Motion of Level Sets by Mean Curvature I

L. C. Evans^{∗}

Department of Mathematics University of Maryland College Park, MD 20742

J. Spruck^{†}

Department of Mathematics University of Massachusetts Amherst, MA 01003

Abstract

to be added later¸

1 Introduction

We set forth in this paper rigorous justification of a new approach for defining and then investigating the evolution of a hypersurface in ^{n} moving according to its mean curvature. This problem has been long studied using parametric methods of differential geometry: see, for instance, Gage []-[], Gage-Hamilton [], Grayson [], Huisken []-[], etc., etc. In this classical setup, we are given at time 0 a smooth hypersurface Γ_{0} which is, say, the connected boundary of a bounded open subset of ^{n}. As time progresses we allow the surface to evolve, by moving each point in the opposite direction to the mean curvature vector, at a velocity equal to (n − 1) times the absolute value of the mean curvature at that point. Assuming this evolution is smooth, we define thereby for each t > 0 a new hypersurface Γ_{t}. The primary problem is then to study geometric properties of {Γ_{t}}_{t>0} in terms of Γ_{0}.

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

†Supported in part by NSF grant DMS-8501952 and DOE grant DE-FG02-86ER250125.

Both authors were also supported by NSF grant DMS-86-10730 at the Institute for Advanced Study.

For the case n = 2 this program has been successfully carried out in great detail: see [], [], etc. For n ≥ 3, however, it is fairly clear that even if Γ_{0} is smooth, a smooth evolution as envisioned above cannot exist beyond some initial time interval. Imagine for instance Γ_{0} to be the boundary of a “dumbbell” shaped region in ^{3}, as illustrated.

In view of [] and numerical calculations of Sethian [], we expect that as time evolves, the surface will smoothly evolve (and shrink) up until a critical time t_{∗} > 0 when the two ends pinch off, as drawn.

After this time, the classical motion via mean curvature is undefined. In addition, if it were possible to define the subsequent motion in some reasonable way, we expect Γ_{t} for t > t_{∗} to comprise two pieces which pull apart at time t_{∗}. If this were so, then Γ_{t} would have changed topological type.

This possibility suggests inherent problems in the classical differential geometric approach of regarding Γ_{0} as a parameterized surface: the parametrization will in general develop singularities .

What is needed is an alternative description of the evolution for all times t > 0, sufficiently general so as to allow for the possible onset of singularities and attendant topological complications. To our knowledge there have been two different such undertakings, by Brakke [] and by Osher-Sethian []. Brakke’s dissertation [] recasts the mean curvature motion problem (even in arbitrary codimension) into the setting of varifold theory from geometric measure theory (cf. Allard []). Brakke defines and then constructs an appropriate generalized varifold solution , which is defined for all time (although it may vanish after a finite time.) He then deduces many geometric properties and under an additional density assumption establishes partial regularity. The principal drawback seems to be the lack of any uniqueness assertion.

A completely different viewpoint is to be found in the paper [] by Osher and Sethian. Their approach, recast slightly, is this. Given the initial hypersurface Γ_{0} as above, select some function g: ^{n} → so that

Γ_{0} = {x ∈ ^{n}|g(x) = 0}. (1.1)

Consider then the parabolic PDE

u_{t} = (δ_{ij} − \frac{u_{x_{i}}u_{x_{j}}}{|Du|^{2}} )u_{x_{i}x_{j}} in ^{n} × [0, ∞) (1.2)

u = g on ^{n} × {t = 0}, (1.3) for the unknown u = u(x, t)(x ∈ ^{n}, t ≥ 0). Now the PDE (1.2) says that each level set of u evolves according to its mean curvature, at least in regions where u is smooth and its spacial gradient Du does not vanish. Consequently, focusing our attention on the set {u = 0}, it seems reasonable in view of (1.1), (1.2) to define Γ_{t} ≡ {x ∈ ^{n} | u(x, t) = 0} (1.4)

for each time t > 0. Osher and Sethian [] introduce various techniques to study (1.2) and related PDE’s numerically, thereby to track computationally the evolution of Γ_{0} into Γ_{t}(t ≥ 0). (Notice by the way that our utilizing (1.1)– (1.3) amounts, in the language of fluid mechanics, to adopting an Eulerian viewpoint, as opposed to the Lagrangian, parametric viewpoint of classical differential geometry ; see also equation (2.2) below).

Our purpose here is to provide theoretical justification for this approach. The undertaking is analytically subtle principally because the mean curvature evolution equation (1.2) is nonlinear, degenerate, and indeed even undefined at points where Du = 0. In addition, it is not so clear that our definition (1.3) is independent of the choice of initial function g verifying (1.1). We will resolve these problems by introducing an appropriate definition of a weak solution for (1.2), inspired by the notion of so-called “viscosity solutions” of nonlinear PDE as in Crandall-Lions [], Crandall-Evans-Lions [], Lions [], Jensen [], etc. We then prove that there exists a unique weak solution of (1.2) and further that definition (1.3) is then independent of the choice of initial function g satisfying (1.1). We additionally check that {Γ_{t}}_{t≥0} so defined agrees with the classical notion of motion via mean curvature, over any time interval for which the latter exists. Finally we employ the PDE (1.2) to deduce assorted geometric properties of {Γ_{t}}_{t≥0}.

The main theoretical advantage of (1.1)–(1.3) as compared with Brakke’s varifold methods seems to us to be the uniqueness assertion: the set Γ_{t} is unambiguously defined by (1.3) once we have a uniqueness assertion for the PDE (1.2). The primary disadvantage is that our techniques work only in codimension one.

We hope to establish in a forthcoming companion paper a partial regularity theorem for {Γ_{t}}_{t≥0}.

Our paper is organized as follows. In Section 2 we motivate and introduce our definition of weak solution for (1.2) and in Section 3 prove the uniqueness of a weak solution. Section 4 establishes the existence of a weak solution to (1.2). In Section 5 we verify the independence of the definition (1.3) on the choice of g. Section 6 contains a consistency check that the definition (1.3) agrees with the classical motion by mean curvature, if and so long as the latter exists. Sections 7 and 8 contain various geometric assertions, examples of pathologies and conjectures.

2 Definition and Elementary Properties of Weak Solution

2.1 Heuristics

We start with a formal derivation of the mean curvature evolution PDE (1.2). For this, suppose temporarily u = u(x, t) is a smooth function whose spatial gradient Du = (u_{x_{1}}, …, u_{x_{n}}) does not vanish in some open region O of ^{n} × (0, ∞). Assume further that each level set of u smoothly evolves according to its mean curvature, as described in §1. We focus our attention onto any one such level set, and for definiteness consider the zero sets

Γ_{t} ≡ {x ∈ ^{n} | u(x, t) = 0} (t ≥ 0). (2.1)

Let ν = ν(x, t) be a smooth unit normal vector field to {Γ_{t}}_{t≥0} in O. Then

\frac{1}{n − 1} div(ν)ν

is the mean curvature vector field. Thus if we fix t ≥ 0, x ∈ Γ_{t} ∩ O, the point x evolves according to the nonautonomous ODE

\{ \begin{matrix}ẋ(s) = −[div(ν)ν] (x(s), s) & (s > t) \\ x(t) = x \end{matrix} (2.2)

As x(s) ∈ Γ_{s} (s ≥ t), we have

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

and so

0 = \frac{d}{ds} u(x(s), s) = −[(Du · ν) div(ν)] (x(s), s) + u_{t}(x(s), s).

Setting s = t, we discover

u_{t} = (Du · ν) div(ν) at (x, t).

Choosing then

ν ≡ \frac{Du}{|Du|}, (2.3)

it follows that

u_{t} = |Du| div(\frac{Du}{|Du|}) = (δ_{ij} − \frac{u_{x_{i}}u_{x_{j}}}{|Du|^{2}} ) u_{x_{i}x_{j}} at (x, t). (2.4)

Similar reasoning demonstrates this PDE to hold throughout the region O.

Now, conversely, assume in some region O u is a smooth solution of (2.4) with Du nonvanishing. Fix t > 0, x ∈ Γ_{t} ∩ O and solve then the ODE (2.2), (2.3). Since u solves (2.4), we deduce as above

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

Consequently the zero sets, and similarly all the level sets, of u evolve in O according to their mean curvatures.

Since the motion of any level set thus depends only upon its own geometry, and not that of any other level set, our PDE (2.4) should be invariant under an arbitrary relabelling of these sets. Thus if Ψ: → is smooth with

Ψ′ ≠ 0

we expect that

v = Ψ(u)

will also be a solution of (2.4) in the region O. A direct calculation verifies this. Hence we see that an arbitrary monotonic function of a solution is still a solution, this in strong contrast to the situation for uniformly parabolic PDE’s. Indeed, we may informally interpret (2.4) as being somehow “uniformly parabolic along each level set”, but being also totally degenerate across different level sets.

2.2 Weak solutions

The foregoing heuristics done with, we turn now to the full mean curvature evolution equation:

u_{t} = (δ_{ij} − \frac{u_{x_{i}}u_{x_{j}}}{|Du|^{2}} ) u_{x_{i}x_{j}} in ^{n} × (0, ∞) (2.5)

u = g on ^{n} × {t = 0}, (2.6)

the function g: ^{n} → being given. We want to define a notion of weak solution to (2.5). Since however, the right hand side of the PDE cannot be put into divergence form, we are not able to define a weak solution by means of formal integration by parts of derivatives onto a smooth test function (as for instance in Bombieri, De Giorgi, Giusti [, Section 1]). We will instead follow Evans[], Lions [], Jensen [], etc. and define our weak solution in terms of pointwise behavior with respect to a smooth test function. The primary difficulty will be to modify extant theory to cover the possibility that Du may vanish.

Definition 2.1 A function u ∈ C( ^{n} × [0, ∞)) ∩ L^{∞}( ^{n} × [0, ∞)) is a weak subsolution of (2.5) provided for each ϕ ∈ C^{∞}( ^{n+1}) if

(2.7) u − ϕ has a local maximum at a point (x_{0}, t_{0}) ∈ ^{n} × (0, ∞)

then

\{ \begin{matrix}ϕ_{t} ≤ (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2}} ) ϕ_{x_{i}x_{j}} at (x_{0}, t_{0}) \\ if Dϕ(x_{0}, t_{0}) ≠ 0, \end{matrix} (2.8)

and

ϕ_{t} ≤ (δ_{ij} − η_{i}η_{j})ϕ_{x_{i}x_{j}} at (x_{0}, t_{0})
for some η ∈ ^{n} with |η| ≤ 1, if Dϕ(x_{0}, t_{0}) = 0.

(2.9)

Definition 2.2 A function u ∈ C( ^{n} × [0, ∞)) ∩ L^{∞}( ^{n} × [0, ∞)) is a weak supersolution of (2.5) provided for each ϕ ∈ C^{∞}( ^{n+1}), if

(2.10) u − ϕ has a local minimum at a point (x_{0}, t_{0}) ∈ ^{n} × (0, ∞)

then

\{ \begin{matrix}ϕ_{t} ≥ (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2}} ) ϕ_{x_{i}x_{j}} at (x_{0}, t_{0}) \\ if Dϕ(x_{0}, t_{0}) ≠ 0, \end{matrix} (2.11)

and

ϕ_{t} ≥ (δ_{ij} − η_{i}η_{j}) ϕ_{x_{i}x_{j}} at (x_{0}, t_{0})
for some η ∈ ^{n} with |η| ≠ 1, if Dϕ(x_{0}, t_{0}) = 0.

(2.12)

Definition 2.3 A function u ∈ C( ^{n} × [0, ∞)) ∩ L^{∞}( ^{n} × [0, ∞)) is a weak solution of (2.5) provided u is both a weak subsolution and a weak supersolution.

As preliminary motivation for these definitions, suppose u is a smooth function on ^{n} × (0, ∞) satisfying

u_{t} ≤ (δ_{ij} − \frac{u_{x_{i}}u_{x_{j}}}{|Du|^{2}} ) u_{x_{i}x_{j}}

whenever Du ≠ 0. Our function u is thus a classical subsolution of (2.5) on {Du ≠ 0}. Suppose now Du(x_{0}, t_{0}) = 0. Assume additionally there are points (x_{k}, t_{k}) → (x_{0}, t_{0}) for which Du(x_{k}, t_{k}) ≠ 0 (k = 1, …). Then

u_{t} ≤ (δ_{ij} − η_{j}^{k}η_{j}^{k})u_{x_{i}x_{j}} at (x_{k}, t_{k})

for η^{k} ≡ \frac{Du(x_{k},t_{k})}{|Du(x_{k},t_{k})|}. Since |η^{k}| = 1 (k = 1, …) we may as necessary pass to a subsequence so that η^{k} → η in ^{n}, |η| = 1. Passing to limits above, we find

u_{t} ≤ (δ_{ij} − η_{i}η_{j})u_{x_{i}x_{j}} at (x_{0}, t_{0}).

If, on the other hand, there do not exist such points {(x_{k}, t_{k})}_{k=1}^{∞}, then Du = 0, and so D^{2}u = 0 and u is a function of t only, near (x_{0}, t_{0}). Moving to the edge of the set {Du = 0}, we see that u is a nonincreasing function of t. Thus

u_{t} ≤ (δ_{ij} − η_{i}η_{j})u_{x_{i}x_{j}} at (x_{0}, t_{0})

for any η ∈ ^{n}.

Further motivation for our definition of weak solution, and in particular, explanation as to why we assume only |η| ≤ 1 in (2.9), (2.12), will be found in subsection 2.4.

2.3 An equivalent definition

We write z = (x, t), z_{0} = (y_{0}, t_{0}).

Definition 2.4 A function u ∈ C( ^{n} × [0, ∞)) ∩ L^{∞}( ^{n} × [0, ∞)) is a weak subsolution of (2.5) if whenever (x_{0}, t_{0}) ∈ ^{n} × (0, ∞) and

\{ \begin{matrix}u(x, t) ≤ u(x_{0}, t_{0}) + p · (x − y_{0}) + q(t − t_{0}) \\ +\frac{1}{2}(z − z_{o})^{T} R(z − z_{0}) + o(|z − z_{0}|^{2}) as z → z_{0}. \end{matrix} (2.13)

for some p ∈ ^{n}, q ∈ , R = (r_{ij}) ∈ S^{n+1×n+1}, then

q ≤ (δ_{ij} − \frac{p_{i}p_{j}}{|p|^{2}} ) r_{ij} if p ≠ 0 (2.14)

and (2.15) q ≤ (δ_{ij} − η_{i}η_{j}) r_{ij} for some |η| ≤ 1, if p = 0.

Definition 2.5 A function u ∈ C( ^{n} × [0, ∞)) ∩ L^{∞}( ^{n} × [0, ∞)) is a weak supersolution of (2.5) if whenever (x_{0}, t_{0}) ∈ ^{n} × (0, ∞) and

\{ \begin{matrix}u(x, t) ≥ u(x_{0}, t_{0}) + p · (x − x_{0}) + q(t − t_{0}) \\ +\frac{1}{2}(z − z_{0})^{T} R(z − z_{0}) + o(|z − z_{0}|^{2}) as z → z_{0} \end{matrix} (2.16)

for some p ∈ ^{n}, q ∈ , R = (r_{ij}) ∈ S^{n+1×n+1}, then

q ≥ (δ_{ij} − \frac{p_{i}p_{j}}{|p|^{2}} ) r_{ij} if p ≠ 0 (2.17)

and (2.18) q ≥ (δ_{ij} − η_{i}η_{j}) r_{ij} for some |η| ≤ 1, if p = 0.

Theorem 2.6 Definitions 2.1 and 2.4 are equivalent, and Definitions 2.2 and 2.3 are equivalent.

2.4 Properties of weak solutions

Theorem 2.7

(i) Assume u_{k} is a weak solution of (2.5) for k = 1, 2, … and (2.13) u_{k} → u boundedly and locally uniformly on ^{n} × [0, ∞). Then u is a weak solution.

(ii) An analogous assertion hold for weak subsolutions and supersolutions.

Proof. 1. Choose ϕ ∈ C^{∞}( ^{n+1}) and suppose first u − ϕ has a strict local maximum at some point (x_{0}, t_{0}) ∈ ^{n} × (0, ∞). As u^{k} → u uniformly near (x_{0}, t_{0}),

(2.19) u_{k} − ϕ has a local maximum at a point (x_{k}, t_{k}) (k = 1, 2, …)

with (x_{k}, t_{k}) → (x_{0}, t_{0}) as k → ∞. (2.20)

Since u_{k} is a weak solution, we have either

ϕ_{t} ≤ (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{i}}}{|Dϕ|^{2}} ) ϕ_{x_{i}x_{j}} at (x_{k}, t_{k}) (2.21)

if Dϕ(x_{k}, t_{k}) ≠ 0, or

ϕ_{t} ≤ (δ_{ij} − η_{i}^{k}η_{j}^{k}) ϕ_{x_{i}x_{j}} at (x_{k}, t_{k}) (2.22)

for some η^{k} ∈ ^{n} with |η^{k}| ≤ 1, if Dϕ(x_{k}, t_{k}) = 0.

2. Assume first Dϕ(x_{0}, t_{0}) ≠ 0. Then Dϕ(x_{k}, t_{k}) ≠ 0 for all large enough k. Hence we may pass to limits in the inequalities (2.21) to discover

ϕ_{t} ≤ (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2}} ), at (x_{0}, t_{0}). (2.23)

3. Next, suppose Dϕ(x_{0}, t_{0}) = 0. We set

ξ^{k} = \{ \begin{matrix}\frac{Dϕ}{|Dϕ|}(x_{k}, t_{k}) & if Dϕ(x_{k}t_{k}) ≠ 0 \\ η^{k} & if Dϕ(x_{k}, t_{k}) = 0 \end{matrix} (2.24)

Passing if necessary to a subsequence we may assume

ξ^{k} → η.

Then |η| ≤ 1. Utilizing now (2.22), we deduce as well

ϕ_{t} ≤ (δ_{ij} − η_{i}η_{j}) ϕ_{x_{i}x_{j}} at (x_{0}, t_{0}). (2.25)

4. If u−ϕ has only a local maximum at (x_{0}, t_{0}) we apply the above argument to

ψ(x, t) ≡ ϕ(x, t) + |x − x_{0}|^{4} + (t − t_{0})^{4},

so that u − ψ has a strict local maximum at (x_{0}, t_{0}). Hence u is a weak subsolution. Similar reasoning verifies that u is a weak supersolution as well. □

Theorem 2.8 Assume u is a weak solution of (2.5) and Ψ: → is continuous. Then

v ≡ Ψ(u)

is a weak solution.

Proof. 1. Assume first Ψ is smooth, with Ψ′ > 0 on . (2.26) Let ϕ ∈ C^{∞}( ^{n+1}) and suppose v−ϕ has a local maximum at (x_{0}, t_{0}). Adding as necessary a constant to ϕ, we may assume

\{ \begin{matrix}v(x_{0}, t_{0}) = ϕ(x_{0}, t_{0}) \\ v(x, t) ≤ ϕ(x, t) & for all (x, t) near (x_{0}, t_{0}). \end{matrix} (2.27)

In view of (2.26)

Φ ≡ Ψ^{−1}

is defined and smooth near u(x_{0}, t_{0}), with Φ′ > 0. (2.28) From (2.27) therefore, we see

\{ \begin{matrix}u(x_{0}, t_{0}) = ψ(x_{0}, t_{0}) \\ u(x, t) ≤ ψ(x, t) & for all (x, t) near (x_{0}, t_{0}) \end{matrix} (2.29)

where ψ ≡ Φ(ϕ). (2.30)

2. Since u is a weak solution we see

ψ_{t} ≤ (δ_{ij} − \frac{ψ_{x_{i}}ψ_{x_{j}}}{|Dψ|^{2}} ) ψ_{x_{i}x_{j}} at (x_{0}, t_{0}) (2.31)

if Dψ(x_{0}, t_{0}) ≠ 0 and ψ_{t} ≤ (δ_{ij} − η_{i}η_{j}) ψ_{x_{i}x_{j}} at x_{0}, t_{0}) (2.32) for some |η| ≤ 1, if Dψ(x_{0}, t_{0}) = 0. Now Dϕ(x_{0}, t_{0}) = 0 if and only if Dψ(x_{0}, t_{0}) = 0. Consequently (2.31) obtains if Dϕ(x_{0}, t_{0}) ≠ 0; in which case we substitute (2.30) to compute

Φ′ϕ_{t} ≤ (δ_{ij} − \frac{(Φ′)^{2}ϕ_{x_{i}}ϕ_{x_{j}}}{(Φ′)^{2} |Dϕ|^{2}} ) (Φ′ϕ_{x_{i}x_{j}} + Φ′′ϕ_{x_{i}}ϕ_{x_{j}}) at (x_{0}, t_{0})

Since Φ′ > 0 we simplify and obtain

ϕ_{t} ≤ (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2}} ) ϕ_{x_{i}x_{j}} at (x_{0}, t_{0}). (2.33)

Suppose on the other hand Dϕ(x_{0}, t_{0}) = 0. Then (2.32) is valid for some |η| ≤ 1. We substitute (2.30) and compute

Φ′ϕ_{t} ≤ (δ_{ij} − η_{i}η_{j}) (Φ′ϕ_{x_{i}x_{j}} + Φ′′ϕ_{x_{i}}ϕ_{x_{j}}) at (x_{0}, t_{0}).

Since Dϕ = 0, the term involving Φ′′ is zero. Thus ϕ_{t} ≤ (δ_{ij} − η_{i}η_{j}) ϕ_{x_{i}x_{j}} at (x_{0}, t_{0}). (2.34) We similarly have the opposite inequalities to (2.33), (2.34) should v−ϕ have a local minimum at (x_{0}, t_{0}).

3. Now assume instead of (2.20) that Ψ′ < 0 on . (2.35) Then

Φ′ < 0 on

as well. Thus (2.21) now implies

\{ \begin{matrix}u(x_{0}, t_{0}) = ψ(x_{0}, t_{0}) \\ u(x, t) ≥ ψ(x, t) & for all (x, t) near (x_{0}, t_{0}) \end{matrix}

Since u is a weak solution, either

ψ_{t} ≥ (δ_{ij} − \frac{ψ_{x_{i}}ψ_{x_{j}}}{|Dψ|^{2}} ) ψ_{x_{i}x_{j}} at (x_{0}, t_{0})

if Dψ(x_{0}, t_{0}) ≠ 0 or

ψ_{t} ≥ (δ_{ij} − η_{i}η_{j}) ψ_{x_{i}x_{j}} at (x_{0}, t_{0})

for some |η| ≤ 1, if Dψ(x_{0}, t_{0}) = 0. Since now Φ′ < 0, we as above deduce either (2.33) or (2.34).

4. We have so far shown that v = Ψ(u) is a weak solution provided Ψ is smooth, with Ψ′ ≠ 0. Approximating and using Theorem 2.2 we draw the same conclusion if

Ψ′ ≥ 0 or Ψ′ ≤ 0 on .

5. Next assume Ψ is smooth and there exist finitely many points −∞ = a_{0} < a_{1}, < a_{2} < … < a_{m} < a_{m+1} = +∞ such that

(2.36) Ψ is monotone on the intervals (a_{j}, a_{j+1}) (j = 0, …, m)

and

(2.37) Ψ is constant on the intervals (a_{j} − σ, a_{j} + σ) (j = 1, …, m)

for some σ > 0.

Suppose v − ϕ has a maximum at (x_{0}, t_{0}). Then

u(x_{0}, t_{0}) ∈ (a_{j} − \frac{σ}{2} , a_{j+1} + \frac{σ}{2} ) for some j ∈ \{0, …, m\}.

As Ψ is monotone on (a_{j} − σ, a_{j+1} + σ) and u is continuous, we can apply steps 1-4 is some neighborhood of (x_{0}, t_{0}) to deduce (2.33) or (2.34). The reverse inequalities similarly obtain if v − ϕ has a minimum.

6. Finally suppose only that Ψ is continuous. We construct a sequence of smooth functions {Ψ^{k}}_{k=1}^{∞} each verifying the structural assumptions (2.36), (2.37) so that

Ψ^{k} → Ψ uniformly on [−||u||_{L^{∞}}, ||u||_{L^{∞}}].

Hence

v^{k} = Ψ^{k}(u) → v ≡ Ψ(u)

boundedly and uniformly. Then Theorem 2.2 asserts v to be a weak solution. □

3 Uniqueness and Comparison of Weak Solutions

3.1 Preliminaries

Our plan, as in Jensen [], Jensen-Lions-Souganidis [], is to regularize using sup and inf convolutions, defined as follows. Assume w: ^{n} × [0, ∞) → is continuous and bounded. Then if ϵ > 0, we write

\begin{aligned}w^{ϵ}(x, t) ≡ & sup\{w(y, s) − \frac{1}{ϵ} (|x − y|^{2} + (t − s)^{2})\}, \\ & y ∈ ^{n} \\ & s ∈ [0, ∞) \end{aligned} (3.1)

\begin{aligned}w_{ϵ}(x, t) ≡ & inf\{w(y, s) + \frac{1}{ϵ} (|x − y|^{2} + (t − s)^{2})\}. \\ & y ∈ ^{n} \\ & s ∈ [0, ∞) \end{aligned} (3.2)

for x ∈ ^{n}, t ∈ [0, ∞). Note that since w is continuous and bounded, the “sup” and “inf” above can be replaced by “max” and “min”.

Lemma 3.1 (Properties of sup and inf convolutions)

There exist constants A,B,C depending only on ||w||_{L^{∞}(R^{n}×[0,∞))}, such that:

(i) w_{ϵ} ≤ w ≤ w^{ϵ} on ^{n} × [0, ∞)

(ii) ||w^{ϵ}, w_{ϵ}||_{L^{∞}(R^{n}×[0,∞))} ≤ A

(iii) If y ∈ ^{n}, s ∈ [0, ∞) and w^{ϵ}(x, t) = w(y, s) − \frac{l}{ϵ} (|x − y|^{2} + (t − s)^{2}),

then |x − y|, |t − s| ≤ Cϵ^{\frac{1}{2}} ≡ σ(ϵ) (3.3)

A similar assertion holds for w_{ϵ}.

(iv) w^{ϵ}, w_{ϵ} → w as ϵ → 0^{+}, uniformly on compact subsets of ^{n} × [0, ∞).

(v) Lip (w^{ϵ}, w_{ϵ}) ≤ \frac{B}{ϵ}.

(vi) The mapping

(x, t) ⟼ w^{ϵ}(x, t) + \frac{1}{ϵ} (|x|^{2} + t^{2})

is convex, and the mapping

(x, t) ⟼ w_{ϵ}(x, t) − \frac{1}{ϵ} (|x|^{2} + t^{2})

is concave.

(vii) Assume w is a weak subsolution of (2.5) in ^{n} × (0, ∞). Then w^{ϵ} is a weak subsolution on ^{n} × (σ(ϵ), ∞). Similarly, if w is a weak supersolution of (2.5), w_{ϵ} is a weak supersolution.

(viii) The function w^{ϵ} is twice differentiable a.e. and satisfies

w_{t}^{ϵ} ≤ (δ_{ij} − \frac{w_{x_{i}}^{ϵ}w_{x_{j}}^{ϵ}}{|Dw^{ϵ}|^{2}} ) w_{x_{i}x_{j}}^{ϵ} (3.4)

at each point of twice differentiability in ^{n}×(σ(ϵ), ∞) where Dw^{ϵ} ≠ 0. Similarly, w_{ϵ} is twice differentiable a.e. and satisfies

w_{ϵt} ≥ (δ_{ij} − \frac{w_{ϵ x_{i}}w_{ϵ x_{j}}}{|Dw^{ϵ}|^{2}} ) w_{ϵ x_{i}x_{j}} (3.5)

at each point of twice differentiability in ^{n} ×(σ(ϵ), ∞) where Dw_{ϵ} ≠ 0.

Proof. 1. Assertions (i) and (ii) are clear from the definitions, for

A = ||w||_{L^{∞}(R^{n}×[0,∞))} .

Statement (iii) follows from (ii), and then (iv) is a consequence of the uniform continuity of w on compact sets. In light of estimate (3.3) we have (v) as well.

2. For each y ∈ ^{n}, s ∈ [0, ∞), the mapping

(x, t) ⟼ w(y, s) − \frac{1}{ϵ} (|x − y|^{2} + (t − s)^{2}) + \frac{1}{ϵ} (|x|^{2} + t^{2})

is affine. Consequently

(x, t) ⟼ sup [w(y, s) − \frac{1}{ϵ} (|x − y|^{2} + (t − s)^{2}) + \frac{1}{ϵ} (|x|^{2} + t^{2})]
y ∈ ^{n}
s ∈ [0, ∞)
= w^{ϵ}(x, t) + \frac{1}{ϵ} (|x|^{2} + t^{2})

is convex.

3. Assume ϕ ∈ C^{∞}( ^{n+1}) and w^{ϵ}−ϕ has a local maximum at a point (x_{0}, t_{0}), with t_{0} > σ(ϵ).

We then employ (3.3) to choose (y_{0}, s_{0}) ∈ ^{n} × (0, ∞) so that

w^{ϵ}(x_{0}, t_{0}) = w(y_{0}, s_{0}) − \frac{1}{ϵ} (|x_{0} − y_{0}|^{2} + (t_{0} − s_{0})^{2}).

Set ψ(x, t) ≡ ϕ(x + x_{0} − y_{0}, t + t_{0} − s_{0}). (3.6)

Since w^{ϵ} − ϕ has a local maximum at (x_{0}, t_{0}) we compute

w(y_{0}, s_{0}) −\frac{1}{ϵ}(|x_{0} − y_{0}|^{2} + (t_{0} − s_{0})^{2}) − ϕ(x_{0}, t_{0})
= w^{ϵ}(x_{0}, t_{0}) − ϕ(x_{0}, t_{0})
≥ w^{ϵ}(x, t) − ϕ(x, t)
≥ w(y, s) − \frac{1}{ϵ}(|x − y|^{2} + (t − s)^{2}) − ϕ(x, t)

for all (x, t) near (x_{0}, t_{0}) and all (y, s) ∈ ^{n} × [0, ∞). Fix (y, s) close to (y_{0}, s_{0}) and set

x = y + x_{0} − y_{0}, t = s + t_{0} − s_{0}

above, to discover

w(y_{0}, s_{0}) − ϕ(x_{0}, t_{0}) ≥ w(y, s) − ϕ(y + x_{0} − y_{0}, s + t_{0} − s_{0}).

Using (3.6) we rewrite this to read

w(y_{0}, s_{0}) − ϕ(y_{0}, s_{0}) ≥ w(y, s) − ψ(y, s)

for all (y, s) near (y_{0}, s_{0}). Hence w − ψ has a local maximum at (y_{0}, s_{0}) and thus

ψ_{t} ≤ (δ_{ij} − \frac{ψ_{x_{i}}ψ_{x_{i}}}{|Dψ|^{2}} ) ψ_{x_{i}x_{j}} at (y_{0}, s_{0})

if Dψ(y_{0}, s_{0}) ≠ 0, and

ψ_{t} ≤ (δ_{ij} − η_{i}η_{j}) ψ_{x_{i}x_{j}} at (y_{0}, s_{0})

for some |η| ≤ 1, if Dψ(y_{0}, s_{0}) = 0. Since

\{ \begin{matrix}Dψ(y_{0}, s_{0}) = Dϕ(x_{0}, t_{0}), ψ_{t}(y_{0}, s_{0}) = ϕ_{t}(x_{0}, t_{0}) \\ D^{2}ϕ(y_{0}, s_{0}) = D^{2}ϕ(x_{0}, t_{0}), \end{matrix}

we immediately obtain

ϕ_{t} ≤ (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2}} ) ϕ_{x_{i}x_{j}} at (x_{0}, t_{0})

or

ϕ_{t} ≤ (δ_{ij} − η_{i}η_{j}) ψ_{x_{i}x_{j}} at (x_{0}, t_{0})

according to whether Dϕ(x_{0}, t_{0}) = 0 or not.

4. Owing to (vi), w^{ϵ}(x, t) + \frac{1}{ϵ}(|x|^{2} + t^{2}) is convex in (x, t) or so is twice differentiable a.e. according to a theorem of Alexandroff (see, e.g., Resetnjak [], Krylov [, Appendix 2], etc.) Thus w^{ϵ} is twice differentiable a.e. □

3.2 Comparison principle, uniqueness

Theorem 3.2 Assume that u is a weak subsolution and v is a weak supersolution of (2.5). Suppose further

u ≤ v on ^{n} × {t = 0}. (3.7)

Finally assume that

\{ \begin{matrix}u and v are constant, with u ≤ v, \\ on ^{n} × [0, ∞) ∩ \{|x| + t ≥ R\} \end{matrix} (3.8)

for some R ≥ 1. Then u ≤ v on ^{n} × [0, ∞). (3.9) In particular, weak solutions of (2.5) are unique. Proof. 1. Should (3.9) fail, then

max_{(x,t)∈R^{n}×[0,∞)} (u − v) ≡ a > 0;

and so for α > 0 small enough,

max_{(x,t)∈R^{n}×[0,∞)} (u − v − α t) ≥ \frac{a}{2} > 0. (3.10)

According to (3.8) we have u^{ϵ} = u, v_{ϵ} = v on {|x| + t ≥ 2R} (3.11) for all small ϵ > 0. Note further u^{ϵ} → u and v_{ϵ} → v uniformly. Consequently if we fix ϵ > 0 small enough,

max_{(x,t)∈R^{n}×[0,∞)} (u^{ϵ} − v_{ϵ} − α t) ≥ \frac{a}{4} > 0. (3.12)

2. For each δ > 0 define for x, y ∈ ^{n}, t, t + s ∈ [0, ∞)

Φ(x, y, t, s) ≡ u^{ϵ}(x + y, t + s) − v_{ϵ}(x, t) − α t − \frac{1}{δ}(|y|^{4} + s^{4}). (3.13)

Owing to (3.12) we see

max_{(x,t),(x+y,t+s)∈R^{n}×[0,∞)} Φ ≥ \frac{a}{4} > 0. (3.14)

Choose now (x_{1}, t_{1}), (x_{1} + y_{1}, t_{1} + s_{1}) ∈ ^{n} × [0, ∞) so that Φ(x_{1}, y_{1}, t_{1}, s_{1}) = max Φ(x, t), (x + y, t + s) ∈ ^{n} × [0, ∞). (3.15) Note in view of (3.11), (3.13) and Lemma 3.1 (ii) that such points exist. Since Φ(x_{1}, y_{1}, t_{1}, s_{1}) > 0, (3.13) implies |y_{1}|, |s_{1}| ≤ Cδ^{1/4} (3.16)

3. We claim next that if ϵ, δ > 0 are fixed small enough, t_{1}, t_{1} + s, > σ(ϵ), (3.17) with σ(ϵ) defined in (3.3). Indeed if t_{1} ≤ σ(ϵ), then

\frac{a}{4} ≤ Φ(x_{1}, y_{1}, t_{1}, s_{1})
≤ u^{ϵ}(x_{1} + y_{1}, t_{1} + s_{1}) − v_{ϵ}(x_{1}, t_{1})
= u(x_{1} + y_{1}, t_{1} + s_{1}) − v(x_{1}, t_{1}) +o(1) as ϵ → 0
= u(x_{1} + y_{1}, s_{1}) − v(x_{1}, 0)  +o(1) as ϵ → 0
= u(x_{1}, 0) − v(x_{1}, 0)  +o(1) as ϵ, δ → 0
≤ o(1)  as ϵ, δ → 0,

where we employed Lemma 3.1 (ii), (3.16), (3.7) and the continuity of u, v. This is a contradition for ϵ, δ > 0 small enough, whence t_{1} > σ(ϵ). Owing to (3.16) we may as necessary adjust δ smaller to ensure (3.17). Hereafter in the proof α, ϵ, δ > 0 are fixed.

According to Lemma 3.1, (vii), (3.18) u^{ϵ} is a weak subsolution of (2.5) near (x_{1} + y_{1}, t_{1} + s_{1}) and (3.19) v_{ϵ} is a weak supersolution of (2.5) near (x_{1}, t_{1})

4. We now demonstrate y_{1} ≠ 0. (3.20)

Assume for contradiction that in fact y_{1} = 0. Then (3.13), (3.15) imply

\{ \begin{matrix}u^{ϵ}(x_{1}, t_{1}, +s_{1}) − v_{ϵ}(x_{1}, t_{1}) − α t_{1} − \frac{1}{δ}s_{1}^{4} \\ ≥ u^{ϵ}(x + y, t + s) − v_{ϵ}(x, t) − α t − \frac{1}{δ}(|y|^{4} + s^{4}) \end{matrix} (3.21)

for all (x, t), (x + y, t + s) ∈ ^{n} × [0, ∞). Put x = x_{1}, t = t_{1}, above and simplify to obtain the inequality

u^{ϵ}(x_{1} + y, t_{1} + s) ≤ u^{ϵ}(x_{1}, t_{1} + s_{1}) + \frac{1}{δ}|y|^{4} + \frac{1}{δ}(s^{4} − s_{1}^{4})

for (x_{1} + y, t_{1} + s) ∈ ^{n} × [0, ∞). Set r = s − s_{1} and rewrite to find

\begin{aligned}u^{ϵ}(x_{1} + y, t_{1} + s_{1} + r) & ≤ u^{ϵ}(x_{1}, t_{1} + s_{1}) + \frac{4}{δ}s_{1}^{3}r + \frac{6}{δ}s_{1}^{2}r^{2} \\ & +O(|r|^{3} + |y|^{4}) as (y, r) → (0, 0). \end{aligned}

Since u^{ϵ} is a weak subsolution near (x_{1} + y_{1}, t_{1} + s_{1}) = (x_{1}, t_{1} + s_{1}) we may invoke (2.7)′, (2.9)′ with x_{0} = x_{1}, t_{0} = t_{1} + s_{1}, p = 0, q = \frac{4}{δ}s_{1}^{3}, r_{n+1,n+1} = \frac{6}{δ}s_{1}^{2}, r_{ij} = 0 otherwise; giving

\frac{4}{δ}s_{1}^{3} ≤ 0. (3.22)

Now go back and insert y = x_{1} − x, s = t_{1} + s_{1} − t into (3.21). This yields after simplifications:

\begin{aligned}v_{ϵ}(x, t) & ≥ v_{ϵ}(x_{1}, t_{1}) + (\frac{4s_{1}^{3}}{δ} − α)(t − t_{1}) − \frac{6}{δ}s_{1}^{2}(t − t_{1})^{2} \\ & +O(|x − x_{1}|^{4} + |t − t_{1}|^{3}) as (x, t) → (x_{1}, t_{1}). \end{aligned}

Now v_{ϵ} is a weak supersolution near (x_{1}, t_{1}). Thus (2.10)′, (2.12)′, with x_{0} = x_{1}, t_{0} = t_{1}, p = 0, q = \frac{4s_{1}^{3}}{δ} − α, r_{n+1,n+1} = −\frac{6s_{1}^{2}}{δ} , r_{ij} = 0 otherwise, imply

\frac{4s_{1}^{3}}{δ} − α ≥ 0. (3.23)

But now we have a contradiction with (3.22), since α > 0. This establishes (3.20).

5. Note next that, in general, if f: ^{m} → is convex, then so is the mapping (w, z) ⟼ f(w + z) on ^{2m}. Consequently Lemma 3.1 (vi) asserts

(x, y, t, s) ⟼ u^{ϵ}(x + y, t + s) + \frac{1}{ϵ}((x + y)^{2} + (t + s)^{2})

to be convex. As

(x, t) ⟼ −v_{ϵ}(x, t) + \frac{1}{ϵ} (|x|^{2} + t^{2})

is convex as well, we see that

(x, y, t, s) ⟼ Φ(x, y, t, s) + C(|x|^{2} + |y|^{2} + t^{2} + s^{2})

is convex near (x_{1}, y_{1}, t_{1}, s_{1}), for some sufficiently large constant C = C(ϵ, δ). Since Φ additionally attains its maximum at (x_{1}, y_{1}, t_{1}, s_{1}) we may invoke Jensen []: there exist points {(x^{k}, y^{k}, t^{k}, s^{k})}_{k=1}^{∞} such that

(x^{k}, y^{k}, t^{k}, s^{k}) → (x_{1}, y_{1}, t_{1}, s_{1}), (3.24)

Φ, u^{ϵ} and v_{ϵ} are each twice differentiable

at (x^{k}, y^{k}, t^{k}, s^{k}) (k = 1, …), (3.25)

D_{x,y,t,s}Φ(x^{k}, y^{k}, t^{k}, s^{k}) → 0, (3.26)

and D_{x,y,t,s}^{2}Φ(x^{k}, y^{k}, t^{k}, s^{k}) ≤ o(1)I_{2n+2} as k → ∞. (3.27)

6. Using (3.13), (3.25), we see

\{ \begin{matrix}D_{x}Φ(x^{k}, y^{k}, t^{k}, s^{k}) & = Du^{ϵ}(x^{k} + y^{k}, t^{k} + s^{k}) − Dv_{ϵ}(x^{k}, t^{k}) \\ & ≡ p^{k} − \overline{p}^{k}. \end{matrix} (3.28)

and

\{ \begin{matrix}D_{y}Φ(x^{k}, y^{k}, t^{k}, s^{k}) & = Du^{ϵ}(x^{k} + y^{k}, t^{k} + s^{k}) − \frac{4}{δ}|y^{k}|^{2}y^{k} \\ & = p^{k} − \frac{4}{δ}|y^{k}|^{2}y^{k} \end{matrix} (3.29)

Since y^{k} → y_{1}, we deduce from (3.26) that

p^{k}, \overline{p}^{k} → \frac{4}{δ}|y_{1}|^{2}y_{1} ≡ p in ^{n}. (3.30)

Assertion (3.20) tells us p ≠ 0 and so p^{k}, \overline{p}^{k} ≠ 0 for large enough k.

Again employing (3.13), (3.25) we note

\{ \begin{matrix}Φ_{t}(x^{k}, y^{k}, t^{k}, s^{k}) & = u_{t}^{ϵ}(x^{k} + y^{k}, t^{k} + s^{k}) − v_{ϵt}(x^{k}, t^{k}) − α \\ & ≡ q^{k} − \overline{q}^{k} − α. \end{matrix} (3.31)

As u^{ϵ} and v_{ϵ} are Lipschitz, we may assume, upon passing to a subsequence and reindexing if necessary, that q^{k} → q, \overline{q}^{k} → \overline{q} in . (3.32) Then (3.26) and (3.31) ensure q − \overline{q} = α. (3.33)

7. Next, (3.13) and (3.25) imply

\begin{aligned}D_{x}^{2}ϕ(x^{k}, y^{k}, t^{k}, s^{k}) & = D^{2}u^{ϵ}(x^{k} + y^{k}, t^{k} + s^{k}) − D^{2}v_{ϵ}(x^{k}, t^{k}) \\ & ≡ R^{k} − \overline{R}^{k}. \end{aligned} (3.34)

Now (3.27) forces R^{k} − \overline{R}^{k} ≤ ϵ_{k}I_{n} (3.35) where ϵ_{k} → 0. Furthermore, Lemma 3.1 (vi) shows

R^{k} ≥ −CI_{n}, \overline{R}^{k} ≤ CI_{n},

for C = C(ϵ). Thus

−CI_{n} ≤ R^{k} ≤ \overline{R}^{k} + ϵ_{k}I_{n} ≤ CI_{n}.

We may consequently suppose, passing as necessary to subsequences, that R^{k} → R, \overline{R}^{k} → \overline{R} in S^{n×n}, (3.36) with R ≤ \overline{R}. (3.37)

8. Now recall (3.25) holds and p_{0}^{k} ≡ Du^{ϵ}(x^{k} + y^{k}, t^{k} + s^{k}), \overline{p}^{k} ≡ Dv_{ϵ}(x^{k}, t^{k}) are non-zero for large k. Since u^{ϵ} is a weak subsolution near (x_{1} + y_{1}, t_{1} + s_{1}) and v_{ϵ} is a weak supersolution near (x_{1}, t_{1}), we thus have

q^{k} ≤ (δ_{ij} − \frac{p_{i}^{k}p_{j}^{k}}{|p^{k}|^{2}} ) r_{ij}^{k}

and

\overline{q}^{k} ≥ (δ_{ij} − \frac{\overline{p}_{i}^{k}\overline{p}_{j}^{k}}{|\overline{p}^{k}|^{2}} ) \overline{r}_{ij}^{k}

for all large k. We send k to infinity, recalling (3.30), (3.32) and (3.36) to obtain

q ≤ (δ_{ij} − \frac{p_{i}p_{j}}{|p|^{2}} ) r_{ij}

and

\overline{q} ≥ (δ_{ij} − \frac{p_{i}p_{j}}{|p|^{2}} ) \overline{r}_{ij}.

Subtract:

q − \overline{q} ≤ (δ_{ij} − \frac{p_{i}p_{j}}{|p|^{2}} ) (r_{ij} − \overline{r}_{ij}).

Now the matrix ((δ_{ij} − \frac{p_{i}p_{j}}{|p|^{2}} )) is nonnegative and R − \overline{R} is nonpositive, by (3.37). Consequently

q − \overline{q} ≤ 0,

a contradiction to (3.33). □

3.3 Contraction property

Theorem 3.3 Assume that u and v are weak solutions of (2.5), such that

\{ \begin{matrix}u and v are constant, with u = v, \\ on ^{n} × [0, ∞) ∩ \{|x| + t ≥ R\} \end{matrix}

(3.38)

for some R > 1. Then

max_{0≤t<∞} ||u(·, t) − v(·, t) ||_{L^{∞}(R^{n})} = ||u(·, 0) − v(·, 0) ||_{L^{∞}(R^{n})}. (3.39)

Proof. Should (3.39) fail, we may assume

max_{(x,t)∈R^{n}×[0,∞)} (u − v) ≡ a > ||u(·, 0) − v(·, 0) ||_{L^{∞}(R^{n})} ≡ b.

Then as in the proof of Theorem 3.2 as above, there exist α, ϵ, δ > 0 such that

max_{(x,t),(x+y,t+s)∈R^{n}×[0,∞)} Φ > b, where

Φ is defined by (3.13). We find a point (x_{1}, y_{1}, t_{1}, s_{1}) verifying (3.15) and check (3.17) is valid provided ϵ, δ > 0 are small enough. The rest of the proof follows that for Theorem 3.2. □

4 Existence of Weak Solutions

4.1 Approximation; geometric interpretation

We turn our attention now to constructing a weak solution of the initial value problem (2.5), (2.6). We will assume that

(4.1) g is constant on ^{n} ∩ {|x| ≥ S}

for some S > 0, and additionally, for the moment at least, g is smooth. Our intention is to approximate (2.5), (2.6) by the PDE

u_{t}^{ϵ} = (δ_{ij} − \frac{u_{x_{i}}^{ϵ}u_{x_{j}}^{ϵ}}{|Du^{ϵ}|^{2} + ϵ^{2}} ) u_{x_{i}x_{j}}^{ϵ} in ^{n} × (0, ∞) (4.2)

u^{ϵ} = g on ^{n} × {t = 0}, (4.3) for 0 < ϵ < 1. (The superscript ϵ here and hereafter is only a label and does not mean the sup − convolution (3.1).)

We interpret (4.2), (4.3) as follows. Assuming for the moment u^{ϵ} = u^{ϵ}(x, t) to be a smooth solution of (4.2), (4.3), write y = (x, x_{n+1}) ∈ ^{n+1} and define v^{ϵ}(y, t) ≡ u^{ϵ}(x, t) − ϵx_{n+1} (4.4)

Then |D_{y}v^{ϵ}|^{2} = |Du^{ϵ}|^{2} + ϵ^{2}. Thus our PDE becomes

v_{t}^{ϵ} = (δ_{ij} − \frac{v_{y_{i}}^{ϵ}v_{y_{j}}^{ϵ}}{|Dv^{ϵ}|^{2}} ) v_{y_{i}y_{j}}^{ϵ} in ^{n+1} × [0, ∞) (4.5)

v^{ϵ} = g^{ϵ} on ^{n+1} × {t = 0}, (4.6)

for

g^{ϵ}(y) ≡ g(x) − ϵx_{n+1}.

As noted in §2, the PDE (4.5) says that each level set of v^{ϵ} evolves according to its mean curvature. This is in particular the case for the zero level sets

Γ_{t}^{ϵ} ≡ {y ∈ ^{n+1} | v^{ϵ}(y, t) = 0}.

But according to (4.4) each Γ_{t}^{ϵ} is a graph:

Γ_{t}^{ϵ} = \{y = (x, x_{n+1}) ∈ ^{n+1} | x_{n+1} = \frac{1}{ϵ} u^{ϵ}(x, t)\},

and Ecker and Huisken [] have shown the evolution of an entire graph by mean curvature remains a smooth entire graph for all time.

Geometrically, if as in §1 we are given Γ_{0} as the boundary of a smooth, bounded, simply connected open set U in ^{n}, we select a smooth function g with g = 0 on Γ_{0}, g < 0 in U, g > 0 in ^{n} − U. Then Γ_{0}^{ϵ} ⊂ ^{n+1} is the graph \{x_{n+1} = \frac{1}{ϵ} g(x)\} as drawn.

For small ϵ, Γ_{0}^{ϵ} roughly approximates the cylinder Γ_{0} × . We may thus hope that for moderate t > 0 and small ϵ > 0, the smooth graph Γ_{t}^{ϵ} will be close to the cylinder Γ_{t} × , Γ_{t} denoting the evolution of Γ_{0} via its mean curvature in ^{n}.

The idea then is that the complicated, possibly singular behavior of {Γ_{t}}_{t≥0} in ^{n} will be approximated by the smooth evolution {Γ_{t}^{ϵ}}_{t≥0} in ^{n+1}, in the sense that for a given t > 0,

Γ_{t}^{ϵ} ≈ Γ_{t} ×

if ϵ > 0 is very small. The illustrations provided make this expectation appear plausible, although there are a number of subtleties.

4.2 Solution of the approximate equations

We now investigate the approximations (4.2), (4.3) analytically.

Theorem 4.1

(i) For each 0 < ϵ < 1 there exists a unique smooth, bounded solution u^{ϵ} of (4.2), (4.3).

(ii) Additionally,

sup_{0<ϵ<1} ||u^{ϵ}, Du^{ϵ}, u_{t}^{ϵ}||_{L^{∞}(R^{n})×[0,∞))} ≤ C||g||_{C^{1,1}(R^{n})} (4.7)

Proof. 1. For each 0 < σ < 1, consider the PDE

u_{t}^{ϵ,σ} = a_{ij}^{ϵ,σ}(Du^{ϵ,σ}) u_{x_{i}x_{j}}^{ϵ,σ} in ^{n} × [0, ∞) (4.8) u^{ϵ,σ} = g on ^{n} × {t = 0}, (4.9) for

a_{ij}^{ϵ,σ}(p) ≡ (1 + σ)δ_{ij} − \frac{p_{i}p_{j}}{|p|^{2} + ϵ^{2}} (p ∈ ^{n}, 1 ≤ i, j ≤ n)

The smooth bounded coefficients {a_{ij}} satisfy also the uniform ellipticity condition

σ|ξ|^{2} ≤ a_{ij}^{ϵ,σ}(p)ξ_{i}ξ_{j}, (ξ ∈ ^{n})

for each p ∈ ^{n}, and consequently classical PDE theory gives the existence of a unique smooth bounded solution u^{ϵ,σ}: see, e.g., Ladyzhenskaja, Solonnikov, Uralcéva []. By the maximum principle, ||u^{ϵ,σ}||_{L^{∞}(R^{n}×[0,∞))} = ||g||_{L^{∞}(R^{n})} (4.10)

2. Now differentiate (4.8) with respect to x_{ℓ}:

u_{x_{ℓ}t}^{ϵ,σ} = a_{ij}^{ϵ,σ}(Du^{ϵ,σ}) u_{x_{ℓ}x_{i}x_{j}}^{ϵ,σ} + a_{ij p_{k}}^{ϵ,σ} (Du^{ϵ,σ})u_{x_{ℓ}x_{k}}^{ϵ,σ}u_{x_{i}x_{j}}^{ϵ,σ}

The maximum principle then implies ||Du^{ϵ,σ}||_{L^{∞}R^{n}×[0,∞))} = ||Dg||_{L^{∞}(R^{n})} (4.11) Similarly ||u_{t}^{ϵ,σ}||_{L^{∞}(R^{n}×[0,∞))} = ||u_{t}(·, 0)||_{L^{∞}(R^{n})} ≤ C||D^{2}g||_{L^{∞}(R^{n})}. (4.12)

3. Since

(1 − \frac{L^{2}}{L^{2} + ϵ^{2}} )|ξ|^{2} ≤ a_{ij}^{ϵ,σ}(p)ξ_{i}ξ_{j} (ξ ∈ ^{n})

provided |p| ≤ L, we deduce from (4.10)-(4.12) and classical estimates that we have bounds, uniform in 0 < σ < 1, on the derivatives of all orders of {u^{ϵ,σ}}_{0<σ<1}. Consequently, uniqueness of the limit implies for each multiindex α,

D^{α}u^{ϵ,σ} → D^{α}u^{ϵ} locally uniformly as σ → 0,

for a smooth function u^{ϵ} solving (4.2), (4.3). Estimate (4.7) follows from (4.10), (4.11), (4.12). □

4.3 Passage to limits

Theorem 4.2 Assume g: ^{n} → is continuous and verifies (4.1). Then there exists a weak solution u of (2.5), (2.6), such that

(4.13) u is constant on ^{n} × [0, ∞) ∩ {|x| + t ≥ R}.

for some R > 0, depending only on the S from (4.1).

Proof. 1. Suppose temporarily g is smooth. Employing estimate (4.7) we can extract a subsequence {u^{ϵ_{k}}}_{k=1}^{∞}, ⊂ {u^{ϵ}}, 0 < ϵ ≤ 1 so that ϵ_{k} → 0 and u^{ϵ_{k}} → u locally uniformly in ^{n}×[0, ∞) for some bounded, Lipschitz function u.

2. We assert now that u is a weak solution of (2.5), (2.6). For this, let ϕ ∈ C^{∞}( ^{n+1}) and suppose u − ϕ has a strict local maximum at a point (x_{0}, t_{0}) ∈ ^{n} × (0, ∞). As u^{ϵ_{k}} → u uniformly near (x_{0}, t_{0}), u^{ϵ_{k}} − ϕ has a local maximum at a point (x_{k}, t_{k}), with

(x_{k}, t_{k}) → (x_{0}, t_{0}) as k → ∞. (4.14)

Since u^{ϵ_{k}} and ϕ are smooth, we have

Du^{ϵ_{k}} = Dϕ, u_{t}^{ϵ_{k}} = ϕ_{t}, D^{2}u^{ϵ_{k}} ≤ D^{2}ϕ at (x_{k}, t_{k})

Thus (4.2) implies

ϕ_{t} − (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2} + ϵ_{k}^{2}} ) ϕ_{x_{i}x_{j}} ≤ 0 at (x_{k}, t_{k}). (4.15)

Suppose first Dϕ(x_{0}, t_{0}) ≠ 0. Then Dϕ(x_{k}, t_{k}) ≠ 0 for large k. We consequently may pass to limits in (4.15), recalling (4.14) to deduce,

ϕ_{t} ≤ (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2}} ) ϕ_{x_{i}x_{j}} at (x_{0}, t_{0}). (4.16)

Next, assume instead Dϕ(x_{0}, t_{0}) = 0. Set

η^{k} ≡ \frac{Dϕ(x_{k}, t_{k})}{(|Dϕ|(x_{k}, t_{k})|^{2} + ϵ_{k}^{2})^{\frac{1}{2}}} ,

so that (4.15) becomes ϕ_{t} ≤ (δ_{ij} − η_{i}^{k}η_{j}^{k}) ϕ_{x_{i}x_{j}} at (x_{k}, t_{k}). (4.17) Since |η^{k}| ≤ 1, we may assume, upon passing to a subsequence and reindexing if necessary, that

η^{k} → η in ^{n}

for some |η| ≤ 1. Sending k to infinity in (4.17), we discover ϕ_{t} ≤ (δ_{ij} − η_{i}η_{j}) ϕ_{x_{i}x_{j}} at (x_{0}, t_{0}). (4.18)

If u − ϕ has a local maximum, but not necessarily a strict local maximum at (x_{0}, t_{0}), we repeat the argument above with ϕ(x, t) replaced by

ϕ̃(x, t) = ϕ(x, t) + |x − x_{0}|^{4} + (t − t_{0})^{4},

again to obtain (4.16) or (4.18).

Consequently, u is a weak subsolution. That u is a weak supersolution follows analogously.

3. Finally we verify u satisfies (4.13). Upon rescaling as necessary, we may as well assume |g| ≤ 1 on ^{n}, g = 0 on ^{n} ∩ {|x| ≥ 1}. (4.19) Consider now the auxilary function (cf. Brakke [ , p.25])

v(x, t) ≡ Ψ(\frac{|x|^{2}}{2} + (n − 1) t) (x ∈ ^{n}, t > 0), (4.20)

for

Ψ(s) = \{ \begin{matrix}0 & (s ≥ 2) \\ (s − 2)^{3} & (0 ≤ s ≤ 2). \end{matrix}

Then Ψ ∈ C^{2}([0, ∞)),

Ψ′(s) = \{ \begin{matrix}0 & (s ≥ 2) \\ 3(s − 2)^{2} & (0 ≤ s ≤ 2), \end{matrix}

Ψ′′(s) = \{ \begin{matrix}0 & (s ≥ 2) \\ 6(s − 2) & (0 ≤ s ≤ 2) \end{matrix}

In particular, |Ψ′′(s)| ≤ C(Ψ′(s))^{1/2} (s ≥ 0) (4.21)

Now

v_{t} − (δ_{ij} − \frac{v_{x_{i}}v_{x_{i}}}{|Dv|^{2}+ϵ^{2}} ) v_{x_{i}x_{j}} = (n − 1)Ψ′ − (δ_{ij} − \frac{(Ψ′)^{2}x_{i}x_{j}}{(Ψ′)^{2}|x|^{2}+ϵ^{2}} ) (Ψ′δ_{ij} + Ψ′′x_{i}x_{j})
= Ψ′[(n − 1) − (δ_{ij} − \frac{(Ψ′)^{2}x_{i}x_{j}}{(Ψ^{′2}|x|^{2}+ϵ^{2}} ) δ_{ij}]
−Ψ′′[(δ_{ij} − \frac{(Ψ′)^{2}x_{i}x_{j}}{(Ψ′)^{2}|x|^{2}+ϵ^{2}} ) x_{i}x_{j}]
≡ A + B

(4.22) We further compute

A = −Ψ′ \frac{ϵ^{2}}{(Ψ′)^{2}|x|^{2} + ϵ^{2}} ≤ 0, (4.23)

since Ψ′ ≥ 0. Furthermore,

|B| = |Ψ′′|\frac{ϵ^{2}|x|^{2}}{(Ψ′)^{2}|x|^{2} + ϵ^{2}} .

Now if |Ψ′| ≤ ϵ, then

|B| ≤ |Ψ′′||x|^{2} ≤ C|Ψ′′| (since Ψ′′ = 0 if |x| ≥ 2)
≤ C(Ψ′)^{\frac{1}{2}} (by (4.21))
≤ Cϵ^{\frac{1}{2}} .

(4.24)

On the other hand if |Ψ′| ≥ ϵ, we have

|B| ≤ |Ψ′′| \frac{ϵ^{2}}{(Ψ′)^{2}} ≤ \frac{Cϵ^{2}}{|Ψ′|^{3/2}} ≤ Cϵ^{1/2} (4.25)

Combining (4.22) - (4.25) we find

v_{t} − (δ_{ij} − \frac{v_{x_{i}}v_{x_{j}}}{|Dx|^{2} + ϵ^{2}} ) v_{x_{i}x_{j}} ≤ Cϵ^{\frac{1}{2}}

and so

w_{t}^{ϵ} ≤ (δ_{ij} − \frac{w_{x_{i}}^{ϵ}w_{x_{j}}^{ϵ}}{|Dw^{ϵ}|^{2} + ϵ^{2}} ) w_{x_{i}x_{j}}^{ϵ} in ^{n} × (0, ∞) (4.26)

for w^{ϵ}(x, t) = v(x, t) − Ctϵ^{\frac{1}{2}} . (4.27)

Now

w^{ϵ}(x, 0) = Ψ (\frac{|x|^{2}}{2} ) = 0 if |x| ≥ 2.

and

w^{ϵ}(x, 0) = Ψ (\frac{|x|^{2}}{2} ) ≤ −1 if |x| ≤ 2.

Consequently, we see from (4.19) that

w^{ϵ} ≤ g on ^{n} × {t = 0}. (4.28)

Applying the maximum principle to (4.2), (4.3), (4.26) and (4.27), we deduce

w^{ϵ} ≤ u^{ϵ} in ^{n} × [0, ∞)

for each 0 < ϵ < 1. Sending ϵ = ϵ_{k} to zero, we then have

Ψ (\frac{|x|^{2}}{2} + (n − 1) t) = v(x, t) ≤ u(x, t)

for all x ∈ ^{n}, t ≥ 0. Thus

u ≥ 0 if \frac{|x|^{2}}{2} + (n + 1) t ≥ 2.

Similarly,

w̃_{t}^{ϵ} ≥ (δ_{ij} − \frac{w̃_{x_{i}}^{ϵ}w̃_{x_{j}}^{ϵ}}{|Dw̃^{ϵ}|^{2} + ϵ^{2}} ) w̃_{x_{i}x_{j}}^{ϵ} in ^{n} × (0, ∞) (4.29)

w̃^{ϵ} ≥ g on ^{n} × (0, ∞), (4.30)

for w̃^{ϵ} ≡ −w^{ϵ}. As above we consequently deduce

u ≤ 0 if \frac{|x|^{2}}{2} + (n + 1) t ≥ 2.

Assertion (4.13) is proved.

4. According to the uniqueness assertion Theorem 3.2, in fact the full limit

lim_{ϵ→0} u^{ϵ} = u

exists. Note also from Theorem 3.3 that

||u − ũ||_{L^{∞}(R^{n}×[0,∞))} = ||g − g̃||_{L^{∞}(R^{n})} , (4.31)

if ũ is the solution built as above for a smooth initial function g̃ verifying (4.1).

Suppose at last g satisfies (4.1), but is only continuous. We select smooth {g^{k}}_{k=1}^{∞}, satisfying (4.1) (for the same S) so that g^{k} → g uniformly on ^{n}. Denote by u^{k} the solution of (2.5), (2.6) constructed above with initial function g^{k}. Utilizing (4.30) we see that the limit

lim_{k→∞} u^{k} = u

exists uniformly on ^{n} × [0, ∞). According to Theorem 2.2 u is a weak solution of (2.5), (2.6). □

5 Definition of the Generalized Evolution by Mean Curvature

We now make precise the definition of the motion {Γ_{t}}_{t>0} for a given initial hypersurface Γ_{0}. In fact, let us assume now only that

(5.1) Γ_{0} is a compact subset of ^{n}.

Choose then any continuous function g: ^{n} → satisfying

Γ_{0} = {x ∈ ^{n} | g(x) = 0} (5.2)

and (5.3) g constant on ^{n} ∩ {|x| ≥ S}

for some S > 0. Utilizing Theorem 3.2, 4.1, we see that there is a unique weak solution of the mean curvature evolution equation

u_{t} = (δ_{ij} − \frac{u_{x_{i}}u_{x_{j}}}{|Du|^{2}} ) u_{x_{i}x_{j}} in ^{n} × (0, ∞) (5.4)

u = g on ^{n} × {t = 0}, (5.5)

with (5.6) u constant on ^{n} × [0, ∞) ∩ {|x| + t ≥ R}

for some R > 0.

Define then the compact set

Γ_{t} ≡ {x ∈ ^{n} | u(x, t) = 0} (5.7)

for each t > 0. We call {Γ_{t}}_{t>0} the generalized evolution by mean curvature of the original compact set Γ_{0}.

We must first verify that {Γ_{t}}_{t>0} is well-defined.

Theorem 5.1 Assume ĝ: ^{n} → is continuous, with

Γ_{0} = {x ∈ ^{n} | ĝ(x) = 0} (5.8)

and

(5.9)  ĝ constant on ^{n} ∩ {|x| ≥ S}.

Suppose û is the unique weak solution of (5.4)-(5.6), with ĝ replacing g. Then

Γ_{t} = {x ∈ ^{n} | û(x, t) = 0} (5.10)

for each t > 0.

Consequently our definition (5.7) does not depend upon the particular choice of initial function g verifying (5.2), (5.3).

Proof. 1. First, we may as well assume g ≥ 0 on ^{n} and thus u ≥ 0, in ^{n} × (0, ∞). Indeed, if g is negative somewhere, we can consider the PDE (5.4)–(5.6) with |g| replacing g, the unique solution of which, owing to Theorems 2.3, 3.2 is |u|. Our definition (5.7) is unchanged if we replace u by |u|. Similarly we may suppose ĝ, û ≥ 0. Set

Γ̂_{t} ≡ {x ∈ ^{n} | û(x, t) = 0} (t ≥ 0).

2. For k = 1, 2, … write E_{0} = ∅ and

E_{k} ≡ \{x ∈ ^{n} | g(x) ≥ \frac{1}{k}\};

so that

E_{1} ⊂ … E_{k} ⊂ E_{k+1} ⊂ …, ^{n} − Γ_{0} = ⋃_{k=1}^{∞} E_{k} (5.11)

Define

a_{k} ≡ max_{R^{n}−E_{k−1}} ĝ > 0 (k = 1, …) (5.12)

Then a_{1} ≥ a_{2} ≥ … and lim_{k→∞}a_{k} = 0, according to (5.8) and (5.11). Next define the continuous function

Ψ: [0, ∞) → [0, ∞)

satisfying Ψ(0) = 0,

Ψ(\frac{1}{k}) = a_{k}  (k = 1, …)
Ψ linear on [\frac{1}{k+1}, \frac{1}{k}] (k = 1, …)

Ψ constant on [1, ∞) Ψ non-decreasing

3. Write

g̃ = Ψ(g), ũ = Ψ(u).

Then ũ solves (5.4)–(5.6), with g̃ replacing g. Now

g̃ = ĝ on Γ_{0}.

Furthermore, if x ∈ E_{k} − E_{k−1}, then

g̃(x) = Ψ(g(x))
≥ Ψ(\frac{1}{k})
= a_{k}

≥ ĝ(x) by (5.12)

Thus g̃ ≥ ĝ on ^{n}. Consequently, Theorem 3.2 asserts

ũ = Ψ(u) ≥ û ≥ 0 on ^{n} × [0, ∞).

Thus if x ∈ Γ_{t}, û(x, t) = 0 and so x ∈ Γ̂_{t}. Hence

Γ_{t} ⊆ Γ̂_{t}.

The opposite inclusion is similarly proved and so

Γ_{t} = Γ̂_{t} for each t > 0.

□

In light of this theorem, we can regard the mappings

Γ_{0} ⟼ Γ_{t} (t ≥ 0)

as defining a time-dependent evolution on the collection K of compact subsets of ^{n}. Let us write M(t)Γ_{0} ≡ Γ_{t} (t ≥ 0) (5.13)

explicitly to display the dependence of Γ_{t} on t and Γ_{0}. Then

M(t): K → K

for each t ≥ 0, and M(0) is the identity operator. We will call {M(t)}_{t≥0} the mean-curvature semigroup on K.

To justify this terminology, let us verify the semigroup property

M(t + s) = M(t)M(s) (t, s ≥ 0). (5.14)

Indeed if t, s > 0 and Γ_{0} ∈ K, choose any continuous function g satisfying (5.2), (5.3). Let u be the corresponding unique weak solution of (5.4)–(5.6). Then M(t + s)Γ_{0} = Γ_{t+s} = {x ∈ ^{n} | u(x, t + s) = 0}. (5.15)

M(s)Γ_{0} = Γ_{s} = {x ∈ ^{n} | u(x, s) = 0}. (5.16)

To compute M(t)Γ_{s} we select any continuous function ĝ so that

Γ_{s} = {x ∈ ^{n} | ĝ(x) = 0} (5.17)

and ĝ is constant outside some large ball. We then find the unique weak solution û of (5.4)–(5.6) (with ĝ replacing g) and set

M(t)Γ_{s} = Γ̂_{t} = {x ∈ ^{n} | û(x, t) = 0}. (5.18)

According to Theorem 5.1, this construction is independent of the particular choice of ĝ satisfying (5.17). In particular, we may as well take

ĝ(x) = u(x, s) (x ∈ ^{n}).

Owing then to the uniqueness of a weak solution to (5.4)-(5.6) we have

û(x, t) = u(x, t + s) (x ∈ ^{n}, t > 0)

Consequently (5.15), (5.18) imply

M(t + s)Γ_{0} = M(t)M(s)Γ_{0},

as required. This establishes (5.14).

Note that we make no assertions concerning continuity of the mapping

(t, Γ_{0}) ⟼ M(t)Γ_{0}.

6 Consistency with Classical Motion by Mean Curvature

We must now check that our generalized evolution by mean curvature agrees with the classical motion, if and so long as the latter exists. Let us therefore suppose for this section that Γ_{0} is a smooth hypersurface, the connected boundary of a bounded open set U_{0} ⊂ ^{n}. According to Hamilton [], Gage-Hamilton [], there exists a time t_{∗} > 0 and a family {Σ_{t}}_{0≤t<t_{∗}} of smooth hypersurfaces evolving from Σ_{0} = Γ_{0} according to classical motion by mean curvature. In particular for each 0 ≤ t < t_{∗}, Σ_{t} is diffeomorphic to Γ_{0}, and is the boundary of an open set U_{t} diffeomorphic to U_{0}. Theorem 6.1 We have

Σ_{t} = Γ_{t} (0 ≤ t < t_{∗})

where {Γ_{t}} is the generalized evolution by mean curvature defined in §5.

Proof. 1. Fix 0 < t_{0} < t_{∗}, and define then for 0 ≤ t ≤ s the (signed) distance function

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

As Σ = ⋃_{0≤t≤t_{0}} Σ_{t} is smooth, d is smooth in the regions

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

and

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

for δ_{0} > 0 sufficiently small.

2. Now for each point (x, t) ∈ Q^{+} there exists a unique point y ∈ Σ_{t} verifying d(x, t) = |x−y|. Consider now near (y, t) the smooth unit vector field ν = Dd pointing from Σ into Q^{+}. Then d_{t}(x, t) = (div ν) (y, t) (6.1) since {Σ_{t}}_{0≤t≤s} is a classical evolution by mean curvature. Additionally, the eigenvalues of D^{2}d(x, t) are (see e.g. [[?], p.355])

\{\frac{−k_{1}}{1 − k_{1}d} , …, \frac{−k_{n−1}}{1 − k_{n−1}d}, 0\} , (6.2)

k_{1}, …, k_{n−1} denoting the principal curvatures of Σ_{t} at the point y, calculated with respect to the unit normal field ν. Thus

Δd(x, t) = − ∑_{i=1}^{n−1} \frac{k_{i}}{1 − k_{i}d} (6.3)

However (div ν) (y, t) = −(k_{1} + ⋯ + k_{n−1}); and so (6.1) (6.3) imply

d_{t} − Δd = (∑_{i=1}^{n−1} \frac{k_{i}^{2}}{1 − k_{i}d}) d at (x, t). (6.4)

Since the quantity

∑ \frac{k_{i}^{2}}{1 − k_{i}d}

is uniformly bounded and d ≥ 0 in Q^{+}, we deduce from (6.4) that \overline{d} ≡ αe^{−λt}d (6.5) satisfies \overline{d}_{t} − Δ\overline{d} ≤ 0 in Q^{+} (6.6) if λ > 0 is fixed large enough and α > 0 (to be selected later.) Furthermore, |Dd|^{2} = |ν|^{2} = 1 and so

d_{x_{i}}d_{x_{i}x_{j}} = 0 in Q^{+}, 1 ≤ j ≤ n.

The function \overline{d} satisfies the same identity, whence (6.6) implies for each ϵ ≥ 0 that

\overline{d}_{t} − (δ_{ij} − \frac{\overline{d}_{x_{i}} \overline{d}_{x_{j}}}{|D\overline{d}|^{2} + ϵ^{2}} ) \overline{d}_{x_{i}x_{j}} ≤ 0 in Q^{+}. (6.7)

We see therefore that \overline{d} is a smooth subsolution of the approximate mean curvature evolution PDE (4.2) in Q^{+}.

3. Choose any Lipschitz function g: ^{n} → so that g(x) = dist(x, Σ_{0}) near Σ_{0}, {g = 0} = Σ_{0} and g(x) is a positive constant for large |x|. For 0 < ϵ < 1 the approximating PDE (4.2) (4.3) then has a continuous solution u^{ϵ} which is smooth in ^{n} × (0, ∞). Additionally we have Γ_{t} = {x ∈ ^{n} | u(x, t) = 0} t ≥ 0. (6.8) Now u = g = δ_{0} > 0 on {(x, 0) | dist(x, Σ_{0}) = dist(x, Γ_{0}) = δ_{0}} and as u is continuous, we thus have

u ≥ \frac{d}{2} > 0 on \{(x, t) | d(x, t) = δ_{0}\} (6.9)

for 0 ≤ t ≤ t_{0}, provided t_{0} > 0 is small enough. Hence (6.9) implies

u^{ϵ} ≥ \frac{d}{4} on \{(x, t) | d(x, t) = δ_{0}\}

for 0 ≤ t ≤ t_{0}, 0 < ϵ ≤ ϵ_{0} if ϵ_{0} > 0 is sufficiently small. Consequently there exists 0 < α < 1 so that u^{ϵ} ≥ \overline{d} on \{(x, t) | d(x, t) = δ_{0}\} (6.10) for 0 ≤ t ≤ t_{0}, 0 < ϵ < ϵ_{0}, \overline{d} defined by (6.5). Since 0 < α < 1, we have u^{ϵ} ≥ \overline{d} on \{(x, 0) | 0 ≤ d(x, t) ≤ δ_{0}\} (6.11) Furthermore, g ≥ 0 implies u^{ϵ} ≥ 0 and so u^{ϵ} ≥ \overline{d} on \{(x, t) | d(x, 0) = 0\} (6.12)

4. Combining (6.10)-(6.12) we see that u^{ϵ} ≥ \overline{d} on the parabolic boundary of Q^{+}. Since \overline{d} solves (6.7) and u^{ϵ} solves (4.2), the maximum principle implies

u^{ϵ} ≥ \overline{d} in Q^{+}

Let ϵ → 0 to deduce

(6.13) u > 0 in the interior of Q^{+}

A similar argument using instead

\overline{d} = −αϵ^{−λt}d

shows (6.14) u > 0 in the interior of Q^{−}.

Since u > 0 in (R^{n} \{dist(x, Σ_{0}) ≤ δ_{0}})×[0, t_{0}], we deduce from (6.13) (6.14) and (6.8) that

Γ_{t} ⊆ Σ_{t} = {x | d(x, t) = 0} 0 ≤ t ≤ t_{0}. (6.15)

5. Now define a new function ĝ: ^{n} → so that ĝ(x) = d(x, 0) (the signed distance function to Σ_{0} = Γ_{0}) near Σ_{0} = Γ_{0}, {ĝ = 0} = Σ_{0}, and ĝ(x) is a positive constant for large |x|. Let û denote the unique weak solution of (2.5) (2.6) (4.13) for this new initial function ĝ. According to Theorem 5.1

Γ_{t} = {x ∈ ^{n} | û(x, t) = 0} (t ≥ 0) (6.16)

Since ĝ < 0 in U_{0} we know by continuity that û < 0 somewhere in U_{t}, provided 0 ≤ t ≤ t_{0} and t_{0} is small. Similarly û > 0 somewhere in ^{n} − \overline{U}_{t} for each 0 ≤ t ≤ t_{0}. Fix any point x_{0} ∈ Σ_{t} and draw a smooth curve C in ^{n}, intersecting Σ_{t} precisely at x_{0} and connecting a point x_{1} ∈ U_{t} where û(x_{1}, t) < 0 to a point x_{2} ∈ ^{n} − \overline{U}_{t} where û(x_{2}, t) > 0. As û is continuous, we must have û(x, t) = 0 for some point x on the curve C. However (6.15) and (6.16) say that the set {û(·, t) = 0} lies in Σ_{t}. Thus û(x_{0}, t) = 0. Since x_{0} denoted any point on Σ_{t} we deduce from (6.15) (6.16) that

Γ_{t} = Σ_{t} if 0 ≤ t ≤ t_{0}. (6.17)

We have consequently demonstrated that the classical motion {Σ_{t}}_{0≤t<t} and the generalized motion {Γ_{t}}_{t≥0} agree at least on some short time interval [0, t_{0}].

6. Write

s ≡ sup_{0≤t<t^{∗}}{t | Γ_{t} = Σ_{t}},

and suppose s < t^{∗}. Then

Γ_{t} = Σ_{t} for all 0 ≤ t < s

and so, using the continuity of u solving (2.5) and (2.6) for g as above, we have

Γ_{s} ⊇ Σ_{s}

On the other hand if x ∈ ^{n}−Σ_{s}, there exists r > 0 so that B(x, r) ⊂ ^{n}−Σ_{t} for all s − ϵ ≤ t ≤ s, ϵ > 0 small enough. Using this we easily deduce x ∉ Γ_{s}. Hence

Γ_{s} = Σ_{s}.

But then applying steps 1–5 we deduce

Γ_{t} = Σ_{t}

for all s ≤ t ≤ s + s_{0} < t^{∗}, if s_{0} > 0 is small enough. This contradicts the definition of s, and so in fact

s = t^{∗}.

□

7 Geometric Properties of Generalized Evolution by Mean Curvature

We devote this section to establishing some elementary properties of the generalized evolution by mean curvature

Γ_{0} ⟼ M(t)Γ_{0} ≡ Γ_{t} (t ≥ 0). (7.1)

for Γ_{0} a compact subset of ^{n}.

7.1 Localization and extinction

First of all, it is known that if Γ_{0} is the sphere ∂B(0, R), then

Γ_{t} = \{ \begin{matrix}∂B(0, R(t)) & if 0 ≤ t < t^{∗} \\ \{0\} & if t = t^{∗} \\ ∅ & if t > t^{∗} \end{matrix} (7.2)

where

R(t) ≡ (R^{2} − 2(n − 1) t)^{\frac{1}{2}} for 0 ≤ t ≤ t^{∗} ≡ \frac{R^{2}}{2(n − 1)} . (7.3)

This assertion follows in our approach by noting

u(x, t) = Ψ (|x|^{2} + 2(n − 1) t)

is a weak solution of (5.4), where Ψ: → is smooth with

\{ \begin{matrix}Ψ′ ≥ 0, Ψ < 0 on [0, R), \\ Ψ > 0 on (R, 3R), Ψ ≡ 1 on [3R, ∞). \end{matrix} (7.4)

By making comparisons with the shrinking sphere (7.2) we derive now some elementary properties of the general motion (7.1). (Cf. Brakke [ , p. 29-30]).

Theorem 7.1 (a) If Γ_{0} ⊂ B (0, R), then

Γ_{t} = ∅ for t > \frac{R^{2}}{2(n − 1)} (7.5)

(b) We have Γ_{t} ⊆ co (Γ_{0}) (t ≥ 0). (7.6)

with co (Γ_{0}) denoting the convex hull of Γ_{0}.

Proof. 1. Assume first Γ_{0} ⊂ B(0, R − ϵ) for some ϵ > 0. Let g: ^{n} → be continuous, with

Γ_{0} = {g = 0}, g = 1 on ^{n} ∩ {|x| ≥ 2R}.

Set

ĝ(x) = Ψ (|x|^{2}),

with Ψ satisfying (7.4) selected so that

ĝ ≤ g on ^{n}.

Then

û ≤ u on ^{n} × [0, ∞),

for û(x, t) = Ψ (|x|^{2} + 2(n − 1) t) and u the weak solution of (5.4)-(5.6). Thus u > 0, and so Γ_{t} = ∅, if t > \frac{R^{2}}{2(n−1)}.

In the general case, replace R by R + ϵ in this argument and send ϵ → 0.

2. Suppose Γ_{0} ⊂ _{+}^{n} = {x_{n} > 0}. Choose R >> 1 so large that

Γ_{0} ⊂ B (Re_{n}, R),

for e_{n} = (0, 0, …, 0, 1). By the argument in step l., we deduce

Γ_{t} ⊂ B (Re_{n}, R(t))

for 0 ≤ t ≤ \frac{R^{2}}{2(n−1)} , R(t) defined as above. In particular,

Γ_{t} ⊆ _{+}^{n} for all t ≥ 0.

Replacing _{+}^{n} in this argument by any open half space containing Γ_{0}, we deduce (7.6). □

7.2 Comparison of different sets moving by mean curvature

Theorem 7.2 Let Γ_{0}, Γ̂_{0} be compact subsets of ^{n} and denote by {Γ_{t}}_{t≥0}, {Γ̂_{t}}_{t≥0} the corresponding generalized motions by mean curvature. Suppose also

Γ_{0} ⊆ Γ̂_{0}. (7.7)

Then

(7.8)  Γ_{t} ⊆ Γ̂_{t}  for each t > 0.

We see therefore that if a compact set Γ_{0} lies within another Γ̂_{0} at time zero, then the subsequent evolution Γ_{t} of Γ_{0} lies within the subsequent evolution Γ̂_{t} of Γ̂_{0}. We will see in §8 that this assertion provides us with a powerful tool for studying specific examples. Proof. Choose continuous functions g, ĝ: ^{n} → [0, ∞) so that

Γ_{0} = {g = 0}, Γ̂_{0} = {ĝ = 0},

and g, ĝ constant on ^{n} ∩ {|x| ≥ S} for some S > 0. Replacing g by g + ĝ if necessary, we may assume

ĝ ≤ g on ^{n}. (7.9)

Now let û, u denote the corresponding weak solutions of (5.4)-(5.6). Then (7.9) implies

0 ≤ û ≤ u on ^{n} × (0, ∞)

Thus x ∈ Γ_{t} implies x ∈ Γ̂_{t}, and so (7.8) is valid. □

Theorem 7.3 Assume Γ_{0}, Γ̂_{0} are nonempty compact sets and {Γ_{t}}_{t≥0}, {Γ̂_{t}}_{t≥0} are the subsequent generalized motions by mean curvature. Then

dist(Γ_{0}, Γ̂_{0}) ≤ dist(Γ_{t}, Γ̂_{t}) (t ≥ 0). (7.10)

By definition dist(Γ_{t}, Γ̂_{t}) = +∞ if Γ_{t} = ∅, Γ̂_{t} = ∅, or both.

Proof. 1. We may assume dist(Γ_{0}, Γ̂_{0}) > 0. Choose g: ^{n} → so that Γ_{0} = {g = 0}, Γ̂_{0} = {g = 1},

g = 2 on ^{n} ∩ {|x| ≥ s} for some S,
Lip(g) = dist(Γ_{0}, Γ̂_{0})^{−1} .

(7.11)

Then Γ_{t} = {u = 0}, Γ̂_{t} = {u = 1}, (7.12) with u denoting the corresponding weak solution of (5.4)–(5.6).

2. From the contraction property, Theorem 3.3, we see that

Lip(u(·, t)) ≤ Lip(g) (t ≥ 0). (7.13)

If Γ_{t} ≠ ∅, Γ̂_{t} ≠ ∅, choose points x ∈ Γ_{t}, x̂ ∈ Γ̂_{t} so that

|x − x̂| = dist(Γ_{t}, Γ̂_{t}).

Then using (7.11)–(7.13) we compute

1 = u(x̂, t) − u(x, t) ≤ Lip(u)|x − x̂| ≤ dist(Γ_{0}, Γ̂_{0})^{−1} dist(Γ_{t}, Γ̂_{t}).

This proves (7.10). □

Inequality (7.10) says in particular that two hypersurfaces evolving under generalized motion by mean curvature do not ever move closer to each other than they were initially. In particular, Γ_{t} ∩ Γ̂_{t} = ∅ for all t > 0 provided Γ_{0} ∩ Γ̂_{0} = ∅. Notice that this property is essential for our approach of representing the evolving surfaces as the level sets of a continuous function.

7.3 Positive mean curvature

Now let us assume that Γ_{0} is a smooth connected hypersurface, the boundary of a bounded open set U ⊂ ^{n}. We will suppose additionally that

div(ν) > 0 on Γ_{0}, (7.14)

ν denoting the outward unit normal vector field to Γ_{0} (extended smoothly to some neighborhood of Γ_{0}). Inequality (7.14) says that Γ_{0} has positive mean curvature with respect to the outer unit normal field. Consequently, if Γ_{0} evolves according to mean curvature, we see from (2.2) that initially at least the motion is directed into U.

We show now that in fact Γ_{t} lies in U for all t ≥ 0, and that Γ_{t} continues to have positive mean curvature, this last statement interpreted in an appropriate weak sense.

Informally our idea is to solve (5.4)–(5.6) by separating variables. Indeed we will show u(x, t) ≡ v(x) − t (x ∈ U, t > 0), (7.15)

where v is the (unique) weak solution of the stationary problem

− (δ_{ij} − \frac{v_{x_{i}}v_{x_{j}}}{|Dv|^{2}} ) v_{x_{i}x_{j}} = 1 in U (7.16)

v = 0 on ∂U = Γ_{0}. (7.17)

We will show that

Γ_{t} = {x ∈ U | v(x) = t} (t ≥ 0), (7.18)

so that Γ_{t} ⊂ U (t ≥ 0) and Γ_{t} = ∅ for t ≥ t^{∗} ≡∥ v ∥_{L^{∞}(U)}. Note also that in any open region where v is smooth and |Dv| ≠ 0, we can rewrite (7.16) to read

div(ν) = \frac{1}{|Dv|} > 0,

for

ν ≡ −\frac{Dv}{|Dv|}

As ν is the outward pointing unit normal field along Γ_{t} ≡ {v = t}, we can informally interpret our PDE (7.16) as implying Γ_{t} to have positive mean curvature for 0 ≤ t < t^{∗}.

To carry out the foregoing program rigorously, let us first define v ∈ C(\overline{U}) to be a weak solution to (7.16) provided for each ϕ ∈ C^{∞}( ^{n}) if

u − ϕ has a local maximum (minimum) at a point x_{0} ∈ U,

then

\{ \begin{matrix}−(δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2}} ) ϕ_{x_{i}x_{j}} ≤ (≥) 1 at x_{0} \\ if Dϕ (x_{0}) ≠ 0 \end{matrix} (7.19)

and

−(δ_{ij} − η_{i}η_{j}) ϕ_{x_{i}x_{j}} ≤ (≥) 1 at x_{0}
for some η ∈ ^{n} with |η| ≤ 1, if Dϕ (x_{0}) = 0.

(7.20)

Theorem 7.4 There exists a unique weak non-negative solution v of (7.16), (7.17). Furthermore, there exist constants A, a > 0 so that

\begin{matrix}a dist(x, Γ_{0}) ≤ v(x) ≤ \sqrt{n} A dist(x, Γ_{0}) x ∈ U \\ |Dv(x)| ≤ \sqrt{n} A \end{matrix} (7.21)

Proof. 1. Similarly to §4, we approximate (7.16), (7.17) by the uniformly elliptic PDE

− (δ_{ij} − \frac{v_{x_{i}}^{ϵ}v_{x_{j}}^{ϵ}}{|Dv^{ϵ}|^{2} + ϵ^{2}} ) v_{x_{i}x_{j}}^{ϵ} = 1 in U (7.22)

v^{ϵ} = 0 on ∂U = Γ_{0} (7.23)

for 0 < ϵ ≤ 1. We will construct upper an lower barriers for (7.22), (7.33) of the form

\overline{v}(x) = λg(d(x)), d(x) = dist(x, Γ_{0})

in a neighborhood U_{0} = {0 < d(x) < 2δ_{0}} of Γ_{0} on which d(x) is smooth. Owing to the mean curvature condition (7.14), d(x) satisfies

− c_{2} ≤ Δ d ≤ −c_{1} < 0 d_{x_{i}}d_{x_{j}}d_{x_{i}x_{j}} ≡ 0 (7.24)

in this region. We then compute using (7.21)

\begin{aligned}M\overline{v} ≡ (δ_{ij} − \frac{\overline{v}_{x_{i}} \overline{v}_{x_{j}}}{|D\overline{v}|^{2}+ϵ^{2}} ) \overline{v}_{x_{i}x_{j}} & = λ(g′Δ d + g′′) − \frac{λ^{3}}{λ^{2}g^{′2}+ϵ^{2}} g^{′2}g′′ \\ & = λg′Δ d + \frac{ϵ^{2}λg′′}{λ^{2}g^{′2}+ϵ^{2}} \end{aligned} (7.25)

Choosing g(t) = δ_{0}^{2} − (t − δ_{0})^{2}, we find from (7.24), (7.25) that M\overline{v} ≥ −cλδ_{0} − 2λ > −1 for λ sufficiently small. Since \overline{v} = 0 on ∂U_{0}, \overline{v} < v^{ϵ} in U_{0} by the maximum principle. In particular, v^{ϵ} > λδ_{0}^{2} on {d(x) = δ_{0}} and so v^{ϵ} > λδ_{0}^{2} on {d(x) > δ_{0}}. These estimates give the lower bound

v^{ϵ}(x) ≥ a d(x) in U

with a independent of ϵ. To obtain the corresponding upper bound, we choose

g(t) = ln(2δ_{0}/2δ_{0} − t)

Then g(t) is convex on [0, 2δ_{0}) and satisfies

g(0) = 0, g′ ≥ \frac{1}{2δ_{0}} , g′′ = g^{′2}, g′(2δ_{0}) = +∞ (7.26)

Again using (7.24)–(7.26), we find

M\overline{v} ≤ −cλ + \frac{ϵ^{2}}{λ} < −1

for λ sufficiently large. Since \frac{∂\overline{v}}{∂v} = +∞ on \{d(x) = 2δ_{0}\} (ν the exterior normal to U_{0}) we find that v^{ϵ} < \overline{v} in U_{0} by a simple variant of the maximum principle. This gives the estimate

v^{ϵ} ≤ A d(x)  for  0 < d(x) < δ_{0}. (7.27)

To complete our preliminary estimates, we observe that (7.27) implies |Dv^{ϵ}| ≤ A on Γ_{0}. By differentiating (7.22) with respect to x_{ℓ}, we see that any derivative v_{x_{ℓ}}^{ϵ} achieves its maximum and minimum on Γ_{0}. Thus

|v_{x_{ℓ}}^{ϵ}| ≤ A in U and so |Dv^{ϵ}| ≤ \sqrt{n} A in U.

In particular v^{ϵ} ≤ \sqrt{n} A d(x) in U_{0}.

2. As a consequence of step 1., we derived the uniform bounds

sup_{0<ϵ≤1} ∥v^{ϵ} ∥_{c^{1}(U)} < ∞

Hence we may extract a subsequence {v^{ϵ_{k}}}_{k=1}^{∞} ⊂ {v^{ϵ}}_{0<ϵ≤1} so that ϵ_{k} → 0 and v^{ϵ_{k}} → v uniformly on \overline{U}. Now as in the proof of Theorem 4.2, we verify that v is a weak solution of (7.16).

3. The uniqueness of this weak solution v will follow from the characterization of {Γ_{t}}_{t≥0} below. □

Theorem 7.5 Let {Γ_{t}}_{t≥0} denote the generalized evolution by mean curvature starting with Γ_{0}. Then

Γ_{t} = {x ∈ U | v(x) = t}

for each t ≥ 0.

Proof. 1. Define u(x, t) ≡ v(x) − t for x ∈ U, t > 0. It is then straightforward to verify that u is a weak solution of

u_{t} = (δ_{ij} − \frac{u_{x_{i}}u_{x_{j}}}{|Du|^{2}} ) u_{x_{i}x_{j}} in U × (0, ∞) (7.28)

set Γ̂_{t} ≡ {x ∈ U | v(x) = t} = {x ∈ U | u(x, t) = 0} (t > 0) (7.29)

2. Now let

ũ(x, t) = |u(x, t)| = |v(x) − t| (x ∈ U, t > 0)

In view of Theorem 2.3 ũ is a weak solution of

\{ \begin{matrix}ũ_{t} = (δ_{ij} − \frac{_{u_{x_{i}}} _{u_{x_{j}}}}{|D_{u}|^{2}} ) ũ_{x_{i}x_{j}} in U × (0, ∞) \\ ũ = t on ∂U × [0, ∞) \\ ũ = v on \overline{U} × \{t = 0\}. \end{matrix} (7.30)

3. Choose any smooth function g: ^{n} → so that

\{ \begin{matrix}Γ_{0} = \{g = 0\}, g ≥ 0 \\ g is constant on ^{n}\{|x| ≥ S\} for some S > 0. \end{matrix}

(7.31)

Let w ≥ 0 be the unique weak solution of

\{ \begin{matrix}w_{t} = (δ_{ij} − \frac{w_{x_{i}}w_{x_{j}}}{|Dw|^{2}} ) w_{x_{i}x_{j}} in ^{n} × (0, ∞) \\ w = g on ^{n} × \{t = 0\}, \end{matrix} (7.32)

so that Γ_{t} = {x ∈ ^{n} | w(x, t) = 0} (t ≥ 0). (7.33)

According to our construction in §4, w is Lipschitz in t, and so

|w(x, t)| ≤ c t (x ∈ Γ_{0}, t > 0). (7.34)

for some constant c.

4. Employing now (7.21), (7.27) and (7.30), we see that

\underline{w} = α w

satisfies

\{ \begin{matrix}\underline{w} ≤ v on \overline{U} × \{t = 0\} \\ \underline{w} ≤ t on ∂U × (0, ∞) \end{matrix} (7.35)

if α > 0 is sufficiently small.

Now the proof of our Comparison Theorem 3.2 can be straightforwardly modified to show from (7.26), (7.28), (7.31) that

\underline{w} ≤ ũ in U × [0, ∞).

Thus x ∈ Γ̂_{t} implies x ∈ Γ_{t}, and so

Γ̂_{t} ⊆ Γ_{t} for t ≥ 0.

Similarly selecting

\overline{w} = β w

for some β >> 1, we deduce

ũ ≤ \overline{w} in U × [0, t_{0}).

Thus

Γ_{t} ⊆ Γ̂_{t} for t ≥ 0.
□

7.4 Convexity

Theorem 7.6 Assume Γ_{0} is the boundary of a smooth convex bounded open set U. Then there exists t^{∗} > 0 such that Γ_{t} is the boundary of a convex, nonempty open set for 0 ≤ t < t^{∗} and Γ_{t} is empty for t > t^{∗}.

Proof. 1. Because of §7.3 it suffices to consider the stationary PDE

(δ_{ij} − \frac{v_{x_{i}} v_{x_{j}}}{|Dv|^{2}} ) v_{x_{i}x_{j}}, = −1 in U, v = 0 on Γ_{0} = ∂U (7.36)

We will show that {x ∈ U | v(x) > A} is convex for 0 ≤ t < t^{∗}, for t^{∗} = sup v. In fact, we will show that \sqrt{v(x)} is concave using ideas of Korevaar [] and Kennington []. The motivation for considering \sqrt{v} comes from the corresponding linear problem Δv = −1.

Formally, if u = \sqrt{v}, with v satisfying (7.36), than u satisfies

(δ_{ij} − \frac{u_{x_{i}}u_{x_{j}}}{|Du|^{2}} ) u_{x_{i}x_{j}} = −\frac{1}{2u} in U (7.37)

This suggests we consider approximations u^{ϵ} = \sqrt{v^{ϵ}} satisfying

\begin{matrix}Mu^{ϵ} = (δ_{ij} − \frac{u_{x_{i}}^{ϵ}u_{x_{j}}^{ϵ}}{|Du^{ϵ}|^{2}+ϵ^{2}} ) u_{x_{i}x_{j}}^{ϵ} = −\frac{1}{2u^{ϵ}}  in U \\ u^{ϵ} = 0  on Γ_{0} \end{matrix} (7.38)

(δ_{ij} − \frac{v_{x_{i}}^{ϵ}v_{x_{j}}^{ϵ}}{|Dv^{ϵ}|^{2} + 4ϵ^{2}v^{ϵ}} ) v_{x_{i}x_{j}}^{ϵ} = \frac{1}{2} \frac{ϵ^{2} |Dv^{ϵ}|^{2}}{|Dv^{ϵ}|^{2} + 4ϵ^{2}v^{ϵ}} − 1 (7.39)

Because the convexity arguments are very sensitive to the form of the equation, we are forced into making a nice approximation u^{ϵ} to (7.37) and then making due with nastier approximations v^{ϵ} to (7.36).

2. We next demonstrate the existence of a solution u^{ϵ} ∈ C^{2}(U) ∩ C^{1/2}(\overline{U}) to (7.38). Consider

\begin{matrix}Mu^{ϵ,δ} ≡ (δ_{ij} − \frac{u_{x_{i}}^{ϵ,δ}u_{x_{j}}^{ϵ,δ}}{|Du^{ϵ,δ}|^{2}+ϵ^{2}} ) u_{x_{i}x_{j}}^{ϵ,δ} = −\frac{1}{2(u^{ϵ,δ}+δ)}  in U \\ u^{ϵ,δ} = 0  on Γ_{0} \end{matrix} (7.40)

Equation (7.40) being uniformly elliptic and strictly coercive in u^{ϵ,δ}, the existence of a solution u^{ϵ,δ} ∈ C^{2+α}\overline{(}U) is well-known [GT].

Choose a large ball B_{R}(P) containing U with dist(P, U) ≥ R/2, and let r = |x − P |.

Set w = (2R − r). Then

Mw + \frac{1}{2(w + δ)} = −\frac{(n − 1)}{r} + \frac{1}{2(2R − r)} ≤ −\frac{(n − 1)}{R} + \frac{1}{3R} < 0

Since w > 0 on ∂U, w > u^{ϵ,δ} in U by the maximum principle. Hence

0 ≤ u^{ϵ,δ} < 2R in U (7.41) with R independent of ϵ, δ.

Next, let w = λ\sqrt{d(x)} in \{0 < d(x) < δ_{0}\}. Using formula 7.25 of §7.3 (with g(t) = \sqrt{t}) we find

Mw + \frac{1}{2(w + δ)} ≤ −\frac{cλ}{\sqrt{d(x)}} + \frac{1}{2(λ\sqrt{d} + δ)} < 0

for λ sufficiently large. If in addition, we choose λ so that λ\sqrt{δ_{0}} ≥ 2R, then w ≥ u^{ϵ,δ} on ∂{0 < d(x) < δ_{0}}, and thus w ≥ u^{ϵ,δ} on {0 < d(x) < δ_{0}} by the maximum principle. In particular

0 ≤ u^{ϵ,δ} ≤ A \sqrt{d(x)} in U (7.42)

with A independent of ϵ, δ.

Estimate (7.42) implies that

|u^{ϵ,δ}(x) − u^{ϵ,δ}(y)| ≤ c|x − y|^{1/2} if x ∈ U, y ∈ Γ_{0} (7.43)

with c independent of ϵ, δ. We show that (7.43) holds for all x, y ∈ Ω by the following well known argument (see e.g. [G]). Given x, y ∈ Ω we set τ = y − x, Ω_{τ} = {z ∈ ^{n}|z − τ ∈ Ω} and u_{τ}^{ϵ,δ}(z) = u^{ϵ,δ}(z − τ). Note that Ω_{τ} is open and non-empty since y ∈ Ω_{τ}. On Ω ∩ Ω_{τ}, both u^{ϵ,δ} and u_{τ}^{ϵ,δ} satisfy (7.40) and hence the difference w = u^{ϵ,δ} − u_{τ}^{ϵ,δ} satisfies a linear elliptic equation of the form Lw + c(x)w = 0 with c(x) ≤ 0. Hence by the maximum principle,

|w(y)| ≤ max_{z∈∂(Ω∩Ω_{τ})} |w(z)|

Since for z ∈ ∂(Ω ∩ Ω_{τ}) either z ∈ ∂Ω or z − τ ∈ ∂Ω we have by (7.43) that

|u^{ϵ,δ}(y) − u^{ϵ,δ}(x)| = |u^{ϵ,δ}(y) − u_{τ}^{ϵ,δ}(y)| ≤ c|x − y|^{1/2} (7.44)

Finally, in order to pass to the limit for a sequence δ_{k} ↘ 0, we need to establish interior estimates for u_{k} = u^{ϵ,δ_{k}}.

||u_{k}||_{C^{2+α}(U′)} ≤ M(ϵ, dist(U′, Γ_{0})) (7.45) with M independent of δ_{k}. By Schauder theory, (7.45) follows from an interior gradient estimate

||Du_{k}||_{L^{∞}(U′)} ≤ C(ϵ, dist(U′, Γ_{0}))

which in turn follows from Theorem 15.5 of [GT]. Therefore, we have established the existence of a (unique) solution u^{ϵ} of (7.38), and in addition

\begin{matrix}0 ≤ u^{ϵ} ≤ A \sqrt{d(x)} \\ |u^{ϵ}(x) − u^{ϵ}(y)| ≤ c|x − y|^{1/2} \\ 0 ≤ u^{ϵ} ≤ 2R \end{matrix} (7.46)

with A, C, R independent of ϵ.

3. Before we proceed to the proof of the concavity of u^{ϵ}, we shall need to establish the lower bound u^{ϵ} ≥ a d(x) (7.47)

with a independent of ϵ.

Consider w = λg(d(x)) in {0 < d(x) < 2δ_{0}} with g(t) = (δ_{0}^{2} −(t−δ_{0})^{2})^{1/2}. Then from formulas (7.24), (7.25) we find

\begin{aligned}Mw & ≥ −\frac{λδ_{0}}{g} (c + \frac{ϵ^{2}δ_{0}}{λ^{2}(d−δ_{0})^{2}+ϵ^{2}(δ_{0}^{2}−(d−δ_{0})^{2})}) \\ & ≥ −\frac{λδ_{0}}{g} (c + \frac{1}{δ_{0}} ) for λ ≥ ϵ \end{aligned}

and so

Mw + \frac{1}{2w} ≥ −\frac{λ}{g} (δ_{0}c + 1) + \frac{1}{2λg} ≥ 0

for ϵ^{2} ≤ λ^{2} = 2(δ_{0}c + 1). With this choice, u^{ϵ} ≥ w in {0 < d(x) < 2δ_{0}}, and as in §7.3 the estimate (7.47) follows easily.

4. We can now show that u^{ϵ} is concave. For x, y ∈ \overline{U} set z = λx+(1−λ)y, λ ∈ (0, 1) fixed. The concavity function of u^{ϵ}, C^{λ}: U × U → is defined by

C^{λ}(x, y) = u^{ϵ}(z) − λu^{ϵ}(x) − (1 − λ)u^{ϵ}(y)

The fundamental concavity maximum principle for C was established by Korevaar [ ] for a large class of equations a_{ij}(Du)u_{ij} = b(x, u, Du). His result states that for b strictly monotone in u and jointly convex in (x, u), then C^{λ} cannot have a negative interior minimum on U × U. In the case at hand, b = −\frac{1}{2u} fails to satisfy Korevaar’s condition. However, Kennington’s improved concavity maximum principle [ , Theorem 3.1] works under the assumption that −\frac{1}{b} is convex. In our case −\frac{1}{b} = u, so the inf of C^{λ} is not attained on U × U.

To complete the proof we must essentially show that u^{ϵ} is concave near Γ_{0}. Since u^{ϵ} = \sqrt{v^{ϵ}} satisfies (7.38) (7.39), it is straightforward to see that v^{ϵ} ∈ C^{2+α}(\overline{U}) and Dv^{ϵ} · ν ≥ a > 0 for ν the interior normal to Γ_{0}. Using the strict convexity of U it is easy to check that

u_{ℓℓ}^{ϵ} = \frac{1}{2\sqrt{v^{ϵ}}} v_{ℓℓ}^{ϵ} − \frac{(v_{ℓ}^{ϵ})^{2}}{4(v^{ϵ})^{3/2}}

is strictly negative near Γ_{0}. It follows easily that C^{λ} ≥ 0 on U × U (for complete details, see Lemma 2.4 of [ ] or the proof of Theorem 3.1 of [CS]). This completes the proof that u^{ϵ} is concave.

5. Since u^{ϵ} is concave, it follows that |Du^{ϵ}| ≠ 0 on each levelset of u^{ϵ} below the maximum of u^{ϵ}. It follows that all these levelsets are smooth convex hypersurfaces.

We claim that these levelsets have uniformly bounded principal curvatures. To see this, it suffices because of the convexity of these levelsets to know that the mean curvature H with respect to the inward normal is uniformly bounded. But

−H|Du^{ϵ}| = (δ_{ij}− \frac{u_{x_{i}}^{ϵ}u_{x_{j}}^{ϵ}}{|Du^{ϵ}|^{2}} ) u_{x_{i}x_{j}}^{ϵ} = −\frac{1}{2u^{ϵ}} −u_{x_{i}}^{ϵ}u_{x_{j}}^{ϵ}u_{x_{i}x_{j}}^{ϵ}(\frac{1}{|Du^{ϵ}|^{2}} − \frac{1}{|Du^{ϵ}|^{2} + ϵ^{2}} ).

Since u^{ϵ} is concave we conclude that

0 ≤ H ≤ \frac{1}{2u^{ϵ}|Du^{ϵ}|}

and therefore H is uniformly bounded on each of the levelsets below the maximum of u^{ϵ}.

6. We complete the proof of Theorem 7.6 by showing that v^{ϵ} → v uniformly on \overline{U}, where v is the unique solution of (7.36) constructed in Theorem 7.4.

Since u^{ϵ} satisfies (7.38), v^{ϵ} satisfies

|v^{ϵ}(x) − v^{ϵ}(y)| ≤ 4R C|x − y|^{1/2} x, y ∈ U.

Hence, we may choose a sequence ϵ_{k} → 0 with

v^{ϵ_{k}} → v uniformly on \overline{U}.

We assert that v is a weak solution of (7.36). As before, it suffices to consider ϕ ∈ C^{∞}( ^{n}) with v − ϕ having a strict local maximum at a point x_{0} ∈ U. As v^{ϵ_{k}} → v uniformly near x_{0}, v^{ϵ_{k}} − ϕ has a local maximum at a point x_{k} with x_{k} → x_{0} as k → ∞.

Since v^{ϵ_{k}} and ϕ are smooth, we have

Dv^{ϵ_{k}} = Dϕ, D^{2}(v^{ϵ_{k}} − ϕ) ≤ 0 at x_{k}.

Thus (7.39) implies

− (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2} + 4ϵ_{k}^{2}v^{ϵ_{k}}} ) ϕ_{x_{i}x_{j}} ≤ −\frac{ϵ_{k}^{2}}{2} \frac{|Dϕ|^{2}}{|Dϕ|^{2} + 4ϵ_{k}^{2}v^{ϵ_{k}}} + 1 (7.48)

at x_{k}. Suppose first Dϕ(x_{0}) ≠ 0. Then Dϕ(x_{k}) ≠ 0 for large k. Consequently we may pass to the limit in (7.48) (since 0 ≤ v^{ϵ_{k}} ≤ 4R^{2}) to deduce,

− (δ_{ij} − \frac{ϕ_{x_{i}}ϕ_{x_{j}}}{|Dϕ|^{2}} ) ϕ_{x_{i}x_{j}} ≤ +1 at x_{0}.

Next, assume instead Dϕ(x_{0}) = 0 and set

η^{k} ≡ \frac{Dϕ(x_{k})}{(|Dϕ(x_{k})|^{2} + 4ϵ_{k}^{2}v^{ϵ_{k}})^{1/2}}

so that (7.48) becomes

− (δ_{ij} − η_{i}^{k}η_{j}^{k}) ϕ_{x_{i}x_{j}} ≤ − \frac{ϵ_{k}^{2}}{2} \frac{|Dϕ|^{2}}{|Dϕ|^{2} + 4ϵ_{k}^{2}v_{k}^{ϵ}} + 1 at x_{k}. (7.49)

Since |η^{k}| ≤ 1 we may pass to a subsequence and reindex if necessary and assume

η_{k} → η in ^{n}

for some |η| ≤ 1. Sending k to infinity in (7.49) we discover

− (δ_{ij} − η_{i}η_{j}) ϕ_{x_{i}x_{j}} ≤ 1 at x_{0}. (7.50)

Consequently v is a weak subsolution. Similarly, we find that v is a weak supersolution and the proof of Theorem 7.6 is complete. □

8 Examples, Pathologies, Conjecture

In this concluding section, we note various odd behavior allowed by our generalized mean curvature flow

Γ_{0} ⟼ M(t)Γ_{0} = Γ_{t} (t ≥ 0)

8.1 Instantaneous extinction

Suppose Σ_{0} is the smooth, connected boundary of a bounded open subset U_{0} ⊂ ^{n}, and let Γ_{0} be a compact subset of Σ_{0}. If Γ_{0} = Σ_{0}, then we know from § 6.1 that at least for small times t > 0 Γ_{t} is the classical evolution via mean curvature.

What happens if Γ_{0} is a proper subset of Σ_{0}?

Theorem 8.1 Assume that Γ_{0} is compact, Γ_{0} ⊆ Σ_{0}, Γ_{0} ≠ Σ_{0}. Then

(8.1)  Γ_{t} = ∅  for each t > 0

If we take Γ_{0} to be, say, Σ_{0} with a small disk removed, we may informally regard (8.1) as asserting Γ_{0} “pops” instantly. In this heuristic interpretation, we may think of Γ_{0} as somehow having so much mean curvature concentrated along its boundary within Σ_{0} that the hole then widens infinitely fast

The proof of Theorem 8.1 will be given later, after the next assertion, which is of independent interest. Assume now that Σ̂_{0} is the smooth connected boundary of a bounded open set Û_{0} ⊂ ^{n} and that Σ̂_{0} ⊂ \overline{U}_{0}, (8.2) where Σ, U are as above. Thus the surface Σ̂_{0} lies within the closed region \overline{U} enveloped by Σ_{0}. Suppose further that Σ̂_{0} ≠ Σ_{0} (8.3) Choose then a time t_{o} > 0 so small that the classical evolutions {Σ_{t}} and {Σ̂_{t}} starting at Σ_{0} and Σ̂_{0}, respectively, exist at least for times 0 ≤ t ≤ t_{0}. Theorem 8.2 We have Σ_{t} ∩ Σ̂_{t} = ∅ for 0 < t ≤ t_{0} (8.4)

We are thus asserting that even if Σ_{0} and Σ̂_{0} coincide except for a very small region:

then for any positive time 0 < t ≤ t_{0} the subsequent evolutions will have completely broken apart:

The point is that the PDE describing evolution by mean curvature is “uniformly parabolic along the surface” and thus admits infinite propagation speed for disturbances.

We will give the proof of Theorem 8.2 (as well as an interesting new proof of the short time existence of classical mean curvature flow) in a separate paper [ ] using the signed distance function d(x, t) that we have introduced in § 6. Suffice it to say here that one can also prove this result by covering Σ_{0} and Σ̂_{0} by overlapping balls small enough so that restricted to each ball ,Σ_{t} and Σ̂_{t} can be written as a graph. Since the mean curvature evolution equation is uniformly parabolic for small t_{0} and Σ̂_{0} ≠ Σ_{0}, in at least one of the balls the surfaces Σ_{t} and Σ̂_{t} must instantly separate. Thus in each ball the surfaces must also separate. Proof of Theorem 8.1. Given Γ_{0} and Σ_{0} as in Theorem 8.1 we may choose a smooth, nearby surface Σ̂_{0} to Σ_{0} satisfying (8.2) (8.3) and

Γ_{0} ⊂ Σ̂_{0}

Owing then to Theorem 7.2 we have

Γ_{t} ⊆ Σ_{t} ∩ Σ̂_{t}

for small t > 0. Assertion (8.1) now follows from (8.4). □

8.2 Development of an interior

The forgoing demonstrates that a “large” initial set Γ_{0} can instantly vanish under the generalized mean curvature flow. An opposite and perhaps more surprising phenomenon is that the set Γ_{t} for t > 0 may develop an interior, even if Γ_{0} had none.

The simplest example occurs if we take Γ_{0} to be the union of the coordinate axes in the plane ^{2}:

(Ignore for the moment that Γ_{0} is not compact and so our theory in § 2.7 in not really applicable.) To discover heuristically at least the subsequent evolution of Γ_{0}, consider instead the simpler figure as drawn.

As for instance in Brakke [ , Figure 3 in the Appendix] we expect this corner to evolve into this shape

for times t > 0. Since Γ_{0} is composed of four rotated copies of this corner, we expect from Theorem 7.2 that Γ_{t} will look like:

This assertion is at variance wtih Brakke [ , Figure 5 in the Appendix]. Our Γ_{t} presumably contains the set

which he draws as one of the (nonunique!) evolutions for Γ_{0}. We conjecture that our Γ_{t} contains all of the evolutions for Γ_{0} allowed by Brakke.

The discussion above can be modified to apply to various compact figures Γ_{0}, to which our theory does apply. We leave it to the reader to provide at least a heuristic proof that the set Γ_{0} ⊂ ^{2} as drawn below will develop an interior.

We conjecture that if Γ_{0} = Σ_{0} is, as above, the boundary of a smooth open set, then Γ_{t} will never have an interior.

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