Motion of Level Sets by Mean Curvature II

L. C. Evans^{∗}

Department of Mathematics

University of California Berkeley, CA 94720

J. Spruck^{†}

Department of Mathematics University of Massachusetts Amherst, MA 01003

Abstract

1 Introduction

We present in this paper a new, elementary, and fairly concise proof of short time existence for the classical motion of a smooth hypersurface evolving according to its mean curvature. In this problem we are given initially a smooth connected hypersurface Γ_{0} which is the boundary of a bounded open set U ⊂ ^{n}. We then allow Γ_{0} to evolve in time into a family of surfaces {Γ_{t}}_{t>0} by moving each point on Γ_{t} (t ≥ 0) in the opposite direction to its mean curvature vector, at a velocity equal to (n−1) times the absolute value of the mean curvature. Our intent is to verify that for small times at least, the classical motion as envisioned in fact exists and is unique. This assertion was first proved by R. Hamilton [4], and we discuss below the relation of our work to his.

∗Supported in part by NSF grant DMS-86-01532.

†Supported in part by NSF grant DMS-85-01952 and DOE grant DE-FG02-86ER250125.

This is a companion to our paper [2], wherein we defined and then studied a generalized notion of evolution via mean curvature, existing for all times and agreeing with the classical motion, if and so long as the latter exists. This generalized evolution is constructed in [2] by first building an appropriately defined unique weak solution of the PDE

u_{t} = (δ_{ij} − \frac{u_{x_{i}}u_{x_{j}}}{|Du|^{2}} )u_{x_{i}x_{j}} in ^{n} × (0, ∞) (1.1)

u = g on ^{n} × {t = 0}. (1.2)

Equation (1.1) says that each level set of u evolves according to its mean curvature, at least in regions when u is smooth and |Du| ≠ 0. Given then Γ_{0} as above, we select a smooth function g:^{n} → for which

Γ_{0} = {x ∈ ^{n} | g(t) = 0}. (1.3)

We next write Γ_{t} ≡ {x ∈ ^{n} | u(x, t) = 0} (t ≥ 0), (1.4)

and call the family of sets {Γ_{t}}_{t≥0} so defined the generalized evolution by mean curvature starting from Γ_{0}. The rest of [2] is devoted to deducing from the PDE (1.1), (1.2) various elementary geometric properties of the flow Γ_{0} ⟼ Γ_{t}(t ≥ 0).

In this paper we return again to the idea of studying a nonlinear PDE, a level set of whose solution evolves via mean curvature. Our idea is first to assume that Γ_{0} develops by classical mean curvature motion, at least for times 0 ≤ t ≤ t_{0}, and then to derive the PDE verified by d, the signed distance function to the surface at each time. This turns out to be a fully nonlinear, uniformly parabolic equation: see (1.9) below. Next we construct for short times a smooth solution of this equation, subject to nonlinear boundary conditions in an appropriate region, and then finally verify that our solution is in fact the signed distance function to a family of smooth surfaces evolving from Γ_{0} by mean curvature motion.

We proceed now to the heuristic derivation of our PDE. Suppose therefore we are given the smooth hypersurface Γ_{0} = ∂U as above, a time t_{0} > 0, and a classical evolution {Γ_{t}}_{0≤t≤t_{0}} of surfaces developing from Γ_{0} by mean curvature flow. Then for each time t in [0, t_{0}] , Γ_{t} is the smooth boundary of a bounded open set U_{t}, which is diffeomorphic to U = U_{0}. Define the signed distance function

(1.5) d(x, t) ≡ \{ \begin{matrix}dist (x, Γ_{t}) & (x ∈ ^{n} − \overline{U}_{t}) \\ −dist (x, Γ_{t}) & (x ∈ U_{t}) \end{matrix}

for x ∈ ^{n}, 0 ≤ t ≤ t_{0}. By assumption Γ ≡ ⋃_{0≤t≤t_{0}} Γ_{t} x{t} is smooth, and thus d is smooth in

Q^{+} ≡ {(x, t) | 0 < t < t_{0}, 0 < d(x, t) < δ_{0}}

and in

Q^{−} = {(x, t) | 0 < t < t_{0}, −δ_{0} < d(x, t) < 0}

for δ_{0} > 0 small enough. Fix any point (x, t) ∈ Q^{+}. Then provided δ_{0} > 0 is sufficiently small, there exists a unique point y ∈ Γ_{t} for which d(x, t) = |x−y|. Let ν = Dd be the smooth unit normal vector field pointing from Γ into Q^{+}. As {Γ_{t}}_{0≤t≤t_{0}} moves via mean curvature, we have

d_{t}(x, t) = div(ν) (y, t).

On the other hand the (unordered) eigenvalues of the matrix D^{2}d(x, t) are

\begin{matrix}λ_{i} = λ_{i}(D^{2}d(x, t)) = −κ_{i}/1 − κ_{i}d(x, t) & (1 ≤ i ≤ n − 1) \\ λ_{n} = λ_{n}(D^{2}d(x, t)) = 0, \end{matrix} (1.6)

{κ_{1}, …, κ_{n−1}} denoting the principle curvatures of Γ_{t} at y, computed with respect to the normal field ν. Inverting (1.6) we compute

κ_{i} = \frac{λ_{i}}{λ_{i}d(x, t) − 1} (1 ≤ i ≤ n − 1).

Since

div(ν) = −(κ_{1} + ⋯ + κ_{n−1}),

we deduce finally

d_{t}(x, t) = f(λ_{1}(D^{2}d(x, t)), ⋯, λ_{n}(D^{2}d(x, t)), d(x, t)) (1.7)

for

f(λ_{1}, ⋯, λ_{n}, z) ≡ ∑_{i=1}^{n} \frac{λ_{i}}{1 − λ_{i}z} (1.8)

The same formula results if (x, t) ∈ Q^{−}.

Now our PDE (1.7) has the general form

d_{t} = F (D^{2}d, d) ; (1.9)

and, as f is symmetric in the variables λ_{1}, … λ_{n}, F is smooth ([1]). Since

\frac{∂f}{∂λ_{i}} = \frac{1}{(1 − λ_{i}d)^{2}} > 0 (1 ≤ i ≤ n) , (1.10)

equation (1.9) is uniformly parabolic ([1]).

Our plan is to study directly the PDE (1.7), (1.8). For this suppose now Γ_{0} is the smooth connected boundary of a bounded open set U ⊂ ^{n}, and let

(1.11) g(x) ≡ \{ \begin{matrix}dist (x, Γ_{0}) & (x ∈ ^{n} − \overline{U}) \\ –dist (x, Γ_{0}) & (x ∈ U) \end{matrix}

be the signed distance function. Fix then δ_{0} > 0 so small that g is smooth within V ≡ {x ∈ ^{n} | −δ_{0} < g < δ_{0}}. (1.12) and write

Q ≡ V × (0, t_{0}), Γ ≡ ∂V × [0, t_{0}].

In § 2 we construct a smooth solution v to the PDE

\{ \begin{matrix}v_{t} = F (D^{2}v, v) & in Q \\ |Dv|^{2} = 1 & on Γ \\ v = g & on V × \{t = 0\} \end{matrix} (1.13)

for some small time t_{0} > 0, our argument being in effect a special case of the Inverse Function Theorem. We then verify in § 3 that

|Dv|^{2} = 1 in Q . (1.14)

Setting

Γ_{t} ≡ {x ∈ V | v(x, t) = 0} (0 ≤ t ≤ t_{0}) ,

we deduce from (1.13) and (1.14) that {Γ_{t}}_{0≤t≤t_{0}} is a smooth evolution by mean curvature and v = d is the corresponding signed distance function.

Section 4 utilizes the PDE (1.7), (1.8) to prove an “instantaneous tearing apart” assertion for close-by evolving surfaces. Some consequences are discussed in [2, § 8].

As noted, R. Hamilton in [4, § 5] has previously established the shorttime existence of the classical mean curvature evolution by studying the degenerate parabolic system describing the parametrized surface. As this system is degenerately parabolic ([4, p. 261]) , Hamilton is forced to employ fairly complicated techniques related to the Moser-Nash Implicit Function Theorem. See also Gage-Hamilton [3, § 2]. Our methods are simpler.

2 Solving the Nonlinear PDE

Our goal in this section is to construct a smooth solution of the PDE (1.13) for δ_{0} > 0 and t_{0} > 0 small enough. Let us first of all select δ_{0} so small that

Mδ_{0} ≤ \frac{1}{4} (2.1)

for

M ≡ max_{V} |D^{2}g|. (2.2)

Set

G ≡ {R ∈ S^{n×n}, z ∈ | |z| < δ_{0}, |R| < 2M}.

Then since (2.1) implies

|λ_{i}(R)z| ≤ |R||z| ≤ \frac{1}{2} (1 ≤ i ≤ n)

if (R, z) ∈ G,

\begin{aligned}F (R, z) & = f(λ_{1}(R), ⋯, λ_{n}(R), z) \\ & = ∑_{i=1}^{n} \frac{λ_{i}(R)}{1−λ_{i}(R)z} \end{aligned} (2.3)

is defined and smooth on G. Arbitrarily extend F off G to be smooth on all of S^{n×n} ×, with |F |, |DF |, |D^{2}F | bounded.

We now check that our PDE is uniformly parabolic near g.

Lemma 2.1 Then exists a constant θ > 0 so that

\frac{∂F}{∂r_{ij}} (R, z)ξ_{i}ξ_{i} ≥ θ|ξ|^{2} (ξ ∈ ^{n}) (2.4)

in V for each (R, z) ∈ G.

Proof. Fix ξ ∈ ^{n} and choose then t > 0 so small that (R + t ξ ⊗ ξ, z) ∈ G. Using Courant’s minimax characterization we deduce that the eigenvalues can be ordered so that

λ_{k}(R + t ξ ⊗ ξ) ≥ λ_{k}(R) (k = 1, ⋯ n).

Consequently

F (R + t ξ ⊗ ξ, z) − F (R, z)

= f(⋯, λ_{k}(R + t ξ ⊗ ξ), ⋯, z) − f(⋯, λ_{k}(R) ⋯, z)

= ∑_{i=1}^{n} ∫_{0}^{1} \frac{∂f}{∂λ_{k}} (⋯, sλ_{k}(R + t ξ ⊗ ξ) + (1 − s)λ_{k}(R), ⋯, z)ds [λ_{k}(R + t ξ ⊗ ξ) − λ_{k}(R)]

≥ θ trace (R + t ξ ⊗ ξ − R)

for some θ > 0 by (1.10). Dividing by t > 0 and sending t → 0^{+}, we arrive at inequality (2.4). □

We will seek our solution v of the PDE (1.13) in the form

v = g + th + w, (2.5)

where h ≡ F (D^{2}g, g) . (2.6)

Then plugging into (1.13) we compute

w_{t} − a_{ij} w_{x_{i}x_{j}} + cw = A(D^{2}w, w, x, t) in Q

for

a_{ij} ≡ \frac{∂F}{∂r_{ij}} (D^{2}g, g), c ≡ \frac{−∂F}{∂z} (D^{2}g, g) (2.7)

and

\begin{aligned}A(R, z, x, t) & ≡ F (D^{2}g + t D^{2}h + R, g + th + z) \\ & −F (D^{2}g, g) − \frac{∂F}{∂r_{ij}} (D^{2}g, g) r_{ij} − \frac{∂F}{∂z} (D^{2}g, g)z. \end{aligned} (2.8)

We next insert (2.5) into the nonlinear boundary condition from the PDE (1.13). Recalling that |Dg|^{2} = 1 and Dg is normal to ∂V we discover

\frac{∂w}{∂ν} = a (Dw, x, t) on Γ ,

when ν is the outer unit normal vector field along V and

a (p, x, t) ≡ \{ \begin{matrix}−\frac{1}{2} |t Dh + p|^{2} − t \frac{∂h}{∂ν} & on \{d = δ_{0}\} \\ \frac{1}{2} |t Dh + p|^{2} − t \frac{∂h}{∂ν} & on \{d = −δ_{0}\}. \end{matrix} (2.9)

Combining everything above, we hereafter seek a smooth function w satisfying

\{ \begin{matrix}w_{t} − a_{ij} w_{x_{i}x_{j}} + cw = A(D^{2}w, w, x, t) & in Q \\ \frac{∂w}{∂ν} = a (Dw, x, t) & on Γ \\ w = 0 & on V × \{t = 0\}. \end{matrix} (2.10)

We intend to solve (2.10) by finding a fixed point of the mapping T defined by inserting a given function into the nonlinear terms A, a and solving the resulting linear PDE. We will work in certain standard parabolic-type Hölder spaces. Fix 0 < α < 1.

We define several norms, following Ladyzenskaja, Solonnikov, Ural’ceva [5, p.7-8]. Set

|u|^{(0)} ≡ sup{|u(x, t)| | (x, t) ∈ Q}
|u|^{(1)} ≡ |u|^{(0)} + |Du|^{(0)}
|u|^{(2)} ≡ |u|^{(1)} + |D^{2}u|^{(0)} + |u_{t}|^{(0)}
⟨u⟩_{x}^{(β)} ≡ sup \{\frac{|u(x,t)−u(y,t)|}{|x−y|^{β}} | (x, t), (y, t) ∈ Q, x ≠ y\}
⟨u⟩_{t}^{(β)} ≡ sup \{\frac{|u(x,t)−u(x,s)|}{|t−s|^{β}} | (x, t), (x, s) ∈ Q, t ≠ s\} ,

for 0 < β < 1. Then

\begin{aligned}∥u∥_{C^{α, \frac{α}{2}} ¯ (Q)} & ≡ |u|^{(0)} + ⟨u⟩_{x}^{(α)} + ⟨u⟩_{t}^{( \frac{α}{2} )} \\ ∥u∥_{C^{1+α, \frac{1+α}{2}} ¯ (Q)} & ≡ |u|^{(1)} + ⟨Du⟩_{x}^{(α)} + ⟨u⟩_{t}^{( \frac{1+α}{2} )} + ⟨Du⟩_{t}^{( \frac{α}{2} )} \\ ∥u∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} & ≡ |u|^{(2)} + ⟨Du⟩_{t}^{( \frac{1+α}{2} )} + ⟨D^{2}u⟩_{x}^{(α)} + ⟨u_{t}⟩_{x}^{(α)} + ⟨D^{2}u⟩_{t}^{( \frac{α}{2} )} + ⟨u_{t}⟩_{t}^{( \frac{α}{2} )}. \end{aligned}

Note

∥Du∥_{C^{1+α, \frac{1+α}{2}} ¯ (Q)}, ∥D^{2}u∥_{C^{α, \frac{α}{2}} ¯ (Q)}, ∥u_{t}∥_{C^{α, \frac{α}{2}} ¯ (Q)} ≤ ∥u∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} .

Finally define

∥u∥_{C^{1+α, \frac{1+α}{2}} (Σ)} ≡ inf \{∥v∥_{C^{1+α, \frac{1+α}{2}} ¯ (Q)} | v = u on Σ\} .

We consider now the linear, uniformly parabolic PDE

\{ \begin{matrix}w_{t} − a_{ij} w_{x_{i}x_{j}} + cw = B & in Q \\ \frac{∂w}{∂ν} = b & on Σ \\ w = 0 & on V × \{t = 0\}. \end{matrix} (2.11)

Suppose as well b = 0 on ∂V × {t = 0} . (2.12)

Lemma 2.2 Assume B ∈ C^{α, \frac{α}{2}} ¯ (Q), b ∈ C^{1+α, \frac{1+α}{2}} (Γ), and the zeroth order compatibility condition (2.12) holds. Then there exists a unique solution w ∈ C^{2+α, \frac{2+α}{2}} ¯ (Q) of (2.11), with the estimate

∥w∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} ≤ C (∥B∥_{C^{α, \frac{α}{2}} ¯ (Q)} + ∥b∥_{C^{1+α, \frac{1+α}{2}} (Γ)}) . (2.13)

The constant C does not depend on t_{0}.

Proof. Except for the last assertion, this is Theorem IV.5.3 in Ladyzenskaja, Solonnikov, Ural’ceva [5, p. 320–321]. To obtain the statement that the constant C remains bounded even for small 0 < t_{0} < 1 , we define

Q̃ = V × (0, 1), Γ̃ = ∂V × (0, 1) ,

B̃(x, t) = \{ \begin{matrix}B(x, t) & if 0 ≤ t ≤ t_{0} \\ B(x, t_{0}) & if t_{0} ≤ t ≤ 1, \end{matrix}

b̃(x, t) = \{ \begin{matrix}b(x, t) & if 0 ≤ t ≤ t_{0} \\ b(x, t_{0}) & if t_{0} ≤ t ≤ 1. \end{matrix}

Then ∥B̃∥_{C^{α, \frac{α}{2}} ˜ (Q)} = ∥B∥_{C^{α, \frac{α}{2}} (Q)}

∥b̃∥_{C^{1+α, \frac{1+α}{2}} (_{Γ})} = ∥b∥_{C^{1+α, \frac{1+α}{2}} (Γ)}.

Let w̃ ∈ C^{2+α, \frac{2+α}{2}} ˜ (Q) solve the PDE (2.11) with Q̃, Γ̃, B̃, b̃ replacing Q, Γ, B, b. The estimate (2.13) then is valid for w̃ with Q̃, Γ̃ replacing Q, Γ. But w̃ = w on Q, by uniqueness for parabolic equations. □

We will henceforth work in the Banach space

X ≡ \{w ∈ C^{2+α, \frac{2+α}{2}} ¯ (Q) | w = 0 on V × \{t = 0\}\}.

Given ŵ ∈ X, we set

\{ \begin{matrix}B(x, t) & ≡ A(D^{2}ŵ, ŵ, x, t) \\ b(x, t) & ≡ a(Dŵ, x, t) \end{matrix} (2.14)

for A, a defined by (2.8), (2.9). Now write

T (ŵ) = w,

when w ∈ X solves the linear PDE (2.11), for B, b as in (2.14). We seek a fixed point of the mapping

T: X → X.

Give r_{0} > 0, set

Y ≡ \{w ∈ X | ∥w∥_{C^{2+α, \frac{2+α}{2}} (Q)} ≤ r_{0}\} .

Lemma 2.3 If t_{0}, r_{0} > 0 are sufficiently small, then

T: Y → Y (2.15)

Proof. 1. Choose any function ŵ ∈ Y, define B, b by (2.14) and let w ∈ X solve (2.11). We must show the inequality ∥ŵ∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} ≤ r_{0} (2.16) implies ∥w∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} ≤ r_{0}, (2.17) provided r_{0}, t_{0} > 0 are adjusted sufficiently small.

2. Now (2.8) implies

A(R, z, x, t) = t\frac{∂F}{∂r_{ij}} (D^{2}g, g)h_{x_{i}x_{j}} + t\frac{∂F}{∂z} (D^{2}g, g)h
+ ∫_{0}^{1}(1 − s)\frac{∂^{2}F}{∂r_{ij}∂r_{kℓ}} (D^{2}g + stD^{2}h + sR, g + sth + sz)ds
×(th_{x_{i}x_{j}} + r_{ij}) (th_{x_{k}x_{ℓ}} + r_{kℓ})
(2.18) + 2 ∫_{0}^{1}(1 − s)\frac{∂^{2}F}{∂r_{ij}∂z} (D^{2}g + stD^{2}h + sR, g + sth + sz)ds
×(th_{x_{i}x_{j}} + r_{ij}) (th + z)
+ ∫_{0}^{1}(1 − s)\frac{∂^{2}F}{∂z^{2}} (D^{2}g + stD^{2}h + sR, g + sth + sz)ds × (th + z)^{2} .

Recall also that ∥uv∥_{C^{α, \frac{α}{2}} ¯ (Q)} ≤ C∥u∥_{C^{α, \frac{α}{2}} ¯ (Q)}∥v∥_{C^{α, \frac{α}{2}} ¯ (Q)} (2.19) for all u, v ∈ C^{α, \frac{α}{2}} ¯ (Q), and ∥t∥_{C^{α, \frac{α}{2}} ¯ (Q)} ≤ Ct_{0}^{1− \frac{α}{2}} . (2.20) Then (2.14), (2.18), (2.20) imply ∥B∥_{C^{α, \frac{α}{2}} ¯ (Q)} ≤ C(r_{0}^{2} + t_{0}^{1− \frac{α}{2}} ). (2.21) Similarly formula (2.9) implies

∥b∥_{C^{1+α, \frac{1+α}{2}} (Γ)} ≤ C(r_{0}^{2} + t0^{\frac{1−α}{2}} ) (2.22)

3. Inserting estimates (2.21), (2.22) into (2.13) we discover

∥w∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} ≤ C(r_{0}^{2} + t0^{\frac{1−α}{2}} ) , (2.23)

the constant C independent of r_{0} and t_{0}. Let

r_{0} ≤ \frac{1}{2C} .

Then (2.23) gives

∥w∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} ≤ \frac{r_{0}}{2} + C t0^{\frac{1−α}{2}} ≤ r_{0}

if t_{0} = t_{0}(r_{0}) is small enough. This verifies inequality (2.17). □

Finally we verify that T: Y → Y is a strict contraction. Lemma 2.4 If r_{0}, t_{0} > 0 are small enough,

∥T (ŵ_{1}) − T (ŵ_{2})∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} ≤ \frac{1}{2}∥ŵ_{1} − ŵ_{2}∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} (2.24)

for all ŵ_{1}, ŵ_{2} ∈ Y. Proof. 1. Choose ŵ_{1}, ŵ_{2} ∈ Y; so that

∥ŵ_{1}, ŵ_{2}∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} ≤ r_{0} .

Set

\{ \begin{matrix}B_{1}(x, t) ≡ A(D^{2}ŵ_{1}, ŵ_{1}, x, t), & B_{2}(x, t) ≡ A(D^{2}ŵ_{2}, ŵ_{2}, x, t) \\ b_{1}(x, t) ≡ a(Dŵ_{1}, x, t), & b_{2}(x, t) ≡ a(Dŵ_{2}, x, t) , \end{matrix} (2.25)

and write

w_{1} ≡ T (ŵ_{1}), w_{2} ≡ T (ŵ_{2})

Thus w_{1} solves the linear PDE (2.11) with B_{1}, b_{1} replacing B, b, and w_{2} solves (2.11) with B_{2}, b_{2} replacing B, b. According to Lemma 2.2 ∥w_{1} − w_{2}∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} ≤ C(∥B_{1} − B_{2}∥_{C^{α, \frac{α}{2}} ¯ (Q)} + ∥b_{1} − b_{2}∥_{C^{1+α, \frac{1+α}{2}} (Γ)}) . (2.26)

2. Recall that ∥Φ(u) − Φ(v)∥_{C^{α, \frac{α}{2}} ¯ (Q)} ≤ (2.27)

C(∥u∥_{C^{α, \frac{α}{2}} ¯ (Q)} + ∥v∥_{C^{α, \frac{α}{2}} ¯ (Q)} + 1)∥u − v∥_{C^{α, \frac{α}{2}} ¯ (Q)}

if u, v ∈ C^{α, \frac{α}{2}} ¯ (Q) and Φ is smooth, with DΦ, D^{2}Φ bounded.

Utilizing then formulas (2.18)–(2.20), (2.27) we compute ∥B_{1} − B_{2}∥_{C^{α, \frac{α}{2}} ¯ (Q)} ≤ C(r_{0} + t_{0}^{1− \frac{α}{2}} )∥ŵ_{1} − ŵ_{2}∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} (2.28) Similarly

∥b_{1} − b_{2}∥_{C^{1+α, \frac{1+α}{2}} (Γ)} ≤ C(r_{0} + t0^{\frac{1−α}{2}} )∥ŵ_{1} − ŵ_{2}∥_{C^{2+α, \frac{2+α}{2}} ¯ (Q)} . (2.29)

Combining (2.26), (2.28), (2.29), we obtain estimate (2.24), provided r_{0}, t_{0} > 0 are small. □

We at last apply Banach’s Fixed Point Theorem to establish Theorem 2.5 If δ_{0}, t_{0} > 0 are sufficiently small,there exists a unique solution v ∈ C^{2+α, \frac{2+α}{2}} ¯ (Q) ∩ C^{∞}(Q) of the PDE (1.13).

In particular (D^{2}v, v) ∈ G in Q.

3 Motion of the Zero Level Set by Mean

Curvature

This section we devote to proving that the sets Γ_{t} ≡ {x ∈ V | v(x, t) = 0} (0 ≤ t ≤ t_{0}) (3.1) are in fact smooth hypersurfaces evolving by mean curvature. Theorem 3.1 We have |Dv|^{2} = 1 in Q . (3.2) Proof. 1. Let w ≡ |Dv|^{2} − 1 ∈ C^{1} ¯ (Q) ∩ C^{∞}(Q). Then w = 0 on Γ (3.3) and w = 0 on V × {t = 0}, (3.4) according to (1.11) and the PDE (1.13).

2. Differentiating (1.13) we compute

v_{t x_{k}} = \frac{∂F}{∂r_{ij}} (D^{2}v, v) v_{x_{i}x_{j}x_{k}} + \frac{∂F}{∂z} (D^{2}v, v) v_{x_{k}}

Thus

\begin{aligned}w_{t} & = 2v_{x_{k}}v_{x_{k}t} \\ & = 2\frac{∂F}{∂r_{ij}} (D^{2}v, v) v_{x_{k}}v_{x_{k}x_{i}x_{j}} + 2\frac{∂F}{∂z} (D^{2}v, v) |Dv|^{2} \\ & = \frac{∂F}{∂r_{ij}} (D^{2}v, v) w_{x_{i}x_{j}} − 2\frac{∂F}{∂r_{ij}} (D^{2}v, v) v_{x_{k}x_{i}}v_{x_{k}x_{j}} + 2\frac{∂F}{∂z} (D^{2}v, v) |Dv|^{2} \end{aligned} (3.5)

Now

F (D^{2}v, v) = f(⋯, λ_{i} (D^{2}v), ⋯, v) in Q.

Thus

\begin{aligned}\frac{∂F}{∂z} (D^{2}v, v) & = \frac{∂f}{∂z} ( ⋯, λ_{i} (D^{2}v), ⋯, v) \\ & = ∑_{i=1}^{n} \frac{λ_{i} (D^{2}v)^{2}}{(1−λ_{i} (D^{2}v)v)^{2}} \end{aligned} (3.6)

On the other hand (see [1])

\frac{∂F}{∂r_{ij}} (D^{2}v) v_{x_{k}x_{i}}v_{x_{k}x_{j}} = ∑_{i=1}^{n} \frac{∂f}{∂λ_{i}} (⋯, λ_{i} (D^{2}v), ⋯, v) λ_{i} (D^{2}v)^{2}
= ∑_{i=1}^{n} \frac{λ_{i} (D^{2}v)^{2}}{(1−λ_{i} (D^{2}v)v)^{2}}
= \frac{∂F}{∂z} (D^{2}v, v) by (3.6) .

Thus (3.5) transforms to read

w_{t} = \frac{∂F}{∂r_{ij}} (D^{2}v, v) w_{x_{i}x_{j}} + 2\frac{∂F}{∂z} (D^{2}v, v) w in Q. (3.7)

In view of Lemma 2.1 this is a uniformly parabolic equation. As (3.3), (3.4) assert w = 0 on the parabolic boundary of Q, we deduce w = 0 everywhere within Q. □

Owing to (3.1) we see that

Γ = \{(x, t) ∈ \overline{Q} | v = 0\}

is a smooth hypersurface in^{n+1}∩Q and each slice Γ_{t} is a smooth hypersurface in V.

Theorem 3.2 The surfaces {Γ_{t}}_{0≤t≤t_{0}} comprise a classical motion by mean curvature starting from Γ_{0}.

Proof. For each fixed t in [0, t_{0}], ν ≡ Dv is a normal unit vector field to Γ_{t} in ^{n}. In addition the PDE (1.13) implies

v_{t} = Δv = div(ν) on Γ_{t}.

However

div(ν) = −(n − 1)H = −(κ_{1} + ⋯ + κ_{n−1}) ,

where κ_{1}, ⋯ κ_{n−1} are the principal curvatures of Γ_{t} computed with respect to ν and

H = \frac{1}{n − 1} (κ_{1} + ⋯ κ_{n−1})

is the mean curvature.

Now fix 0 ≤ t ≤ t_{0}, x ∈ Γ_{t}, and evolve the point x according to the nonautonomous ODE

\{ \begin{matrix}ẋ(s) = −[div(ν)ν] (x(s), s) & (s > t) \\ x(t) = x . \end{matrix}

Then

\frac{d}{ds} v(x(s), s) = −[(Dv · ν) div(ν)] (x(s), s) + v_{t}(x(s), s) = −div(ν) + v_{t} = 0.

Thus

v(x(s), s) = 0 (s > t)

and this implies {Γ_{t}}_{0≤t≤t_{0}} is evolving by mean curvature. □

4 Instantaneous Tearing Apart

As a further application of the PDE (1.7)–(1.9) we prove in this section that two distinct smooth surfaces evoloving by mean curvature must instantly tear completely apart, even if they initially touched on a large set.

More precisely, suppose as above Γ_{0} is the smooth connected boundary of the bounded open set U ⊂ ^{n}. Assume also that Γ̂_{0} is the smooth connected boundary of a bounded open subset Û ⊂ ^{n} and that

Γ̂_{0} ⊂ \overline{U}. (4.1)

Thus the surface Γ̂_{0} lies within the closed region \overline{U} bounded by Γ_{0}. Suppose further that Γ̂_{0} ≠ Γ_{0}. (4.2) Choose next t_{0} > 0 so small that the classical evolutions {Γ_{t}} and {Γ̂_{t}} starting at Γ_{0} and Γ̂_{0}, respectively, exist at least for times 0 ≤ t ≤ t_{0}.

Theorem 4.1 We have

Γ_{t} ∩ Γ̂_{t} = ∅ for 0 < t ≤ t_{0}. (4.3)

We see therefore that the two evolutions by mean curvature instantly completely “tear apart”. The point is that the PDE describing evolution by mean curvature is “uniformly parabolic along the surface” and this admits infinite propagation speed for disturbances. See [2, §8] for some geometric consequences. Proof. 1. Fix δ_{0} so small that the signed distance function

d(x, t) ≡ \{ \begin{matrix}–dist(x, Γ_{t}) & if x ∈ U_{t} \\ dist(x, Γ_{t}) & if x ∈ ^{n} − U_{t} \end{matrix}

is smooth in

Q ≡ {(x, t) | −δ_{0} ≤ d(x, t) ≤ δ_{0}, 0 ≤ t ≤ t_{0}}.

(Here, U_{t} is the bounded region enveloped by Γ_{t}). As above we have d_{t} = F (D^{2}d, d) in Q. (4.4) Similarly the signed distance

d̂(x, t) ≡ \{ \begin{matrix}–dist(x, tΓ̂_{t}) & if x ∈ Û_{t} \\ dist(x, Γ̂_{t}) & if x ∈ ^{n} − Û_{t} \end{matrix}

is smooth in Q, provided δ_{0}, t_{0} > 0 are sufficiently small, and Γ̂_{0} is close enough to Γ_{0}. (We denote by Û_{t} the region bounded by Γ̂_{t}). Also d̂_{t} = F (D^{2}d̂, d̂) in Q. (4.5) as well.

2. Now since (4.1) holds, we have

Γ̂_{t} ⊆ \overline{U}_{t}

for each 0 ≤ t ≤ t_{0}. Thus

d̂ ≥ d in Q.

3. Set

w ≡ d̂− d.

Then w ≥ 0 in Q, w ≠ 0 on Q ∩ {t = 0}, (4.6) and from (4.4), (4.5) it follows that w_{t} = a_{ij} w_{x_{i}x_{j}} + cw in Q (4.7) for

a_{ij} ≡ ∫_{0}^{1} \frac{∂F}{∂r_{ij}} (t D^{2}d̂+ (1 − t) D^{2}d, td̂+ (1 − t)d)dt

and

c ≡ ∫_{0}^{1} \frac{∂F}{∂z} (t D^{2}d̂+ (1 − t) D^{2}d, td̂+ (1 − t)d)dt .

According to Lemma 2.1 the coefficients ((a_{ij})) are uniformly positive definite if δ_{0}, t_{0} are small enough and Γ̂_{0} is close enough to Γ_{0}. From (4.6), (4.7) and the strong maximum principle we deduce that

(4.8) w > 0 in the interior of Q.

Assertion (4.3) follows.

4. In the general case that Γ̂_{0} verfies (4.1), (4.2), but is not necessarily everywhere close to Γ_{0}, we interpolate a smooth surface Γ̃_{0} between Γ_{0} and Γ̂_{0}, so that Γ̃_{0} satisfies (4.1), (4.2) and is close to Γ_{0}. Then, by steps 1–3,

Γ̃_{t} ∩ Γ_{t} = ∅

for all small t > 0, {Γ̃_{t}} denoting the classical evolution from Γ̃_{0}. As Γ̃_{t} lies within the region bounded by Γ_{t} and Γ̂_{t} lies in the region bounded by Γ̃_{t}, assertion (4.3) follows in this case as well. □

References

[1] L. A. Caffarelli, L. Nirenberg and J. Spruck, The Dirichlet problem for nonlinear second order elliptic equations III: Functions of the eigenvalues of the Hessian, Acta Math. 155 (1985), 261–301.

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

[3] M. Gage and R. S. Hamilton, The heat equation shrinking convex plane curves, J. Differential Geometry 23 (1986), 69–96.

[4] R. S. Hamilton, Three manifolds with positive Ricci curvature, J. Differential Geometry 17 (1982), 255–306.

[5] O. A. Ladyzhenskaja, V. A. Solonnikov and N. N. Ural’tseva, Linear and quasilinear equations of parabolic type, American Mathematical Society, Providence, R.I. 1968.