anisotropic surface measures as limits of volume fractions€¦ · anisotropic surface measures as...

$: Anisotropic surface measures as limits of volume fractions€¦ · Anisotropic surface measures as limits of volume fractions Giovanni E. Comi (SNS) work in collaboration with L$
Anisotropic surface measures aslimits of volume fractions

Giovanni E. Comi (SNS)

work in collaboration with

L. Ambrosio

XXVII Convegno Nazionale di Calcolo delle Variazioni

Levico Terme

Febraury, 6-10, 2017

L. Ambrosio, G. E. Comi, Anisotropic surface measures as limits of volumefractions, preprint.

G. E. Comi (SNS) Limits of volume fractions Febraury, 6-10, 2017 1 / 21

Plan

1 Motivations and main result

2 Proof of the theorem

3 Variants and examples


Ambrosio, Bourgain, Brezis and Figalli’s paper

Ambrosio, Bourgain, Brezis and Figalli recently studied a newcharacterization of the perimeter of a set in Rn by considering thefollowing functionals originating from a BMO-type seminorm

Iε(f ) = εn−1 supGε

∑Q′∈Gε

−∫

Q′|f (x)− −

∫Q′

f | dx , (1)

where Gε is any disjoint collection of ε-cubes Q′ with arbitraryorientation and cardinality not exceeding ε1−n.In particular, they studied the case f = 1A; that is, the characteristicfunction of a measurable set A, and proved that

limε→0

Iε(1A) = 12 min1,P(A). (2)


Ambrosio, Bourgain, Brezis and Figalli’s paper

In particular, removing the cardinality bound, the scaling invarianceeasily implies that

limε→0

Iε(1A) =

12P(A) if 1A ∈ BV+∞ if othewise

.


New approach

Our research consisted in looking for general covering sets.Let C be a bounded connected open set with Lipschitz boundary,0 ∈ C . We define

HCε (A) := εn−1 sup

Hε

∑C ′∈Hε

−∫

C ′|1A(x)− −

∫C ′

1A| dx (3)

= εn−1 supHε

∑C ′∈Hε

2 |C′ ∩ A||C ′ \ A||C ′|2 ,

where Hε is any disjoint family of translations C ′ of the set εC withno bounds on cardinality.


Main result

Theorem

There exists ϕC : Sn−1 → (0,+∞), bounded and lowersemicontinuous, such that, for any set of finite perimeter A, one has

limε→0

HCε (A) =

∫FA

ϕC (νA(x)) dH n−1(x), (4)

where FA and νA are respectively the reduced boundary of A and themeasure theoretic unit interior normal to FA. Moreover, if A ismeasurable and P(A) =∞, one has

limε→0

HCε (A) = +∞. (5)


The case A is not a set of finite perimeter

There exists a constant k(C , n) such that

HBε (A) ≤ k(C , n)HC

ε (A) (6)

where B = B(0, 1) is the unit ball. In addition, there exists a constantcn such that

HBε (A) ≥ cnIε(1A) (7)

and we recall that, if P(A) =∞,

limε→0

Iε(1A) = +∞. (8)

Hence, the result follows from (6), (7) and (8).


Properties of H±We fix a set A of finite perimeter and we localize Hε to an open set Ω,in order to define the following increasing set functions on open sets

HC+(A,Ω) := lim sup

ε→0HCε (A,Ω), (9)

HC−(A,Ω) := lim inf

ε→0HCε (A,Ω). (10)

Such functions are clearly translation invariant: for any τ ∈ Rn,H±(A + τ,Ω + τ) = H±(A,Ω).Since C is an open bounded connected set with Lipschitz boundary,we have the following relative isoperimetric inequality: there exists aconstant γ = γ(C) such that

|C ∩ A||C \ A||C |2 ≤ γP(A,C). (11)

By scaling, we get an upper bound for H+:H+(A,Ω) ≤ 2γP(A,Ω). (12)


Properties of H±

1 Homogeneity: for any t > 0, Htε(tA, tΩ) = tn−1Hε(A,Ω) andH±(tA, tΩ) = tn−1H±(A,Ω).

2 Superadditivity of H−: Hε(A,Ω1 ∪ Ω2) = Hε(A,Ω1) + Hε(A,Ω2) wheneverΩ1 ∩ Ω2 = ∅ implies

H−(A,Ω1 ∪ Ω2) ≥ H−(A,Ω1) + H−(A,Ω2). (13)

3 Almost subadditivity of H+ (WLOG diam(C) = 1):

Hε(A,Ω1 ∪ Ω2) ≤ Hε(A, Iε(Ω1)) + Hε(A, Iε(Ω2)), (14)

for any open set Ω1,Ω2, where It(Ω) := x ∈ Rn : dist(x ,Ω) < t. Hence,for any open sets Wi ⊃ Iδ(Ωi ), i = 1, 2, for some δ > 0, we get

H+(A,Ω1 ∪ Ω2) ≤ H+(A,W1) + H+(A,W2). (15)


Definition of ϕ±

We set

ϕ+(ν) := H+(Sν ,Qν),ϕ−(ν) := H−(Sν ,Qν),

where ν ∈ Sn−1, Sν := x ∈ Rn : x · ν ≥ 0 and Qν a unit cubecentered in the origin having one face orthogonal to ν and bisected bythe hyperplane ∂Sν .


First properties of ϕ±

Using the relative isoperimetric inequality, it is easy to find the upperbound ϕ+ ≤ 2γ. On the other hand, using comparison arguments, itis also possible to show that ϕ− ≥ c(n,C) > 0.In addition, using the homogeneity and the additivity of Hε and ascaling argument with cubes it is possible to show that

ϕ−(ν) = supε>0

Hε(Sν ,Qν).

From this it follows easily that ϕ− is lower semicontinuous and that

ϕ−(ν) ≤ ϕ+(ν) = lim supε→0

Hε(Sν ,Qν) ≤ supε>0

Hε(Sν ,Qν) = ϕ−(ν).

Hence, ϕ− = ϕ+ and we can define

ϕ(ν) := limε→0

Hε(Sν ,Qν).


Derivation theorems

TheoremLet E be a set of finite perimeter and νE be its measure theoreticinterior normal. Then, for H n−1-a.e. x ∈ FE , we have

D−P H−(x) := lim infr→0

H−(E ,QνE (x)(x , r))P(E ,QνE (x)(x , r)) ≥ ϕ(νE (x)), (16)

D+P H+(x) := lim sup

r→0

H+(E ,QνE (x)(x , r))P(E ,QνE (x)(x , r)) ≤ ϕ(νE (x)). (17)

In particular, it follows that

D−P H−(x) = D+P H+(x) = ϕ(νE (x)) for H n−1-a.e. x ∈ FE .


Inner regular envelope

To proceed, we apply an argument similar to the classical densitytheorems for measures to the nondecreasing set functions H±, forwhich we need Vitali covering theorem for cubes and properties whichreplace the additivity.Indeed, H− is superadditive, but H+(E , ·) is not a subadditive setfunction on the family of open sets, so we consider

H∗+(E ,Ω) := supH+(E ,Ω′) : Ω′ b Ω,

the inner regular envelope, which is actually σ-subadditive.


Density theorems

TheoremFor any Borel set B ⊂ FE and t > 0, we have that

lim infr→0

H−(E ,QνE (x)(x , r))P(E ,QνE (x)(x , r)) ≥ t (18)

for all x ∈ B implies H−(E ,U) ≥ tH n−1(B) for any open set U ⊃ B.On the other hand, we have that

lim supr→0

H+(E ,QνE (x)(x , r))P(E ,QνE (x)(x , r)) ≤ t (19)

for all x ∈ B implies H∗+(E ,U) ≤ tP(E ,U) + 2γP(E ,U \ B) for anyopen set U ⊃ B.


Proof of the main theorem in the rectifiablecase

We use the previous results to adapt the classical proofs of thedifferentiation theorem for Radon measures to the nondecreasing setfunctions H±(E , ·).The key idea is to partition FE in a suitable way and then use thedensity theorems. The superadditivity of H−(E , ·) ensures a lowerestimate and the σ-superadditivity of H∗+(E , ·) an upper estimate, sothat we obtain∫

FEϕ(νE ) dH n−1 ≤ H−(E ,Rn) ≤ H∗+(E ,Rn) ≤

∫FE

ϕ(νE ) dH n−1.

It is easy to show that H∗+(E ,Rn) = H+(E ,Rn), and so we conclude.


The case C is the unit ball

Let C to be the unit ball B(0, 1), then

ϕ ≡ ξn,

a constant depending only on the space dimension. Indeed, thefunctionals Hε and H± are rotationally invariant. We obtain an upperestimate using the relative isoperimetric inequality in the ball withsharp constant. For the lower estimate, we choose a covering ofdisjoint ε-balls which can stay inside Qν and are bisected by ∂Sν .Thus, this problem is related to the Kepler’s problem, which consistsin looking for the optimal fraction ρn ∈ (0, 1] of the volume of then-dimensional unit cube covered by finite unions of disjoint balls withthe same radius ε, as ε→ 0. Hence, we get

ρn−1

2ωn−1≤ ξn ≤

12ωn−1

. (20)


The isotropic case

If we redefine Hε in an isotropic way; that is, allowing for anyorientation of the sets C ′ in the covering, we clearly get the rotationalinvariance for the modified functionals H iso

ε and so the associatedfunction ϕiso is a constant ξ(C). In these cases the main result forsets of finite perimeter follows directly from the density theorems withB = FE .In particular, if C = Q and we allow for arbitrary orientation, we have

ϕQ ≡ 12 ,

which implies the result of Ambrosio, Brezis, Bourgain and Figalli.


A 2-dimensional anisotropic case

Since the anisotropic perimeter∫FA ϕ(νA) dH n−1 is lower

semicontinuous w.r.t. the convergence in measure if and only if ϕ isthe restriction to the unit sphere of a positively 1-homogeneous andconvex function, it would be desirable to find conditions on C (weakerthan the isotropy of the ball) for which this happens. The problem isnontrivial since even the unit square in R2 gives rise to a function ϕnot satisfying this condition.Indeed, if C is the unit cube Q = (0, 1)2, then

ϕQ(ν) =

23

√23 |ν1||ν2| if 27

32 |ν2| ≤ |ν1| ≤ 3227 |ν2|

‖ν‖∞2 if |ν1| ≤ 27

32 |ν2| or |ν1| ≥ 3227 |ν2|

.


A 2-dimensional anisotropic case

The function Φ(x) := |x |ϕQ(

x|x |

)is not convex, indeed, the

upper-right corner to the set Φ(x) ≤ 2 is

(courtesy of A. Nikishova)


A variant

One may define a family of functionals similar to Hε allowing fordifferent dilations of the set C under a fixed level ε > 0:

Hε(A,Ω) := supGε

∑C ′∈Gε

2(ε(C ′))n−1 |C ′ ∩ A||C ′ \ A||C ′|2 , (21)

where C ′ = ε(C ′)(C + a), for some translation vector a, and Gε is adisjoint family inside Ω of translations of the set ηC , for anyη ∈ (0, ε]. Since these functionals satisfy the same properties of Hε

and H±, we can define the functions ϕ±(ν) := H±(Sν ,Qν) and showan analogous version of the main theorem for them.If we take C to be the unit ball, then ϕ is a constant, since Hε isrotation invariant, and

ϕ ≡ 12ωn−1

,

since arbitrarily small radii are allowed.G. E. Comi (SNS) Limits of volume fractions Febraury, 6-10, 2017 20 / 21

Thank you for your attention!


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