ANISOTROPIC TOTAL VARIATION FLOW OFNON-DIVERGENCE TYPE ON A HIGHER DIMENSIONAL
TORUS
MI-HO GIGA, YOSHIKAZU GIGA, AND NORBERT POŽÁR

Abstract. We extend the theory of viscosity solutions to a class of very
singular nonlinear parabolic problems of non-divergence form in a periodic
domain of an arbitrary dimension with diffusion given by an anisotropic total
variation energy. We give a proof of a comparison principle, an outline of a
proof of the stability under approximation by regularized parabolic problems,and an existence theorem for general continuous initial data, which extend theresults recently obtained by the authors.

1. Introduction
The goal of this note is the announcement of the results in [37] and their extension
to smooth anisotropic total variation energies. Furthermore, we give a slightly
different exposition of the technically demanding proof of the comparison theorem.
In an arbitrary dimension n ≥ 1 we consider the following problem for a function
u(x, t) : Tn × (0, T ) → R on the torus Tn := Rn \ Zn for some T > 0:
(1.1)

ut + F (∇u, div ∂W (∇u)) = 0

in Q := Tn × (0, T ),with the initial condition(1.2)

on Tn .

u|t=0 = u0

In this paper we assume that
(1.3)

W ∈ C 2 (Rn \ {0}),W 2 is strictly convex,

and that W is a convex one-homogeneous function, positive outside of the origin,
i.e., there exists a positive constant λ0 such that
(1.4)

W (ap) = aW (p) ≥ λ0 a |p|for all p ∈ Rn , a ≥ 0.

Furthermore, we assume that F : Rn × R → R is a continuous function, nonincreasing in the second variable, i.e.,(1.5)

for ξ, η ∈ R, ξ ≥ η, p ∈ Rn .

F (p, ξ) ≤ F (p, η)This makes the operator in (1.1) degenerate parabolic.
The symbol ∂W denotes the subdifferential of W . In general, the subdifferentialof a convex lower semi-continuous function ϕ on a Hilbert space H endowed with a2010 Mathematics Subject Classification. 35K67, 35D40, 35K55, 35B51, 35K93 .
Key words and phrases. phase transitions, curvature flows, crystalline mean curvature,anisotropic total variation flow, viscosity solutions, comparison theorems.
12

M.-H. GIGA, Y. GIGA, AND N. POŽÁR

scalar product h·, ·iH is defined as the set∂ϕ(x) := {v ∈ H : ϕ(x + h) − ϕ(x) ≥ hh, viH for all h ∈ H}

x ∈ H.

Since W is not differentiable at the origin, ∂W (0) is not a singleton and therefore
an extra care has to be taken when defining the meaning of the term div ∂W (∇u).
In fact, we shall understand the term div ∂W (∇u) through the subdifferential of
the anisotropic total variation energy on the Hilbert space L2 (Tn ),(R
W (∇ψ) ψ ∈ L2 (Tn ) ∩ BV (Tn ),Tn(1.6)
E(ψ) :=+∞
ψ ∈ L2 (Tn ) \ BV (Tn ),where BV (Tn ) is the space of functions of bounded variation on Tn . We shall clarify
this relation in Section 2. Let us introduce the domain of the subdifferential ∂E ofthe energy E on L2 (Tn ), namely
D(∂E) := ψ ∈ L2 (Tn ) : ∂E(ψ) 6= ∅ .
The problem (1.1) can be written more rigorously as(1.7)

ut + F (∇u, −∂ 0 E(u(·, t))) = 0,

where ∂ 0 E is the minimal section (canonical restriction) of the subdifferential ∂E
defined for ψ ∈ D(∂E) as∂ 0 E(ψ) ∈ ∂E(ψ) such that∂ 0 E(ψ) L2 (Tn ) =min kvkL2 (Tn ) .

v∈∂E(ψ)Clearly ∂ 0 E(ψ) is well-defined and unique since ∂E(ψ) is a nonempty closed convex
subset of L2 (Tn ) whenever ψ ∈ D(∂E).
Motivation. The prototypical example of (1.1) is the total variation flow [5]∇uut = div
(1.8)
,
|∇u|
n
o
∇u
since ∂W (∇u) = |∇u|for W (p) = |p| when ∇u 6= 0, or more generally theanisotropic total variation flow [3,4,55]. This problem also explains the interpretationof div ∂W (∇u) as the minimal section of −∂E(u). Indeed, problem (1.8) is formallythe subdifferential inclusion(ut ∈ −∂E(u(t))
t > 0,
2nu(0) = u0 ∈ L (T ).
The theory of monotone operators due to Kōmura [54] and Brézis [17] yields theexistence of a unique solution u ∈ C([0, T ], L2 (Tn )) that is moreover, for all t ∈ (0, T ),right-differentiable, u(t) ∈ D(∂E) andd+ u(t) = −∂ 0 E(u(t))
for t ∈ (0, T ).
dt
Nevertheless, our main motivation for the study of problem (1.1) in its general
non-divergence form comes from the models of crystal growth. Let us outline how
problem (1.1) can be heuristically derived as the graph formulation of the motionof a surface by the anisotropic crystalline curvature of a particular form. Following
the notation of [9, 10, 14], we consider the surface energy functionalZ(1.9)
F(Γ) :=φ◦ (ν) dHnΓANISOTROPIC TOTAL VARIATION FLOW

3

that measures the surface energy of the surface Γ = ∂K ⊂ Rn+1 of a body K ⊂ Rn+1
with the unit outer normal vector ν. Here Hn is the n-dimensional Hausdorff measureand φ◦ is a convex one-homogenous function positive outside of the origin given as(1.10)φ◦ (η) = W (−p) + |ηn+1 |for all η = (p, ηn+1 ) ∈ Rn+1 .

The Wulff shape of this surface energy is the one-level set

Wulff φ := η ∈ Rn+1 : φ(η) ≤ 1of the dual function
φ(ξ) := sup ξ · η : η ∈ Rn+1 , φ◦ (η) ≤ 1 .
Note that this makes φ◦ the support function of Wulff φ . A simple computationshows thatφ(ξ) = max {W ◦ (−x), |ξn+1 |}ξ = (x, ξn+1 ) ∈ Rn × R,

whereW ◦ (x) := sup {x · p : p ∈ Rn , W (p) ≤ 1}.
Setting
W := {x ∈ Rn : W ◦ (x) ≤ 1},
we observe that the Wulff shape of φ◦ is a cylinder of length 2 with base −W, thatis,
Wulff φ = (−W) × [−1, 1].
The assumptions (1.3) and (1.4) on W guarantee that W ◦ also satisfies (1.3) and(1.4) (possibly with a different λ0 ). In particular, W is a strictly convex, compact
set with a C 2 boundary containing the origin in its interior.
The first variation of the functional F in (1.9) is called the crystalline mean
curvature [14, 15](1.11)

κφ := − divφ,τ nmin
φ ,where divφ,τ is the tangential divergence on Γ with respect to φ, introduced in [14],and nminis a so-called Cahn-Hoffman vector field on Γ that minimizes the norm ofφ
divφ,τ nφ in L2 (Γ) with weight φ◦ (νΓ (ξ)) among all Cahn-Hoffman vector fields nφ .
A Cahn-Hoffman vector field is any vector field on Γ that satisfies nφ (ξ) ∈ ∂φ◦ (ν(ξ))
where ν(ξ) is the unit outer normal vector of K at ξ. Since φ◦ is not differentiable
everywhere, the vector field nmin
might not be unique but divφ,τ nmin
is uniqueφφ[14, 43]. We use a sign convention different from [14] so that κφ equals to theconventional mean curvature in the direction of ν when φ(ξ) = |ξ|.
Consider now a surface Γ(t) ⊂ Rn+1 , t ≥ 0, that can be expressed as the graphof a sufficiently smooth Zn -periodic function u : Tn × R → R:
Γ(t) = {(x, u(x, t)) : x ∈ Rn }t ≥ 0,

which is the boundary of the (crystal) body K(t) := {(x, ξn+1 ) : ξn+1 < u(x, t)}. In
the graph case, ν(ξ) for ξ ∈ Γ(t) has the simple form [40](−∇u, 1)ν(x, u(x, t)) = q.
2
1 + |∇u|4

M.-H. GIGA, Y. GIGA, AND N. POŽÁR

Using the definition of φ◦ in (1.10), we have the expression∂φ◦ (p, ηn+1 ) = {(x, 1) : x ∈ −∂W (−p)}

for ηn+1 > 0.

Therefore nφ (ξ) = (−zW (x), 1) for some vector field zW (x) ∈ ∂W (∇u(x, t)), x ∈ Rn ,and the expression (1.11) reduces for graphs Γ(t) tominκφ = div zW(x),min
where div is the divergence on Tn and zW
minimizes the L2 -norm of div zW among
all vector fields zW (x) ∈ ∂W (∇u(x)) a.e. such that div zW ∈ L2 (Tn ). It turns outthat −κφ coincides with the minimal section of the total variation energy (1.6), see
Section 2.2, and therefore we shall formally write

κφ = div ∂W (∇u)(= −∂ 0 E(u(·, t))).

The motion of Γ(t) by the crystalline mean curvature κφ can be written as(1.12)

V = κφ ,where V is the normal velocity of Γ(t) that can be expressed in terms of thederivatives of u as [40]
ut
.
V =q
2
1 + |∇u|Thus we can formally rewrite (1.12) for graphs asq2
ut = 1 + |∇u| div ∂W (∇u),
which is not of divergence form, but obviously can be cast in the form of (1.1).
Literature overview. The motion by anisotropic crystalline mean curvature
has attracted significant attention due to its importance in modeling of crystal
growth. The majority of articles follow one of the three main approaches: polygonal,
variational and viscosity.
The polygonal approach relies on the relatively simple expression of the anisotropic
crystalline curvature κφ for curves in a two-dimensional plane. In fact, the quantity
κφ is constant on the flat line segments that are parallel to the facets of theWulff shape Wulff φ , and is inversely proportional to the length of the line segment.
Therefore when the Wulff shape Wulff φ is a convex polygon, i.e., the energy densityφ◦ is “crystalline”, it is possible to define the evolution of polygonal curves with sides
parallel to the facets of Wulff φ . This special family of solutions, often referred to asa crystalline flow or a crystalline motion, was introduced in [6, 58]. The validity ofthis approach is limited in higher dimensions [42] because the quantity κφ might not
be constant or even continuous on the facets and facet-breaking and facet-bending
phenomena might occur [13, 16]. For a further development see also [51].
The variational approach applies only to problems with a divergence structure.
One then understands κφ as a subdifferential of the corresponding singular interfacial
energy. It was shown in [24, 28] that in such case the crystalline motion can beinterpreted as the evolution given by the abstract theory of monotone operators[17,54]. In this approach, the crystalline motion can be approximated by an evolution
by smooth energies and vice-versa, or by a crystalline algorithm [46, 47].
As we explained above, the curvature κφ might not be constant or even continuous
on the facets of bodies in dimension higher than two, and facet breaking or bending

ANISOTROPIC TOTAL VARIATION FLOW

5

might occur [13, 16]. In fact, κφ is in general only bounded and of bounded variationon the facets [14, 15] and a nontrivial obstacle problem has to be solved to calculateκφ [16, 43]. The facets with constant curvature κφ are called calibrable [16]. The
convex calibrable sets were first characterized in two dimensions by E. Giusti [48]in the isotropic case W (p) = |p|. That result was extended recently to higher
dimensions in [2], and to anisotropic norms in [19]. The concept of calibrable sets is
related to the so-called Cheeger sets [1, 20, 52].
This suggests that the crystalline flow cannot be restricted in dimensions higher
than two to bodies with facets parallel to the facets of the Wulff shape Wulff φ anda more general class of solutions is necessary. A notion of generalized solutions
and a comparison principle was established through an approximation by reactiondiffusion equations in [11, 12] for Vν = φκφ . However, the existence is known only forconvex compact initial data [10]. Even in two dimensions, if there is a nonuniform
driving force c the abstract theory suggests that κφ + c might not be constant on
the facets [30]. This situation is important because c is often non-constant in themodels of crystal growth. However, if one allows to include bent polygons withfree boundaries corresponding to the endpoints of a facet, it is possible to give arather explicit solution [41, 44, 45]. In the graph case in one-dimension, there isalso an approach that defines solutions via an original definition of compositionof multivalued operators that allows the study of the evolution of facets and theregularity of solutions for a general class of initial data under a non-uniform driving
force c [53, 56].
Viscosity solutions. The third approach based on the theory of viscosity solutions
is the approach taken in this paper. The merit is that one can prove existence anduniqueness in a general class of continuous functions without requiring a divergencestructure of the problem, only relying on the comparison principle. The review paper[34] compares the viscosity and variational approaches for equation of divergenceform.
Since the operator in (1.1) has a parabolic structure, it can be expected thatany reasonable class of solutions of the problem satisfies a comparison principle.
In particular, (1.1) should fall in the scope of the theory of viscosity solutions.
Unfortunately, the conventional theory of degenerate parabolic equations does not
apply to (1.1) because of the strong singularity of the operator div ∂W (∇ψ) on
the facets of ψ, that is, whenever ∇ψ = 0. Suppose that ψ ∈ C 2 (U ) in an open
set U ⊂ Rn and ∇ψ 6= 0 in U . Then ∂W (∇ψ(x)) is a singleton for x ∈ U anddiv ∂W (∇ψ) can be expressed as[div ∂W (∇ψ)] (x) = k ∇ψ(x), ∇2 ψ(x) ,
where


k(p, X) := trace ∇2 W (p)Xp ∈ Rn \ {0}, X ∈ S n .

Here S n is the set of symmetric n × n-matrices. Since W is positively onehomogeneous, ∇2 [W (ap)] = a−1 ∇2 W (p) for a > 0 and p ∈ Rn \ {0} and therefore

  
1
p
(1.13)k(p, X) =trace ∇2 W
X .
|p|
|p|
We observe that k(p, X) is unbounded as p → 0, and, in fact, at p = 0 the diffusion isso strong that the operator div ∂W (∇ψ) becomes a nonlocal operator that depends
on the shape and size of the facet of ψ. For this reason an equation with such

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M.-H. GIGA, Y. GIGA, AND N. POŽÁR

operator is often called a very singular diffusion equation [28, 35]. If the singularity ofthe operator k(p, X) is relatively weak at p = 0 so that the operator is still local, as
q−2
in the case of the q-Laplace equation ut − div(|∇u|∇u) = 0 for 1 < q < 2, whichqcorresponds to W (p) = |p| /q in our notation, the theory of viscosity solutions
can be extended [40, 49, 50, 57]. Note that the level set formulation of the motion
by the mean curvature can also be written in the form of (1.1) with W (p) = |p|and F (p, ξ) = − |p| ξ. However, the singularity of k(p, X) in (1.13) is canceled out
by |p| in F (p, ξ) and the operator is bounded as p → 0 [21, 26]. There has been a
considerable effort to extend the theory of viscosity solutions to the problem (1.1)
with a positively one-homogeneous W and a general continuous F satisfying only
the monotonicity assumption (1.5). Until recently, however, the results have been
restricted to the one-dimensional case [29, 33, 36, 38] or to related level set equations
for evolving planar curves [32, 33]; see also the review paper [39].
In the recent paper [37], we extended the theory of viscosity solutions to problem(1.1) with W (p) = |p|. In the present paper, we shall generalize this result to anarbitrary W that satisfies the assumptions above.
Main results. We introduce a notion of viscosity solutions for problem (1.1) andprove the following well-posedness result, which is an extension of the main resultin [37].
Theorem 1.1 (Main theorem). Suppose that a continuous function F : Rn ×R → R
is degenerate elliptic in the sense of (1.5), and that W : Rn → R satisfies (1.3)
and (1.4). Then the initial value problem (1.1)–(1.2) with u0 ∈ C(Tn ) has a uniqueglobal viscosity solution u ∈ C(Tn × [0, ∞)). If additionally u0 ∈ Lip(Tn ), i.e., u0
is a periodic Lipschitz function, then u(·, t) ∈ Lip(Tn ) for all t ≥ 0 andk∇u(·, t)k∞ ≤ k∇u0 k∞

for t ≥ 0.

As in [37], the uniqueness of solutions will be established via a comparison
principle, and the existence of solutions is verified by showing the stability ofsolutions under approximation by regularized problems. As a corollary, we see thatin the case of the standard anisotropic total variation flow equation our viscosity
solutions coincide with the semigroup (weak) solutions given by the theory ofmonotone operators.
Viscosity solutions are defined as those functions that admit a comparison principlewith a class of test functions, which are sufficiently regular functions to which theoperator in (1.1) can be applied directly. The difficult task is the crafting of an
appropriate class of such test functions that is on one hand large enough so that theviscosity solutions can be shown to be unique, by the means of proving a comparison
principle, and on the other hand small enough so that the proof of existence ispossible for any given sufficiently regular initial data.
As the computation above suggests, we can evaluate the operator div ∂W (∇ψ)
at a point x0 whenever ψ ∈ C 2 (Ux0 ) and ∇ψ(x0 ) 6= 0. Thus arbitrary sufficiently
smooth functions ϕ(x, t) with ∇ϕ 6= 0 serve as test functions.
However, the situation is much more delicate at places where the gradient of
the solution vanishes, that is, on the facets. The main difficulty stems from therestriction that the operator div ∂W (∇ψ) = −∂ 0 E(ψ) is only defined for functionsψ ∈ D(∂E). Fortunately, a simple class of what we call faceted functions is available
and we are able to show that such functions belong to D(∂E) under some regularity

ANISOTROPIC TOTAL VARIATION FLOW

7

assumptions on the shape of the facet. The main tool is the characterization of thesubdifferential ∂E(ψ) of a Lipschitz function ψ (Corollary 2.3). Namely, a function
belongs to ∂E(ψ) if it is the distributive divergence of a vector field that pointwise
almost everywhere belongs to the sets ∂W (∇ψ(x)). To construct a Lipschitz faceted
function, we start from a pair of sets that satisfy certain regularity conditions andcharacterize the facet. This characterization follows from the simple observationthat any facet of a continuous function ψ can be uniquely described by a pair ofdisjoint open sets {ψ > a} and {ψ < a} for some a ∈ R. The quantity −∂ 0 E iswell-defined for such faceted functions, and, moreover, if two pairs are ordered in a
specific sense, the values of −∂ 0 E are also ordered on the intersection of the facets.
In contrast to [37], we do not introduce the quantity Λ, which we refer to as
the nonlocal curvature of a facet there. This makes the current exposition morestraightforward.
The definition of viscosity solutions (Definition 3.3) then contains the classicaltest with smooth test functions when the gradient of the solution is nonzero, and a
new faceted test with a class of faceted test functions. In the faceted test we only
evaluate the essential infima and suprema of −∂ 0 E over balls of small radius andthus obtain a pointwise quantity. Furthermore, to facilitate the proof of stability
and existence, we require that the faceted test function can be shifted in an arbitrary
direction by a small amount, that is, we say that the faceted test function is in
general position.
The proof of the comparison principle (Theorem 4.1) follows the standard doublingof-variables argument with an important twist. Suppose that u and v are viscosity
solutions of (1.1) such that u(·, 0) ≤ v(·, 0). We introduce an extra parameterζ ∈ Tn and investigate the ζ-dependence of the maxima of the functions2Φζ (x, t, y, s; ε) := u(x, t) − v(y, s) −|x − y − ζ|
− S(t, s; ε)2εover (x, t, y, s) ∈ Tn × [0, T ] × Tn × [0, T ] and a fixed parameter ε > 0. The time
penalization S(t, s; ε) is defined in Section 4. This device was developed in [29],but its history goes back to [21, 49]. In particular, by varying ζ, we increase thechange that some maximum will occur at a point (x, t, y, s) such that x − y − ζ 6= 0
and the standard construction of a test function for the classical test with nonzero
gradient is available [22, 40]. If all maxima of Φζ for all small ζ happen to lie at
points (x, t, y, s) such that x − y − ζ = 0, we get extra information about the shapeof u and v at their contact point. To be more specific, u and v must have someflatness and therefore there is enough room for finding two ordered smooth pairsthat can be used to construct ordered faceted test functions for both u and v.
The existence of solutions (Theorem 5.4) follows from the stability under approximation by regularized degenerate parabolic problems (Theorem 5.3) for which thestandard theory of viscosity solutions applies [22]. We regularize (1.1) through anapproximation of W by a decreasing sequence of strongly convex smooth functions
0
Wm , m ≥ 1, with a quadratic growth at
R infinity, so that the subdifferential −∂ Em
of the corresponding energy Em (ψ) := Wm (∇u) is a uniformly elliptic quasi-linear
differential operator.
Since we approximate a nonlocal problem by local problems, the main difficultymaterializes while passing through the limit in the definition of viscosity solutions.
More precisely, when we apply the regularized operator to a (smooth) faceted test

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M.-H. GIGA, Y. GIGA, AND N. POŽÁR

function, we recover only local information that is independent of the overall shapeof the facet, while in the limit the shape of the facet is very important.
To recover the nonlocal information, we perturb the test function ϕ(x, t) =ψ(x) + g(t) by one step of the implicit Euler approximation of the anisotropic
total variation flow with time-step a > 0, that is, by finding the solution ψa of theresolvent problem
ψa = (I + a∂E)−1 ψ.
By solving the resolvent problem for the regularized energy Em ,
ψa,m = (I + a∂Em )−1 ψ,
we obtain a smooth perturbed test function ϕa,m (x, t) = ψa,m (x) + g(t) for the
regularized problem that contains the missing nonlocal information. This type ofapproximation has two advantages. Firstly, ψa is uniformly approximated by ψa,m
as m → ∞ for a fixed a and so is ψ by ψa as a → 0. Secondly, if ψ ∈ D(∂E) thenthe function −∂ 0 E(ψ) is approximated in L2 (Tn ) as a → 0+ by the ratio (ψa − ψ)/a.
This is the main ingredient in the proof of stability.
To finish the proof of existence, we have to show that the limit of solutions of theregularized problem has the correct initial data. This is done by a comparison withbarriers at t = 0. However, it is necessary to construct barriers depending on m. As
in [29] and [37], we use the convex conjugates of Wm , but with a more robust cutoffof large gradients that requieres neither one-dimensionality nor radial symmetry of
Wm .
Outline. This paper consists of the following parts. First, in Section 2, we discuss
the interpretation of the term div ∂W (∇ψ) ∼ −∂ 0 E(ψ) for a class of functions ψthat have flat parts, the so-called facets. This will be then used in Section 3 tointroduce viscosity solutions of problem (1.1) and a suitable class of test functions.
Once the solutions are defined, we establish a comparison principle in Section 4.
The paper is concluded in Section 5 with a brief discussion of stability of (1.1) underapproximation by regularized problems, which provides, as a corollary, the existenceof solutions.
2. Nonlocal curvature
The main challenge for developing a reasonable theory of viscosity solutions is theselection of an appropriate class of test functions. In particular, a special care has
to be taken when the gradient of a solution vanishes. In such a case, the solution
should have a facet, i.e., it should be constant on a closed neighborhood of the point.
Functions that have such facets will be called faceted functions.
In this section we will investigate the value of the term div ∂W (∇ψ) ∼ −∂ 0 E(ψ)
on facets of faceted functions. It turns out that such facets can be described by apair of disjoint open sets, which characterize the convexity and concavity of thefunctions at the facet boundary. The understanding of the term −∂ 0 E(ψ) is further
complicated by the fact that it is a nonlocal quantity on facets. Motivated by the
motion by crystalline mean curvature, we shall refer to this term as the nonlocal
curvature, in particular if this term is evaluated on a facet. Instead of evaluating
it directly, we approximate it via a resolvent problem for the energy E. This bothyields a comparison principle for −∂ 0 E(ψ) and a way how the approximate it via
regularized energies in the proof of existence.

ANISOTROPIC TOTAL VARIATION FLOW

9

In contrast to [37], we do not define the quantity Λ which we called nonlocal
curvature there and showed that it is independent of the choice of support functionof a given pair. The proof of this fact is quite technical, but it is extendable to thecurrent context. However, this quantity is not necessary for definition of viscosity
solutions and we choose a more direct approach here.
2.1. Torus. We consider the total variation energy for periodic functions on Rn .
These functions can be identified with functions on the n-dimensional torus Tn :=Rn /Zn . The set Tn is the set of all equivalency classes {x + Zn : x ∈ Rn } with theinduced metric and topology, namely(2.1)

dist(x, y) := distRn (x + Zn , y + Zn ),|x| := dist(x, 0) = infn |x + k|Rn ,k∈Znfor x, y ∈ T . Consequently, the open ball Br (x) centered at x ∈ Tn of radiusr > 0 is defined as Br (x) := {y ∈ Tn : |x − y| < r}. Note that Br (x) has a smoothboundary if r < 1/2.
2.2. Subdifferential of the total variation energy. Function u is called afunction of bounded variation and said to belong to BV (Tn ) if u ∈ L1 (Tn ) and its
gradient Du in the sense of distributions is a vector valued Radon measure withfinite total variation on Tn .
To characterize the subdifferential of E, we need a pairing between functions ofbounded variations and vector fields with L2 divergence that was studied in [7] (seealso [27]) for bounded domains in Rn . The modification for Tn is straightforward.
We recall the definition of the space of vector fields

X2 (Tn ) := z ∈ L∞ (Tn ; Rn ) : div z ∈ L2 (Tn ) .
It was also shown in [7] that for any z ∈ X2 (Tn ) and u ∈ BV (Tn ) ∩ L2 (Tn ) we candefine a Radon measure (z, Du) on Tn asZZh(z, Du), ϕi := −uϕ div z −uz · ∇ϕϕ ∈ C ∞ (Tn ).
Tn

Tn

The following characterization of the subdifferential of energy E was proved in[55] on subsets of Rn , but a modification for Tn is straightforward.
Proposition 2.1. Let u, v ∈ L2 (Tn ). Then v ∈ ∂E(u) if and only if u ∈ BV (Tn )and there exists a vector field z ∈ X2 (Tn ) such that z(x) ∈ ∂W (∇u(x)) a.e.,(z, Du) = W (Du) as measures in Tn and v = − div z.
Remark
2.2. Since W is one-homogeneous, we can define the measure W (Du) := W (∇u)+
 s 
D u
W |D
|Ds u| for any u ∈ BV (Tn ), where ∇u is the absolutely continuous part of Dus u|with respect to the Lebesgue measure and Ds u is the singular part.

However, for our purposes we only need the characterization of the subdifferential
for Lipschitz test functions, in which case we get the following simpler corollary.
Corollary 2.3. Let u ∈ Lip(Tn ) and v ∈ L2 (Tn ). Then v ∈ ∂E(u) if and only
if there exists a vector field z ∈ X2 (Tn ) such that z(x) ∈ ∂W (∇u(x)) a.e. andv = − div z.
Remark 2.4. It is clear from Corollary 2.3 that if ψ ∈ Lip(Tn ) and v ∈ ∂E(ψ) then forany positive constants α, β > 0 we have v ∈ ∂E(ψ̂) whereψ̂ = α[ψ]+ − β[ψ]−10M.-H. GIGA, Y. GIGA, AND N. POŽÁR

where [s]± := max(±s, 0); see [18, Remark 3.2]. In particular, ∂E(ψ) = ∂E(ψ̂). This is aconsequence of the one-homogeneity and convexity of W which imply that ∂W (p) = ∂W (ap)and ∂W (p) ⊂ ∂W (0) for all p ∈ Rn , a > 0.

2.3. General facets. By P we shall denote all ordered pairs of disjoint subsets of
Tn . Additionally, (P, ) will be a partially ordered set with ordering(A− , A+ )  (B− , B+ )⇔A+ ⊂ B+ and B− ⊂ A−

for (A− , A+ ), (B− , B+ ) ∈ P. We will also denote the reversal by−(A− , A+ ) := (A+ , A− ).
By definition, if (A− , A+ )  (B− , B+ ) then −(B− , B+ )  −(A− , A+ ).
Definition 2.5. A pair (A− , A+ ) ∈ P is open if both sets A± are open.
We say that ψ ∈ Lip(Tn ) is a support function of an open pair (A− , A+ ) ∈ P if
> 0 in A+ ,
ψ = 0 in (A− ∪ A+ )c ,
< 0 in A− .
On the other hand, for any function ψ on Tn we define its pair (not necessarily
open)
Pair(ψ) := ({x : ψ(x) < 0}, {x : ψ(x) > 0}).
Remark 2.6. If ψ is a support function of an open pair (A− , A+ ) ∈ P then −ψ is a
support function of the open pair −(A− , A+ ) := (A+ , A− ). With this notation we havePair(ψ) = − Pair(−ψ)
for any function ψ.
Example 2.7. For any open pair (A− , A+ ) ∈ P the function
ψ(x) := dist(x, Ac+ ) − dist(x, Ac− )is a support function of (A− , A+ ).

Definition 2.8. We say that an open pair (A− , A+ ) ∈ P is a smooth pair if(i) dist(A− , A+ ) > 0,(ii) ∂A− ∈ C ∞ and ∂A+ ∈ C ∞ .
Note that this definition allows for A− and/or A+ to be empty as dist is +∞ by
definition when one of the sets is empty.
Definition 2.9. We say that an open pair (A− , A+ ) ∈ P is an admissible pair if
there exists a support function ψ of (A− , A+ ) such that ψ ∈ D(∂E).
We shall show that every pair in P can be approximated in Hausdorff distanceby a smooth pair, and in turn that every smooth pair is an admissible pair.
The main tool in the construction will be the generalized ρ-neighborhood of a set
A, defined asρ > 0,
A + B ρ (0)ρ
U (A) := A
ρ = 0,
nx ∈ T : B ρ (x) ⊂ A
ρ < 0,
where G + H := {x + y : x ∈ G, y ∈ H} denotes the Minkowski sum of sets andB ρ (x) is the closed ball of radius ρ centered at x. In image analysis it is often

ANISOTROPIC TOTAL VARIATION FLOW

11

written as U ρ (A) = A ⊕ B ρ (0) for ρ > 0 and U ρ (A) = A B |ρ| (0) for ρ < 0, where⊕ denotes the Minkowski addition anddenotes the Minkowski decomposition.
In morphology, ⊕ is called dilation andis called erosion. We collect the basic
properties of U ρ in the following proposition; its proof is quite straightforward.
(a) U −ρ (A) ⊂ A ⊂ U ρ (A) for ρ > 0.

Proposition 2.10.
(b) (complement)
(2.2)

c

(U ρ (A)) = U −ρ (Ac )for any set A ⊂ Tn and ρ ∈ R(c) (monotonicity)U ρ (A1 ) ⊂ U ρ (A2 )for A1 ⊂ A2 ⊂ Tn and ρ ∈ R.

(d) U ρ (A1 ∩ A2 ) ⊂ U ρ (A1 ) ∩ U ρ (A2 ) for all ρ ∈ R, with equality for ρ ≤ 0.
(e) U r (U ρ (A)) ⊂ U r+ρ (A) for r ≥ 0 and ρ ∈ R; equality holds if ρ ≥ 0.
(f ) For any ρ ∈ R, we have U ρ (A1 ) ⊂ A2 if and only if A1 ⊂ U −ρ (A2 ).
For a set A ⊂ Tn we introduce the signed distance functiondA (x) := dist(x, A) − dist(x, Ac ).
We observe that
int U ρ (A) = {x ∈ Tn : dA (x) < ρ},U ρ (A) = {x ∈ Tn : dA (x) ≤ ρ}
for all ρ ∈ R.
For pair (A− , A+ ) ∈ P we define the ρ-neighborhood as
U ρ (A− , A+ ) := (U −ρ (A− ), U ρ (A+ )).
Clearly
U −ρ (A− , A+ )  (A− , A+ )  U ρ (A− , A+ )

ρ ≥ 0.

The following lemma was proved in [37].
Lemma 2.11. For any set A ⊂ Rn and constants ρ1 , ρ2 , 0 < ρ1 < ρ2 , there exist
open sets G− , G+ ⊂ Rn with smooth boundaries such thatU −ρ2 (A) ⊂ G− ⊂ U −ρ1 (A) ⊂ A ⊂ U ρ1 (A) ⊂ G+ ⊂ U ρ2 (A).
Using the previous lemma, we can show that any pair in P can be approximatedin Hausdorff distance by a smooth pair.
Proposition 2.12. Let (A− , A+ ) ∈ P be a pair and let 0 ≤ ρ1 < ρ2 . Then there
exists a smooth pair (G− , G+ ) ∈ P such that
(2.3)

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

Proof. Let us set δ := (ρ2 − ρ1 )/3 > 0. We apply Lemma 2.11 to the set A+ andobtain a smooth set G+ such thatU ρ1 (A+ ) ⊂ G+ ⊂ U ρ1 +δ (A+ ).
Then we apply Lemma 2.11 to the set A− and obtain a smooth set and G− suchthatU −ρ2 (A− ) ⊂ G− ⊂ U −ρ2 +δ (A− ).
We claim that dist(G− , G+ ) ≥ δ. Indeed, we can assume that both G− and G+ arenonempty and we choose any x ∈ G+ , y ∈ G− and z ∈ A+ . Since by definition

12

M.-H. GIGA, Y. GIGA, AND N. POŽÁR

of G− we have dist(y, Ac− ) ≥ ρ2 − δ and z ∈ A+ ⊂ Ac− , clearly dist(y, z) ≥ ρ2 − δ.
Thereforeρ1 + 2δ = ρ2 − δ ≤ dist(y, z) ≤ dist(y, x) + dist(x, z).
Since inf z∈A+ dist(x, z) = dist(x, A+ ) ≤ ρ1 + δ by the definition of G+ , we conclude
that dist(G− , G+ ) = inf y∈G− inf x∈G+ dist(x, y) ≥ δ.
Therefore (G− , G+ ) is a smooth pair and by construction (2.3) holds.
Finally, every smooth pair is an admissible pair.
Proposition 2.13. Suppose that (G− , G+ ) ∈ P is a smooth pair. Then there exists
a support function ψ of (G− , G+ ) such that ψ ∈ D(∂E).
Proof. Since ∂G± is smooth and Tn is compact, there exists δ± such that dG± issmooth in the set x : dG± < δ± ; see [23]. Let us takeδ :=1min {δ− , δ+ , dist(G− , G+ )} > 0.
3

Introduce the cutoff functions χ ∈ Lip(R) and θ ∈ Cc∞ (R) such that
χ(s) := max(0, min(δ, s))and θ(s) ∈ [0, 1] with θ(s) = 1 on [0, δ] and θ(s) = 0 on R \ (−δ, 2δ).
We defineψ(x) := χ(dGc+ (x)) − χ(dGc− (x)) = min δ, dist(x, Gc+ ) − min δ, dist(x, Gc− )and a vector fieldz(x) = θ(dGc+ (x))∂ 0 W (∇dGc+ (x)) + θ(dGc− (x))∂ 0 W (−∇dGc− (x)).
Clearly ψ ∈ Lip(Tn ), z ∈ Lip(Tn ) and ψ is a support function of (G− , G+ ).
It is also easy to see that z(x) ∈ ∂W (∇φ(x)) for a.e. x ∈ Tn . In particular,− div z ∈ ∂E(ψ) and therefore ψ ∈ D(∂E) by Corollary 2.3.

2.4. Resolvent equation. It is possible to approximate the minimal section of thesubdifferential −∂ 0 E via a resolvent problem on Tn . That is, for given ψ ∈ L2 (Tn )and a > 0 find ψa ∈ L2 (Tn ) that satisfies(2.4)

ψa + a∂E(ψa ) 3 ψ.

The standard theory of calculus of variations yields that this problem has a uniquesolution ψa ∈ D(∂E); see [25]. We have the following well-known result [8, 25].
Proposition 2.14. If ψ ∈ D(∂E) thenψa − ψ
→ −∂ 0 E(ψ)
ain L2 (Tn ) as a → 0,

where ψa is the unique solution of (2.4)
Moreover, a comparison theorem for (2.4) was proved in [18].
Proposition 2.15. Let ψa1 , ψa2 ∈ L2 (Tn ) be two solutions of (2.4) with a > 0 andright-hand sides ψ 1 , ψ 2 ∈ L∞ (Tn ), respectively. If ψ 1 ≤ ψ 2 then ψa1 ≤ ψa2 .

ANISOTROPIC TOTAL VARIATION FLOW

13

2.5. Monotonicity of nonlocal curvatures. First, we state a useful lemma forgenerating support functions in the domain of the subdifferential ∂E given an
admissible pair and an upper semi-continuous function.
Lemma 2.16. Let θ ∈ U SC(Tn ) and let (G− , G+ ) := Pair(θ). Suppose that
(H− , H+ ) ∈ P is an admissible pair and that there exists δ > 0 such that
(G− , G+ )  U −δ (H− , H+ ).
Then there exists a support function ψ of (H− , H+ ) such that ψ ∈ D(∂E) andθ≤ψ

on Tn .

If, moreover, ψ̂ ∈ D(∂E) is a support function of (H− , H+ ), we can take ψ suchthat −∂ 0 E(ψ) = −∂ 0 E(ψ̂).
Proof. Since (H− , H+ ) is an admissible facet, there exists a support function ψH ∈D(∂E). By the definition of (G− , G+ ) and ψH , we immediately have that θ ≤ ψHcon Gc+ ∩ H−
. We will modify the function ψH on the rest of Tn to guarantee thatthe ordering holds on the whole Tn . From the strict ordering of the pairs by δ > 0,
we immediately get
G+ ⊂ H+ ,H− ⊂ G− .

We define a new support function of (H− , H+ ) as
ψ(x) := α[ψH ]+ − β[ψH ]− ,where α and β are given positive constants specified below and [·]+ and [·]− arethe positive and negative parts. ψ is still a support function of (H− , H+ ) andRemark 2.4 yields that ψ ∈ D(E).
We shall determine the constants α and β. If G+ = ∅ then θ ≤ ψH on G+
trivially and we set α = 1. Otherwise, by compactness, semi-continuity and thedefinition of support functions, we havemaxTn θα :=> 0.
minG+ ψH
Similarly, if H− = ∅ we set β = 1, otherwise
maxH− θβ :=> 0.
minTn ψH
We observe that such a choice of α and β guarantees that
(2.5)

θ≤ψ

on Tn .

Finally, we can take ψH = ψ̂. Then Remark 2.4 yields that −∂ 0 E(ψ) =−∂ 0 E(ψH ).

The following monotonicity result plays the role of a comparison principle foradmissible pairs. The analogous result in [37] was stated for ordered smooth pairs,
and thanks to this extra regularity we did not need to assume that the pairs areordered strictly.
Proposition 2.17. Suppose that (G− , G+ ) ∈ P and (H− , H+ ) ∈ P are two open
pairs that are moreover strictly ordered, i.e., there exists δ > 0 such thatU δ (G− , G+ )  (H− , H+ ).

14

M.-H. GIGA, Y. GIGA, AND N. POŽÁR

Then for any support function ψG of (G− , G+ ) and any support function ψH of(H− , H+ ) such that ψG , ψH ∈ D(∂E) we have−∂ 0 E(ψG ) ≤ −∂ 0 E(ψH )cca.e. on Gc− ∩ Gc+ ∩ H−
∩ H+.

Proof. We apply the comparison principle for the resolvent problem (2.4); it is also
possible to use the evolution equation as in [42].
Let us denote the intersection of the facets as D,cc
D := Gc− ∩ Gc+ ∩ H−
∩ H+.

We can assume that ψG ≤ ψH . Indeed, if this ordering does not hold we replace
ψH with the function ψ provided by Lemma 2.16 applied with θ = ψG and ψ̂ = ψHsince −∂ 0 E(ψ) = −∂ 0 E(ψH ).
Clearly, the support functions coincide with zero on the intersection of the facets,
i.e.,
(2.6)

ψG = ψH = 0

on D.

For each a > 0, we find the solution ψai of the resolvent problem (2.4) withright-hand side ψi , i = G, H. Due to the L2 convergence in Proposition 2.14, we
can find a subsequence ak → 0 as k → ∞ such that (ψai k − ψi )/ak → −∂ 0 E(ψi ) a.e.
on Tn as k → ∞ for i = G, H.
The comparison principle, Theorem 2.15, and (2.5) imply that ψaGk ≤ ψaHk .
Moreover, by (2.6), ψai k − ψ i = ψai k on D for all k. Therefore
ψaGkk→∞ akψH≤ lim ak = −∂ 0 E(ψH )k→∞ ak−∂ 0 E(ψG ) = lim

a.e. in D

and the comparison principle for −∂ 0 E is established.



3. Viscosity solutions
This section finally introduces viscosity solutions of (1.1). As in the previous
work [37], it is necessary to separately define test functions for the zero gradientof a solution and the nonzero gradient. In this section we work on the parabolic
cylinder Q := Tn × (0, T ) for some T > 0.
Definition 3.1. Let (A− , A+ ) ∈ P be a smooth pair and let x̂ ∈ Tn \ A− ∪ A+ .
Function ϕ(x, t) = ψ(x) + g(t), where ψ ∈ Lip(Tn ) and g ∈ C 1 (R), is called anadmissible faceted test function at x̂ with a pair (A− , A+ ) if ψ ∈ D(∂E) and ψ is a
support function of the pair (A− , A+ ).
Definition 3.2. We say that an admissible faceted function ϕ at x̂ with a pair(A− , A+ ) is in a general position of radius η > 0 with respect to u : Q → R at(x̂, t̂) ∈ Q if B η (x̂) ⊂ Tn \ A− ∪ A+ andu(x, t) −infh∈B η (0)ϕ(x − h, t) ≤ u(x̂, t̂) − ϕ(x̂, t̂)for all x ∈ Tn , t ∈ [t̂ − η, t̂ + η].

Definition 3.3 (Viscosity solutions). An upper semi-continuous function u : Q → R
is a viscosity subsolution of (1.1) if the following holds:

ANISOTROPIC TOTAL VARIATION FLOW

15(i) ( faceted test) If ϕ(x, t) = ψ(x) + g(t) is an admissible faceted test function
such that ϕ is in general position of radius η with respect to u at a point(x̂, t̂) ∈ Q then there exists δ ∈ (0, η) such that
ϕt (x̂, t̂) + F 0, ess inf −∂ 0 E(ψ) ≤ 0.
Bδ (x̂)(ii)

2,1( conventional test) If ϕ ∈ Cx,t(U ) in a neighborhood U ⊂ Q of a point (x̂, t̂),such that u − ϕ has a local maximum at (x̂, t̂) and |∇ϕ| (x̂, t̂) 6= 0, thenϕt (x̂, t̂) + F ∇ϕ(x̂, t̂), k(∇ϕ(x̂, t̂), ∇2 ϕ(x̂, t̂)) ≤ 0,
where ∇2 is the Hessian and


(3.1)
k(p, X) := trace (∇2 W )(p)Xfor p ∈ Rn \ {0}, X ∈ S n ,so that k(∇ϕ(x̂, t̂), ∇2 ϕ(x̂, t̂)) = [div(∇W )(∇ψ)] (x̂, t̂). Here S n is the set ofn × n-symmetric matrices.
A viscosity supersolution can be defined similarly as a lower semi-continuousfunction, replacing maximum by minimum, ≤ by ≥, and ess inf by ess sup. Furthermore, in (i) ϕ must be such that −ϕ is in a general position of radius η with respect
to −u (see also Remark 2.6).
Function u is a viscosity solution if it is both a subsolution and supersolution.
The next result indicates that it is possible to find an admissible test function in
general position for a given upper semi-continuous function u given an admissible
facet that is in general position with respect to the facet of u.
Lemma 3.4. Suppose that (H− , H+ ) ∈ P is an admissible pair, and let u ∈ U SC(Q)be a bounded upper semi-continuous function on Q := Tn × (0, T ) for some T > 0,and let g ∈ C 1 (R). Moreover, let (x̂, t̂) ∈ Q be a point such that x̂ ∈ Tn \ H− ∪ H+ .
Suppose that there is δ > 0 such that
Pair(u(·, t) − u(x̂, t̂) − g(t))  U −δ (H− , H+ )for t ∈ (t̂ − δ, t̂ + δ).

Then there exists a support function ψ ∈ D(∂E) of (H− , H+ ) and η > 0 suchthat ϕ(x, t) = ψ(x) + g(t) is an admissible faceted test function at (x̂, t̂) with pair
(H− , H+ ) in a general position of radius η with respect to u at a point (x̂, t̂).
Proof. Let us first set
1
min δ, dist(x̂, H+ ∪ H− ) .
2Then we introduce the function θ by
η :=θ(x) :=supsupu(x + h, t) − u(x̂, t̂) − g(t).

h∈B η (0) t∈[t̂−η,t̂+η]Clearly θ ∈ U SC(Tn ). Observe that, since B η (x̂) ∈ Tn \ H+ ∪ H− by the definitionof η, the function ϕ(x, t) = ψ(x) + g(t) is in general position of radius η with respect
to u at the point (x̂, t̂) if and only ifθ≤ψ

on Tn .

But such a function ψ ∈ D(∂E) is provided by Lemma 2.16.

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M.-H. GIGA, Y. GIGA, AND N. POŽÁR

4. Comparison principle
In this section we will establish the comparison principle for viscosity solutions
introduced in Definition 3.3. We will fix the spacetime cylinder Q := Tn × (0, T ).
Theorem 4.1 (Comparison). Let u and v be respectively a bounded viscosity subsolution and a viscosity supersolution of (1.1) on Q. If u ≤ v at t = 0 then u ≤ v onQ.
We shall give a slightly different exposition of the proof of the theorem than the
one that appears in [37], but the method is identical.
We perform a variation of the doubling-of-variables procedure: we define
w(x, t, y, s) := u(x, t) − v(y, s),and, for a positive constant ε > 0 and point ζ ∈ Tn , the functions
2|x − y − ζ|
+ S(t, s; ε),2ε2|t − s|εεS(t, s; ε) :=++,2ε
T −t T −s
where |x − y − ζ| was defined in (2.1).
We analyze the maxima of functions
Ψζ (x, t, y, s; ε) :=Φζ (x, t, y, s; ε) := w(x, t, y, s) − Ψζ (x, t, y, s; ε)for ζ ∈ Tn .

Following [29], we define the maximum of Φζ`(ζ; ε) = max Φζ (·; ε)Q×Q

and the sets of points of maximum of Φζ , over Q × Q
A(ζ; ε) := arg max Φζ (·; ε) := (x, t, y, s) ∈ Q × Q : Φζ (x, t, y, s; ε) = `(ζ; ε) .
Q×Q

Suppose that the comparison principle, Theorem 4.1, does not hold, that is,
suppose that
m0 := sup [u − v] > 0.
Q

We have the following proposition.
Proposition 4.2. There exists ε0 > 0 such that for all ε ∈ (0, ε0 ) we haveA(ζ; ε) ⊂ Q × Q

for all |ζ| ≤ κ(ε),

12where κ(ε) := 18 (m0 ε) . Moreover,1|x − y − ζ| ≤ (M ε) 2 ,1|t − s| ≤ (M ε) 2 ,for all (x, t, y, s) ∈ A(ζ; ε),where M := supQ×Q w < ∞.
Proof. See [29, Proposition 7.1, Remark 7.2].

1
2

In the view of Proposition 4.2, we fix one ε ∈ (0, ε0 ) such that (M ε) < 14 forthe rest of the proof and drop the dependence of the formulas below on ε for thesake of clarity. Moreover, we introduce
κ(ε)
λ :=
.
2

ANISOTROPIC TOTAL VARIATION FLOW

17

Again, following [29], we define the set of gradients


x−y−ζ
B(ζ) :=: (x, t, y, s) ∈ A(ζ) ⊂ Rn ,εwhere x − y − ζ is interpreted as a vector in (− 41 , 14 )n ⊂ Rn .
The situation can be divided into two cases:
Case I. B(ζ) = {0} for all |ζ| ≤ κ(ε).
Case II. There exists ζ ∈ Tn and p ∈ B(ζ) such that |ζ| ≤ κ(ε) and p 6= 0.
4.1. Case I. This is the less standard case since it is necessary to construct admissible faceted test functions for the faceted test in the definition of viscosity solutions.
We have B(ζ) = {0} for all |ζ| ≤ κ(ε). In this case, we apply the constancy lemmathat was presented in [29, Lemma 7.5].
Lemma 4.3 (Constancy lemma). Let K be a compact set in RN for some N > 1and let h be a real-valued upper semi-continuous function on K. Let φ be a C 2
function on Rd with 1 ≤ d < N . Let G be a bounded domain in Rd . For each ζ ∈ G
assume that there is a maximizer (rζ , ρζ ) ∈ K ofHζ (r, ρ) = h(r, ρ) − φ(r − ζ)over K such that ∇φ(rζ − ζ) = 0. Then,hφ (ζ) = sup {Hζ (r, ρ) : (r, ρ) ∈ K}
is constant on G.
We apply Lemma 4.3 with the following parameters:
N = 2n + 2,

d = n,

ρ = (y, t, s) ∈ Tn × R × R,

K = {(x − y, y, t, s) : (x, y) ∈ Tn × Tn , (t, s) ∈ [0, T ] × [0, T ]},G = B2λ (0),
2

|r|
.
2ε
K can be treated as a compact subset of Rn in a straightforward way. We infer that`(ζ) = hφ (ζ) is constant for |ζ| ≤ λ.
Therefore we have also an ordering analogous to [29, Corollary 7.9], which yields
the crucial estimate.
h(r, ρ) = w(r + y, t, y, s) − S(t, s),φ(r) =

Lemma 4.4. Let (x̂, t̂, x̂, ŝ) ∈ A(0). Thenu(x, t) − v(y, s) − S(t, s) ≤ u(x̂, t̂) − v(x̂, ŝ) − S(t̂, ŝ)for all s, t ∈ (0, T ) and x, y ∈ Tn such that |x − y| ≤ λ := κ(ε)/2.
From now on, we fix (x̂, t̂, x̂, ŝ) ∈ A(0) and set
α := u(x̂, t̂),As in [37], we introduce the closed sets
U := x : u(x, t̂) ≥ α ,β := v(x̂, ŝ).
V := {x : v(x, ŝ) ≤ β},which will be used to generate strictly ordered smooth facets. To accomplish that,
let us now for simplicity set r := λ/10. Furthermore, define the closed sets
X := (U r (U ))c ,Y := (U r (V ))c .

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M.-H. GIGA, Y. GIGA, AND N. POŽÁR

Since dist(U, X) = dist(V, Y ) = r, the definition of U and V and the semi-continuityof u and v imply that there exists δ > 0 such that
(4.1a)u(x, t) − α + S(t̂, ŝ) − S(t, ŝ) < 0,x ∈ X, t ∈ [t̂ − δ, t̂ + δ],(4.1b)

v(x, t) − β + S(t̂, t) − S(t̂, ŝ) > 0,x ∈ Y, t ∈ [ŝ − δ, ŝ + δ].

Moreover, the estimate from Lemma 4.4 and the definition of U and V implies that
(4.2a)u(x, t) − α + S(t̂, ŝ) − S(t, ŝ) ≤ 0,x ∈ U λ (V ), t ∈ (0, T ),(4.2b)

v(x, t) − β + S(t̂, t) − S(t̂, ŝ) ≥ 0,x ∈ U λ (U ), t ∈ (0, T ).

This suggests introducing
gu (t) := S(t, ŝ) − S(t̂, ŝ),gv (t) := S(t̂, ŝ) − S(t̂, t)

and the pairs
Pu (t) := Pair(u(·, t) − α − gu (t)),
Pv (t) := Pair(v(·, t) − β − gv (t)).
If we denote by −Pv (t) the reversed pair of Pv (t), we infer from (4.1) and (4.2) inparticular that
(4.3a)Pu (t)  ((U r (U ))c , U r (U ) \ U λ (V )) =: Ru ,(4.3b)−Pv (t)  ((U r (V ))c , U r (V ) \ U λ (U )) =: Rv .

We define the pairs
Su := (U c , U \ U λ−3r (V )),Sv := (V c , V \ U λ−3r (U )).

Since Su , Sv ∈ P, Proposition 2.12 implies that there exist smooth pairs (U− , U+ )and (V− , V+ ) such that
(4.4a)

U 2r (Su )  (U− , U+ )  U 3r (Su ),(4.4b)

U 2r (Sv )  (V− , V+ )  U 3r (Sv ).

Before proving Lemma 4.6 below, we give the following trivial estimate.
Lemma 4.5. Suppose that G, H ⊂ Tn . ThenU ρ (G) \ U ρ (H) ⊂ U ρ (G \ H)for any ρ > 0.

Proof. Suppose that x ∈ U ρ (G) \ U ρ (H). Then there exists y ∈ G such thatx ∈ B ρ (y). In particular, y ∈/ H and therefore y ∈ G \ H, which implies thatx ∈ U ρ (G \ H).
Lemma 4.6. The pair (U− , U+ ) and the pair (V− , V+ ) have the following properties:(a) The pairs are strictly ordered in the sense(4.5)

U r (U− , U+ )  (V+ , V− ) = −(V− , V+ ).

(b) The contact point x̂ lies in the interior of the intersection of the facets, that
is,
(4.6)

c
c
B r (x̂) ⊂ U−
∩ U+
∩ V−c ∩ V+c .

(c) The pairs are in general position with respect to Ru and Rv , i.e.,(4.7)

U r (Ru )  (U− , U+ ),U r (Rv )  (V− , V+ ).

ANISOTROPIC TOTAL VARIATION FLOW

19

Proof. Let us recall the properties of U ρ in Proposition 2.10. To show (a), firstestimate using (4.4a) and (2.2)(4.8)

U r (U+ ) ⊂ U r (U 3r (U \ U λ−3r (V ))) ⊂ U 3r (U 3r−λ (V c )) ⊂ U 6r−λ (V c ).

On the other hand, since 6r − λ = −4r < −3r, we have from (4.4b)
U 6r−λ (V c ) ⊂ U −3r (V c ) ⊂ V− .
Combining these two estimates we get U r (U+ ) ⊂ V− . Symmetric estimates showthat U r (V+ ) ⊂ U− , i.e., V+ ⊂ U −r (U− ). Consequently, (4.5) follows.
For (b), we first realize that by definition x̂ ∈ U ∩ V and therefore B r (x̂) ⊂U r (U ) ∩ U r (V ). But (4.4a) yields
U− ⊂ U −2r (U c ) = (U 2r (U ))c .
Similarly, (4.8) implies
U+ ⊂ U 6r−λ (V c ) ⊂ U −2r (V c ) = (U 2r (V ))c .
Symmetric estimates hold for V± and (b) follows.
To show (c), we estimate using Lemma 4.5
U r (Ru )  (U −2r (U c ), U 2r (U ) \ U λ−r (V )) (U −2r (U c ), U 2r (U \ U λ−3r (V ))) = U 2r (Su )  (U− , U+ ).
The statement for (V− , V+ ) is analogous.

We can finally finish the construction for Case II using the estimates in Lemma 4.6.
Indeed, the estimate (4.7), recalling the definitions of Ru and Rv in (4.3), is all thatis necessary to apply Lemma 3.4, with an obvious modification for v. Then we haveϕu (x, t) = ψu (x) + gu (t) (resp. ϕv (x, t) = ψv (x) + gv (t)), an admissible faceted test
function at (x̂, t̂) (resp. (x̂, ŝ)) with facet (U− , U+ ) (resp. (V+ , V− ) = −(V− , V+ )).
Moreover, ϕu is in general position with respect to u at (x̂, t̂) and −ϕv is in general
position with respect to −v at (x̂, ŝ), both with some radius η > 0. Since the facets
are strictly ordered (4.5), the monotonicity Proposition 2.17 and (4.6) yield
ess inf −∂ 0 E(ψu ) ≤ ess sup −∂ 0 E(ψv ) .
(4.9)
B η (x̂)B η (x̂)By definition of viscosity solutions, we have(gu )t (t̂) + F!
 0
0, ess inf −∂ E(ψu ) ≤ 0,
B η (x̂)(gv )t (ŝ) + F!
 0
0, ess sup −∂ E(ψv ) ≥ 0.
B η (x̂)Subtracting these two inequalities and using (4.9) with the ellipticity of F (1.5), we
arrive at
!
 0

ε
ε
0<
+
+ F 0, ess inf −∂ E(ψu )(T − ŝ)2(T − t̂)2
B η (x̂)!
 0

− F 0, ess sup −∂ E(ψv ) ≤ 0,
B η (x̂)a contradiction. Therefore we conclude that Case I cannot occur.

20

M.-H. GIGA, Y. GIGA, AND N. POŽÁR

4.2. Case II. This the more classical case since there exists ζ ∈ Tn and p ∈ B(ζ)
such that |ζ| ≤ κ(ε) and p 6= 0, and we only need to construct a smooth test function
for the classical test in the definition of viscosity solutions. Here we refer the reader
to [29, 37]. We again arrive at a contradiction, yielding that Case II cannot occur
either. Therefore the comparison principle Theorem 4.1 holds.
5. Existence of solutions via stability
5.1. Stability. In this section we discuss the stability of solution of (1.1) under anapproximation by regularized problems.
Suppose that {Wm }m∈N is a decreasing sequence of C 2 functions on Rn thatconverge locally uniformly to W and such that the functions Wm satisfy2
a−1
m I ≤ ∇ Wm (p) ≤ am I

for all p ∈ Rn , m ∈ N

and some sequence of positive numbers am .
1
. Define the
Example 5.1. Let φ 1 be the standard mollifier with support of radius mmsmoothingWm (p) = (W ∗ φ 1 )(p) +
m

1
|p|2
m

p ∈ Rn .

By convexity we have Wm > W (clearly true for p 6= 0, and at p = 0 we use (1.4)),
1
I and Wε ↓ W as ε → 0 locally uniformly. The bound on ∇2 Wm
Wm ∈ C ∞ , ∇2 Wm ≥ mfrom above follows from the one-homogeneity of W which yields ∇2 W (ap) = a−1 ∇2 W (p)for a > 0.

Let us introduce the regularized energies(R
Wm (∇ψ) ψ ∈ H 2 (Tn ),Tn
Em (ψ) :=+∞ψ ∈ L2 (Tn ) \ H 2 (Tn ),where H 2 (Tn ) is the standard Sobolev space.
We shall approximate the problem (1.1) by a sequence of problems(5.1)

ut + F (∇u, −∂ 0 Em (u(·, t))) = 0,

with initial data
(5.2)

u|t=0 = u0 .

We have the following proposition proved in [37].
Proposition 5.2.
(a) Em form a decreasing sequence of proper convex lower semi-continuous functionals on L2 (Tn ) and E = (inf m Em )∗ , the lower semi-continuous envelope ofinf m Em in L2 (Tn ).
(b) The subdifferential ∂Em is a singleton for all ψ ∈ D(∂Em ) = H 2 (Tn ) and its
canonical restriction can be expressed as


(5.3)
−∂ 0 Em (ψ) = div [(∇Wm )(∇ψ)] = trace (∇2 Wm )(∇ψ)∇2 ψa.e.
(c) Due to the ellipticity of F , the problem (5.1) is a degenerate parabolic problem
that has a unique global viscosity solution for given continuous initial datau0 ∈ C(Tn ).

ANISOTROPIC TOTAL VARIATION FLOW21

The main theorem of this section is the stability of solutions of (1.1) with respect
to the half-relaxed limits?-limsup um (x, t) := lim sup
m→∞

sup

sup um (y, s),

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

m→∞

Theorem 5.3 (Stability). Let um be a sequence of viscosity subsolutions of (5.1)
on Tn × [0, ∞), and let u = ?-limsupm→∞ um . Assume that u < +∞ in Tn × [0, ∞).
Then u is a viscosity subsolution of (1.1).
Similarly, u = ?-liminf m→∞ um is a viscosity supersolution of (1.1) provided thatum is a sequence of viscosity supersolutions of (5.1) and u > −∞.
The proof is the same as in [37] and we shall skip it here.
5.2. Existence. We shall use the stability theorem to prove the following existence
result.
Theorem 5.4 (Existence). If F is continuous and degenerate elliptic (1.5), andW satisfies (1.3) and (1.4), and u0 ∈ C(Tn ), there exists a unique solution u ∈C(Tn × [0, ∞)) of (1.1) with the initial data u0 . Furthermore, if u0 ∈ Lip(Tn ) thenk∇u(·, t)k∞ ≤ k∇u0 k∞ .
The proof of the theorem will proceed in three steps: 1) due to the stability, byfinding the solution um of the problem (5.1) for all m ≥ 1, we can find a subsolution
u and a supersolution u of (1.1); 2) a barrier argument at t = 0 shows that u and uhave the correct initial data u0 ; and 3) the comparison principle shows that u = u
is the unique viscosity solution of (1.1), and the Lipschitz estimate holds.
Before giving a proof of the existence theorem, we construct barriers for step 2.
Since the operator (5.3) degenerates at points where ∇u = 0 as m → ∞, it seems
to be necessary to construct barriers that depend on m. We will use the Wulff
functions for energy Em ; these were previously considered in the proof of stability
for general equations of the type (5.1) in one-dimensional setting in [31] and in theisotropic setting in [37]. However, the construction is slightly more complicated in
the anisotropic case in higher dimension. Since the operator F in (1.1) depends
on the derivative of the solutions, we have to construct test functions that have
uniformly bounded space derivatives as m → ∞. However, the derivatives of theWulff functions for Em blow up as m → ∞. Therefore we have to cut off large
derivatives. This was done in [31, 37] by a simple idea that can be only applied forone-dimensional or radially symmetric Wulff functions. Here we present a different
idea that relies on the modification of Wm directly using the properties of the
Legendre-Fenchel transform.
For a convex proper function φ : Rn → (−∞, +∞] we define its convex conjugate?
φ via the Legendre-Fenchel transform as
φ? (x) := sup [x · p − φ(p)] .
p∈Rn

It is well-known that φ? is also convex and that, if φ is also lower semi-continuous,φ?? = φ.
We give the proof of the following lemma for completeness.

22

M.-H. GIGA, Y. GIGA, AND N. POŽÁR

Lemma 5.5. Let Ω ⊂ Rn be a non-empty bounded convex open set and let φ ∈LSC(Rn ) be a convex function on Rn such that φ ∈ C 2 (Ω), φ = ∞ on Rn \ Ω andφ is strictly convex in Ω, i.e., ∇2 φ > 0 in Ω.
Then φ? ∈ C 2 (Rn ) ∩ Lip(Rn ),

−1
(5.4)
∇φ? (x) ∈ Ω and ∇2 φ? (x) = ∇2 φ(∇φ? (x))>0x ∈ Rn .
Proof. Since x · p − φ(p) is upper semi-continuous and x · p − φ(p) = −∞ on ∂Ω, thesupremum in the definition is for every x ∈ Rn attained at a point p ∈ Ω such that∇φ(p) = x. Additionally, p is unique due to the strict convexity, and the function p(x)is C 1 by the inverse function theorem. If we differentiate φ? (x) = x · p(x) − W (p(x))
we get ∇φ? (x) = p(x) ∈ Ω. Thus ∇φ? is the inverse map of ∇φ and the inverse
function theorem implies the expression for ∇2 φ? (x).
Let ψ : Rn → (−∞, ∞] be a lower semi-continuous convex function such thatψ ∈ C ∞ (B1 (0)) and ψ(p) = ∞ for |p| ≥ 1 and ψ(0) = 0. Note that the semicontinuity implies that ψ(p) → ∞ as |p| → 1− . For given positive constants m, A, q,
we define p
Wm;A,q (p) := A Wm (p) + qψ
− Wm (0) .
q
We also define the quasilinear differential operators Lm : C 2 (Rn ) → R for m ∈ N
motivated by the expression for −∂ 0 Em in (5.3) as
Lm (u)(x) := trace (∇2 Wm )(∇u(x))∇2 u(x)u ∈ C 2 (Rn ).
∗Functions Wm;A,q, the conjugates of Wm;A,q , approximate the Wulff functions∗
Wm of the energies Em and we summarize their properties in the following lemma.
∗
Lemma 5.6. For any m, A, q positive, Wm;A,q
are strictly convex, nonnegative, C 2nfunctions on R and?
∇Wm;A,q(x) < q,?
0 < Lm (Wm;A,q)(x) ≤ A−1 n

x ∈ Rn .

Proof. Strict convexity and regularity follows from Lemma 5.5. In particular, we?
observe that Ω = Bq (0) and hence ∇Wm;A,q∈ Bq (0). Nonnegativity is also obvious.
n?
Let x ∈ R and set p = ∇Wm;A,q (x). Since ψ in the definition of Wm;A,q isconvex and thus ∇2 ψ ≥ 0 on B1 (0), (5.4) yields?
0 < ∇2 Wm;A,q(x) = (∇2 Wm;A,q (p))−1 −11p= A−1 ∇2 Wm (p) + (∇2 ψ)
q
q
 2
−1
−1
≤A
∇ Wm (p).

We also recall that if M, N ≥ 0 then also trace M N ≥ 0. Therefore?
?
Lm (Wm;A,q)(x) = trace (∇2 Wm )(p)∇2 Wm;A,q(x) ≤ A−1 trace I = A−1 n.
?
Similarly Lm (Wm;A,q)(x) > 0.

Now we define the barriers?
φm;A,q (x, t) := βA,q t + Wm;A,q(x),?
φm;A,q (x, t) := −βA,q t − Wm;A,q
(−x),



ANISOTROPIC TOTAL VARIATION FLOW

23

where
(5.5)

βA,q :=supsup|F (p, ξ)| + 1 < ∞.

p∈Bq (0) |ξ|≤A−1 n

Corollary 5.7. For any m, A, q > 0 the function φm;A,q is a classical supersolutionof (5.1) on Rn and the function φm;A,q is a classical subsolution of (5.1) on Rn .
Proof. The corollary follows from Lemma 5.6 and the definition of βA,q in (5.5).
Additionally, we observe that if u ∈ C 2 (Rn ) and v(x) = −u(−x) thenLm (v)(x) = −Lm (u)(−x).
?
Finally, we observe that Wm;A,q
can be bound from below away from the origin.

Lemma 5.8. For any δ, K > 0 there exist m0 , A, q > 0 such that?
Wm;A,q(x) ≥ 2K

for all x, |x| ≥ δ, and m ≥ m0 .

Proof. Let us defineµ := sup [W (p) + ψ(p)] ∈ (0, ∞).

(5.6)|p|=1/2

Now we set
8K
δ
,
q :=
.
8µ
δ
By the locally uniform convergence of Wm → W , we can find m0 > 0 such thatA :=sup |Wm (p) − Wm (0) − W (p)| ≤ qµm > m0 .

|p|=q/2x
Now for any x such that |x| ≥ δ and any m > m0 , setting p = 2q |x|, we estimate?
Wm;A,q(x) ≥ x · p − Wm;A,q (p)

 

q
p
= |x| − A Wm (p) + qψ
− Wm (0)
2
q

 

q
p
≥ |x| − A W (p) + qψ
+ qµ
2
q
  
 

p
p
q
= |x| − Aq W
+ψ
+µ
2
q
q
q
≥ |x| − 2Aqµ ≥ 2K,
2
where we used the one-homogeneity (1.4) of W and (5.6).

With the constructed barriers, we are able to finish the proof of the existence
theorem.
Proof of Theorem 5.4. Let Wm be a sequence that approximates W as in Section 5.1.
By Proposition 5.2, the approximate problem (5.1) with initial data u0 has a uniquecontinuous solution um on Tn × [0, ∞). Since function (x, t) 7→ F (0, 0)t + α isa solution of (5.1) for any m and α ∈ R, um are locally uniformly bounded by
the comparison principle. Therefore the stability result, Theorem 5.3, yields thatu = ?-limsupm→∞ um is a subsolution of (1.1) and u = ?-liminf m→∞ um is asupersolution of (1.1). Clearly u ≤ u.

24

M.-H. GIGA, Y. GIGA, AND N. POŽÁR

We are left to prove that u(x, 0) ≤ u0 ≤ u(x, 0) since then the comparison
principle, Theorem 4.1, yields that u = u on Tn × [0, ∞) and u = u is the uniquesolution of (1.1) with initial data u0 .
Let us thus set K := supTn |u0 | < ∞ and choose ξ ∈ Tn and ε > 0. We shall showthat u(ξ, 0) ≤ u0 (ξ)+2ε. By continuity, there exists δ > 0 such that u0 (x) ≤ u0 (ξ)+ε
for x ∈ Bδ (ξ). Let m0 , A and q be the constants given by Lemma 5.8 and defineφm (x, t) := infn φm;A,q (x + k − ξ, t) + u0 (ξ) + ε.
k∈Z

Observe that φm is a viscosity supersolution of (5.1) for every m. Moreover, byLemma 5.8 and (5.6), and the choice of the parameters, u0 ≤ φm (·, 0) on Tn for
all m > m0 . Therefore the comparison principle yields um ≤ φm on Tn × [0, ∞).
Finally, it is easy to observe that φm (ξ, 0) ≤ u0 (ξ) + 2ε for all sufficiently large m.
Since φm are q-Lipschitz continuous in space by Lemma 5.6, we haveum (x, t) ≤ φm (x, t) ≤ βA,q t + q |x − ξ| + u0 (ξ) + 2ε
for all large m. Hence u(ξ, 0) ≤ u0 (ξ) + 2ε.
Since ε was arbitrary, we conclude that u(ξ, 0) ≤ u0 . A similar argument with φyields u(ξ, 0) ≥ u0 .
A standard argument yields the Lipschitz continuity of the solution.
Acknowledgments. The work of the second author is partly supported by Japan
Society for the Promotion of Science through grants Kiban (S) 21224001, Kiban (A)23244015 and Houga 25610025.
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