Алексеев Г.В. Классические методы математической физики. Часть 2
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ε
v(x
0
) = 0
v(x
0
) = 0
u(x
0
) = u
1
(x
0
) x
0
u
K u
1
u x
0
x
0
u
x
|u(x)| ≤ µ(x) l n
1
|x − x
0
|
.
µ(x) → 0 x → x
0
u
x
0
u
x
0
x
0
u Ω
Ω
u
Ω u
Ω u Ω
M = sup
x∈Ω
u(x)
u Ω
x
1
Ω R Ω
x
1
R v
ε
(x, x
1
) = M + ε ln(R/|x −
x
1
|) Ω
δ
Ω
x
1
δ
u(x) < v
ε
(x) δ
u(x) < v
ε
(x) Ω
δ
ε u(x) < M
x Ω
δ
u(x) > m
m = inf
x∈Γ
u(x)
n
n > 2
|u(x)| ≤ µ(x)|x − x
0
|
2−n
µ(x) → 0 x → x
0
R
n
E
n
(·, y) R
n
E
n
(x, y ) =
1
ω
n
|x − y|
n−2
n ≥ 3
E
2
(x, y ) =
1
2π
ln
1
|x − y|
= −
1
2π
ln |x − y |
n = 2.
ω
n
R
n
ω
n
= 4π
n = 3
E
3
(x, y) =
1
4π|x − y|
.
∂
∂x
i
E
n
(x, y) = −
∂
∂y
i
E
n
(x, y), ∇
x
E
n
(x, y) = −∇
y
E
n
(x, y).
(Ω, ρ) Ω R
n
Γ ρ
Ω
Ω R
n
ρ Ω |ρ(y)| ≤ M ∀y ∈ Ω
n = 2
(Ω, ρ) u : R
n
→ R
x ∈ R
n
u(x) = [Aρ](x) ≡
Z
Ω
E
n
(x, y )ρ(y)dy.
x ∈ Ω
e
Ω
e
≡ R
n
\Ω Ω
x ∈ Ω E
n
(·, y) x 6= y
R
n
\{y} u
Ω
e
Ω
e
∆u = 0.
E
n
n ≥ 3
u(x) |x| → ∞ O(|x|
2−n
)
|u(x)| ≤
C
|x|
n−2
|x| → ∞,
C u Ω n = 2
u(x) =
1
2π
Z
Ω
ln
1
|x − y|
ρ(y)dy = −
1
2π
Z
Ω
ln|x − y|ρ(y)dy
u(x) = −
ln|x|
2π
Z
Ω
ρ(y)dy +
1
2π
Z
Ω
ln
|x|
|x − y|
ρ(y)dy.
ln
|x|
|x − y|
→ 0
x → ∞
y ∈ Ω
|x| → ∞
x → ∞
(Ω, ρ) R
2
|x| → ∞ ρ
Z
Ω
ρ(y)dy = 0.
u(x)
|x| → ∞
u
R
n
R
n
u Ω
e
x ∈ Ω
n
u
R
n
x
i
x ∈
R
n
x
i
∂u(x)
∂x
i
=
Z
Ω
∂
∂x
i
E
n
(x, y)ρ(y)dy, x ∈ R
n
, i = 1, 2, ..., n.
∂u(x)
∂x
i
=
1
4π
Z
Ω
∂
∂x
i
1
|x − y|
ρ(y)dy = −
1
4π
Z
Ω
x
i
− y
i
|x − y|
3
ρ(y)dy, i = 1, 2, 3,
x Ω
u Ω
e
x ∈ Ω
x = (x
1
, x
2
, x
3
) ∈ Ω
x
′
= (x
1
+∆x
1
, x
2
, x
3
) x
′
∈ Ω
∆
x
1
u(x
′
)
∆x
1
=
u(x
′
) − u(x)
∆x
1
α =
∆
x
1
u(x)
∆x
1
−
1
4π
Z
Ω
y
1
− x
1
|x − y|
3
ρ(y)dy.
α ∆x
1
→ 0
ε > 0 B
δ
(x)
δ = δ(ε)
u(x) = u
1
(x) + u
2
(x) ≡
1
4π
Z
B
δ
(x)∩Ω
ρ(y)
|x − y|
dy +
1
4π
Z
Ω\
B
δ
(x)
ρ(y)
|x − y|
dy ,
u(x
′
) = u
1
(x
′
) + u
2
(x
′
) =
1
4π
Z
B
δ
(x)∩Ω
ρ(y)
|x
′
− y|
dy +
1
4π
Z
Ω\
B
δ
(x)
ρ(y)
|x
′
− y|
dy .
α =
u
1
(x
′
) − u
1
(x)
∆x
1
−
1
4π
Z
B
δ
(x)∩Ω
y
1
− x
1
|x − y|
3
ρ(y)dy
+
+
u
2
(x
′
) − u
2
(x)
∆x
1
−
1
4π
Z
Ω\
B
δ
(x)
y
1
− x
1
|x − y|
3
ρ(y)dy
.
ε/3
δ = δ(ε)
xx
′
y x x
′
y
|x − y| |x
′
− y| |∆x
1
|
|x
′
− y| − |x − y| < |∆x
1
|.
|ab| ≤ 1/2(a
2
+ b
2
)
u
1
(x
′
) − u
1
(x)
∆x
1
=
1
4π|∆x
1
|
Z
B
δ
(x)∩Ω
|x
′
− y| − |x − y|
|x
′
− y||x − y|
ρ(y)dy
≤
≤
M
8π
Z
B
δ
(x)
1
|x
′
− y|
2
+
1
|x − y|
2
dy .
R
Ω
dy
|x−y|
2
ε/3
δ = δ(ε)
u
2
(x)
x
|∆x
1
| < δ
′
δ
′
u
2
(x
′
) − u
2
(x)
∆x
1
−
1
4π
Z
Ω\
B
δ
(x)
y
1
− x
1
|x − y|
3
ρ(y)dy
<
ε
3
.
|∆x
1
| |α| ≤ ε
i = 1
i = 2
x ∈ Ω
Ω u ∈ C
1
(Ω)
n
(i), ( ii)
u = Aρ
R
n
ρ
Ω Ω
ρ ∈ C(Ω) ∩ C
1
(Ω) |∇ρ| ≤ M
1
< ∞ Ω
u = Aρ C
2
(Ω)
Ω
x ∈ Ω
Γ Ω
Γ Ω
∂u/∂x
i
∂E
n
/∂x
i
= −∂E
n
/∂y
i
∂u(x)
∂x
i
= −
Z
Ω
∂
∂y
i
E
n
(x, y)ρ(y)dy =
=
Z
Ω
E
n
(x, y)
∂ρ(y)
∂y
i
dy −
Z
Γ
E
n
(x, y)ρ(y)cos(n
y
, y
i
)dσ
y
.
n
y
Γ y
dσ
y
y
x ∈ Ω
E
n
(x, y) x 6= y ∈ Γ
x
i
∂ρ/∂y
i
Ω
∂
∂x
i
Z
Ω
E
n
(x, y)
∂ρ(y)
∂y
i
dy =
Z
Ω
∂E
n
(x, y)
∂x
i
∂ρ(y)
∂y
i
dy .
u ∈ C
2
(Ω)
∂
2
u(x)
∂x
2
i
=
Z
Ω
∂E
n
(x, y)
∂x
i
∂ρ(y)
∂y
i
dy −
Z
Γ
∂E
n
(x, y)
∂x
i
ρ(y)cos(n
y
, y
i
)dσ
y
=
= −
Z
Ω
∂E
n
(x, y)
∂y
i
∂ρ(y)
∂y
i
dy+
Z
Γ
∂E
n
(x, y)
∂y
i
ρ(y)cos(n
y
, y
i
)dσ
y
, x ∈ Ω.
Γ Ω
Ω
′
⊂ Ω
Γ
′
x
∂u(x)
∂x
i
= −
Z
Ω
′
∂
∂y
i
E
n
(x, y)ρ(y)dy −
Z
Ω\Ω
′
∂
∂y
i
E
n
(x, y)ρ(y)dy.
∂u(x)
∂x
i
=
Z
Ω
′
E
n
(x, y)
∂ρ(y)
∂y
i
dy −
−
Z
Γ
′
E
n
(x, y)ρ(y)cos(n
′
y
, y
i
)dσ
′
y
−
Z
Ω\Ω
′
∂
∂y
i
E
n
(x, y)ρ(y)dy.
dσ
′
y
Γ
′
n
′
y
Γ
′
y ∈ Γ
′
x ∈ Ω
′
x
x ∈ Ω
′
u ∈ C
2
(Ω
′
)
Ω
′
Ω
u ∈ C
2
(Ω)
∂
2
u/∂x
2
i
Γ Ω
u 6∈ C
2
(Ω)
∆u(x) = −ρ(x)
x ∈ Ω
Γ Ω
i E ≡ E
3
∆u(x) = −
Z
Ω
3
X
i=1
∂ρ(y)
∂y
i
∂E(x, y)
∂y
i
dy +
Z
Γ
3
X
i=1
∂E(x, y)
∂y
i
ρ(y)cos(n
y
, y
i
)dσ
y
=
= −li m
ε→0
Z
Ω
ε
3
X
i=1
∂ρ(y)
∂y
i
∂E(x, y)
∂y
i
dy +
Z
Γ
∂E(x, y)
∂n
y
ρ(y)dσ
y
.
Ω
ε
≡ Ω
ε
(x) = {y ∈ Ω : |y − x| > ε} = Ω \ B
ε
(x)
E ∆
y
E(x, y) = 0 ∀y ∈ Ω
ε
(x)
3
X
i=1
∂ρ
∂y
i
∂E
∂y
i
=
3
X
i=1
∂
∂y
i
ρ
∂E
∂y
i
Ω
ε
(x).
Ω
ε
Z
Ω
ε
3
X
i=1
∂ρ(y)
∂y
i
∂E(x, y)
∂y
i
dy =
3
X
i=1
Z
Ω
ε
∂
∂y
i
ρ(y)
∂E(x, y)
∂y
i
dy =
=
Z
Γ
ρ(y)
∂E(x, y)
∂n
y
dσ
y
+
Z
Γ
ε
ρ(y)
∂E(x, y)
∂n
y
dσ
y
.
Γ
ε
B
ε
(x) n
y
Ω
ε
n
y
= (x − y)/|x − y| Γ
ε
(x)
∂E(x, y)
∂n
y
=
1
4π
∇
y
1
|x − y|
· n
y
=
(x − y) · n
y
4π|x − y|
3
=
1
4π|x − y|
2
Γ
ε
(x).
∆u(x) = −lim
ε→0
Z
Γ
ε
ρ(y)
∂E(x, y)
∂n
y
dσ
y
= −li m
ε→0
Z
|y−x|=ε
ρ(y)dσ
y
4π|y − x|
2
=
= −lim
ε→0
Z
|y−x|=ε
ρ(y) − ρ(x)
4πε
2
dσ
y
− ρ(x) lim
ε→0
Z
|y−x|=ε
dσ
y
4πε
2
.
R
|y−x|=ε
dσ
y
= 4πε
2
∀ε > 0
|ρ(y) − ρ(x)| ≤ sup
y∈
Ω
|∇ρ(y)||y − x| ≤ M
1
|y − x| ∀x ∈
Ω
Z
Γ
ε
ρ(y) − ρ(x)
4πε
2
dσ
y
≤
M
1
4π
Z
Γ
ε
|x − y|
ε
2
dσ
y
≤ M
1
ε
ε → 0.
ε → 0
Ω
x
0
∈ Ω u(x)
u(x) =
Z
Ω
R
E(x, y)ρ(y)dy +
Z
B
R
(x
0
)
E(x, y)ρ(y)dy , x ∈ B
R
(x
0
),
B
R
(x
0
) Ω Ω
R
≡ Ω
R
(x
0
) = Ω \ B
R
(x
0
)
B
R
(x
0
)
x 6= y ∈ Ω
R
(i), ( ii)
u = Aρ C
1
(R
n
) n ≥
3
Ω
e
Ω Ω
e
(iii)
u Ω C
2
(Ω)
x ∈ Ω
u Ω
f : Ω ⊂ R
n
→ R
Ω α > 0
L x ∈ Ω y ∈ Ω
|f(x) − f(y)| ≤ L|x − y |
α
C
α
(Ω)
Ω α ∈ (0, 1]
f ∈ C
α
(Ω) Ω C
α
(Ω) ⊂ C(Ω)
f ∈ C
α
(Ω)
kfk
C
α
(
Ω)
= kfk
C(Ω)
+ kfk
H
α
(Ω)
≡ kfk
C(Ω)
+ sup
x,y∈
Ω
x6=y
|f(x) − f(y)|
|x − y|
α
.
C
α
(Ω)
C
m,α
(Ω) C
m
(Ω)
Ω m
Ω α C
m,α
(Ω)
k · k
C
m,α
(
Ω)
kfk
C
m,α
(
Ω)
= kfk
C
m
(Ω)
+
X
|α|=m
∂
α
f
∂x
α
1
1
...∂x
α
n
n
H
α
(
Ω)
.
C
m,α
(Ω)
C
m,α
(Ω) C
m
(Ω)
α Ω
K Ω
Ω
e
C
m,α
(Ω
e
)
C
m
(Ω
e
) f m
α
Ω
e
∩ B
R
Ω
e
= Ω
e
∪ Γ B
R
R
C
m,α
(Ω
e
)
C
m
(Ω
e
) f m
α Ω
e
K ⊂ Ω
e
u Ω Ω
e
u Ω Ω
e
u
+
= u|
Ω
u
−
= u|
Ω
e
Ω
′
Ω Ω
′
⊂⊂ Ω Ω
′
⊂ Ω
′
ρ ∈ C
α
(Ω) 0 < α < 1
′
u
+
∈ C
2,α
(Ω)
u
−
∈ C
2,α
(Ω
e
) Ω
′
⊂⊂ Ω
kuk
C
2,α
(
Ω
′
)
≤ Ckρk
C
α
(Ω)
.
C Ω
′
n α ρ
C
2,α
u
Ω
Ω
′
Γ
u
′
Γ
u
Γ
Ω Ω
e
Ω Ω
e
n = 2 Γ
C
l,λ
l ∈ N 0 ≤ λ ≤ 1
B
r
(x
0
) x
0
∈ Γ F
x
0
(x) = 0
F
x
0
∈ C
l,λ
(B
r
(x
0
)) l ≥ 1 gradF
x
0
6= 0