数理解析研究所講究録
第 898 巻 1995 年 104-112

104

On Instability in Geometric Evolution Equations

Yoshikazu Giga* and Kazuyuki Yama-uchi
Department of Mathematics, Hokkaido University, Sapporo 060, JapanABSTRACT. A general parabolic evolution equation is considered for a closed hypersurface inEuclidean space. All stationary solutions are shown to be Lyapunov unstable if the normal
velocity of a hypersurface depends only on its normal and second fundamental form and isindependent of its position. Instability of time periodic solution is also discussed.

1. Introduction
This is a preliminary note. We consider the initial value problem of an evolution of a closed

hypersurface$\Gamma_{t}$in$\mathbb{R}^{n}$$V=f(n, -A)$ .
Here

$n$

is an inward unit normal vector field on$\Gamma_{t}$

(1)

and

$V$

is normal velocity in the directionof $n;A=-$ dn denotes the second fundamental form. We shall prove that all stationary
solutions$S$of (1) is Lyapunov unstable provided that (1) is (nondegenerate) parabolic.

This generalized a recent work of Ei and Yanagida [EY] where they assumed that$f$defends

on A only through its mean curvature. Their method is completely different from ours.

They linearized equation around stationary solution and appeal to spectral analysis. Theirmethod applies to the equation depending on space variable but invarimt under translation. We simply use a distance function of$S$

and appeals to the maximum principle. We

believe our proof is simpler than theirs for this problem.

Our method also applies to instability of the periodic solution of$V=f(t, n, -A)$ .

where$f$

is time periodic. We show that periodic solutions

fundamental form vanishes somewhere on$S_{t}$(2)$S_{t}$

are unstable unless second

for all .
$i$*Partially supported by the Inamori Foundation
Typeset by$\mathcal{A}_{\mathcal{M}^{S}}$105

2. Parabolic evolution equations

We formulate our equations as in [GGl]. Let $E$ be a bundle over the sphere $S^{n-1}$ of theform$E=\{(\overline{p}, Q_{\overline{p}}(X))\in S^{n-1}\cross S_{n};X\in S_{n}\}$with$Q_{\overline{p}}(X)=R_{\overline{p}}XR_{\overline{p}}$anddenotes the space of theis the projection orthogonal to . Here

$R_{\overline{p}}=I-\overline{p}\otimes\overline{p};R_{\overline{p}}$$n\cross n$$\overline{p}$real symmetric matrices. By the standard Euclidean metric

the bundle $E$ is identified with the tensor bundle $TS^{n-1}\otimes T^{*}S^{n-1}$ over $S^{n-1}$ . Leta function from $[0, \infty)\cross E$ toSuppose that a hypersurface$\nabla u\neq 0$on$\Gamma$$\Gamma$$S_{n}$$\mathbb{R}$. We shall always assume thatis given as a zero level set of

$u$

in

$f$

be

is at least continuous.

$f$such that the gradient$\mathbb{R}^{n}$

and $n=\nabla u/|\nabla u|$ . Then as in [GGl], the second fundamental form (in thedirection of n) is of the form
$A=-Q_{\overline{p}}(\nabla^{2}u)/|\nabla u|$

where$\nabla^{2}u$denotes the Hessian of$u$

$-$

with

$\overline{p}=\nabla u/|\nabla u|$,(3)

in space variables.

We recall the notion of parabolicity of the equation(4)$V=f(t, n, -A)$for evolving hypersurface$\Gamma_{t}$. It is convenient to introduce the level set equation(5)$u_{t}+F_{f}(t, \nabla u, \nabla^{2}u)=0$

with$F_{f}(t,p,X)=|p|f(t,p/|p|, Q_{\overline{p}}(X)/|p|)$This equation is uniquely determined if each level set of

$u$

.

(6)

moves by (4) and a super level

set $u>c$ is “inside” the level set $u=c$ . Sign in (6) are different from those in [CGG]
because our convention of$n$is opposite.

We say (4) is strictly parabolic (uniformly in t) if for each $M>0$ there is $\mu>0$ suchthat
$F_{f}(t,\overline{p},X+Y)-F_{f}(t,\overline{p},X)\leq-\mu$

trace$(Q_{\overline{p}}(Y))$

(7)

106

for all $Y\geq 0,$ $|X|\leq M,$$|\overline{p}|=1,$$t\in[0, \infty)$, where $|X|$ is the operator norm of $X$ as aselfadjoint operator. If (7) holds for $\mu=0$ we say (4) is (degenerate) parabolic. A level set
method [CGG], [ES] provides a unique global generalized solution. The following version
is taken from [GGl].

2.1. Unique global existence. Suppose that (4) is parabolic. Letof a bounded open set instarting from .

$\mathbb{R}^{n}$be the boundary$\Gamma_{0}$. Then there is a unique generalized solution$\{\Gamma_{t}\}_{t\geq 0}$of (4)$\Gamma_{0}$

If

$f$

andtion$\Sigma_{t}$$\Gamma_{0}$are smooth enough and (4) is parabolic, there is a local-in-time classical solu-(see e.g. [GG2]). Moreover$\Sigma_{t}$agrees with$\Gamma_{t}$as far as the former exists$[G_{\backslash }G2]$(seealso [ES] for the mean curvature flow). So our generalized solution is a natural extentionof classical solution.

3. Instability of stationary solution

We say that$C^{2}$

hypersurface$S$

is stationary for$V=f(n, -A)$

if $f(n, -A)=0$ on

$S$

. Let

$U(\alpha)$denote a tubular neighborhood of$U(\alpha)=\{x\in \mathbb{R}^{n}$$S$

there is $\delta>0$ such that a (generalized) solutionof$\alpha’$of$S$$\Gamma_{0}$is contained insuch that every point ofand dented by reach$S$$U(\alpha’)$$S$of the form

; dist $(x,$ $S)<\alpha\}$ ,where dist denotes the distance. We say that$t>0$ provided that(8)$U(\epsilon)$

is Lyapunov stable for (8) if for each$\Gamma_{t}$. If not,with initial data$S$

stays in$\Gamma_{0}$is called unstable. If

has a unique nearest point on $S,$$\alpha$for all

is a supremum

is called the reach.

3.1. Instability Theorem.Suppose that (8) is strictly parabolic. Let$C^{2}$$\alpha$$U(\delta)$$\epsilon>0$

hypersurface of (8) such that $S=\partial D$ for some open set $D$ in$S$$\mathbb{R}^{n}$be a stationary. Suppose that107

reach $S=\alpha_{0}>0,$ $infs|A|=\sigma>0$ and that $|A|$ is bounded on$S$(if$S$is not compact).

For $0<\alpha<\alpha_{0}$ let$\Gamma^{\alpha}=\{x\in D$Let$)$$\Gamma_{t}^{\alpha}$; dist $(x,$ $S)=\alpha\}$ .

be a (generalized) solution of ( $8j$ starting from$\Gamma^{\alpha}$(9). Then there are$\alpha_{1}(0<\alpha_{1}<\alpha_{0}$and $c_{0}=c_{0}(f, \sigma, S)>0$ such that

distfor all $t>0$$(\Gamma_{t}^{\alpha}, S)\geq(\alpha+c_{0}t)\wedge\alpha_{1}$(10)where $a \wedge b=\min(a, b)$ . The same inequality holds if $D$ in (9) is replaced by its complement.

3.2. Corollary. Suppose that$S$

is a stationary

vanishing second fundamental form. Then

$S$$C^{2}$closed hypersurface for (8) with non-is (Lypunov) unstable provided that (8) isstrictly parabolic.

Of course Lyapunov instability follows from (10).

3.3. Remark on noncompact surface.Following formula for distance function is a key

for the proof of Theorem 3.1. Let$v$

be the signed distance function of , i. e.,$S$$v(x)=\{\begin{array}{l}dist (x, S) for x\in D- dist (x, S) otherwise.\end{array}$

If

$S$

is$C^{2}$, then so is$d$which is proved in [GT; \S 14].

Even for a non bounded open set generalized solution can be constructed by levelset
method; see Ilmanen [I] and Ishiiand Souganidis [IS].

1083.4. Lemma.For a general$C^{2}$

hypersurface $S=\partial D$ with an open set $D$$\nabla^{2}v(y)(I-v(y)\nabla^{2}v(y))^{-1}=\nabla^{2}v(x)$,$y=x+vn,$ $x\in S$for $|v|<$ reach .
$S$This is also a key in [GG2], where the local existence of classical solution is proved for(2) by a level set method.

3.5. Proof of Theorem 3.1. We set$w(t, x)=v(x)-\rho(t)$Our goal is to take $c_{0}>0$ so that$w$with $\rho(t)=\alpha+c_{0}t$ .

is a supersolution of the level set equation of (8):$u_{t}+F(\nabla u, \nabla^{2}u)=0$

in a set$U_{+}(\alpha_{1})\backslash U_{+}(\alpha/2)$$\{w\geq 0\}$Since$c_{0}$$F=F_{f}$, where$U_{+}(\alpha)=\{x\in D$

If such a,; dist $(x,$ $S)<\alpha\}$ .

exists, comparison principle ([CGG], [GGl]) implies that$\Gamma_{t}^{\alpha}$

is contained in. This yields (10).
$S$

is stationary, we see$F(\nabla v, \nabla^{2}v)=0$

on

$S$.

By Lemma 3.4 this yields$F(\nabla v, \nabla^{2}v(I-v\nabla^{2}v)^{-1})=0$in$U(\alpha_{0})$.

This implies$w_{t}+F(\nabla w, \nabla^{2}w)=-c_{0}+F(\overline{p},X)-F(\overline{p}_{2}X(I-vX)^{-1})$

in$U_{+}(\alpha_{0})$

.

(11)

109

with

$\overline{p}=\nabla v,$

We take$X=\nabla^{2}v$$\alpha_{1}>0$.

small so thatin$|\nabla^{2}v|\geq\sigma/2$$U_{+}(\alpha_{1})$$X\overline{p}\otimes\overline{p}=0$

(12)

.

(13)

in$(I-v\nabla^{2}v)^{-1}\geq I/2$

Since

(by $inf|A|$ ),$U_{+}(\alpha_{1})$by $|\nabla v|=1$ , we see$Q_{\overline{p}}(X^{2}(I-rX))^{-1}=X^{2}(I-rX)^{-1}=- \frac{1}{v}(X-X(I-vX)^{-1})$

By parabolicity$F(\overline{p}_{2}X)-F(\overline{p}_{2}X(I-vX)^{-1})\geq\mu v$$=\mu v$

with

$M= \sup_{U_{+}(\alpha_{1})}|\nabla^{2}v|$

trace $Q_{\overline{p}}[X^{2}(I-rX)^{-1}]$

trace $X^{2}(I-rX)^{-1}$on$U_{+}(\alpha_{1})$.

. Using (12), (13) we see
trace$X^{2}(I-rX)^{-1} \geq\frac{1}{2}(\frac{\sigma}{2})^{2}=c_{1}$which yields$F(\overline{p}, X)-F(\overline{p}, X(I-vX)^{-1})\geq\mu c_{1}v$

If we set $c_{0}=\mu c_{1}\alpha/2$ , from (11) it follows thatset equation of (8) in$U_{+}(\alpha_{1})\backslash \overline{U_{+}(\alpha/2)}$

on$U_{+}(\alpha_{1})$.

is a classical supersolution of the level$w$.

The proof for the last statement is parallel so is omitted.

3.6. General instability Theorem.

For (8) there is no stable stationary

hypersurface provided that (8) is strictly parabolic and that$f$is$C^{1}$$C^{2}$

closed.

Proof. By Corollary 2.2 we may assume that there is a point on at which$(n, A)=(\overline{p}_{0}, O),$ $f(n, A)=(\overline{p}, O)$

for some$\overline{p}_{0}\in S^{n-1}$. Since$S$

is stationary,$f(\overline{p}_{0}, O)=0$.

The following lemma implies the nonexistence of closed stationary solution, so the proof

is complete.

110

3.7. Nonexistence Lemma. Suppose that (8) is strictly parabolic andthat $f(\overline{p}_{0}, O)=O$ for some$\overline{p}_{0}\in S^{n-1}$. Then there is no stationary$C^{2}$$f$is. Suppose$C^{1}$

closed hypersurface

for (8)Proof. Let$S$

be a stationary$C^{2}$

closed hypersurface. Since$S$

is compact, there is a half

space $H$ such that
$H=\{x+c\in \mathbb{R}^{n}; x\cdot\overline{p}_{0}\geq 0\}$Note thattouch$\partial H$$S\subset H$

with $c\in S$ .

is a stationary solution of (8) since $f(\overline{p}_{0}, O)=0\}$ . Since (8) is strictly$\partial H$parabolic and,is$f$$C^{1}$

we may apply the strong maximum principle and conclude$S$

for $t>0$ . This contradicts the existence of stationay closed hypersuface

3.8. Remark.If$f$

is$C^{1}$in $A$ , the parabolicty is equivalent to say that$\partial f/\partial A$

cannot$S$.

is positive

definite.

4. Instability of periodic solutions

We consider$V=f(t, n, -A)$ ,where$f$:$[0, T]\cross Earrow \mathbb{R}$is continuous and T-periodic, $i.e$ . $f(t,\overline{p}, -A)=f(t+T,\overline{p}, -A)$ .

We say $S_{t}(-\infty<t<\infty)$ is a T-periodic$C^{2,1}$$S_{t}$

implies thatis now aLetin space and$C^{1}$$C^{2,1}$solution of (14) such that $S_{t}=S_{t+T}$ wherein time. Note that the signed distance function$v$offunction.

$C^{2,1}$$U(\alpha, t)$$C^{2}$(14)denote the -tubular neighborhood of$\alpha$$\epsilon>0$

always stays in$U(\epsilon, t)$$S_{t}$. We say$S_{t}$

is Lyapunov stable forthere is $\delta>0$ such that a generalized solution(14) if for each

In some cases$S_{t}$

for all $t>t_{0}$ provided thatis called$S_{t}$$\Gamma_{t_{0}}\subset U(\delta, t_{0})$is called T-periodic even if$S_{t}$$\Gamma_{t}$with$\Gamma_{t}|_{t=t_{0}}=\Gamma_{0}$.

is T-periodic up to T-periodictranslation $a(t)$ , i. e., $S_{t+T}=S_{t}+a(t)$ . Here $a(t)$ is asuumed to have the form$a(t)=b(t+T)-b(t)$111

for some$C^{1}$

function $b(t)$ . If we set$\Sigma_{t}=S_{t}-b(t)$then$\Sigma_{t}=\Sigma_{t+T}$

and$\Sigma_{t}$solves$V=f+n\cdot b(t)$ .

Thus by T-periodic solution we shall always mean T-periodic with no ambiguity of translation.

4.1. Instability Theorem.Suppose that (14) is strictly parabolic. Let

be a T-periodic

solution of (8) of closed hypersurfaces surrounding an bounded open set $D$ in$C^{2}$

Suppose that $\inf_{S_{t}}|A|=a(t)\not\equiv O$ . Let$0<\alpha<\alpha_{0},$Let$\Gamma_{t}^{\alpha}$$S_{t}$$,$

denote the minimum of reach; dist $(x,$$S_{t}$.

in . For$t$

for $t>t_{0}$ .

$S_{t_{0}}=\alpha\}$be a (generalized) solution of (14) with$\alpha_{1}(S_{t})(0<\alpha_{1}<\alpha_{0})$a$\alpha_{0}>0$$\mathbb{R}^{n}$$t_{0}\in \mathbb{R}$$\Gamma^{\alpha}=\{x\in D$$f,$$S_{t}$$\Gamma_{t}^{\alpha}=\Gamma^{\alpha}$

and nonnegative T-periodic function

at $t=t_{0}$ . Then there are$c_{0}(t)(\not\equiv 0)$

depending only onsuch thatdist$( \Gamma_{t}^{\alpha}, S_{t})\geq(\alpha+\int_{0}^{t}c_{0}(\tau)d\tau)\wedge\alpha_{1}$The same inequality holds if $D$ in the definition of$\Gamma^{\alpha}$

for all $t>0$ .

replaced by$\mathbb{R}^{n}\backslash D$.

Proof. As in the proof of Theorem 3.1 we set$w(x,t)=v(x,t)-\rho(t)$ ,and choose $c_{0}’>0$ in suitable way so that(14). Here

$v$

$w$

$\rho(t)=c_{0}’\int_{0}^{t}a(\tau)^{2}d\tau$,is a supersolution of the level set equation ofdenotes the signed distance function of$S_{t}$.

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4.2. Corollary.Suppose that (14) is strictly parabolic. Ifof (14) consisting of closed hypersurfaces, thena function of time.

$S_{t}$$S_{t}$

is a T-periodic

is Lyapunov unstable if$C^{2,1}$solution$\inf_{S_{t}}|A|\not\equiv 0$asThis is an imediate from Theorem 4.1.

References[CGG] Y.-G. Chen, Y. Giga and S. Goto, Uniqueness and existence of viscosity solu-tions of generalized mean curvature flow equations, J.Defferential Geometry 33, 749-786,(1991).
[EY] S.-I. Ei and E. Yanagida, Stability of stationary interfaces in a generalized mean

curvature flow, preprint.
[ES] L.C. Evans and J. Spruck, Motion of level sets by mean curvature I, J. Defferential

Geometry 33, 635-681, (1991).
[GGl] Y. Giga and S. Goto, Motion of hypersurfaces and geometric equations, J.Math. So$($44, $99- 111$ , (1992).
[GG2] Y. Giga and S. Goto, Geometric evolutions of phase boundaries, “On the evolutionof phase boundaries”,(eds. M.E. Gurtin, G.B. McFadden), IMA volumes in mathematics
and its applications 43 (1992), pp. 40, 443-470, (1991).
[GT] D. Gilbarg and N.S. Tmdinger, Elliptic Partial Differential Equations of Second

Order,$2^{nd}$Ed., Springer-Verlag, New york 1983.

[I] T. Ilmanen, Generalized flow of sets by mean curvature on a manifold, Indiana Univ.

Math. J. 41 (1992), 671-705.
[IS] H. Ishii and P. Souganidis, forthco ming