MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE:
A SURVEYYOSHIKAZU GIGA AND NORBERT POŽÁR
Abstract. We consider a class of anisotropic curvature flows called
crystalline curvature flows. We present a survey on this class of flows
with special emphasis on the well-posedness of its initial value problem.

Contents
1. Introduction
2. Some models
3. Polygonal flow
4. Explicit solutions
5. Approach by the theory of maximal monotone operators
5.1. Abstract theory
5.2. Calibrability and Cheeger sets
5.3. Curvature-like quantity
5.4. Comparison and approximation
6. Approach by the theory of viscosity solutions
6.1. Definition of viscosity solutions
6.2. Comparison principle
6.3. Existence of solutions
6.4. Convergence of various approximations
7. Approach by distance functions
8. Some numerics
9. Volume-preserving and fourth-order problems
9.1. Volume preserving flow
9.2. Fourth-order problem
Appendix A. A direct proof of properties of Ψ
Acknowledgment

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Date: June 2, 2022 (git: 2022-06-01, 31a5450).
2010 Mathematics Subject Classification. Primary 35K93; Secondary 35B51.
Key words and phrases. crystalline mean curvature, comparison principle, total
variation.
YG is supported by JSPS through grants No. 19H00639 (Kiban A), No. 18H05323(Kaitaku), No. 17H01091. NP is supported by JSPS KAKENHI Wakate Grant (No.
18K13440).
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Y. GIGA AND N. POŽÁR

References

67

1. Introduction
The famous mean curvature flow was introduced by W. W. Mullins [136]to model the motion of an antiphase grain boundary in annealing metals.
Its governing equation is called the mean curvature equation and it is anequation for a one-parameter family of hypersurfaces {Γt } (an evolving hypersurface) in Rn which imposes that the normal velocity V equals the mean
curvature κ, i.e.,V = κ onΓt ;here, the curvature and the velocity is taken in the direction of the normal
vector field ν of Γt . This equation can be interpreted as a steepest descentflow of the surface area. In materials science the surface area is considered
as an interfacial energy of the grain boundary. It is quite natural to consider
anisotropic effects. For this purpose, one considers the anisotropic interfacial
energyZ
I(Γ) = σ(ν) dHn−1 ,Γwhere σ is a given positive function called the interfacial energy density;
here, dHn−1 is the surface area element of a hypersurface Γ. Its first variation is called the anisotropic mean curvature denoted by κσ ; this is often
called the weighted mean curvature. If one replaces the mean curvature by
the anisotropic mean curvature in the mean curvature flow equation, theresulting equation is of the form
(1.1)

V = κσonΓt .

In general, this equation may not be parabolic even if σ is smooth. We
consider the one-homogeneous extension of σ in Rn and still denote it by σ,
i.e.,
(1.2)

σ(p) = |p|σ(p/|p|),p ∈ Rn \ {0}.

If σ is convex, the equation (1.1) is at least degenerate parabolic. Although
the problem when σ is not convex is interesting, we do not touch this problem in this paper. The reader is referred to [22] for such an ill-posed problem.
The anisotropic mean curvature flow can be considered as the mean curvature flow in a Minkowski metric or a Finsler metric. In this case, V should
be replaced by the Minkowski normal velocity. If one uses the Euclidean

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

3

normal velocity, it is of the form
V = σκσ ;see [30] for this perspective.
The curvature flow is not restricted to the form (1.1). For second-ordermodels, a general form of the flow is(1.3)

V = g(ν, κσ )with g non-decreasing in the second variable. A typical example in thermodynamics is
V = M (ν)(κσ + C)
with mobility M (ν) > 0 and a driving force C, where C is a constant[99], [10]. There are several other examples when g is nonlinear in κσ . For
example,V = |κσ |α−1 κσwith some positive number α. We shall discuss these examples in Section 2.
For later convenience, we say that σ : Rn → [0, ∞) is an anisotropy if σ
is positively one-homogeneous, convex and σ > 0 outside the origin. Notethat we do not assume that σ is even. By definition, σ satisfies (1.2) andthe Frank diagram
Fσ = {p ∈ Rn | σ(p) ≤ 1}
is bounded, convex and contains the origin as an interior point.
For many applications, especially in low temperature physics, it is often
considered the case that σ is not C 1 . An extreme case is that the anisotropy
σ is (purely) crystalline, i.e., σ is piecewise linear so that Fσ is a convex polytope. A crystalline mean curvature flow is formally (1.3) when anisotropy
σ is crystalline. In mathematical community, it was introduced by J. E.
Taylor [156] and independently by S. B. Angenent and M. E. Gurtin [10]around 1990.
One might be curious on the value of κσ when σ is crystalline. To motivate it we consider an anisotropic isoperimetric problem of the form
“Find a shape D in Rn with fixed volume which minimizes the surface energy I(Γ) with Γ = ∂D.”
This problem was first studied by Wulff [167] and it turns out that theminimizer is the Wulff shape\Wσ ={x ∈ Rn | x · m ≤ σ(m)} ,|m|=1

which is the polar of Fσ . This has been proved in quite general setting; seee.g. [155], [62]. For recent developments related to optimal transport theory,

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Y. GIGA AND N. POŽÁR

see [61]. Note that if σ is crystalline so that Fσ is a polytope, then Wσ is alsoa polytope. For a smooth anisotropy, one observes that the anisotropic κσ on
the surface of Wσ is a non-zero constant, and so Wσ plays the same role as aball for the usual curvature. More precisely, if one takes ν inward, κσ = n−1.
If σ is crystalline, then Wσ is a polytope. Nevertheless, κσ should not be
zero. This simple observation shows that the value κσ cannot be determinedby infinitesimal quantities like tangent and second fundamental form of thesurface. We say that (1.3) is a crystalline (mean) curvature flow (equation)if σ is crystalline.
We now consider a simple formal example of a crystalline curvature flow
for a graph-like curve. For later convenience, we write the equation (1.1)
when Γt is given as the graph of a function w = w(x′ , t), i.e., xn = w(x′ , t)
for x = (x′ , xn ) ∈ Rn , x′ ∈ Rn−1 . The upward normal velocity is given aswt,
V =(1 + |∇′ w|2 )1/2
where wt = ∂w/∂t, ∇′ w = (∂x1 w, . . . , ∂xn−1 w), ∂xj = ∂/∂xj , wxj = ∂xj w.
The anisotropic mean curvature is formally of the form
κσ = − divΓt ζ(ν) with

ζ(ν) = (∇p σ)(ν),

where ∇p σ denotes the gradient of σ, i.e., ∇p σ = (∂p1 σ, . . . , ∂pn σ) for
anisotropy σ = σ(p1 , . . . , pn ). The divergence divΓt denotes the surface divergence, i.e.,
divΓt X = trace (I − ν ⊗ ν)∇X ;here, we extend X in a tubular neighborhood of Γt in a suitable way and∇X denotes its Jacobi matrix. This value is independent of the extension;see e.g. [83]. In our setting,


n−1
X
∂
∂σ
(ν)
divΓt ζ(ν) =∂xℓ ∂pℓℓ=1where ν = (−∇′ w, 1)/ (1 + |∇′ w|2 )1/2. Indeed,

n
X

n
X

∂
νi νj
trace(ν ⊗ ν∇ζ) =(∂pj σ)(ν) =νi νj (∂pj ∂pℓ σ)(ν)∂xj ν ℓ = 0
∂xi
i,j=1
i,j,ℓ=1
Pnsince j=1 νj ∂pj ((∂pℓ σ)(ν)) = 0 by positively zero-homogeneity1 of ∂pℓ σ.
Moreover, since ∂pℓ σ(ν) is independent of xn , we have the desired identity.
1Let s be a real number. A function f allowing values ±∞ defined in a vector space V
is called positively s-homogeneous if f (λv) = λs f (v) holds for all λ > 0 and v ∈ V . The
function f is allowed to take the value +∞ especially when V is an infinite dimensional
space. Indeed, in Section 5.1, we consider a total variation type energy in L2 space, which
is an example of positively one-homogeneous but take the value +∞ somewhere.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

p2
1

−1

5

x2
1

1

p1

−1

−1
Fσ

1

x1

−1
Wσ

Figure 1. The Frank diagram and the Wulff shape forσ(p) = |p1 | + |p2 |.
If Γt is a graph-curve in R2 , then− divΓt ζ = −∂x1

∂σ(−wx , 1) ,∂p1

since ∇p σ is positively zero-homogeneous.
We now observe that (1.1) is formally of the form


∂σ
wt
(−wx1 , 1) .
(1.4)
= −∂x1(1 + wx21 )1/2
∂p1
If σ(p) = |p|, then
∂σ
p1
(p) =
∂p1
|p|

so that

wx1∂σ(−wx1 , 1) = −,∂p1(1 + wx21 )1/2

which yields a curve-shortening equation for a graph-like curve Γt : x2 =w(x1 , t), i.e.,


wx1
wt
wx1 x1
.
=
∂
or
w
=
x
t
1
(1 + wx21 )1/2
(1 + wx21 )1/2
1 + wx21
We are interested in the case when σ is crystalline. Let us consider
σ(p) = |p1 | + |p2 |so that the Frank diagram Fσ is a square whose vertices are (±1, 0) and(0, ±1); see Figure 1 for Fσ and the corresponding Wulff shape Wσ . The∂σderivative ∂pmust be interpreted in a suitable generalized sense, for ex1
ample, as the subdifferential of σ; see (1.6) below. Then (1.4) becomes
wt = (1 + wx21 )1/2 ∂x1 (sgn wx1 ). Since it is expected that ∂x1 (sgn wx1 ) = 0
away from the set {wx1 = 0}, it is formally equivalent to(1.5)

wt = ∂x1 (sgn wx1 ),where sgn p1 = p1 /|p1 |. This equation is a total variation flow equation in
one-dimensional setting. If one calculates the right-hand side formally, then

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Y. GIGA AND N. POŽÁR

(1.5) iswt = 2δ(wx1 )wx1 x1 ,where δ denotes Dirac’s delta. This shows∂x1 (sgn wx1 ) = (1 + wx21 )1/2 ∂x1 (sgn wx1 ).
However, the quantity δ(wx1 ) is undefined because it is a pull-back of thedelta measure although it suggests the diffusion coeﬀicient equals zero if
wx1 is not equal to zero. In other words, the place where wx1 is not zero
does not move. To see the speed where wx1 is zero, let us consider a special(Lipschitz) profile x2 = w0 (x1 ) which takes the minimum value on [a, b] andw0x1 > 0 (resp. w0x1 < 0) in x1 > b (resp. x1 < a), where a < b (Figure 2).
We try to move this function by (1.5). Since it is natural to assume that
x2
a

b

x1
w0

Figure 2. The graph of w0 .
the speed equals zero outside [a, b], the important thing is to calculate thespeed on [a, b]. Here we put the ansatz:“The speed wt on [a, b] is spatially constant.”
In other words, a flat part (called facet) stays as a facet and no bending nor
facet splitting occurs.
We integrate (1.5) in a neighborhood of [a, b], i.e., (a − ε, b + ε) withsmall ε > 0 and obtain at t = 0
Z b+ε
Z b+ε
wt dx =∂x1 (sgn w0x1 )dx = sgn w0x1 (b + ε) − sgn w0x1 (a − ε)
a+ε

a+ε

= 1 − (−1) = 2.
By our ansatz, the left-hand side is of the formwt (b − a)as ε → 0. Thus, we obtainwt = 2/(b − a).
The right-hand side is a nonlocal quantity and this is a one-dimensionalversion of the Cheeger ratio Hn−1 (∂Ω)/Ln (Ω) defined for a domain Ω in Rn ,where Ln (Ω) denotes the Lebesgue measure of Ω while Hn−1 (∂Ω) denotes

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

7

the (n−1)-dimensional Hausdorff measure of the boundary ∂Ω of Ω. We now
observe that the crystalline curvature should be determined by a semilocal
quantity like Cheeger ratio if one assumes the ansatz.
In a one-dimensional setting, this ansatz is justified in the sense that sucha profile is approximated by a solution of a uniformly parabolic equation
which approximates the equation (1.5). For example, order-preserving property called comparison principle is expected to hold. However, in a higher
dimensional setting, as we see later, this ansatz is no longer appropriate.
For example, this ansatz violates the comparison principle.
For curve evolutions, using this ansatz J. E. Taylor [156] and independently S. B. Angenent and M. E. Gurtin [10] introduced a special classof polygonal curves called admissible. We say that an oriented polygon isadmissible if the orientation (normal ν) of each facet (edge) is one of thatin ∂Wσ and the orientation of adjacent facets should be adjacent in ∂Wσ .
Here Wσ is the Wulff shape associated with anisotropy σ and it is a convex
polygon if σ is crystalline. If the second condition (called adjacency condition) is not required, one expects that new facets may be created becauseof a strong curvature effect. We shall discuss this point in Section 3. Let{Γt } be a smooth family of admissible polygons. In other words, vertices of
Γt are assumed to move C 1 in time t. The motion of vertices is completely
determined by the crystalline flow equation (1.3). Here, κσ of each facetwith normal ν is assumed to be equal to χ∆/L, where L is the length of the
facet and ∆ is the length of the facet of Wσ with normal ν; χ takes valuesin {−1, 0, +1} depending upon convexity near the facet. Since L depends
upon vertices, combining these equations, a system of ordinary differential
equations (ODEs) for vertices or lengths is obtained. Its initial value problem is uniquely solvable at least when g is (locally) Lipschitz continuous.
For later convenience, we say that {Γt } is a crystalline flow if Γt is a smoothfamily of admissible polygons satisfying the system of these ODEs. However, there is a chance that in finite time a facet disappears. Fortunately,
in many cases at the time when a facet disappears, Γt is still admissible so
one is able to continue to solve the system of ODEs with fewer facets. Thisapproach is simple and it is easy to compute the crystalline flow [156], [160],[157]. Moreover, it satisfies the desired property like comparison principle
which says that if one admissible polygon encloses another, then the corresponding crystalline flow starting from these polygons keeps this order; see[160], [85].

8

Y. GIGA AND N. POŽÁR

There is another approach based on the theory of maximal monotone
operators initiated by Y. Kōmura [123] and developed by H. Brezis [31] andothers in late 1960s and 1970s. A basic theory asserts the unique global-intime solvability of the initial value problem for the gradient flow equation
whose “energy” E is a convex, lower semicontinuous functional in a Hilbert
space H equipped with an inner product ⟨·, ·⟩ so that ∥f ∥2H = ⟨f, f ⟩. Moreprecisely, it is a solvability for the system wt ∈ −∂E(w) where ∂E(w) is thesubdifferential of E at w, which is an extended notion of a differential of E.
It is defined as(1.6)∂E(w) = {f ∈ H | E(w + h) − E(w) ≥ ⟨f, h⟩ for all h ∈ H} .

Note that E may not be differentiable so that ∂E(w) may not be a singleton.
However, the solution is unique and it “knows” how to grow even though theevolution law looks ambiguous. Actually, the solution is right differentiablein time and its speed equals to the minimal section (canonical restriction)∂ 0 E(w) of ∂E(w), i.e.,∂ 0 E(w) = argmin ∥f ∥H f ∈ ∂E(w) ,which is uniquely determined. In [63], it is shown that if {Γt } is given as thegraph of a periodic function of one variable, then the equation V = M (ν)κσ
can be written as a gradient flow system. Moreover, the speed given by thegeneral theory is the same as the one given in the ansatz on a facet. Thissuggests the approach by [156], [10] is quite natural. In fact, it is shownin [63] that the crystalline flow is obtained as a limit of approximate solutions solving a usual uniformly parabolic problem approximating the original problem. This justifies the ansatz for curve evolution. The proof is basedon a general convergence theory for gradient flow systems developed by [33]and [165]. To apply the theory, it suﬀices to prove that the approximatingenergy E ε converges to E in the sense of Mosco, i.e., it satisfies(i) sequential lower semicontinuity under weak topology:(1.7a)

E(w) ≤ lim E ε (wε ) for wε ⇀ w (as ε → 0);ε↓0(ii) existence of a strong recovery sequence: for any v ∈ H, there isvε → v as ε → 0 such that
(1.7b)

E(v) = lim E ε (vε ).
ε↓0

The nonlocal property of the speed related to a total-variation-type singularenergy was also observed in [102].

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

9

If the flow equation is written as a gradient flow of a convex, lower
semicontinuous functional in a Hilbert space, one is able to calculate thespeed by calculating the minimal section. It is a kind of obstacle problem
as we will see later. Reflecting this idea, G. Bellettini, M. Novaga and M.
Paolini [25] gave an example that the speed of a facet may not be a constanton a facet. In other words, the quantity κσ may not be a constant on a facet
since otherwise it would contradict a comparison principle. Later, they gave
a characterization of non-constancy of κσ on a facet depending on shape. To
illustrate the problem, let us consider a closely related problem: the total
variation flow equation(1.8)

wt = div (∇w/|∇w|)

on an n-dimensional torus Tn = Πni=1 (R/ωi Z), ωi > 0 (i = 1, . . . , n). Except
Section 9, we shall assume ωi = 1 for simplicity. It can be interpreted as a
gradient flow of the total variation energy

Z
Z
1
n
n
w div z dx |z(x)| ≤ 1, z ∈ C (T , R )E[w] =|∇w| := supTn

Tn

for an L2 function w. We set the energy E in the Hilbert space H = L2 (Tn )such that E = E. Then, it is not diﬀicult to see that E is convex and lower
semicontinuous in H = L2 (Tn ). The problem (1.8) should be interpreted as
wt ∈ −∂E(w)and there is a unique solution starting from w0 ∈ H = L2 (Tn ). The speed isgiven as the minimal section and we are interested in the value. We restrict
ourselves to a facet where w is “convex” in its neighborhood. We fix t > 0
and let w take its minimum on a facet, i.e.,n
F = x ∈ T w(x, t) = minn w(y, t) .
y∈TAssume that the boundary of F is smooth. Then it turns out that−∂ 0 E(w) F = div z,Z| div ζ| dx ζ · νF = 1 on ∂F, |ζ| ≤ 1 in F .
2

z = argmin
F

Here νF is the exterior unit normal of F . This is a convex minimization problem but it is of obstacle type because of the constraint |ζ| ≤ 1. Although theminimizer is not unique, div z is uniquely determined. The characterizationof the minimal section is nontrivial but it can be done for the total variation
flow equation. For a detailed explanation, the reader is referred to a very
nice book by F. Andreu-Vaillo, V. Caselles and J. M. Mazón [7]. If div z is

10

Y. GIGA AND N. POŽÁR

constant, we say that F is calibrable. There are several necessary and sufficient conditions; see e.g. [26] for the curvature flow. We shall discuss this
topic in Section 5. If it is calibrable, then div z must be the Cheeger ratio,i.e., div z = Hn−1 (∂F )/Ln (F ). Indeed, integration by parts yields
ZZn(div z)L (F ) =div z dx =z · νF dHn−1 = Hn−1 (∂F ).
F∂F

In general, div z ∈ L∞ ∩BV but may be discontinuous as shown in [27], [28].
Since there may exist non-calibrable facets, it took a long time to constructa solution in a general setting. G. Bellettini and M. Novaga [24] introduced
a notion of a solution based on distance function reflecting the variational
structure and proved its uniqueness. However, its existence is only provedfor convex initial data [18]. It is quite recent that the well-posedness problem
is settled by two groups through the level-set method, which is the main
topic of this survey.
Although there are several approaches to solve the problems by now, they
are roughly classified into three main ones. The first approach is to consider
a special class of evolving polygons by reducing the problem to a system of
ODEs we discussed before. This approach is valid only for curve evolution.
The second approach is a variational approach. A simple way is to apply
the theory of maximal monotone operators which is restricted for the graphcase but it has an advantage to apply to a higher order crystalline flow,for example crystalline surface diffusion equation for a graph-like surface.
The reader is referred to [74] for this topic as well as Section 9.2. A variantof this variational approach involving a distance function yields a global
well-posedness for convex sets as mentioned before [18].
The third approach is a viscosity approach. This is based on the theory ofviscosity solutions, which was originally introduced to characterize the valuefunction of a control problem as a solution of a Hamilton-Jacobi equations;see [47]. The notion of a viscosity solution is based on a comparison principle
for the second-order elliptic or parabolic equations which can be degenerate.
It does not depend on a variational structure. However, since the crystalline
flow is non-local, one needs to adjust the theory. This is not trivial evenfor an evolution of a curve. In the case of graph-like curves, i.e., Γt is given
as a graph w = w(x1 , t), the notion of a viscosity solution was adjusted
for general crystalline flow when w is periodic in x1 [68], [70]. It can beapproximated by a smoother problem as proved in [71].
This viscosity approach was later extended to a closed curve by adjusting the level-set method [73], [72]. The original level-set method based on

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

11

Figure 3. Example of a fattening in the crystalline flow for
σ(p) = |p1 | + |p2 | with initial curve Γ0 given by the figure-8shaped solid line. The set Γt immediately (t > 0) fattens. The
gray area denotes Γt at t = 0.8t∗ where t∗ is the extinction
time of the individual squares.
the theory of viscosity solution was introduced by [60], [45] for the mean
curvature flow equations. The idea of the original level-set method for themean curvature flow V = κ is to consider its level set flow equation


∇u
=0
ut − |∇u| div|∇u|which requires that each level set moves by V = κ. For a given initial
hypersurface Γ0 , one constructs a continuous function u0 such that Γ0 is
the zero level set of u0 and solves the level-set flow equation globally-intime and sets Γt as the zero level set of the solution. A unique solvability
is guaranteed by the theory of viscosity solutions. Moreover, Γt is uniquelydetermined by Γ0 . However, as already pointed out in [60], Γt may haveinterior even if Γ0 has no interior, Figure 3; see also [83]. This phenomenon
is called fattening and from the point of an evolution of hypersurfaces this isconsidered a non-uniqueness phenomenon. A basic merit of this approach is
to handle a topological change. The generalized solution Γt of course agrees
with a smooth solution if the latter exists though the proof is less trivial[60], [84]. For a general theory of the level-set method for smooth anisotropy,see [45] or a book [83]. The level-set method itself was introduced by [145]for numerical study and independently by [143] to explain a scaling law of
V = κ. For the development of the numerical approach, see [150] and [144].
It took quite a long time to extend this theory to evolution of a hypersurface mainly because the crystalline curvature κσ may not be a constant ona facet. A first breakthrough is done by [81], where the viscosity theory was
extended to a total-variation-flow-like equation; see also [80]. Later it was
extended to level-set flow equations, [91], [92], and to the case when there is aspatially inhomogeneous driving force term [93]. In the meanwhile, another12

Y. GIGA AND N. POŽÁR

approach to construct a level-set flow based on distance functions whichgoes back to [151] was developed independently. In fact, A. Chambolle, M.
Morini and M. Ponsiglione [42] constructed a level-set flow for V = σκσ forvery general σ containing crystalline σ as a special case. With M. Novaga
they even extended their approach in [40] for more general equations withmobility and spatially inhomogeneous driving force term. In both theories,
the theory of maximal monotone operators is reflected in some sense. In thepurely viscosity approach by [91], [92], [93], the value κσ is defined as theminimal section of the crystalline interfacial energy. In the approach by [42],[40], the distance function from the zero level-set of a solution is interpreted
as a supersolution of the original gradient flow of the form ut ∈ −∂E(u),where E is an anisotropic total variation energy with density σ.
We warn the reader that the value κσ is not determined completely by
the facet F if the problem is spatially inhomogeneous as pointed out by [93].
If there is a non-constant driving force C = C(x), then κσ +C is not just thesum of the two quantities. See [93] for more details and futher references.
We do not intend to cover all topics related to well-posedness for a crystalline flow. Several interesting topics like a crystalline multi-phase curvature flow are missing in this paper. For a multi-phase crystalline flow, see[21].
This paper is organized as follows. In Section 2, we give several model
equations for curvature flow equations involving a crystalline curvature. In
Section 3, an evolution of a polygon is discussed. In Section 4, some explicit
solutions such as self-similar solutions are discussed. In Section 5, we give
an approach by the theory of maximal monotone operators. In Section 6,
we give an approach based on viscosity solutions. In Section 7, we give anapproach based on distance functions. In Section 8, some numerics are given.
In Section 9, examples of a fourth-order problem and a volume-preserving
flow are discussed.
2. Some models
We begin with second-order models in materials sciences. There is anaxiomatic derivation of evolution laws of phase-interfaces involving bulk
energy and surface energy with constitutive relation compatible with thermodynamical laws in [10], [99]. Its explicit form isb(ν, V )V = κσ − fwithb(ν, V ) ≥ 0

where f is a driving force term coming from bulk interface difference whichis assumed to be a constant in [10], [99]. The function b is called a kinetic

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

13

coeﬀicient. If b(ν, V ) is independent of V and positive, then it is reduced to
V = M (ν)(κσ + C)
with C = −f , M (ν) = b(ν)−1 . If b(ν, V ) is taken so thatb(ν, V )V = log(1 + V )with f = 0, this is nothing but the model of thermal grooving of a surface
due to evaporation-condensation proposed by W. W. Mullins [137]. Here isa way of derivation. The Gibbs-Thomson law readslog(p/p0 ) = β(−κσ )with a positive constant β > 0. Here p is the pressure and p0 is the atmospheric pressure. The evolution law is
V = M (ν)(p0 − p).
If M (ν) ≡ 1, p0 = 1, then one gets
(2.1)

V = 1 − exp(−βκσ ).

If the right-hand side is linearized around κσ = 0, we get V = βκσ . See
the discussion by N. Hamamuki [101]. As we will see later in this section,a model similar to V = κσ was introduced by H. Spohn [152] when σ is akind of crystalline anisotropy to model evaporation-condensation below theroughening temperature.
Another source of equations stems from image processing. An axiomatic
derivation is provided by [4]. For curve evolution, equation(2.2)

V = |κσ |α−1 , κσα>0is important especially with α = 1/3, where the evolution law is invariant
under aﬀine transform (not only under rotation, dilation and translation)
when σ is isotropic. In higher dimensional case, the corresponding equation
should be V = K 1/(n+1) where K is the Gauss curvature not the mean curvature. A crystalline Gaussian curvature flow V = Kσ has been studied toapproximate the Gaussian curvature flow; see e.g. [163]. However, we do nottouch this topic in this paper. There are many examples of curvature flows(see e.g. [83, Chapter 1]). In the case that the mean curvature is involved
like the inverse mean curvature flow equation, it is easy to generalize
V = −1/κσ .
If σ is isotropic, then the equation was used to prove the positive mass
conjecture [105] since the Geroch mass is monotone under this flow.

14

Y. GIGA AND N. POŽÁR

We note that the total variation flow
wt = div′ (∇′ w/|∇′ w|)
can be understood as a particular case of V = M (ν)κσ as discussed in the
introduction for evolution of graph-like curves. If an evolving surface Γt isgiven as the graph of w = w(x′ , t), x′ ∈ Rn−1 , the total variation flow for w
can be written as
V = M (ν)κσ
with
σ(p) = |p′ | + |pn | withM (ν) = νnwithp = (p′ , pn ),ν = (ν ′ , νn ).

Here ν ′ = −∇′ w/ (1 + |∇′ w|2 ) and νn = 1/ (1 + |∇′ w|2 ) .
The model proposed by H. Spohn [152] is almost the same. Here w
denotes the height of the crystal surface at x′ and at time t. It is of the form
1/2

1/2

wt = div′ (∇′ w/|∇′ w|) + β div′ (|∇′ w|∇′ w) ,where β > 0 is a constant. If one writes it in the form of a surface evolution,
it is
V = M (ν)κσ
with σ(p) = |p′ | + β|p′ |3 / (3|pn |2 ) + |pn |.
There are several fourth-order models. For relaxation of crystal surface,a fourth-order total variation type equation is proposed by [152]. Its explicit
form iswt = −∆′ (div (∇′ w/|∇′ w|) + β div (|∇′ w|∇′ w)) ,where ∆′ denotes the Laplacian in x′ variable, i.e., ∆′ = div′ grad′ = ∇′ · ∇′ .
This equation is derived as a continuum limit of models describing motion ofsteps on crystal surface as discussed in [142], where a numerical simulation
is given. This model describing step-motion is microscopic in the directionof height but macroscopic in the horizontal direction. We refer the reader toa nice review article by R. V. Kohn [120] on this issue. Of course, if β = 0,
this is nothing but the fourth-order total variation flow. This is popularfor image processing. For example, Osher-Solé-Vese [146] model gives the
fourth-order total variation flow of the form
ut = −∆ div (∇u/|∇u|) + λ(f − u)for λ > 0, and given f . See also [56] for such a flow, where the well-posednessof the equation is proved by using the Galerkin method. For relaxation phenomena, W. W. Mullins [137] introduced a surface diffusion flow equation;

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

15

see also [35] for derivation. It is of the form
V = − divΓt j,

j = − gradΓt ρkµ, µ = κσ ,Twhere T is a given temperature and ρ0 is an equilibrium density; k is apositive constant. The quantity j is the mass flux and µ is the chemical
potential. The resulting equation islog(ρ/ρ0 ) =(2.3)

V = ∆Γt exp(−kκσ /T ),∆Γ = divΓ gradΓ ;here, ∆Γ denotes the Laplace-Beltrami operator on the surface Γ. We shallset k = 1, T = 1 for simplicity of presentation to get
V = ∆Γt exp(−κσ ).
If one linearizes around κσ = 0, the resulting equation is
V = −∆Γt κσ .
If V is replaced by an upward velocity and ∆Γt is replaced by ∆′ for thegraph of w, then the equation becomes the fourth-order total variation flow
if σ(p) = |p′ |, i.e.,
wt = −∆′ (div′ (∇′ w/|∇′ w|)) .
One significant property of the surface diffusion flow is the preserving
property of the volume (area) enclosed by Γt . This is not the case for thesecond-order problem. However, one is able to consider a volume-preserving
crystalline curvature flow, which is a nonlocal equation. For example, thevolume-preserving version of (1.1) is of the formZ
1
V = κσ − n−1
κσ dHn−1
H (Γt ) ΓtR
so that Γt V dHn−1 = 0. See Section 9.1 for more discussion of the volumepreserving problem.
3. Polygonal flow
In this section, we consider a special class of a polygonal flow called
admissible introduced by J. Taylor [156] and S. B. Angenent and M. E.
Gurtin [10] for a planar purely crystalline curvature flow equation.
Admissible polygonal flow. We first introduce a special class of polygonalflows associated to a purely crystalline anisotropy σ. Let Wσ denote theWulff shape corresponding to σ. Since the anisotropy σ is purely crystalline,
Wσ is a bounded, convex polygon containing the origin as an interior point,
but it does not need to be symmetric with respect to the origin since σ is

16

Y. GIGA AND N. POŽÁR

nS
−1
0
S

+1

Figure 4. Value of χ based on the convexity/concavity near
the facets.
not assumed to be even. Let N be a finite subset of the unit circle so that
it is the set of all orientations (exterior normals) of edges on the boundary∂Wσ of Wσ . We call N the set of admissible directions. This set can be
written as
with nk = (cos θk , sin θk )N = {nk }mk=1
with 0 ≤ θ1 < · · · < θm < 2π. The set Θ = {θk }m
k=1 is called the setof admissible angles, which is considered as a subset in T = R/2πZ. For
example, θ1 , θm−1 are adjacent to θm . We say that an (oriented) polygon isadmissible if(i) (direction condition) the orientation of each facet (edge) is in N ;(ii) (adjacence condition) the angles of orientations of adjacent facets
should be adjacent.
An evolving polygon {Γt }t∈I is an admissible polygonal flow if Γt is anadmissible polygon for t ∈ I and the motion of all vertices is C 1 in timet ∈ I, where I is a time interval.
Crystalline curvature. Since the Wulff shape is a substitute of the unit
disk, it is natural to postulate that κσ = −1 on ∂Wσ . Let ∆(n) denote thelength of a facet (edge) of ∂Wσ whose orientation equals n. For a generaladmissible polygon Γ, let S denote one of its facets. By the ansatz for curve
evolution, κσ on S must be a constant and its value must be a kind of
Cheeger ratio. In our setting on S with orientation nS , it is natural toassignκσ = χ∆(nS )/L,
where L is the length of the facet S and χ is a transition number, i.e.,χ = +1 (resp. −1) if Γ is convex (concave) in the direction of nS near S,
and otherwise χ = 0; see Figure 4. By this definition, κσ = −1 on ∂Wσ sincenS is taken outward from Wσ ; this is the outward curvature. We measure
the curvature by comparing with the Wulff shape, which is consistent withthe definition of the usual curvature by the inverse of the radius of theosculating circle called a circle of curvature. This quantity κσ is often calleda crystalline curvature.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

17

Vj+1

Vj−1
sin φjVj+1
sin φj+1

φj

Vj
Vj cot φj+1

φj+1

Vj−1

Vj cot φj

Figure 5. Situation near facet j with nj pointing up.
Derivation of a system of ODEs. Let {Γt }t∈I be an admissible polygonal
flow such that for t ∈ I, Γt is an ℓ-polygon consisting of facets {Sj (t)}ℓj=1
numbered counterclockwise and vertices of Sj (t) whose motion is C 1 in time.
Let Vj (t) denote the normal speed of Sj (t) in the direction of the orientation
nj of Sj (t). We consider a general form of the equation(3.1)

V = g(ν, κσ )with g non-decreasing in the second variable so that the problem is at least
degenerate parabolic. For an admissible polygonal flow, this equation isformally reduced to(3.2)

Vj (t) = g (nj , χj ∆(nj )/Lj (t)) ,j = 1, . . . , ℓ,where Lj (t) is the length of Sj (t) and χj is the transition number of Sj (t).
By an elementary geometry Fig. 5, we observe that

(3.3)

dLj (t)1
Vj−1 (t) + (cot φj + cot φj+1 )Vj (t)=−dtsin φj1−
Vj+1 (t), j = 1, . . . , ℓ,sin φj+1

where φj = θj − θj−1 and θj is the angle of nj , i.e.,nj = (cos θj , sin θj ).
We use the convention that the indices are considered modulo ℓ, i.e., we
identify θℓ+j = θj . We conclude (3.2) and (3.3) to get a system of ℓ ODEs
for Lj ’s. The initial value problem of this system is locally-in-time solvablefor example when g is C 1 in the second variable. The resulting admissible
polygonal flow is called a crystalline flow. This idea is introduced by J.
Taylor [156] for V = σκσ and S. B. Angenent and M. E. Gurtin [10] for
V = M (ν)(κσ + C); both examples are introduced in Section 2.
Starting from a general polygon. If one considers a polygon whose

18

Y. GIGA AND N. POŽÁR

nB
nB

nC
nA

Γt
nA

∂Wσ

Γ
Figure 6. Example of an evolution of a initial polygon Γwith facets with nonadjecent orientations.
orientation belongs to N but violates the adjacence condition, it is expectedthat new facets with “missing directions” are created from a corner. To be
more precise, let us consider the equation
V = κσ .
We consider adjacent facets SA , SB of a polygon Γ whose angles θA , θB oforientation nA , nB are not adjacent; see Figure 6. In other words, there are
missing admissible angles between θA and θB . If SA and SB do not move,i.e., they are stationary, it is expected that there is a unique self-similar
expanding crystalline flow which converges to SA ∪ SB as time tends tozero. The unique existence of such a self-similar expanding crystalline flow
has been claimed in a pioneering work by J. Taylor [160, Proposition 2.2(1)]. However, unfortunately, the proof skips over important details. Here,by self-similar we mean that the flow Γt is of the form Γt = t1/2 Γ∗ with some
admissible Γ∗ ; we here assume that the vertex connecting SA and SB is theorigin by translation. Note that Γ∗ may not be a part of the Wulff shape
as observed in a numerical calculation [103]. The unique existence of sucha self-similar expanding solution is proved in D. Campbell [36] in the casethat Wσ is a regular polygon and in [77] for general Wσ . This problem isreduced to solving a system of algebraic equations and methods presented
in [36] and [77] are quite different. Approximating by such a self-similar
expanding solution, one is able to construct an expanding solution even if
SA and SB are moving. This is carried out by Y. Ochiai [141] for V = κσand is extended to an equation including V = M (ν)(κσ + C) by R. Kuroda[124]; see also [78] for a complete proof for more general equations.
Although there is a large number of articles studying crystalline flows,
this type of facet creation problems are not discussed frequently. A facet
creation problem was observed in [70] and further developed in [132], [133],[134] mostly for graph-like solutions. However, the number of newly createdfacets in one point is just one. This aspect is quite different from works by

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

19[160], [36], [77], [141], [124], [78], where several facets are created from one
point (corner).
On the other hand, it is not diﬀicult to handle the case when the direction condition is violated. In this case, we just regard ∆(n) = 0 for suchdirections. Such a facet is preserved at least for a short time, so we may callsuch a polygonal flow satisfying “adjacence condition” a weakly admissible
polygonal flow [67].
Behavior of convex crystalline flow. If the initial polygon is a convex (admissible) polygon, the behavior of a solution (crystalline flow) has
been well-studied for V = M (ν)κσ . It is easy to see that the convexity ispreserved.
(i) The case when M is parallel to σ, i.e., M (ν) = cσ(ν) with somec > 0. It is easy to see that there always exists a self-similar solution
shrinking to a point whose profile is the Wulff shape Wσ . By a spatialtranslation, this solution can be written as Γt = (2c)1/2 (T −t)1/2 ∂Wσ ,where T is the extinction time. The uniqueness of a self-similar solution is proved when the Wulff shape Wσ is symmetric with respect
to the origin and the number of its vertices is larger than four in[153]; in the case Wσ is a parallelogram, all parallelograms shrink
self-similarly; see the next section for an explicit solution. Moreover,
it is shown in [153] that all convex solutions shrink asymptoticallysimilarly to the self-similar solution. These results are parallel tothose for conventional curve shortening flow as established in [64],[65], [50], [49].
(ii) The case where M is unrelated to σ. In this case, the situation is complicated as discussed in [154], [9]. In [9] a rather complete picture isgiven. We first consider the orientation-free case i.e., M (n)∆(n) =M (−n)∆(−n) for n ∈ N where N = −N . In this case, there are
two possibilities. Either phenomenon similar to (i) occurs or there
is no self-similar shrinking solution and the isoperimetric ratio ofa solution may tend infinity [9]. Moreover, in the second case it isshown in [9] that the minimal length of facets at time t behaves like{(T − t)/ log(t − t)}1/2 or (T − t)β , 1/2 < β < 1 as t tends to T ,where T is the extinction time. For a self-similar solution, the length
should behave like (T − t)1/2 so, in the second case, it is shorter thanthat of a self-similar solution. This has a strong contrast comparedto the conventional orientation-free anisotropic curvature flow, where
all flows shrink in a self-similar way. This indicates that a qualitative

20

Y. GIGA AND N. POŽÁR

property of a solution may differ from the conventional curve shorting equation depending upon the Wulff shape. If the motion is notorientation-free, it is shown in [115] that a crystalline flow may not
become convex. There also exists a non-convex self-similar shrinking
solution when the Wulff shape is a square or a regular triangle for
V = M κσ with M unrelated to σ which is not orientation-free [115].
We next consider the equation V = |κσ |α−1 κσ for α > 0. The situation
depends on the value of α. We have discussed the case α = 1. In the caseα ≥ 1, it is shown in [72] that there is no degenerate pinching at theextinction time T . By degenerate pinching we mean that two parallel facets
touch with positive length at the extinction time. For α > 1, all (convex)
solutions shrink to a point in a self-similar way like (i) [9]. If α < 1, a
degenerate pinching may happen and there is a solution whose enclosed area
tends to zero but the limit of the length remains positive [9]. For α < 1,there also exists a non-convex self-similar solution for V = M |κσ |α−1 κσ even
if the equation is orientation-free [115].
We now consider the case when the initial polygon does not fulfill the
direction condition but satisfies the adjacency condition with interpretation that ∆(ni ) = 0 for a non-admissible direction of the initial polygon
and that ni belongs to N . In [169] a quite general results are established.
The equation considered there is V = g(ν, κσ ) with g(ν, 0) = 0 which is
non-decreasing and locally Lipschitz in the second variable. By solving thesystem of ODEs, we see that the number of facets is unchanged during
a short time. At some time either at least one of the facets with a nonadmissible direction disappears or the whole evolution shrinks to a point[169].
Behavior of a general admissible polygon. If the initial polygon is admissible but not convex, it must have an inflection facet, i.e., a facet with
χ = 0. There is a crystalline flow with such initial data until the length ofsome facet tends to zero. It is already proved in [160, Theorem 3.2] that for
the equation V = σκσ , at such occasion only at most two adjacent inflection
facets disappear unless the flow shrinks to a point. However, the proof thereis rather sketchy. In [107] a full proof is given when Wσ is a regular polygon
with an even number of facets. The resulting polygon at the time when
infection facets disappear stays admissible, so one can extend a solution asa crystalline flow until it loses another facet. We are able to complete thisprocedure until it shrinks to a point. Such an extended flow is called an
extended crystalline flow.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

21

For the curve shortening equation V = κ, it is shown that the solution(flow) becomes convex in finite time [98]. It seems that the corresponding result is not established even when Wσ is symmetric with respect to the origin
and the equation is V = σκσ . To the best of our knowledge, the (extended)
crystalline flow (after losing several inflection facets) becomes almost convex in the sense that all facets have positive crystalline curvature possibly
except two adjacent inflection facets for V = σκσ with symmetric Wσ asshown in [108].
Equations with a driving force term. We next consider the equation
V = M (ν)(κσ + C), where C is a constant. This equation is sometimes
called the eikonal curvature flow equation. There are several new phenomena in this equation compared to the case C = 0. For example, this motion
certainly depends on the orientation. If C is taken positive and ν is taken
outwards, it grows to the whole plane in infinite time provided that thepolygon is suﬀiciently large. The large time behavior of an (extended) crystalline flow is studied in [75] with special emphasis on the anisotropic effectof mobility M and σ. For V -shaped initial data, its evolution was studiedin [110], [111]. A crystalline flow is also applied to the study of a growth ofspirals since the work of [112], which is further developed in [113]. Various
methods for the numerical computation of the crystalline flow are comparedin [114].
A few remarks on consistency and stability. If the initial data is given
as the graph of a periodic piecewise linear function, for V = M (ν)κσ thetheory of maximal monotone operators applies to construct a solution [63].
This notion of a solution is consistent with the (extended) crystalline flow;
see also [52] where a numerical scheme based on a variational inequality isgiven.
Note that the crystalline flow satisfies a comparison principle or an orderpreserving property. It reads that if an admissible polygon Γa is enclosed
by another admissible polygon Γb , then the corresponding crystalline flows{Γat } and {Γbt } starting from Γa and Γb , respectively, have the same property, i.e., {Γbt } encloses {Γat } as far as both exist; see [160] and [85]. This iseasily seen by comparing their crystalline curvatures. Based on this property, one is able to establish a notion of viscosity solutions. This was first
introduced in the case where Γt is given as the graph of a periodic function [70]; see also [68] and its consistency with an (extended) crystalline
flow already discussed in [67]. Moreover, their solution can be obtained asa limit of a smoother problem, i.e., the problems where σ is smooth [71].

22

Y. GIGA AND N. POŽÁR

This stability property applies for a variational solution [63]. Note also thatin both frameworks solutions with a smooth σ can be approximated by acrystalline flow [63], [71]. This gives a numerical algorithm to solve a smoothanisotropic curvature flow or even the heat equation by approximating itby crystalline flows. This topic is studied in [63], [96], [71] for a graph-like
solution. In [96] a convergence rate is also given. The approach by viscosity
solution is extended to closed curves through a level-set method [73] andits consistency is discussed in [72]. The stability is also discussed in [73].
Among other results, a solution with a smooth σ can be approximated byextended crystalline flows. It is proved for V = κ in [95] for convex curves
with convergence rate and in [107] for a general curve. In [73] such stability is discussed for a general equation V = g(ν, κσ ). Note that it is also
shown in [73] that an extended crystalline flow is a limit of flows of problems
with smooth anisotropy. More precisely, if Wσ is close in the sense of theHausdorff distance, the solution must be close (up to fattening).
When one discusses consistency for equations with driving force term
like the eikonal-curvature flow V = M (ν)(κσ + C), one should be careful
to handle corners. If we consider just the eikonal equation V = C > 0 fora bounded convex polygon, it is expected that the solution will be rounded
following the Huygens principle. To preserve corners, one has to restrict
the mobility M (ν). We consider a general equation V = g(ν, κσ ). Let us
explain the corner preserving condition explicitly stated in [75, Lemma 4.1,
Lemma 4.2]. We say that g satisfies the corner preserving condition if foreach nk ∈ N
1(g(nk , 0) sin ψk+1 + g(nk+1 ) sin ψk )g(m, 0) =sin φk+1
for all m = (cos θ, sin θ) with θk < θ < θk+1 , where φk+1 = θk+1 − θk and ψk(resp. ψk+1 ) is the angle between nk (nk+1 ) and m so that φk+1 = ψk+1 +ψk .
Geometrically speaking, this condition can be written as
Ak ⊂ x ∈ R2 | x · m ≤ g(m, 0), m = (cos θ, sin θ), θk < θ < θk+1 ⊂ Bkwith
Ak = Hk ∩Hk+1 , Bk = Hk ∪Hk+1 , Hk+j = x ∈ R2 | x · nk+j ≤ g(nk+j , 0) .
If Γ is convex with outward orientation, we only need the inclusion of Ak .
In other words, in the above identity the equality should be replaced by≥ so that g(m, 0) is always larger than the right-hand side. This condition
says that in the corner all segments whose orientation is between that offacets forming the corner move faster than corner facets for V = g(ν, 0).

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY23

R0 ∂WσFigure 7. Timesteps Γit∗ /20 , i = 0, 1, . . ., of a homethetic
Wulff shape solution of V = σκσ . Note that even though κσ
is a constant on Γt , the edges further from the origin movefaster due to the mobility factor σ and the solution is homothetic.
This condition is first pointed out explicitly by [82] and independently by[100]. It is stated in a different from in [67]. The geometric version is foundin [72]; however, unfortunately, the definition of Bk was mistyped.
We shall postpone the definition of viscosity solutions to Section 6. We
note that the theory covers a wide range of σ not necessarily purely crystalline in planar case for general equation V = g(ν, κσ ) including (2.1), (2.2),
while in higher dimension, for such setting it is limited for purely crystalline
σ for general equations; see Section 6. In [73] it is only assumed that Fσ isconvex, C 2 except finitely many vertices and the curvature is bounded.
Although the approach by admissible polygonal flow is convenient tostudy planar curvature flow equations, it is limited because it implicitly
requires that the speed of a facet is spatially constant. For example, even in
R2 if one considers the equation with spatially inhomogeneous driving force
like
V = M (ν) (κσ + f (x)) ,then it is not appropriate to assign the speed of a facet as a spatially constantto obtain a comparison principle. For a graph-like solution with special M ,as a variational solution several facet-breaking solutions are given in [69].
4. Explicit solutions
In this section we give examples of a few interesting explicit solutions toillustrate the behavior of the equations.
The simplest solution of the crystalline mean curvature flow is the homothetic (self-similar) solution starting from the Wulff shape, Fig. 7, that canbe translated and scaled. Rotations are of course not allowed. As noted in24

Y. GIGA AND N. POŽÁR

the introduction, the crystalline mean curvature on the surface of the Wulff
shape Wσ is the constant n − 1; here, the orientation (normal) is taken
inward. ThereforeqΩt =R02 − 2(n − 1)t Wσ

is a solution of the crystalline mean curvature flow V = σ(ν)κσ for any
R02
R0 > 0 on the interval t ∈ [0, t∗ ), where t∗ = 2(n−1)
is the extinction time.
Note the factor σ(ν) in the velocity law. The (inner) normal velocity ofR(t)Wσ at a boundary point x with inner unit normal ν is −R′ (t)x · ν =−R′ (t)σ(ν)R(t).
One might ask whether the above solutions are the only homothetic
solutions of the flow. This is however not always the case as the following
simple example illustrates. We consider n = 2 and the anisotropy σ(p) =|p1 | + |p2 | = ∥p∥1 . Let Ω0 = (−a, a) ×q(−b, b) be a rectangle for some a > 0,2
t is a solution of both V = σκσb > 0. Then Ωt = R(t)Ω0 for R(t) = 1 − aband V = κσ . In n = 2, the uniqueness of the Wulff shape homothetic
solution was proved by Stancu [153] for even anisotropies σ when (so thatthe problem is orientation-free) the Wulff shape Wσ is not a quadrilateralas mentioned in Section 3 (i).
A related question is whether a solution starting from an arbitrary convex initial data will asymptotically approach the homothetic Wulff shapesolution as in the case of the usual mean curvature flow. As mentioned in
Section 3 (i), this was shown again by Stancu [154] in n = 2 for even nonquadrilateral anisotropies. The situation is much more complex in n > 2
and is studied in [139].
By an interpretation different from Section 3 (ii), we also mention thatfor V = σκσ examples of non-convex homothetic solutions in n = 2 givenin [115] for non-even anisotropies σ, that is, σ(p) ̸= σ(−p) for some p. Thisshows that one cannot in general expect that a non-convex connected initial
shape will become convex before the extinction time.
In dimensions n > 2 the situation is more complex and nonzero genus
explicit homothetic solutions are known. For example, for σ(p) = ∥p∥1 acube with a square-shaped hole along each axis is a homothetic solution,
Figure 8. See [148] for more details.
For examples of solutions of the related total variation flow see for example [130, Sec. 5].

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

25

Figure 8. Sponge-like homothetic solution of the crystalline
mean curvature flow in dimension 3 with σ(p) = ∥p∥1 .
5. Approach by the theory of maximal monotone operators
5.1. Abstract theory. In this section we introduce the crystalline mean
curvature as a solution of a certain minimization problem. This interpretation is based on the theory of maximal monotone operators of Kōmura [123]and Brezis [31].
Let us give a motivation for this point of view. It is natural to expect thatthe crystalline mean curvature flow with anisotropy σ can be approximatedby anisotropic mean curvature flow with smooth anisotropies σm so that
σm → σ in some sense.
If Γ is a C 2 surface in Rn and σm is a C 2 smooth anisotropy, theanisotropic mean curvature Γ at x ∈ Γ is given as div ∇σm (∇u)(x) =tr[∇2 σm (∇u(x))∇2 u(x)], where u is any C 2 level set function of Γ in theneighborhood of x with ∇u(x) ̸= 0.
Consider now the anisotropic total variation flow
ut − div ∇σm (∇u) = 0
on L2 (Tn ), Tn = Rn /Zn , or more rigorously,ut ∈ −∂Em (u),

(5.1)
where
(5.2)

(R
Em (v) :=σm (∇v) dx v ∈ BV (Tn ) ∩ L2 (Tn ),+∞otherwise,Tn

is the anisotropic total variation energy.
Since Em is a convex lower semicontinuous functional on the Hilbert space2
L (Tn ) with a dense domain, the operator ∂Em is a maximal monotone
operator and the anisotropic total variation flow (5.1) has a unique solution
for any initial data in L2 (Tn ).
Suppose now that (σm )m≥1 is a sequence of C 2 anisotropies that monotonically converge to the crystalline anisotropy σ. Then it is known [12] that26

Y. GIGA AND N. POŽÁR

Em → E in the sense of Mosco convergence, see (1.7), where(Rv ∈ BV (Tn ) ∩ L2 (Tn ),n σ(∇v) dx
T
E(v) :=+∞
otherwise.
But Mosco convergence implies the convergence of the nonlinear semigroups[12, Theorem 3.26] and [32, Theorem 3.2]: the solutions of (5.1) locallyuniformly converge to the unique solution u : [0, ∞) → L2 (Tn ) of
 du ∈ −∂E(u(t)),
t > 0,(5.3)
dt
u(0) = u0
for any initial data u0 ∈ L2 (Tn ).
As we will see below, ∂E(v) is in general multivalued even if ∇v ̸= 0
for typical crystalline mean curvature evolutions. Nevertheless, the uniquesolution of (5.3) is right-differentiable at every t > 0, ∂E(u(t)) ̸= ∅ andd+ u/dt(t) = −∂ 0 E(u(t)) for t > 0, where ∂ 0 E(v) is the canonical restriction
or minimal section of the subdifferential ∂E(v), i.e., the unique element of∂E(v) ⊂ L2 (Tn ) with minimal norm.
This strongly suggests that we should use −∂ 0 E(v) as the definition of thecrystalline mean curvature to hope to obtain stability under approximation
by anisotropic mean curvature flow.
Fortunately the characterization of ∂E is well understood even for rather
general σ = σ(x, p), see [130] for example.
We include the proof here for completeness for σ = σ(p) and space2
L (Tn ). We need to introduce a number of definitions.
The functional E(u) can be defined in two equivalent ways. The first one
is a generalization of the definition of the total variation,
 Z

◦
1
n
n
E(u) := sup −u div z dx σ (z) ≤ 1, z ∈ C (T , R ) ,u ∈ L2 (Tn ).
Tn

Note the minus sign since σ is not assumed even. The function σ ◦ is the
support function of the Frank diagram Fσ = {σ ≤ 1}, i.e.,σ ◦ (x) = sup {x · p | p ∈ Fσ }so that Wσ = {σ ◦ ≤ 1}. This E is clearly a convex, positively one-homogeneous,
lower semi-continuous functional on L2 (Tn ). It is known [5] that it is therelaxation (closure or lower semicontinuous envelope) of the functional(Rσ(∇u) dx, u ∈ W 1,1 (Tn ) ∩ L2 (Tn ),Tn
J(u) :=+∞,
otherwise.
In fact E(u) = J(u) for any u ∈ W 1,1 (Tn ) ∩ L2 (Tn ). While [5] deals with L1
relaxation, it is not diﬀicult to see that E is also the L2 relaxation. Indeed,MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

27

R
Du
) d|Du| for any u ∈ BV (Tn ) and E(u) = +∞we have E(u) = Tn σ( |Du|Du
is the Radon-Nikodym derivative offor u ∈ L2 (Tn ) \ BV (Tn ), where |Du|
the Radon measure Du with respect to its total variation [130]. Then bystandard approximation of u ∈ BV (Tn ) ∩ L2 (Tn ) there is a sequence uk ∈W 1,1 (Tn ) ∩ L2 (Tn ) converging to u in L2 such that |Duk | → |Du|. For suchsequence we have E(u) = limk→+∞ E(uk ) = limk→+∞ J(uk ) by Reshetnyak’s
continuity theorem [149]. We conclude that E is the L2 relaxation of J.
To characterize the subdifferential, we here present a simplification of
the proof in [130], which itself is based on the unpublished note of F. Alter;
see also [7] for the proof when σ(p) = |p| and more details. The idea is basedon the characterization of the subdifferential using the polar of E defined as(u, v)◦u∈H ,E (v) := sup {(u, v) | u ∈ H, E(u) ≤ 1} = supE(u)where we set H := L2 (Tn ) the Hilbert space with the L2 -inner productR(u, v) = uv dx. In the formula we use the convention 00 = 0, a0 = +∞ fora
any a > 0 and +∞= 0 for any a ∈ R. Since E is positively one-homogeneous,convex and lower semicontinuous, we have the following standard characterization [7, Lemma 1.7]:(5.4)

v ∈ ∂E(u)⇔E ◦ (v) ≤ 1 and (v, u) = E(u).

We will show that E ◦ coincides with the functional(5.5)

Ψ(v) := inf {∥σ ◦ (z)∥∞ | v = − div z, z ∈ L∞ (Tn )},v ∈ H.

The equality v = − div z is understood in the sense of distributions: thefunction −v is the distributional divergence of z. For the discussion of properties of Ψ, see Lemma A.1 in Appendix A below. In particular, Ψ is again aconvex, positively one-homogeneous, lower semicontinuous functional. For
any such functional we have (Ψ◦ )◦ = Ψ [7, Proposition 1.6]. Moreover, if
Ψ(v) < ∞ the infimum is attained by a vector field and hence it is a minimum.
Theorem 5.1. The equality E ◦ ≡ Ψ holds.
Proof. ≤ : Take v ∈ H with Ψ(v) < ∞ and fix z ∈ L∞ (Tn ) with div z = −v.
Since E is the lower semicontinuous envelope of J, for any u ∈ H with
E(u) ≤ 1 there is a sequence (uk )k ⊂ W 1,1 (Tn ) ∩ H with uk → u in H andJ(uk ) = E(uk ) → E(u). We haveZZ
(uk , v) = z · ∇uk dx ≤ σ ◦ (z)σ(∇uk ) dx
Z◦≤ ∥σ (z)∥∞ σ(∇uk ) dx = ∥σ ◦ (z)∥∞ E(uk ).

28

Y. GIGA AND N. POŽÁR

In the limit k → ∞ we obtain(u, v) ≤ ∥σ ◦ (z)∥∞for all u ∈ H with E(u) ≤ 1.

Thus by definition of Ψ we deduce E ◦ (v) ≤ Ψ(v).
≥ : Fix u ∈ H. By definition we have
 Z

◦
1
n
n
E(u) = sup − u div z dx σ (z) ≤ 1, z ∈ C (T , R )(u, − div z)(u, − div z)≤ sup◦z∈C 1 Ψ(− div z)
z∈C 1 ∥σ (z)∥∞◦≤ Ψ (u),= sup

where we again use 0/0 := 0. We deduce E ◦ ≥ (Ψ◦ )◦ = Ψ.

□We now have the following characterization of the subdifferential for
Lipschitz functions. For the general characterization we refer the reader to
Theorem 12 in [130].
Corollary 5.2. Let u ∈ Lip(Tn ) and v ∈ L2 (Tn ). The following are equivalent:• v ∈ ∂E(u)• there exists z ∈ L∞ (Tn ) with v = − div z such that z ∈ ∂σ(∇u) a.e.
Proof. ⇒ : v ∈ ∂E(u) implies that E ◦ (v) ≤ 1 and E(u) = (u, v). In particular by Theorem 5.1 there exists a vector field z ∈ L∞ (Tn ) with v = − div z
and ∥σ ◦ (z)∥ = E ◦ (v) ≤ 1. We have∇u · z ≤ σ(∇u)σ ◦ (z) ≤ σ(∇u)

a.e.

However, E(u) = (u, v) and thereforeZZZσ(∇u) dx = uv dx = ∇u · z dx,
and so we can deduce that ∇u·z = σ(∇u) a.e., which coupled with σ ◦ (z) ≤ 1
a.e. implies z ∈ ∂σ(∇u) a.e.
⇐ : The opposite implication can be proved by reversing the above
steps.
□The vector fields z play a central role and we define
X 2 (U ) := z ∈ L∞ (U ) | div z ∈ L2 (U ) ,for U ⊂ Rn open or U = Tn , following [11]. The vector fields that characterize the subdifferential are often called Cahn-Hoffman vector fields andwe define(5.6)
CH(u; U ) := z ∈ X 2 (U ) | z ∈ ∂σ(∇u) a.e.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

29

for any u ∈ Lip(U ). Note that if U = Tn , by Corollary 5.2−∂E(u) = div CH(u; Tn ) := {div z | z ∈ CH(u; Tn )}.
Recall that this is a closed convex set, but it might be empty.
Since the set ∂E(u) is in general not a singleton, we need to determine
how to select a value that gives a reasonable generalization of the anisotropic
mean curvature to the crystalline case. The theory of maximal monotone
operators suggests that we should choose the unique element of −∂E(u) withthe smallest L2 -norm. We will denote this element −∂ 0 E(u) if ∂E(u) ̸= ∅,since it is the projection of the origin 0 on the convex closed set −∂E(u).
Example 5.3. Suppose that σ ∈ C 2 (Rn \ {0}) and u ∈ C 2 (Tn ). Let x ∈ Tn
with ∇u(x) ̸= 0. Then ∂σ(∇u) = {∇σ(∇u)} in the neighborhood of x andtherefore if z ∈ CH(u; Tn ) we necessarily have div z(x) = div ∇σ(∇u)(x).
As was shown in the introduction, the element −∂ 0 E(u) is a solutionof a minimization problem with an n-dimensional obstacle z ∈ ∂σ(∇u).
The value of the minimizer div zmin can depend nonlocally on u whenever∂σ(∇u) is not a singleton, as is illustrated in the introduction. However thisnonlocality is restricted to “flat” parts of u. Those correspond to facets andedges of the evolving crystal. The following technical “patching” lemma wasproved in [91, Lemma 2.8]. Let 1E denote the characteristic function of E,
i.e., 1E (x) = 1 for x ∈ E and 1E (x) = 0 for x ∈/ E.
Lemma 5.4. Let σ : Rn → R be a positively one-homogeneous convex
function. Let U1 , U2 be two open subsets of Rn and ψi ∈ Lip(Ui ) two
Lipschitz functions. Let δ > 0 and set G := {x ∈ U1 | |ψ1 (x)| < δ}. Supposethat G ⊂ U1 ∩ U2 and ψ1 = ψ2 on G. If zi ∈ CH(ψi ; Ui ) are two Cahn–
Hoffman vector fields, thenz := z1 1U1 \G + z2 1Gis also a Cahn–Hoffman vector field z ∈ CH(ψ1 ; U1 ), anddiv z = div z1 1U1 \G + div z2 1G .
We add the following simple observation that follows from |{0 < |ψ1 | < δ}| →0 as δ → 0. Note that we still need ψ1 = ψ2 on a neighborhood of {ψ1 = 0}.
Corollary 5.5. Under the assumptions of Lemma 5.4,
z := z1 1U1 \{ψ1 =0} + z2 1{ψ1 =0}
is also a Cahn–Hoffman vector field z ∈ CH(ψ1 ; U1 ), anddiv z = div z1 1U1 \{ψ1 =0} + div z2 1{ψ1 =0} .

30

Y. GIGA AND N. POŽÁR

The above lemma shows that we can isolate div zmin on a neighborhoodof {ψ = 0}. This is necessary to have some locality of the crystalline mean
curvature which allows us to localize the construction of test functions to agiven facet.
We conclude this section by an important way of approximating thevalues ∂ 0 E(ψ). Let us now fix the domain Tn for simplicity. For a givenψ ∈ L2 (Tn ) and a > 0, we consider the resolvent problem(5.7)

v + a∂E(v) ∋ ψ

in the unknown v ∈ L2 (Tn ). This can be viewed as the implicit Euler discretization of the gradient flow (5.3). It is also the Euler–Lagrange equationof the minimization problem
arg min∥v − ψ∥2L2 (Tn )v∈L2 (Tn )a+ E(v),

which appears in an important discrete approximation of the crystalline
mean curvature flow, Chambolle’s scheme discussed in Section 6.4.
We have the following standard existence and approximation result thatis valid for any convex proper lower semi-continuous functional like E, seefor example [12].
Proposition 5.6. For every ψ ∈ L2 (Tn ) and a > 0 the resolvent problem(5.7) has a unique solution ψa ∈ L2 (Tn ) and ψa → ψ as L2 (Tn ).
If furthermore ∂E(ψ) ̸= ∅, thenψa − ψ
→ −∂ 0 E(ψ)
ain L2 (Tn ) as a → 0.

The solutions also satisfy a comparison principle, see [38] for a proof.
Proposition 5.7. If ψ 1 , ψ 2 ∈ L2 (Tn ) are two right-hand sides with ψ 1 ≤ ψ 2
and a > 0, we have ψa1 ≤ ψa2 where ψa1 and ψa2 are the respective solutionsof (5.7).
5.2. Calibrability and Cheeger sets. As we already briefly mentioned
in the introduction, the minimization problem one needs to solve to findthe value ∂ 0 E(ψ) for a given ψ has interesting connections to the so-called
Cheeger problem for sets. For a given open set U ⊂ Rn , define the Cheeger
constant as
P (F )n: F Borel ⊂ U, L (F ) ∈ (0, ∞) ,h(U ) := inf
Ln (F )
where P (F ) = E(1F ) is the anisotropic perimeter of F . Usually the isotropicσ(ν) = 1 is considered, in which case this is just the usual perimeter equal toMOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

31)
Hn−1 (∂F ) for suﬀiciently regular sets. A set F ⊂ U such that LPn(F= h(U )
(F )is called a Cheeger set of U . If U itself is a Cheeger set of U , it is called
self-Cheeger. Finding the value h(U ) or characterizing the Cheeger subsetsof U is then often referred to as the Cheeger problem. For a recent reviewof this topic see [126].
In the current note, the question whether a given set U is self-Cheeger
is closely related to the questions whether the value of ∂ 0 E(ψ) is constanton a facet U of ψ. If ∂ ◦ E(ψ) is constant on a given facet, the facet is called
calibrable or σ-calibrable, see [26] and also [20].
We point out that this notion of calibrability is slightly weaker than thenotion used in the context of total variation flows [3, 126]. There an open
bounded set U is called calibrable if the total variation flow (5.3) with initial)
data 1U has the unique solution a(t)1U with a(t) = max(1 − LPn(Ut, 0). This(U )0ctherefore implies that ∂ E(1U ) is constant on U and on U .
We use the former notion of calibrability. The following theorem in amore general setting (but still only in dimension n = 2), including nonuniform forcing, was proved in [26, Th. 6.1]. See also [6] for further developments.

Theorem 5.8. Let n = 2 and let σ be an even anisotropy on R2 , σ(p) =σ(−p). Suppose that ψ ∈ Lip(R2 ) is such that CH(ψ; R2 ) is nonempty.
Let U be a bounded connected component of int {ψ = 0}. The following are
equivalent:(i) U is calibrable (∂ 0 E(ψ) is constant on U )(ii) for any F ⊂ U of finite perimeter(5.8)

SP (F )SP (U )≥.
L2 (F )
L2 (U )
Here SP (F ) is the signed perimeter of F defined using the reduced
boundary ∂ ∗ F as
Z
Z
1
SP (F ) =σ(ν) dH −σ(ν) dH1 ,∂ ∗ F+∂ ∗ F−

with ∂ ∗ F− := {x ∈ ∂ ∗ F ∩ ∂ ∗ U : νU (x) · ∇ψ(x) < 0} and ∂ ∗ F+ :=∂ ∗ F \ ∂ ∗ F− .
)(U )is a generalization of the usual Cheeger ratio LPn(UtoThe quantity SP
Ln (U )
(U )facets: sets with signed boundary whose sign is determined by whether thesurface at the boundary point is convex or concave in the normal directionof the facet; see also Section 5.3 below for a notion of facet.

32

Y. GIGA AND N. POŽÁR

To illustrate the proof of (i) ⇒ (ii) in a simplified setting, consider now
a Lipschitz function ψ ∈ Lip(Rn ) whose int {ψ = 0} is a simply connectedbounded open set U ⊂ Rn with Lipschitz boundary. Let us also for simplicity
assume that we can define ∇ψ ̸= 0 on ∂U Hn−1 -a.e. as the limit of ∇ψ fromc
U . Suppose that there exists a vector field z ∈ L∞ (Rn ) ∩ C(Rn ) with
div z ∈ L2 (Rn ) and z ∈ ∂σ(∇ψ) a.e. that is suﬀiciently regular so that thefollowing calculation can be justified and assume that div z = λ on U forsome λ. Then the divergence theorem yields
Z
ZnλL (U ) =div z dx =z · ν Hn−1 .
U

∂U

∇ψ
∇ψ
We observe that ν = |∇ψ|on ∂U+ and ν = − |∇ψ|on ∂U− . Since z ∈ ∂σ(∇ψ),

we have z · ν = ±σ(ν) on ∂U± . We have
Z
Z
Z
n−1
n−1
z·ν H .=
σ(ν) dH−∂U∂U+

σ(ν) dHn−1 = SP (U ).

∂U−

(U )
. However, for any Lipschitz subset F of U we
In particular, λ = SP
Ln (U )
have
Z
Z
Z
Z
n−1
n−1
div z dx =z · ν dH=z · ν dH+z · ν dHn−1 .
F∂F \∂U∂F

Using the estimateZZn−1z · ν dH≤∂F \∂U∂F ∩∂UZ◦

σ (z)σ(ν) dH

n−1∂F \∂U≤σ(ν) dHn−1 ,∂F \∂U(F )(U )we deduce that SP≥ λ = SP.
Ln (F )Ln (U )
However, it seems that the proof of (ii) ⇒ (i) in Theorem 5.8 is available
only for n = 2. We expect it to be valid in arbitrary dimension.
Let us give a well-known example of a facet that breaks immediately inthe evolution.

2
B
2

1
2

A

1
Figure 9.
Example 5.9. Let n = 2 and σ(p) = ∥p∥1 = |p1 | + |p2 |. Consider the set
C = A ∪ B with A = [−1, 0] × [−1, 1] and B = [0, 1] × [ 12 , 1], see Figure 9,
and let ψ(x) = dist(x, C). It is well-known that C considered as a facet of

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY33

ψ is not calibrable and breaks into two facets A and B moving at different
speeds. See [25] for the computation in the crystalline flow case and [130,Sec. 5] for the explicit computation in the anisotropic total variation flowcase. [130] shows that the solution of the anisotropic total variation flow
(5.3) with initial data u0 = 1C is given asu(x, t) = max(1 − 3t, 0)1A + max(1 − 4t, 0)1B .
Let us set U = int C. In terms of Theorem 5.8 note that SP (U ) = P (U ) = 8
and L2 (U ) = 52 , yielding a Cheeger ratio SP (U )/L2 (U ) = 16
= 3 + 15 , while5
A has a Cheeger ratio SP (A)/L2 (A) = 26 = 3, violating (5.8). U therefore
cannot be calibrable.
5.3. Curvature-like quantity. The characterization of the subdifferential of the anisotropic total variation and the localization of the canonicalrestriction −∂ 0 E motivates the following definition of the crystalline mean
curvature. To allow for a forced mean curvature flow, we need to include
the forcing into the definition. We follow the notation in [93].
Suppose that U ⊂ Rn is an open set and ψ ∈ Lip(U ). If CH(ψ; U )defined in (5.6) is nonempty we define the σ ◦ -(L2 ) divergence of ψ for any
f ∈ L2 (U ) as
Λf [ψ] := div zmin − f

on {ψ = 0}

where zmin is a minimizer of ∥div z − f ∥L2 (U ) on CH(ψ; U ), that is, div zminis the projection of f onto div CH(ψ; U ). Since div CH(ψ; U ) is closed convex, the value div zmin is unique, but zmin might not be.
One might wonder whether the value of Λf [ψ] depends on the choice U ,but thanks to the patching Lemma 5.4 that is not the case. For details see[91, Prop. 4.10].
Remark 5.10. Note that since ∂σ is positively 0-homogeneous, Λf [tψ] does
not depend on t > 0, and in fact for any Lipschitz function θ : R → R with
θ(0) = 0 and θ′ (s) > 0 for a.e. s we have Λf [θ ◦ ψ] = Λf [ψ]. Indeed, by thechain rule for the Lipschitz functions ∇(θ ◦ ψ)(x) = θ′ (ψ(x))∇ψ(x) a.e. if
we interpret the right-hand side as 0 when ∇ψ = 0. The 0-homogeneity of∇σ implies that CH(ψ; U ) = CH(θ ◦ ψ; U ).
We also note the scaling invarianceΛf [ψ](x) = a−1 Λaf (a·) [ψ(a·)](ax),thanks to which we can always assume that U ⊂ (− 21 , 12 )n .
In general, Λ0 is only BV and can be discontinuous as was shown in [27],[28]. Finding the value of Λf [ψ] explicitly in dimensions n ≥ 2 is in general

34

Y. GIGA AND N. POŽÁR

diﬀicult. However, if {ψ = 0} has a suﬀiciently regular boundary and there
is a vector field in CH(ψ; U ) with constant divergence on {ψ = 0}, thenΛf [ψ] can be found as the ratio of the signed anisotropic perimeter and thearea (volume) of the facet. Such facets are referred to as calibrable. Even
though this is well-known in the literature, we have not found a statementthat applies precisely to our setting and therefore we present it here with aproof.
Lemma 5.11. Let U ⊂ Rn be bounded open set. Suppose that ψ ∈ Lip(U ),|ψ| > 0 on ∂U and there exists δ0 > 0 such that |∇ψ| > 0 a.e. on{0 < |ψ| < δ0 } and the sets {ψ < δ}, {−ψ < δ} are Lipschitz regular forδ ∈ (0, δ0 ), andZZ
n−1σ(±ν) dH→σ(±ν) dHn−1
as δ → 0,∂{±ψ<δ}∂{±ψ<0}where ν is the outer unit normal to the respective sets. If there exists zC ∈CH(ψ; U ), that satisfies div zC = C a.e. on {ψ = 0} for some constantC ∈ R, then
(5.9)

R

Λ0 [ψ] = C =∂{ψ≤0}

Rσ(ν) dHn−1 − ∂{ψ≥0} σ(−ν) dHn−1|{ψ = 0}|a. e. on {ψ = 0}.

If ψ is non-positive and σ = 1, then this number C is the Cheeger ratioof the set {ψ = 0} if the boundary ∂{ψ ≥ 0} is Lipschitz. Note that we
invoke only approximability of surface energy by that of Lipschitz regular
set and do not assume Lipschitz regularity of ∂{ψ ≥ 0} itself.
Proof. Due to the existence of zC we know that Λ0 [ψ] is well-defined. Let
us first prove that for all z ∈ CH(ψ; U ) we have
(5.10)
ZZ
div z dx =

{ψ=0}

Z
σ(ν) dHn−1−∂{ψ≤0}

σ(−ν) dHn−1 = C|{ψ = 0}|.
∂{ψ≥0}

The characterization of ∂σ in (5.4) yields z·∇ψ = σ(∇ψ) a.e. on {0 < |ψ| < δ0 }.
For ε > 0 let ηε be the standard mollifier with radius ε and let zε := z ∗ ηε ,where we extend z by 0 outside U . We havezε · ∇ψ → σ(∇ψ)

a.e. in U ,div zε → div z

in L2 (U ),as ε → 0. The divergence theorem givesZZ
div zε dx ={|ψ|<δ}∂{|ψ|<δ}zε · ν dHn−1 .

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

35

∇ψ
∇ψ
By the coarea formula, ν = |∇ψ|Hn−1 -a.e. on ∂{ψ < δ} and ν = − |∇ψ|Hn−1 -a.e. on ∂{ψ > −δ} for a.e. δ ∈ (0, δ0 ). After sending ε → 0, thedominated convergence theorem yields for a.e. δ ∈ (0, δ0 )
Z
Z
Z
n−1
div z dx =σ(ν) dH−σ(−ν) dHn−1 .
{|ψ|<δ}∂{ψ<δ}∂{ψ>−δ}Sending δ → 0 along a sequence leads to (5.10). We recover the second
equality in (5.10) by recalling that zC ∈ CH(ψ; U ) satisfies div zC = C a.e.
on {ψ = 0}.
R
R
Let us write A = {ψ = 0}. For v = div zmin we have A v dx = A C dxby (5.10) and therefore
Z
Z
Z
Z
2
2
2
(5.11)
v dx =
C dx + (v − C) dx ≥C 2 dx.
A

A

A

A

By patching Corollary 5.5, the vector fieldz̃ = zC 1A + zmin 1U \A
is also a Cahn-Hoffman vector field with
div z̃ = div zC 1A + div zmin 1U \Aa.e. in U.

Therefore (5.11) implies that ∥div zmin ∥L2 (U ) ≥ ∥div z̃∥L2 (U ) and we conclude
that div z̃ is minimizing. By uniqueness, div zmin = C a.e. on A.
□
Let us conclude with a few examples of simple useful facets for which we
can compute Λ0 explicitly.
Example 5.12. Wulff facet.
For R > 0 consider ψ(x) := max(σ ◦ (x)−R, 0). We have {ψ = 0} = RWσ .
Take U to be a suﬀiciently large open ball containing RWσ and consider
the vector field(x,σ ◦ (x) ≤ R,
z(x) := R x, otherwise.
σ ◦ (x)Clearly z ∈ L∞ (U ) and div z ∈ L2 (U ) with(n,σ ◦ (x) ≤ R,R
div z = n−1, otherwise.
σ ◦ (x)
It is easy to check that z ∈ ∂σ(∇ψ) a.e. Therefore CH(ψ; U ) ̸= ∅ andΛ0 [ψ] = Rn on {ψ = 0} by Lemma 5.11.
Example 5.13. Wulff facet with a “concentric” hole; Fig. 10. This example
is useful in calculating the crystalline curvature of the exterior facets of the
sponge-like solution in Figure 8.

36

Y. GIGA AND N. POŽÁR

R∂Wσ

r∂Wσ
+

+

0

Figure 10. Wulff facet with a hole in Example 5.13, withsign of ψ indicated.
Suppose that σ ◦ is even, i.e., σ ◦ (−x) = σ ◦ (x) for all x. Consider 0 < r <R and the function
ψ(x) := max(r − σ ◦ (x), 0, σ ◦ (x) − R),so that {ψ = 0} = RWσ \ int rWσ . Let us seta :=Rn−1 rn−1 (R + r),
Rn − r nb :=Rn−1 + rn−1
Rn − r n

We claim that the vector fieldx
0 < σ ◦ (x) ≤ r,− σ◦ (x) ,z(x) :=− a(σ ◦ (x))−n + b x, r < σ ◦ (x) < R, x ,σ ◦ (x) ≥ R,σ ◦ (x)is a Cahn–Hoffman vector field for ψ on any U away from x = 0. To see that,
we consider g(s) := (−as−n + b)s. We note that g(r) = −1 and g(R) = 1,
and g is increasing on s > 0 which yields −1 < g(s) < 1 for r < s < R. By
the assumption that σ ◦ is even, we haveσ ◦ (z(x)) = |g(σ ◦ (x))| ≤ 1

r < σ ◦ (x) < R.

This, by the characterization of the subdifferential, for example (5.4), implies that z(x) ∈ ∂σ(0) = ∂σ(∇ψ(x)) for r < σ ◦ (x) < R. For other x theinclusion z(x) ∈ ∂σ(∇ψ(x)) a.e. is obvious.
We also see that z is in fact locally Lipschitz continuous away fromx = 0. Therefore div z ∈ L2 (U ) for any U away from x = 0 and hencez ∈ CH(ψ; U ).
A direct computation using x · ∇σ ◦ (x) = σ ◦ (x) yields that almost everywhere
 n−1
◦

− σ◦ (x) , 0 < σ (x) ≤ r,
div z = nb,r < σ ◦ (x) < R,
 n−1 ,σ ◦ (x) ≥ R.
σ ◦ (x)MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY
n−1

37

n−1

In particular, Λ0 [ψ] = nb = n R Rn +r
by Lemma 5.11, matching the for−rnmula (5.9).
Note that in the limit r → 0+, z converges to the vector field in Example 5.12.
Example 5.14. Convex-concave facet; Fig. 10 with negative sign in thehole.
Consider 0 < r < R and the function
ψ(x) := min(σ ◦ (x) − r, max(0, σ ◦ (x) − R)),so that again {ψ = 0} = RWσ \ int rWσ , but this time ψ < 0 in rWσ . The
vector fieldxz(x) = ◦σ (x)is a Cahn–Hoffman vector field for ψ on any U away from 0.
But we can be more precise as in Example 5.13. Let us set
Rn−1 rn−1 (R − r)
Rn−1 − rn−1,b:=
Rn − r n
Rn − r n
and consider the vector field x
0 < σ ◦ (x) ≤ r, σ◦ (x) ,z(x) :=a(σ ◦ (x))−n + b x, r < σ ◦ (x) < R, x ,σ ◦ (x) ≥ R.
σ ◦ (x)a :=This is a locally Lipschitz continuous vector field away from x = 0. Moreover, div z = nb almost everywhere for r < σ ◦ (x) < R.
Let us check that it is a Cahn–Hoffman vector field. The inclusion z(x) ∈∂σ(∇ψ(x)) is clear for σ ◦ (x) < r and R < σ ◦ (x). Since ψ(x) ≡ 0 forr < σ ◦ (x) < R, we only need to check that σ ◦ (z) ≤ 1 by (5.4).
The function g(s) := (as−n + b)s is convex on s > 0 with minimum atnb)−1/n with value g(ŝ) = n−1
bŝ > 0. Therefore as−n + b > 0 forŝ = ( (n−1)as > 0 and we haveσ ◦ (z(x)) = g(σ ◦ (x))

r < σ ◦ (x) < R.

Since g(r) = g(R) = 1, by convexity of g we conclude that σ ◦ (z(x)) ≤ 1
for all x ̸= 0. Therefore z is a Cahn–Hoffman vector field with constantdivergence on the facet {ψ = 0} and hence by Lemma 5.11 we have Λ0 [ψ] =nb.
5.4. Comparison and approximation. We start with the comparison
principle for the σ ◦ -(L2 ) divergence. Here sign s = −1, 0, 1 if s < 0, s = 0,or s > 0 respectively.

38

Y. GIGA AND N. POŽÁR

Theorem 5.15. Let σ be a convex, positively one-homogeneous function
on Rn that is positive away from 0. Let U be an open subset of Rn and let
ψi ∈ Lip(U ) with {ψi = 0} compact subset of U , for i = 1, 2. Let fi ∈ L2 (U )for i = 1, 2. If CH(ψi ; U ) ̸= ∅ for i = 1, 2 andsign ψ1 ≤ sign ψ2 ,f 1 ≥ f2then
Λf1 [ψ1 ] ≤ Λf2 [ψ2 ]a.e. on {ψ1 = 0} ∩ {ψ2 = 0}.

Before we proceed with the proof, we recall here a technical lemma provedin [91, Lemma 4.13]. It is a variant of a result for continuous functions
established in [45, 60]; see also [83, Lemma 4.2.9].
Lemma 5.16. Suppose that ψ and φ are two nonnegative periodic Lipschitz
functions on Rn , such that {ψ = 0} ⊂ {φ = 0}. Then there exists a Lipschitz
continuous function θ : [0, ∞) → [0, ∞) such that θ(0) = 0, θ(s) > 0 for
s > 0 and θ′ (s) > 0 for almost every s > 0 and we haveθ◦φ≤ψ

on Rn .

Proof of Theorem 5.15. For simplicity, we assume f1 = f2 = 0. We canassume that U is connected and by scaling that U ⊂ (− 21 , 21 ). By making U
smaller if necessary, we may assume that min∂U |ψi | > 0.
We can modify ψi away from {ψi = 0} to make it nonzero constant near
∂U and then extend it using this constant periodically so that ψi ∈ Lip(Tn )and CH(ψi ; Tn ) ̸= ∅ without changing the value of Λ0 [ψi ] on {ψi = 0}. If
n = 1 we might have to do an even extension first if the sign differs on inf U
and sup U .
By Lemma 5.16, we can also find θ1 , θ2 ∈ Lip(R) with θi (0) = 0 and′θi > 0 a.e. so that θ1 ◦ ψ1 ≤ θ2 ◦ ψ2 everywhere. Since by Remark 5.10
we have CH(θi ◦ ψi ; Tn ) = CH(ψi ; Tn ), we can replace ψi with θi ◦ ψi andassume that ψ1 ≤ ψ2 on Tn .
We then have Λ0 [ψi ] = −∂ 0 E(ψi ) on {ψi = 0} by the characterization of
the subdifferential in Corollary 5.2. Due to the comparison Proposition 5.7,
we have ψ1 ≤ ψ2 implies ψ1,a ≤ ψ2,a for the solutions of the resolvent
problem (5.7). On {ψ1 = 0} ∩ {ψ2 = 0} we have
ψ2,a − ψ2
ψ1,a − ψ1≤,aawhich using the convergence in Proposition 5.6 and sending a → 0 implies−∂ 0 E(ψ1 ) ≤ −∂ 0 E(ψ2 ) a.e. on {ψ1 = 0} ∩ {ψ2 = 0}.
□

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

39

Abstract facets. The comparison principle for Λf [ψ] implies that the valueon {ψ = 0} depends only on f and sign ψ. We define the relation ∼ on
F = {ξ | ξ : Rn → R} the set of all real-valued function on Rn asξ1 ∼ ξ2⇔sign ξ1 = sign ξ2 .

This relation is an equivalence relation on F. We refer to its equivalenceclasses [ξ] := {ψ | ψ ∼ ξ} ⊂ F as (abstract) facets. We write [ξ1 ] ⪯ [ξ2 ]when sign ξ1 ≤ sign ξ2 and this relation defines a partial order on the set of
all facets F/ ∼:= {[ξ] | ξ ∈ F }.
Cahn–Hoffman facet. We say that a facet [ξ] is a σ ◦ -(L2 ) Cahn–Hoffman
facet if {ξ = 0} is compact and there are an open set U ⊂ Rn , {ξ = 0} ⊂ U
and a Lipschitz function ψ ∈ [ξ] such that CH(ψ; U ) ̸= ∅. The facets in
Examples 5.12–5.14 are Cahn–Hoffman. It is not clear that this regularity
condition can be compared with the notion of σ 0 -regularity, or Lipschitz
σ 0 -regularity in the literature [24, 25, 27]
Proposition 5.17. For σ ◦ -(L2 ) Cahn–Hoffman facets [χ1 ] and [χ2 ] andfunctions fi ∈ L2 ({χ1 = 0} ∪ {χ2 = 0}) we have[χ1 ] ⪯ [χ2 ],f1 ≥ f2 a.e.

implies
Λf1 [χ1 ] ≤ Λf2 [χ2 ] a.e. on {χ1 = 0} ∩ {χ2 = 0}.
We will use σ ◦ -(L2 ) Cahn-Hoffman facets to build test functions for viscosity solutions of the crystalline mean curvature flow and so we need tomake sure there are enough of them. In fact, any facet with bounded zeroset can be approximated by σ ◦ -(L2 ) Cahn-Hoffman facets monotonically arbitrarily close in the Hausdorff distance. The following theorem was provenin [81] for σ the Euclidean norm, and in [92] in full generality.
Theorem 5.18. Let χ be an n-dimensional facet with {χ = 0} bounded andσ an anisotropy. Given ρ > 0 there exists a σ ◦ -(L2 ) Cahn-Hoffman facet χ̃such that χ(x) ≤ χ̃(x) ≤ sup|x−y|≤ρ χ(y) for x ∈ Rn .
6. Approach by the theory of viscosity solutions
In this section we introduce a notion of viscosity solutions for nonlinear partial differential equations that include the very singular term
div ∇σ(∇u)−f that represents an anisotropic mean curvature with forcing.
For the definition of the anisotropic mean curvature we use the quantity
Λf that was introduced in Section 5.3. It is important to note that if f
depends on x, the term div ∇σ(∇u) − f must be carefully defined together

40

Y. GIGA AND N. POŽÁR

and f cannot be added separately. Heuristically, the anisotropic mean curvature flow prefers flat facets in the singular directions of σ even in thepresence of nonuniform forcing, and so the full quantity div ∇σ(∇u) − fshould be constant on facets. If we considered the forcing f separately inthe definition of a viscosity solution, the comparison principle would still bevalid however we would have a problem with stability in the approximation
by regularized problems and ultimately we could not establish existence ofsolutions. For a counterexample to existence see [93, Sec. 6].
6.1. Definition of viscosity solutions. If σ ∈ C 2 (Rn \ {0}), it only has asingularity at p = 0 and we have everything we need to define the viscosity
solution for (6.4). The following is the notion of the viscosity solution introduced in [81, 80] assuming that F does not depend on x and t and there
is no forcing term.
Definition 6.1. An upper semicontinuous function u on Rn × (0, ∞) is aviscosity subsolution of(6.1)

ut + F (∇u, div ∇σ(∇u)) = 0

if the following two conditions hold:(i) (conventional test) If φ ∈ C 2 near (x̂, t̂), ∇φ(x̂, t̂) ̸= 0 and u − φ has
a local maximum at (x̂, t̂), then
(6.2)

φt (x̂, t̂) + F (∇φ(x̂, t̂), div ∇σ(∇φ)(x̂, t̂)) ≤ 0.

(ii) (faceted test) If φ(x, t) = ψ(x)+g(t) with g ∈ C 1 (R) and ψ ∈ Lip(Rn )so that [ψ] is a σ ◦ -(L2 ) Cahn-Hoffman facet, x̂ ∈ int {ψ = 0}, u −φ(· − h) has a global maximum at (x̂, t̂) for all |h| small, then there
exists δ > 0 such that
(6.3)

g ′ (t̂) + F (0, ess inf Λ0 [ψ]) ≤ 0.
Bδ (x̂)A lower semi-continuous function is a viscosity supersolution if it satisfies the above two conditions with maximum, ≤ and ess inf Λ replaced byminimum, ≥ and ess sup Λ, respectively.
Let us remark that in [81, 80] the facet test was restricted to test functions where the facet [ψ] has a smooth boundary. However, this is notessential as was observed in later papers.
As you can see, we need to reduce the class of test functions testingat points where ∇u = 0 to be even able to define a reasonable value ofdiv ∇σ(∇φ).

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

41

To include a forcing term f that depends on the x variable, we can follow[93] to modify the above definition. We introduceΛf [ξ](x) := lim ess inf Λf [ξ],δ→0+ Bδ (x)Λf [ξ](x) := lim ess sup Λf [ξ],δ→0+Bδ (x)

on the interior of {ξ = 0}, which are well-defined and finite by the comparison principle with Wulff facets in Example 5.12 as long as f is locallybounded. In fact, in this case Λf [ξ] is lower semi-continuous while Λf [ξ] is
upper semi-continuous.
Then we can define a viscosity subsolution of the PDE
ut + F (x, t, ∇u, div ∇σ(∇u) − f ) = 0(6.4)

following the above definition, but replacing (6.2) withφt (x̂, t̂) + F (x̂, t̂, ∇φ(x̂, t̂), div ∇σ(∇φ)(x̂, t̂) − f (x̂, t̂)) ≤ 0.
and (6.3) with
(6.5)

g ′ (t̂) + F (x̂, t̂, 0, Λf [ψ](x̂)) ≤ 0.

This latter condition is slightly weaker than (6.3) used in [81, 80], andallows for the proof of stability to handle a non-constant driving force f .
If the anisotropy σ has singularities other than at p = 0, the faceted test
has to be extended to those gradients of the solution. However, the singular
set of σ might be in general very complicated and it is not clear how todefine a viscosity solution for a general convex anisotropy σ (or a convex
function σ) except in one dimension.
Therefore we restrict our attention to crystalline anisotropies: an anisotropy σ is called crystalline if it is a maximum of a finite number oflinear functions. In this case, the structure of singularities of σ is relatively
simple. The “kind” of singularity is determined by the dimension of thesubdifferential ∂σ(p), which corresponds to the expected dimension of the
facet in the direction p. We introduce the following orthogonal decomposition of the space Rn . For a fixed gradient p̂ ∈ Rn , define Z to be the linear
subspace of Rn parallel to the aﬀine hull of ∂σ(p̂), see Figure 11. In otherwords, Z is the smallest linear subspace such that ∂σ(p̂) ⊂ Z + ξ for someξ ∈ Rn . Set k := dim ∂σ(p̂) := dim Z. We have an orthogonal decomposition Rn = Z ⊕ Z ⊥ . We fix orthonormal bases of Z, Z ⊥ which give two linear
isometries T : Rk → Z and T⊥ : Rn−k → Z ⊥ . This allows us to write anyx ∈ Rn uniquely as x = T x′ + T⊥ x′′ for some x′ ∈ Rk and x′′ ∈ Rn−k . For
k = 0, k = n we take x = x′′ and x = x′ , respectively. If we denote theadjoint of T as T ∗ and of T⊥ as T⊥∗ , we have x′ = T ∗ x and x′′ = T⊥∗ x.

42

Y. GIGA AND N. POŽÁR

Z ⊥ + p̂
Z
p̂
0
σ=1

Figure 11. An illustration of the orthogonal decomposition
R2 = Z ⊕ Z ⊥ at p̂ that lies on a “one-dimensional” singularityof σ. The thick polygon is the boundary ∂Fσ = {σ = 1}, andthe solid rays from the origin indicate where dim ∂σ = 1.
Using the above decomposition, we can “slice” the anisotropy σ to extract
only the part that contains the singularity by introducingσ(p̂ + λT w) − σ(p̂)
,
λ→0+
λ

σp̂sl (w) := limw ∈ Rk .

This sliced function is again positively one-homogeneous and so we canintroduce a curvature-like quantity Λp̂,f [ψ] for (σp̂sl )◦ -(L2 ) Cahn–Hoffman
facets [ψ] on Rk and f ∈ Rk .
Let us give the definition of viscosity solution assuming that f ≡ 0 thatappeared in [92, Def. 4.7] with F independent of x and t and σ purely
crystalline.
Definition 6.2. An upper semicontinuous function u on Rn × (0, ∞) is aviscosity subsolution of(6.6)

ut + F (∇u, div ∇σ(∇u)) = 0

if whenever p̂ ∈ Rn , x̂ ∈ Rn , t̂ ∈ (0, T ) and φ is a stratified test function
φ(x, t) = ψ(x′ ) + θ(x′′ ) + p̂ · x + g(t) with g ∈ C 1 (R), θ ∈ C 1 (Rn−k ) satisfying∇θ(x̂′′ ) = 0, and ψ ∈ Lip(Rk ) so that [ψ] is a (σp̂sl )◦ -(L2 ) Cahn-Hoffman
facet, x̂′ ∈ int {ψ = 0}, and u − φ(· − h) has a global maximum at (x̂, t̂) forall h = T h′ with |h′ | small, then
(6.7)

g ′ (t̂) + F (p̂, ess inf Λp̂,0 [ψ]) ≤ 0.
Bδ (x̂)A lower semi-continuous function is a viscosity supersolution if it satisfies the above two conditions with maximum, ≤ and ess inf replaced byminimum, ≥ and ess sup, respectively.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

43

When a forcing f that depends on the x variable is involved, the condition(6.7) has to be weakened as in (6.5), replacing ess inf Λp̂,0 by Λp̂,fˆ, fˆ(w) =f (x̂+T w), for the stability with respect to an approximation by regularized
problems to hold. See [93, Def. 2.7] for more details.
6.2. Comparison principle. In this section we review the comparison
principle for the PDE (6.4). There are a few different versions available
depending on the assumptions on F , σ and f .
Let us first suppose that f ≡ 0. The comparison theorem was first provedin [81, 80] in the setting of an anisotropic total variation flow with smoothanisotropy σ ∈ C 2 (Rn \ {0}) with σ 2 strictly convex and F independent of
the x and t variables in the sense of Definition 6.1 on the torus Tn = Rn /Zn .
We follow [80, Th. 4.1].
Theorem 6.3. Let σ ∈ C 2 (Rn \{0} be an anisotropy and F ∈ C(Rn ×R) be
nonincreasing in the second variable. Let u and v be respectively a boundedviscosity subsolution and a viscosity supersolution of (6.6) in the sense of
Definition 6.1 on Tn × [0, T ]. If u ≤ v at t = 0 then u ≤ v on Tn × (0, T ).
When σ is crystalline while there is still no forcing, f ≡ 0, and F does notdepend on x and t, the following comparison principle for viscosity solutionsin the sense of Definition 6.2 was proved in [91, 92]. We follow the statementin [92, Th. 1.4]. Instead of a torus, the solutions are assumed to be constantoutside of a ball.
Theorem 6.4. Let σ be a crystalline anisotropy and F ∈ C(Rn × R)
be nonincreasing in the second variable, F (0, 0) = 0. Suppose that u is aviscosity subsolution and v is a viscosity supersolution of (6.6) on Rn ×(0, T )in the sense of Definition 6.2 and that there are constants R > 0, a ≤ bsuch that u = a and v = b on (Rn \ BR (0)) × (0, T ). Then if u ≤ v at t = 0,then u ≤ v on Rn × (0, T ).
If we consider a forcing term f that depends on x and t and solutionsof (6.4), an analogous comparison principle was proven in [93, Th. 3.1].
However, in this case at least one of u or v must be continuous, and either
it is also Lipschitz, or further regularity of F must be assumed. For details
see [93].
For simplicity of exposition, we present here the proof of a comparison
theorem in a simplified setting of a stationary problem. We consider the44

Y. GIGA AND N. POŽÁR

pstationary equation with σ(p) = |p|, ∇σ(p) = |p|, that reads


∇u
= f,
(6.8)
u − div|∇u|where f ∈ C(Rn ) is given. This equation is of elliptic type. The definitionof viscosity solution is naturally modified to the following.
An upper semicontinuous function u is a viscosity subsolution of (6.8) if• If φ ∈ C 2 near x̂, ∇φ(x̂) ̸= 0 and u − φ has a local maximum at x̂,
then


∇φ
u(x̂) − div(x̂) ≤ f (x̂).
|∇φ|• If φ ∈ Lip so that [φ] is a σ ◦ -(L2 ) Cahn-Hoffman facet, x̂ ∈ int {φ = 0},
u − φ(· − h) has a global maximum at x̂ for all |h| small, thenu(x̂) − Λf [φ](x̂) ≤ 0.
A definition of a supersolution and a solution can be modified analogously.
Recall that





∇φ
1
∇φ ⊗ ∇φ2
div=
tr I −
∇φ .
|∇φ|
|∇φ|
|∇φ|2
Remark 6.5. Note that the proof in this section applies with a small modification to general problems



∇u
F u, ∇u, div |∇u|
= 0,
where F : R × Rn × R → R is a continuous function, satisfying monotonicity
F (r, p, ξ) ≤ F (r, p, η)for ξ ≥ η,

F (r, p, ξ) ≤ F (s, p, ξ) − µ(s − r)

for r < s,where µ > 0 is a constant. We write the proof for F (r, p, ξ) := r − ξ.
Theorem 6.6. Suppose that u and v are a viscosity subsolution and a
viscosity supersolution on Rn , respectively. Furthermore, assume that u andv are bounded. If there exist constants R > 0 and a ≤ b such that u ≡ a andv ≡ b on Rn \ BR (0) then u ≤ v everywhere.
To show this theorem, we assume that the conclusion does not hold andm0 := sup(u − v) = max(u − v) > 0.
To reach a contradiction, we double variables and for ζ ∈ Rn we consider
the function|x − y − ζ|2.
Φζ (x, y) = u(x) − v(y) −2εMOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

45

We consider the maximum of Φζ as a function of ζ, that is,ℓ(ζ) = sup Φζ .
x,y

It is convenient to introduce the set of points of maxima
A(ζ) := arg max Φζ := {(x, y) | Φζ (x, y) = ℓ(ζ)}
and the set of gradients of |x−y−ζ|
at these points
2ε


x−y−ζ
B(ζ) :=| (x, y) ∈ A(ζ) .
εThe parameter ε > 0 determines how much we penalize x ̸= y. We havethe following standard estimate on |x − y|; see [70]. We give a proof forcompleteness.
√
Lemma 6.7. There is C > 0 such that for all ε > 0, |ζ| ≤ m0 ε we have√
(6.9)
|x − y| ≤ C ε
for all (x, y) ∈ A(ζ).
2

Furthermoreℓ(ζ) ≥

m0.
2

Proof. First note that
m0
m0
|ζ|2
≥ m0 −=.
2ε22x
Let M be a bound on u(x) − v(y). Then for (x, y) ∈ A(ζ) we obtainℓ(ζ) ≥ sup(u(x) − v(x)) −M−|x − y − ζ|2|x − y − ζ|2m0≥ u(x) − v(y) −= ℓ(ζ) ≥
> 0.
2ε
2ε
2

Hence
|x − y| ≤√2M ε + |ζ| ≤√2M ε +√m0 ε.
□

Let ωf be the modulus of continuity of f , that is, ω : [0, ∞) → [0, ∞),ω(0) = 0, ω nondecreasing such that |f (x) − f (y)| ≤ ωf (|x − y|) for all x, y.
√Let ε0 > 0 be such that ωf (C ε0 ) < m40 , where C is from (6.9). We consider
two possible cases:√
1. There exist 0 < ε < ε0 and |ζ| ≤ m0 ε such that B(ζ) ̸= {0}.
√2. B(ζ) = {0} for all |ζ| ≤ m0 ε, 0 < ε < ε0 .
Case 1. We can fix ε, ζ and (x, y) ∈ A(ζ) such that 0 < ε < ε0 ,√|ζ| ≤ m0 ε and p := x−y−ζ̸= 0. Note that this choice together withεLemma 6.7 implies√√m0|f (x) − f (y)| ≤ ωf (|x − y|) ≤ ωf (C ε) ≤ ωf (C ε0 ) ≤(6.10),
4

46

Y. GIGA AND N. POŽÁR

and|x − y − ζ|2m0= ℓ(ζ) ≥.
2ε2
Since the operator is smooth near ∇u ̸= 0, we are well within the classicalviscosity solution framework for continuous operators. In particular, we canuse the now standard maximum principle for semicontinuous functions, see
the exposition in [47] or [83]. Therefore there exist symmetric matrices X, Y
with X ≤ Y , sequences xn → x, yn → y and sequences of C 2 functions φn ,
ψn such that u−φn has a local maximum at xn , v −ψn has a local minimum
at yn , andu(x) − v(y) ≥ u(x) − v(y) −(xn , u(xn ), ∇φn (xn ), ∇2 φn (xn )) → (x, u(x), p, X),(yn , v(yn ), ∇ψn (yn ), ∇2 ψn (yn )) → (y, u(y), p, Y ).
From the definition of viscosity solution we deduce
∇φn ⊗ ∇φn
1
2
∇ φn ≤ f
tr I −u−|∇φn ||∇φn |21∇ψn ⊗ ∇ψn2v−
tr I −∇ ψn ≥ f|∇ψn ||∇ψn |2

at xn
at yn .

In the limit n → ∞, continuity and (6.10) yield

 
1
p⊗p
u(x) −
tr I −
X ≤ f (x)
|p|
|p|2

 
1
p⊗p
m0
m0
≤ v(y) −tr I −Y +.
≤ f (y) +
2
4
|p|
|p|
4
On the other hand, u(x) ≥ v(y) + m20 and X ≤ Y imply


 
 
p⊗p
1
p⊗p
m0
1
tr I −< u(x) −tr I −Y +X .
v(y) −
|p|
|p|2
4
|p|
|p|2
We reach a contradiction.
Case 2. Since we cannot find any maximum of Φζ at which the “gradient” of u nor v is nonzero, we need to construct admissible faceted test
functions for the faceted test in the definition of viscosity solution.
The extra parameter |ζ| provides a little bit of space to construct thesefaceted test functions. The following “constancy” lemma was proven in amore general settings in [70, Lemma 7.5]. We include the proof in our simple
setting for completeness.
Lemma 6.8 (Constancy). Let G ⊂ Rn be a closed ball. If for all ζ ∈ Gthere exists (x, y) ∈ A(ζ) such that x − y − ζ = 0 then ℓ(ζ) is constant on
G.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

47

Proof. Take ζ, µ ∈ G and (x, y) ∈ A(ζ) with x − y − ζ = 0. In particular,ℓ(ζ) = u(x) − v(y). From the definition of ℓ,ℓ(µ) ≥ u(x) − v(y) −|x − y − µ|2|x − y − µ|2= ℓ(ζ) −.
2ε
2ε

Since x − y − ζ = 0, we have|x − y − µ|2 = |x − y − ζ + ζ − µ|2 = |ζ − µ|2 ,yielding|ζ − µ|2ℓ(µ) − ℓ(ζ) ≥ −,2εand, by symmetry,|ζ − µ|2
for all ζ, µ ∈ G.
2ε
We conclude that ℓ(ζ) = ℓ(µ) for all ζ, µ ∈ G.
|ℓ(µ) − ℓ(ζ)| ≤□Since in Case 2 we have B(ζ) = {0} for all ζ ∈ G := B √m0 ε , we canchoose (x̂, ŷ) ∈ A(0) with x̂ − ŷ = 0 and the above lemma yields(6.11)
u(x) − v(y) ≤ ℓ(x − y) = ℓ(0) = u(x̂) − v(x̂)√
Let us set λ = m0 ε. Definingηu = sign(u − u(x̂))

for |x − y| ≤

√

m0 ε.

ηv = sign(v − v(x̂)),the inequality (6.11) yields
sup ηu ≤ inf ηv

for all x.

B λ/2 (x)B λ/2 (x)By the density result Theorem 5.18, there are admissible facets χu andχv satisfying
sup ηu ≤ χu (x) ≤ sup ηu ≤ inf ηv (x) ≤ χv ≤ inf ηv .
B λ/4 (x)B λ/2 (x)B λ/2 (x)B λ/4 (x)Clearly χu = χv = 0 on Bλ/4 (x̂). By the comparison principle for the curvature operator, Proposition 5.17,
ess inf Λf [χu ] ≤ ess sup Λf [χv ],Bλ/4 (x̂)

Bλ/4 (x̂)

which implies(6.12)

Λf [χu ](x̂) ≤ Λf [χv ](x̂).

Let us choose an admissible support function φu ∈ Lip ∩ [χu ]. Since u isbounded and upper semicontinuous, we can multiply the positive part of φu

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Y. GIGA AND N. POŽÁR

by a large positive constant, and the negative part of φu by a small positive
constant, if necessary, to guarantee thatu ≤ φu (· − h) + u(x̂)

for |h| < λ8 .

Note that the equality is attained at x̂ as φu = 0 on Bλ/4 (x̂). Therefore φuis an admissible faceted test function for the viscosity solution test andu(x̂) − Λf [φu ](x̂) ≤ 0.
Similarly, we can find φv ∈ Lip ∩ [χv ] withv(x̂) − Λf [φv ](x̂) ≥ 0.
Thus, recalling (6.12), we haveu(x̂) ≤ Λf [φu ](x̂) ≤ Λf [φv ] ≤ v(x̂) ≤ u(x̂) − m0 < u(x̂),
a contradiction. This finishes the proof of Theorem 6.6.
6.3. Existence of solutions. The existence of viscosity solutions is usually
established using Perron’s method: the largest subsolution of the problem is
automatically a solution. However, the operation of taking a supremum ofa class of viscosity solutions requires a stability property whose validity isunclear for the viscosity solutions considered here in dimensions n ≥ 2. In
one dimension, Perron’s method was used to construct viscosity solutionsfor (6.4) in [79]. This however requires a careful treatment of the nonlocal
anisotropic curvature.
The main issue with the stability required for the supremum of subsolutions to be a subsolution is the discontinuity of the value of Λf [ψ] whena facet bends or breaks. In the standard proof of this stability, it is crucial
to localize by replacing a test function φ by another so that u − φ can beassumed to have a strict local maximum (or minimum). Due to the discontinuity of our operator Λf with respect to such bending, this tool is notavailable.
The approach that was taken in [81, 91] is via stability with respect toapproximation by problems with regularized σ. In particular, we consider
two ways of approximating a crystalline σ:2(a) σm ∈ C 2 (Rn ) with a−1
m I ≤ ∇ σm ≤ am I for some am > 0, σm is adecreasing sequence with σm → σ locally uniformly.
2is strictly(b) σm are anisotropies with σm ∈ C 2 (Rn \ {0}) such that σmconvex and σm → σ locally uniformly.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

49

However, for various reasons related to the regularity of the solutions of
the approximating problems, we need to assume that F does not dependon the x variable, and consider solutions of the regularized problems(6.13)

ut + F (t, div ∇σm (∇u) − f ) = 0.

Since σm are C 2 and convex, the classical theory of viscosity solutions applies, including the unique existence of solutions for given bounded continuous initial data.
We have the following stability result when approximating using (a), see[93, Th. 4.1] or [91, Th. 8.1], which resembles the usual stability of viscosity
solutions in the classical theory. Let lim sup∗ (resp. lim inf ∗ ) denote therelaxed upper limit (lower) limit defined by(lim sup∗ um )(x, t) = lim sup uk (y, s) |x − y| + |t − s| + 1/k < 1/mm→∞(lim inf ∗ um )(x, t) = − (lim sup(−um )) (x, t)
for a sequence {um } of functions on Rn × [0, ∞).
Theorem 6.9. Let σ be a crystalline anisotropy and assume that F does
not depend on the x variable and f ∈ C(Rn × R) is Lipschitz continuous inspace, uniformly in time. If {um } is a locally bounded sequence of viscosity
solutions of (6.13) with σm as in (a) above, then lim sup∗m→∞ um is a viscosity
subsolution of (6.4), and lim inf ∗m→∞ um is a viscosity supersolution of (6.4).
The main idea of the proof of Theorem 6.9 is inspired by the perturbed
test function method due to Evans [58]. Let us for simplicity assume thatf ≡ 0. The crystalline mean curvature, or specifically the operator Λ0 [ψ] isnonlocal on the facets of ψ. In contrast, the elliptic operators div ∇σm (∇ψ)
are local and they are in fact zero on the facets of ψ. To recover the nonlocal
information in the limit m → ∞, we perturb the test function ψ using
a sequence ψm of uniformly converging C 2 functions ψm → ψ, such thatdiv ∇σm (∇ψm ) approximates the value of Λ0 [ψ] in a suitable sense at thecontact point.
Such approximation is available via the resolvent problem for the regularized energy Em , with σ replaced by σm in (5.2). For a given a > 0 andψ ∈ L2 (Tn ), there exists a unique solution ψa,m ∈ L2 (Tn ) of
ψa,m + a∂Em (ψa,m ) ∋ ψ.
If ψ ∈ Lip(Tn ), then ψa,m is Lipschitz uniformly in a and m by the comparison principle like Proposition 5.7 and translation invariance of the operator, and in fact it is C 2 by the elliptic regularity theory. Since Em Moscoconverges to E (see (1.7) for the definition), we have a convergence of the

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Y. GIGA AND N. POŽÁR

resolvent solutions ψa,m → ψa in L2 (Tn ), see [12], and hence uniformly
by the uniform Lipschitz continuity. Using Proposition 5.6, we can deducethat ψa,m uniformly approximate ψ as a → 0 and then m → ∞. Functions
ψa,m are used to build test functions for the regularized problem, and allow
us to deduce that lim sup∗m→∞ um is a viscosity subsolution of (6.4), andlim inf ∗m→∞ um is a viscosity supersolution of (6.4).
Approximation using (b) is relevant when considering the crystalline
mean curvature flow as a limit of a smooth anisotropic mean curvature
flow. To prove the stability for (b), we use the stability Theorem 6.9 toapproximate each σm by a sequence of C 2 functions σm,δ and therefore weneed to know that a given solution um can be approximated by a sequenceof solutions um,δ with this anisotropy. This is known for example when um
have continuous bounded initial data. We have the following stability result,[93, Th. 4.4].
Theorem 6.10. Let σ, F and f be as in Theorem 6.9. Let T > 0 and let um
be a locally bounded sequence of viscosity solutions of (6.13) on Rn × (0, T )with σm as in (b) with initial data um (·, 0) = u0,m , where u0,m ∈ C(Rn ) are
uniformly bounded. Then lim sup∗m→∞ um is a viscosity subsolution of (6.4),
and lim inf ∗m→∞ um is a viscosity supersolution of (6.4).
Now with the stability with respect to approximation by the regularized
problems established, and the comparison principle discussed in Section 6.2,
we can follow the standard idea to show existence of (6.4) for given initial
data when the operator F does not depend on the x variable. For given
bounded uniformly continuous initial data, we take um solutions of the regularized problem with initial data u0 from Theorem 6.10. By using barriers
at t = 0, we can show that the limits satisfy
lim inf ∗ um t=0 ≥ u0 ,m→∞

lim sup∗ um t=0 ≤ u0 .
m→∞

From the comparison principle for (6.4) we immediately have
lim sup∗ um ≤ lim inf ∗ um .
m→∞

m→∞

This implies that both limits are equal, the convergence is locally uniform,
and the limit is a viscosity solution of (6.4).
If the forcing f depends on x, there is an additional diﬀiculty that thecomparison principle for semi-continuous solutions is not available, see [93,Sec. 3]. The comparison principle established in [93] requires that at least
one of the solutions is continuous. Fortunately, for operators F that comefrom the level set formulation of geometric motions one can prove uniform

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

51

Lipschitz bounds in space and uniform Hölder bounds in time on the approximating sequence um for Lipschitz initial data u0 , see [93, Sec. 5]. Therefore
the convergence um is locally uniform for subsequences and the limits area priori continuous. In particular, the restricted comparison principle applies and existence of solutions can be established. We have the following
existence theorem, [93, Th. 1.1].
Theorem 6.11. Assume that g ∈ C(S n−1 × R) is Lipschitz continuous in
the second variable uniformly in the first variable and non-decreasing in thesecond variable, σ is a crystalline anisotropy and f ∈ C(Rn ×R) is Lipschitz
continuous in space uniformly in time. Then there is a unique global-in-timelevel set flow to
V = g(ν, κσ + f (x, t))when the initial hypersurface is compact.
Remark 6.12. If f is constant, then the global Lipschitz continuity of F
is unnecessary [91], [92]. In particular, it applies to (2.1). In the case n = 2,
it applies to a general anisotropy under a slightly different definition of asolution [73]. Note that the level set equation for V = κσ isut = |∇u| div ∇σ(∇u)
so that each level set of u moves by V = κσ . The level set flow is a level setof a viscosity solution u. Its uniqueness (up to fattening) is guaranteed by
the comparison principle and an invariance under a change of the dependent variable u (representing its level sets) together with Lemma 5.16. This
procedure is standard for a level set flow; see e.g. [83]. The terminology ofthe level set flow here is different from that in Section 7.
6.4. Convergence of various approximations. It is well-known thatthe solution of the mean curvature flow equation is approximated by thatof the Allen-Cahn equation; see [131], [34], [44], [59]. Anisotropic versionof the Allen-Cahn equation is introduced by [129], which is an L2 -gradient
flow ofZ 
1
12
Fε (v) =σ(∇v) + 2 (W (v) − ελF (v)) dx.
2εRnHere, W (v) is a double-well potential typically W (v) = (v 2 − 1)2 /2 andF (v) = Cv with constant C for simplicity. The parameter λ > 0 should bechosen in a suitable way. In an explicit form, the anisotropic Allen-Cahn
equation reads
1(6.14)
β(∇v)vt − div (σ(∇v)ζ(∇v)) + 2 (W ′ (v) − ελC) = 0ε52

Y. GIGA AND N. POŽÁR

with some kinetic coeﬀicient β > 0 which is positively one-homogeneous;here ζ(p) = ∇p σ(p). If one considers an L2 gradient flow of F , the equation(6.14) with β ≡ 1 is obtained. The extra multiplier comes from the modelling to include anisotropy of mobility. For a given closed interface Γ0 , weconsider a function v0ε which converges in a suitable way to −1 in an open
set surrounded by Γ0 and to 1 outside the closure of the open set. It is expected that the solution of the anisotropic Allen-Cahn equation with initial
data v0ε converges to 1 inside an open set surrounded by Γt and −1 outside
Γt and this open set, where Γt is a (generalized) solution to the interface
equationβ(n)V = σ(n)(κσ − C).
(Here λ should be taken as λ = 2/3 if W (v) = (v 2 − 1)2 /2.) Formal asymptotic analysis is carried out by [129], [166] and [29], which derives theinterface equation. For smooth anisotropy with β ≡ 1, the convergence isestablished by [54] when the solution of the interface equation is smooth,
here W is taken as double-obstacle type, for example, W (v) = 1 − v 2 in|v| ≤ 1 and W (v) = ∞ for |v| > 1. This result is extended when Γt is a generalized solution (a level-set solution allowing fattening). In [90] it is shownthat such convergence is uniform in σ provided that the Frank diagram Fσ
is bounded by a ball both from inside and outside. It does not depend on
regularity of σ.
For crystalline σ under β ≡ 1, the convergence with some rate is established for planar crystalline flow [23]. It is somewhat extended to higher
dimension for a special class of solutions of the interface equation; its existence is not clear [24]. Several explicit examples of convergence are givenby [161]. One of the reasons why β ≡ 1 is assumed is that the notion of solutions for the Allen-Cahn equation is unclear. Maybe a viscosity approach
will resolve this issue.
Since our solution for the interface equation for crystalline σ is obtained
as a limit of smoother problems as in the previous subsection, combining
uniform convergence with respect to σ we are able to prove the convergenceas ε → 0 by approximating β and σ by smooth function; see [90, Theorem2.4]. Note that in two dimensional case, the stability was proved in [73].
Another typical way to approximate a solution is what is called Chambolle’s scheme introduced by [39]. We here give its anisotropic version [38],[43]. We consider
V = M (ν)κσ .

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

53

We set the support function of the polar of 1/M (Frank diagram of M ) by
M 0 , i.e.,
M 0 (x) := sup x · p |p| ≤ 1/M (p/|p|) .
Here M is assumed to be positive on S n−1 . The function M 0 is convex, positively 1-homogeneous in Rn and it is positive outside the origin. However,it may not satisfy the symmetry M (x) = M (−x) so that distM0 (x, y) =M 0 (x − y) is a non-symmetric distance. For a given bounded set E0 in Rn ,let dM 0 (x, E0 ) denote its anisotropic signed distance, i.e.,dM 0 (x, E0 ) := distM 0 (x, E0 ) − distM 0 (x, E0c ),x ∈ Rn ,where
distM 0 (x, E0 ) := inf distM 0 (x, y).
y∈E0

We next consider an energy functional of the form

Z 
1
2
Jh (v, E0 ) =σ(∇v) + |v − dM 0 | dx
2hΩ
for a domain Ω containing E0 with a small parameter h > 0. This value isfinite in L2 (Ω) ∩ BV (Ω) so we regard Jh as a lower semicontinuous convex
functional on L2 (Ω) by interpreting its value equal to ∞ on L2 (Ω)\BV (Ω).
It admits a unique minimizer w = argmin Jh . We introduce the operator ThasTh (E0 ) = {x ∈ Rn | w(x) ≤ 0} .
An approximate flow is defined by applying the above step iteratively as(6.15)⌊t/h⌋

E h (t) = Th

(E0 ),

where ⌊s⌋ denotes its integer part of s > 0. We expect that E h converges
to the level-set solution of V = M (ν)κσ as h → 0, for example, in the
Hausdorff distance sense uniformly in t ∈ [0, T ] with finite T . Let us give avery heuristic argument. We consider the isotropic case V = κ so that M = 1
and σ(p) = |p|. Then the minimizer w satisfies the resolvent equation
∇w
w−d
− div
= 0,
h
|∇w|
where d denotes the Euclidean signed distance of E0 . This is the implicit
Euler scheme for the total variation flow. The signed distance function satand it is expected that theisfies |∇d| = 1 on the interface Γt so V ≈ w−dhzero level of w approximates the solution Γt .
The isotropic case of this scheme was first introduced in [39], which givesa monotone way to realize the time discrete scheme proposed by [2]; see also[128]. In [39] L1 convergence: E h (t) → E(t) on [0, T ], where E(t) is the levelset solution of V = κ (starting from a closed set E0 with E0 = int E0 ) was

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Y. GIGA AND N. POŽÁR

established provided that no fattening phenomena occur. Its anisotropic
extension is done by [38] in the case when E0 is convex and compact under
the assumption that σ/M is constant on S n−1 ; see [43] for non-convex initial
data; here anisotropy is assumed to be smooth. In [18] for a non-smooth σ
including crystalline, a unique solution for V = σκσ is constructed when
E0 is convex and compact by defining a solution by the distance function.
For smooth anisotropy for a bounded nonconvex initial data, the Hausdorffconvergence is proved in [57], where they prove locally uniform convergenceof an associated function


[t/h]
uh (x, t) = Sh u0 (x)with(S h u0 )(x) = sup µ ∈ R x ∈ Th ({x ∈ Rn | u0 (x) ≥ µ}) .
Although it is remarked in [57] and [43], the case when σ and M are unrelated is not discussed in detail. In [106] a proof based on the distancefunction is given for several choices of σ and M and general initial data notnecessarily compact mostly for smooth case. However, it is also shown in[106] that if the solution of crystalline anisotropy has a stability property weare able to prove the convergence of Chambolle’s scheme by approximating
M and σ. Since at that time, the stability was only available in two dimensional case [73], convergence result in [106] looks limited but it applies togeneral dimension at least for purely crystalline anisotropy since the stability holds for general dimension as discussed in the previous subsection.
The reason why M and σ are approximated by a smoother one in Chambolle’s scheme in [106] seems to avoid analysis for the resolvent equation for
non-smooth M and σ, so it seems that it is not substantial.
In the next section we discuss a notion of solutions based on distancefunctions to the evolving surface that can be showed to be the limits of thediscrete evolutions (6.15) given by Chambolle’s scheme, see Theorem 7.7.
7. Approach by distance functions
In this section we discuss an alternative approach to defining a notionof solutions of the crystalline mean curvature flow that appeared in a seriesof papers by Chambolle, Morini, Novaga and Ponsiglione [42, 40, 41]. The
main idea is to require that the distance function to an evolving set is a
sub/supersolution of a related partial differential equation in the sense of
distributions.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

55

This approach applies to a form of the crystalline mean curvature flowthat is linear in the curvature term:
V = M (ν)(κσ − f ).

(7.1)

However, both σ and M can be arbitrary anisotropies, not necessarily crystalline. For simplicity, we will assume that both σ and M are even, thatis, σ(p) = σ(−p) and M (p) = M (−p) for all p ∈ Rn . This restriction
however does not appear to be essential. Moreover, the initial data E 0
can be an unbounded closed set, and the forcing term needs to be onlyf ∈ L∞ (Rn × (0, T )) with f (·, t) Lipschitz uniformly in t.
The distance function must be adapted to the mobility M . As in [40] forany norm η we denotedistη (x, E) := inf η(x − y),y∈E

E ⊂ Rn .

Note that distη (x, ∅) = +∞.
Let En ⊂ Rn be a sequence of closed sets and E ⊂ Rn a closed set.
K
We say that En converges to E in Kuratowski sense, and write En → E,
if distη (·, En ) → distη (·, E) locally uniformly in Rn for some norm η. It iseasy to see that if this converges for one norm, it converges for all norms.
The following definition appeared in [40].
Definition 7.1. Let E 0 ⊆ Rn be a closed set. Let E be a closed set in
Rn × [0, +∞) and for each t ≥ 0 define E(t) := {x ∈ Rn : (x, t) ∈ E}. We
say that E is a superflow of (7.1) with initial datum E 0 if:(a) E(0) ⊆ E 0 ,K(b) E(s) → E(t) as s ↗ t for all t > 0,(c) If E(t) = ∅ for some t ≥ 0, then E(s) = ∅ for all s > t.
(d) Set T ∗ := inf {t > 0 : E(s) = ∅ for s ≥ t}, and◦d(x, t) := distM (x, E(t))

for all (x, t) ∈ Rn × (0, T ∗ ) \ E.

Then there exists K > 0 such that the inequality(7.2)

dt ≥ div z + f − Kdholds in the distributional sense in Rn × (0, T ∗ ) \ E for a suitablez ∈ L∞ (Rn × (0, T ∗ )) such that z ∈ ∂σ(∇d) a.e., div z is a Radon
measure in Rn × (0, T ∗ ) \ E, and(div z)+ ∈ L∞ ({(x, t) ∈ Rn × (0, T ∗ ) : d(x, t) ≥ δ})for every δ ∈ (0, 1).

An open set A ⊂ Rn × [0, +∞) is a subflow of (7.1) with initial datum
E 0 if Ac is a superflow of (7.1) with f replaced by −f and with initial datum(int E ◦ )c .

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Y. GIGA AND N. POŽÁR

A closed set E ⊂ Rn × [0, +∞) is a solution of (7.1) with initial datum
E 0 if it is a superflow and if int E is a subflow, both with initial datum E 0 .
The condition (b) is meant to prevent a possibility that E expands discontinuously, for example a bubble closing up, which cannot be ruled outby (7.2).
Note that K is related to the Lipschitz constant of f with respect to the
distance induced by M . In fact, in the smooth case σ, M , M ◦ ∈ C 2 (Rn \{0}),f continuous, then E is a superflow in the sense of Definition 7.1 if and only
if −1E is a viscosity supersolution of the level set equationut = M (∇u)(div ∇σ(∇u) + f ),in Rn × (0, T ∗ ]; see [40, Lemma 2.6]. For viscosity supersolution −1E we cantake K = Lip(f ) in (7.2).
We cannot in general expect uniqueness of a solution in the sense of
Definition 7.1 since there may occur fattening phenomena. The comparison
principle between superflows and subflows requires a positive distance between initial data and therefore by itself does not provide uniqueness. The
following theorem appeared in [40].
Theorem 7.2 (c.f. [40, Theorem 2.7]). Let E be a superflow with initial
datum E 0 and F be a subflow with initial datum F 0 in the sense of Defini◦tion 7.1. If distM (E 0 , (F 0 )c ) =: δ > 0, then◦distM (E(t), F (t)c ) ≥ δe−Kt

for all t ≥ 0,

where K > 0 is the constant in (7.2) for both E and F .
To obtain uniqueness, [40] introduce the associated level-set flow.
Definition 7.3. Let u0 be a uniformly continuous function on Rn . We saythat a lower semicontinuous function u : Rn × [0, ∞) → R is a level-set
supersolution corresponding to (7.1) with initial datum u0 if u(·, 0) ≥ u0
and if for a.e. λ ∈ R the closed sublevel set {u ≤ λ} is a superflow of (7.1)
in the sense of Definition 7.1 with initial datum {u0 ≤ λ}.
Similarly, an upper semicontinuous function u : Rn × [0, ∞) → R is alevel-set subsolution corresponding to (7.1) with initial datum u0 if −u isa level-set supersolution in the previous sense, with initial datum −u0 andwith f replaced by −f .
A continuous function u : Rn × [0, ∞) → R is a level-set solution corresponding to (7.1) with initial datum u0 if it is both a level-set supersolution
and level-set subsolution with the same initial datum.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

57

Our terminology here is different from that in [83, Chapter 5]. A superflow here is called a set-theoretic supersolution in [83]. A level set supersolution in [83] is a superflow given by sublevel set of a continuous level-set
supersolution.
The following comparison theorem was proven in [40].
Theorem 7.4 (c.f. [40, Theorem 2.5]). Let u0 , v 0 be uniformly continuousfunctions on Rn and let u, v be respectively a level-set subsolution with initial
datum u0 and a level-set supersolution with initial datum v 0 , in the sense of
Definition 7.3. If u0 ≤ v 0 then u ≤ v.
The main idea using Theorem 7.2 to prove Theorem 7.4 is that due to theuniform continuity, the superflow {u ≥ λ1 } and the superflow {v ≤ λ2 } forλ1 > λ2 are initially separated by a positive distance so that Theorem 7.2
applies.
It remains to establish the existence of the level-set solutions. In thesmooth case, the notion in the sense of Definition 7.3 is equivalent to thestandard notion of viscosity solutions. In general, an approximation by asequence of smooth anisotropies Mn , σn and a stability result establishedin [41, Theorem 2.8] allows to construct a level-set solution as the limit ofviscosity solutions. However, the stability result requires that the approximating sequence Mn is uniformly σn regular, that is, it is required that there
exists ε0 > 0 such that
Mn = M0,n + ε0 σnfor all n for some convex functions M0,n . Or equivalently, the Wulff shapes
WMn is must satisfy interior Wσ condition uniformly in n. Intuitively, if M◦
is σ regular the level sets of d := distM (·, E) have σ-curvature bounded by
C/d for some constant C > 0.
In particular, this stability result is only able to construct level-set solutions in the sense of Definition 7.3 if M is σ-regular. Therefore the authorsof [41] propose a definition of a solution via approximation.
Definition 7.5 (c.f. [41, Definition 3.6]). A continuous function u : Rn ×
[0, ∞) → R is a solution via approximation to the level set flow corresponding to (7.1) with initial datum u0 if there exists a sequence {Mn } of σ-regularmobilities such that Mn → M and, denoting un the unique level-set solutionof (7.1) with mobility Mn and initial datum u0 , we have un → u locallyuniformly in Rn × [0, ∞).
Such a solution always exists and is independent of the approximating
sequence {Mn }.

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Y. GIGA AND N. POŽÁR

Theorem 7.6 (c.f. [41, Theorem 3.7]). Let u0 be a uniformly continuousfunction on Rn . There exists a unique solution u in the sense of Definition 7.5 with initial datum u0 .
Alternatively, the level-set flow solutions in Definition 7.3 and the solutions via approximation Definition 7.5 can be constructed using a minimizing movement scheme; see [40] and the discussion in Section 6.4. To be
more precise, for given initial data u0 one can define the level set discrete
evolution uh : Rn × R → R asuh (x, t) := inf {λ ∈ R : x ∈ Eλ,h (t)},where Eλ,h (t) is the discrete evolution given by Chambolle’s scheme in (6.15)
with E0 := {u0 ≤ λ}. The following result was proved in [40, Th. 5.7].
Theorem 7.7. Let u0 be a uniformly continuous function on Rn . The uniquesolution of (7.1) in Theorem 7.6 is the locally uniform limit in Rn × [0, +∞)as h → 0+ of the level set minimizing movements uh .
Here are the types of solutions that are currently available if velocity
law is linear in curvature, i.e., of the form (7.1), and the initial data u0 isconstant outside of a bounded ball:• σ smooth, M arbitrary: classical viscosity solutions [45]• σ purely crystalline, M arbitrary: crystalline viscosity solutions [93]• σ arbitrary, M is σ-regular: level-set solutions [40, 41]• M , σ arbitrary: solutions via approximation [40, 41]
If the velocity law is not linear in curvature, only the viscosity solutions
are currently available. On the other hand, the latter two notions apply also
to general uniformly continuous initial data.
If the law is linear in the curvature, σ is purely crystalline and u0 isconstant outside of a large ball, so that the notions of crystalline viscosity
solutions and solutions via approximation both apply, they also give thesame solutions. This can be seen by applying stability properties under theapproximation of σ by smooth σn .
Notion of solutionsσM
2classical viscosity solutions [45]Cany+
crystalline viscosity solutions [93]purely crystalline
any+level-set solutions [40, 41]any
σ-regular
solutions via approximation [40, 41]any
anyany+: allows any nonnegative function, not just anisotropies.

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

59

8. Some numerics
The study of the crystalline mean curvature flow using numerical methods goes back to the seminal work of J. E. Taylor, who developed the crystalline algorithm based on the polygonal flow in Section 3 in both two andthree dimensions [157, 159], including spiral growth in two dimensions andobservation of possible facet breaking in three dimensions. Examples of facet
breaking were further numerically investigated in [138].
In higher dimension, the crystalline algorithm is limited to evolutions inwhich topological changes or facet breaking do not occur, or the result of
facet breaking can be computed and produces facets with somewhat simple
topology. In a more general situation, the level set method is popular to
track the evolution past singularities. However, the level set equation forthe crystalline mean curvature is rather singular and so its direct use islimited.
An anisotropic version of the Allen–Cahn equation was used to approximate the crystalline mean curvature flow in three dimensions in [147]. In
particular, an example of facet bending was demonstrated.
A. Chambolle reformulated the minimizing movements scheme of [2]and [128] for anisotropic mean curvature flow in terms of the signed distance function as the level set function and proposed a numerical method
to solve the resulting minimization problem in [39] (see Section 6.4 for moredetails). In [140] it was observed that the minimization problem in Chambolle’s scheme can be solved eﬀiciently using the split-Bregman method forthe total variation minimization [97], and presented computational results
for two dimensional crystalline mean curvature flow. However, the method
easily generalizes to any dimension; see [148] for computational results forthree dimensional evolutions.
It is also possible to regularize the crystalline anisotropy and consider
the almost-crystalline but smooth anisotropic mean curvature flow, withmany numerical methods available. One way to approximate the smoothanisotropic mean curvature flow numerically is using the Allen–Cahn equation (6.14) with double obstacle potentials (see §6.4) [13, 14, 15]. For estimates of the Allen–Cahn approximation see for example [53]. Anotherpossibility is to track the evolving surface explicitly using a parametric approach [51, 16, 17].
For an extensive review of the early numerical approaches see [48].

60

Y. GIGA AND N. POŽÁR

9. Volume-preserving and fourth-order problems
9.1. Volume preserving flow. In many applications it is important toimpose that the volume of the set surrounded by the evolving surface ispreserved. Examples include crystal growth, droplet motion and bubbles.
A common way to achieve this for the mean curvature flow is to add a
Lagrange multiplier to the velocity law. Consider a family of hypersurfaces{Γt } with Γt = ∂Ωt for some evolving set {Ωt } that evolves with the velocitylaw
V = g(ν, κσ + λ)on Γt = ∂Ωt .

Here the forcing term λ = λ(t) is chosen so that|Ωt | = |Ω0 |t ≥ 0.

If {∂Ωt } is suﬀiciently smooth, we haveZd|Ωt | =
V dHn−1 ,dt∂Ωtand λ(t) must be chosen so thatZg(ν, κσ + λ(t)) dHn−1 = 0,

t ≥ 0.

∂Ωt

In general, the regularity of λ is not clear.
The problem has been studied in the case of linear dependence on κσ ,V = M (ν)(κσ + λ).
For convex initial data, the existence of solutions and convergence to the
Wulff shape Wσ was shown in [8] for smooth σ, and in [19] for nonsmoothσ, generalizing the classical result for the isotropic mean curvature flowof [104]. For a planar crystalline flow, a similar result has been proved by[168]. Moreover, it approximates corresponding smooth problems as provedin [164].
For general initial data, the existence of solutions still remains mostly
open. In the isotropic case, global existence results are available under acertain energy convergence assumption [135, 125].
One can also consider initial data for which topological changes do notoccur like star-shaped sets in the isotropic case [118] or sets that satisfy acertain reflection symmetry property in the anisotropic case including somecrystalline flow [119].

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

61

9.2. Fourth-order problem. We begin with a fourth-order model to describe a relaxation process of a crystal surface by surface diffusion under
the roughening temperature, which is proposed by [152] as mentioned in
Section 2. It is explicitly written aswt = −∆ div (∇w/|∇w|) + β div (|∇w|∇w)
with β > 0, where w(x, t) represents the height of a crystal at x and at time
t. Fortunately, this can be handled by the theory of maximal monotone op1(Tn ) denote the space of average-free H 1 functions
erators [74], [86]. Let Hav
equipped with the inner productn Z
X(f, g)1 :=∂xi f ∂xi g dx.
i=1

In other words,
1(Tn ) =
Hav

Tn





Z
f ∈ L (T )
2

n

1/2
∥f ∥H 1 = (f, f )H 1 < ∞,f dx = 0 .
Tn

It is of course a Hilbert space. This space is densely embedded in


Z
n
2
n
2
f dx = 0 .
Lav (T ) = f ∈ L (T )Tn

−1
1
. The canonical(under L2 pairing) is denoted by Hav
The dual space of Hav−1
1
isomorphism from Hav to Hav is denoted by −∆ and it agrees with the−1(Tn ) is a Hilbert
usual minus Laplacian for distributions. The space Hav
space equipped with the inner product(f, g)−1 := (−∆)−1 f, g ,−1−11(Tn ) is. This Hav
and Hav
where ⟨ , ⟩ denotes a canonical pairing of Hav
our basic Hilbert space. We set energy
ZZ
βEβ,p (w) :=|∇w| +|∇w|p dx
p Tn
Tn−1(Tn ), i.e.,with p > 1, β ≥ 0. We consider the gradient flow of Eβ,p in Hav(9.1)

wt ∈ −∂Eβ,p (w).

Formally, this is an equation
p−2
wt = −∆ div (∇w/|∇w|) + β div |∇w| ∇w .
If β = 0, this is nothing but the fourth-order total variation flow. A general
theory guarantees the global-in-time existence of a solution to (9.1) with
−1(Tn ) since Eβ,p is a lowerβ ≥ 0, p > 1 for any initial data w0 ∈ Hav−1(Tn ). The important difference
semicontinuous convex functional on Hav
between second-order and fourth-order is that in the latter the comparison

62

Y. GIGA AND N. POŽÁR

principle fails. Here is an example for the case β = 0, which implies thatthe comparison principle should not hold.
Theorem 9.1 ([74]). For the fourth-order total variation flow (9.1) (β = 0),
the solution may become discontinuous in space even if the initial data is
Lipschitz continuous.
In [74], this is proved by giving an explicit example for n = 1, which works
for general n. For the second-order problem, the comparison principle yields
Lipschitz preserving property. Indeed, if the initial data w0 is L-Lipschitz,then
w0 (x) ≤ w0 (x + h) + Lh =: w0h .
The solution starting with w0h is w(x+h, t)+Lh. If the comparison principle
were valid, we would havew(x, t) ≤ w(x + h, t) + Lh.
Similarly,
w(x, t) ≥ w(x + h, t) − Lh,so we would have |w(x, t) − w(x + h, t)| ≤ Lh. Theorem 9.1 shows that thecomparison principle fails for (9.1) with β = 0.
Note that for β > 0, w(·, t) is spatially continuous for n = 1 sinceEβ,p (w) < ∞ implies continuity.
−1(Tn ) orThere is a characterization of the subdifferential ∂Eβ,p in Hav
similar space see [116], [117] for β > 0 and [86] for β = 0. The minimal
section is also calculated in [116] and [74] in the case n = 1; for radial casewith β > 0, see [117]. There are a few differences between second-order andfourth-order problem. First, the value of ∂ ◦ Eβ,p on a facet is not determinedin a neighborhood of a facet in fourth-order problem. This is in some sense−1. Second, theexpected because of a “nonlocal property” of a norm on Hav◦
value of ∂ Eβ,p may contain δ-type function (n = 1), which yields instantdiscontinuity of a solution in Theorem 9.1.
Of course, there are several common properties between second-orderand fourth-order problems. For example, the solution will stop to move infinite time. In fourth-order problems, it is only known for n = 1, 2, 3, 4. LetT∗ (w0 ) be the extinction time of the solution of (9.1), i.e.,T∗ (w0 ) = sup {t ∈ R | w(x, t) ̸≡ 0} .
Theorem 9.2 ([86]). Let w be the solution of (9.1) with initial data w0 ∈−1
. There exists a constant C depending only on ωi and n (Tn = Πni=1 (R/ωi Z))
Hav

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

63(independent of dilation) such that−1for n = 4
T∗ (w0 ) ≤ C∥w0 ∥Hav


!1/α
α
a∥w0 ∥H −1∥w0 ∥X 
av
T∗ (w0 ) ≤
1+
− 1
a
C∥w0 ∥αXfor1 ≤ n ≤ 4, 1 ≤ p ≤ ∞with θ ∈ 12 , 1 satisfying 1 + n2 = θ(n − 1) + (1 − θ)(3 + n/p), wherea = (ω1 · · · ωN )1/p , α = 2 − 1/p and ∥w0 ∥X = ∥(−∆)−1 w0 ∥Ẇ −1,p .
Here, Ẇ −1,p is the dual of the homogeneous Sobolev space Ẇ 1,p , i.e.,Z∞nf φ dx φ ∈ C (T ), ∥∇φ∥Lp′ ≤ 1 , 1/p+1/p′ = 1.
∥f ∥Ẇ −1,p = supTn

The proof for n = 4 is easy, so we give it here for β = 0; the case β > 0
can be proved essentially in the same way. We multiply the equation
wt = (−∆) div (∇w/|∇w|)
with (−∆)−1 w and integrate in space to get a dissipation identity
Z
1d
2
|∇w|
∥w∥Hav
(9.2)
−1 =
2 dt
Tn
since (u, v)−1 = ⟨(−∆)−1 u, v⟩. In the case n = 4 and θ = 1, by the Sobolev
and the Calderón-Zygmund inequality for ∇(−∆)−1/2 , we have
−1 =(−∆)−1/2 w L2 ≤ A′ (−∆)−1/2 w Lp ≤ Ap ∥w∥Lp ,∥w∥Hav

1/2 = 1/p−1/4

for some constants A′ and Ap . Again by the Sobolev inequality, there is aconstant S satisfying
Z∥w∥L4/3 ≤ S

|∇w|.

Tn

We now conclude thatZ
−1 ≤ A4/3 S∥w∥Hav

Tn|∇w|.

Thus we conclude
1d
−1
−1 ,
∥w∥2Hav∥w∥Hav−1 ≤ −(A4/3 S)
2 dt−1 with C = A4/3 S. For general case, we
which yields T∗ (w0 ) ≤ C∥w0 ∥Hav
establish an interpolation inequality
θ
Z
1−θ
−1
−1 ≤ C|∇w|(−∆) w Ẇ −1,p∥w∥HavTn

and a rough growth estimate for a weaker normd(−∆)−1 w Ẇ −1,p ≤ a1/p .
dt

64

Y. GIGA AND N. POŽÁR

We then apply these inequalities to the dissipation identity (9.2) to get thedesired estimate. For details, see [86], [87]. Combining a dissipation identity,
an interpolation inequality and a growth of a weaker norm is also a key idea
to estimate the coarsening rate in a surface diffusion flow as studied in [121].
There are several numerical studies for the above fourth-order singular
diffusion equations. A numerical computation for β > 0, p = 3 is done by[122]. Their numerical scheme regularizes the singularity. A duality based
numerical scheme which applies the forward-backward splitting has been
proposed in [89]. A Bregman method is adjusted to the fourth-order problem
by [94], where the singularity at ∇w = 0 is not regularized.
We are interested in a polygonal flow by surface diffusion. Formally, atypical example is V = −∆κσ when σ is crystalline. In [37] evolution by
polygonal flow is proposed and there are several numerical tests. However,there is no general notion for a solution of closed curves. It is not clear whatclass of polygonal flows is preserved during evolution. Recently, in [76] itis shown that there is a special class of periodic piecewise linear graph-likecurves which is preserved under the evolution provided that the problem iswritten as a gradient flow of a lower semicontinuous convex function.
If the dependence on κσ is nonlinear like in (2.3), no notion of a generalsolution is known. By studying a special solution of (2.3), a new phenomenon
is found in [127] with discussion on a relation with a step motion. There isa numerical work to calculate (2.3) in [46].
Recently, the fourth-order total variation flow equation is studied in Rn[88]. In this case, the definition of a solution itself is not direct, especially forn = 1, 2. A notion of calibrability is introduced there, and it is shown thatall balls are calibrable. However, unlike in the second-order total variation
flow, the outside of a ball is calibrable if and only if n ̸= 2. If n ̸= 2, allannuli are calibrable. However, in the case n = 2, an annulus is calibrable if
and only if it is suﬀiciently thin. Moreover, for n = 1, 2, the solution stays
non-zero for all time if the initial data is a characteristic function of a ball;the finite extinction property does not hold in the case of Rn (n = 1, 2) [88].

Appendix A. A direct proof of properties of ΨIn this section we give a direct proof of the convexity and lower semicontinuity of Ψ used in Theorem 5.1, defined in (5.5) as
Ψ(v) := inf {∥σ ◦ (z)∥∞ | v = − div z, z ∈ L∞ (Tn )},v ∈ L2 (Tn ).

MOTION BY CRYSTALLINE-LIKE MEAN CURVATURE: A SURVEY

65

Lemma A.1. Ψ is convex and lower semicontinuous. Moreover, if Ψ(v) <∞ for some v ∈ L2 (Tn ) then the infimum is attained and therefore it is aminimum.
Proof. Step 1 We first show the lower semicontinuity of Ψ. Suppose thatvk → v in L2 (Tn ) and L := lim inf k→∞ Ψ(vk ) < ∞. By selecting a subsequence (not relabeled), we can assume Ψ(vk ) < ∞ for all k and limk→∞ Ψ(vk ) =L. By definition of Ψ we can fix zk with vk = − div zk and ∥σ ◦ (zk )∥ <Ψ(vk ) + k1 for every k. Since ∥zk ∥∞ ≤ C ∥σ ◦ (zk )∥∞ for some C > 0, (zk )k∈N
is bounded in L∞ and hence we can take a further subsequence (not relabeled) such that zk → z weakly* in L∞ to some vector field z. Since
vk = − div zk , we haveZZ
zk · ∇ϕ dx = vk ϕ dx
for all k ∈ N, ψ ∈ Cc∞ (Tn ).
Sending k → ∞ we see that v = − div z in the sense of distributions. By
Lemma A.2 below, we haveΨ(v) ≤ ∥σ ◦ (z)∥ ≤ lim inf ∥σ ◦ (zk )∥ = lim inf Ψ(vk ).
k→∞

k→∞

Step 2 The fact that the infimum is attained when Ψ(v) < ∞ canbe proved by a direct method of the calculus of variations. If (zk )k∈N is
a minimizing sequence, as discussed in Step 1, by compactness we may
assume that it has an L∞ weak* limit z by taking a subsequence. Step 1
also shows the lower semicontinuity in this weak* topology. Thus, z must
be a minimizer.
Step 3 Convexity of Ψ is straightforward. Take v1 , v2 with Ψ(vi ) < ∞.
By Step 2, we can choose minimizing zi with vi = − div zi and Ψ(vi ) =∥σ ◦ (zi )∥∞ for i = 1, 2. We have− div(tz1 + (1 − t)z2 ) = tv1 + (1 − t)v2 ,and by convexity of ∥σ ◦ (z)∥ in z, we haveΨ(tv1 + (1 − t)v2 ) ≤ ∥σ ◦ (tz1 + (1 − t)z2 )∥∞≤ t ∥σ ◦ (z1 )∥∞ + (1 − t) ∥σ ◦ (z2 )∥∞= tΨ(v1 ) + (1 − t)Ψ(v2 ).
□This finishes the proof.

In the proof above we used the following weak* lower semicontinuity of∥σ (z)∥∞ .
◦Lemma A.2. If zk → z weakly* in L∞ (Tn ; Rn ) then∥σ ◦ (z)∥∞ ≤ lim inf ∥σ ◦ (zk )∥∞ .
n→∞

66

Y. GIGA AND N. POŽÁR

Proof. Step 1 : We first show the following claim: If z ∈ L∞ (Tn ; Rn ) then

Z
1
n
n
◦
z · p dx : p ∈ L (T ; R ), ∥σ(p)∥1 ≤ 1 .
∥σ (z)∥∞ = sup
To show this, we start with the observation that σ ◦ (z) = sup {z · p : σ(p) ≤ 1} =supm z · pm , where {pm }m∈N is a dense subset of {p : σ(p) ≤ 1}.
Clearly ∥σ ◦ (z)∥ < ∞. Fix now ε > 0 and define the measurable set[
A := {x : σ ◦ (z(x)) > ∥σ ◦ (z)∥∞ − ε} =
Am ,
m

with
Am = {x : z(x) · pm > ∥σ ◦ (z)∥∞ − ε}.
By definition of the L∞ norm we have |A| > 0 and so there exists m suchthat |Am | > 0. Define1p(x) :=
pm 1Am (x).
|Am |R
Clearly σ(p(x)) dx = σ(pm ) ≤ 1 andZZ
1z · p dx =z · pm dx ≥ ∥σ ◦ (z)∥∞ − ε|Am | Amby definition of Am . Since ε > 0 was arbitrary, the claim is proved.
Step 2 : For any p ∈ L1 (Tn ; Rn ) we haveZZ
zk · p dx ≤ σ ◦ (zk )σ(p) dx ≤ ∥σ ◦ (zk )∥∞ ∥σ(p)∥1
As zk → z weakly* in L∞ , we deduceZ
z · p dx ≤ ∥σ(p)∥1 lim inf ∥σ ◦ (zk )∥∞n→∞

By claim in Step 1, we have◦∥σ (z)∥∞ =sup∥σ(p)∥1 ≤1Zz · p dx ≤ lim inf ∥σ ◦ (zk )∥∞ .
□This finishes the proof.

Remark A.3. Note that Tn can be replaced by any open set U ⊂ Rn in theabove lemmas. If |U | = ∞, one might have to replace Am in the definitionof p(x) in Step 1 of proof of Lemma A.2 by its subset of a finite positive
measure.
Acknowledgment
The authors are grateful to the anonymous referee for careful reading of
the manuscript and valuable comments.

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67

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Graduate School of Mathematical Sciences, The University of Tokyo,
3-8-1 Komaba, Meguro-ku, Tokyo 153-8914, Japan
Email address: labgiga@ms.u-tokyo.ac.jp
Faculty of Mathematics and Physics, Institute of Science and Engineering, Kanazawa University, Kakuma, Kanazawa 920-1192, Japan
Email address: npozar@se.kanazawa-u.ac.jp