Anisotropic and Crystalline mean curvature flows · Anisotropic and Crystalline mean curvature ow An anisotropic mean curvature ow is a tube of sets E(t)with boundary evolving with

Anisotropic and Crystalline mean curvature flows

Antonin Chambolle

CMAP, Ecole Polytechnique, CNRS, Palaiseau, France

“The Role of Metrics in the Theory of Partial Differential Equations”,11th Seasonal Institute of the MSJ, Hokkaido Univ, Sapporo

2-13 July 2018

Introduction

The goal of these lectures is to provide the audience with an introduction to

singular curvature flows, by reviewing some recent and less recent works with

G. Bellettini (Trieste), V. Caselles, M. Morini (Parma), M. Ponsiglione

(Roma I), M. Novaga (Pisa)

Outline

I Introduction to anisotropic / crystalline curvature flows

I Approximation algorithms

I Strong existence in dimension 2

I A framework for higher dimension: Construction of a solution,Comparison

I Extensions

Anisotropic surface tensions

Given a set E ⊂ Rd and a surface tension ϕ, which is a convex,one-homogeneous and even norm, we consider the anisotropicperimeter

Perϕ(E ) :=

ˆ∂E∩Ω

ϕ(νE )dHd−1.

(In these lectures in general Ω = Rd .)As is standard, this is extended to finite perimeter (“Caccioppoli”) setsby the dual formula:

Perϕ(E ) = sup

Ê

divψ dx : ψ ∈ C∞c (Ω;Rd), ϕ(ψ) ≤ 1

.

Anisotropic surface tensions

More generally, we will also encounter the anisotropic total variation:

ϕ(Du)(Ω) =

ˆΩ

ϕ(Du) = sup

û divψ dx : ψ ∈ C∞c (Ω;Rd), ϕ(ψ) ≤ 1

.

which is finite if and only if u ∈ BV (Ω), that is when Du is a boundedRadon measure, and obviously

Perϕ(E ) = ϕ(DχE )(Ω).

Indeed E is a set with finite perimeter if and only if χE is BV , and

DχE = −νEHd−1 ∂∗E

(νE outer normal, ∂∗ “reduced boundary”).

The anisotropy and its polar

Let us introduce the polar norm:

ϕ(x) := supϕ(ξ)≤1

ξ · x .

By convex duality,

ϕ(ξ) = ϕ(ξ) := supϕ(x)≤1

ξ · x .

Equivalently, in terms of (Legendre-)Fenchel dual (or convex conjugate):

(1

2ϕ2)∗(x) = sup

ξx · ξ − 1

2ϕ2(ξ) =

1

2ϕ(x)2

and of course, ( 12ϕ2)∗ = 1

2ϕ2.

The anisotropy and its polar: important remarks

• Euler’s identity (which stems from 1-homogeneity) reads

ϕ(ξ) = x · ξ

for x ∈ ∂ϕ(ξ) = x : ϕ(η) ≥ ϕ(ξ) + x · (η − ξ)∀η. One also checksthat

∂ϕ(0) = ϕ ≤ 1,

∂ϕ(ξ) = x ∈ ϕ ≤ 1 : x · ξ = ϕ(ξ)

and in particular if x 6= 0 and x ∈ ∂ϕ(ξ), ϕ(x) = 1.

The anisotropy and its polar: important remarks

• Since ( 12ϕ

2)∗ = 12ϕ2, in particular ϕ∂ϕ and ϕ∂ϕ are inverse

maximal monotone operators in Rd . This means that

x ∈ ∂ϕ(ξ)⇔ xϕ(ξ) ∈ ϕ(ξ)∂ϕ(ξ)

⇔ ξ ∈ ϕ(xϕ(ξ))∂ϕ(xϕ(ξ)) = ϕ(ξ)ϕ(x)∂ϕ(x)

hence x ∈ ∂ϕ(ξ)⇔ ξ/ϕ(ξ) ∈ ∂ϕ(x), unless ξ = 0.

• Similar statements hold switching ϕ and ϕ, obviously.

The Wulff shape

We introduce Wϕ = ϕ ≤ 1 (= ∂ϕ(0)), which is called the Wulffshape.It was conjectured by Wulff (1901) that it is the unique minimizer of thePerϕ-perimeter with a mass constraint. There are many proofs of this(Dinghas 1944, Taylor 1978, uniqueness by Fonseca and Muller 1990,anisotropic symmetrizations: Alvino Ferone Lions Trombetti 1997, VanSchaftingen 2006..., OT: Figalli-Maggi-Pratelli 2010, after Gromov 1986)

The Wulff shape - isoperimetry

A formal OT based proof is simple (Figalli-Maggi-Pratelli 10): Let E be aset, ψ its (convex) Brenier map to the Wulff shape ϕ ≤ R = RWϕ ofsame volume. This means that ∇ψ transports χE to χϕ≤R, or

Ê

f (∇ψ(x))dx =

ˆϕ≤R

f (y)dy

for any f , in particular detD2ψ = 1 a.e. in E .

Here R = (|E |/|Wϕ|)1/d .

The Wulff shape - isoperimetry

Then, on one hand,Ê

∆ψdx =

Ê

div∇ψdx =

ˆ∂E

∇ψ · ν ≤ R

ˆ∂E

ϕ(ν) = RPerϕ(E )

since ϕ(ν) ≥ ν · (∇ψ/R) on the boundary, as ϕ(∇ψ/R) ≤ 1.On the other hand, since D2ψ is a nonnegative symmetric matrix, in E ,

1 = (detD2ψ)1/d ≤ 1

dTrD2ψ =

∆ψ

d

(with equality iff the eigenvalues are identical, that is if D2ψ = Id) sothat ˆ

E

∆ψdx ≥ d |E |.

We find that d |E | ≤ RPerϕ(E ), with possible equality only if E = W ,and indeed it is easy to check that equality holds in this case (exactly likefor the Euclidean ball).

First variation of Perϕ

The first variation is easy to derive for smooth sets.

Let for instance E be a bounded C 2 set and ϕ be a C 2 anisotropy. LetΦt be smooth diffeomorphisms with

limt→0

Φt(x)− Φ0(x)

t= ψ(x), ψ ∈ C∞c (Ω;Rd).

We set Et = Φt(E ) and compute limt→0(Perϕ(Et)− Perϕ(E ))/t.


Let u be a C 2 function such that E = u ≤ 0 and the level 0 isnoncritical (for instance, introducing

dE (x) = dist (x ,E )− dist (x ,E c)

the signed distance function to ∂E , u can be defined as dE near theboundary (or more precisely a smooth truncation of dE ).

Then νE = ∇u/|∇u| on ∂E . Observe in addition thatφ(νE ) = ∇φ(νE ) · νE (Euler’s identity). Define z(x) = ∇ϕ(∇u/|∇u|),which is a C 1 vector field with ϕ(z(x)) ≤ 1 everywhere.


Then one has for t small

Perϕ(E ) =

ˆ∂E

z · νEdHd−1, Perϕ(Et) ≥ˆ∂Et

z · νEtdHd−1.

Hence

Perϕ(Et)− Perϕ(E )

t≥ −1

t

ˆz · (DχEt − DχE ) =

ˆdiv z

χEt − χE

t

so that

lim inft→0


t≥ˆ∂E

div z ψ · νE dHd−1

[Remark: as measures, we have the convergence

(χEt − χE )/t = (χE Φ−1t − χE )/t

∗ −ψ · DχE = ψ · νEHd−1 ∂∗E ]


The reverse inequality is almost as easy: in fact, one can build ut → u inC 2 such that Et = ut ≤ 0 (for instance with the same smoothtruncation of dEt ). Then, switching the role of Et and E we arrive to

Perϕ(E )− Perϕ(Et)

t≥ −1

t

ˆzt · (DχE − DχEt ) = −

ˆdiv zt

χEt − χE

t

where zt = ∇ϕ(∇ut)→ ∇ϕ(∇u) = z as t → 0 (uniformly), so thateventually,

lim supt→0


t≤ˆ∂E

div z ψ · νE dHd−1

First variation of Perϕ: anisotropic curvature

Introducingκϕ = div∇ϕ(∇u).

(where E = u ≤ 0, u quite arbitrary otherwise: in fact as usual theabove expression only depends on the normal vector to ∂E ) the aboveexpression for the first variation is

ˆ∂E

κϕ(x)ψ(x) · νE (x)dHd−1.

[Remark: see Bellettini-Paolini, Hokkaido Mathematical Journal Vol. 25(1996) p. 537–566 for more complete derivations and definitions.]

Comparison

We observe that if E ⊂ F , x ∈ ∂E ∩ ∂F are smooth enough, thenκϕ(x ,F ) ≤ κϕ(x ,E ) (ellipticity, easy to show in the smooth case, mustbe interpreted correctly otherwise).

Anisotropic and Crystalline mean curvature flow

An anisotropic mean curvature flow is a tube of sets E (t) with boundaryevolving with a speed proportional to −κϕ (or − a nondecreasingfunction of κϕ). In particular, it enjoys a comparison principle.

The techniques we describe in these lectures deal with the more specificcase of a motion driven by

V = −ψ(νE )(κϕ(x) + g(x , t))

with: ψ convex, one-homogeneous ([inverse] “mobility”), and g Lipschitzin space, bounded (forcing).


Anisotropic mean curvature flows evolutions are well defined

I if ϕ is smooth enough (and ψ = | · |, Almgren-Taylor-Wang, 1993);

I in the level set sense: Chen-Giga-Goto (1991) show existence for

ut + H(x ,Du,D2u) = 0

in the viscosity sense, where H is “geometric” (meaning that alllevel sets evolve independently with same law); for anisotropicmotion a typical H has the form

H(x ,Du,D2u) = ψ(Du)(D2ϕ(Du) : D2u + g).


The level set sense and viscosity solutions (Evans-Spruck,Chen-Giga-Goto, Giga-Goto-Ishii-Sato 92 for unbounded evolutions...):

I Definition is classical, by contacts with smooth functions which mustsatisfy a differential inequality;

I This makes proofs of existence easy, by Perron’s method;

I Uniqueness is more difficult.

Crystalline curvature flow

What if ϕ is not smooth?

The crystalline case is the case where ϕ ≤ 1 is a polytope, that is, ϕ isthe surface tension of a crystal. In this case, ∇ϕ(ν) is a priori not welldefined. We should use a selection of the subgradient ∂ϕ(ν):“κϕ ∈ div ∂ϕ(∇u)”.

I loss of ellipticity / solutions are not expected to be regular in theclassical sense [not a problem for viscosity solutions];

I infinite diffusion / the motion should be non-local.


In the purely crystalline case, the Wulff shape is a polytope and oneknows how to define mean curvature flows

I in 2D if the initial set is a polygon with facets parallel to the faces ofW (system of ODEs, cf Almgren-Taylor 95, Giga-Gurtin 96...);

I in 2D for a short time if the initial set has a “interior/exterior Wulffshape conditions” (C.-Novaga, 2012/15);

I in any dimension if the initial set is convex(Bellettini-Caselles-C-Novaga, 2006);


I in 2D in the “viscosity sense” adapted to crystalline motions(Giga-Giga, 2001)

I in 3D and more, viscosity: Giga-Pozar 2016 (preprint 2014) and2018 (preprint 2017).

I here, a different approach is introduced, based on a distributionalformulation.

An important advantage of the viscosity approach is that it solves anyequation of the form VN = −F (ν, κϕ) (with F nondecreasing wr κ).However, up to now, it needs the the anisotropy to be be purelycrystalline.

Here, the form VN = −ψ(ν)(κϕ + g(x , t)) is considered, yet for any ϕ.

Crystalline curvature flow in 2 dimensions

We give an insight of how a flow can be defined in 2D. We consider thecase where ϕ is purely crystalline, and a set E whose boundaries areparallel to the facets of the Wulff shape.



Remark: in case the boundaries are not parallel to the facets of the Wulffshape (and E is not necessarily faceted) one could have points withinfinite curvature and large parts of the boundary with zero curvature.


Remark: in case the boundaries are not parallel to the facets of the Wulffshape (and E is not necessarily faceted) one could have points withinfinite curvature and large parts of the boundary with zero curvature.

An example1

Crystallisation of Karl Marx.

1Computed using a diffusion/thresholding scheme

A tool to build variational evolutions

Almgren-Taylor-Wang (1993) [Luckhaus-Sturzenhecker 1993] propose an“algorithm” to build (variational) curvature flows.

Idea: one starts from E 0. We pick a time step h > 0, and for E0

(temporarily a compact set), we define E n+1 from E n, n ≥ 0 by solving

minE

Perϕ(E ) +1

h

Ê4E n

dist (x , ∂E )dx .

They then let Eh(t) = E bt/hc, which is piecewise constant in time.

The Euler-Lagrange equation is (remember dE n is the signed distancefunction to ∂E n):

dE n(x) = −hκϕ(x ,E n+1)

which corresponds to an implicit time discretization of the flow. Itremains to pass to the limit h→ 0...

An equivalent problem

It is shown that this problem can be solved equivalently by solving

minu

ˆϕ(Du) +

ˆ(u − dE n)2

2hdx

and letting then E n+1 = u ≤ 0.More generally (so that in particular one does not need to assume thatE 0 is compact), one should consider the Euler-Lagrange equation of thisproblem in Rd . It reads

− h div z + u = dE n ,

z ∈ ∂ϕ(∇u) a.e.

Then one lets E n+1 = u ≤ 0.

Remark: total variation minimization in Rd

We consider the equation, for f ∈ L∞loc(Rd) (Lp, p > d would work).

− h div z + u = f ,

z ∈ ∂ϕ(∇u) a.e.

The second line should in fact be written ϕ(z) ≤ 1, z · Du = ϕ(Du) asmeasures. However we will usually consider Lipschitz RHS terms f , inwhich case the solution u is also Lipschitz. We claim this equation has aunique solution in BVloc ∩ Lploc for p > d .


Indeed, assume un is any solution of the equation (for h = 1) in Bn,n ≥ 1: then in Bn ∩ Bm = Bminm,n,

−div (zn − zm) + un − um = 0

Multiply this by ηp(x)Ψ(un − um) with Ψ(t) = [t+]p−1, where η hassupport in Bminn,m, we obtain

ˆ(zn − zm) · [Ψ′(un − um)(Dun − Dum)ηp(x) + pΨ(un − um)ηp−1∇η]

+

ˆ(un − um)[ηpΨ(un − um)]dx = 0.

The first part is nonnegative. Hence:


Hence:

ˆ[η(un − um)+]p ≤ p

ˆ((zn − zm) · ∇η)[(un − um)+]p−1ηp−1

≤ Cp‖∇η‖Lp‖(un − um)+η‖1−1/pLp , (1)

where C = diamWϕ, and it follows

‖η(un − um)+‖Lp ≤ Cp‖∇η‖Lp .

Consider now 2R ≤ minm, n, η = (2− |x/R|)+ ∧ 1, this implies

‖(un − um)+‖Lp(BR ) ≤ C ′pRd/p−1.

We conclude that (un)n is a Cauchy sequence in any ball if p > d .

Remark: we also have proved that the solution u which is built in thisway enjoys a comparison principle.

An explicit solution

Consider now the function f = ϕ. If ϕ = ϕ = | · | (Euclidean case),then the solution is explicit (and radial). One solves

−h(z ′(r) +d − 1

rz) + u = r

where z = 1 whenever u is increasing, and |z | ≤ 1 whenever u isconstant. An explicit solution is easily found and is given by

u(r) = r + hd − 1

r, z = 1

for r ≥ rh =√h(d + 1),

u(r) =2d√h√

d + 1, z = 1−

(r

h√d + 1

− 1

)2

.


It turns out that it is easy to show that for a general ϕ,ϕ, the solution of

−hdiv z + u = ϕ, z ∈ ∂ϕ(∇u)

has exactly the same form (now with r = ϕ(x)).

In particular, if one replaces in the scheme the signed distance function

dE (x) = dist (x ,E )− dist (x ,E c)

with the ϕ-signed distance function

dϕ

E (x) := infy∈E

ϕ(x − y)− infy 6∈E

ϕ(x − y)

one finds that a Wulff shape decreases self-similarly with the scheme.


Indeed, if E n = ϕ ≤ R = RWϕ, then E n+1 = u ≤ 0,for u which solves the problem with f (x) = dϕ

E 0 = ϕ(x)− R, so thatE n+1 = Th(R)Wϕ with

Th(R)− hd − 1

Th(R)− R = 0 ⇔ Th(R)− R

h= − d − 1

Th(R).

This is an implicit Euler scheme for R = (d − 1)/R, whose solution isR(t) =

√R2

0 − 2(d − 1)t for t ≤ R20/(2(d − 1)) and 0 for larger t.


So clearly, in the limit h→ 0, we find an evolving Wulff shapeE (t) = R(t)Wϕ, precisely with this law

R(t) =√R2

0 − 2(d − 1)t, t ≤ t∗(R) =R2

0

2(d − 1).

This corresponds to a motion

VN = −ϕ(ν)κϕ (MCF )

which is usually referred to as the motion with “natural” mobility ϕ(ν)(it is natural in the mathematical sense). We will focus mostly on thisparticular case and explain in the end how to consider other mobilities.

(Change of mobility)

Remark: Replacing dE with the ψ-distance function in the scheme:

dψ

E (x) := infy∈E

ψ(x − y)− infy 6∈E

ψ(x − y)

means that at each step, we solve the equation

dψ

En (x) = −hκϕ(x ,E n+1).

If y ∈ ∂E n is a point which realises dψ

En (x) = ±ψ(x − y), one has that∇ψ(x − y) is a normal to ∂E n. Denote ν = ∇ψ(x − y)/|∇ψ(x − y)|, then

ψ(ν) =1

|∇ψ(x − y)| , ±dψ

En (x) = ψ(x−y) = (x−y)·∇ψ(x−y) =(x − y) · νψ(ν)

Hence, along the normal vector ν, the displacement is

VN = −hψ(νEn )κϕ(x ,E n+1).

Strong evolutions in 2D

(With M. Novaga, Pisa. The original result is in fact with a possiblydiscontinuous forcing term, we state it in a simpler form to give the mainideas of the construction.)

Theorem Let E0 ⊂ R2 be a set with compact boundary satisfying bothan inner and outer RWϕ-condition. Then, there exists T > 0 (T ∼ R2)and a flow E (t) for 0 ≤ t ≤ T which solves (MCF ) with E (0) = E0.

More precisely, there is R ′ > 0, a neighborhood U of⋃

0≤t≤T such thatE (t) satisfies the inner and outer R ′Wϕ-condition and∣∣∣∣∣∂dϕ

E

dt− div z

∣∣∣∣∣ = O(dϕ

E ),

where z ∈ ∂ϕ(∇dϕ

E ).

The meaning of an inner/outer RWϕ-condition is obvious in case theanisotropy is regular, a bit tricky in the crystalline case.


The idea (especially in this simplified form) is very simple.

Consider E a set with a rWϕ-condition, and denote ThE the evolution bythe ATW scheme with step h > 0. Assume first the anisotropy is smooth,then each point x ∈ ∂E is tangent to a set ϕ(· − x) ≤ r and both setshave a common normal ∇ϕ(x − x). One has∇ϕ(∇ϕ(x − x)) = (x − x)/ϕ(x − x) = (x − x)/r , in other words

x = x ± r∇ϕ(νE ).


As we have seen from the explicit solution, the set ϕ(· − x) ≤ r isevolved by the scheme as ϕ(· − x) ≤ rh with

rh = r − h1

rh≈ r − h

1

r

for h small enough. All in all, we deduce by comparison that

∂(ThE ) ⊂

dist ϕ(x , ∂E ) ≤ h

1

rh

But from the equation

−h div z + u = dϕ

E

and using ∂ThE = ∂u ≤ 0, we obtain that |div z | ≤ (d − 1)/rh: hencethe mean curvature of the new set is bounded by the curvature of rhWϕ.


In 2D, it turns out that this is enough to show that the new set has arhWϕ-condition from both sides. Iterating this, we get the theorem.

Actually, we get it only in the smooth case, however, the scheme of theproofs is made so that no estimate depends on the smoothness of ϕ,ϕ

itself, so it easily passes to the crystalline limit.

In the convex case, similar techniques have been used to show existenceand uniqueness of a flow in any dimension (Caselles-C-Bellettini-Novaga,2006)

Weak evolutions in any dimension

We now describe a recent approach to defining weak crystalline curvatureflows in any dimension. This is a joint work with Massimiliano Morini(Parma) and Marcello Ponsiglione (Roma I), then extended to moregeneral situations with M. Novaga (Pisa).Let us start, once more, from the ATW construction. Given E0

2, h > 0we define Eh as before.

2not necessarily bounded or with bounded boundary.

Weak evolutions in any dimension

We observe, then, than up to a subsequence,⋃

t≥0 Ehk (t)× tconverges in the Kuratowski sense, in Rd × R+ to some space-timeclosed “tube” E ⊂ Rd × R+.

That is, if xk ∈ Ehk (tk), any limit point of (xk , tk) is in E , and converselyfor any (x , t) ∈ E , there is (xk , tk) with xk ∈ Ehk (tk) converging to (x , t),or, equivalently, the distance to Ehk converges to the distance to E ,locally uniformly.

Let us show this is also true, in general, for the slices at time t.

Estimate for the distance

Assume x 6∈ Eh(t). Letting R = dist ϕ(x ,Eh(t)), we have that

ϕ(· − x) < R ∩ Eh(t) = ∅.

As before, we deduce

ϕ(· − x) < Rh ∩ Eh(t + h) = ∅,

for Rh = R − h(d − 1)/Rh. Hence, dist ϕ(x ,Eh(t + h)) ≥ Rh.

Iterating, we find that for s ≥ t.

dist ϕ(x ,Eh(t + s))2 ' dist ϕ

(x ,Eh(t))2 − 2(d − 1)s

Estimate for the distance

This means that dh(t, x) := dist ϕ(x ,Eh(t)) is “mostly” nondecreasing

in time. On the other hand, for fixed t, it is a compact family (uniformlyequicontinuous).Hence we can use Helly’s selection theorem and find an at mostcountable set N ⊂ (0,+∞) and a subsequence (still denoted hk) suchthat for all t 6∈ N ,

dhk (t)→ d(t) = dist ϕ(x ,E (t))

locally uniformly, or, in other words, Ehk (t)→ E (t) in the Kuratowskisense.

An equation for the limiting motion

Now, we return to the definition of our scheme. The set Eh(t + h) isu ≤ 0 where u solves

−h div z + u = dϕ

Eh(t), z ∈ ∂ϕ(∇u).

Remark that clearly (by its definition) for any x , x ′

dϕ

E (x)− dϕ

E (x ′) ≤ ϕ(x − x ′)

so that uy := u(· − y)− ϕ(y) is a subsolution: by translationalinvariance, letting also zy := z(· − y), we have

−h div zy + uy = dϕ

Eh(t)(· − y)− ϕ(y) ≤ dϕ

Eh(t).

By comparison we deduce that also

u(x)− u(x ′) ≤ ϕ(x − x ′)


For x ′ ∈ ∂Eh(t + h) = ∂u ≤ 0, it means

u(x) ≤ ϕ(x − x ′)

and minimizing with respect to x ′ we deduce

u ≤ dist ϕ(x ,Eh(t + h)) = dh(t + h).


Hence the equation yields, when dh(t) ≥ 0,

dh(t + h)− dh(t)

h− div z ≥

u − dϕ

Eh(t)

h− div z = 0.

Defining zh in an obvious way, and assuming (zhk )k converges (weak-∗ inL∞(Rd ;Wϕ)) to a limit z , we deduce in the limit that

∂d

∂t≥ div z

in the distributional sense (or as measures) in [Rd × (0,∞)] \ E .

Characterizing the limit z

It is natural to think that the limit z ∈ ∂ϕ(∇d) a.e.This is however not elementary without further information. One knowsthat z ∈Wϕ a.e. by construction (as a limit of zh in this convex set).Hence, one needs to show that z · ∇d = ϕ(∇d) a.e., and, even, as thereverse inequality is clear, that

z · ∇d ≥ ϕ(∇d) a.e. in E c

In the sequel we admit that uh(t) (also defined in an obvious way) hasthe same limit as dh: for t 6∈ N , maxuh(t), 0 → d(t) locally uniformly.


Let η ∈ C∞c (E c ;R+), in particular dhk , uhk > 0 in the support of η for klarge enough. On one hand (using zhk ∈ ∂ϕ(∇uhk ))

ˆ ˆϕ(∇d)ηdxdt ≤ lim inf

k

ˆ ˆϕ(∇uhk )η dxdt

= lim infk

ˆ ˆ(zhk · ∇uhk )η dxdt.

On the other hand, we haveˆ ˆ

(zhk · ∇uhk )η dxdt = −ˆ ˆ

uhk zhk · ∇η dxdt −ˆ ˆ

ηuhk div zhk dxdt

and

−ˆ ˆ

uhk zhk · ∇η dxdt → −ˆ ˆ

d z · ∇η dxdt.


If we can also show that

−ˆ ˆ

ηuhk div zhkdxdt → −ˆ ˆ

η d div z dxdt, (∗)

then we will deduce

ˆ ˆϕ(∇d)η dxdt ≤ −

ˆ ˆd z · ∇η dxdt −

ˆ ˆη d div z dxdt

=

ˆ ˆηz · ∇d dxdt

showing that z · ∇d ≥ ϕ(∇d) a.e., hence z ∈ ∂ϕ(∇d) a.e. in E c .

However, (∗) does not hold without further knowledge. We need anestimate on div zh.

Estimate on div z

We have all the tools to provide a natural estimate on div zh. Observethat

dEh(t)(x) ≤ dist ϕ(x ,Eh(t)) = min

y∈Eh(t)ϕ(x − y)

hence by comparison

uh(t+h, x) ≤ miny∈Eh(t)

ϕ(x − y) + h d−1

ϕ(x−y) if ϕ(x − y) ≥√

h(d + 1),2d√h√

d+1else

and in particular, if dϕ(x ,Eh(t)) > 0 and h is small enough, we obtain

choosing a y ∈ Eh(t) closest to x :

uh(t + h, x) ≤ dist ϕ(x ,Eh(t)) + h

d − 1

dist ϕ(x ,Eh(t)).

Estimate on div z

One obtains:

uh(t + h, x) ≤ dist ϕ(x ,Eh(t)) + h

d − 1

dist ϕ(x ,Eh(t)).

from which it follows, thanks to the equation

−h div zh(t + h) + uh(t + h) = dϕ

Eh(t),

div zh(t + h) ≤ d − 1

dist ϕ(x ,Eh(t)),

at least whenever dist ϕ(x ,Eh(t)) &

√h.

Conclusion

We return to the termˆ ˆ(zhk ·∇uhk )η dxdt =

ˆ ˆ(zhk ·∇(uhk−d))η dxdt+

ˆ ˆ(zhk ·∇d)η dxdt

and observe first thatˆ ˆ(zhk · ∇d)η dxdt →

ˆ ˆ(z · ∇d)η dxdt,

since zhk∗ z in L∞.

For each t, we let mk(t) = minspt η(·,t)(uk − d),Mk(t) = maxspt η(·,t)(uk − d). (Observe that these quantities go to zeroexcept for t ∈ N .)

Conclusion

Then,

ˆ ˆ(zhk · ∇(uhk − d))η dxdt =

ˆ ˆ(zhk · ∇(uhk − d −mk(t)))η dxdt

= −ˆ ˆ

(uhk − d −mk(t))[ηdiv zhk + zhk · ∇η]dxdt

For k large enough, dist ϕ(sptη(·, t),Eh(t)) ≥ δ > 0, hence

div zhk ≤ (d − 1)/δ in the above integrals. Hence

lim infk

ˆ ˆ(zhk · ∇(uhk − d))η dxdt

≥ lim infk−ˆ ˆ

(uhk − d −mk(t))(d − 1)η

δdxdt

+ limk−ˆ ˆ

(uhk − d −mk(t))(zhk · ∇η)dxdt = 0.

Conclusion

In the same way, replacing uhk − d with uhk − d −Mk ≤ 0, we show that

lim supk

ˆ ˆ(zhk · ∇(uhk − d))η dxdt ≤ 0

and it follows that (∗) holds, and that z ∈ ∂ϕ(∇d) a.e. in E c .

We have shown that E satisfies the properties summarized in thefollowing definition of a “supersolution”.[And, symmetrically, A, defined as the complement of a Kuratowski limitof the complements of

⋃t≥0 Ehk (t)× t, is a “subsolution”.]

Super/subsolutions

Definition [CMP17] A (closed) “tube” E ⊆ Rd × [0,+∞) is asupersolution starting from the initial E 0 if

a. E (0) ⊆ E 0;

b. E (t) = ∅ ⇒ E (s) = ∅ if s > t;

c. E is (Kuratowski) left-continuous;

d. For d = dist ϕ(x ,E ), there exists z ∈ ∂ϕ(∇d) with

∂td ≥ div z

in the distributional sense in Rd × (0,T ∗) \ E where T ∗ ≤ +∞ isthe extinction time of E , moreover (for t ≤ T ∗)

(div z)+ ∈ L∞(d > δ)

for any δ > 0

A subsolution is an open tube A such that Ac is a supersolution startingfrom (E 0)c .

Uniqueness?

So what the ATW construction shows so far, is the existence of suchflows, as limits of discrete flows, for any ϕ. The question is obviously theuniqueness.

On the other hand, this definition is far less pleasant than more standardgeneralized solutions such as in the viscosity sense (using discontinuousv.s. or the distance function, cf Barles-Soner-Souganidis (93),Ambrosio-Soner (96)), which ensure that a max of subsolutions remains asolution (here this is a priori quite unclear).

However, if existence is a bit harder, uniqueness should be easier...

Comparison

Theorem Let E be a supersolution with initial datum E 0 and F be asubsolution with initial datum F 0 ⊃ E 0, and assume∆ = dist ϕ

(E 0, (F 0)c) > 0. Then dist ϕ

(E (t),F (t)c) ≥ ∆ for any

t ≥ 0.

The proof is by parabolic comparison. Indeed, if d(x , t) = dist ϕ

(x ,E(t)) and

d ′(x , t) = dist ϕ

(x ,F c(t)) then between E and F

∂td ≥ div ∂ϕ(∇d), ∂td′ ≥ div ∂ϕ(∇d ′),

and one has d + d ′ ≥ ∆ at t = 0. Using a priori estimate on the speed at

which d , d ′ decrease, one can control also d + d ′ on a parabolic boundary of a

small tube, and obtain the comparison inside this tube. As d , d ′ are distance

function it yields global comparison.

Comparison: proof

The proof is easy but a bit too technical to be described in details here.The idea is as follows: we build, in the strip between E and F c , functionsd and d ′ with the following properties:

I near the boundaries ∂E and ∂F c , d + d ′ ≥ ∆ for all time t ≤ t0;

I near the middle of the strip, d = d and d ′ = d ′.

To ensure the first property, we use an estimate on the decay of d , d ′.

Comparison: proof

Lemma Let E be a supersolution and d = dist ϕ(·,E ). Let (x , t) with

d(x , t) = R > 0. Then for s ≤ cR2, d(x , t + s) ≥ R − c ′√s for some

c , c ′ depending only on d .

The proof, which we do not detail, is based on the fact that by definition,d is a supersolution (in E c) of

∂u

∂t= div z , ϕ(z) ≤ 1, z · Du = ϕ(Du)

which is the ϕ-total variation flow. We compare then d with the solutionstarting from

v0 = R − 4

3ϕ(x − x)

which is explicit and decays as in the lemma.

Comparison: proof

Thanks to this lemma, we know how to modify d , d ′ to build d , d ′: Welet, for some ε > 0,

d = maxd , ε+ c ′√t,

d ′ = maxd ′, ε+ c ′√t,

so that at t = 0, d + d ′ ≥ ∆, and near the boundaries, d + d ′ ≥ ∆ + ε;and thanks to the Lemma the latter remains true for some time.

In addition, one always have z ∈ ∂ϕ(∇d), etc, since ∇d ∈ 0,∇d a.e.

Eventually, one still has that ∂t d ≥ div z and ∂t d′ ≥ div z ′.

Because where d 6= d , then ∂t d = c ′/(2√t) which is arbitrarily large for

small time, while, if we are at distance from d = 0, div z is bounded(by assumption).

Comparison: proof

It is now easy to show, by parabolic comparison for sub/super solutionsof the total variation flow, that for some time t0 depending only on ∆,d + d ′ ≥ ∆. But this implies that dist ϕ

(E (t0),F (t0)c) ≥ ∆. It remains

to iterate until extinction time of either F or E , if finite.

Conclusion: uniqueness

Strictly speaking, we cannot assert the uniqueness of the limit startingfrom one set E 0, due to the well-known “fattening phenomenon”.To provide uniqueness, we need to embed an evolution in a continuousfamily of evolutions, and obtain uniqueness for almost all initial data.This is usually done with a level set formulation:

I We consider u0 bounded, uniformly continuous;

I We evolve each level set by the flow;

I A superflow starting from one level set must always stay inside asubflow starting from a larger level set,

I hence the evolving u(t) is unique (and the geometric evolution isunique for all levels but an at most countable number).

ExampleThe following example illustrate a facet-breaking phenomenon observedby Bellettini, Novaga, Paolini (99).[The ATW scheme is solved numerically by a parametric maximal flow

algorithm.]

Evolution with a cubic anisotropy

ExampleThe following example illustrate a facet-breaking phenomenon observedby Bellettini, Novaga, Paolini (99).[The ATW scheme is solved numerically by a parametric maximal flow

algorithm.]

Zoom on the facet breaking at step 1

Extensions

Let ψ be a norm (1-homogeneous, convex, symmetric), g a boundedfunction, Lipschitz in space, and consider now the equation:

VN = ψ(ν)(κϕ + g).

Of particular interest is the case ψ(·) = | · | considered by Gurtin, Taylor,ATW, etc.

Extensions

Definition A (closed) “tube” E ⊆ Rd × [0,+∞) is a supersolutionstarting from the initial E 0 if

a. E (0) ⊆ E 0;

b. E (t) = ∅ ⇒ E (s) = ∅ if s > t;

c. E is (Kuratowski) left-continuous;

d. For d = dist ψ(x ,E ), there exists z ∈ ∂ϕ(∇d) with, for some

M ≥ 0,∂td ≥ div z + g −Md

in the distributional sense in Rd × (0,T ∗) \ E where T ∗ is theextinction time of E , moreover (for t ≤ T ∗)

(div z)+ ∈ L∞(d > δ)

for any δ > 0

A subsolution is an open tube A such that Ac is a supersolution startingfrom (E 0)c .

Extensions

I Uniqueness is as before (same comparison Thm, with nowdist ψ

(E (t),F (t)c) ≥ ∆e−Mt);

I Existence is almost the same with one difficulty.

Compatible anisotropies

Now the equation which is solved at step n to find E n+1 = u ≤ 0 is

−hdiv z + u = dψ

E n , z ∈ ∂ϕ(∇u).

One crucial step which was needed, both in the comparison proof and forpassing to the limit in the ATW scheme, was an estimate of div z . Which

was following from an estimate on (u − dϕ

E n )/h, using barriers for the

level sets of dϕ

with Wulff shapes and explicit solutions.

The question now, is whether this estimate is still valid when dϕ

isreplaced with dψ

. A priori, the level sets of dψ

cannot be “swept” by

Wulff shapes, unless we assume a compatibility condition.

Compatible anisotropies

We say that ψ is “ϕ-regular” if for some ε > 0 and some norm ψ0,ψ = ψ0 + εϕ. Equivalently, Wψ = ψ ≤ 1 = Wψ0 + εWϕ.

Compatible anisotropiesIn that case, one can show that

div z .d − 1

εdist ψ(x ,E n)

and proceed as before.

Incompatible anisotropies

So, what can we do in case ψ,ϕ are incompatible? Typically, if ψ = | · |,ϕ crystalline, as in the original paper of Almgren-Taylor-Wang.The obvious idea is to approximate first ψ by letting ψε = ψ + εϕ, andthen send ε→ 0: does it converge? (Yes.)


In the framework of ATW, we have the following result, which forsimplification we state without forcing term g .

Lemma Let ψ1, ψ2 with ψi ≤ βϕ and

ψ2 ≤ ψ1 ≤ (1 + δ)ψ2

for β > 0, δ > 0 small. Let E ⊂ F with dist ϕ(E ,F c) = ∆ > 0. Let T1,h

be the ATW scheme with mobility ψ1 and T2,h be the ATW scheme withmobility ψ2 and forcing term −c0δ/∆ (where c0 depends only β), thenfor h small enough,

dist ϕ(T k

1,hE , (Tk2,hF )c) ≥ ∆

for all k.

In other words, the change of mobility is compensated by a forcing termproportional to δ/∆.


A corollary is that if uε(t) is the level-set solution starting from aB.U.C. function u0, for the mobility ψε = ψ + εϕ, uε is a Cauchysequence in L∞(Rd) and converges to a limit, which in addition is shownnot to depend from the sequence approximating ψ.

Another (non immediate) corollary is that the ATW scheme, at least foralmost all level sets of u0 as initial data, converges to this unique limit,even if the anisotropies are incompatible.

A simpler construction for this unique crystalline flow can be made byapproximation with viscosity solutions, however this simpler approachdoes not yield any information on the ATW scheme[C-Morini-Novaga-Ponsiglione 2017/18]

Distributional vs Viscosity solutions

Lemma Assume ϕ,ψ, ψ are C 2 and that the forcing g is continuous.Then E is a supersolution/flow in the sense of the definition if and only if−χE is a viscosity supersolution of

ut = ψ(∇u)(div∇ϕ(∇u) + g).

The proof is a bit technical. The requested regularity of ψ,ψ, whichseems not natural, is to ensure that the level sets of dist ψ

to a C 2 set

are also C 2 [it might be just a technical assumption].

A comparison result with different mobilities

In the viscosity setting, we have an equivalent of the Lemma statedabove for the ATW scheme, with a simpler proof.Lemma Assume |ψ1 − ψ2| ≤ δψ2. Then if E is a superflow forVN = −ψ1(ν)κϕ, F a subflow for −ψ2(ν)(κϕ − c0δ/∆) where∆ = dist ϕ

(E (0),F (0)c), then for all t ≥ 0 (until possible extinction of

one set), dist ϕ(E (t),F (t)c) ≥ ∆.

The proof is classical, using a doubling variable method and theJensen-Ishii Lemma. One first regularizes the flows by adding to E andF c (∆/4)Wϕ, so that their curvatures become bounded at points ofminimal distance (by 4(d − 1)/∆). Formally, this allows to bound thespeed difference ψ1(ν)κϕ − ψ2(ν)κϕ.


Corollary Again, consider E0 ⊂ F0 with dist ϕ(E0,F

c0 ) = ∆ > 0, and

E ,F with E (0) = E0, F (0) = F0 and

I −χE is a viscosity supersolution of ut = ψ1(∇u)(div∇ϕ(∇u) + g),

I −χF is a viscosity subsolution of ut = ψ2(∇u)(div∇ϕ(∇u) + g)

Then

dist ϕ(E (t),F (t)c) ≥ ∆e−βLt − δ(2c0/∆ + ||g ||∞)

1− e−βLt

L

where L is the Lipschitz constant of g in the ϕ norm.


The proof is based on a splitting argument: for ε > 0 small one solvesuεt = 2ψ1(∇uε)div∇ϕ(∇uε) 2jε < t ≤ (2j + 1)ε ,

uεt = 2ψ1(∇uε)ffl 2(j+1)ε

2jεg(x , s)ds (2j + 1)ε ≤ t ≤ 2(j + 1)ε.

with uε(0) = −χE0 , andvεt = 2ψ2(∇vε)(div∇ϕ(∇vε)− 2c0

δ∆ ) 2jε < t ≤ (2j + 1)ε ,

vεt = 2ψ2(∇uε)(ffl 2(j+1)ε

2jεg(x , s)ds + 2c0

δ∆

)(2j + 1)ε ≤ t ≤ 2(j + 1)ε.

On [2jε, (2j + 1)ε], one uses the previous comparison result. On[(2j + 1)ε, (2j + 2)ε] (first order motion), one uses classical comparisonresults. Then, one sends ε→ 0 and uses a homogenization result (Barles2006) allows to get the global estimate.

Conclusion

Now, to solve the problem with a general ϕ,ψ, g : one first finds asequence of smooth ϕn, ψn, gn (with smooth ψn) converging to ϕ,ψ, g .One solves the level set equation in the viscosity sense to build anevolution un, and then pass to the limit. Thanks to the corollary, onegets that un is a Cauchy sequence in L∞, hence it converges. One alsocan deduce that the limit does not depend on the approximation.

I When applicable [g = g(t), ϕ purely crystalline] one obtains asolution in the sense of Giga-Pozar (2017).

I When applicable [ϕ,ψ compatible, in which case one should makesure ϕn, ψn are uniformly compatible], one obtains a solution in thedistributional sense;

I In all cases the limit is unique. But what equation is solved?

I Nonlinear extensions?

Thank you for your attention

Conclusion

Now, to solve the problem with a general ϕ,ψ, g : one first finds asequence of smooth ϕn, ψn, gn (with smooth ψn) converging to ϕ,ψ, g .One solves the level set equation in the viscosity sense to build anevolution un, and then pass to the limit. Thanks to the corollary, onegets that un is a Cauchy sequence in L∞, hence it converges. One alsocan deduce that the limit does not depend on the approximation.

I When applicable [g = g(t), ϕ purely crystalline] one obtains asolution in the sense of Giga-Pozar (2017).

I When applicable [ϕ,ψ compatible, in which case one should makesure ϕn, ψn are uniformly compatible], one obtains a solution in thedistributional sense;

I In all cases the limit is unique. But what equation is solved?

I Nonlinear extensions?

Thank you for your attention

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