Giga Y. Surface Evolution Equations: A Level Set Approach

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48 Chapter 1. Surface evolution equations

In this book we consider degenerate parabolic equations so we do not need

to distinguish strongly geometricity and geometricity. We shall mainly use the

geometricity to describe the scaling property of level set equations.

It is sometimes convenient to extend the notion of the geometricity to general

second order operators to handle boundary value problems.

Deﬁnition 1.6.13. Let E be a real-valued function deﬁned in a dense subset W of

×S

. Assume that (q, Y ) ∈ W implies (λq, λY +σq⊗q) ∈ W for λ>0,σ∈ R.

We say that E is geometric on E =0ifE(q, Y ) ≤ 0(resp.E(q, Y ) ≥ 0) implies

E(λq, λY + σq⊗ q) ≤ 0(resp.E(λq, λY + σq⊗ q) ≥ 0) for all λ>0, σ ∈ R .

Let E(z,·, ·) be a real-valued function deﬁned in W with z ∈O,whereO is

a locally compact subset of R

. We say an equation

E(z, Du, D

u)=0,z∈O

is geometric in O if E(z,·, ·)isgeometriconE =0forallz ∈O.HereDu =

(∂u/∂z

)

i=1

, D

u =(∂

u/∂z

∂z

)

1≤i,j≤d

It is strightforward to see that

E(q, Y )=τ + F (p, X),q=(τ,p) ∈ R × (R

\{0}),Y= {0}⊕X, X ∈ S

with W = R × (R

\{0}) × S

, d = N + 1 is geometric on E =0ifandonly

if F is geometric. In particular a level set equation (1.6.2) is always a geometric

equation in the sense of Deﬁnition 1.6.13. Thanks to Theorems 1.6.4 and 1.6.12,

if an equation

+ F (x, t, ∇u, ∇

u)=0, (x, t) ∈ Ω × (0,T)

is geometric in Ω × (0,T), then it is a level set equation of some surface evolution

equation (1.6.1) provided that F (x, t, ·, ·) is degenerate elliptic for all (x, t) ∈

Ω × (0,T), where Ω is a domain in R

. (We say that the preceeding equation

is degenerate parabolic if F is degenerate elliptic.)

Remark 1.6.14. We have introduced the notion of geometricity for E since it is

convenient to handle boundary value problems. We consider

+ F (x, t, ∇u, ∇

u)=0, in Ω × (0,T)

with, for example, the boundary condition (1.6.14):

∂u

∂ν

+ z|∇u| =0 on∂Ω × (0,T),

where |z| < 1, z ∈ R.Asweseein§2.3, to study this problem it is natural to

indroduce

E(x, τ, p, X)=



τ + F (p, X),x∈ Ω,

(τ + F (p, X)) ∧ ( ν, p  + z|p|),x∈ ∂Ω

1.6. Level set equations 49

where a ∧ b =min(a, b) and regard the boundary value problem as

E(x, u

, ∇u, ∇

u)=0 inΩ × (0,T).

For this operator, it is easy to see that E is geometric at E = 0 so that the equation

is geometric. We may replace (1.6.14) by a more general ﬁrst order boundary

condition

B(x, t, ∇u)=0

with B(x, t, λp)=λB(x, t, p),λ>0sothatE is geometric at E =0;hereE is

deﬁned by

E(x, t, τ, p, X)=(τ + F (p, X)) ∧ B(x, t, p).

In Chapter 4 we shall see how important the geometric property of equations

are. The main observation is that if u solves a geometric equation so does θ ◦

u = θ(u) with nondecreasing θ,whereθ ◦ u denotes the composition of functions.

This indicates that a geometric equation is invariant under coordinate change of

dependent variables. Although we postpone a rigorous proof for general solutions,

we here formally indicate how geometricity yields such a property. Assume that u

solves a geometric equation

+ F (∇u, ∇

u)=0

and θ



≥ 0. For v = θ ◦ u we calculate

+ F (∇v,∇

= θ



(u)u

+ F (θ



(u)∇u, θ



(u)∇

u + θ



(u)∇u ⊗∇u)

= θ



(u)(u

+ F (∇u, ∇

u)) = 0

by geometricity of F . This invariance property is natural if we recall that a ge-

ometric equation is the level set equation of some surface evolution equation so

that motion of each level set of solutions is independent of other levels and the

value of levels.

1.6.5 Singularities in level set equations

Regularity of f in (1.6.1) of course reﬂects to F

in (1.6.4). Here is a trivial

observation.

Proposition 1.6.15. Let f be a real-valued function deﬁned in E given by (1.6.15).

The associate function F

deﬁned by (1.6.16) is continuous in (R

\{0}) × S

and only if f is continuous in E.

Examples of equations (1.6.1) with f continuous in its variables includes

(1.5.2), (1.5.13), (1.5.14) with continuous h, β > 0,c with γ (provided that it is

outside the origin) as well as the mean curvature ﬂow equation. Examples

also include (1.5.9)–(1.5.12) but these equations may not be degenerate parabolic

50 Chapter 1. Surface evolution equations

so we rather consider modiﬁed equations (1.6.22), (1.6.23) instead of them. By

Theorem 1.6.10 (and its following paragraph) these equations (1.6.22), (1.6.23)

can be written in the form of (1.6.1) with continuous f . So for such an f the

associate function F

is continuous in (R

\{0}) × S

We next study the magnitude of singularity of F

(p, X)nearp =0.Forthis

purpose we introduce the upper semicontinuous envelope

∗

: R

× S

→ R ∪{∞}

of F deﬁned on (R

\{0}) × S

by setting

∗

(p, X) = lim

ε↓0

sup{F (q, Y ); |q − p|≤ε, ||X − Y ||

≤ ε}.

The lower semicontinuous envelope F

∗

is deﬁned by F

∗

= −(−F )

∗

with values in

R ∪{−∞}.IfF is continuous in (R

\{0}) × S

,then

F (p, X)=F

∗

(p, X)=F

∗

(p, X)for(p, X) ∈ (R

\{0}) × S

If F is geometric and degenerate elliptic, so is F

∗

and F

∗

Lemma 1.6.16. Assume that F is continuous (R

\{0}) × S

and that F is

geometric and degenerate elliptic. Let M and m denote

M =sup{F (p, −I); |p|≤1,p=0},

m =inf{F (p, I); |p|≤1,p=0}.

Then the following three conditions are equivalent:

(a) F

∗

(0,O) < ∞ (resp. F

∗

(0,O) > −∞),

(b) M<∞ (resp. m>−∞),

∗

(0,O)=0 (resp. F

∗

(0,O)=0).

Proof. Let |X| denote the operator norm of X as a self-adjoint operator. In other

words |X| equals the largest modulus of eigenvalues of X. Since the Hilbert–

Schmidt norm ||X||

is equivalent to the operator norm |X| for ﬁnite dimensional

,wemayreplace||X − Y ||

by |X − Y | in the deﬁnition of F

∗

.Weﬁrstnote

that |X|≤ε implies

−εI ≤ X ≤ εI

for ε>0. Since F is degenerate elliptic, we observe that

sup

|X|≤ε

F (p, X) ≤ F (p, −εI),p=0.

The converse inequality is trivial since |−εI| = ε.Wethusobservethat

sup

|p|≤ε

p=0

sup

|X|≤ε

F (p, X)= sup

|p|≤ε

p=0

F (p, −εI)=ε sup

|p|≤ε

p=0

F (p/ε, −I)=εM

since F is geometric. Thus the equivalence of (a), (b), (c) is clear for F

∗

. The proof

for F

∗

is symmetric. 

1.6. Level set equations 51

Remark 1.6.17. Even if F depends on (x, t) ∈ Ω × (0,T) the same proof shows

that

∗

(x, t, 0,O) = lim

ε↓0

(εM

∗

(x, t)),

with M

∗

(x, t)=sup{F (x, t, p, −I); |p|≤1,p=0},

provided that F is continuous in its variables and that F (x, t, ·, ·)isgeometric

and degenerate elliptic for all (x, t). Here F

∗

denotes the upper semicontinuous

envelope as a function of (x, t, p, Y ) (see for deﬁnition §2.1.1). Again if M

∗

(x, t) <

∞,thenF

∗

(x, t, 0,O) = 0 and of course if m

∗

(x, t) > −∞,thenF

∗

(x, t, 0,O)=0,

where m is deﬁned in the same way as M by replacing sup by inf.

Notice that the condition

−∞ <F

∗

(x, t, 0,O)=F

∗

(x, t, 0,O) < ∞

is equivalent to saying that F can be continuously extended to (x, t, 0,O).

Proposition 1.6.18. Let f be a real-valued continuous function deﬁned in E by

(1.6.15). Assume that f is degenerate elliptic. Then the associate function F

deﬁned by (1.6.16) can be continuously extended to (0,O) with value zero if and

only if

inf

0<ρ≤1

ρ inf

|p|=1

f(−p, −R

I/ρ) > −∞, sup

0<ρ≤1

ρ sup

|p|=1

f(−p, R

I/ρ) < +∞.

(1.6.24)

(The ﬁrst (second) quantity equals −M (resp. −m) deﬁned in Lemma 1.6.16 with

F = F

This follows from Lemma 1.6.16 since geometricity of F

implies

M =sup{F (p, −I); |p|≤1,p=0}

=sup|p|F (p/|p|, −I/|p|); |p|≤1,p=0}

= − inf

0<ρ<1

ρ inf

|p|=1

f(−p, −R

I/ρ)

and a similar expression is valid for m.

The condition (1.6.24) is a growth restriction of f = f(n, ∇n)in∇n.It

roughly says that f grows either linearly or sublinearly in ∇n as |∇n|→∞.For

example if f is positively homogeneously of degree 1 in the second variable, i.e.,

f(p, λZ)=λf(p, Z)forλ>0, (p, Z) ∈ E then (1.6.24) is fulﬁlled. If we write

(1.5.2) in the form of (1.6.1), then evidently f satisﬁes (1.6.24) (with constant c)

since f is linear in ∇n.Inparticular,f corresponding to the mean curvature ﬂow

equation fulﬁlls (1.6.24). The condition (1.6.24) is also fulﬁlled for (1.5.13) and

(1.5.14) provided that

lim

z→∞

|h(z)|/|z| < ∞; here, lim denotes limsup.

Both estimates of (1.6.24) are violated for f corresponding to (1.6.22) (for

2 ≤ m ≤ N − 1) and (1.6.23) (for 1 ≤ <m≤ N − 1,− 1 <m); if m =  − 1,

52 Chapter 1. Surface evolution equations

the growth condition (1.6.24) is fulﬁlled for (1.6.23). Note that if both inequalities

in (1.6.24) are, violated, then

∗

(0,O)=−∞,F

∗

(0,O)=+∞

by Lemma 1.6.16.

1.7 Exact solutions

We give here some explicit solutions mainly for the mean curvature ﬂow equation

(1.5.4) and its anisotropic version (1.5.2).

1.7.1 Mean curvature ﬂow equation

Shrinking sphere. As expected there is a solution Γ

of (1.5.4) that is a family of

spheres of radius R(t) centered at the origin. The equation (1.5.4) is now of form

−dR/dt =(N − 1)/R (1.7.1)

since the left-hand side is the inward velocity and the right hand side is the inward

mean curvature; note that for the sphere of radius R all principal curvatures are

1/R. Integrating (1.7.1), we see

R(t)=(R

− 2(N − 1)t)

1/2

(1.7.2)

with R(0) = R

> 0. Thus an evolving sphere

= {x ∈ R

; |x| = R(t)} (1.7.3)

solves (1.5.4) if (and only if) R(t) is of form (1.7.2). Note that if t>t

∗

/(2(N − 1)), then R(t) is not well deﬁned as a real number. It is natural to

interpret that Γ

becomes empty after the time t

∗

when Γ

shrinkstoapoint.

Similarly for the Gaussian curvature ﬂow equation (1.5.9) there is a shrinking

sphere solution Γ

of form (1.7.3) provided that R(t)solves

−dR/dt =1/R

N−1

(1.7.4)

instead of (1.7.1). Integrating (1.7.4) yields

R(t)=(R

− Nt)

1/N

instead of (1.7.2). For more general equation (1.5.12) it is still easy to ﬁnd a

shrinking sphere solution although we do not give its explicit form here.

Shrinking cylinders. We consider a little bit more general evolving hypersurface

called a cylinder:

= {(x

,...,x

j+1

,...,x

); (x

j+1

+ ···+ x

)

1/2

= R(t)};(1.7.5)

1.7. Exact solutions 53

of course Γ

is a sphere if j = 0. The equation (1.5.4) is interpreted as

−dR/dt =(N − 1 − j)/R (1.7.6)

which generalizes (1.7.1); note that κ

= ···= κ

N−1−j

=1/R, κ

N−j

= κ

N−j+1

···= κ

N−1

= 0. Solving (1.7.6) to get

R(t)=(R

− 2(N − 1 − j)t)

1/2

. (1.7.7)

Thus an evolving cylinder Γ

of (1.7.5) solves the mean curvature ﬂow equation if

and only if R(t) is given by (1.7.7).

Since the Gaussian curvature of a cylinder of form (1.7.5) with j ≥ 1isalways

zero, any cylinder is a stationary solution of the Gaussian curvature ﬂow equation

(1.5.9).

Intheseexactsolutionswenoticethattheshapeofanevolvinghypersurface

is independent of time up to dilation. Such a solution is often called a self-similar

solution. We here give a rigorous deﬁnition of self-similarity.

Deﬁnition 1.7.1. Let Γ

be an evolving hypersurface t ∈ I where I is a time

interval. We say that Γ

is self-similar if there is x

∈ R

and a hypersurface Γ

(independent of time) in R

such that for some λ = λ(t), Γ

is of form

= {λ(t)(x − x

)+x

; x − x

∈ Γ}.

For example the evolving cylinder Γ

given (1.7.5) is self-similar. If (1.7.7) is

fulﬁlled, then it is a self-similar solution of (1.5.4).

Level set approach. It is sometimes convenient to ﬁnd a self-similar solution by

using level set equations. We seek a solution of the level set equation (1.6.7) of

form

u(x, t)=−(t + ζ(r)),r= |x|

with nondecreasing ζ.Thenζ(r(x)) must solve

1=|∇

ζ(r)|div



∇ζ(r)

|∇ζ(r)|



= ζ



(r)|∇

r|div(

∇r

|∇r|

Since ∂r/∂x

= x

/r and ∂/∂x

/r)=(δ

− x

−1

, this implies

1=ζ



(r)(N − 1)/r.

By normalizing ζ(0) = 0 we get ζ(r)=r

/(2(N − 1)). Thus

u(x, t)=−(t + |x|

/2(N − 1))

solves (1.6.7) at least outside the place where ∇u = 0. Each level set gives a

shirinking sphere solution of (1.5.4). Similarly, we see

u(x, t)=−(t +



=j+1



/(2(N − 1)))

54 Chapter 1. Surface evolution equations

solves (1.6.7) at least formally (see Remark 4.3.8). Each level set of u gives a

shirinking cylindrical solution of (1.5.4). In these problems −u also solves (1.6.7).

This reﬂects the fact that the evolution law (1.5.4) is independent of the choice of

the orientation n.

1.7.2 Anisotropic version

We shall consider a class of anisotropic curvature ﬂow equations (1.5.2) with con-

stant c which includes the mean curvature ﬂow equation as a special case. We try

to ﬁnd a self-similar shrinking solution similar to a sphere. A typical property of

a sphere is that its mean curvature is constant on the sphere. (If an (embedded)

closed hypersurface has constant mean curvature, then it must be a sphere by a

result of A. D. Aleksandrov(1956).) We shall seek a hypersurface whose anisotropic

mean curvature is constant.

Wulﬀ shape. Let γ

be a surface energy. We say that

W =



|q|=1

{x ∈ R

; x, q≤γ

(q)}

is the Wulﬀ shape associated with γ

.Ifγ

≡ 1, W is a unit sphere.

()q

Figure 1.4: Wulﬀ shape

According to Wulﬀ’s theorem, W minimizes the surface energy



∂D

(n)dσ

among all sets D with the same volume as W. Such a minimizer is unique up to

translation. In other words W is a unique solution of the anisotropic isoperimetric

problem.

1.7. Exact solutions 55

The Wulﬀ shape is also characterized by the zero set of the conjugate convex

function Γ of γ deﬁned as



(x)=sup{x, q−γ(q); q ∈ R

where γ is given by (1.3.7). The function γ



is convex even if γ is not convex. Since

γ is positively homogeneous of degree 1, we see that γ



is the indicator function of

W, i.e., γ



=0onW and γ



=+∞ outside W.Indeed,ifx ∈W,thenx, q−γ(q)

attains zero so γ



(x)=0.Ifx/∈W,thenc = x, q−γ>0forsomeq satisfying

|q| =1.Sinceγ is positively homogeneous of degree 1,

x, q−γ(λq)=λc

for all λ>0. This implies γ



(x)=∞.

Since γ



is convex and lower semicontinuous, W is convex and closed. Since

the surface energy density γ

is always assumed to be positive, W contains the

origin as an interior point. The next lemma shows that the boundary of a Wulﬀ

shape substitutes for the role of a sphere for anisotropic mean curvature. Let P

be the Minkowski function of W deﬁned by

P (x)=inf{λ ∈ (0, ∞); x/λ ∈W}.

Clearly, P is positively homogeneous of degree 1 and convex. Since W contains

theoriginasaninteriorpoint,P is deﬁned in all of R

. Clearly,

W = {x ∈ R

; P (x) ≤ 1}.

Lemma 1.7.2 (Anisotropic mean curvature of the Wulﬀ shape). Assume that sur-

face energy density γ

> 0 is C

(m ≥ 2) in the sense that γ is C

outside the

origin. Assume that γ satisﬁes a strict convexity assumption: ∇

γ(p)+p ⊗ p>O

for p =0or equivalently,



i=1



j=1

∂

∂p

(p)η

> 0 for all η, p ∈ R

\{0} with η, p =0,

where η =(η

,...,η

). Then the boundary Γ=∂W of the Wulﬀ shape associated

to W is C

(so that P is C

outside the origin). Moreover,

γ(∇P (x)) = 1 in R

\{0}, (1.7.8)

ξ(∇P (x)/|∇P(x)|)=ξ(∇P (x)) = x/P (x),x∈ R

,x=0, (1.7.9)

∇



|x|

P (x)



|x|

 =0,x∈ R

,x=0, (1.7.10)

where ξ is the Cahn–Hoﬀman vector of γ, i.e., ξ = ∇γ.Inparticularξ(n(x)) =

x, x ∈ Γ so that the anisotropic mean curvature h in the direction of n equals

−(N − 1),wheren is the outward unit normal vector ﬁeld of Γ.

56 Chapter 1. Surface evolution equations

(())Px

[



()

x

Figure 1.5: Cahn-Hoﬀman vector

We postpone the proof in the next subsection. We seek a self-similar solution

of (1.5.2), with a ≥ 0whenc is a constant, by the level set approach as for (1.5.4).

We set

u(x, t)=−(t + ζ(P )) (1.7.11)

with nondecreasing ζ deﬁned on [0, ∞), where P is the Minkowski function of W.

The level set equation of (1.5.2) is (1.6.10) or equivalently

−|∇u|(−a div ξ(−∇u)+c)/β(−∇u/|∇u|)=0

by the identity (1.4.13). For the special form of u in (1.7.11), this equation is

equivalent to

1=ζ



(P )|∇P |(a div ξ(∇P ) − c)/β(∇P/|∇P |). (1.7.12)

Using (1.7.9) and (1.7.10), we have

div ξ(∇P )=div(x/P )=(|x|/P )div(x/|x|)+∇

|x|

 =(N − 1)/P +0.

Thus if βγ

is a constant function with βγ

=1/σ

on S

N−1

, then, by (1.7.8), the

identity (1.7.12) is equivalent to

1=ζ



(P )σ

(a(N − 1)/P − c). (1.7.13)

Regarding P as an independent variable, (1.7.13) is an ordinary diﬀerential equa-

tion for ζ. It is easy to integrate (1.7.13). If c ≤ 0, then a solution of (1.7.13) is a

strictly increasing function of form

ζ(ρ)=



(a(N − 1) − cτ)

dτ (1.7.14)

1.7. Exact solutions 57

by normalizing ζ(0) = 0. For example, if c =0witha>0, then ζ(ρ)=

/(2σ

a(N − 1)). If a =0andc<0, ζ(ρ)=−ρ/σ

c.Wethusobservethat

u of (1.7.11) with ζ of form (1.7.14) solves the level set equation of (1.5.2) at least

where ∇u = 0. Since each level set of u solves (1.5.2) at least where ∇u =0,

= {x ∈ R

; P (x)=R(t)} (1.7.15)

solves (1.5.2) provided that

R(t)=ζ

−1

(ζ(R(0)) − t), (1.7.16)

where ζ

−1

denotes the inverse function of ζ. This generalizes (1.7.2). If c =0,Γ

disappears in a ﬁnite time; it disappears at t = ζ(R(0)). If c>0, then we cannot

ﬁnd an increasing function ζ solving (1.7.13). This corresponds to the phenomena

that there is a growing solution of form (1.7.15) to (1.5.2) with c>0whenn is

taken outward.

We give another way to ﬁnd a self-similar solution to see these properties

depending on c. We argue as in the same way to derive (1.7.1) from (1.5.4). We

note that

R(t)=|∇P (x)|n,x,x∈ Γ

since R(t)=P (x)=x, ∇P (x) by the homogeneity of P .Wealsonotethat

dR/dt may not equal the outward normal velocity V of Γ

given by (1.7.15) but

dR/dt = V/n,x/R(t).

By homogeneity of γ we see

ξ(n), n = γ(n);

indeed diﬀerentiating γ(λp)=λγ(p)inλ and setting λ = 1 yields the Euler

equation

ξ(p),p = γ(p). (1.7.17)

By (1.7.9) this yields

γ(n)=

R(t)

, n

so we obtain

dR/dt = V/γ

(n).

If βγ

=1/σ

, the equation (1.5.2) becomes

V = σ

(n)(−a div

ξ(n)+c). (1.7.18)

This yields

dR/dt = σ

(−a(N − 1)/R + c)(1.7.19)