Алексеев Г.В. Классические методы математической физики. Часть 2
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g ∈ C(Γ) u
Ω
e
u C
2
(Ω
e
) ∩ C(Ω
e
)
Ω
e
Ω
e
Ω Ω
e
= R
n
\Ω
|u(x)| = o(1) |x| → ∞
n ≥ 3
|u(x)| = O(1) |x| → ∞
Ω ⊂ R
2
u(x)
x → ∞
u x
Ω
e
Ω
e
R
n
n ≥ 3 R
2
Ω
e
⊂ R
3
u ∈ C
2
(Ω
e
)
Ω
e
= R
3
\Ω Ω
e
u Ω
e
|u(x)| ≤ max
x∈Γ
|u(x)| ∀x ∈ Ω
e
, Γ = ∂Ω.
M = max
x∈Γ
|u(x)| B
R
R Ω
R Ω ⊂ B
R
|u(x)| < M ∀x ∈ R
3
\B
R
.
Ω
e
Ω
R
= Ω
e
∩ B
R
= B
R
\Ω.
u ∂Ω
R
= Γ ∪ Γ
R
Γ
R
=
∂B
R
|u(x)| ≤ M,
M u
Ω
R
|u(x)| < M Γ
R
Ω
R
Ω
u(x) = 1/|x| B
1
R
3
\B
1
|x| > 1 −1/|x|
u ∈ C
2
(Ω) ∩ C(Ω)
u ∈ C
2
(Ω
e
) ∩ C(Ω
e
) R
3
u
1
u
2
. u =
u
2
− u
1
Ω
Γ u(x) ≡ 0 Ω ⇒
u
1
= u
2
Ω
u
1
u
2
u = u
2
−u
1
∆u = 0 Ω
e
Ω
e
Ω
e
Γ
u(x) ≡ 0 Ω
e
⇒ u
1
= u
2
Ω
e
R
n
n ≥ 3
u ∈ C
2
(Ω) ∩C
0
(Ω)
u ∈ C
2
(Ω
e
) ∩ C(Ω
e
) R
2
u
1
u
2
u = u
2
− u
1
Ω
e
= Ω
e
∪ Γ Γ
Ω
e
|u(x)| ≤ M ∀x ∈ Ω
e
x
0
∈ Ω
e
Ω
e
Ω Ω x
0
x
0
Γ
r
r Ω Γ
R
Ω x
0
Ω
R
Γ
Γ
R
Ω
R
v
R
v
R
(x, x
0
) = M
ln(|x − x
0
|/r)
ln(R/r)
, x ∈ Ω
R
.
§1
v
R
Ω
R
Ω
R
M
Γ
R
Γ
x ∈ Γ
|x −x
0
| > r v
R
Ω
R
u ∂Ω
R
= Γ ∪ Γ
R
|u(x)| ≤ v
R
(x, x
0
),
v
R
u ∂Ω
R
x ∈ Ω
R
R v
R
v
R
(x
0
, x
0
) → 0 R → ∞.
u(x
0
) = 0
x
0
∈ Ω
e
u(x) = 0 Ω
e
⇒ u
1
= u
2
Ω
e
Ω
e
u R
2
Γ
R
K
R
K
R
\{0}
u
Ω
e
= R
2
\Ω Ω
e
u Ω
e
|u(x)| ≤ max
x∈Γ
|u(x)| ∀x ∈ Ω
e
,
u
lim
x→∞
u(x) = u
∞
,
|
∂u(x)
∂x
| = O(
1
|x|
2
), |
∂u(x)
∂y
| = O(
1
|x|
2
), |x| → ∞.
R
3
R
3
Ω = Ω
a
= {(x, y) ∈ R
2
: x
2
+ y
2
< a
2
} a, Ω
e
= R
2
\Ω
∆u = 0
Ω
u = g Γ
a
= ∂Ω,
Ω
e
u(x, y) = O(1) (x
2
+ y
2
) → ∞.
ρ, ϕ
∆u ≡
1
ρ
∂
∂ρ
ρ
∂u
∂ρ
+
1
ρ
2
∂
2
u
∂ϕ
2
= 0.
u(ρ, ϕ) = R(ρ)Φ(ϕ).
ρ(ρR
′
)
′
R
= −
Φ
′′
Φ
= λ,
λ
R Φ
ρ(ρR
′
)
′
− λR = 0 (0, a),
Φ
′′
+ λΦ = 0 (0, 2π).
u Φ
Φ(0) = Φ(2π).
λ
λ
k
Φ
k
λ
k
= k
2
, Φ
k
(ϕ) = a
k
coskϕ + b
k
sinkϕ, k = 0, 1, 2, ...
k
λ k
2
ρ
2
R
′′
+ ρR
′
− k
2
R = 0.
k > 0
R(ρ) = ρ
µ
.
ρ
µ
µ
2
= k
2
=⇒ µ = ±k.
k > 0 ρ
k
ρ
−k
k = 0 1 lnρ
u Ω
Ω
e
Ω
R
R
0
(ρ) = 1, R
k
(ρ) = ρ
k
, k ≥ 1
R
0
(ρ) = 1, R
k
(ρ) = ρ
−k
, k ≥ 1
2π ϕ
u
k
(ρ, ϕ) = ρ
k
(a
k
coskϕ + b
k
sinkϕ) , k = 0, 1, ...,
u
k
(ρ, ϕ) = ρ
−k
(a
k
coskϕ + b
k
sinkϕ), k = 0, 1, ...,
u(ρ, ϕ) =
∞
X
k=0
ρ
k
(a
k
coskϕ + b
k
sinkϕ)
u(ρ, ϕ) =
∞
X
k=0
ρ
−k
(a
k
coskϕ + b
k
sinkϕ)
2π ϕ
r ϕ Ω
Ω
e
a
k
b
k
ρ → a
u(a, ϕ) =
∞
X
k=0
a
k
(a
k
coskϕ + b
k
sinkϕ) = g(ϕ).
g
(i) g ∈ C
0
[0, 2π] g(0) = g(2π)
g
g(ϕ) =
α
0
2
+
∞
X
k=1
(α
k
coskϕ + β
k
sinkϕ),
α
0
=
1
π
2π
Z
0
g(ψ)dψ, α
k
=
1
π
2π
Z
0
g(ψ)coskψdψ, β
k
=
1
π
2π
Z
0
g(ψ)sinkψdψ, k = 1, 2, ... .
a
k
b
k
a
0
= α
0
/2 a
k
= α
k
/a
k
b
k
= β
k
/a
k
k = 1, 2, ...
u(ρ, ϕ) =
α
0
2
+
∞
X
k=1
u
k
(ρ, ϕ) ≡
α
0
2
+
∞
X
k=1
ρ
a
k
(α
k
coskϕ + β
k
sinkϕ).
u
u(ρ, ϕ) =
α
0
2
+
∞
X
k=1
a
ρ
k
(α
k
coskϕ + β
k
sinkϕ) .
Ω Ω
e
Ω
ρ
0
= {(ρ, ϕ) : 0 ≤ ρ ≤ ρ
0
, ϕ ∈ [0, 2π)} ρ
0
< a
X
k
∂u
k
∂ρ
,
X
k
∂
2
u
k
∂ρ
2
,
X
k
∂u
k
∂ϕ
,
X
k
∂
2
u
k
∂ϕ
2
,
(i) α
k
β
k
g
|α
0
|, |α
k
| |β
k
| < M = const ∀k = 1, 2, ... .
|coskϕ| ≤ 1 |si nkϕ| ≤ 1
Ω
ρ
0
M
2
+ 2M
X
k
ρ
0
a
k
,
2M
a
X
k
k
ρ
0
a
(k−1)
,
2M
a
2
X
k
k(k −1)
ρ
0
a
(k−2)
,
2M
X
k
k
ρ
0
a
k
, 2M
X
k
k
2
ρ
0
a
k
.
ρ ϕ ρ < a
Ω
Ω
|α
0
| + Σ
∞
k=1
(|α
k
| + |β
k
|),
Ω
|u
k
(ρ, ϕ)| ≤ |α
k
| + |β
k
| Ω k = 1, 2, ...
(i)
g [0 , 2π] g
′
˜α
k
˜
β
k
g
′
α
k
β
k
g
˜α
k
≡
1
π
2π
Z
0
g
′
(ϕ)coskϕdϕ = k
1
π
2π
Z
0
g(ϕ) sinkϕdϕ = kβ
k
, k = 1, 2, ...,
˜
β
k
≡
1
π
2π
Z
0
g
′
(ϕ)sinkϕdϕ = −k
1
π
2π
Z
0
g(ϕ) cos kϕdϕ = −kα
k
, k = 1, 2, ... .
|α
k
| + |β
k
| = (|˜α
k
| + |
˜
β
k
|)/k
∞
X
k=1
(
|˜α
k
|
k
+
|
˜
β
k
|
k
)
.
|˜α
k
|
k
≤
1
2
˜α
2
k
+
1
k
2
,
|
˜
β
k
|
k
≤
1
2
˜
β
2
k
+
1
k
2
Σ
∞
k=1
(˜α
2
k
+
˜
β
2
k
)
Σ
∞
k=1
(1/k
2
)
g
′
(i)
Ω
(ii) Ω
Ω
e
Ω
(ii)
Ω
e
g
u(ρ, ϕ) =
1
π
2π
Z
0
g(ψ)
(
1
2
+
∞
X
k=1
ρ
a
k
(coskψcoskϕ + sinkψsinkϕ)
)
dψ =
=
1
π
2π
Z
0
g(ψ)
(
1
2
+
∞
X
k=1
ρ
a
k
cosk(ϕ − ψ)
)
dψ.
t ≡ ρ/a < 1
Σ
∞
k=1
γ
k
=
γ
1 − γ
|γ| < 1,
1
2
+ Σ
∞
k=1
t
k
cosk(ϕ − ψ) =
1
2
+
1
2
Σ
∞
k=1
t
k
h
e
ik(ϕ−ψ)
+ e
−ik(ϕ−ψ)
i
=
=
1
2
n
1 + Σ
∞
k=1
h
(te
i(ϕ−ψ)
)
k
+ (te
−i(ϕ−ψ)
)
k
io
=
=
1
2
1 +
te
i(ϕ−ψ)
1 − te
i(ϕ−ψ)
+
te
−i(ϕ−ψ)
1 − te
−i(ϕ−ψ)
=
=
1
2
·
1 − te
−i(ϕ−ψ)
− te
i(ϕ−ψ)
+ t
2
+ te
i(ϕ−ψ)
− t
2
+ te
−i(ϕ−ψ)
− t
2
(1 − te
i(ϕ−ψ)
)(1 − te
−i(ϕ−ψ)
)
=
=
1
2
·
1 − t
2
1 − 2tcos(ϕ − ψ) + t
2
, |t| < 1.
u(ρ, ϕ) =
1
2π
2π
Z
0
a
2
− ρ
2
a
2
− 2aρcos(ϕ − ψ) + ρ
2
g(ψ)dψ.
k(ρ, ϕ; a, ψ) ≡
a
2
− ρ
2
a
2
− 2aρcos(ϕ − ψ) + ρ
2
k
ρ < a ρ > a k(ρ, ϕ; a, ψ) > 0 ρ < a 2aρ < a
2
+ ρ
2
ρ 6= a
x = (ρ, ϕ) ∈ Ω y = (a, ψ) ∈ Γ
a
ϕ, ψ ∈ [0, 2π)
r = r
xy
x y
r
2
xy
= |x − y|
2
= a
2
− 2aρcos(ϕ − ψ) + ρ
2
.
s = aψ (ds = adψ)
u(x) =
1
2πa
Z
Γ
a
(a
2
− ρ
2
)
r
2
g(y)ds
y
=
1
2πa
Z
Γ
a
(a
2
− |x|
2
)
|x − y|
2
g(y)ds
y
, ρ = |x|.
Ω
g ∈ C(Γ
a
)
g ∈ C(Γ
a
)
Ω
g
lim
x→x
0
u(x) = g(x
0
) ∀x
0
∈ Γ
a
, lim
ρ→a
ϕ→ϕ
0
u(ρ, ϕ) = g(ϕ
0
) ∀ϕ
0
∈ [0, 2π),