Lecture 8
A Priori Estimates for Poisson’s Equation.
�
Recall that N f = Ω Γ(x − y)f (y)dy is called the Newtonian Potential of f .
Proposition 1 Suppose Ω is bounded domain, f ∈ L1 (Ω), and ω = N f is the Newto­
nian potential of f . Then ω ∈ C 1 (Rn ) and
�
Di ω(x) =
Di Γ(x − y)f (y)dy, ∀x ∈ Ω.
Ω
Proof: Γ = C|x|2−n =⇒ |DΓ| ≤ C|x|1−n , therefore
�
v(x) =
Di Γ(x − y)f (y)dy
Ω
�
is well defined. (|v(x)| ≤ �f �L∞ Ω |Di Γ|dy ≤ C�f �L∞ .)
Define η� (t) to be C ∞ function with properties: (1) η� (t) = 0 for t < �; (2) η� (t) = 1
for t > 2�; (3) 0 ≤ η� (t) ≤ 1; (4) |Dη� | ≤ 2� . Define ω� (x) to be
�
ω� (x) =
Γ(x − y)η� (|x − y |)f (y)dy.
Ω
Then ω� (x) ∈
C1
and ω� (x) −→ ω uniformly.
�
(Di Γ(x − y) − Di (Γ(x − y)η� (|x − y |)))f (y)dy
v(x) − Di ω� (x) =
�Ω
Di ((1 − η� (|x − y |))Γ(x − y))f (y)dy
=
Ω
�
Di ((1 − η� (|x − y |))Γ(x − y))f (y)dy
=
|x−y|≤2�
So
�
2
( |Γ(x − y)| + |Di Γ(x − y)|)dy
|x−y|≤2� �
�
�
2
|Γ(z)|dz +
|Di Γ(z)|dz)
≤ sup |f |(
� |z|≤2�
|z|≤2�
�
�
2
C
C
≤ sup |f |(
dz +
dz)
n−2
n−1
� |z |≤2� |z|
|z|≤2� |z|
�
�
2
≤ C sup |f |(
rdr +
dr)
� r≤�
r≤�
|v(x) − Di ω� (x)| ≤ sup |f |
≤ C · � sup |f |.
Now we have ω� → ω and Di ω� → v uniformly on compact subsets as � → 0, thus
ω ∈ C 1 (Rn ) and Di ω = v.
�
1
Theorem 1 Let u ∈ C 2 (Ω), f ∈ L∞ (Ω) and Δu = f in Ω. Then for any compact
subdomain Ω� ⊂⊂ Ω,
�u�C 1 (Ω� ) ≤ C(�u�C 0 (Ω) + �f �L∞ (Ω) ).
�
Proof: Let ω be the Newtonian potential of f , i.e. ω(y) = Ω Γ(x − y)f (x)dx. Then
from Green’s representation formula,
�
∂
∂u
v(y) = u(y) − ω(y) =
u(x)
Γ(x − y) − Γ(x − y) dσx
∂νx
∂ν
∂Ω
is a harmonic function. So
�
�
�ω�C 0 (Ω� ) = sup | Γ(x − y)f (x)dx| ≤ �f �L∞ (Ω) sup
y∈Ω�
y∈Ω�
Ω
Ω
C
ds ≤ C�f �L∞ (Ω) ,
|x − y |n−2
and
�
Di Γ(y − x)f (x)ds
�
≤ �f �L∞ (Ω)
Di Γ(y − x)ds
Di ω(y) =
Ω
Ω
≤ �f �L∞ (Ω)
C
ds
|x − y |
n−1
= C�f �L∞ (Ω)
Thus �Di ω�C 0 (Ω) ≤ C�f �L∞ (Ω) , and so
�ω�C 1 (Ω) ≤ C�f �L∞ (Ω) .
Since v is harmonic, we have
�Dv�C 0 (Ω� ) ≤ C�v�C 0 (Ω)
≤ C(�u�C 0 (Ω) + �ω�C 0 (Ω) )
≤ C(�u�C 0 (Ω) + �f �L∞ (Ω) ).
Thus
�u�C 1 (Ω� ) ≤ �v�C 1 (Ω� ) + �ω�C 1 (Ω� )
≤ C(�u�C 0 (Ω) + �f �L∞ (Ω) ).
More over, one can show that for any 0 ≤ α < 1,
�u�C 1,α (Ω� ) ≤ C(�u�C 0 + �f �L∞ ).
This is not true for α = 1.
2
�
But if f ∈ C α (Ω), then
�u�C 2 (Ω� ) ≤ C(�u�0C + �f �L∞ (Ω) )
and
�u�C 2,α (Ω� ) ≤ C(�u�αC + �f �L∞ (Ω) ).
C 1,α estimate for Newtonian Potential (Ω Bounded)
�
�
ω(x) =
Γ(x − y)f (y)dy =⇒ Di ω(x) =
Di Γ(x − y)f (y)dy.
Ω
Ω
Theorem 2 If f ∈ L∞ , then ω ∈ C 1,α (Ω).
Proof: Take x, x ∈ Ω, let δ = |x − x|, and ξ = 12 (x + x).
�
Di ω(x) − Di Ω(x) = (Di Γ(x − y) − Di Γ(x − y))f (y)dy
Ω
�
≤ �f �L∞ (Ω)
|Di Γ(x − y) − Di Γ(x − y)|dy
Ω
�
≤ �f �L∞ (Ω) (
|Di Γ(x − y) − Di Γ(x − y)|dy
Bδ (ξ)
�
|Di Γ(x − y) − Di Γ(x − y)|dy)
+
Ω−Bδ (ξ)
= �f �L∞ (Ω) (I + II),
where
�
�
|Di Γ(x − y)|dy +
I≤
|Di Γ(x − y)|dy
Bδ (ξ)
Bδ (ξ)
�
�
|Di Γ(x − y)|dy +
≤
B 3δ (x)
|Di Γ(x − y)|dy
B 3δ (x)
�2
2
≤C
B 3δ (x)
1
dy +
|x − y|n−1
2
�
B 3δ (x)
1
dy
|x − y)|n−1
2
3δ
≤C·
2
= C |x − x0 |,
and
�
|Di Γ(x − y) − Di Γ(x − y)|dy
II =
Ω−Bδ (ξ)
�
≤ |x − x|
�
≤C ·δ
|DDi Γ(x̂ − y)|dy
Ω−Bδ (ξ)
|y−ξ|≥δ
1
dy
|ˆ
x − y |n
3
Since we have
|y − ξ | ≤ |y − x
ˆ| + |ˆ
x − ξ|
δ
≤ |y − x|
ˆ +
2
1
≤ |y − x|
ˆ + |y − ξ |
2
1
=⇒ |y − ξ | ≤ |y − x̂|.
2
Thus
�
1
II ≤ C · δ
dy
|
y
−
ξ |n
|y−ξ|≥δ
� R
1
≤C ·δ
dr
δ r
≤ C · δ(log R − log δ)
≤ C ·
δ(log R + cδ α−1 )
(� − log δ ≤ Cδ α−1 f or 0 ≤ α < 1)
= Cδ + Cδ α ≤ Cδ α
= C |x − x|α .
Combine the above results, we get ω ∈ C 1,α (Ω).
Further more, we get the C 1,α estimate
�ω�C 1,α (Ω) ≤ C�f �L∞ (Ω) , 0 ≤ α < 1.
�
Just as last time, this implies
Corollary 1 Suppose Δu = f, f ∈ L∞ (Ω), Ω� ⊂⊂ Ω, then for 0 ≤ α < 1,
�u�C 1,α (Ω� ) ≤ C(�u�C 0 (Ω) + �f �L∞ (Ω) ).
Remark 1 Look at the above proof and assume f ∈ Lp (Ω).
�
Di ω(x) − Di ω(x) = (Di Γ(x − y) − Di Γ(x − y))f (y)dy
Ω
�
�
≤ { |Di Γ(x − y) − Di Γ(x − y)|q dy}1/q { |f (y)|p dy}1/p
Ω
Ω
1 1
p
+ = 1 =⇒ q =
.
p q
p−1
In the 2nd part of the above proof, we had
�
II ≤ { ||x − x||DD
ˆ
x − y)||q }1/q
i Γ(ˆ
Ω �
1
)1/q
≤ C · δ(
nq
|y−ξ|≥δ |y − ξ |
≤ C · δ · (δ −nq+n )1/q
=C ·δ
1−n+ n
q
.
4
Let α = 1 − n + n p−1
p , then p(α − 1 + n) = pn − n, i.e. p =
If Δu = f, f ∈
Lp (Ω), p
=
n
1−α ,
Ω�
n
1−α ,
⊂⊂ Ω, then
�u�C 1,α (Ω� ) ≤ C(�f �Lp (Ω) + �u�C 0 (Ω) ).
Later we will show
�u�W 2,p (Ω� ) ≤ C(�f �Lp (Ω) + �u�C 0 (Ω) ).
So C 1,α estimate follows by Sobolev Embedding theorem.
5
we have: