Алексеев Г.В. Классические методы математической физики. Часть 2
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n ≥ 3
g ∈ C(Γ) n = 2
−µ(x) + 2
Z
Γ
µ(y)
∂E(y, x)
∂n
x
dσ
y
= 0
µ
0
(x) = 1
ϕ ∈ C( Γ)
µ
0
R
Γ
g(x)dσ = 0
ˆϕ
R
Γ
ˆϕ(x)dσ = 0
ϕ(x) = ˆϕ(x) + Cϕ
0
(x) C
ϕ
0
−ϕ(x) + 2
Z
Γ
ϕ(y)
∂E(x, y)
∂n
x
dσ
y
= 0, x ∈ Γ.
ϕ
0
u V ϕ
ϕ ∈ C(Γ) V ϕ
Ω ϕ u = V ϕ
R
n
ϕ
Z
Γ
ϕ(y)E(x, y)dσ
y
= g(x), x ∈ Γ.
V ϕ
V ϕ
A
∂
∂n
x
Z
Ω
ψ(y)
∂E(x, y)
∂n
y
dσ
y
= g(x), x ∈ Γ.
ψ
y → x |x − y|
−3
Ω ⊂ R
3
u
∆u = −f Ω,
Γ = ∂Ω
[Bu](x) ≡ α(x)u(x) + β(x)
∂u(x)
∂n
= g(x), x ∈ Γ.
f Ω
u α β g Γ
g u
§2 u : R
3
→ R
u(x) = [Af](x) ≡
Z
Ω
E(x, y)f(y)dy,
E(x, y) = E
3
(x, y) = 1/(4π|x − y|)
§2 x ∈ Ω
α β g
Ω G(x, y) x ∈ Ω y ∈ Ω
y ∈ Ω
G(x, y) x ∈ Ω
Ω
G(x, y) = E(x, y) + v(x, y ) ≡
1
4π|x − y|
+ v(x, y),
v(x, y) x ∈ Ω
Ω C
1
(Ω)
G(x, y)
α(x)G(x, y) + β( x)
∂G(x, y)
∂n
x
= 0, x ∈ Γ.
y ∈ Ω
Ω v
Γ
α(x)v(x, y) + β(x)
∂v(x, y)
∂n
x
= −α(x)E(x, y) −β(x)
∂E(x, y)
∂n
x
, x ∈ Γ, y ∈ Ω.
y ∈ Ω
Γ Ω α β
Γ
G
G v y ∈ Ω
∆
x
G(x, y) = −δ(x, y) ∆
x
v(x, y) = 0
δ(x, y) δ y ∈ Ω
< ∆
x
G(x, y), ϕ >= − < δ(x, y), ϕ > ∀ϕ ∈ D(Ω)
y ∈ Ω
x ∈ Ω x ∆
G v x D(Ω)
Ω
G(x, y)
x ∈ Ω
y Ω
Γ
α ≡ 1 β ≡ 0 G Γ
Γ
1/(4π|x−y|)
y ∈ Ω
Γ
Ω
Ω R
3
Ω
Z
Ω
(v∆u − u∆v)dy =
Z
Γ
v
∂u
∂n
− u
∂v
∂n
dσ
u(x) = −
Z
Ω
1
4π|x − y|
∆u(y)dy+
1
4π
Z
Γ
1
|x − y|
∂u(y)
∂n
y
− u(y)
∂
∂n
y
1
|x − y|
dσ.
u v
Ω
α(x) = 1, β(x) ≡ 0 ∀x ∈ Γ; β(x) 6= 0 ∀x ∈ Γ
G
Ω×Ω
G(x, y) = G(y, x) ∀x, y ∈ Ω, x 6= y
x
1
x
2
Ω
G(x
1
, x
2
) = G(x
2
, x
1
).
S
1
S
2
ε x
1
x
2
Ω
ε
u
1
(x) = G(x, x
1
) =
1
4π|x − x
1
|
+ v( x, x
1
),
u
2
(x) = G(x, x
2
) =
1
4π|x − x
2
|
+ v( x, x
2
),
Ω
ε
= Ω\(B
ε
(x
1
) ∪ B
ε
(x
2
))
S
Γ
Ω
S
x
1
x
2
1
2
u
1
u
2
Ω
ε
Ω
ε
u
1
u
2
C
2
(Ω
ε
) ∩ C
1
(Ω
ε
)
∆u
1
= ∆u
2
= 0 Ω
Z
Γ
u
1
∂u
2
∂n
− u
2
∂u
1
∂n
dσ ≡
Z
Γ
u
1
∂u
2
∂n
+ ηu
2
−
−u
2
∂u
1
∂n
+ ηu
1
dσ,
η = α/β
Z
S
1
u
2
∂u
1
∂n
− u
1
∂u
2
∂n
dσ = −
Z
S
2
u
2
∂u
1
∂n
− u
1
∂u
2
∂n
dσ.
R
S
1
dσ = 4πε
2
u
2
(x
ε
)
∂u
1
(x
ε
)
∂n
− u
1
(x
ε
)
∂u
2
(x
ε
)
∂n
4πε
2
, x
ε
∈ S
1
,
x
ε
u
2
∂v(·, x
1
)/∂n
v(·, x
1
)∂u
2
/∂n
S
1
ε → 0
u
1
(x
ε
)
∂u
2
(x
ε
)
∂n
4πε
2
=
1
4πε|x
ε
− x
1
|
+ v(x
ε
, x
1
)
∂u
2
(x
ε
)
∂n
4πε
2
→ 0
ε → 0,
u
2
(x
ε
)
∂u
1
(x
ε
)
∂n
4πε
2
=
−
∂
∂r
1
4πr
|
r=ε
+
∂v(x
ε
, x
1
)
∂n
4πε
2
u
2
(x
ε
) =
=
1
4πε
2
+
∂v(x
ε
, x
1
)
∂n
u
2
(x
ε
)4πε
2
→ u
2
(x
1
) = G(x
1
, x
2
)
ε → 0.
ε → 0 G(x
1
, x
2
)
ε → 0
G(x
2
, x
1
)
x
1
x
2
G(x, y)
y x
y x
v
G v
x ∈ Ω G v
y ∈ Ω
∆
y
G(x, y) = −δ(x, y),
∆
y
v(x, y) = 0,
v y Ω
C
1
(Ω)
x ∈ Ω G v
α(y)G(x, y) + β(y)
∂G(x, y)
∂n
= 0, y ∈ Γ,
α(y)v( x, y ) + β(y)
∂v(x, y)
∂n
= −α(y)
1
4π|x − y|
−β(y)
∂
∂n
y
1
4π|x − y|
, y ∈ Γ.
v(x, y) y
C
2
(Ω) ∩ C
1
(Ω) x ∈ Ω
x y
v(x, y)
u C
2
(Ω)∩
C
1
(
Ω) v(x, y)
x ∈ Ω y Ω
u v(x, ·)
∆u = −f, ∆
y
v = 0 Ω ∀x ∈ Ω ,
0 =
Z
Ω
v(x, y)f(y)dy +
Z
Γ
v(x, y)
∂u(y)
∂n
y
− u(y)
∂v(x, y)
∂n
y
dσ
y
.
u x ∈ Ω
∆u = −f
u(x) =
Z
Ω
G(x, y)f(y)dy +
Z
Γ
G(x, y)
∂u(y)
∂n
y
− u(y)
∂G(x, y)
∂n
y
dσ
y
.
α(x) ≡ 1 β(x) ≡ 0 v
G(x, y) = 0 y ∈ Γ ∀x ∈ Ω.
G
u(x) =
Z
Ω
G(x, y)f(y)dy −
Z
Γ
∂G(x, y)
∂n
g(y)dσ
y
.
α(x) ≡ 0 β(x) ≡ 1 v
∂G(x, y)
∂n
y
= 0
y ∈ Γ ∀x ∈ Ω.
G
u(x) =
Z
Ω
G(x, y)f(y)dy +
Z
Γ
G(x, y)g(y)dσ
y
.
β(x) = 1 α(x) v
∂G(x, y)
∂n
+ α(y)G(x, y) = 0
y ∈ Γ ∀x ∈ Ω.
G
u(x) =
Z
Ω
G(x, y)f(y)dy +
Z
Γ
G(x, y)g(y)dσ
y
.
G(x, y)
G(x, y)
G
v
Ω
Γ = ∂Ω
v
f
g
G
Γ
v(·, y)
C
1
(Ω)
v ∈ C( Ω) v
u v
α(x) β(x) g( x) f (x)
C
2
(Ω) ∩ C
1
(Ω)
G(x, y)
Z
Ω
G(x, y)f(y)dy ,
Z
Γ
G(x, y)g(y)dσ
y
,
Z
Γ
∂G(x, y)
∂n
g(y)dσ
y
,
Z
Ω
E
3
(x, y)f(y)dx,
Z
Γ
E
3
(x, y)g(y)dσ,
Z
Γ
∂
∂n
E
3
(x, y)g(y)dσ.
ϕ
α(x)ϕ(x) =
Z
b
a
K(x, y)ϕ(y)dy + f(x), x ∈ [a, b],
α f K(·, ·) ϕ
K : [a, b] × [a, b] → R ( C)
f
α(x) ≡ 1
ϕ(x) =
Z
b
a
K(x, y)ϕ(y)dy + f(x).
f = 0
α = 0
Z
b
a
K(x, y)ϕ(y)dy = −f (x),
A
[Aϕ](x) =
Z
b
a
K(x, y)ϕ(y)dy.
ϕ(x) − [Aϕ] (x) = f.
λ
ϕ(x) − λ[Aϕ](x) = f.
λ λ ϕ
ϕ(x) − λ[Aϕ](x) = 0,