VISCOSITY SOLUTIONS FOR THE LEVEL SET FORMULATION
OF THE CRYSTALLINE MEAN CURVATURE FLOWNORBERT POŽÁR

Abstract. In this short note we announce the results of [25], coauthored with
Y. Giga (University of Tokyo), concerning a new notion of solutions for thelevel set formulation of the crystalline mean curvature flow of surfaces in three
dimensions. We establish a comparison principle, the stability with respect toan approximation by regularized problems, and the unique existence of thelevel set flow for arbitrary bounded three dimensional crystals. Most of ourapproach applies to an arbitrary dimension, except for the construction of
faceted test functions.

This note is based on a recent joint work [25] with Y. Giga (University of Tokyo).
Here we present a simplified exposition of our results concerning the level set
formulation of the crystalline mean curvature flow in three dimensions that appearedin [25], but we omit the details and proofs and instead refer the reader to that
paper.
1. Introduction
The crystalline mean curvature flow was proposed independently by S. B. Angenent and M. E. Gurtin [3] and by J. Taylor [35] to model the anisotropic motionof antiphase boundaries in material science. It can be regarded as an example of amean curvature flow under a Finsler metric [12]. Many properties of the crystalline
mean curvature were already established by the original authors, and there is alarge amount of literature devoted to its study. However, even the local-in-time
unique existence of solutions had been a long-standing open problem. For a full
bibliography we refer the reader to [6, 22, 25]. It was our aim in [25] to address thisissue for surfaces in three dimensions.
Let n ∈ N be the dimension. The anisotropic mean curvature κσ : Γ → R is
defined as the first variation of the surface energy functionalZ
F(Γ) :=
σ(ν) dHn−1 ,Γnwhere Γ ⊂ R is a smooth closed oriented surface with the unit outer normal
ν, σ : S n−1 → (0, ∞) is the surface energy density defined on the unit sphereS n−1 := {x ∈ Rn : |x| = 1} and Hn−1 is the (n − 1)-dimensional Hausdorff measure.
We will for convenience suppose that σ is defined on Rn as the one-homogeneous
Date: March 11, 2016 (git: 2016-03-11, dbb9cde).
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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functionσ(p) = |p|σ

p
|p|


,

p ∈ Rn \ {0},and σ(0) = 0. If σ is C 2 away from the origin such that σ 2 is strictly convex, it canbe shown that(1.1)

κσ = − divΓ (∇p σ(ν)),

where divΓ is the surface divergence operator on Γ [21] and ∇p σ is the gradientof σ, itself a zero-homogeneous function. If σ(p) = |p| then κσ is nothing but thestandard mean curvature.
In this note we are primarily concerned with the crystalline case: we assume that
σ is a convex, positively one-homogeneous, piece-wise linear function, zero only atthe origin, and we refer to such σ as crystalline. These functions (not necessarily
one-homogeneous) are also known as polyhedral functions in convex analysis [32].
We then call κσ the crystalline (mean) curvature of Γ. Note that (1.1) cannot
be used to define κσ anymore and κσ must be understood in a variational sense[6, 10, 11]. We will give details in Section 2.
Now consider a family of closed surfaces {Γt }t≥0 evolving under the surface
velocity law(1.2)

V = f (ν, κσ ),where V , ν, and κσ are respectively the normal velocity of Γt in the direction of ν,
the outer unit normal vector to Γt , and the crystalline curvature of Γt . We assume
that f : S n−1 × R → R is a continuous function, non-decreasing in the second
variable. A typical example is the crystalline curvature flow
(1.3)

V = κσ .

A characteristic feature of the crystalline mean curvature flow (1.2) with crystalline
σ is the appearance of flat parts of the surface in the directions in which σ is notdifferentiable, the so-called crystal facets. On these facets, κσ is a nonlocal quantity.
In general it should be a constant so that facets move with constant velocity andare preserved. This is indeed the case in two dimensions, however, it is known thatκσ might have jumps and then facet breaking or bending occurs in three dimensions[9]. This makes analysis of the flow rather difficult.
In dimensions n ≥ 3, even if σ(p) = |p| and (1.3) is the standard mean curvature
flow, the solution of (1.3) might develop singularities like pinch off in finite time,even when starting with a smooth initial surface. Therefore a suitable notion ofa generalized solution is necessary. There are a few standard approaches to thesolutions of (1.3) in dimensions three and higher. One of the first ones was a varifold
solution introduced by K. Brakke [14], further developed by [27] and [34]. However,this approach seems to be still limited to the isotropic mean curvature flow.
Another approach uses the minimizing movements scheme developed by Almgren,
Taylor and Wang [1] and Luckhaus and Sturzenhecker [30]. This approach provides
a global-in-time solution, but uniqueness was shown only in very specific cases likeconvex initial data [7, 8, 16]. After the work on our paper [25] had been finished, we
found out about a result of Chambolle, Morini, Ponsiglione [17] where the authors
define a weak solution of(1.4)

V = σ(ν)κσ

VISCOSITY SOLUTIONS FOR CRYSTALLINE MEAN CURVATURE FLOW

3

for crystalline σ in terms of the distance function to the evolving surface in thespirit of [33], but in the distributional sense that appeared in [16], and prove a
comparison principle in an arbitrary dimension. The existence is then shown using
the Almgren-Taylor-Wang scheme. However, their approach seems to be currently
restricted to the very specific form of the equation (1.4) where the mobility is
proportional to σ and does not apply to equations like (1.3) or V = κσ + 1.
We take the level set method approach. It was introduced independently byChen, Giga and Goto [18] and Evans and Spruck [19] for geometric motions likethe anisotropic mean curvature flow (1.2). The idea is to take a function u : Rn ×
[0, ∞) → R whose zero level set is equal to Γt , that is, Γt = {x ∈ Rn : u(x, t) = 0}.
Then u is a solution of a geometric partial differential equation (PDE) of parabolic
type. It turns out that one can generalize the theory of viscosity solutions into thiscontext and prove the well-posedness of the initial value problem. The zero level setof any solution of the PDE is then a generalized solution of (1.2). It is unique if thezero level set of u does not fatten, that is, if its Lebesgue measure is zero. The main
merit of this approach, in contrast to the above variational approaches, is that itallows for very general f in (1.2).
To introduce the level set formulation of (1.2), we will use the one-homogeneous
extension of f ,


p
(1.5)
F (p, ξ) := −|p|f − , ξ .
|p|Suppose that u : Rn × [0, ∞) → R is a function whose every level set moves withthe velocity law (1.2), in the sense that V = ut /|∇u| and ν = −∇u/|∇u|. Here∇u := (∂u/∂x1 , . . . , ∂u/∂xn ) is the space gradient and ut = ∂u/∂t. Then u formallysatisfies(1.6)

ut + F (∇u, div ∂W (∇u)) = 0

in Rn × (0, ∞),where W (p) := σ(−p). ∂W denotes the subdifferential of W , which in the caseof differentiable W is just the gradient of W . We explain how to understand thequantity div ∂W (∇u) in Section 2 below. Since ∂W has jumps as W is piece-wiselinear, div ∂W (∇u) has a very strong singularity which makes this operator nonlocal.
Therefore (1.6) is not a usual PDE. We say that (1.6) is a level set formulation of
the surface evolution problem (1.2).
In [22, 23], we have together with M.-H. Giga and Y. Giga succeeded in extending
the theory of viscosity solutions to the equation (1.6) in an arbitrary dimension
with W being a positively one-homogeneous function so that the level set {W ≤ 1}
is a smooth strictly convex set containing the origin in its interior. Then (1.6) is ananisotropic total variation flow of non-divergence type. It corresponds to a graphformulation of the crystalline curvature flow with an anisotropy σ(ν) = |ν 0 | + |νn |,where |ν 0 | is the Euclidean norm on Rn−1 , and ν = (ν 0 , νn ), ν 0 ∈ Rn−1 .
The current note summarizes our most recent work [25], coauthored with Y. Giga.
For a general crystalline W we introduced a new notion of viscosity solutions for(1.6) and for dimensions n = 2, 3 proved the comparison principle Theorem 4.1,
stability Theorem 5.2 and well-posedness of the initial value problem. Given an
initial Γ0 that is a boundary of a bounded open set in Rn , n = 2, 3, if u0 is acontinuous function, constant outside of a bounded set, with zero level set Γ0 and{Γt }t≥0 are the zero level sets of the unique viscosity solution u of (1.6) with initial
data u0 , Γt := {x : u(x, t) = 0}, we call {Γt }t≥0 the level set flow of (1.2). Using

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the standard arguments, it can be shown that {Γt }t≥0 is independent of the choiceof u0 . However, it is known that even if Γ0 is a surface, the level set Γt might fatten
and thus might not be a surface for some t > 0 [18, 21]. We have the following
well-posedness theorem (stated for dimension n = 3 for simplicity).
1.1. Theorem (Well-posedness). Let n = 3 and let σ be crystalline in R3 . Assume
that f = f (p, ξ) is continuous on S 2 × R3 and non-decreasing in the second variable.
Furthermore, assume that f (p, ξ)/(|ξ| + 1) is bounded in S 2 × R3 . If Γ0 is theboundary of a bounded open set in R3 then there exists a global unique level set flow{Γt }t≥0 of (1.2).
Note that the assumption of linear growth of f in its second variable is only forsimplicity, and can be removed using the techniques in [21, 26, 28]. We also should
point out that it is rather straightforward to allow for f depending on time. On theother hand, the dependence of f on the space variable is much more subtle, and amodification of the definition of κσ itself is probably necessary, as in [24].
Now we consider the approximation of (1.2) by problems(1.7)

V = f (ν, κσε ),nwhere σε : R → R are convex positively one-homogeneous surface energy densities
that are smooth on Rn \ {0}. We can prove the following stability result (again
stated for dimension n = 3).
1.2. Theorem (Stability). Let n = 3. Under the assumption of Theorem 1.1, let u
be a viscosity solution of (1.2) with initial data u0 ∈ C(R3 ) such that u0 (x) = −c
for |x| ≥ R with some R and c > 0. Assume that σε is smooth in R3 \ {0}, convex
and one-homogeneous and σε → σ uniformly on S 2 . Let uε be a viscosity solutionof (1.2) with W = Wε (p) = σε (−p), with initial data uε0 such that uε0 (x) = −c for|x| ≥ R. Assume that uε0 → u0 uniformly. Then uε converges locally uniformly to u
in R3 × [0, ∞).
Outline. The discussion of our approach follows. Section 2 covers the definition of
crystalline curvature on facets. This is then used to introduce the viscosity solutionsin Section 3. We then briefly explain the comparison principle and stability in
Section 4 and Section 5, respectively. Finally, we present a simple example of test
functions in Section 6.
2. Crystalline curvature
We take a variational approach to understanding the crystalline curvature κσ fora crystalline energy density σ [6, 10, 11]. This is motivated by viewing the mean
curvature as the quantity that drives the gradient flow of the surface energy F
with respect to the L2 -metric on the surface. Consider the classical gradient flow
problem for some convex lower-semicontinuous energy E : H → R, formulated as
the subdifferential inclusiond(2.1)
u(t) ∈ −∂E(u(t)),
dtwhere H is a Hilbert space with inner product (·, ·) and ∂E(ψ) for ψ ∈ H denotes
the subdifferential of E defined as∂E(ψ) := {v ∈ H : E(ψ + h) − E(ψ) ≥ (h, v) for all h ∈ H}.

VISCOSITY SOLUTIONS FOR CRYSTALLINE MEAN CURVATURE FLOW

5This set is closed and convex, but possibly nonempty. However, it is known [15, 29]
that the right derivative of the solution of (2.1) equals to −∂ 0 E(u(t)), where ∂ 0 E(ψ)
is the minimal section (canonical restriction) of the subdifferential,∂ 0 E(ψ) := arg min kvkH .
v∈∂E(ψ)That is, it is the unique element of the subdifferential with the smallest norm. In
particular, ∂E(u(t)) is nonempty for every t > 0.
We therefore interpret the quantity − div ∂W (∇u) in (1.6) as the minimal sectionof the subdifferential of the total variation energy E : L2 (Ω) → R defined as(R
W (Dψ) dx, ψ ∈ L2 (Ω) ∩ BV (Ω),Ω
(2.2)
E(ψ) :=+∞,
otherwise,for some Ω ⊂ Rn , where BV (Ω) is the space of functions of bounded variation on Ω.
The definition of E has to be clarified when Dψ is merely a measure: we therefore
understand E as the closure (that is, the lower semi-continuous envelope) of thetotal variational functional defined for functions on W 1,1 (Ω) ∩ L2 (Ω) [13, 20]. In
particular, note that the functional defined only for W 1,1 -functions is not lower
semi-continuous and we need to consider BV -functions.
To avoid issues with boundary values, we will always consider only Ω with theflat torus topology. In other words, we only apply E to periodic functions. It turns
out that this is sufficient for our purposes since even though the crystalline mean
curvature is a nonlocal quantity, its nonlocality is restricted to facets (flat parts) of
the surface. Furthermore, we only need to deal with bounded facets.
We shall define the crystalline curvature on facets (flat parts) of Lipschitz functions.
For that we need a suitable description of facets that is given by the notion ofpairs that we have introduced in [22, 23, 25]. A simple closed (compact) set in Rn
is insufficient to specify a facet because we also need the information about the
behavior of the function in a neighborhood of the facet to be able to define thecrystalline mean curvature.
2.1. Definition. Let k ∈ N. We say that (A− , A+ ) is a bounded open pair in Rk
if A± ⊂ Rk are open sets, A− ∩ A+ = ∅, and either Ac− or Ac+ is bounded.
We say that Ac− ∩ Ac+ is the facet of the bounded open pair (A− , A+ ).
Let (A− , A+ ) and (B− , B+ ) be two bounded open pairs in Rk . We say that
(A− , A+ )  (B− , B+ )if
B− ⊂ A−andA+ ⊂ B + .

The relation  introduces a partial order on the collection of bounded open pairs.
For the purpose of perturbation of facets, we introduce the notion of generalized
neighborhood of sets in Rk and of bounded open pairs.
2.2. Definition (cf. [22, 25]). For any set E ⊂ Rk and ρ ∈ R the generalized
neighborhood is defined asρ > 0,
E + B ρ (0)U ρ (E) := E
ρ = 0,
ρ < 0,x ∈ E : B |ρ| (x) ⊂ E

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where Bρ (x) := y ∈ Rk : |y − x| < ρ is the open ball and B ρ (x) is its closure.
For a bounded open pair (A− , A+ ) in Rk we introduce the generalized neighborhoodU ρ (A− , A+ ) := U −ρ (A− ), U ρ (A+ ) .
Finally, to precisely state what we mean by a facet of a function, we introduce
the support functions of pairs.
2.3. Definition (cf. [22, 25]). A Lipschitz function ψ ∈ Lip(Rk ) is called a support
function of a bounded open pair (A− , A+ ) in Rk if
> 0 x ∈ A+ ,ψ(x) = 0 x ∈ Ac− ∩ Ac+ ,< 0 x ∈ A− .
The crystalline mean curvature is given as the divergence of a certain vector
field, the Cahn-Hoffman vector field, that lives on the facet, and whose divergence
minimizes the L2 -norm among all possible Cahn-Hoffman vector fields [6, 10, 11].
We make the following definition: given p̂ ∈ Rn , we define V to be the smallest
subspace of Rn such that V + ξ contains ∂W (p̂) for some ξ ∈ Rn . Note that V + ξ
is the affine hull of ∂W (p̂), denoted as aff ∂W (p̂).
Set k = dim V , U = V ⊥ and let TV : Rk → V , TU : Rn−k → U be isomorphisms.
For x ∈ Rn we use x0 , x00 to denote the unique points x0 ∈ Rk , x00 ∈ Rn−k such thatx = TV x0 + TU x00 .

(2.3)

Note that if k = 0 then W is differentiable at p̂. However, we are interested in thecase k ≥ 1 when W has a “corner” at p̂ and the crystal has a facet in this direction.
It turns out that W is locally affine in the directions in U , and has a “corner” inthe directions in V . We therefore want to only consider its behavior in V . This iswhere the sliced energy density comes in. We introduce Wp̂sl : Rk → R as
W (p̂ + λTV p0 ) − W (p̂)λ→0λWp̂sl (p0 ) := lim

for p0 ∈ Rk .

It is well known that Wp̂sl is a positively one-homogeneous convex function, see [25]for details.
This energy density then induces the sliced energy functional Ep̂sl : L2 (Γ0 ) → R
given by(R
Wp̂sl (Dψ) dx, ψ ∈ L2 (Γ0 ) ∩ BV (Γ0 ),Γ0Ep̂sl (ψ) :=+∞,otherwise.
We consider Γ0 to be a k-dimensional flat torus, so that ψ is a periodic function on
Rk .
With these definitions at hand, we can finally define the crystalline mean curvatureon k-dimensional facets with slope p̂ as the minimal section of the subdifferentialof the sliced energy Ep̂sl . The subdifferential can be characterized as divergencesof certain vector fields (Cahn-Hoffman vector fields). Let therefore Γ0 be a kdimensional flat torus. We introduce the set of Cahn-Hoffman vector fields for a
Lipschitz function ψ ∈ Lip(Γ0 ) asCHp̂sl (ψ; Γ0 ) := z ∈ X2 (Γ0 ) : z(x) ∈ ∂Wp̂sl (∇ψ(x)) a.e. x ∈ Γ0 .

VISCOSITY SOLUTIONS FOR CRYSTALLINE MEAN CURVATURE FLOW

7

Using the notation from [4], the space X2 (Γ0 ) is the space of bounded vector fields
with L2 divergence,X2 (Γ0 ) := z ∈ L∞ (Γ0 ; Rk ) : div z ∈ L2 (Γ0 ) .
The subdifferential of Ep̂sl then can be characterized for ψ ∈ Lip(Γ0 ) [2, 31] as∂Ep̂sl (ψ) = − div CHp̂sl (ψ; Γ0 ) := − div z : z ∈ CHp̂sl (ψ; Γ0 ) .
Suppose that (A− , A+ ) is a bounded open pair in Rk and ψ ∈ Lip(Rk ) is its
support function. Since the facet Ac− ∩ Ac+ is bounded, we can find L > 0 sufficiently
large so that Ac− ∩ Ac+ ⊂ (−L/2, L/2)k . Then we can find an 2L-periodic Lipschitz
function ψ̂ that ψ̂ = ψ on a neighborhood of the facet Ac− ∩ Ac+ and ψ̂ 6= 0 on[−L, L]k \ (Ac− ∩ Ac+ ). We set Γ0 = [−L, L)k with the flat torus topology, so thatψ̂ ∈ Lip(Γ0 ).
Now if there exists such a periodic function ψ̂ for which CHp̂sl (ψ̂; Γ0 ) is nonempty,
we say that ψ is a p̂-admissible support function of pair (A− , A+ ) and we writeψ ∈ D(Λp̂ ). We then define the crystalline mean curvature on the facet Ac− ∩ Ac+
for the support function ψ asΛp̂ [ψ] := −∂ 0 Ep̂sl (ψ̂; Γ0 )on Ac− ∩ Ac+ .

In other words, the crystalline mean curvature Λp̂ [ψ] is given as the divergence of
the Cahn-Hoffman vector field for ψ̂ that minimizes the L2 -norm of the divergence.
It takes some work to prove that Λp̂ is well-defined, in particular, that it does
not depend on the choice of the periodic extension ψ̂. We can also show thata Cahn-Hoffman vector field exists for some periodic extension if and only if a
Cahn-Hoffman vector field exists for ψ on the neighborhood of the facet Ac− ∩ Ac+ .
For details we refer the reader to [25].
It is easy to see that any vector field zmin ∈ CHp̂sl (ψ̂; Γ0 ) that minimizes theL2 -norm of div z should satisfy the Euler-Lagrange equation ∇(div zmin ) = 0 on thefacet Ac− ∩ Ac+ . If such a vector field exists, then Λp̂ [ψ] = div zmin is a constant on
the facet. However, it is known that there might not be any such vector field, inwhich case Λp̂ [ψ] is not a constant on the facet, and in fact it might have jumps.
This leads to facet breaking or bending. In general, Λp̂ [ψ] is only a BV-function[10, 11]. This is not a problem for our viscosity solutions, however, we have to becareful when trying to use the value of Λp̂ [ψ] at a point. This is one of the reasonsfor giving the faceted test in (3.2) in terms of the essential infimum or supremum ofΛp̂ [ψ] over a small ball.
A pair that has at least one p̂-admissible support function is called p̂-admissible.
2.4. Definition. Let p̂ ∈ Rn and k = dim aff ∂W (p̂). A bounded open pair (A− , A+ )in Rk is called p̂-admissible if there exists a support function ψ of this pair suchthat ψ ∈ D(Λp̂ ).
It turns out that the crystalline mean curvature on a facet of an admissible pairdoes not depend on the choice of the admissible support function. This follows fromthe comparison principle for the crystalline curvature Λp̂ .
2.5. Proposition (Comparison principle for Λp̂ ). Let p̂ ∈ Rn and k = dim aff ∂W (p̂).
Suppose that (A1,− , A1,+ ) and (A2,− , A2,+ ) are two p̂-admissible pairs in Rk . If the

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pairs are ordered in the sense of(A1,− , A1,+ )  (A2,− , A2,+ ),then for any two p̂-admissible support functions ψ1 and ψ2 of the respective pairs
we haveΛp̂ [ψ1 ](x) ≤ Λp̂ [ψ2 ](x)(2.4)

a.e. x ∈ Ac1,− ∩ Ac1,+ ∩ Ac2,− ∩ Ac2,+ .

This result follows from the comparison principle for the resolvent problems
for the corresponding energy Ep̂sl , and the resolvent approximation of the minimal
section of the subdifferential. For details again see [25].
3. Viscosity solutions
Viscosity solutions are defined as the functions that satisfy a comparison principlewith a certain class of sufficiently “regular” test functions for which the differential
equation can be understood in the classical sense. The particular choice of test
functions is crucial for the well-posedness of the Cauchy problem for (1.6). Notethat smooth functions are not sufficient for this purpose since they might not be
admissible in the sense of Section 2 and therefore we cannot define the crystalline
curvature for them. For example, consider a function in one dimension that has astrict local minimum or maximum. Therefore the test functions from the following
definition play a crucial role. Recall the notation x0 , x00 from (2.3).
3.1. Definition. Let (x̂, t̂) ∈ Rn × R and p̂ ∈ Rn , V ⊂ Rn be the subspace parallel
to aff ∂W (p̂), U = V ⊥ , k = dim V . We say that a function ϕ(x, t) is a stratified
faceted test function at (x̂, t̂) with gradient p̂ ifϕ(x, t) = ψ̄ (x0 − x̂0 ) + f (x00 − x̂00 ) + p̂ · x + g(t),where• ψ̄ : Rk → R is a support function of a p̂-admissible pair (A− , A+ ) in Rk
with 0 ∈ int(Ac− ∩ Ac+ ) and ψ̄ ∈ D(Λp̂ ),• f ∈ C 2 (Rn−k ), f (0) = 0 and ∇f (0) = 0,
• g ∈ C 1 (R).
With this notion of test functions, we define viscosity solutions on Q := Rn ×
(0, ∞) in a straightforward way, while carefully extending the notion of “local
minima/maxima” of u − ϕ to allow for the proof of both the comparison principle
and the stability.
3.2. Definition. An upper semi-continuous function u : Q → R is a viscosity
subsolution of (1.6) if the following hold:(i) (faceted test) Let ϕ be a stratified faceted test function at (x̂, t̂) ∈ Q withgradient p̂ ∈ Rn \ {0} and pair (A− , A+ ). Then if there is ρ > 0 such thatu(x + w, t) − ϕ(x, t) ≤ u(x̂, t̂) − ϕ(x̂, t̂)(3.1)
for all0|w | ≤ ρ, w00 = 0,

and x0 − x̂0 ∈ U ρ (Ac− ∩ Ac+ ), |x00 − x̂00 | ≤ ρ, t − t̂ ≤ ρ,

then there exists δ > 0 such that Bδ (x̂0 ) ⊂ int(Ac− ∩ Ac+ ) and(3.2)ϕt (x̂, t̂) + F (p̂, ess inf Λp̂ [ψ̄]) ≤ 0.
Bδ (0)

VISCOSITY SOLUTIONS FOR CRYSTALLINE MEAN CURVATURE FLOW

9(i-cf) (curvature-free test) Let g ∈ C 1 (R), ϕ(x, t) := g(t) and suppose that u − ϕhas a local maximum at (x̂, t̂). Theng 0 (t̂) + F (0, 0) = g 0 (t̂) ≤ 0.
(ii) (off-facet test) Let ϕ ∈ C 1 (U) where U is a neighborhood of some point(x̂, t̂) ∈ Q and suppose that dim ∂W (∇ϕ(x̂, t̂)) = 0. If u − ϕ has a local
maximum at (x̂, t̂) thenϕt (x̂, t̂) + F (∇ϕ(x̂, t̂), 0) ≤ 0.
Viscosity supersolutions are defined analogously by reversing the appropriate
inequalities.
A continuous function that is both a subsolution and a supersolution is called aviscosity solution.
4. Comparison principle
We formulate the comparison principle theorem for solutions on the whole Rn thatare constant outside of a compact set. This allows us to prove the theorem without
having to deal with unbounded facets. Since the crystalline curvature quantity isnonlocal, this can lead to potential technical difficulties that we are trying to avoid.
On the other hand, it does not significantly diminish the applicability of our result
since the statement still covers the evolution of arbitrary bounded crystals.
4.1. Theorem (Comparison principle). Let W : Rn → R be a positively onehomogeneous convex polyhedral function such that the conclusion of Proposition 4.2
holds for 1 ≤ k ≤ n − 1, and let F be as defined in (1.5). Suppose that u and v are a
subsolution and a supersolution of (1.6) on Rn × [0, T ] for some T > 0, respectively.
Moreover, suppose that there exist a compact set K ⊂ Rn and constants cu ≤ cv
such that u ≡ cu , v ≡ cv on (Rn \ K) × [0, T ]. Then u(·, 0) ≤ v(·, 0) on Rn impliesu ≤ v on Rn × [0, T ].
The requirement that both u and v are equal to constants outside of a compact
set guarantees that they might cross only inside this compact set and also that wewill need to only construct stratified faceted test functions with bounded facets
described by bounded open pairs.
To prove the comparison principle, we need to be able to construct stratified
faceted test functions at a contact point of the subsolution and the supersolution.
This unfortunately appears to be a nontrivial problem. Currently we only know how
to construct such test functions in one and two dimensions, which still allows us toprove the comparison principle for the level set flow of a three dimensional crystal.
The existence of such test functions is asserted by the following proposition. It is adensity result, stating that any bounded open pair can be approximated arbitrarily
well in the Hausdorff distance by an admissible pair. We expect this to hold in any
dimension, but we have been unable to prove it in dimensions k ≥ 3.
4.2. Proposition. Let W : Rn → R, n ∈ N, be a polyhedral convex function finite
everywhere. Suppose that p̂ ∈ Rn such that dim ∂W (p̂) = k for k = 1 or 2. Thenfor any bounded open pair (A− , A+ ) in Rk and any 0 ≤ ρ1 < ρ2 there exists ap̂-admissible pair (G− , G+ ) satisfying(4.1)

U ρ1 (A− , A+ )  (G− , G+ )  U ρ2 (A− , A+ ),10

N. POŽÁR

To prove the comparison principle, Theorem 4.1, we follow the standard doublingof variables argument with an extra parameter. That is, we study the maxima of
the function2|x − y − ζ|− Sε (t, s),2εover Q × Q := Rn × [0, T ] × Rn × [0, T ] for arbitrary ζ ∈ Rn and sufficiently small
ε > 0 whereΦζ,ε (x, t, y, s) := u(x, t) − v(y, s) −2|t − s|
ε
ε
+
+
.
2ε
T −t T −s
If the comparison theorem fails, there exists a point where u > v and this will
imply that Φζ,ε also has a positive maximum for small ε > 0 for all sufficiently small
ζ. We fix one such small ε > 0 and define the maximum of Φζ,ε ,(4.2)

Sε (t, s) :=`(ζ) := max Φζ,ε .
Q×Q

We observe that since W is a polyhedral convex function, the space Rn canbe decomposed into a finite union of relatively open convex sets on which thesubdifferential of W is constant [25, 32]. The Baire category theorem then allows
us to choose one of these sets Ξ and a small open ball B2λ (ζ0 ) ⊂ Rn such that forany ζ ∈ B2λ (ζ0 ) there always exists a point of maximum (x̂, t̂, ŷ, ŝ) of Φζ,ε withthe property that x̂ − ŷ − ζ ∈ Ξ. This has consequences for the behavior of `(ζ),
the maximum of Φζ,ε , when changing ζ in the direction orthogonal to aff Ξ. We
denote this subspace orthogonal to aff Ξ by V , and it happens that V is parallel toaff ∂W (p) for p ∈ Ξ. Specifically, the function `(ζ) is affine in (ζ0 + V ) ∩ B2λ (ζ0 )with the slope given by the projection of some element of Ξ into V . Since W is
positively one-homogeneous, 0 ∈ aff Ξ. Therefore ` is constant on (ζ0 + V ) ∩ B2λ (ζ0 ).
This fact has a very important implication, namely, there exists p̂ ∈ Ξ such thatthe functions u and v have certain flatness property in the directions in V in aneighborhood of the maximum of Φζ,ε and gradient p̂. Now we proceed dependingon the dimension of V . In particular, if 0 < dim V < n, using Proposition 4.2
we can find two ordered p̂-admissible pairs in Rk , k = dim V , with p̂-admissible
support functions that belong to D(Λp̂ ). Moreover, we can build stratified faceted
test functions in the sense of Definition 3.1 that are ordered with respect to u andv as in (3.1). Since they then have ordered values of Λp̂ due to the comparison
principle for Λp̂ , Proposition 2.5, we obtain a contradiction with the faceted test
(3.2) in the definition of viscosity solutions, Definition 3.2. The cases dim V = 0
and dim V = n yield contradiction with the other conditions in Definition 3.2. We
conclude that u ≤ v.
5. Stability
There are two natural ways how to regularize the problem (1.6), each with different
merits. On the one hand, we can take the PDE point of view and approximate W bysmooth energy densities with quadratic growth, yielding a uniformly elliptic operatorψ 7→ div ∂W (∇ψ) and turning the approximating PDE into a degenerate parabolic
equation that falls into the scope of the classical viscosity solution theory. On theother hand, since (1.6) is the level set formulation of a surface evolution equation(1.2), we want to show that the evolution can be approximated by a evolution withsmooth anisotropic mean curvature, that is, we approximate W by a sequence of

VISCOSITY SOLUTIONS FOR CRYSTALLINE MEAN CURVATURE FLOW

11

positively one-homogeneous surface energy densities with smooth uniformly convex
level sets. We obtain stability results in both cases.
Let us first recall the definition of half-relaxed limits (semi-continuous limits) ofa sequence of functions {um }m∈N :?-limsup um (x, t) := lim sup
m→∞

sup

sup um (y, s),

k→∞ m>k |y−x|< 1 |t−s|< 1kk?-liminf um (x, t) := − ?-limsup − um (x, t) .
m→∞

m→∞

Recall that both limits are equal if and only if the convergence is uniform.
We have the following two theorems on stability in either mode of approximation.
5.1. Theorem. Suppose that {Wm }m∈N ⊂ C 2 (Rn ) is a decreasing sequence of
convex functions such that Wm & W as m → ∞ locally uniformly. Moreover,2
suppose that there exist positive numbers am such that a−1
m I ≤ ∇p Wm (p) ≤ am I forn
all p ∈ R , m ∈ N, where I is the n × n identity matrix. Let {um }m∈N be a locallybounded sequence of viscosity solutions of(5.1)

ut + F (∇u, tr[(∇2p Wm )(∇u)∇2 u]) = 0.

Then ?-limsupm→∞ um is a viscosity subsolution of (1.6) and ?-liminf m→∞ um is aviscosity supersolution of (1.6).
5.2. Theorem. Suppose that F is of the form given in (1.5) and that {Wm }n∈N ⊂C(Rn )∩C 2 (Rn \{0}) are positively one-homogeneous functions with bounded, strictly
convex sub-level sets {Wm ≤ 1} such that Wm ⇒ W uniformly on B 1 (0). Let um
be for each m ∈ N the unique viscosity solution of(
ut + F (∇u, div ∇p Wm (∇u)) = 0,
in Rn × (0, ∞),u(·, 0) = u0,m , in Rn ,where u0,m ∈ C(Rn ) are uniformly bounded. Then ?-limsupm→∞ um is a viscosity
subsolution of (1.6) and ?-liminf m→∞ um is a viscosity supersolution of (1.6).
The classical viscosity solutions are stable with respect to the half-relaxed limits,and the above stability theorems are natural extensions of this stability. However,there is a significant difference between the stratified faceted test functions for(1.6) and the smooth test functions for the regularized problems. Another majordifficulty in proving the stability is the nature of the differential operators: we areapproximating a nonlocal operator by a sequence of local operators. Therefore
there needs to be a way how to recover the nonlocal information about the shapeof the facet that determines the value of the crystalline curvature Λp̂ from thefundamentally local regularized problems.
We overcome this difficulty by adapting the standard perturbed test function
method due to Evans. Suppose that we have a stratified faceted test function ψ forthe faceted test in Definition 3.2. Since this function is in general only Lipschitz, itcannot be used as a test function for the regularized problem (5.1). And even if thisfunction were C 2 , application of the elliptic operator tr[(∇2p Wm )(∇ψ)∇2 ψ] would
yield zero on the facet (flat part) of ψ for any m. We therefore perturb
this test
Rfunction by solving the resolvent problem for the energy Em (ψ) := Wm (∇ψ) dx

12

N. POŽÁR

and the total variational energy (2.2) in the class of periodic functions,ψa,m + a∂Em (ψa,m ) 3 ψ,
ψa + a∂E(ψa ) 3 ψ,
for a > 0. This is indeed a uniform perturbation in the sense that for subsequencesm → ∞ and then a → 0 the solutions converge uniformly ψa,m → ψa and ψa → ψ
due to uniform Lipschitz estimates for the resolvent problems. By elliptic regularity
ψa,m ∈ C 2,α and therefore it is a test function for the regularized problem. Moreover,by the standard result [5], we have (ψa − ψ)/a → −∂ 0 E(ψ) in L2 . This provides themissing nonlocal information and allows us to pass in the limit m → ∞ and provethat the uniform limit of solutions of the regularized problems (5.1) is a solutionsof the crystalline curvature flow (1.6). Heuristically, the resolvent problem is theone step of the implicit Euler discretization for the gradient flow of the energy E
and will therefore carry the information about the crystalline curvature Λp̂ which is
given as the subdifferential of E.
To prove Theorem 5.2, the stability with respect to one-homogeneous Wm , weuse the above idea with an extra regularization step by uniformly elliptic operators.
The stability results together with the comparison principle, using barriers at
t = 0, yield the well-posedness of (1.6) for arbitrary continuous initial data, constantoutside of some compact set, in three dimensions.
6. Whole space test function
In this section we mention an important example of “test functions” that are
useful to prove some properties of solutions.
Let W : Rn → R be a positively one-homogeneous convex function, zero only atthe origin. We define the polar of W as
W ◦ (x) = sup {x · p : W (p) ≤ 1}.
It is easy to see that W ◦ is again a positively one-homogeneous convex function,
zero only at the origin. Moreover,W ◦◦ = W.
The Wulff shape associated with W is defined as
W := {x ∈ Rn : W ◦ (x) ≤ 1}.
W is a compact subset of Rn containing the origin in its interior.
For given c > 0, defineψ(x) := max {W ◦ (x) − c, 0}.
ψ is clearly a Lipschitz continuous function on Rn . It is in fact a support functionof the pair (∅, cW c ).
We shall show that ψ is an admissible support function and compute the crystalline
curvature Λ0 on its facet cW.
Define the vector field(xx ∈ cW,xc,z(x) :==xc◦
max(c, W (x))
W ◦ (x) , x ∈ cW .

VISCOSITY SOLUTIONS FOR CRYSTALLINE MEAN CURVATURE FLOW

13

Note that z is Lipschitz continuous. By Lemma 6.1 and Lemma 6.2 below we see thatz(x) ∈ ∂W (∇ψ(x)) for almost every x ∈ Rn (that is, wherever ψ is differentiable).
On the other hand, div z ∈ L2loc (Rn ) andn
div z ≡ ,on cW.
c
Since div z is constant on the facet, we conclude the crystalline mean curvature isthe constantnΛ0 (ψ) ≡ ,on cW.
cn−1(c∂W)
Note that as the size of the facet increases, the curvature decreases as H |cW|
∼ 1c ,
the ratio of the surface area of the facet boundary to the facet’s volume.
We now state the two technical lemmas that were used above.

6.1. Lemma. If p =6 0 and x ∈ ∂W (p) then W ◦ (x) = 1 and x · p = W (p). Similarly,
if x =
6 0 and p ∈ ∂W ◦ (x) then W (p) = 1 and x · p = W ◦ (x). Suppose now thatx 6= 0 and p 6= 0. Thenxp
∈ ∂W (p) ⇔∈ ∂W ◦ (x).
W ◦ (x)W (p)Proof. Write x̂ = W ◦x(x) and p̂ = Wp(p) . We have W ◦ (x̂) = 1, W (p̂) = 1 andtherefore W ∗ (x̂) = 0, W ◦∗ (p̂) = 0. Since W and W ◦ are one-homogeneous, wehave ∂W (p̂) = ∂W (p) and ∂W (x̂) = ∂W (x). Suppose that x̂ ∈ ∂W (p) = ∂W (p̂).
Then by the above and the characterization of the subdifferential, for instance[32, Theorem 23.5], we have
1 = W (p̂) + W ∗ (x̂) = x̂ · p̂.
ThereforeW ◦ (x̂) + W ◦∗ (p̂) = 1 ≤ x̂ · p̂,which by the characterization of the subdifferential yields p̂ ∈ ∂W ◦ (x̂) = ∂W ◦ (x).
The other direction can be proved by reversing the steps.
6.2. Lemma. Suppose that W is positively one-homogeneous convex function on
Rd . Then ∂W (p) ⊂ ∂W (0) for any p ∈ Rd . We also have (x − y) ⊥ p for any
x, y ∈ ∂W (p) and any p ∈ Rd .
Proof. The proof is straightforward, see [32] for details.



Acknowledgments. The author is partially supported by KAKENHI Grant Number 26800068 (Wakate B) of Japan Society for the Promotion of Science.
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Kanazawa University, Kakuma town, Kanazawa, Ishikawa 920-1192, Japan.
E-mail address: npozar@se.kanazawa-u.ac.jp