Giga Y. Surface Evolution Equations: A Level Set Approach

Подождите немного. Документ загружается.

18 Chapter 1. Surface evolution equations

In any case if n is a unit normal vector ﬁeld around x

,then

Γ={τ ∈ R

; τ,n(x) =0}

for x ∈ Γ around x

. We shall often suppress the words ‘vector ﬁeld’. If a hyper-

surface Γ is a topological boundary ∂D of a domain D, the unit normal (vector

ﬁeld) pointing outward from D is called the outward unit normal (vector ﬁeld).

Evolving hypersurface. Suppose that Γ

is a set in R

depending on the time

variable t. We say that a family {Γ

} or simply Γ

is a (C

2m,m

) evolving hypersur-

face around (x

)(withx

∈ Γ

)ifthereisaC

2m,m

(m ≥ 1) function u(x, t)

deﬁned for t

− δ<t<t

+ δ, x ∈ U for some δ>0 and some neighborhood U

of x

in R

such that

∩ U = {x ∈ U; u(x, t)=0} (1.1.3)

and that the spatial gradient ∇u of u does not vanish on Γ

. (This is a level set

representation of Γ

.) Here by a C

2m,m

function we mean that derivatives ∇

(k)

∂

are continuous for k +2h ≤ 2m,where∂

denotes the h-th diﬀerentiation in the

time variable and ∇

(k)

denotes the k-th diﬀerentiation in the space variables. If

u can be taken C

∞

, i.e., C

2m,m

for all m ≥ 1, we just say that Γ

is a smoothly

evolving hypersurface around (x

). If Γ

is a C

2m,m

(resp. smoothly) evolving

hypersurface around all (x, t) with x ∈ Γ

and t belonging to an interval I,wesay

that Γ

is a C

2m,m

(resp. smoothly) evolving hypersurface on I. In this chapter we

always assume that Γ

is a smoothly evolving hypersurface in some time interval

unless otherwise claimed.

1.2 Normal velocity

Let n be a unit normal vector ﬁeld of Γ

so that it depends on time t smoothly.

The reasonable quantity which describes the motion of Γ

is a normal velocity,

that is the speed in the direction of n. Note that there is a chance that each point

of Γ

moves but the set Γ

is independent of time like a rotating sphere.

Deﬁnition 1.2.1.Letx

be a point of Γ

.Letx(t)bea(C

) curve deﬁned on

− δ, t

+ δ)forsomeδ>0 such that x(t)isapointonΓ

and x(t

)=x

.The

quantity

V =



), n



is called the normal velocity at x

of Γ

at the time t

in the direction of n.

As we will see later V is independent of the choice of curve x(t). We shall

give various expressions of V .

Level set representation. Suppose that Γ

is represented by (1.1.3). We shall

calculate the normal velocity V at (x

) in the direction of n deﬁned by

n(x

)=−

∇u(x

)

|∇u(x

1.2. Normal velo city 19

Let x(t)beacurveinR

on Γ

with x(t

)=x

. Since Γ

is the zero level set

of u(·,t), u(x(t),t)=0fort near t

. Diﬀerentiate u(x(t),t)int and evaluate at

)toget



), ∇u(x

)



=0,

where u

= ∂

u. Recalling n = −∇u/|∇u|,weobtain

V =

)

|∇u(x

. (1.2.1)

Clearly, this shows that V is independent of the choice of x(t).

Graph. We shall give a formula for V when Γ

is represented by a graph of a

function. By rotating coodinates we may assume that Γ

is expressed as

= {x

= g(x



,t); x



∈ R

N−1

}

around (x

), where g(x



)=x

and x



=(x

,...,x

N−1

). If n is taken upward,

is given as a zero level set of

u(x, t)=−x

+ g(x



,t)

with n = −∇u/|∇u| as in (1.1.2). Then by (1.2.1) we see that

V =



)

(1 + |∇



g(x



)

1/2

. (1.2.2)

Axisymmetric surface. Suppose now that Γ

is obtained by rotating the graph

of a function ϕ(x

,t) around the x

-axis and that x

is not on the axis. In other

words, around (x

)Γ

is of the form

⎧

⎪

⎨

⎪

⎩

r = ϕ(x

,t); r =

⎛

⎝



j=2

⎞

⎠

1/2

⎫

⎪

⎬

⎪

⎭

Since Γ

is regarded as the zero level set of

u(x, t)=−r + ϕ(x

,t),

the normal velocity V in the direction of

n =

(−ϕ

),x



/|x



(1 + (ϕ

))

)

1/2

= −

∇u(x

)

|∇u(x

(1.2.3)

at (x

)isoftheform

V =

)

(1 + (ϕ

))

)

1/2

(1.2.4)

20 Chapter 1. Surface evolution equations

where x



=(x

,...,x

)andϕ

= ∂ϕ/∂x

Rescaled motion. If we would like to study the behavior of Γ

near x

∗

∈ R

t tends to t

∗

with t<t

∗

, we often magnify Γ

near (x

∗

) by rescaling. There

are of course several ways to rescale but here we only give a typical example. Let

(y,s) be deﬁned by

y =(t

∗

− t)

−1/2

(x − x

∗

),s= −log(t

∗

− t)(1.2.5)

for t<t

∗

.Let

V denote the normal velocity of

= {y ∈ R

; y =(t

∗

− t)

−1/2

(x − x

∗

),x∈ Γ

}

at (y

), where s is regarded as the new time variable. Here the unit normal

vector

n(y

)of

is taken so that its direction is the same as the unit normal

n(x

)ofΓ

at x

,where(x

) is determined by (1.2.5) by setting (y, s)=

). Let V be the normal velocity of Γ

at x

in the direction of n(x

Then

V = e

−s

V +

 y

n(y

) . (1.2.6)

Note that the behavior of Γ

as t tends to t

∗

with t<t

∗

corresponds to the large

time behavior of

by (1.2.5).

To see this formula we use level set representation of Γ

near (x

). Suppose

that Γ

is represented by (1.1.3). We may assume n(x

)isofform

n(x

)=−

∇u(x

)

|∇u(x

We set

w(y, s)=u((t

∗

− t)

1/2

y + x

∗

,t),s= − log(t

∗

− t)

so that

is represented as a zero level set of w near (y

). Since

n(y

)=−

∇w(y

)

|∇w(y

the formula (1.2.1) yields

V =

)

|∇w(y

A direct calculation shows that



 y, ∇w 



)=e

−s

), ∇w(y

)=e

−s

∇u(x

These two identities together with representation of V ,

V yields (1.2.6).

If we consider a little bit more general rescaling than (1.2.5) of form

y =(t

∗

− t)

−α

(x − x

∗

),s= − log(t

∗

− t)withα>0,

then

V = e

−(1−α)s

V + α  y

n(y

) 

instead of (1.2.6).

1.3. Curvatures 21

1.3 Curvatures

Let Γ be a hypersurface in R

. For a point x

let τ be a tangent vector of Γ at

.LetX be a (C

) vector ﬁeld on Γ around x

, i.e., let X be a C

function from

ΓtoR

around x

.HereC

means that X canbeextendedtoaC

function in

a neighborhood of x

in R

. The vector ﬁeld X need not be tangential to Γ. Let

ζ be a curve on Γ that satisﬁes

ζ(0) = x

dζ

(0) = τ.

A tangential derivative in the direction of τ is deﬁned by

X)(x

(X(ζ(t))|

t=0

If X is extended to a neighborhood of x

in R

,weobservethat

X)(x

)=(τ ·∇)X =



j=1

∂

∂x

X, τ =(τ

,...,τ

This shows that the operator D

is independent of the choice of the curve ζ.By

the deﬁnition of D

the quantity (τ ·∇)X is independent of the extension of X

outside Γ. Thus the operator D

is well deﬁned for any vector ﬁeld on Γ.

Second fundamental form. Suppose now that Γ is a (C

) hypersurface around x on

Γ. Let n be a unit normal vector ﬁeld around x. From the level set representation

it is clear that n is C

on Γ. For each τ ∈ T

Γweset

Aτ = −(D

n)(x) ∈ R

Since |n| =1,

Aτ,n(x) = −

(|n|

)=0

so that Aτ ∈ T

Γ. The linear operator A = A

from T

Γ into itself is called

the Weingarten map (in the direction of n(x)). The bilinear form on T

Γ × T

associated with A deﬁned by

(τ,η)= Aτ, η  (1.3.1)

is called the second fundamental form (in the direction of n(x)) at x ∈ Γ.

To see the geometric meaning of B

for τ, η ∈ T

Γletφ be a function from

a neighborhood of the origin of R

to Γ ⊂ R

that satisﬁes

∂φ

∂x

(0, 0) = τ,

∂φ

∂x

(0, 0) = η, φ(0, 0) = x.

22 Chapter 1. Surface evolution equations

Since



n(φ(x

)),

∂φ

∂x

)



=0 (1.3.2)

near (x

)=(0, 0) ∈ R

, diﬀerentiating in x

and evaluating at zero yields



∂φ

∂x

(0, 0) ·∇



∂φ

∂x

(0, 0)





∂

∂x

(0, 0)



=0,

 (η ·∇)n,τ +



∂

∂x

(0, 0)



=0.

By deﬁnition this yields

(η, τ)=



∂

∂x

(0, 0)



. (1.3.3)

In particular B

is a symmetric bilinear form and the Weingarten map A is a

symmetric linear operator. Thus its eigenvalues are all real and called principal

curvatures of Γ at x (in the direction of n(x)). The principal curvatures are denoted

,...,κ

N−1

. Diﬀerentiating (1.3.2) in x

and evaluating at zero, we get

(τ, τ)=



∂

∂x

(0, 0)



instead of (1.3.3). If |τ | = 1, this quantity is called the normal curvature of Γ in

the direction of τ.

T *

H*

curvature of the curve

at equals

the normal curvature

in the direction of

*

Figure 1.2: Normal curvature

Surfaces of higher codimension. As well known the second fundamental form is

deﬁned even for any embedded manifold M in R

whose dimension k is strictly

1.3. Curvatures 23

less than N −1 (i.e., codimension N −k is strictly greater than 1). We brieﬂy review

the deﬁnition since it is almost the same as (1.3.1). For the tangent space T

of M at x let N

M denote its orthogonal complement in R

.ThespaceN

M is

called the normal space of M at x.SinceM has dimension k, the dimensions of

M and N

M equal k and N − k, respectively. Let n

,...,n

N−k

be (C

) vector

ﬁelds near x ∈ M such that {n

(z); 1 ≤ i ≤ N − k} is an orthonormal basis of

M for every z near x.Thesecond fundamental form of M at x is deﬁned by

(τ, η)=−

N−k



j=1

 (D

)(x),η n

(x),τ,η∈ T

as a mapping

: T

M × T

M → N

As in the same way we derived (1.3.3), we have

(η, τ)=π



∂

∂x

(0, 0)



with

∂φ

∂x

(0, 0) = τ,

∂φ

∂x

(0, 0) = η, φ(0, 0) = x,

where π denotes the orthogonal projection from R

onto N

M.InparticularB

is symmetric and B

is independent of the choice of n

,...,n

N−k

forming an

orthonormal basis of N

M for z near x. Note that the deﬁnition does not require

k<N− 1. If k = N − 1, then, as expected

 B

(τ,η), n  = B

(τ,η),

where B

is the second fundamental form in the direction of n. This shows that

is a natural generalization of B

Surface divergence. Let X be a C

vector ﬁeld on a hypersurface Γ. For x ∈ Γ

let {τ



;1≤  ≤ N − 1} be an orthonormal basis of T

Γ. The surface divergence

of X is denoted by div

X andisdeﬁnedby

(div

X)(x)=

N−1



=1

 (D



X)(x),τ



. (1.3.4)

If we extend X around Γ so that ∇X is well deﬁned as an N × N matrix, then

(1.3.4) yields

(div

X)(x)=trace



N−1



=1



⊗ τ





(∇X)(x)(1.3.5)

since D

X =(τ ·∇)X,where⊗ denotes the tensor product of N -vectors and trace

denotes the trace of an N × N matrix. The deﬁnition of div

X is independent of

24 Chapter 1. Surface evolution equations

the extension of X, so the right-hand side of (1.3.5) is independent of the extension

of X.Since

−

N−1



=1



⊗ τ



= n(x) ⊗ n(x)

as matrices, the identity (1.3.5) is rewritten as

(div

X)(x)=trace{(I − n(x) ⊗ n(x))(∇X(x))}, (1.3.6)

where I denotes the N × N unit matrix. From (1.3.6) it follows that the surface

divergence is deﬁned independent of the choice of orthonormal basis of T

Γand

the orientation n.

The surface divergence div

X is also deﬁned for a vector ﬁeld X on an

embedded manifold M of dimension k in R

whose codimension N − k>1. It is

deﬁned as (1.3.4), i.e.,

(div

X)(x)=



=1

(D

 X)(x),τ



,

where {τ



;1≤  ≤ k} is an orthonormal basis of T

M. As for a hypersurface,

div

X is independent of the choice of orthonormal basis of T

M which can be

proved directly from (1.3.5) with N − 1 replaced by k.

Mean curvature. Let n be a unit normal vector ﬁeld around x on a (C

)hyper-

surface Γ. Let H be the sum of all principal curvatures κ

(x),...,κ

N−1

(x)atx in

the direction of n, i.e.,

H = κ

(x)+···+ κ

N−1

(x).

We say that H is the mean curvature (of Γ) at x (in the direction of n). We do not

take the average of principal curvatures although many authors since the time of

Gauss have taken the average to deﬁne the mean curvature. Since H is the trace

of the Weingarten map,

H =

N−1



=1

(τ



,τ



)=−

N−1



=1



n(x),τ



)=−(div

n)(x),

where τ



( =1,...,m− 1) is an orthonormal basis of T

Γ. If N =2,H is simply

called the curvature at x ∈ Γ (in the direction of n) and is denoted κ.

Even if a manifold M in R

has higher codimension, or its dimension k<

N − 1, the mean curvature is deﬁned as a vector. We say that

H =

N−1



=1

(τ



,τ



) ∈ N

1.3. Curvatures 25

is the mean curvature vector at x ∈ M. By deﬁnition of the second fundamental

form and the surface divergence

H = −

N−k



i=1

(div

where n

(1 ≤ i ≤ N − k) is the same as in the deﬁnition of B

.Ifk = N − 1, then

by deﬁnition

H =  H, n ,

where H is the mean curvature in the direction of n.SinceN

Γ is one dimensional,

the mean curvature H has all information of the mean curvature vector if we take

the sign of H into account.

Symmetric curvatures. Let κ

(x),...,κ

N−1

(x) be principal curvatures at x ∈ Γ

(in the direction of n), where Γ is a (C

)hypersurfaceinR

. We shall consider

elementary symmetric polynomials of principal curvatures. Let e

(1 ≤ m ≤ N −1)

denote the m-th elementary symmetric polynomial of λ

,...,λ

N−1

, i.e.,

(λ

,...,λ

N−1



···λ

where the sum is taken over all integers i

,...,i

that satisfy 1 ≤ i

< ···<

≤ N − 1. In particular

(λ

,...,λ

N−1

)=λ

+ ···+ λ

N−1

(λ

,...,λ

N−1

)=λ

· λ

···λ

N−1

Clearly, the mean curvature is of the form

H = e

(κ

,...,κ

N−1

The quantity

K = e

N−1

(κ

,...,κ

N−1

)

is called the Gaussian curvature. In general we say that

= e

(κ

,...,κ

N−1

)

is the m-th symmetric curvature. Sometimes we consider a little more complicated

curvature deﬁned by the ratio H



with m>.ThequantityH

N−1

N−2

for

N ≥ 3 is called the harmonic curvature since

N−1

N−2



N−1



i=1



−1

Anisotropic curvatures. It is well known that the mean curvature H is the change

ratio of surface area of Γ per change of volume of D enclosed by Γ, where Γ is a (C

)

26 Chapter 1. Surface evolution equations

hypersurface in R

. If the hypersurface Γ has anisotropic structure depending on

its normal direction, it is natural to consider surface energy instead of area. Let

be a positive function deﬁned on the unit sphere

N−1



p ∈ R

; |p| =1



We extend γ

on R

so that

γ(p)=γ

(p/|p|) |p|. (1.3.7)

Clearly, γ(p)ispositively homogeneous of degree 1, , i.e.,

γ(λp)=λγ(p) for all λ>0,p∈ R

. (1.3.8)

Conversely, γ satisfying (1.3.8) with γ|

N −1 = γ

is always expressed by (1.3.7).

The surface energy of Γ with surface energy density γ

is deﬁned by



(n)dσ

where dσ denotes the surface element. Of course, this is nothing but a surface area

if γ

≡ 1. We shall deﬁne a quantity (called the anisotropic mean curvature) which

describes the change ratio of surface energy per change of volume of D enclosed

by Γ as a generalization of the mean curvature.

For a given surface energy density γ

let ξ be the gradient of the homogeniza-

tion γ of γ

given by (1.3.7), i.e., ξ = ∇γ. The vector ξ is called the Cahn–Hoﬀman

vector of γ

.Wesay

h(x)=−(div

ξ(n))(x)

is the anisotropic (or weighted) mean curvature of Γ at x (in the direction of n)

with respect to surface energy density γ

.Todeﬁneh as a continuous function

we need to assume that γ is C

outside the origin. Of course, if γ

≡ 1, then

ξ(p)=p/|p| so that ξ(n)=n. Thus the anisotropic mean curvature agrees with

usual mean curvature when γ

≡ 1 as expected.

1.4 Expression of curvature tensors

Let Γ ⊂ R

be a (C

) hypersurface around x

∈ Γ. Let A = A

denote the

Weingarten map in the direction of n(x

), where n is a unit normal vector ﬁeld

on Γ around x

. We shall give various expressions of A and curvatures.

Level set representation. Suppose that Γ is represented by (1.1.1) with

n(x

)=−

∇u(x

)

|∇u(x

. (1.4.1)

1.4. Expression of curvature tensors 27

Then for τ ∈ T

Γ,

τ = −(D

n)(x

)=(τ ·∇)

∇u

|∇u|



(τ ·∇)∇u −(τ ·∇)∇u, ∇u 

∇u

|∇u|



at x = x

. (1.4.2)

It is convenient to introduce the orthogonal projection Π

from T

to T

Γ.

Its matrix expression is

n(x

)

= I − n(x

) ⊗ n(x

)

so that

ζ = R

n(x

)

ζ, ζ ∈ T

= R

where ζ is regarded as a column vector. Using the notation

= I − p ⊗ p/|p|

for p ∈ R

,p=0, (1.4.3)

we observe from (1.4.2) that

τ =

|∇u|

(∇

u)τ at x

with p = ∇u(x

where τ is regarded as a column vector. Since

τ = τ, p = ∇u(x

)

by deﬁnition, we see

τ =

|∇u(x

(∇

u(x

))τ at x

with

(X)=R

, (1.4.4)

where X is an n × n real symmetric matrix. Although R

∇

u may not be sym-

metric, Q

(∇

u) is now symmetric. It is not diﬃcult to see that the symmetric

operator

from T

to T

deﬁned by

ζ =

|∇u(x

∇u(x

)

(∇

u(x

))ζ, ζ ∈ T

= R

(1.4.5)

is a unique linear operator with the property that

ζ = A

ζ.

We often identify the Weingarten map A

. By deﬁnition

is given by

a direct sum of operators

= A

⊕ 0