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Rm+1
Γ
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m
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∇φ(x) 6= 0
x ∈ Γ
{x : φ(x) < 0}
;
κ
Γ
:J
Γ
n
n
Γ
n=
;
{x : φ(x) > 0}
∇φ
|∇φ|
and κ = ∇ ·
k≥2
∇φ
.
|∇φ|
k=2
b@ % b@
φ : Rm+1 × I →
I ⊂ R
po
t
φ(x, t)
R
C1
Γ (t) = {x : φ(x, t) = 0}
t 7→ Γ (t)
t 7→ Γ (t)
φ
α : I → Rm+1
C
2
'q
∇φ 6= 0
φ(α(t), t) ≡ 0
∂φ
∂φ
+ α̇ · ∇φ = 0 or
+ vn |∇φ| = 0 ,
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∂t
˙ = d/dt
r
stO
R
vn = α̇ · n
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m
m=1
φ
φ
Z
Γ
R
E(Γ ) = Γ 1 dσ
Z
|∇φ|δ(φ) dx =
F (φ, ∇φ) dx ,
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E(φ) =
„
…X∇φ
†
Rm+1
Rm+1
x
E(φ)
E(Γ )
pƒ
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|∇φ|δ(φ) dx
m+1
X ∂ ∂F ∂F
−
=0 ,
∂φ
∂xi ∂φxi
i=1
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κδ(φ) = 0
+H
∂φ
= ±κδ(φ) ,
∂t
E
φ
δ(φ)
|∇φ|
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¾G¾GY[Y[N1N ¾4?B*A[¾GbMN]NP?BA@Ã¥¾¨ÙUyÓSR¥?MNPDGÄFN]`*U?*FV¾GUZF `MF A@N ¾GDGJMJ¥DR¥R¥ÃSFÄÀšN]NPR¬DG¿KNP?BA@L ¾GFVJ*?BÇL§î F A[bÃSUEJB¾GJ?mbMbMF F `M`MNaNPAã?bMÃ[D4AS?MÄiR¥¾GF F NPJMA@AS¾E?BFL A@¾GNPDGÓ[DGN¾4?*¾GF JMA:\ JBÂ
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XH
‰
Š
4]
∂φ
= κ|∇φ| .
∂t
O
δ(φ)
@O 4[ 4O @]
M
Γ ⊂ Rm+1
Γ ∈M
m
9=FE
dE
Γ0
m
Γ0
M
Γ (0) = Γ0
Γ0 ∼ Γ
∼
Œ
M
|∇φ|
E:M →R
Rm+1
Γ0
t 7→ Γ (t)
φ : Rm+1 × I → R
Γ (t) = {x : φ(x, t)}
Γ (t) = Γ 0
t>0
Γ0 ∼ Γ Γ0
Γ
0
Γ
∼
Γ
Γ ∼ Γ
Γ 00 ∼ Γ 0
Γ 00 ∼ Γ
∼
Rm+1
M = {Γ 0 : Γ 0 ∼ Γ0 }
Γ ∈M
M
‹D
D
M
Γ ∈M
 0 Γ0
− < t < x0 ∈ Γ 0
Γ (0) = Γ0
φ = φ(x, t)
Γ (t) = {x : φ(x, t) = 0}
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í
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Z
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Z Z
d
∇φ
E(φ + sψ)
=
· ∇ψ δ(φ) dx +
|∇φ|δ 0 (φ)ψ dx
ds
|∇φ|
s=0
Z
Z
∇φ
= (−ψ) ∇ ·
δ(φ) dx = (−ψ)κ δ(φ) dx ,
|∇φ|
dE(Γ )v =
v = −ψ/|∇φ|
φ(x) + sψ(x)
>
|∇φ|/|∇φ|
dσ = |∇φ|δ(φ) dx
v
Z
Z
−ψ
dE(Γ )v = (
)κ dσ =
v κ dσ = hv, κiΓ .
|∇φ|
Γ
φs (x) =
−ψ
∇E = κ
φ(x, t) = 0
−(∂φ/∂t)/|∇φ| = −∇E = −κ
∂φ/∂t = κ|∇φ|
δ(φ)
φ
f :R→R
φ
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δ(φ)
f (0) = 0
|∇φ|
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g(x, n) = v(x) · n
∇E = ∇ · [gn + g n] = ∇ · [v − (v · n) n + (v · n) n] = ∇ · v ,
∂φ/∂t = (∇ · v)|∇φ|
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I : R2 → R
Γ
E(Γ ) =
Z
f (|∇I|) dσ ,
Γ
f : R+ → R+
F
f
g(x, n) = f (|∇I(x)|)
n
∇E =
∇ · (f (|∇I|)n)
i
h
i
∂φ h
= ∇ · (f (|∇I|)n) |∇φ| = f (|∇I|) κ + ∇f (|∇I|) · n |∇φ| .
∂t
‡
¸‹¹
R
E(Γ ) = Γ d(x) dσ
∇E = ∇d · n + dκ
d(x) : Rm+1 → R
14¾ 9½ p B"*147
R
ΓRd(x) dσ
Γ
F (Γ )
,
G(Γ )
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E(Γ ) =
(
∇G = κ
Γ
dσ
=
∇F = ∇d · n + dκ
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∇E =
∇d · n + κ(d − E(Γ ))
(∇d · n + dκ)G(Γ ) − F (Γ )κ
=
,
G(Γ )2
G(Γ )
G(Γ ) > 0
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Γ d(x) : Rm+1 → R
Ω
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E(Γ ) =
H(·)
(‡
Z
d(x) dx =
Ω
Z
d(x)(1 − H(φ)) dx ,
9=FE
E
Z
d
d
dE(Γ )v =
E(φ + sψ)
=
(1 − H(φ + sψ))
d(x) dx
ds
ds
s=0
s=0
Z
Z
−ψ
= (−ψ)δ(φ)d(x) dx = (
)d(x)|∇φ|δ(φ) dx
|∇φ|
Z
=
v d(x) dσ = hv, d(x)iΓ ,
Γ
í Û @î
n
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