Giga Y. Surface Evolution Equations: A Level Set Approach

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28 Chapter 1. Surface evolution equations

corresponding to the decomposition of the tangent space R

= T

of form

= T

Γ ⊕ N

Γ,

where N

Γ denotes the normal vector space at x

. Thus the eigenvalues of

consist of principal curvatures κ

,...,κ

N−1

and 0.

We shall derive the level set representation of various curvatures from (1.4.5).

The mean curvature H at x

∈ Γ in the direction of n(x

) (deﬁned by (1.4.1)) is

H = κ

+ ···+ κ

N−1

+0=trace

|p|

traceQ

(X)withp = ∇u(x

),X= ∇

u(x

)

Since R

= R

by (1.4.3) and trace (R

) = trace(R

X), H is of the form

H =

|∇u(x

trace



I −

∇u(x

) ⊗∇u(x

)

|∇u(x



∇

u(x

)



|∇u|

⎛

⎝

⎛

⎝

∆u −



1≤1,j≤N

|∇u|

⎞

⎠

⎞

⎠

at x = x

, (1.4.6)

where u

= ∂u/∂x

, u

= ∂

u/∂x

∂x

and ∆u =



j=1

.Form-th sym-

metric curvature with m ≤ N − 1bytheformof

we see

= e

(κ

,...,κ

N−1

, 0)

where κ

,...,κ

N−1

, 0 are eigenvalues of Q

(X)/|p| with p=∇u(x

), X =∇

u(x

since

(κ

,...,κ

N−1

, 0) = e

(κ

,...,κ

N−1

If m = N − 1, then the Gaussian curvature K is of the form

K = e

N−1

(κ

,...,κ

N−1

, 0) = e

N−1

(κ

,...,κ

N−1

)=e

(κ

,...,κ

N−1

, 1).

This observation gives a simple representation of the Gaussian curvature

K =det



(X)

|p|

p ⊗ p

|p|



,p= ∇u(x

),X= ∇

u(x

)(1.4.7)

since the eigenvalues of Q

(X)/|p| + p ⊗ p/|p|

are κ

,...,κ

N−1

, 1. If we arrange

≤···≤ κ

N−1

, κ

(1 ≤ i ≤ N − 1) is written as

= k

(p, X),p= ∇u(x

) =0,X= ∇

u(x

)

where k

(p, X) ≤ k

(p, X) ≤ ··· ≤ k

N−1

(p, X) are the eigenvalues of the linear

operator Q

(X)/|p| the orthogonal complement of the vector p. By this expression

it is possible to express the m-th symmetric curvature by u.

1.4. Expression of curvature tensors 29

There is another way to derive the level set representation of the mean cur-

vature and anisotropic mean curvature without using the representation (1.4.5)

but using surface divergence. We set m = −∇u/|∇u| and note that the vector

ﬁeld m is deﬁned not only on Γ but also on some neighborhood of Γ near x

.Itis

an extension of n in a neighborhood of Γ. By (1.3.6) we observe that

H = −div

n = −trace((I − n(x

) ⊗ n(x

))(∇m)(x

))

= −trace(∇m(x

)) + trace(n(x

) ⊗ n(x

)(∇m)(x

))

=div



∇u

|∇u|





1≤i, j≤N

)(

∂

∂x

)(x

), (1.4.8)

where n =(n

,...,n

), m =(m

,...,m

). Here div denotes the divergence of a

vector ﬁeld on R

.Sincem is a unit vector ﬁeld near x



i=1

)



∂

∂x





∂

∂x



i=1



∂

∂x

1=0.

Thus the last term of (1.4.8) disappears and we obtain

H =



div



∇u

|∇u|



)(1.4.9)

which is the same as (1.4.6).

We shall give a level set representation of anisotropic mean curvature. We

recall (1.3.6) to calculate

h = −div

ξ(n)=−trace{(I − n(x

) ⊗ n(x

))(∇ξ(m))(x

)}

= −trace(∇ξ(m))(x

)+trace{(n(x

) ⊗ n(x

))(∇ξ(m))(x

)}. (1.4.10)

The second term is of form



1≤i,k,≤N

)

∂ξ

∂p



(n(x

))

∂m



∂x

)

with ξ =(ξ

,...,ξ

). Since ξ = ∇γ and γ is positively homogeneous of degree 1

so that

γ(λp)=λγ(p),λ>0,

diﬀerentiating in p



yields

λξ



(λp)=λξ



(p)orξ



(λp)=ξ



(p)for1≤  ≤ N.

In other words ξ



is positively homogeneous of degree zero. Diﬀerentiating in λ

and setting λ = 1 yields the Euler equation



i=1

∂ξ



∂p

(p)=0, 1 ≤  ≤ N. (1.4.11)

30 Chapter 1. Surface evolution equations

Since ∂ξ



/∂p

= ∂ξ

/∂p



, the second term of (1.4.10) can be rewritten as



1≤i,j,≤N

)

∂ξ



∂p

(n(x

))

∂m



∂x

)



1≤i,j,≤N





i=1

)

∂ξ



∂p

(n(x

))



∂m



∂x

)=0 by(1.4.11).

Thus we have

h = −(div ξ(m))(x

); (1.4.12)

note that this formula holds for any extension m of n in a tubular neighborhood of

Γnearx

since we do not use the property |m| =1.Instead,ξ should be a Cahn–

Hoﬀman vector. If γ(p)=|p| so that ξ = p/|p|, we recover the formula (1.4.9) with

m = −∇u/|∇u|.Sinceξ is homogeneous of degree zero, (1.4.12) yields

h = −div(ξ(−∇u))(x

)



1≤i,j≤N

∂

∂p

(−∇u(x

))

∂

∂x

). (1.4.13)

(Of course if γ(p)=|p|, this again yields (1.4.9).) By (1.4.11) we see R

∇

γ(p)=

∇

γ(p)=∇

γ(p)R

. From (1.4.13) it now follows that

h =

|∇u(x

trace



∇

γ(n(x

))Q

n(x

)

(∇

u)(x

)



(1.4.14)

with n(x

)=−∇u(x

)/|∇u(x

)| since ∂

γ/∂p

∂p

is homogeneous of degree −1

so that

∂

∂p

(−∇u)=

|∇u|

∂

∂p



−

∇u

|∇u|



, 1 ≤ i, j ≤ N.

Using the second fundamental form in (1.4.14) we have

h =trace(∇

γ(n(x

))

). (1.4.15)

The formula (1.4.15) may explain a reason why h is sometimes called a weighted

mean curvature.

Graph representation. It is easy to derive a formula for the second fundamen-

tal form from (1.4.5) when Γ is given as the graph of a function. By rotating

coordinates we may assume that Γ is expressed as

Γ={x

= g(x



),x



∈ R

N−1

}

around x

∈ Γ, where g(x



)=x

.Ifn is taken upward, then Γ is given as the

zero level set of

u(x)=−x

+ g(x



)

1.4. Expression of curvature tensors 31

with n = −∇u/|∇u| which is the same as (1.1.2). Plugging in (1.4.5) we obtain a

formula of

written by g. A general formula is complicated so we do not give it

here. However if ∇



g(x



)=0sothatn(x

)=(0, 0,...,0, 1), then the expression

is simple. Indeed, since ∇u(x

)=−n(x

), we see

∇u(x

)

(∇

u(x

)) =



∇

2

g(x



Thus we obtain

ζ =



∇

2

g(x

)ζ





,ζ



∈ R

N−1

with ζ =(ζ



,ζ

) ∈ R

. (1.4.16)

We shall calculate the mean curvature H at x

in the direction of n.Weplug

u in (1.4.9) to get

H =



div





∇





1+|∇







∂

∂x



−1



1+|∇



)





div





∇





1+|∇







), (1.4.17)

where div



denotes the divergence in x



variables.

Axisymmetric surface. Suppose that Γ is obtained by rotating the graph of a

function ϕ(x

) around the x

-axis and that x

∈ Γ is not on the axis. Around

∈ Γ the hypersurface Γ is of form

Γ=

⎧

⎪

⎨

⎪

⎩

r = ϕ(x

); r =

⎛

⎝



j=2

⎞

⎠

1/2

⎫

⎪

⎬

⎪

⎭

If n is taken outward from x

-axis, i.e., n is given by (1.2.3), then Γ is given as

the zero level set of

u(x

,...,x

)=−r + ϕ(x

)

with n = −∇u/|∇u| (around x

). We obtain a formula for

in terms of ϕ by

plugging above u into (1.4.5). However, we do not give its explicit formula. Here

we only calculate the mean curvature H at x

in the direction of n.Weplugu in

(1.4.9) and use the formula (1.2.3) to get

H =

∂

∂x



(1 + ϕ

)

1/2



−

(1 + ϕ

)

1/2



j=2

∂

∂x

(1 + ϕ

)

3/2

−

(1 + ϕ

)

1/2

N − 2

at x = x

=(x



) (1.4.18)

32 Chapter 1. Surface evolution equations

where r = |x



| with x



=(x

,...,x

Gradient of normal vector ﬁelds Let n be a unit normal vector ﬁeld of Γ around

.Letm be a (C

) extension of n to a tubular neighborhood of Γ around x

Since

τ = −((τ ·∇)m)(x

for τ ∈ T

Γ,

ζ = −((Π

ζ ·∇)m)(x

)

= −((ζ ·∇)m)(x

)+ ζ,n(x

) (n(x

) ·∇)m(x

This implies that the matrix expression of

(with respect to the standard

Euclidean basis) equals

−

∂

∂x

)+n

(x)(n(x

) ·∇)m

), 1 ≤ i, j ≤ N

(which should be symmetric since

is symmetric). If the extension m has the

property that

((n(x

) ·∇)m)(x

)=0, (1.4.19)

then one may identify −∇m(x

)by

. We shall use the notation ∇n by extend-

ing n to a tubular neighborhood of Γ around x

such that (1.4.19) holds. By this

interpretation −∇n =

, so we shall often identify second fundamental form

with −∇n.

We conclude this section by studying the range of ∇n.LetS

denote the

space of all N × N real symmetric matrices.

Lemma 1.4.1. Let x

be a point in R

.Foreachp ∈ S

N−1

and X ∈ S

there is

a smooth hypersurface Γ around x

with the property that n(x

)=p, ∇n(x

(X),wheren isaunitnormalvectorﬁeldonΓ around x

and is extended to

a tubular neighborhood of Γ satisfying ((n(x

) ·∇)n)(x

)=0.

Proof. We may assume p =(0, 0,...,1) by rotation of coordinates and x

=0by

translation. The matrix Q

(X)isoftheform

⎛

⎜

⎝

0 ··· 0

⎞

⎟

⎠

with Y ∈ S

N−1

.Weset

g(x



)=−

 Yx



 for x



∈ R

N−1

to get g(0) = 0, ∇



g(0) = 0, ∇

2

g(0) = −Y .Since∇n(x

)=−

, the formula

(1.4.16) yields

∇n(x

)=Q

(X). 

1.5. Examples of surface evolution equations 33

1.5 Examples of surface evolution equations

We give general examples of equations of an evolving hypersurface whose normal

velocity V is determined by its normals and second fundamental forms. In general

such an evolution equation is of the form

V = f(x, t, n, ∇n)(1.5.1)

on an evolving hypersurface Γ

,wheref is a given function and n is a unit normal

vector ﬁeld of Γ

. For consistency with the literature we take the negative of the

second fundamental form A of Γ

as an independent variable of f and denote it

by ∇n.HereV is the velocity in the direction of n.

1.5.1 General evolutions of isothermal interfaces

For an evolving hypersurface Γ

we consider

β(n)V = −a div

ξ(n)+c(x, t). (1.5.2)

Here ξ is the Cahn–Hoﬀman vector of a surface energy density γ

: S

N−1

→ (0, ∞)

and β is a given positive function on S

N−1

, a is a nonnegative constant and c is

a given function. We always assume that N ≥ 2ifa = 0 since the curvature term

equals zero when N = 1. As we observed in (1.4.15), this equation is an example of

(1.5.1). If c is independent of x and a = 1, (1.5.2) is often used to describe motion

of isothermal interface; there c is regarded as bulk free energy diﬀerence such as

temperature diﬀerence in both phases. The function 1/β is called the mobility.It

may be again anisotropic in the sense that it depends on the direction of normals.

The mobility is determined by microstructure of the hypersurface. Sometimes it

is proportional to γ

, i.e., βγ

is constant independent of n but in general 1/β is

not necessarily proportional to γ

. The equation (1.5.2) has an energy structure.

Indeed, we set

G(Γ) =



aγ

(n)dσ +



cdx (1.5.3)

for a hypersurface Γ surrounding D, then (1.5.2) is of the form

β(n)V = −δG/δΓ

where δG/δΓ

denotes the change ratio of (free energy) G per change of volume

of D in the direction of n.Heren is taken outward from D. In other words (1.5.2)

is a gradient ﬂow of G.

Mean curvature ﬂow equation. The equation (1.5.2) includes several interesting

important examples as a special case. If the mobility and the surface energy density

is isotropic with no driving force c and a = 1, then (1.5.2) becomes

V = H (1.5.4)

34 Chapter 1. Surface evolution equations

by rescaling time if necessary (or taking β ≡ γ

≡ 1, a =1,c = 0). This equation

is called the mean curvature ﬂow equation.IfN =2sothatΓ

is a curve, (1.5.4)

is called the curve shortening equation.Ifκ denotes the curvature in the direction

of n, the curve shortening equation is of the form

V = κ. (1.5.5)

For the mean curvature ﬂow equation the energy G in (1.5.3) is the surface area

of Γ. Thus (1.5.4) gives a deformation so that decrease ratio of area is steepest.

This is why (1.5.5) is called the curve shortening equation. Note that the equation

(1.5.4) is invariant under the change of orientation n. In other words the evolution

law (1.5.4) is the same even if we replace n by −n.

Hamilton–Jacobi equations. If a = 0, then (1.5.2) becomes

β(n)V = c(x, t). (1.5.6)

This equation is regarded as a special form of the Hamilton–Jacobi equation.

Indeed, if Γ

is represented by the graph of a function, i.e., x

= g(x



,t), then by

(1.1.2) and (1.2.2) the equation (1.5.6) is of the form

+ H(x



,t,g,∇



g)=0 (1.5.7)

with

H(x



,t,r,p



)=−c(x



,r,t)µ(p



)/β(−p



/µ(p



),µ(p



)

−1

),µ(p



)=(1+|p



)

1/2

The equation (1.5.7) is a ﬁrst order equation and the curvature plays no role in

(1.5.6).

1.5.2 Evolution by principal curvatures

In the mean curvature ﬂow equation the normal velocity depends only on principal

curvatures of hypersurfaces. There are several other examples of form

V = g(κ

,...,κ

N−1

; n), (1.5.8)

where g is a given function of principal curvatures of κ

,...,κ

N−1

and n.Inthe

mean curvature ﬂow equation, g is independent of n and is taken as the ﬁrst

elementary symmetric polynomial e

If g is taken as e

N−1

so that e

N−1

(κ

,...,κ

N−1

) equals the Gaussian curva-

ture K, (1.5.8) becomes the Gaussian curvature ﬂow equation

V = K. (1.5.9)

Of course if N = 2, then this equation becomes the curve shortening equation

V = κ. More generally, for each 1 ≤ m ≤ N − 1,

V = H

(1.5.10)

1.5. Examples of surface evolution equations 35

is called the m-th symmetric curvature ﬂow equation,whereH

is the m-th sym-

metric curvature e

(κ

,...,κ

N−1

). The equation

V = H

N−1

N−2

(N ≥ 3) (1.5.11)

is called the harmonic curvature ﬂow equation, which is a special case of

V = H



. (1.5.12)

All equations (1.5.9)-(1.5.12) are examples of (1.5.8). As we see later, we impose

the restriction <min (1.5.12) so that the equation is parabolic at least for convex

surfaces. Note that in general the evolution law (1.5.9) or (1.5.10) (with even m)

may depend on the choice of orientation n. For a closed evolving hypersurface Γ

we take the inward normal vector ﬁeld as n so that the sphere shrinks as time

develops. The same remark applies to (1.5.12). We use this convention when we

consider the evolution by principle curvatures (1.5.10) with even m and (1.5.12)

with even m − .

1.5.3 Other examples

In general the right-hand side of (1.5.2) is not proportional to the velocity. A

natural generalization is

V = h(−a div

ξ(n)+c, n)(1.5.13)

where h is nondecreasing in the ﬁrst variable and h(0, n)=0.IfN =2sothatΓ

is a curve in the plane and h(σ, n)=σ

1/3

with c =0,a =1,γ

≡ 1, then (1.5.13)

becomes

V =(κ

)

1/3

(1.5.14)

where σ

=max(σ, 0). The equation (1.5.14) is called the aﬃne curvature ﬂow

equation since the equation is invariant under aﬃne transformation.

1.5.4 Boundary conditions

It often happens that a hypersurface Γ

moves in a domain Ω in R

and the

geometric boundary of Γ

intersects the boundary ∂Ω of Ω. In this case in addition

to the equation (1.5.1) in Ω we have to impose the boundary condition so that

evolution is determined by equations. We give here typical examples of them. We

assume that the boundary ∂ΩisatleastaC

hypersurface in R

.Letν be the

unit normal vector ﬁeld of ∂ΩoutwardfromΩ.Letn be the unit normal vector

ﬁeld of a smooth hypersurface {Γ

} in Ω.

Neumann boundary condition. This condition imposes

 ν, n  =0

36 Chapter 1. Surface evolution equations

on the intersection of Γ

and ∂Ω. Geometrically speaking, Γ

intersects ∂Ω orthog-

onally, i.e., Γ

⊥∂Ω.

Prescribed contact angle boundary condition. Let z be a given real-valued contin-

uous function on ∂Ω that satisﬁes |z| < 1on∂Ω. The prescribed contact boundary

condition imposes

 ν, n  = z

on the intersection of Γ

and ∂Ω. Of course if z = 0, this condition is exactly

the Neumann boundary condition. Although the orientation of Γ

is irrelevant to

describe the Neumann boundary condition, the prescribed angle condition depends

on the orientation of Γ

. In the literature the prescribed contact angle condition

is often referred to as the Neumann boundary condition.

cosz

Figure 1.3: Prescribed contact angle

Dirichlet boundary condition. Let S be a given codimension 2-closed surface in

. The Dirichlet condition imposes that the geometric boundary of Γ

always

equals S. This condition is so far not easy to treat in a level set method so we do

not discuss this problem much in this book.

1.6 Level set equations

For a given surface evolution equation we shall introduce its level set equation.

We shall study various properties of level set equations.

1.6.1 Examples

We consider a surface evolution equation

V = f(x, t, n, ∇n)(1.6.1)

1.6. Level set equations 37

on an evolving hypersurface Γ

in a domain Ω in R

.Heref(x, t, ·, ·)for

(x, t) ∈ Ω × [0,T] is a given real-valued function deﬁned in

E = {(p, Q

(X)); p ∈ S

N−1

,X∈ S

By Lemma 1.4.1 the set E is a natural space so that (n, ∇n) lives. We say an

equation

(x, t)+F (x, t, ∇u(x, t), ∇

u(x, t)) = 0 for all (x, t) ∈ Ω × (0,T)(1.6.2)

is a level set equation of (1.6.1) if for each level set

= {x; u(x, t)=constant}

of a C

2,1

solution u of (1.6.2) near (x

) ∈ Ω satisﬁes (1.6.1) at (x

)provided

that ∇u(x

) = 0 and that the orientation n is chosen so that n(x

−∇u(x

) /|∇u(x

)|,wherex

∈ Γ

.HereF is a real-valued function deﬁned

in Ω × [0,T] × (R

\{0}) × S

Such an equation is uniquely determined by (1.6.1). Indeed, using the level

set representation of V,n and ∇n, (1.6.1) is of form

|∇u|

= f



x, t, −

∇u

|∇u|

, −

|∇u|

∇u

(∇



on the evolving hypersurface. The representation of V,n and ∇n is the same if the

hypersurface is a c-level set instead of the zero level set of u.Thus,theequation

−|∇u|f



x, t, −

∇u

|∇u|

, −

|∇u|

∇u

(∇



is the unique level set equation. It may be rewritten as

+ F

(x, t, ∇u, ∇

u)=0 (1.6.3)

with

(x, t, p, X)=−|p|f



x, t, −

|p|

, −

|p|

(X)



(1.6.4)

for p ∈ R

\{0}, X ∈ S

,(x, t) ∈ Ω × [0,T]. Note that the function F

is not

deﬁned for p = 0 in general as we will see in following examples.

Level set mean curvature ﬂow equation. If (1.6.1) is the mean curvature ﬂow

equation (1.5.4) : V = H, then by (1.2.1) and (1.4.6), the level set equation is

− ∆u +



1≤i, j≤N

|∇u|

=0 (1.6.5)