p (Ω1 ), Ω1 ( Ω}
Facts:
(i) H k,p (Ω) and H0k,p (Ω) are complete Banach spaces by definition
k,p
k,p
(ii) We introduce a topology on Hloc
(Ω) as follows: uk → u in Hloc
if kuk − ukH k,p (Ω1 ) → 0,
∀ Ω1 ( Ω
(iii) C ∞ (Ω) is dense in H k,p (Ω)
Notice that (iii) =⇒ ∀u ∈ H k,p (Ω), ∃ui → u in H k,p , with ui ∈ C ∞ (Ω)
Q: Is there a systematic construction of ui ?
∞
n
A: Yes,
R via a “mollification”. We need a function ϕ ∈ Cc (R ), with 0 ≤ ϕ ≤ 1, supp(ϕ) ⊆ B1 (0)
and Rn ϕ = 1 where ϕ is radially symmetric (ie. ϕ(x) = ϕ(kxk)). We can visualize ϕ as a “bump” (see
figure below).
1
2
Now, for u ∈ L1loc (Ω) and x ∈ Ω ≡ {y ∈ Ω | dist(y, ∂Ω) ≥ } (see figure above), define:
Z x − y −n
u (x) =
ϕ
u(y)dn y
R
The function ϕ = −n ϕ(x/) satisfies supp(ϕ ) ⊆ B (0), 0 ≤ ϕ ≤ −n and ϕ (x)dx = 1. u is
the “mollification” of u.
Theorem: If u ∈ L1loc (Ω) and u is as defined, then:
1) a. u ∈ C ∞ (Ω ) ∀. Here, ∇α (u ) = (∇α ϕ ) ∗ u where ∗ denotes a convolution.
b. If supp(u) ⊆ K ( Ω then supp(u ) ⊆ {y ∈ Ω | dist(y, K) ≤ } for sufficiently small.
(see figure below)
2) u ∈ C 0 (Ω) =⇒ u → u uniformly on Ω1 ( Ω.
3) u ∈ Lploc (Ω) =⇒ u → u in Lploc . (ku kLp (Ω1 ) ≤ kukLp (Ω1 ) )
4) u → u a.e.
Homework Prove 1) – 3).
Let us examine claim 1).
Z
x−y
u (x) =
−n ϕ
u(y)dn y
Rn
Z
∂u (x) justification
∂
x−y
=
−n
ϕ
u(y)dn y
∂x
∂x
Rn
Next: An alternate characterization
Definition: u ∈ Lp (Ω), α a multi-index. Dα u is the “αth weak derivative of u” if
Z
Z
(Dα u)ϕ = (−1)|α|
u∇α ϕ ∀ϕ ∈ Cc∞ (Ω)
Ω
Ω
Note 1. This is not saying that Dα u = ∇α u because u ∈ Lp (Ω)
Note 2. The requirement is just that the object Dα y “behaves like” ∇α u under integration by parts.
3
Note 3. Many functions have weak derivatives.
R
If u ∈ C k (Ω), then ∇α u exists ∀α such that |α| ≤ k. Integrating by parts, Ω ∇α uϕ =
R
(−1)|α| u∇α ϕ. Thus, for u ∈ C k (Ω), ∇α u = Dα u and the weak derivative generalizes the
ordinary derivative.
Claim: Suppose u ∈ C 0 (Ω) is a Lipschitz function (ie. ∃L with |u(x) − u(y)| ≤ L|x − y| ∀x, y ∈ Ω.
L is called the Lipschitz constant.) Then u has a weak derivative, and Du = ∇u whenever it exists.
Homework Prove the claim. Also, what about when ∇u does not exist? What is Du?
Now, the alternate characterization.
Definition: W k,p (Ω) = {u ∈ Lploc (Ω) | Dα u exists for |α| ≤ k, and Dα u ∈ Lp (Ω)}
k,p
Definition: Wloc
(Ω) = {u ∈ Lploc (Ω) | u|Ω1 ∈ W k,p (Ω1 ) ∀Ω1 ( Ω}
Aside: Why do we have H k,p and W k,p spaces? H k,p has nice density properties while W k,p has nice
analytic features — u ∈ W k,p have weak derivatives, W k,p is more general than C k so that W k,p is
good for PDEs.
Goal: Prove that if Ω is a “nice” domain, then H k,p (Ω) = W k,p (Ω).
Proposition: W k,p (Ω) is a Banach space {also reflexive (for p > 1) and separable (for p < ∞). This is
a consequence of the fact that Lp is reflexive if p > 1 and separable if p < ∞.}
Proof: Show that W k,p (Ω) is complete. Suppose that ui ∈ W k,p (Ω) is Cauchy in the W k,p norm
Pk
kui kW k,p = l=0 kDl ui kLp . Then Dα ui are all Cauchy in Lp . Thus, there is a subsequence ui0 such
that Dα ui0 → v α in Lp for some v α ∈ Lp (Ω).
We have ui0 → v ∈ Lp and we need to show v α = Dα v. Choose ϕ ∈ Cc∞ (Ω). Then,
Z
Z
v α ϕ = lim Dα ui ϕ
Z
|α|
= lim(−1)
u i ∇α ϕ
Z
= (−1)|α| v∇α ϕ
Since this holds ∀ϕ, v α = Dα v
Note that the previous calculation says that if u ∈ H k,p (Ω) then there exists a sequence ui ∈ C ∞ (Ω)
such that ui → u in H k,p (Ω) =⇒ ∇α ui are Cauchy in W k,p norm. Thus, the H k,p (Ω) and W k,p (Ω)
norms are the same on C ∞ functions and v α = Dα u so that H k,p (Ω) ⊆ W k,p (Ω).
k,p
k,p
Proposition: Hloc
(Ω) = Wloc
(Ω)
k,p
Proof: ⊆ is already done. We look at ⊇ by taking u ∈ Wloc
(Ω). Use mollification. Let Ω1 ( Ω, take
u|Ω1 and extend it by setting it equal to zero outside of Ω1 . Call this extension ū : Ω1 → R. Mollify ū:
Z
x − y −n n
d y,
ū (x) = ū(y)ϕ
where < 0 < dist(Ω1 , ∂Ω)/2 (Aside: Ω1 ( supp(ū ) ( Ω)
Fact: ū → ū in Lp on K ( supp(ū0 ).
Homework Fact: ∇α ū → Dα ū in Lp on K ( supp(ū0 ).
In particular, for K = Ω1 and 0 ≤ |α| ≤ k,
∇α ū |Ω1
Lp
→ (Dα ū)|Ω1
= (Dα u)|Ω1
Thus, ū → u on Ω1 , ū ∈ C ∞ (Ω). That is, u ∈ H k,p (Ω1 ).