Алексеев Г.В. Классические методы математической физики. Часть 2
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µ
Z
Γ
µ(y)dσ
y
= 0.
R
2
n = 2
Γ µ
v C
2
(B
ε
)
v(y) = 0,
∂v(y)
∂n
= µ(y), y ∈ Γ
′
ε
,
Z
Γ
′
ε
µ(y)ln|x − y|dσ
y
+
Z
S
′
ε
ln|x − y|
∂v(y)
∂n
y
− v(y)
∂
∂n
y
ln|x − y|
dσ
y
+
+q(x)v(x) =
Z
B
′
ε
ln|x − y|∆
y
v(y)dy, x ∈ B
ε
(x
0
).
x
0
∈ Γ
u(x
0
) ≡ −
1
2π
Z
Γ
ln|x
0
−y|µ(y)dσ
y
= −
1
2π
lim
ε→0
Z
Γ
′′
ε
(x
0
)
ln|x
0
−y|µ(y)dσ
y
.
B
ε/2
(x
0
)
Γ
u(x) = −
1
2π
Z
Γ
′′
ε
µ(y)ln|x − y|dσ
y
−
−
1
2π
Z
S
′
ε
v(y)
∂
∂n
y
ln|x − y| − ln|x − y|
∂v(y)
∂n
y
dσ
y
−
−
1
2π
Z
B
′
ε
ln|x − y|∆
y
v(y)dy +
1
2π
q(x)v(x).
x ∈ B
ε/2
(x
0
)
C
1
(R
2
)
v
Γ
′
ε
qv v B
ε
(x
0
) x → x
0
qv B
ε
(x
0
)
Γ
Γ
∂u
+
(x
0
)
∂n
−
∂u(x
0
)
∂n
=
1
2
µ(x
0
), x
0
∈ Γ,
∂u
−
(x
0
)
∂n
−
∂u(x
0
)
∂n
= −
1
2
µ(x
0
), x
0
∈ Γ,
∂u
+
(x
0
)
∂n
−
∂u
−
(x
0
)
∂n
= µ(x
0
), x
0
∈ Γ.
∂u(x
0
)/∂n
∂u(x
0
)
∂n
=
1
2π
Z
Γ
(y − x
0
) · n(x
0
)
|x
0
− y|
2
µ(y)dσ
y
=
= lim
ε→0
1
2π
Z
Γ
′′
ε
(x
0
)
(y − x
0
) · n(x
0
)
|x
0
− y|
2
µ(y)dσ
y
.
(i) (ii)
u(x
0
) : Γ → R
x
0
∈ Γ C
1
(Γ)
(i), ( ii)
x Ω Ω
e
x
0
∈ Γ ∂u/∂n
n ≥ 3
Γ Ω µ
S
x ∈ Γ
n = n(x) r
0
> 0
x ∈ Γ Γ∪B
r
0
(x) Γ ∪B
r
0
(x)
x Γ
n
x
λ ≤ 1 Γ
λ ≤ 1 C > 0
|n(x) − n (y)| ≤ C|x − y|
λ
∀x, y ∈ Γ.
C
1
C
2
λ ≤ 1
˜
C
1,λ
C
2
⊂
˜
C
1,λ
⊂ C
1
λ ≤ 1
Ω R
n
Γ
˜
C
1,λ
0 < λ < 1
µ ∈ C(Γ)
x ∈ R
n
x ∈ Γ
u(x) = O(|x|
1−n
) |x| → ∞.
R
n
\ Γ u
u ∈ C
∞
(R
n
\ Γ)
x
Γ u
u
+
(x
0
) u
−
(x
0
)
x
0
∈ Γ
u
+
(x
0
) = −
1
2
µ(x
0
) +
Z
Γ
µ(y)
∂
∂n
y
E(x
0
, y)dσ
y
,
u
−
(x
0
) =
1
2
µ(x
0
) +
Z
Γ
µ(y)
∂
∂n
y
E(x
0
, y)dσ
y
,
u
+
(x
0
) − u
−
(x
0
) = −µ(x
0
) ∀x
0
∈ Γ.
Z
Γ
µ(y)
∂
∂n
y
E(x
0
, y)dσ
y
x
0
∈ Γ
C(Γ) u
+
∈ C(Ω) u
−
∈ C(Ω
e
)
µ ∈ C
α
(Γ) α ≤ λ u
+
∈ C
α
(Ω) u
−
∈
C
α
(
Ω
e
) x
0
∈ Γ
C
α
(Γ)
x ∈ R
n
x ∈ Γ n ≥ 3
u(x) = O
|x|
2−n
|x| → ∞.
R
n
\Γ u
u ∈ C
∞
(R
n
\Γ)
x
u ∈ C
α
(R
n
) ∀α ∈ [0, λ)
Γ n
Γ x
0
∈ Γ
∂u
+
(x
0
)/∂n ∂u
−
(x
0
)/∂n
Γ
∂u
+
(x
0
)
∂n
=
∂u(x
0
)
∂n
+
1
2
µ(x
0
), x
0
∈ Γ,
∂u
−
(x
0
)
∂n
=
∂u(x
0
)
∂n
−
1
2
µ(x
0
), x
0
∈ Γ.
∂u(x
0
)/∂n
x
0
∈ Γ
∂u(x
0
)
∂n
=
Z
Γ
µ(y)
∂
∂n
x
E(x
0
, y)dσ
y
.
∂u
+
(x
0
)
∂n
−
∂u
−
(x
0
)
∂n
= µ(x
0
) ∀x
0
∈ Γ,
∂u(x
0
)/∂n ∈ C(Γ)
(jj) µ ∈ C
α
(Γ) α ≤ λ gr adu
+
gradu
−
Ω Ω
e
gradu
+
∈ C
α
(Ω) gradu
−
∈ C
α
(Ω
e
) ∂u(x
0
)/∂n ∈ C
α
(Γ)
(j) (jj) Γ ∈ C
k+1,α
0 < α < 1 µ ∈ C
l,α
(Γ) l
k 0 ≤ l ≤ k u
+
∈ C
l+1,α
(Ω) u
−
∈ C
l+1,α
(Ω
e
)
Γ Ω
∆u = 0
Ω Γ Ω
e
= R
n
\ Ω
Ω Ω
u ∈ C
2
(Ω)∩
C(
Ω) Ω
u = g(x) Γ.
u ∈ C
2
(Ω
e
) ∩
C(
Ω
e
) Ω
e
u(x) = O
|x|
2−n
|x| → ∞.
u ∈ C
2
(Ω)∩
C
1
(
Ω) Ω
∂u
∂n
= g(x)
Γ.
u ∈ C
2
(Ω
e
) ∩
C
1
(
Ω
e
) Ω
e
u
u(x) = [W ψ] (x) ≡
Z
Γ
ψ(y)
∂E(x, y)
∂n
y
dσ
y
ψ ∈ C(Γ)
ψ ∈ C(Γ)
u
Ω u ∈ C
∞
(Ω) ∩ C
0
(Ω) ψ
x
0
∈ Γ
x → x
0
lim
x→x
0
,
x∈Ω
u(x) ≡ u
+
(x
0
) = −
1
2
ψ(x
0
) +
Z
Γ
ψ(y)
∂E(x
0
, y)
∂n
y
dσ
y
.
u
+
(x
0
) = g(x
0
), x
0
∈ Γ.
ψ(x) − 2
Z
Γ
ψ(y)
∂E(x, y)
∂n
y
dσ
y
= −2g(x) , x ∈ Γ.
ψ ∈ C( Γ)
g ∈ C(Γ)
ψ ∈ C(Γ) ψ
ψ
ψ ∈ C(Γ)
n ≥ 3 0 6∈ Ω
e
u(x) = [W ψ] (x) + A|x|
2−n
,
A |x|
2−n
R
n
x = 0
Ω
e
ψ ∈ C(Γ)
A
ψ
A
ψ
ψ(x) + 2
Z
Γ
ψ(y)
∂E(x, y)
∂n
y
dσ
y
= 2g(x) −
2A
|x|
n−2
, x ∈ Γ.
A
R
n
n ≥ 3 ψ ∈ C(Γ)
A
ψ(x) + 2
Z
Γ
ψ(y)
∂E(x, y)
∂n
y
dσ
y
= 0, x ∈ Γ,
ψ(x) + 2
Z
Γ
ψ(y)
∂E(y, x)
∂n
y
dσ
y
= 0, x ∈ Γ,
ψ
0
= 1
ψ ∈ C(Γ)
ψ
0
Z
Γ
g(x) −
A
|x|
n−2
dσ = 0.
A
ψ
u |x| → ∞ O(|x|
2−n
)
[W ψ](x) O(|x|
1−n
) |x| → ∞
Ω
e
u W ψ
W ψ A/|x|
n−2
O(|x|
−1
) |x| → ∞
n = 2
u
u
u(x) = [V ϕ] (x) + A =
Z
Γ
ϕ(y)E(x, y)dσ
y
+ A
ϕ
A
ϕ ∈ C(Γ) V ϕ
Ω
ϕ ∈ C(Γ)
x
0
∈ Γ
x
0
∂u
+
(x
0
)
∂n
=
1
2
ϕ(x
0
) +
Z
Γ
ϕ(y)
∂E(x
0
, y)
∂n
x
dσ
y
.
∂u
+
(x
0
)
∂n
= g(x
0
).
ϕ(x) + 2
Z
Γ
ϕ(y)
∂E(x, y)
∂n
x
dσ
y
= 2g(x), x ∈ Γ.
ϕ ∈ C(Γ)
ϕ ∈ C(Γ)
µ(x) + 2
Z
Γ
µ(y)
∂E(y, x)
∂n
x
dσ
y
= 0, x ∈ Γ
µ
0
= 1
Z
Γ
µ
0
(x)g( x)dσ ≡
Z
Γ
g(x)dσ = 0.
ϕ(x) =
ˆϕ(x)+Cϕ
0
(x)
ˆϕ ϕ
0
ϕ(x) + 2
Z
Γ
ϕ(y)
∂E(x, y)
∂n
x
dσ
y
= 0, x ∈ Γ,
u(x) = [V ϕ] (x) =
Z
Γ
ϕ(y)E(x, y)dσ
y
ϕ ∈ C(Γ)
∂u
−
(x
0
)/∂n x
0
∂u
−
(x
0
)
∂n
= −
1
2
ϕ(x
0
) +
Z
Γ
ϕ(y)
∂E(x
0
, y)
∂n
x
dσ
y
,
ϕ
−ϕ(x) + 2
Z
Γ
ϕ(y)
∂E(x, y)
∂n
x
dσ
y
= 2g(x), x ∈ Γ.
ϕ ∈ C(Γ)