APPROXIMATION OF GENERAL FACETS BY REGULAR FACETSWITH RESPECT TO ANISOTROPIC TOTAL VARIATION ENERGIESAND ITS APPLICATION TO THE CRYSTALLINE MEAN CURVATURE
FLOWYOSHIKAZU GIGA AND NORBERT POŽÁR
Abstract. We show that every bounded subset of an Euclidean space can be approximatedby a set that admits a certain vector field, the so-called Cahn-Hoﬀman vector field, thatis subordinate to a given anisotropic metric and has a square-integrable divergence. Moregenerally, we introduce a concept of facets as a kind of directed sets, and show that they
can be approximated in a similar manner.
We use this approximation to construct test functions necessary to prove the comparison
principle for viscosity solutions of the level set formulation of the crystalline mean curvature
flow that were recently introduced by the authors. As a consequence, we obtain the wellposedness of the viscosity solutions in an arbitrary dimension, which extends the validity of
the result in the previous paper.

1. Introduction
In this note we consider the question of approximating general compact subsets of an
Euclidean space by sets whose boundary regularity is similar to the regularity of the Wulﬀ
shape of a given anisotropic norm. This regularity is related to the existence of the so-called
Cahn-Hoﬀman vector field with an L2 divergence on the set.
To give a more specific example of what we have in mind, let n ≥ 1 be a dimension,
A ⊂ Rn be a compact set and ρ > 0. Then there exists a compact set G ⊃ A with a smoothboundary such that dH (∂G, ∂A) < ρ, and a Lipschitz continuous vector field z : G → Rnsuch that |z(x)| ≤ 1, z · ν = 1 on ∂G. Here ν is the unit outer normal vector on ∂G and dHis the Hausdorﬀ distance with respect to the Euclidean norm. For this result see [GGP2].
Such a vector field is an example of a Cahn-Hoﬀman vector field for the isotropic metricgiven by the Euclidean norm |·|. The unit ball {z : |z| ≤ 1} is its Wulﬀ shape. Such vector
fields are important for evaluating the anisotropic curvature of a surface or the subdiﬀerentialof a total variation energy. Unfortunately, no Cahn-Hoﬀman vector field might exist for agiven set, in which case the anisotropic curvature cannot be evaluated. However, the above
result allows us to approximate arbitrary sets by sets for which the anisotropic curvature iswell-defined.
The goal of the present note is to extend this result to general anisotropic norm-likefunctions (non necessarily symmetric) W : Rn → [0, ∞). We assume that W is (i) convex,(ii) positively one-homogeneous, W (tp) = tW (p) for all p ∈ Rn , t ≥ 0, and (iii) positive
definite, W (p) = 0 if and only if p = 0. We will call such a function W an anisotropy. Given
Date: July 30, 2017.
2010 Mathematics Subject Classification. 35K67, 35D40, 35K55, 35B51, 35K93.
Key words and phrases. crystalline mean curvature flow, level set method, viscosity solutions, comparison
theorems.
1

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Y. GIGA AND N. POŽÁR

W , its polar W ◦ is defined as(1.1)

W ◦ (x) := sup {x · p : p ∈ Rn , W (p) ≤ 1}.

W ◦ is also an anisotropy, see [R]. The Wulﬀ shape W of W is given as the set
W := {x ∈ Rn : W ◦ (x) ≤ 1}.
By the positive definiteness of W ◦ , W is a compact set containing 0 in its interior. We
claim that any compact set A can be approximated by a larger set G with the properties
as above but with respect to the anisotropy W . However, we need to relax the notion ofboundary values of z. We will show that for every ρ > 0 there exists a compact set G ⊃ A,
dH (∂G, ∂A) < ρ, an open set U ⊃ G, a Lipschitz continuous function ψ on U , ψ ≥ 0,
ψ(x) = 0 if and only if x ∈ G, and a vector field z ∈ L∞ (U, Rn ) with div z ∈ L2 (U ) suchthat z ∈ W, z · ∇ψ = W (∇ψ) a.e. on U . Note that the last condition on the vector fieldz is equivalent to requiring z ∈ ∂W (∇ψ) a.e. on U , where ∂W is the subdiﬀerential of W
with respect to the Euclidean inner product x · y, given as(1.2)∂W (p) := {ξ ∈ Rn : W (p + h) − W (p) ≥ ξ · h, h ∈ Rn } = {ξ ∈ Rn : ξ ∈ W, ξ · p = W (p)}.
As we will see later, the approximation discussed so far is related to the surfaces of convex
or concave solid bodies. To handle general solid bodies, we want to control the direction inwhich the vector field z flows through the boundary of the set A. Therefore we introduce
a concept of directionality of a set. In particular, we specify which components of thecomplement are sources and which are sinks of the vector field. We will call such directed
sets facets.
Definition 1.1. Let n ≥ 0 be the dimension. A compact set A ⊂ Rn together with a directionχ ∈ C(Ac , {−1, 1}) is called an n-dimensional facet, and we write it as (A, χ). We will set
χ(x) = 0 for x ∈ A.
The direction χ introduces an order on the family of all facets: (A, χ) ⪯ (G, χ̃) if χ ≤ χ̃.
If χ, χ̃ ≥ 0, we have (A, χ) ⪯ (G, χ̃) if and only if A ⊂ G. We now introduce “regular” facetsthat admit a certain vector field subordinate to the anisotropy W .
Definition 1.2. We say that (A, χ) is a W ◦ -(L2 ) Cahn-Hoﬀman facet if there exist an open
set U ⊂ Rn , a Lipschitz function ψ on U and a vector field z ∈ L∞ (U, Rn ), div z ∈ L2 (U ),such that A ⊂ U , sign ψ = χ on U , z(x) ∈ ∂W (∇ψ(x)) for a.e. x ∈ U . Here we define
sign s to be −1, 0, 1 if s < 0, s = 0 or s > 0, respectively.
If A has a Lipschitz boundary, we would like to say that (A, χ) is a W ◦ -(L2 ) Cahn-Hoﬀman
facet if there exists a vector field z ∈ L∞ (A, Rn ), div z ∈ L2 (A) such that z ∈ W on A andz(x) ∈ ∂W (χ(x+ν0+)ν(x)) on ∂A, where χ(x+ν0+) is the limit of χ at x from the direction
outside of A and ν(x) is the unit outer normal to A at x. However, we currently do notknow if this is equivalent to Definition 1.2.
Such vector fields z are usually referred to as Cahn-Hoﬀman vector fields. We are mainlyinterested in the divergence of these vector fields, in particular in the one that is minimalin the L2 -norm on A. We call such an L2 -function the W ◦ -(L2 ) minimal divergence of the
facet (A, χ), Definition 4.2. We will see later, Proposition 4.1, that this minimal divergenceis unique and depends only on A and χ.

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3

The main claim of this paper is that an arbitrary n-dimensional facet (A, χ) can be approximated by a Cahn-Hoﬀman facet (G, χ̃) consistently with the direction, that is, (A, χ) ⪯(G, χ̃).
Theorem 1.3. Let (A, χ) be an n-dimensional facet and W an anisotropy. Given ρ > 0
there exists a W ◦ -(L2 ) Cahn-Hoﬀman facet (G, χ̃) such that χ(x) ≤ χ̃(x) ≤ sup|x−y|≤ρ χ(y)for x ∈ Rn .
The main application of the approximation result Theorem 1.3 is the recently developed
notion of viscosity solutions of the level set formulation of a crystalline mean curvature flowby the authors in [GP]. In particular, the viscosity solutions of the initial value problem{
ut + F (∇u, div(∇W (∇u))) = 0, in Rn × (0, ∞),(1.3)
u|t=0 = u0 , in Rn ,were introduced. Here F : Rn × R → R is assumed to be a continuous function that isnonincreasing in the second variable, and the initial data u0 is assumed to be continuous andconstant outside of a compact set. Finally, W is a crystalline anisotropy, that is, a piece-wiselinear anisotropy.
This problem can be derived as the level set formulation of the motion of a set {Et }t≥0 by
the crystalline mean curvature flow
(1.4)

V = f (ν, κW ),where V is the normal velocity of the surface ∂Et , ν is its outer unit normal vector andκW = − div∂Et (∇W (ν)) is the anisotropic (crystalline) mean curvature of ∂Et . The function f : S n−1 × R → R is assumed to be continuous and nondecreasing in the second
variable, where S n−1 denotes the (n − 1)-dimensional unit sphere. The idea of the level set
method is to introduce an auxiliary function u : Rn × [0, ∞) → R whose every sub-level set
t 7→ {x : u(x, t) < c}, c ∈ R, evolves under the crystalline mean curvature flow (1.4). Thisfunction then satisfies the initial value problem (1.3) with an appropriate u0 , typically sout∇uthat E0 = {x : u0 (x) < 0}. Since V = − |∇u|, ν = |∇u|and div∂Et (∇W (ν)) = div(∇W (∇u)),psee [G], we deduce that F (p, ξ) = |p|f ( |p| , −ξ). In [GP], the viscosity solutions of (1.3) wereshown to satisfy the comparison principle in three dimensions, and the well-posedness of thisproblem was established.
Theorem 1.3 allows us to extend the result of [GP] to an arbitrary dimension. For a detailed
discussion of the notion of viscosity solutions for (1.3) see Section 4.2. Heuristically, the key
idea is to define the viscosity solution using the comparison principle with test functions
that have flat parts that are W ◦ -(L2 ) Cahn-Hoﬀman facets, and then interpret the operator
div(∇W (∇u)) as the W ◦ -(L2 ) minimal divergence of the facet. This is consistent with thetheory of monotone operators when (1.3) is in a divergence form. The approximation resultin Theorem 1.3 implies that the family of these test functions is suﬃciently large. We havethe following results.
Theorem 1.4 (Comparison principle). Let W : Rn → R be a crystalline anisotropy, andF ∈ C(Rn × R) be nonincreasing in the second variable, F (0, 0) = 0. Suppose that u isan upper semicontinuous function and v is a lower semicontinuous function, u and v are a
viscosity subsolution and a viscosity supersolution, respectively, of (1.3) in Rn × (0, T ) forsome T > 0, and that there exist a compact set K ⊂ Rn and constants cu , cv ∈ R such

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Y. GIGA AND N. POŽÁR

that u ≡ cu ≤ cv ≡ v on (Rn \ K) × [0, T ]. Then u(·, 0) ≤ v(·, 0) on Rn implies u ≤ v on
Rn × (0, T ).
Theorem 1.5 (Stability). Let W : Rn → R be a crystalline anisotropy, and F ∈ C(Rn × R)
be nonincreasing in the second variable. Let um , m ∈ N, be a sequence of viscosity solutionsof (1.3) with W = Wm and initial data u0,m . Suppose that the sequence u0,m is locallyuniformly bounded. If either
2(a) Wm ∈ C 2 (Rn ), a−1
m ≤ ∇ Wm ≤ am for some am > 0, and Wm ↘ W , or(b) Wm are smooth anisotropies, that is, Wm is an anisotropy, Wm ∈ C 2 (Rn \ {0}) and{p : Wm (p) ≤ 1} is strictly convex, such that Wm → W locally uniformly,thenū(x, t) =lim supum (y, s) andu(x, t) =(y,s,m)→(x,t,∞)lim inf(y,s,m)→(x,t,∞)um (y, s)

are respectively a viscosity subsolution and a viscosity supersolution of
ut + F (∇u, div(∇W (∇u))) = 0

in Rn × (0, ∞).

Theorem 1.6 (Well-posedness). Let W : Rn → R be a crystalline anisotropy, and F ∈C(Rn × R) be nonincreasing in the second variable, F (0, 0) = 0. Suppose that u0 ∈ C(Rn )and that there exist a compact set K ⊂ Rn and a constant c ∈ R such that u0 = c on Rn \ K.
Then there exists a unique viscosity solution of (1.3) such that u(x, t) = c on Rn \ Kt forsome compact set Kt , t ≥ 0.
p, −ξ) coming from the level set formulation of the crystalline
Note that F (p, ξ) = |p|f ( |p|mean curvature flow (1.4) satisfies automatically F (0, 0) = 0. In fact, F (0, ξ) = 0 for allξ ∈ R. Therefore the above theorem implies the global unique existence (up to fattening of
the set {x : u(x, t) = 0}) of the crystalline mean curvature flow for any initial bounded set
E0 .

Theorem 1.7. Let W : Rn → R be a crystalline anisotropy, and f ∈ C(S n−1 × R) be
nondecreasing in the second variable. For every bounded open set E0 ⊂ Rn there exists a
unique (open) evolution {Et }t≥0 of the crystalline mean curvature flow V = f (ν, κW ).
1.1. Relationship to the anisotropic total variation energy. In this section we relate
the above approximation result to the problem of evaluating the subdiﬀerential of a total
variation energy functional. Let W be an anisotropy and let Ω be either a smooth domain
in Rn or the torus Tn := Rn \ Zn . Consider the anisotropic total variation energy functional
E : L2 (Ω) → R defined as{∫
W (Dψ) dx, ψ ∈ BV (Ω) ∩ L2 (Ω),Ω
(1.5)
E(ψ) :=+∞,
otherwise.
Here Dψ is the distributional gradient of a function of bounded variation, whose space isdenoted as BV (Ω). In general, Dψ is a vector-valued Radon measure. Therefore E isunderstood as the relaxation (closure or lower semicontinuous envelope) of the energy (1.5)
defined for W 1,1 (Ω) functions. It is known that this relaxation is equivalent to defining
W (Dψ) as the measure
( s )
Dψ
n
|Ds ψ| ,W (∇ψ)L + Ws|D ψ|APPROXIMATION BY REGULAR FACETS

5

where we decompose Dψ into to the absolutely continuous part ∇ψLn and the singular partDs ψ
Ds ψ, with respect to the n-dimensional Lebesgue measure Ln . |Ds ψ| denotes the RadonNikodým derivative.
2
The subdiﬀerential ∂E : L2 (Ω) → 2L (Ω) is defined as the set-valued mapping
{
}
(1.6)
∂E(ψ) := v ∈ L2 (Ω) : E(ψ + h) − E(ψ) ≥ (v, h), h ∈ L2 (Ω) ,∫where (v, h) := Ω vh dx is the L2 -inner product. ∂E(ψ) is a closed convex, possibly empty
set. The domain of the subdiﬀerential is defined as D(∂E) := {ψ ∈ L2 (Ω) : ∂E(ψ) ̸= ∅}.
The characterization of ∂E is well-known [M, AVCM]. Following [A1], we introduce theset of bounded vector fields with L2 divergence{}
X2 (Ω) := z ∈ L∞ (Ω; Rn ) : div z ∈ L2 (Ω) .
For ψ ∈ Lip(Ω), the set ∂E(ψ) can be characterized as∂E(ψ) = {div z : z ∈ X2 (Ω), z ∈ ∂W (∇ψ) a.e., [z · ν] = 0 on ∂Ω},
where ∂W is the subdiﬀerential of W with respect to the inner product on Rn , introducedin (1.2). The symbol [z · ν] denotes the boundary trace of a vector field in X2 (Ω), see [A1].
The vector fields z are usually called Cahn-Hoﬀman vector fields, and we shall denote their
set as(1.7)

CH(ψ) := {z ∈ X2 (Ω) : z ∈ ∂W (∇ψ) a.e.},ψ ∈ Lip(Ω).

Note that ∂E(ψ) might be empty even for smooth ψ. For example, consider the function(∑n2 )1/2
n
|p
|
ψ(x)
=
sin(2πx)onTwithanisotropy
W
(p)
=
∥p∥
:=
, or W (p) = ∥p∥1 :=
i
1
2
i=1
∑n
|p
|.
In
this
case,
every
possiblecandidatevector
field
z
with
z(x)
∈ ∂W (∇ψ) a.e. will
i=1 i
}
{
1
have
a jump{} discontinuity in z1 across the sets x1 = 4 , where ψ attains its maximum, and3x1 = 4 , where it attains its minimum. This is a serious diﬃculty for using the operator ∂E
in the theory of viscosity solutions, which usually rely on evaluating the diﬀerential operator
on a class of suﬃciently smooth test functions.
Theorem 1.3 allows us to approximate any function by a function with nonempty ∂Earbitrarily close in the Hausdorﬀ distance of the respective positive and negative sets. Statedin terms of the functions ψ, we get the following theorem.
Theorem 1.8. Let W be an anisotropy and Ω be either Tn or a bounded domain with
Lipschitz boundary in Rn . If ψ is a Lipschitz function on Ω with a compact zero set, forevery ρ > 0 there exists a Lipschitz function ψ̃ on Ω such that ∂E(ψ̃) ̸= ∅ andsign ψ(x) ≤ sign ψ̃(x) ≤ sup sign ψ(y),x ∈ Ω.

y∈Ω|y−x|≤ρ

Note that any Lipschitz function ψ with a compact zero set naturally corresponds to a
facet(A, χ) := ({x : ψ(x) = 0}, sign ψ).
Intuitively, it is related to the surface facet (flat part of the surface) of the (n+1)-dimensional
solid body given by the epigraph of ψ, epi ψ := {(x, xn+1 ) ∈ Rn+1 : xn+1 ≥ ψ(x)}. If χ ≥ 0or χ ≤ 0, the solid body is respectively convex or concave in a neighborhood of the surface
facet.

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Y. GIGA AND N. POŽÁR

1.2. Literature overview. Crystalline mean curvature was introduced independently by
S. Angenent and M. E. Gurtin [AG] and J. E. Taylor [T] to model the growth of small crystals.
It is the first variation of the surface energy given by a crystalline anisotropy with respect tothe change of volume. A crystalline anisotropy refers to an anisotropic surface energy density
σ whose unit ball is a convex polytope. When extended positively one-homogeneously to thefull space, the anisotropy σ is a convex piece-wise linear function. Due to the singularity ofthe surface energy density, the crystalline mean curvature is a nonlocal quantity on the flatparts, or facets, of the crystal surface. As with smooth anisotropic curvatures, the crystalline
mean curvature can be evaluated as the surface divergence of a so-called Cahn-Hoﬀman vector
field on the surface. A Cahn-Hoﬀman vector field is a selection of the subdiﬀerential of theanisotropy evaluated at the outer normal vector on a given surface whose surface divergence
has a certain regularity. However, for crystalline anisotropies, even smooth surfaces might
not have a Cahn-Hoﬀman vector field whose surface divergence is a function. Therefore there
has been a lot eﬀort to characterize sets with surfaces that admit reasonable Cahn-Hoﬀman
vector fields. In particular the various notions of ϕ-regular sets were introduced (ϕ being thepolar of the surface energy density σ) [BN, BNP1, BNP2, BNP3, B1] and the related notionof rBϕ -condition [BCCN2], Bϕ being the Wulﬀ shape W.
A crystalline mean curvature flow or a motion by the crystalline mean curvature is anevolution of sets such that the normal velocity of the surface is proportional to the crystalline
mean curvature. Already the standard mean curvature flow is known to develop singularities
even when starting from smooth initial data, and therefore a weak notion of solutions isnecessary. We mention the varifold solutions initiated by K. Brakke [B2,I1,TT], that however
apply only to the isotropic mean curvature flow, and the level set method approach [OS,CGG,
ES] that can be generalized to the anisotropic mean curvature with a smooth anisotropy as
already done in [CGG]. The extension to a crystalline anisotropy is not straightforward evenin the case of a curve evolution because of the non-local nature of the crystalline curvature[GG3]. For a more detailed overview of the literature see [GP].
Most of the attempts at defining a reasonable notion of solutions for the crystalline mean
curvature flow have required some kind of regularity of the evolution so that the crystalline
curvature can be evaluated, such as a ϕ-regular flow [BN, BCCN1] or a rBϕ -regular flow
[BCCN2].
Recently, A. Chambolle, M. Morini and M. Ponsiglione [CMP] established a unique global
solvability of the flow V = σ(ν)κσ for arbitrary convex σ and not necessarily bounded initial
data, in an arbitrary dimension. They introduce a notion of solutions of this flow via ananisotropic sign distance function in the spirit of H. M. Soner [S], but in a distributional sensethat appeared in [CC], prove a comparison principle, and use the minimizing movements
algorithm of Chambolle [C] to construct a solution. This result has been recently improvedin [CMNP] to cover V = ψ(ν)(κσ + f (x, t)), where ψ is a convex anisotropy and f is a given
Lipschitz function. They also prove, see [CMNP, Remark 6.1], that their notion of solutions
coincides with the notion studied in this paper, whenever both apply, in arbitrary dimension
thanks to Theorem 1.5. We mention the result of K. Ishii [I2], who shows that Chambolle’s
minimizing movements algorithm converges to the viscosity solution of the crystalline mean
curvature flow in two dimensions.
We take a diﬀerent approach using the ideas of the theory of viscosity solutions [GG1,
GG3, GGP1, GGP2]. The level set formulation of the crystalline mean curvature flow was

APPROXIMATION BY REGULAR FACETS

7

introduced by the authors in [GP]. Viscosity solutions are defined via the comparison principle by testing a candidate for a solution by an appropriate class of regular test functions.
The main advantage of this approach is that it does not require the solution itself to haveany a priori regularity besides continuity. Moreover, testing a solution by a test function is arather local concept. Since the crystalline mean curvature might be nonlocal on flat parts ofthe evolving surface, we can localize the construction of test functions to the neighborhoodof such flat parts, called (surface) facets. By choosing a local coordinate system so that thesurface facet is given as a part of a function graph where the function is equal to zero, we
are at a situation covered by Theorem 1.3. Our notion of facet introduced in Definition 1.1
then corresponds to the set A where the surface facet is located in this coordinate systemand directions in which the surface rises above (χ = +1) and falls below (χ = −1) the
surface facet. The anisotropy Wp̂sl introduced in (4.2) captures the local lower-dimensional
behavior of the full anisotropy W in the direction of the surface facet. A Cahn-Hoﬀman
vector field for (A, χ) then corresponds to a Cahn-Hoﬀman vector field on the surface in aneighborhood of the surface facet. In general, we consider surface facets to be all the flatfeatures of a surface with various dimensions, including edges (n = 1) and (planar) facets(n = 2), etc. Their dimension then guides the choice of the ambient space for the facet(A, χ). The approximation result Theorem 1.3 and its corollary in Theorem 1.8 allow us
to construct a large family of test functions at any facet of a crystal. This then yields thecomparison principle for viscosity solutions and the well-posedness of the level set formulation of the crystalline mean curvature flow [GP]. In [GP], we showed Theorem 1.8 by adirect construction for n = 1, 2 and piece-wise linear anisotropies W and therefore we could
deduce the well-posedness of the crystalline mean curvature flow in dimensions n ≤ 3. The
generalization in this paper then automatically extends the results of [GP] to any dimension.
We can also slightly simplify the definition of viscosity solutions, see Definition 4.8. Let us
mention that in [GP] we described a facet by a pair of open sets (A− , A+ ). They are related
to the notion of facet (A, χ) from Definition 1.1 as A± = {x : χ(x) = ±1}.
Let us conclude with a comment on the requirement that W is purely crystalline. The
simple structure is essential for the construction of “localized” and “stratified” faceted test
functions. We believe that it is possible to extend this approach to anisotropies W thatare crystalline in some directions but smooth in others such as σ(p) = |p1 | + |p2 |2 . In thiscase, the second derivatives must be handled in the proof of the comparison principle, likely
using a variant of Ishii’s lemma. Other generalizations such as x-dependence of F seem also
possible. This is a subject of a future work.
Outline. The construction of the facet (G, χ̃) for Theorem 1.3 will be performed first by
reducing the situation to the case of a simple set in Section 2 using the gradient flow of thetotal variation energy on a torus Tn , and then combining it to produce a facet in Section 3.
In Section 4 we outline an important application of Theorem 1.3 to the theory of viscosity
solutions for the level set formulation of the crystalline mean curvature problems and discuss
the proofs of Theorems 1.4–1.7.
2. Approximability of a single set
We start the proof of Theorem 1.3 with a simple facet (A, χ) with χ ≤ 0. Note that weneed to construct the Cahn-Hoﬀman vector field only in a neighborhood of the boundary of
the approximating facet since 0 ∈ W = ∂W (0). We can thus always extend the vector fieldby 0 in the interior of the facet away from the boundary. To construct the approximating

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Y. GIGA AND N. POŽÁR

set, we will use the gradient flow of the anisotropic total variation energy (1.5) on the torus
Tn := Rn /Zn .
Let us thus consider a single open set D ⊂ Tn with a nonempty boundary. Let d : Tn → R
be the signed distance function to ∂D (in the torus topology of Tn ) induced by W with d > 0
in D. Recall that{
inf y∈Dc W ◦ (x − y),x ∈ D,(2.1)
d(x) := inf c W ◦ (x − y) − inf W ◦ (y − x) =y∈Dy∈D− inf y∈D W ◦ (y − x), x ∈ Dc ,where we treat D ⊂ Rn as periodic in the sense that if x ∈ D then x + Zn ⊂ D, and W ◦ isthe polar of W defined in (1.1). Recall that W ◦ is again an anisotropy, see [R], and therefore
also Lipschitz continuous. In particular, d is a Lipschitz continuous function on Tn .
Fix δ > 0 such that
(2.2)

1 1
{x ∈ Rn : W ◦ (x) ≤ 4δ} = 4δW ⊂ (− , )n .
2 2

We consider the gradient flow (or diﬀerential inclusion) on Tn
 dφ (t) ∈ −∂E(φ(t)),
t > 0,(2.3)
dt
 φ(0) = max(−δ, min(d, δ)),where the subdiﬀerential ∂E was defined in (1.6) with Ω = Tn .
It is well-known [B3, B4] that a unique solution φ ∈ C([0, T ]; L2 (Tn )) exists (for any+
T > 0) and it is right diﬀerentiable for all t ∈ (0, T ). Moreover ddtφ (t) = −∂ 0 E(φ(t)), where∂ 0 E(ψ) is called the minimal section (also the canonical restriction) of ∂E(ψ), and it is theelement of ∂E(ψ) with the smallest L2 -norm. In particular, φ(t) ∈ D(∂E) for all t > 0.
Such well-posedness goes back to the work of Y. Komura [K], where initial data is assumed
to be in D(∂E). Finally, φ(t) is Lipschitz for all t > 0 since φ(0) is Lipschitz and (2.3)
has a comparison principle and is translationally invariant. The comparison principle (andLipschitz continuity of φ in space) can be established, for instance, by approximating theenergy E in the Mosco sense by uniformly elliptic energies, for which (2.3) is just a uniformly
parabolic PDE. Mosco convergence then implies the convergence of the resolvent problems,see [GP] for details. Alternatively, multiply the diﬀerence of equations (2.3) for two solutions
u and v by the positive part of the diﬀerence, (u − v)+ := max(u − v, 0), and integrate byparts.
We want to prove thatdH (∂{x : ±φ(x, t) > 0}, ∂D) → 0

as t → 0+,

where dH is the Hausdorﬀ distance with respect to the usual Euclidean metric.
To show this, we will compare φ with barriers of the form}{ψc (x) := min min(max(W ◦ (x − k) − c, − 2c ), δ) : k ∈ Zn ,for 0 < c < δ. These will control the expansion of the set {x : φ(x, t) = 0}, which is equal
to ∂D at t = 0.

APPROXIMATION BY REGULAR FACETS

9

We claim that ψc (x) + M t is a supersolution of (2.3) for M = M (c) > 0 large enough. To
see this, introduce the vector field for x ∈ (− 21 , 12 )n as
 2x
W ◦ (x) < 2c ,c,c
≤ W ◦ (x) < c + 2δ,z(x) := W ◦x(x) ,2
 x max(1 + c+2δ−W ◦ (x) , 0), otherwise,W ◦ (x)δand then extend it periodically to Rn . If 0 < c < δ, z = 0 in a neighborhood of the boundaryof (− 12 , 21 )n by (2.2).
We claim that z is a Cahn-Hoﬀman vector field for ψc , that is, z ∈ CH(ψc ). Indeed,working only on the unit cell x ∈ (− 21 , 12 )n , note that W ◦ (z) ≤ 1, with equality if and only
if 2c ≤ W ◦ (x) ≤ c + 2δ. Therefore z ∈ ∂W (0) on Rn , and thus z(x) ∈ ∂W (∇ψ(x)) = ∂W (0)for W ◦ (x) < 2c and W ◦ (x) > c + δ. Since W ◦x(x) ∈ ∂W (p) if p ∈ ∂W ◦ (x), we have thatz ∈ ∂W (∇ψc (x)) whenever W ◦ is diﬀerentiable at x and 2c < W ◦ (x) < c + δ. As z is
Lipschitz, div z ∈ L∞ (Tn ) ⊂ L2 (Tn ) and so z is a Cahn-Hoﬀman vector field for ψc .
Let us set M := ∥div z∥L∞ and f := M −div z ∈ L2 (Tn ). Observe that (x, t) 7→ ψc (x)+M t
is the unique solution ϕ ∈ C([0, ∞); L2 (Tn )) of dϕ (t) ∈ −∂E(ϕ(t)) + f, t > 0,
(2.4)
dt
 ϕ(0) = ψ ,cand that f ≥ 0, and thus it is a supersolution of (2.3).
Lemma 2.1. Let φ be the unique solution of (2.3). For every ρ > 0 there exists τ > 0 suchthat for all x ∈ D with dist(x, ∂D) > ρ we have φ(x, t) > 0 for all 0 ≤ t < τ , and for allx ∈ Dc with dist(x, ∂D) > ρ we have φ(x, t) < 0 for all 0 ≤ t < τ . Here dist is the usual
Euclidean distance.
Proof. Take c ∈ (0, δ) such that W ◦ (x) > c for |x| ≥ ρ, M = M (c) so that (x, t) 7→ ψc (x)+M tc. We claim that by definition of d in (2.1), for any
is a supersolution of (2.3), and τ = 2Mcx0 ∈ D such that dist(x0 , ∂D) > ρ, we haveφ(·, 0) ≤ ψc (· − x0 )in Tn .

To see this for one such x0 , we only need to show W ◦ (x − x0 ) ≥ d(x) + c for x ∈ Rn . Ifcx ∈ D, then |x − x0 | > ρ and hence W ◦ (x − x0 ) > c. Set y := W ◦ (x−x(x − x0 ) + x0 . Note0)◦cthat W (y − x0 ) = c and therefore |y − x0 | ≤ ρ. In particular, y ∈ D . Then by (2.1)()c◦d(x) ≤ W (x − y) = 1 − ◦W ◦ (x − x0 ) = W ◦ (x − x0 ) − c.
W (x − x0 )On the other hand, if x ∈ Dc , convexity and positive one-homogeneity yields for all y ∈ D
W ◦ (y − x) + W ◦ (x − x0 ) ≥ W ◦ (y − x0 ) > c.
Taking the infimum over y ∈ D, we get from (2.1)
W ◦ (x − x0 ) − c ≥ d(x).
Fix thus one x0 as above. By the comparison principle for (2.4),φ(x0 , t) ≤ ψc (0) + M t < 0,c
In particular, φ(x0 ) < 0 for 0 ≤ t < 2M.

0 ≤ t < τ.

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Y. GIGA AND N. POŽÁR

We can argue similarly for x0 ∈ D using the subsolution −ψc (−x) − M t.

□

The lemma above allows us to control the speed of the boundary of {±φ > 0}.
Corollary 2.2. For every ρ > 0 there exists τ > 0 such thatdH (∂D, ∂{φ(·, t) > 0}) ≤ ρ,

for 0 ≤ t < τ ,dH (∂D, ∂{φ(·, t) < 0}) ≤ ρ,

where dH is the Hausdorﬀ distance with respect to the usual Euclidean metric.
Proof. Given ρ > 0, by compactness there exists η ∈ (0, ρ) such that for all x ∈ ∂D there existy ∈ D, z ∈ Dc , such that dist(y, x) ≤ ρ, dist(z, x) ≤ ρ while dist(y, ∂D) > η, dist(z, ∂D) > η.
Take τ > 0 from Lemma 2.1 for ρ = η, and fix t ∈ (0, τ ). Let us show the inequality forΓt := ∂{φ(·, t) > 0}, the second one is analogous.
First, let x ∈ Γt , which by continuity implies φ(x, t) = 0. By Lemma 2.1 we havedist(x, ∂D) ≤ η < ρ.
Now suppose that x ∈ ∂D. By the choice of η there exist y ∈ D, z ∈ Dc with theproperties above. By Lemma 2.1 we have φ(y, t) > 0 and φ(z, t) < 0 for 0 ≤ t < τ . The
convexity of the norm yields dist(x, ∂Γt ) ≤ ρ since the line segment connecting y with z
must contain a point in ∂Γt .
□3. Construction of a Cahn-Hoffman facet
To prove Theorem 1.3, let us fix a facet (A, χ) such that A ⊂ (− 14 , 41 )n . Indeed, an
arbitrary facet can be reduced to this case by simple scaling since the anisotropy is positively
one-homogeneous. We can also assume that ρ < 41 . We will use the result of Section 2
to construct the approximating Cahn-Hoﬀman facet. By continuity of χ on Ac , χ is aconstant on Rn \ (− 14 , 14 )n . Let us assume that χ = −1 on this set. The other case can be◦handled similarly. Defining the periodic function χ(x) := max {χ(y) : y ∈ x + Zn }, we see◦◦that χ = χ on (− 21 , 12 )n . Set H+ = {x ∈ Rn : sup|y−x|< ρ4 χ(y) = +1} and H− = {x ∈ Rn :◦sup|y−x|< 3ρ χ(y) ≥ 0}. Sets H− and H+ are open subsets of Tn . Moreover H+ ⊂ H− and4dist(H+ , H−c ) = dist(∂H− , ∂H+ ) ≥ ρ2 .
Define the open cube U = (− 12 , 21 )n ⊂ Rn . For one fixed δ > 0 as in Section 2, let φ+ be
the solution of (2.3) for set D = H+ , and φ− be the solution D = H− . If ∂H+ is empty, thatis, when H+ is empty, we set φ+ ≡ 0, z + ≡ 0 below. Similarly if ∂H− is empty, implying
H− = Tn , we set φ− ≡ −1, z − ≡ 0. Applying Lemma 2.1 with ρ = ρ8 , we have τ > 0 such
that
{
}
◦
◦
{χ = 1} ⊂ φ+ (·, t) > 0 ⊂ {x : sup χ(y) = 1},(3.1)|y−x|≤ρ{}◦{χ ≥ 0} ⊂ φ− (·, t) ≥ 0 ⊂ {x : sup χ(y) ≥ 0},◦

0 ≤ t < τ,|y−x|≤ρ

and
(3.2)

{
} {
}
ρ
dist( φ+ (·, t) > 0 , φ− (·, t) < 0 ) ≥ ,4

and finally(3.3)

φ− (·, t) < 0 on ∂U,

0 ≤ t < τ.

0 ≤ t < τ,

APPROXIMATION BY REGULAR FACETSLet us fix t ∈ (0, τ ) and set{}
G− := x ∈ U : φ− (x, t) < 0 ∪ U c ,11{}
G+ := x ∈ U : φ+ (x, t) > 0 .

By the continuity of φ± and (3.3), G± ⊂ Rn are open.
We define a facet by setting G = Gc− ∩ Gc+ ⊂ U , χ̃ = +1 on G+ and χ̃ = −1 on G− . Then
(G, χ̃) is a facet in the sense of Definition 1.1 and χ(x) ≤ χ̃(x) ≤ sup|y−x|≤ρ χ(y).
To finish the proof of Theorem 1.3, we need to show that it is actually a W ◦ -(L2 ) CahnHoﬀman facet by finding a Cahn-Hoﬀman vector field on its neighborhood U . Since φ± are
solutions of (2.3) and for t > 0 we have that φ± (·, t) ∈ D(∂E) are Lipschitz, there exist
Cahn-Hoﬀman vector fields z ± ∈ X2 (Tn ) for φ± (·, t), respectively. We define the function
ψ : U → R as+
φ (x, t), x ∈ G+ ,ψ(x) := (φ+ )+ (x, t) − (φ− )− (x, t) = 0,x ∈ G,
φ− (x, t), x ∈ G ∩ U,−
where (s)+ := max(s, 0) and (s)− := max(−s, 0) denote the positive and negative parts,
respectively. The functions φ± are Lipschitz continuous, and so by (3.2) ψ is well-definedand Lipschitz continuous function on U .
Set η = dist(G− , G+ )/4. Let us introduce the vector fieldz(x) := ξ + (x)z + (x) + ξ − (x)z − (x),where ξ ± ∈ C ∞ (Rn ) are cut-oﬀ functions such that ξ ± (x) = 1 when dist(x, G± ) ≤ η, ξ ± (x) =0 when dist(x, G± ) ≥ 2η and 0 ≤ ξ ± ≤ 1 otherwise. Clearly z ∈ X2 (U ). Note thatξ − + ξ + ≤ 1. Since ψ ≡ 0 on G = {ψ = 0}, we see that z ∈ ∂W (∇ψ) on U as ∂W (p) is
a convex set. Therefore z is a Cahn-Hoﬀman vector field on U , and (G, χ̃) is a W ◦ -(L2 )Cahn-Hoﬀman facet. The proof of Theorem 1.3 is complete.
4. Level set crystalline mean curvature flow
In this section, we explain the application of the approximation result of Theorem 1.3 tothe theory of viscosity solutions of the crystalline mean curvature flow problems. Specifically,
let us consider the initial value problem (1.3),{
ut + F (∇u, div(∇W (∇u))) = 0
in Rn × (0, ∞),(4.1)
u|t=0 = u0 , in Rn .
The anisotropy W : Rn → R is now assumed to be piece-wise linear. In convex analysis,
convex piece-wise linear functions are also known as polyhedral functions [R]. We call suchanisotropies crystalline. The nonlinearity F ∈ C(Rn × R) is assumed to be nonincreasing inthe second variable,
F (p, ξ) ≤ F (p, η),p ∈ Rn , ξ ≥ η.

Thanks to this assumption, the problem (4.1) has a comparison principle structure.
Problem (4.1) appears as the level set formulation of an anisotropic mean curvature
flow, specifically the crystalline mean curvature flow [GP]. It can be also thought of as
an anisotropic total variation flow of non-divergence form. Of particular interest is the singular operator div(∇W (∇u)), which is interpreted as the minimal section (also canonicalrestriction) −∂ 0 E(u) of the subdiﬀerential ∂E(u) defined in (1.6). Since the problem has a

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Y. GIGA AND N. POŽÁR

comparison principle structure, it falls within the scope of the theory of viscosity solutions.
However, the extension of the theory to problems like (4.1) is quite nontrivial. In [GP], theauthors succeeded in defining a reasonable notion of viscosity solutions for (4.1). However,
due to the diﬃculty of construction test functions like those in Theorem 1.8 in dimensionsn ≥ 3, the well-posedness was limited to dimensions n ≤ 3. Theorem 1.3 now provides asuﬃciently large class of test functions and thus the results of [GP] apply to an arbitrary
dimension. We will outline this in the rest of this section.
4.1. Crystalline curvature. Let us review the notion of the crystalline curvature of afacet. Recall the definition (1.7) of the set CH(ψ) of Cahn-Hoﬀman vector fields for a given
Lipschitz continuous function on an open set U ⊂ Rn . The set of all divergences of suchvector fields div CH(ψ) := {div z : z ∈ CH(ψ)} is a closed convex subset of L2 (U ). If it isnonempty, there exists a unique element with the minimal L2 norm. We denote it Λ[ψ],{}Λ[ψ] := div zmin ,∥div zmin ∥L2 (U ) = inf ∥div z∥L2 (U ) : z ∈ CH(ψ) .
In [GP], we proved the following comparison principle for the quantity Λ.
Proposition 4.1 ([GP, Proposition 4.12]). If ψ1 , ψ2 are two Lipschitz functions on an open
set U such their zero sets are compact subsets of U ⊂ Rn , and Λ[ψi ], i = 1, 2, are well-defined,
thensign ψ1 ≤ sign ψ2

on U

impliesΛ[ψ1 ] ≤ Λ[ψ2 ],a. e. on {ψ1 = 0} ∩ {ψ2 = 0}.

This allows us to introduce the W ◦ -(L2 ) minimal divergence of a W ◦ -(L2 ) Cahn-Hoﬀman
facet.
Definition 4.2. Given a W ◦ -(L2 ) Cahn-Hoﬀman facet (A, χ), that is, a facet for which
there exists an open set U ⊃ A, Lipschitz function ψ on U with sign ψ = χ, and a CahnHoﬀman vector field z ∈ CH(ψ), we define the W ◦ -(L2 ) minimal divergence of the facet(A, χ), Λ[(A, χ)] ∈ L2 (A), asΛ[(A, χ)] := Λ[ψ]on A.

By Proposition 4.1, this definition is independent of the choice of U and ψ.
Note that Proposition 4.1 implies a comparison for Λ of facets: if (F1 , χ1 ), (F2 , χ2 ) are two
W ◦ -(L2 ) Cahn-Hoﬀman facets that are ordered, (F1 , χ1 ) ⪯ (F2 , χ2 ) in the sense of χ1 ≤ χ2 ,we haveΛ[(F1 , χ1 )] ≤ Λ[(F2 , χ2 )] a.e. on F1 ∩ F2 .
Example 4.3. Let n = 1 and W (p) = |p|. Then the facet (A, χ) with A = {a}, χ ∈ C(R \{a}, {−1, 1}) is always Cahn-Hoﬀman if and only if χ(a−) ̸= χ(a+).
Facet (A, χ) with A = [a, b], a < b, χ ∈ C(Ac , {−1, 1}) is Cahn-Hoﬀman. The minimal divergenceis constant on the facet, Λ[(A, χ)] = χ(a−)+χ(b+)
on A. It is inversely proportional to the length ofb−athe facet.
For n = 0, R0 = {0} and the facet ({0}, χ) is always Cahn-Hoﬀman with Λ[(A, χ)] = 0.

APPROXIMATION BY REGULAR FACETS

13

Example 4.4. In an arbitrary dimension for any anisotropy W , the rescaled Wulﬀ shape A :=cW = {x : W ◦ (x) ≤ c}, χ = 1 on Ac forms a W ◦ -(L2 ) Cahn-Hoﬀman facet (A, χ) for any c > 0 with
Λ[(A, χ)] = nc . This can be seen easily by taking ψ(x) = max(W ◦ (x) − c, 0), z(x) = max(Wx◦ (x),c) .
Example 4.5. For n = 2 and W (p) = |p1 | + |p2 |, a rather thorough characterization of Λ[(A, χ)]for axes-aligned polygons A is available in [LMM]. In particular, if Λ[(A, χ)] is a constant, then1
1 (∂A∩{χ=−1})
Λ[(A, χ)] = H (∂A∩{χ=+1})−H, where H1 is the one-dimensional Hausdorﬀ measure.
|A|
Example 4.6. There is an interesting relationship between Λ[(A, χ)] and the Cheeger problem,(Ω)
among all subsets,that is, the problem of finding the subset Ω ⊂ A that minimizes the ratio P|Ω|
where P (Ω) is the perimeter of Ω. In [BNP4] it was shown that a facet (A, χ) with A ⊂ R2 convex
and χ = +1 on Ac has Λ[(A, χ)] constant on A if and only if A is a solution of the Cheeger problem
on A with the W ◦ -perimeter.

4.2. Review of the notion of viscosity solutions. Viscosity solutions are defined as continuous functions that satisfy the comparison (maximum) principle with a class of suﬃciently
regular test functions. This way we do not need to assume any further regularity about thecandidate for a solution, but we have to choose a class of test functions that is suﬃciently
large. As was pointed out in the introduction, the operator div(∇W (∇·)) might not bewell-defined even for smooth functions. We therefore restrict the family of test functions to
only the stratified admissible test functions defined below.
Before the definition, let us recall convenient coordinates for the space Rn , introduced in[GP]. For a fixed p̂ ∈ Rn , we define Z1 ⊂ Rn to be the subspace parallel to the aﬃne hullaﬀ ∂W (p̂) of ∂W (p̂), that is, the smallest subspace such that Z1 +ξ ⊃ ∂W (p̂) for some ξ ∈ Rn .
Its dimension is then the dimension of the subdiﬀerential ∂W (p̂), k = dim ∂W (p̂) := dim Z1 .
Let Z2 := Z1⊥ be the orthogonal subspace. Then Rn = Z1 ⊕Z2 , Z1 is isometrically-isomorphic
to Rk and Z2 is isometrically-isomorphic to Rn−k . Fixing such isometries T1 : Rk → Z1 ,T2 : Rn−k → Z2 , we can write every x ∈ Rn uniquely in terms of (x′ , x′′ ), x = T1 x′ + T2 x′′ ,where x′ ∈ Rk and x′′ ∈ Rn−k . If k = 0 or k = n, we simply take x = x′′ or x = x′ ,respectively.
We also introduce the convex, positively one-homogeneous function Wp̂sl : Rk → R as
(4.2)

W (p̂ + λT1 w) − W (p̂)
,
λ→0+
λ

Wp̂sl (w) := limw ∈ Rk .

This function represents the infinitesimal structure of W near p̂, sliced in the direction of Z1 .
Analogously to (1.5), we define the functional Ep̂sl : L2 (Ω) → R.
With p̂, k as above, we say that a function ψ ∈ Lip(Rk ) is a p̂-admissible support function
if ({ψ = 0}, sign ψ) is a (Wp̂sl )◦ -(L2 ) Cahn-Hoﬀman facet. For a p̂-admissible support function
ψ, we define the nonlocal curvature-like operator
Λp̂ [ψ](x) := Λ[({ψ = 0}, sign ψ)],x ∈ {ψ = 0},

where Λ, defined in Definition 4.2, is used with the anisotropy W = Wp̂sl .
Definition 4.7. Let p̂ ∈ Rn , (x̂, t̂) ∈ Rn × R, k := dim ∂W (p̂). We say that φ(x, t) =ψ(x′ ) + f (x′′ ) + p̂ · x + g(t) is an admissible stratified faceted test function at (x̂, t̂) withslope p̂ if f ∈ C 1 (Rn−k ), ∇f (x̂′′ ) = 0, g ∈ C 1 (R), and ψ ∈ Lip(Rk ) is a p̂-admissible support
function with x̂′ ∈ int {ψ = 0}. Note that if k = 0, we have φ(x, t) = f (x) + g(t) for somef ∈ C 1 (Rn ), g ∈ C 1 (R).

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Y. GIGA AND N. POŽÁR

Definition 4.8 (Viscosity solution, cf. [GP, Definition 5.2]). We say that an upper semicontinuous function u is a viscosity subsolution of ut +F (∇u, div(∇W (∇u))) = 0 on Rn ×(0, T ),T > 0, if for any p̂ ∈ Rn , x̂ ∈ Rn , t̂ ∈ (0, T ) and any admissible stratified faceted test function φ at (x̂, t̂) with slope p̂ of the form φ(x, t) = ψ(x′ ) + f (x′′ ) + p̂ · x + g(t) such that thefunction u − φ(· − h) has a global maximum on Rn × (0, T ) at (x̂, t̂) for all suﬃciently smallh′ ∈ Rk and h′′ = 0, there exists δ > 0 such that
(4.3)

gt (t̂) + F (p̂, ess inf
Λp̂ [ψ]) ≤ 0.
′
Bδ (x̂ )Viscosity supersolutions are defined analogously as lower semicontinuous functions, replacing a global maximum with a global minimum, ess inf with ess sup, and reversing theinequality in (4.3).
A continuous function that is both a viscosity subsolution and a viscosity supersolution iscalled a viscosity solution.
Remark 4.9. Definition 4.8 at p̂ = 0 is a natural extension of the definition that appeared in an
earlier paper [GGP1] for anisotropies W smooth outside of the origin. In that case, if appropriate tests are given at p̂ ̸= 0 where W is smooth, the definition is equivalent to the definition ofF-solutions [G]. In fact, this equivalence can be proved along the lines of the proof of [G, Proposition 2.2.8], where one has to replace |x − y|4 by an appropriate function whose zero set consists of
the Wulﬀ shape of W .
Note that we do not need to consider the special test for “curvature-free” directions, p̂ = 0, as in[GP] due to our ability to construct admissible stratified facet functions in an arbitrary dimension.

Let us also state the invariance of solutions of the level set equation with respect torelabeling.
Proposition 4.10. Suppose that u is a viscosity subsolution/supersolution of the level set
formulation of the crystalline mean curvature flow V = v(ν, κW ) for some v ∈ C(S n−1 × R),
nondecreasing in the second variable. If θ ∈ C 2 (R) and inf θ′ > 0 then θ ◦ u is also a viscosity
subsolution/supersolution, respectively.
The diﬃculty in proving this proposition lies in the fact that θ ◦ φ might not be a test
function even if φ is since θ ◦ φ is not stratified in general. However, it is possible perturb
the test function while preserving ordering. The following lemma is a simpler version of theconstruction in [GG3, Proposition 3.3] and we skip its proof.
Lemma 4.11. Let φ(x, y) = ψ(x) + h(y) with ψ, h Lipschitz continuous on Rk and Rm ,respectively, such that φ(0, 0) = ψ(0) = h(0) = 0 and {ψ = 0} is compact. Let θ ∈ C 2 (R)with θ′ > 0 and θ(0) = 0. Set µ := θ′ (0) > 0. Then there exist δ, K > 0 such thatθ(φ(x, y)) ≥ µ(ψ(x) − 21 |ψ(x)| + h(y)) − K|y|2
for all |y| < δ, dist(x, {ψ = 0}) < δ. Note that sign s = sign(s − 21 |s|).
Proof of Proposition 4.10. We show the statement only for a supersolution. Suppose thatφ(x, t) = ψ(x′ ) + f (x′′ ) + p̂ · x + g(t) is a test function at (x̂, t̂) for θ ◦ u. We may assume
that (x̂, t̂) = (0, 0) and φ(0, 0) = 0. Note that p̂ · x = p̂′′ · x′′ since W is positively onehomogeneous. Applying Lemma 4.11 with θ̂ := θ−1 − θ−1 (0), and by extending the functions
properly away from a neighborhood of {ψ = 0} × {0}, we can find Lipschitz continuous ψ̂with sign ψ̂ = sign ψ and diﬀerentiable fˆ, ĝ with ∇fˆ(0) = 0 and ĝ ′ (0) = g ′ (0) such thatφ̂(x, t) := ψ̂(x′ ) + µ(fˆ(x′′ ) + p̂ · x + ĝ(t)) ≤ θ−1 (φ(x, t))
for all (x, t),APPROXIMATION BY REGULAR FACETS

15

and φ̂(0, 0) = θ−1 (0), where µ := (θ−1 )′ (0) > 0. In particular, since ∂W (µp̂) = ∂W (p̂), we
see that φ̂ is an admissible stratified faceted test function at (0, 0) for the supersolution testp̂
, − ess inf Bδ (0) Λµp̂ [ψ̂]) ≥ 0 for some δ > 0 (or µĝt (0) ≥ 0
of u. Therefore µĝ ′ (0) − µ|p̂|v( |p̂|
if p̂ = 0). But sign ψ̂ = sign ψ yields Λµp̂ [ψ̂] = Λp̂ [ψ], and we conclude that φ satisfies thesupersolution condition.
□
Remark 4.12. We currently do not know if it is possible to extend Proposition 4.10 to only
continuous nondecreasing θ. Such a generalization is usually established by an approximation ofθ by smooth functions, relying on the stability of viscosity solutions with respect to the uniform
convergence, see [G], for example. However, we have not established such stability for the viscosity
solutions of (1.3). Formally, such operations might expose facets whose structure is unclear, andit seems diﬃcult to perturb faceted test functions to avoid testing such facets. However, it is still
possible to show the uniqueness of the set evolution, see Section 4.4.3.

4.3. Comparison principle. One of the main results of [GP] was the comparison principle
Theorem 1.4, assuming Theorem 1.8. We shall give a brief sketch of the proof of Theorem 1.4
for the reader’s convenience.
Due to the nonlocality of the problem, we require that u and v are constant outside of acompact set to avoid technical issues with unbounded facets. However, it is an interestingquestion how to handle such solutions as well as boundary conditions. This will be addressedin a future work.
To show Theorem 1.4, we follow a variant of the standard proof by contradiction. The
problem (4.1) has features of a second order problem in that similarly to Ishii’s lemma that isused to construct test functions with ordered second derivatives even for only semicontinuous
solutions in the usual argument, we need to show the existence of ordered Cahn-Hoﬀman
facets at a contact point. In this article we only give the outline of the proof, and show how
to apply Theorem 1.3. For full details, see [GP].
We thus suppose that we have a viscosity subsolution u and a viscosity supersolution v
satisfying the hypothesis of Theorem 1.4, but for which the conclusion does not hold, thatis,m0 := sup(u − v) > 0,Qwith Q := Rn × (0, T ).
Then we follow the standard doubling-of-variables argument with an extra parameter.
That is, for ε > 0, ζ ∈ Rn , we study the maxima of the functionsΦζ,ε (x, t, y, s) := u(x, t) − v(y, s) −|x − y − ζ|2− Sε (t, s),2εover Q × Q = Rn × [0, T ] × Rn × [0, T ], where|t − s|2εεSε (t, s) :=++.
2εT −t T −s
The introduction of ζ helps “flatten” the profile of u and v near the point of maximum ofΦζ,ε that we are interested in, see Corollary 4.17 below.
We have the following important observation.
Proposition 4.13 (cf. [GG1]). There exists ε0 > 0 such that for all ε ∈ (0, ε0 ), |ζ| ≤ κ =κ(ε), Φζ,ε does not attain its maximum on the boundary of Q × Q.

16

Y. GIGA AND N. POŽÁR

Therefore, in what follows, we fix one such ε > 0 from Proposition 4.13 and κ = κ(ε). We
then write Φζ = Φζ,ε and S = Sε .
Now we follow the notation from [GG1, GP]. We define the maximum of Φζ asℓ(ζ) := max Φζ ,Q×Qand the set of the points of maximum{}
A(ζ) := arg max Φζ := (x, t, y, s) ∈ Q × Q : Φζ (x, t, y, s) = ℓ(ζ) .
Q×Q

Moreover, the set of gradients at maximum will be denoted as
{
}
x−y−ζ
B(ζ) :=: (x, t, y, s) ∈ A(ζ) .
εWe have the following compactness property.
Proposition 4.14 (cf. [GP, Proposition 7.3]). The graphs of A(ζ) and B(ζ) over |ζ| ≤ κare compact.
This compactness and the simple structure of ∂W , piece-wise constant on relatively open
convex sets, allow us to use the Baire category theorem to find a direction in which u and v
have certain flatness.
Proposition 4.15 (cf. [GP, Proposition 7.4]). There exists a set Ξ ∈ Rn , a vector ζ̂ ∈ Rn
and λ > 0 such that |ζ̂| + 2λ < κ, ∂W (p) is independent of p ∈ Ξ, andB(ζ) ∩ Ξ ̸= ∅for all |ζ − ζ̂| < 2λ.

The set Ξ can be taken as a relatively open convex set in its aﬃne hull aﬀ Ξ, which isorthogonal to aﬀ ∂W (p), p ∈ Ξ.
In other words, the proposition implies the existence of a set Ξ ⊂ Rn and a point ζ̂ ∈ Rn
such that for any ζ ∈ Rn close to ζ̂ there exists a point of maximum (x̂, t̂, ŷ, ŝ) ∈ A(ζ) of∈ Ξ. Noting that Φζ (x, t, ŷ, ŝ) ≤ Φζ (x̂, t̂, ŷ, ŝ), we see that x̂−ŷ−ζis theΦζ such that x̂−ŷ−ζεε2+ S(t, ŝ) touching u at (x̂, t̂). A similar
gradient of the smooth test function (x, t) 7→ |x−ŷ−ζ|2εreasoning applies for v. Since this gradient always falls into Ξ, we recover some flatness of uand v in the directions orthogonal to aﬀ Ξ, in the sense of the following lemma.

Lemma 4.16 (cf. [GP, Lemma 7.6]). Suppose that there exist p̂, ζ̂ ∈ Rn , a subspace Z2 ⊂ Rn
and λ > 0 such that |ζ̂| + 2λ < κ andB(ζ) ∩ (p̂ + Z2 ) ̸= ∅for all |ζ − ζ̂| < 2λ.

Thenℓ(ζ) − p̂ · ζ ≡ const

for all ζ ∈ ζ̂ + Z1 , |ζ − ζ̂| < 2λ,where Z1 := Z2⊥ .
The following can be derived about the behavior of u and v from the above lemma. Recall
that x′ ∈ Rk is identified with the orthogonal projection of x onto Z1 and x′′ ∈ Rn−k withthe projection onto Z2 .

APPROXIMATION BY REGULAR FACETS

17

Corollary 4.17 (cf. [GP, Corollary 7.7]). Suppose that we have p̂, ζ̂, λ, Z1 and Z2 as inLemma 4.16. Defineθ(x, t, y, s) := u(x, t) − v(y, s) −|x′′ − y ′′ − ζ̂ ′′ |2− p̂′ · (x′ − y ′ − ζ̂ ′ ) − S(t, s).
2ε
′

′

′

Then for any (x̂, t̂, ŷ, ŝ) ∈ A(ζ̂) such that x̂ −ŷε −ζ̂ = p̂′ we haveθ(x, t, y, s) ≤ θ(x̂, t̂, ŷ, ŝ) for (x, t), (y, s) ∈ Q, |x′ − y ′ − (x̂′ − ŷ ′ )| ≤ λ.
Now we want to construct admissible stratified faceted test functions for u and v to reacha contradiction with the definition of a viscosity solution. Therefore for the rest of the proof,
we fix Ξ, ζ̂ and λ from Proposition 4.15, and a point of maximum (x̂, t̂, ŷ, ŝ) ∈ A(ζ̂) suchζ̂that p̂ := x̂−ŷ−∈ Ξ. Given that ∂W (p) is independent of p ∈ Ξ, we set Z1 ⊂ Rn parallelεto the aﬃne hull aﬀ ∂W (p), p ∈ Ξ. The convexity of W implies that Ξ − p̂ ⊂ Z2 := Z1⊥ , see[GP, Proposition 3.1], and therefore Lemma 4.16 and Corollary 4.17 apply.
Let us set k := dim ∂W (p̂) = dim Z1 = n − dim Z2 . We need to construct two p̂-admissible
support functions on Rk ordered so that the comparison principle in Proposition 4.1 applies,
and which we can use to build the admissible faceted test functions at (x̂, t̂) for u and at(ŷ, ŝ) for v, with slope p̂. Since this is trivial if k = 0, we will from now assume that k ≥ 1.
4.3.1. Facet construction. We first observe that p̂ ⊥ Z1 since W is positively one-homogeneousand therefore p̂′ = 0. We need to find two facets to which to apply Theorem 1.3. We follow[GP]. First we define the functionsû(w) := u(T1 w + x̂, t̂) − u(x̂, t̂),w ∈ Rk ,v̂(w) := v(T1 w + ŷ, ŝ) − v(ŷ, ŝ),and their level sets{}Û := w ∈ Rk : û(w) ≥ 0 ,{}V̂ := w ∈ Rk : v̂(w) ≤ 0 .

Lemma 4.18. Sets Û and V̂ are closed with compact boundary, and at least one is compact.
Proof. Closedness follows from semicontinuity of û and v̂. Since supQ (u − v) > 0 and u ≤ v
outside of a bounded set, we have u(x̂, t̂) > v(ŷ, ŝ) and therefore û < v̂ outside of a bounded
set. Since û and v̂ are also both constant outside of a bounded set, Û and V̂ cannot be both
unbounded, and their boundaries must be compact. The lemma follows.
□
For convenience, we set|x′′ − ŷ ′′ − ζ̂ ′′ |2 |x̂′′ − ŷ ′′ − ζ̂ ′′ |2−+ S(t, ŝ) − S(t̂, ŝ),2ε2ε|x̂′′ − ŷ ′′ − ζ̂ ′′ |2 |x̂′′ − y ′′ − ζ̂ ′′ |2
ξv (y ′′ , s) :=−+ S(t̂, ŝ) − S(t̂, s),2ε2εξu (x′′ , t) :=x′′ , y ′′ ∈ Rn−k , t, s ∈ R. From Corollary 4.17 we deduce(4.4)

u(x, t) − u(x̂, t̂) − ξu (x′′ , t) ≤ 0, for (x, t) ∈ Q, dist(x′ − x̂′ , V̂ ) ≤ λ,v(y, s) − v(ŷ, ŝ) − ξv (y ′′ , s) ≥ 0, for (y, s) ∈ Q, dist(y ′ − ŷ ′ , Û ) ≤ λ.

18

Y. GIGA AND N. POŽÁR

Fu

2r

Fv

χv = −1χv = 1

Û
χu =

χu =

0

V̂

1

−1

Figure 1. An illustration of facets near the contact point 0. Sets Û andV̂ are drawn with dashed lines. Sets Fu and Fv are then composed of the
thick line with distance exactly 2r from Û and V̂ , respectively, as well as thegray regions (lighter for Fu , medium for Fv ). The dark gray region is Fu ∩ Fv .
Finally, the boundaries of the regular facets Gu and Gv are indicated by dottedlines.
For reasons that shall become apparent in the proof of Lemma 4.19 below, we set r := λ5 .
From the definition of Û , V̂ , semicontinuity of u, v, and the fact that u and v are constantoutside of a bounded set, there exists δ > 0 such that
(4.5)

u(x, t) − u(x̂, t̂) − ξu (x′′ , t) < 0 if dist(x′ − x̂′ , Û ) ≥ r, |x′′ − x̂′′ | ≤ δ, |t − t̂| ≤ δ,v(y, s) − v(ŷ, ŝ) − ξv (y ′′ , s) > 0 if dist(y ′ − ŷ ′ , V̂ ) ≥ r, |y ′′ − ŷ ′′ | ≤ δ, |s − ŝ| ≤ δ.

We now introduce k-dimensional facets (Fu , χu ) and (Fv , χv ). Fu and χu ∈ C(Fuc , {−1, 1})
are defined as(4.6)

Fu := {w ∈ Rk : dist(w, Û ) ≤ 2r and (dist(w, Û ) ≥ 2r or dist(w, V̂ ) ≤ λ − r)},
dist(w, Û ) < 2r and dist(w, V̂ ) > λ − r,1,χu (w) := −1, dist(w, Û ) > 2r,
0,otherwise,and Fv , χv ∈ C(Fvc , {−1, 1}) are defined analogously, swapping Û with V̂ . See Figure 1 for avisual explanation of this construction. Note that Fu and Fv are compact due to Lemma 4.18.
They also contain all the points with distance 2r from Û and V̂ , respectively, so that χu , χv
as defined in (4.6) are continuous on their complements.
Applying Theorem 1.3 with ρ = r, for anisotropy Wp̂sl , we obtain a Wp̂sl,◦ -(L2 ) CahnHoﬀman facet (Gu , χ̃u ) for the facet (Fu , χu ). Similarly, with anisotropy p 7→ Wp̂sl (−p), we
obtain a facet (Gv , χ̃v ) for (Fv , χv ). Note that (Gv , −χ̃v ) is a Wp̂sl,◦ -(L2 ) Cahn-Hoﬀman facet.

APPROXIMATION BY REGULAR FACETS

19

To finish the construction of the test functions, we set for x′ , y ′ ∈ Rk[]ũ(x′ − x̂′ ) := supsup u(x, t) − u(x̂, t̂) − ξu (x′′ , t) ,|x′′ −x̂′′ |≤δ |t−t̂|≤δṽ(y ′ − ŷ ′ ) :=inf

inf [v(y, s) − v(ŷ, ŝ) − ξv (y ′′ , s)] .

|y ′′ −ŷ ′′ |≤δ |s−ŝ|≤δ

Lemma 4.19. The facets (Gu , χ̃u ) and (Gv , −χ̃v ) have the following properties:(a) The facets are ordered, and ordered with respect to the “facets” of ũ and ṽ, namely,(4.7)

sup sign ũ(y) ≤ χ̃u (x) ≤ −χ̃v (x) ≤ inf sign ṽ(y),|y−x|≤r|y−x|≤rx ∈ Rn .

(b) The origin 0 lies in the interior of the intersection of the facets, i.e.,
B r (0) ⊂ Gu ∩ Gv .
Proof. The lemma was previously proved in [GGP1, Lemma 4.6], using a diﬀerent notation.
For the reader’s convenience, we present a self-contained proof using the new notion of facets.
To simplify the notation, we write sup|y−x|≤r χ(y) as (supr χ)(x), and analogously for inf r .
For (a), note that by construction from Theorem 1.3, χu ≤ χ̃u ≤ supr χu and inf r (−χv ) ≤−χ̃v ≤ −χv . Therefore the first inequality in (4.7) will follow from supr sign ũ ≤ χu , thesecond will follow from supr χu ≤ inf r (−χv ) and the third from −χv ≤ inf r sign ṽ. See also
Figure 1.
To show supr χu ≤ inf r (−χv ), we show the equivalent χu (w) ≤ −χv (z) for all |z − w| ≤ 2r.
Fix therefore w, z with |z − w| ≤ 2r. When χu (w) = −1, there is nothing to show since−χv ≥ −1. If χu (w) = 0 then by (4.6) dist(w, Û ) ≤ 2r, and by the triangle inequality
dist(z, Û ) ≤ 4r ≤ λ − r and therefore −χv (z) ≥ 0 = χu (w). Similarly, if χu (w) = 1 then by(4.6) dist(w, V̂ ) > λ − r, by the triangle inequality dist(z, V̂ ) > λ − 3r ≥ 2r and therefore−χv (z) = 1 = χu (w). We conclude that χu (w) ≤ −χv (z) for all w, z, |z − w| ≤ 2r.
To show supr sign ũ ≤ χu , fix w, z with |z − w| ≤ r. If ũ(w) < 0, then automaticallysign ũ(w) = −1 ≤ χu (z). If ũ(w) = 0, then by (4.5) dist(w, Û ) < r. By the triangle
inequality, dist(z, Û ) < 2r and so by (4.6) χu (z) ≥ 0 = sign ũ(w). Finally, if ũ(w) > 0, by(4.4) dist(w, V̂ ) > λ and by (4.5) dist(w, Û ) < r. Therefore the triangle inequality implies
dist(z, Û ) < 2r and dist(z, V̂ ) > λ − r, and so by (4.6) χu (z) = 1 = sign ũ(w). The proof of−χv ≤ inf r sign ṽ is analogous.
For (b), recall that 0 ∈ Û ∩ V̂ . Therefore by the triangle inequality dist(w, Û ) ≤ r ≤ 2r
and dist(w, V̂ ) ≤ r ≤ λ − r for any |w| ≤ r. In particular χu (w) = 0 by (4.6). Furthermore,
for any z, |z − w| ≤ r, dist(z, Û ) ≤ 2r, dist(z, V̂ ) ≤ 2r ≤ λ − r and therefore χu (z) = 0. We
conclude that 0 = χu (w) ≤ χ̃u (w) ≤ (supr χu )(w) = 0, in particular, w ∈ Gu . An analogous
argument implies w ∈ Gv .
□Let us now set m := sup{ũ(w) : dist(w, {χ̃u = −1}) ≤ 2r } and define()
2 sup ũψu (x) := max
dist(x, Gu )χ̃u (x), m .
rThen due to the ordering in Lemma 4.19(a) and the fact that ũ is a constant outside of abounded set, m < 0, and ũ(· − w) ≤ ψu for all |w| ≤ 2r . Note that sign ψu = χ̃u . In a similar
way we can construct ψv such that ψv ≤ ṽ(· − w) for all |w| ≤ 2r .

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Y. GIGA AND N. POŽÁR

The functions φu (x, t) := ψu (x′ − x̂′ ) + ξu (x′′ , t) and φv := ψv (x′ − ŷ ′ ) + ξv (x′′ , t) areadmissible stratified faceted test functions at (x̂, t̂) and (ŷ, ŝ), respectively, with slope p̂, inthe sense of Definition 4.7. Since u is bounded above and ψu is bounded, by modifying andsmoothly extending ξu for |x′′ − x̂′′ | > 2δ , |t − t̂| > 2δ , we can assume that u − φu (· − h) has
a global maximum at (x̂, t̂) in Q for all suﬃciently small h′ , |h′ | ≤ 2r , and h′′ = 0. A similarreasoning applies to φv . Therefore φu and φv are test functions in the sense of Definition 4.8.
From the definition of viscosity solutions, Definition 4.8, we infer that for some r̃ ∈ (0, r),()(ξu )t (t̂) + F p̂, ess inf [Λp̂ [ψu ]] ≤ 0,Br̃ (0)()(4.8)(ξv )t (ŝ) + F

p̂, ess sup [Λp̂ [ψv ]]≥ 0.

Br̃ (0)On the other hand, Lemma 4.19(a–b) and the comparison principle Proposition 4.1 imply(4.9)

ess inf [Λp̂ [ψu ]] ≤ ess sup [Λp̂ [ψv ]] ,Br̃ (0)Br̃ (0)and therefore by subtracting the inequalities in (4.8) and applying the ellipticity of F we
obtain()()εε+ F p̂, ess inf [Λp̂ [ψu ]] − F p̂, ess sup [Λp̂ [ψv ]] ≤ 0,+
0<Br̃ (0)(T − t̂)2 (T − ŝ)2Br̃ (0)a contradiction. This finishes the proof of the comparison principle, Theorem 1.4.
4.4. Stability and existence of solutions. With the comparison principle, Theorem 1.4,
valid in any dimension, the stability with respect to an approximation by regularized problems, Theorem 1.5, and the well-posedness, Theorem 1.6, proved originally in [GP] immediately generalize to all dimensions. Let us give a brief outline of the approach, for all details
see [GP]. The argument in the simplified form presented below basically first appeared in[GGP2].
In the standard viscosity theory, the existence of solutions usually follows from Perron’s
method: the largest subsolution (or the smallest supersolution) turns out to be a solution of
the problem, see [CIL, G]. However, it is not clear whether Perron’s method can be used forthe crystalline curvature problem (1.3) due to the very strong nonlocality of the curvature
operator, except in one dimension [GG1, GG3] when the speed of a facet is constant. In casewhen the speed of the facet is not constant, this approach seems to be diﬃcult except in a
one-dimensional setting, where Perron’s method is applied to construct a graph-like solution
with non-uniform driving force term by careful classification of speed profile of each facet[GGN]. To be more specific, the standard shift (Ishii’s shift) of a faceted test function tocreate a larger subsolution when the largest subsolution fails to be a supersolution and reacha contradiction cannot be performed unless the crystalline curvature Λ is constant on thefacet. Therefore we use the stability of solutions with respect to regularization of W in placeof Perron’s method.
4.4.1. Stability. We consider two modes of regularization of W :2(a) Wm ∈ C 2 (Rn ), a−1
m ≤ ∇ Wm ≤ am for some am > 0, and Wm ↘ W , or(b) Wm are smooth anisotropies, that is, Wm is an anisotropy, Wm ∈ C 2 (Rn \ {0}) and{p : Wm (p) ≤ 1} is strictly convex, such that Wm → W locally uniformly.

APPROXIMATION BY REGULAR FACETS

21

The regularization (a) yields a sequence of degenerate parabolic problems that are within theclassical viscosity theory [CIL], while (b) regularizes the crystalline curvature operator bysmooth anisotropic curvatures studied in [CGG] when F comes from a level set formulation,or in [GGP2, GGP1] for general F .
Note that both regularizations produce local problems, except at ∇um = 0 in (b) . The
main diﬃculty in proving the stability property with respect to the approximation is therefore
again caused by the nonlocality of the crystalline curvature operator. In the limit m → ∞,the nonlocal information contained in Λ must be recovered. This is achieved with the helpof a variant of the perturbed test function method.
Suppose therefore that we approximate W by a sequence of regular Wm as in (a) above
and obtain a sequence of solutions um of the regularized problems. We want to show thatu(x, t) = lim sup(y,s,m)→(x,t,∞) um (y, s) is a subsolution of (4.1) by verifying Definition 4.8.
Let φ be an admissible stratified faceted test function at a point (x̂, t̂) with slope p̂ satisfying
the assumptions in Definition 4.8. We need to show (4.3).
To simplify the explanation, let us assume that k = n so that p̂ = 0, Z1 = Rn andφ(x, t) = ψ(x) + g(t), where ψ ∈ Lip(Rn ) is 0-admissible support function. By definition,(A, χ) := ({ψ = 0}, sign ψ) is a (W0sl )◦ -(L2 ) Cahn-Hoﬀman facet. For the treatment of thegeneral case, see the details in [GP]. By adding constants and a translation, we can assume
that (x̂, t̂) = (0, 0) and u(x̂, t̂) = u(0, 0) = 0, g(t̂) = g(0) = 0. By Definition 4.8, we may
assume that u − φ(· − h) has a global maximum at (x̂, t̂) = (0, 0) for all |h| ≤ ρ for someρ > 0, and ψ(x) = 0 ⇔ x ∈ A for |x| ≤ ρ.
There are now a few diﬃculties with trying to follow the standard stability argument for
viscosity solutions. The first is the smoothness of φ. We need at least C 2 -regularity in space
to be able to use φ as a test function for the approximate problems called m-problems. Let
us therefore assume that φ is in fact smooth so that this is not an issue. The second problem
arises when we try to find a subsequence ml → ∞ and points of maxima (xl , tl ) of uml − φsuch that (xl , tl ) → (x̂, t̂) = (0, 0). Such a sequence exists in general only if u − φ has a strict
maximum at (0, 0). In the standard argument, this is ensured by a smooth perturbation ofφ, for instance by adding a term like |x|4 + |t|2 to φ. Suppose that we have such a sequence(xml , tml ) of maxima converging to (0, 0). Then by the definition of the viscosity solution of
the m-problem, we have[φt + F (∇φ, div(∇Wml (∇φ)))](xml , tml ) ≤ 0,

for all l.

As div(∇Wm (∇φ)) = tr[(∇2 Wm (∇φ))∇2 φ] and ∇φ(0, 0) = 0, there is little hope that thislocal quantity will converge to anything useful due to the singularity of W at 0, much lessto Λp̂ [ψ](x̂), which is nonlocal.
The key idea is to introduce a uniform perturbation of φ that depends on m, so that it
captures the necessary nonlocal information. This basic scheme was introduced with great
success in the viscosity theory by Evans [E], where it is called the perturbed test function
method. In fact, this idea was actually carried out in the one-dimensional setting, where
the test function is taken essentially as Wm◦ [GG2]. However, in higher dimensional case,
one has to test with more functions, which requires a new idea for a choice of test functions
depending on m. Such an idea has first appeared in [GGP2]. We shall sketch it below.
As −Λp̂ (ψ) coincides on the facet with the minimizing element of the subdiﬀerential of
the anisotropic total variation energy at ψ, it can be approximated by its resolvent problem.
This problem is equivalent to performing one step of the implicit Euler discretization of the

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Y. GIGA AND N. POŽÁR

anisotropic total variation flow. To have compactness, we modify ψ far away from them
facet and rescale so that it is Zn -periodic. For given a > 0, we find the unique solutions
ψa , ψa,m ∈ L2 (Tn ) of the resolvent problems
ψa − ψ
ψa,m − ψ∈ −∂Em (ψa,m ),∈ −∂E(ψa ),aawhere E and Em are the energies defined in (1.5) with Ω = Tn = Rn /Zn . It is possible tomodify ψ away from the facet in such a way that ψ ∈ Lip(Tn ) and ∂E(ψ) ̸= ∅ since ψ is a0-admissible support function.
The resolvent problems have a number of very useful properties. Since Wm are smooth, theproblem for ψa,m is a quasilinear elliptic problem. Therefore ψa,m ∈ C 2 (Tn ) by the elliptic
regularity. Moreover, due to the monotone convergence of Wm → W , Em converges in Mosco
sense to E, which implies the resolvent convergence ψa,m → ψa in L2 (Tn ) as m → ∞ [A2].
By the comparison principle and the translation invariance of the resolvent problems (4.10),∥∇ψa,m ∥∞ ≤ ∥∇ψ∥∞ . Therefore ψa,m → ψa uniformly and ∥∇ψa ∥∞ ≤ ∥∇ψ∥∞ . Finally,
since ∂E(ψ) ̸= ∅, ψa → ψ uniformly and ψaa−ψ → −∂ 0 E(ψ) := div zmin in L2 (Tn ) as a → 0+.
Recall that Λ0 [ψ] = −∂ 0 E(ψ) on the facet of ψ. This construction yields perturbed test
functions φa,m (x, t) = ψa,m (x) + g(t) and φa (x, t) = ψa (x) + g(t).
We now turn our attention to a compact neighborhood of the facet of ψ,(4.10)

O := {x : dist(x, A) ≤ ρ}.
We assume that we have modified ψ above only far away from the facet (A, χ) so that thevalue of ψ(· − w) does not change on O for all |w| ≤ ρ. For convenience, we defineū(x) := sup u(x, t) − g(t).
|t|≤ρNote that ū − ψ(· − h) ≤ 0 on O for |w| ≤ ρ, with equality at x = 0.
We set δ := ρ5 and define the critical set
N := {x ∈ O : ū(x) ≥ 0, ψ(x − w) ≤ 0 for some |w| ≤ δ}.
We can deduce that [GP, Corollary 8.3]
ū(x) ≤ 0,

ψ(x − z) ≥ 0,

for all dist(x, N ) ≤ 3δ, |z| ≤ δ,

dist(N, ∂O) ≥ 4δ.
In particular, we see that for any |z| ≤ δ and any α > 0 all maxima of ū − αψ(· − z)
in {x : dist(x, N ) ≤ 3δ} lie in N . We can therefore make the Lipschitz constant ∥∇ψ∥∞
arbitrarily small by multiplying ψ by small α > 0 in the sequel. By adding |t|2 to g(t) if
necessary, we may assume that all maxima of u − φ(· − z, ·) are located at t = 0.
For a > 0 let us fix za such that |za | ≤ δ and ψa (za ) = min|w|≤δ ψa (w). This choice will
become important later.
By the above consideration and the uniform convergence of ψa → ψ, there exists a0 > 0such that all the maxima of u − φa (· − za , ·) in M 3δ lie in M δ , whereM s := {(x, t) : dist(x, N ) ≤ s, |t| ≤ s}.
Now for every a ∈ (0, a0 ), by the uniform convergence of ψa,m → ψa and properties ofhalf-relaxed limits, there exist a point (xa , ta ) ∈ M δ of maximum of u − φa (· − za , ·) andsequences ml → ∞, (xl , tl ) → (xa , ta ) as l → ∞, where (xl , tl ) is a point of maximum of
uml − φa,ml (xl − za , tl ). By the uniform Lipschitz continuity of ψa,m , we can assume that

APPROXIMATION BY REGULAR FACETS

23

there exists pa ∈ Rn , |pa | ≤ ∥∇ψ∥∞ , such that ∇ψa,ml (xl − za ) → pa as l → ∞. Since φa,m
are smooth, the definition of viscosity solution um implies
g ′ (tl ) + F (∇ψa,ml , div(∇Wml (∇ψa,ml )))(xl − za ) ≤ 0.
ψ−ψ

As div(∇Wml (∇ψa,ml )) = −∂ 0 Eml (ψa,ml ) = a,mal , the uniform convergence implies in thelimit l → ∞)( ψa − ψ(xa − za ) ≤ 0.
g ′ (ta ) + F pa ,a
The point za was chosen above in such a way that a geometric lemma, [GP, Lemma 8.5],
impliesψa − ψ
ψa − ψ
(xa − za ) ≤ min
(w).
|w|≤δ
a
a
Monotonicity of F in the second variable thus yields()
ψa − ψ
g ′ (ta ) + F pa , min(w) ≤ 0.
|w|≤δaNote that ta → 0 as a → 0+. By finding a sequence of aj → 0+ such that there exist p ∈ Rn ,ψa −ψ|p| ≤ ∥∇ψ∥∞ with paj → p and limj→∞ min|w|≤δ aj j (w) = lim inf a→0+ min|w|≤δ ψaa−ψ (w),
we deduce thatg ′ (0) + F (p, lim inf min

a→0+ |w|≤δψa − ψ
(w)) ≤ 0.
a

Since ψaa−ψ → Λ0 [ψ](w) in L2 (Bδ (0)), we conclude that
ess inf Λ0 [ψ](w) ≥ lim inf min|w|≤δ

a→0+ |w|≤δψa − ψ
(w).
a

The monotonicity of F therefore yields
g ′ (0) + F (p, ess inf Λ0 [ψ](w)) ≤ 0.
|w|≤δ

Finally, we recall that we can assume that |p| ≤ ∥∇ψ∥∞ is arbitrarily small. Continuity of
F in the first variable yield the final conclusion that (4.3) is satisfied.
The full argument is more technically involved. In particular, we need to decompose
the space Rn into the direct sum of Z1 and Z2 as explained in Section 4.2, and treat them
independently. In essence, the resolvents ψaa−ψ do not depend on the directions parallel to Z2 ,[GP, Lemma 3.9], and therefore we can treat these directions as we treat time in the above
simplified argument. A symmetric argument implies that u(x, t) = lim inf (y,s,m)→(x,t,∞) um is
a viscosity supersolution.
To address the stability with respect to positively one-homogeneous approximations Wmin (b) above, we approximate Wm by a sequence Wmδ as in (a) and modify the previous
stability argument. In particular, we have solutions uδm besides um , and uδm ⇒ um locallyuniformly as δ → 0+ due to the stability results in [GGP2, GGP1].

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Y. GIGA AND N. POŽÁR

4.4.2. Existence. Existence of solutions of the equation (4.1) is now a standard consequenceof the stability property. Fixing initial data u0 and an approximating sequence of onehomogeneous Wm so that the stability result Theorem 1.5 holds, we get a sequence ofviscosity solutions of (4.1) with the anisotropy Wm and initial data u0 . The existence ofsuch solutions follows from the standard theory of viscosity solutions [CGG, G] if F comesfrom the level set formulation of (1.4), or from [GGP1] in the general case. By the stability result Theorem 1.5, the half-relaxed limits ū(x, t) = lim sup(y,s,m)→(x,t,∞) um (y, s) andu(x, t) = lim inf (y,s,m)→(x,t,∞) um (y, s) are respectively a subsolution and a supersolution of(4.1), without the initial data. Clearly u ≤ ū. To show the other inequality, we use the comparison theorem Theorem 1.4 after we show that ū and u attain the correct initial data u0 andthat they are equal to a constant outside of a compact set at each time. This can be done via
the comparison principle for the regularized problems in a rather standard way using translations of the barriers for the solutions um of the type (x, t) 7→ min(max(Wm◦ (x), c1 ), c2 ) + ct
for appropriate constants c1 , c2 , c, where Wm◦ is the polar of Wm . Since Wm → W locallyuniformly, Wm◦ → W ◦ locally uniformly as well. The existence part of the well-posedness theorem, Theorem 1.6, is established. The uniqueness is a direct consequence of the comparison
principle, Theorem 1.4.
4.4.3. Uniqueness of the crystalline mean curvature flow. Let us conclude the paper by giving
an outline of the proof of Theorem 1.7. This is standard for the regular mean curvature flow,
see for instance [G, Theorem 4.2.8]. We use the stability property Theorem 1.5 here. For
given bounded open set E0 ⊂ Rn , we can define the evolution {Et }t≥0 by considering an
initial data u0 ∈ C(Rn ) with {x : u0 (x) < 0} = E0 such that for some C > 0 we have u0 = C
for all |x| large. Then by Theorem 1.6 there is a unique viscosity solution u of (1.3) withp, −ξ). We set Et := {x : u(x, t) < 0}.
F (p, ξ) = |p|f ( |p|To prove the independence of the evolution {Et }t≥0 on the choice of u0 , let us consider thesolution v with another initial data v0 for E0 . It is well known, for example [G, Lemma 4.2.9],that there exists θ ∈ C(R), nondecreasing, with θ(s) = 0 for s ≥ 0 and θ(s) < 0 for s < 0such that θ ◦u0 ≤ v0 . Let um , vm be a sequence of approximations of u and v with initial datau0 and v0 , respectively, as in Theorem 1.5(b). By the invariance property for the regularized
problems, we know that θ ◦ um are again viscosity solutions of the regularized problems withinitial data θ ◦ u0 . In particular θ ◦ um ≤ vm by the comparison principle, and in the limitθ ◦ u ≤ v. We conclude that {x : v(x, t) < 0} ⊂ {x : u(x, t) < 0} for all t ≥ 0. By switching
the roles of u and v, we deduce the opposite inclusion. The uniqueness of {Et }t≥0 follows.
Acknowledgments. The work of the first author is partly supported by Japan Society for
the Promotion of Science (JSPS) through grants No. 26220702 (Kiban S), No. 17H01091(Kiban A) and No. 16H03948 (Kiban B). The work of the second author is partially supported
by JSPS KAKENHI Grant No. 26800068 (Wakate B).
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27(Y. Giga) Graduate School of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba
Meguro-ku, Tokyo 153-8914, Japan.
(N. Požár) Faculty of Mathematics and Physics, Institute of Science and Engineering,
Kanazawa University, Kakuma town, Kanazawa, Ishikawa 920-1192, Japan.
E-mail address: npozar@se.kanazawa-u.ac.jp