Giga Y. Surface Evolution Equations: A Level Set Approach
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238 Chapter 5. Set-theoretic approach
By this observation one can compare D and E with E
−
I
for I starting at t
∗
.By
definition of barrier solutions and translation invariance we see
dist(E(t),∂U
ε
(t)) ≥ dist(E(t
∗
),∂U
0ε
),
dist(
U
ε
(t),D
c
(t)) ≥ dist(U
0ε
,D
c
(t
∗
))
for t ∈ I. This together with (ii)
yields
dist(E(t),D
c
(t)) ≥ dist(E(t
∗
),D
c
(t
∗
)) − ε.
Since I is independent of ε, this yields
dist(E(t),D
c
(t)) ≥ dist(E(t
∗
),D
c
(t
∗
))
for t ∈ [t
∗
,t
1
). This contradicts the definition of t
∗
so t
∗
= T ; the proof is now
complete.
Remark 5.5.3. (i) In various situations a local existence of solutions for (5.1.1)
has been proved. For the mean curvature flow equation it has been proved by G.
Huisken (1984). Existence time t
1
has been estimated by
t
1
≥ t
0
+ c
N
/M
2
,
where M is a bound of the second fundamental form and c
N
depends only on
the dimension. Our theorem apparently does not assume parabolicity (f2) but to
establish (LE) we need some parabolicity.
(ii) Theorem 5.5.2 together with Lemma 5.1.9 yields (CP (ii)) for the level set equa-
tions without using comparison results in Chapter 3. Note that our local existence
(LE) does not require uniqueness of smooth solutions. Conceptually speaking, ex-
istence result (LE) implies the comparison principle, which yields the existence of
a continuous solution by Perron’s method.
5.6 Notes and comments
A set-theoretic interpretation of solutions was introduced by H. M. Soner (1993)
for the anisotropic curvature flow equation with no explicit dependence on the spa-
tial variables and the time variable. However, his definition is based on (signed)
distance functions instead of characteristic functions used in Definition 5.1.1. Our
definition seems new although characteristic functions are often used to describe
typical properties of generalized evolutions for example in G. Barles, H. M. Soner
and P. Souganidis (1993). It turns out that our definition agrees with the one by
distance functions as stated in Theorem 5.1.7. For the mean curvature flow equa-
tion K. Ishii and H. M. Soner (1999) called a set satisfying Theorem 5.1.2 (i) a
weak solution. They compared weak subsolutions with set-theoretic subsolutions
based on distance functions. In fact, based on results by H. M. Soner (1993) (§7,
5.6. Notes and comments 239
§14) they stated Corollary 5.1.8 for the mean curvature flow equation. Our Theo-
rem 5.1.2 gives a geometric interpretation of our set-theoretic solutions. Although
Theorem 5.1.2 follows from Theorem 5.1.7 once Corollary 5.1.8 is proved, it is not
stated in the literature as far as the author knows. We give a direct proof without
appealing to distance functions.
Consistency. It is clear that smooth solution Γ ⊂ R
N
× [0,T
0
) is a set-theoretic
solution of (5.1.1) thanks to Corollary 5.1.5. However, it is not very obvious that
Γ is also a level set solution or a generalized interface evolution, since one has
to prove that there is no fattening. The consistency of a smooth solution (with
smooth initial data) with an interface evolution has been proved first by L. C.
Evans and J. Spruck (1991) for the mean curvature flow equation and later by
Y. Giga and S. Goto (1992b) for general equations (5.1.1) satisfying (f1) and (f2)
when F is independent of x and grows linearly in ∇n. The proof can be extended
for all (5.1.1) satisfying (f1) and (f2) without further restrictions. The major task
is to construct sub- and supersolutions of the level set equation whose zero level
set is the evolving surface by using a distance function. We do not present the
detail.
Our characterization of solutions of level set equation (Theorem 5.1.6) is
important to bridge the set-theoretic approach to the level set approach. The
proof of the ‘only if’ part is standard and this implication is often used in the
literature, for example, in G. Barles, H. M. Soner and P. E. Souganidis (1993).
However, the converse seems to be unfamiliar.
Our Theorem 5.1.7 characterizes set-theoretic solutions by distance functions.
The ‘only if’ part has been proved by G. Barles, H. M. Soner and P. E. Souganidis
(1993) by representing distance functions by sup-convolutions. The proof given
here is direct and does not appeal to sup-convolutions. The converse is contained in
Theorem 5.1.6 when F in (5.1.4) is independent of the spatial variable. Otherwise
we need a little bit of extra work as mentioned in the proof.
It is clear that the comparison principle for level set equations implies the
comparison principle for set-theoretic solutions. We here note the converse is also
easy to prove. We have given two kinds of such statements (Lemma 5.1.9 and
5.1.10) for bounded sets and general sets. It seems that there is no literature
containing both lemmas.
Several similar problems for evolution equations for h in Example 5.2.1 were
studied for example by N. Ishimura (1995) (the Gaussian curvature flow equation)
and by Y. Kohsaka (2001) (the curve-shortening-like equation with free bound-
aries).
Nonuniquenss of set-theoretic solutions of V = k with a given initial data
was first observed by L. C. Evans and J. Spruck (1991), where they described this
phenomena as ‘fattening’ of level sets. Example 5.2.1 is taken from H. M. Soner
(1993). Several criteria on initial data for uniqueness, described in §4.5, were given
240 Chapter 5. Set-theoretic approach
in G. Barles, H. M. Soner and P. E. Souganidis (1993), H. M. Soner and P. E.
Souganidis (1993), S. Altschuler, S. Angenent and Y. Giga (1995). The latter two
works show that axisymmetric evolution for the mean curvature flow is regular.
Examples of nonuniqueness were given for various equations by several authors
including Y. Giga, S. Goto and H. Ishii (1992), G. Barles, H. M. Soner and P. E.
Souganidis (1993), G. Bellettini and M. Paolini (1994) and Y. Giga (1995b) even
if initial data has a smooth boundary. For the mean curvature flow equations with
N ≥ 3, examples of nonuniqueness for smooth initial data is also proved by S.
Angenent,T.IlmanenandJ.J.L.Vel´azquez (2002) for 4 ≤ N ≤ 7,N =8and
numerically conjectured by S. B. Angenent, D. L. Chopp and T. Ilmanen (1995).
A survey article by B. White (2002) includes an example of nonuniqueness for
N =3.
The notion of level set solutions is introduced to describe generalized evolu-
tions constructed in Chapter 4 from the set-theoretic point of view.
The notion of barrier solutions was first introduced by T. Ilmanen (1993a)
and by E. De Giorgi (1990), and developed by G. Bellettini and M. Paolini (1995)
and L. Ambrosio and H. M. Soner (1996). These works include the statement that
minimal barrier is equivalent to generalized open evolution (Corollary 5.3.4) for
several equations. (For general geometric equations see the work of G. Bellettini
and M. Novaga (1998)). See also a review article by L. Ambrosio in the lecture
note by L. Ambrosio and N. Dancer (2000). These results depend on comparison
principles. However, the equivalence of a barrier supersolution and a set-theoretic
supersolution seems to be new. We note we did not use the separation lemma
(Lemma 5.5.1) nor comparison principles for the proof of this last statement. This
characterization provides a method to prove the comparison results for level set
equations via local existence of classical solutions of surface evolution equations.
The notion of barrier solutions is also useful to prove convergence of internal
layers of the Allen–Cahn equation to the mean curvature flow. In fact, G. Bar-
les and P. E. Souganidis (1998) introduced a kind of barrier solutions to solve
such problems. In particular, they proved the global-in-time convergence for the
homogeneous Neumann problem which was only known for convex domains (M.
Katsoulakis, G. T. Kossioris and F. Reitich (1995)). This idea is further developed
by G. Barles and F. Da Lio (2003) to handle oblique type boundary conditions.
Approximation. There are several ways to approximate solutions of level set
equations or generalized interface evolution itself. We list several related articles
(which apply the mean curvature flow equation) without detailed explanation.
(i) Finite difference approximations. M. G. Crandall and P. L. Lions (1996)
proves the convergence of their schemes (see a paper by K. Deckelnick (2000) for
further development). The stability of a more intuitive scheme has been proved
in Y.-G. Chen, Y. Giga, T. Hitaka and M. Honma (1994). There is also a finite
element approximation to calculate solutions of level set equations. The reader is
5.6. Notes and comments 241
referred to papers by K. Deckelnik and G. Dzuik (2003a), (2003b) for convergence
of the scheme and A. Handloviˇcov´a and K. Mikula (2002) for related schemes.
(ii) Bence–Merriman–Osher scheme. This is a kind of filter proposed by J.
Bence, B. Merriman and S. Osher (1992). The convergence has been proved by
G. Barles and C. Georgelin (1995) and independently by L. C. Evans (1993). It
is further developed by H. Ishii (1995) F. Cao (1998) and by H. Ishii, G. E. Pires
and P. E. Souganidis (1999). Recently, convergence rate is also studied by K. Ishii
(2004).
(iii) A variational scheme. This is proposed by F. Almgren, J. E. Taylor and
L. Wang (1993) and by I. Fonseca and M. A. Katsoulakis (1995). For further
development see a paper by S. Luckhaus and T. Sturzenhecker (1995).
(iv) Approximation by interacting particle systems. This relates the descrip-
tion of macroscopic model and microscopic models. It has been studied by M.
Katsoulakis and P. E. Souganidis (1994).
The reader is referred to a review by C. M. Elliott (1997) and a book by F. Cao
(2003) for other approximations. Note that the above approximation works for
generalized evolution which is not necessarily smooth.
Regularity. There are several attempts to prove some regularity of generalized
interface evolutions for the mean curvature flow equation. For example, L. C.
Evans and J. Spruck (1992b) proved that the evolution fulfills the clearing-out
lemma which is important for the Brakke flow. As an application they estimate
the extinction time from above by using the area of initial data. An estimate from
below is given by Y. Giga and K. Yama-uchi (1993). They also prove that if a part
of interface evolution is given as the graph of a continuous function, the part of
evolution is represented as a graph-like solution of the equation. Such a solution
has been proved by them to be regular. The proof of this reduction is simplified
by M.-H. Giga and Y. Giga (2001).
Furthermore, L. C. Evans and J. Spruck (1995) prove that almost every level
set of solutions of the level set mean curvature flow equation is a Brakke type
flow. This is generalized by Y. Tonegawa (2000) for anisotropic curvature flow
equations. For monotone evolutions the size of the singular sets of generalized
interface evolution Γ is well-estimated. In fact, B. White (2000) proved that the
singular set of Γ ⊂ R
N
× [0, ∞) has the parabolic Hausdorff dimension at most
N − 2. This is optimal since a torus shrinking to a ring has N − 2 dimensional
singularity.
Nonlocal problems. In this book we restrict ourselves to interface equations
(1.5.1) whose normal velocity is determined by infinitesimal quantities of an evolv-
ing hypersurface, except the notion by crystalline curvature ; see a paper by M.-H.
Giga and Y. Giga (2001). We list here several other motions whose normal velocity
is determined by nonlocal quantities of an evolving hypersurface. If the problem
has some order preserving structure, there is a chance to establish a level set
242 Chapter 5. Set-theoretic approach
method for that problem. The examples include works of D. Adalsteinsson, L. C.
Evans and H. Ishii (1997), B. Andrews and M. Feldman (2002), P. Cardaliaguet
and E. Rouy (2004), P. Cardaliaguet (2000), (2001), P. Cardaliaguet and O. Ley
(2004), F. Da Lio, C. I. Kim and D. Slepˇcev (2004), C. I. Kim (2003), (2004), H.
Ishii and T. Mikami (2004a). The scope of problems include etching and deposition
problems, the Hele–Shaw problem as well as some shape optimization problems.
The barrier approach is somewhat useful, especially when the usual comparison
principle is difficult to establish directly. See for example a paper by M.-H. Giga
and Y. Giga (1998b) and a recent paper by P. Cardaliaguet and O. Ley (2004).
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