Алексеев Г.В. Классические методы математической физики. Часть 2
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B
δ
0
(x
0
) δ
0
x
0
x
0
∈ Ω ε > 0
δ
0
= δ
0
(ε) J x ∈ B
δ
0
(x
0
) ∩Ω
Z
ω
δ
∩Ω
F (x, y)ρ(y)dy
< ε
ω
δ
δ ≤ δ
0
x
0
x x
0
δ
U(x
0
)
x
0
C > 0 λ < 3
|F (x, y)| ≤
C
|x − y|
λ
∀x, y ∈ U(x
0
) ∩
Ω, x 6= y .
x
0
B
δ
0
(x
0
)
x
0
ω
δ
δ ≤ δ
0
x
0
x ∈ B
δ
0
(x
0
)
Z
ω
δ∩ Ω
F (x, y)ρ(y)dy
≤
Z
ω
δ
∩Ω
|F (x, y)||ρ(y) |dy
≤ CM
Z
B
2δ
0
(x)
dy
|x − y|
λ
.
B
2δ
0
(x) 2δ
0
x
r, θ, ϕ x r = |x − y|
Z
B
2δ
0
(x)
dy
|x − y|
λ
=
2π
Z
0
dϕ
π
Z
0
sinθdθ
2δ
0
Z
0
r
2
r
λ
dr = 4π
2δ
0
Z
0
r
2−λ
dr =
4π
3 − λ
(2δ
0
)
3−λ
.
δ ≤ δ
0
Z
ω
δ
∩Ω
F (x, y)ρ(y)dy
≤
4π
3 − λ
(2δ
0
)
3−λ
.
λ < 3
ε δ
0
> 0
x, y ∈ Ω ⊂ R
2
λ < 2
R
3
Z
Γ
F (x, y)ρ(y)dσ
y
, x ∈ Γ,
Γ
λ = 1
∂
∂x
i
1
|x − y|
≡
|x
i
− y
i
|
|x − y|
3
≤
1
|x − y|
2
, x 6= y, i = 1, 2, 3,
λ = 2
x ∈ Ω
F
λ =
1
e
1
= n
y
F (x, y) ≡
1
4π
∂
∂n
y
1
|x − y|
=
1
4π
x − y
|x − y|
3
· n
y
=⇒ |F (x, y )| ≤
1
4π
1
|x − y|
2
.
λ = 2
x ∈ Γ
n
n
x
λ n n
λ n − 1 n
E
n
n
Γ n = 2
λ
λ
|ln |x − y|| ≤ C|x − y|
−λ
∀x, y ∈ Q, x 6= y,
Q R
2
λ ∈ (0, 1) C = C(Q)
x
0
∈ Ω
x
0
ε > 0
δ > δ(ε) x ∈ Ω
|x − x
0
| < δ
|J(x) − J(x
0
)| < ε.
B
δ
≡ B
δ
(x
0
)
J(x) J(x
0
) Ω∩B
δ
(x
0
)
Ω\B
δ
(x
0
)
|J(x) − J(x
0
)| =
Z
Ω
[F (x, y) − F (x
0
, y)]ρ(y)dy
≤
≤
Z
Ω\B
δ
[F (x, y) − F (x
0
, y)]ρ(y)dy
+
Z
Ω∩B
δ
F (x, y)ρ(y)dy
+
Z
Ω∩B
δ
F (x
0
, y)ρ(y )dy
.
δ > 0
ε/3
x
0
|x − x
0
| < δ δ
′
< (δ/2)
|x −x
0
| < δ
′
F y ∈ Ω\B
δ
ε/3 x
x
0
Γ n = 2
C
l
l ∈ N B
r
(x
0
) x
0
∈ Γ
F
x
0
(x) = 0 F
x
0
∈ C
l
(B
r
(x
0
)) l ≥ 1
F
′
x
0
6= 0 l ≥ 1
Ω
Γ
Γ
i
Γ
i
Γ
Ω R
3
P, Q, R : Ω → R
Ω
Ω
Γ
P, Q, R ∈ C
1
(Ω) ∩ C(Ω)
Ω
P, Q R
Z
Ω
∂P
∂x
+
∂Q
∂y
+
∂R
∂z
dx =
Z
Γ
(P cos α + Q cos β + R cos γ) dσ,
x = x y z σ
α, β γ n Γ
i, j k
Z
Ω
divv
x =
Z
Γ
v
n
σ ≡
Z
Γ
v · n
σ, v
n
≡ v · n.
v v
v = P i + Qj + Rk, divv =
∂P
∂x
+
∂Q
∂y
+
∂R
∂z
.
P = uv, Q = 0, R = 0
Z
Ω
∂
∂x
(uv)dx ≡
Z
Ω
∂u
∂x
v + u
∂v
∂x
dx =
Z
Γ
uv cos αdσ.
x x
i
Z
Ω
∂u
∂x
i
vdx = −
Z
Ω
u
∂v
∂x
i
dx +
Z
Γ
uv cos(n, x
i
)dσ,
R
3
P = u∂v/∂x Q = u∂v/∂y R = u∂v/∂z
Z
Ω
u∆v
x =
Z
Γ
u
∂v
∂n
σ −
Z
Ω
∇u · ∇v x.
∇u · ∇v =
∂u
∂x
∂v
∂x
+
∂u
∂y
∂v
∂y
+
∂u
∂z
∂v
∂z
,
∂v/∂n
∂v
∂n
= ∇v · n =
∂v
∂x
cos α +
∂v
∂y
cos β +
∂v
∂z
cos γ
Γ.
u v
Z
Ω
v∆u
x =
Z
Γ
v
∂u
∂n
σ −
Z
Ω
∇u · ∇v x.
Z
Ω
(u∆v −v∆ u)
x =
Z
Γ
u
∂v
∂n
− v
∂u
∂n
σ.
Ω
u v
u, v ∈ C
1
(Ω) ∩ C(Ω)
(2a) Ω u v
u ∈ C
1
(Ω) ∩ C(Ω) v ∈ C
2
(Ω) ∩ C
1
(Ω)
u ∈ C
2
(Ω) ∩ C
1
(Ω) v ∈ C
1
(Ω) ∩ C(Ω)
(2a)
u, v ∈ C
2
(Ω) ∩ C
1
(Ω)
Ω
′
u ∈ C
1
(Ω)
′
u ∈ C
1
(Ω) v ∈ C
2
(Ω)
′
u ∈ C
2
(Ω) v ∈ C
1
(Ω)
′
u, v ∈ C
2
(Ω)
Γ
Ω
Ω
R
2
R
n
S R
3
S
( ) S
Γ
( ) P, Q, R
S
( ), ( )
Z
S
∂R
∂y
−
∂Q
∂z
y z +
∂P
∂z
−
∂R
∂x
z x +
∂Q
∂x
−
∂P
∂y
x y =
=
I
Γ
P
x + Q y + R z,
Γ
S
Z
S
rotv · n
σ =
I
Γ
v · t s.
t Γ
S s Γ v · t
v
Γ rotv
v
rotv =
∂R
∂y
−
∂Q
∂z
i +
∂P
∂z
−
∂R
∂x
j +
∂Q
∂x
−
∂P
∂y
k.
rotv
S
v Γ
rotv
S v Γ
S
S Γ
Γ S
Γ Γ
Γ
Γ rotv
S Γ
S
Ω x, y R = 0 P Q
z
Z
Ω
∂Q
∂x
−
∂P
∂y
x y =
I
Γ
P x + Q y,
Ω
(2a) P Q Ω
Ω Ω
P Q
Γ
Ω
v : Ω → R
v
rotv = (∂Q/∂x−∂P/∂y) k
x, y ∂Q/∂x − ∂P/∂y
P Q
v
R
Ω
rotvdxdy =
H
Γ
v · tds
Ω ∆ ∇
∂/∂n
R
3
R
2
R
n
R
n
x
1
, x
2
, ..., x
n
∆u =
∂
2
u
∂x
2
1
+
∂
2
u
∂x
2
2
+ . . . +
∂
2
u
∂x
2
n
, ∇u =
∂u
∂x
1
,
∂u
∂x
2
, . . . ,
∂u
∂x
n
,
divv =
∂v
1
∂x
1
+
∂v
2
∂x
2
+ . . . +
∂v
n
∂x
n
,
∂u
∂n
=
∂u
∂x
1
n
1
+
∂u
∂x
2
n
2
+ . . . +
∂u
∂x
n
n
n
.
v
1
, v
2
, ..., v
n
n
1
, n
2
, ..., n
n
v n
R
3
R
2
R
n
n ≥ 4
n
C
2
Ω R
3
Γ, x
0
∈ Ω u ∈ C
2
(Ω) ∩ C
1
(Ω)
x
0
∈ Ω
Z
Ω
∆u(x)
x
|x − x
0
|
.
u(x
0
) =
1
4π
Z
Γ
1
|x − x
0
|
∂u(x)
∂n
x
− u(x)
∂
∂n
x
1
|x − x
0
|
σ −
1
4π
Z
Ω
∆u(x)
|x − x
0
|
x,
C
2
u ∈ C
2
(Ω)
Ω B
ε
(x
0
) ⊂ Ω ε
x
0
Ω
ε
S
ε
B
ε
(x
0
)
u v = 1/r
r = |x − x
0
| Ω
ε
v Ω
ε
Z
Ω
ε
∆u
r
x =
Z
Γ
1
r
∂u
∂n
x
− u
∂
∂n
x
1
r
σ
x
+
Z
S
ε
1
r
∂u
∂n
x
− u
∂
∂n
x
1
r
σ
x
.
n Ω
ε
∂u/∂n
n x σ
x
x x
0
ε → 0
Ω
ε
ε → 0
−4πu(x
0
)
S
ε
n x
x − x
0
r = |x − x
0
| = ε
∂
∂n
x
1
r
= −
∂
∂r
1
r
=
1
r
2
=⇒
∂
∂n
x
1
r
S
ε
=
1
ε
2
,
I
1
(x
0
) ≡
Z
S
ε
u
∂
∂n
x
1
r
σ =
1
ε
2
Z
S
ε
u
σ.
u ∈ C
1
(Ω)
S
ε
4πε
2
I
1
(x
0
) = 4πu(x
ε
)
x
ε
S
ε
ε → 0
u
lim
ε→0
I
1
(x
0
) ≡ lim
ε→0
Z
S
ε
u
∂
∂n
x
1
r
σ = 4πu(x
0
).
u ∈ C
1
(Ω) C > 0 |∂u/∂n| ≤
C
Ω
Z
S
ε
1
r
∂u
∂n
x
σ
≤ C
1
ε
Z
S
ε
σ = 4πCε → 0 ε → 0.