MINIMAL SEGMENTATIONS FORTHE MUMFORD-SHAH FUNCTIONAL

Guy David, Universite de Paris-Sud (Orsay)a l’IHP, Janvier 08

Main theme: Regularity properties of the singularset for minimizers, and a few techniques of analysis orelementary geometric measure theory to get them.

1. The Mumford-Shah functional

We are given a simple domain Ω ⊂ Rn, a boundedfunction g ∈ L∞(Ω), and we set

(1) Jg(u,K) = Hn−1(K)+∫

Ω\K|∇u|2+

∫Ω\K|u−g|2

for (u,K) ∈ A, the set of acceptable pairs (u,K) suchthat K ⊂ Ω is closed in Ω, and u ∈ W 1,2(Ω \K) hasone derivative in L2 on Ω \K.

Here Hn−1(K), the Hausdorff measure, is the cor-rect analogue of (n − 1)-dimensional surface measureof K, defined as soon as K ⊂ Rn is Borel-measurable.

Introduced by Mumford and Shah (≤ 1989), atleast in dimension n = 2, for image segmentation. Wasalso considered as a tool for modelling cracks whenn = 3.

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(1) Jg(u,K) = Hn−1(K)+∫

Ω\K|∇u|2+

∫Ω\K|u−g|2

For image segmentation, Ω is a screen, g is a givenimage, and u defines a segmentation for g. If (u,K)minimizes J , u should give a good compromise be-tween three constraints:- u− g should be small- u is simple (varies slowly), but may have jumps alonga singular set K (which we see as describing edges inthe picture), but- K is not too complicated.

Comments:- Segmentation 6= compression: it is also fine if u andK only give some simplified idea of g.- We could give different weights to the three terms,but the difference can be scaled out by multiplying uand g by a constant, and composing with a dilation- Lots of variants exist, but often with a term likeHn−1(K).- This works fine because, as conjectured by Mumfordand Shah, K is automatically regular (instead of justbeing short) when (u,K) is a minimal pair.- Good and bad thing: automatic and context free!

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(1) Jg(u,K) = Hn−1(K)+∫

Ω\K|∇u|2+

∫Ω\K|u−g|2

Empirical considerations

Assuming that minimizers exist, what can we expect?

Making K larger allows more jumps for u, hence helpsreduce the tension and make

∫Ω\K |∇u|

2 smaller. Butthis is only efficient if K has good local separationproperties, which:- forces Hn−1(K ∩B(x, r)) to be of the order of rn−1

(more would be inefficient, less would allow too muchpassage; see later)- gives the homogeneity of the problem: when there is areal competition between the first two terms in B(x, r),r small, we expect both terms to give contributions ofroughly rn−1 in B(x, r)- shows that the third term, which contributes lessthan Crn in B(x, r), plays a minor role locally.

In addition, the expected separation properties ofK are a main reason why K is regular.

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The Mumford-Shah conjecture

Observe that if g and K are given, it is easy to

minimize∫

Ω\K|∇u|2+

∫Ω\K|u−g|2 with respect to u.

This is a convex problem, there is a unique solution u,and u is better than C1 away from K (elliptic equation∆u = u− g). So the question is K (and u near K).

First, Mumford and Shah conjectured the exis-tence of minimal pairs. True (see page 6).

But the (main) Mumford-Shah conjecture is thefollowing: if (u,K) is a reduced minimizer for Jg indimension 2, then K is a finite union of C1 curves,which can only meet by sets of 3 and with 120 angles∗.

- See the next page for “reduced”.- C1 implies more when g is regular. Up to analytic[Koch-Leoni-Morini].- If Ω is nice, regularity near ∂Ω is known [Maddalena-Solimini ; D.-Leger]. Otherwise, make the conclusionlocal in Ω.- K is allowed to have tips (but incidentally, we stilldo not know whether they can really occur).- Hence (u,K) looks like a decent segmentation, exceptthat T -junctions are destroyed at small scales∗.- In dimension 3, some regularity is known, but we donot know a precise conjectural list of local behaviours.

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Definition. The pair (u,K) is called reduced when,if K ⊂ K is a proper closed subset of K, the functionu has no extension u ∈W 1,2(Ω \ K).

Given (u,K) ∈ A, say with Hn−1(K) < +∞ (andhence |K| = 0), we can always find K ′ ⊂ K such thatu ∈W 1,2(Ω \K ′) and (u,K ′) is reduced.

Then (u,K ′) is equivalent to (or even better than)(u,K) for Jg. So it is enough to consider reduced pairs.We shall do that.

This allows a better description of K: we avoidproblems that come from adding to K an ugly set ofvanishing Hn−1-measure.

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Existence of minimal segmentations isa theorem of Ambrosio and De Giorgi-Carriero-Leaci.

The “stupid way” (taking a minimizing sequence(uj ,Kj) and letting Kj tend to a limit K through asubsequence) does not work trivially, becauseHn−1(K)could be much larger than the limit of the Hn−1(Kj).Think about dotted lines∗.

The proof uses a weak formulation of Jg in thesubclass SBV ⊂ BV of functions of bounded varia-tion, where u ∈ SBV , and K = Ku is now the singularset of u (not necessarily closed).

It uses two facts: the nice compactness propertiesof BV extend to SBV ; and minimizers for the SBVanalogue of Jg are so regular that Hn−1(Ku\Ku) = 0,so they also provide minimal pairs for Jg.

There is also a direct proof, based on the concen-tration property of Dal Maso, Morel, and Solimini. ByDMS when n = 2, Maddalna and Solimini for n large.[Maybe two words about it later.]

No uniqueness in general because there may bea brutal change of strategy (a circle vanishes∗) or byrupture of symmetry (checkerboard∗).Even less continuous dependence on parameters.But could it be that uniqueness is generic (in g)?

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Local almost-minimizers

Easy to check: if (u,K) is a reduced minimizer for Jh,then it is a (reduced) almost minimizer, with gaugefunction h(r) = C||g||2∞r.

Definition. A (local) almost minimizer with gaugefunction h is a pair (u,K) ∈ A such that, whenever(u, K) ∈ A coincides with (u,K) out of some ball B =B(x, r) (such that B ⊂ Ω),

Hn−1(K ∩B) +∫

Ω∩B\K|∇u|2

≤ Hn−1(K ∩B) +∫

Ω∩B\K|∇u|2 + h(r)rn−1.

Proof: By a cut-off argument, ||u||∞ ≤ ||g||∞ and wecan assume that ||u||∞ ≤ ||g||∞, so

LHS ≤ Jg(u,K) ≤ Jg(u, K)

= Hn−1(K ∩B) +∫

Ω∩B\K|∇u|2 +

∫Ω∩B\K

|u− g|2

≤ RHS.

Comments: other definitions exist; nice way to saythat in Jg the third term matters less at small scales.

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2. Regularity properties of K

From now on, (u,K) is a reduced (local) almostminimizer, with gauge function h (nondecreasing, andsuch that limr→0 h(r) = 0).

We shall mostly worry about local properties (farfrom ∂Ω), and often in dimension 2.

We start with the trivial estimate:

(2) Hn−1(K ∩B(x, r)) +∫

Ω∩B(x,r)\K|∇u|2 ≤ Crn−1

for r ≤ 1 (and if B = B(x, r) ⊂ Ω).

Proof: just try (u, K) = (u,K) out of B, K∩B = ∂B,and u = 0 in B. Here and below, C depends on n andh, not on (u,K).

Next, K is locally Ahlfors-regular:

(3) C−1rn−1 ≤ Hn−1(K ∩B(x, r)) ≤ Crn−1

for x ∈ K and r < 1 such that B(x, r) ⊂ Ω.

Proof by Dal Maso, Morel, Solimini 89 (n = 2)and Carriero, Leaci (n ≥ 2). Idea: if K is too thin, itcannot separate enough to release the tension. Esti-mate the loss in energy when we remove a piece of K:integrate by parts and estimate the jump of u.

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Local Ahlfors regularity is not so easy to get, but veryuseful. It allows us to use analysis techniques likeCarleson measures.

Often a good idea on spaces of homogeneous type:define functions on the space of balls

(4) ∆ =

(x, r) ∈ K × (0, 1] ; B(x, r) ⊂ Ω.

Let us measure the normalized local energy with

(5) ω(x, r) = r1−n∫B(x,r)\K

|∇u|2

for (x, r) ∈ ∆, and its Lp generalization for 1 ≤ p ≤ 2

(6) ωp(x, r) = r 1rn

∫B(x,r)\K

|∇u|p 2

p

Note that ω(x, r) = ω2(x, r) ≤ C by the trivialestimate, and then ωp(x, r) ≤ C by Holder.

But ωp(x, r) is often much smaller, to the point ofbeing integrable against the locally infinite invariantmeasure dHn−1(x)drr . So we can trade the optimalpower aganist better integrability:

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[D-Semmes, 96]: for 1 ≤ p < 2, there exists Cp ≥ 0such that for (x, r) ∈ ∆,∫

y∈B(x,r/2)

∫0<t<r/2

ωp(y, t)dHn−1(y)dt

t≤ Cprn−1.

Thus ωp(x, r)dHn−1(x)dr

r is a Carleson measure on ∆.The result is interesting but the proof is not: use thetrivial bound, Holder, Fubini, and the local Ahlfors-regularity to compute interior integrals.

Corollary: for each 1 ≤ p < 2 and ε > 0, there existsC(ε, p) such that, for every (x, r) ∈ ∆, we can find(y, t) ∈ ∆, with y ∈ K ∩ B(x, r/2) and C(ε, p)−1r ≤t ≤ r/2, and ωp(y, t) ≤ ε.

Thus each ball contains not-much-smaller good balls.Proof by Chebyshev: otherwise the integral above is

≥∫y∈B(x,r/2)

∫C(ε,p)−1r<t<r/2

εdHn−1(y)dt

t

≥ εHn−1(K ∩B(x, r/2))∫C(ε,p)−1r<t<r/2

dt

t

≥ C−1εrn−1 log(C(ε, p)/2) > Cprn−1

if C(ε, p) is large enough [a contradiction].

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The concentration property: For each small τ > 0,there exists C(τ) such that, for all (x, r) ∈ ∆ withh(r) ≤ C(τ)−1, we can find (y, t) ∈ ∆, such that y ∈K ∩B(x, r/2), C(τ)−1r ≤ t ≤ r/2, and

(7) Hn−1(K ∩B(y, t)) ≥ (1− τ)Hn−1(P ∩B(y, t)),

where P is any hyperplane through y.

Thus K has almost optimal density in B(y, t).Theorem of Dal Maso, Morel, Solimini when n = 2,Maddalena and Solimini when n > 2.I like it because of the following lowersemicontinuityresult from [DMS]:Let Kj be a sequence of closed sets that satisfy theconcentration property with uniform constants C(τ).Suppose that Kj converges to the closed set K, lo-cally in Ω for the Hausdorff distance∗. Then

(8) Hn−1(K ∩ U) ≤ lim infj→+∞

Hn−1(Kj ∩ U)

for U ⊂ Ω open.

Comments:- Not true without assumption (dotted lines)- Useful for producing minimizers (see later twice?)- Proof by definition of Hn−1 and coverings!

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Proof when n = 2[Advertisement for Carleson measures].Let τ and (x, r) ∈ ∆ be given. Pick p < 2 close to 2and ε > 0 very small (chosen later), and let (y, t) beas in the corollary with Chebyshev. Thus

(9) ωp(y, t) = t 1tn

∫B(y,t)\K

|∇u|p 2

p ≤ ε

(here with n = 2, so the power of t is 1− 1p ).

We want to check that H1(K ∩B(y, t)) ≥ 2(1− τ) t.Enough to check that K meets ∂B(y, ρ) at least twicefor most ρ ∈ (0, t).For instance for all ρ > τ

2 t such that

(10)∫∂B(y,ρ)\K

|∇u|p ≤ C(τ)εp2 ρ1− p

2 .

We suppose it does not and construct a better com-petitor (u, K).

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Cover K∩∂B(y, ρ) with an arc Z of ∂B(y, ρ) of lengthρ

2C , with C as in the Ahlfors-regularity condition (3)).

Set K = [K ∪ Z] \ B(y, ρ). We pay H1(Z) = ρ2C , but

we win H1(K ∩B(y, ρ)) ≥ C−1ρ ≥ 2H1(Z) by (3).The main point is that by (10), we can find an exten-sion u of u∣∣∂B(y,ρ)\Z

to B(y, ρ), with

(11)∫B(y,ρ)

|∇u|2 ≤ C(τ)ερ.

[First estimate the jump across Z, then extend linearlyacross Z, then use the Poisson kernel.]

Then (u, K) ∈ A, and coincides with (u,K) out ofB(y, t). The definition of local almost minimizer shouldyield

H1(K ∩B(y, ρ)) ≤ H1(Z) +∫B(y,ρ)

|∇u|2 + rh(r)

≤ H1(Z) + C(τ)ερ+ rh(r)

a contradiction if ε and h(r) are small enough.

Already here, the fact that K ∩ ∂B(y, ρ) oftenhas at least 2 points allows K to separate B(y, t) intoregions. We’ll see this again in the next proof.

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Uniform rectifiability when n=2

Theorem [D.-Semmes]. For (x, r) ∈ ∆, there is aregular curve Γ ⊂ B(x, r), with constant ≤ C, suchthat K ∩B(x, r/2) ⊂ Γ.

Definition. A regular curve is a (connected) curve Γsuch that

length(Γ ∩B(x, r)) ≤ Cr whenever 0 < r < diam(Γ).

But we could also have taken Ahlfors-regular connectedsets. The point is that regular curves are almost as niceas Lipschitz graphs or even C1 curves.Another way to say state the theorem: K is locallyuniformly rectifiable.

A slightly stronger property, which also makes senseand holds in every dimension n ≥ 2, is that K locallycontains big pieces of Lipschitz graphs, i.e., thatThere exist constants τ > 0 and C ≥ 0 such that, forall (x, r) ∈ ∆, there is a C-Lipschitz graph G ⊂ Rnsuch that Hn−1(K ∩G ∩B(x, r)) ≥ τrn−1.

No proof for n > 2 here, no definition of uniform rec-tifiability when n > 2, or further advertisement foruniform rectifiability. Again, separation plays a bigrole in the proof.

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When n = 2, the theorem follows from thisMain Lemma (big pieces of connected sets).There exists C > 0 such that for (x, r) ∈ ∆ such thath(r) ≤ C−1, there is a connected set F ⊂ B(x, r) withH1(F ) ≤ Cr and H1(K ∩ F ∩B(x, r)) ≥ C−1r.

Please trust: once we know this, the theorem is a con-sequence of local Ahlfors regularity, iterations, gluing,and optimizing.

Proof of the main lemma. Pick p < 2 close to 2, andε > 0 small. By the corollary with Carleson meaures,we can find y ∈ K ∩B(x, r/2) and t ∈ [C−1, r/2] suchthat ωp(y, t) ≤ ε.By Chebyshev, we can choose ρ ∈ (t/2, t) such that

(12)∫∂B(y,ρ)\K

|∇u|p ≤ Cεp2 ρ1− p

2

as for (10) above. Since H1(K ∩ B(y, t)) ≤ 7t by thetrivial estimate, we can also arrange that

(13) K ∩ ∂B(y, ρ) has at less than 20 points.

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Denote by Jj , 1 ≤ j ≤ L, the components of∂B(y, ρ) \K, and by mk the mean value of u on Jk.

Claim. we can find j and k 6= j such that|Jj | ≥ C−1

1 ρ, |Jk| ≥ C−11 ρ, and |mj −mk| ≥ C−1

2 ρ1/2.

Proof of claim.∗ Otherwise, cover K ∩ ∂B(y, ρ) andthe short arcs Jj by a union Z of arcs of length C−1

1 ρ,and with H1(Z) ≤ 100C−1

1 ρ < H1(K ∩ B(y, ρ))(if C1 is large enough).Then use (12) to extend u∣∣∂B(y,ρ)\Z

first to ∂B(y, ρ),

and then to B(y, ρ) with small energy∫B(y,ρ)

|∇u|2

(if C2 is large enough).Get a contradiction as for the concentration property.

Now let j and k be as in the claim.By (12), |u(z) −mj | ≤ (10C2)−1ρ1/2 for z ∈ Jj , andsimilarly |u(z)−mk| ≤ (10C2)−1ρ1/2 for z ∈ Jk.Suppose mj < mk. There is an interval [a, b] in themiddle of [mj ,mk], with b− a ≥ (2C2)−1ρ1/2 and

(14) u(z) < a < b < u(w) for z ∈ Jj and w ∈ Jk.

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(14) u(z) < a < b < u(w) for z ∈ Jj and w ∈ Jk.

Let us apply the co-area formula to (a smoothmodification of) u in B(y, t)\K. For t ∈ R, denote byΓt =

z ∈ B(y, t) ; u(z) = t

the level set. Then∫

t

H1(Γt) dt ≤∫B(y,t)

|∇u| = t3/2ω1(y, t)

≤ Ct3/2ωp(y, t) ≤ Ct3/2ε

by Holder and our choice of (y, t).By Chebyshev, we can find s ∈ [a, b] such that

(15) H1(Γs) ≤ Cεt.

By (14), Γs separates Jj from Jk in B(y, t) \ K.Then K ∪ Γs separates Jj from Jk in B(y, t).

By “elementary 2d-topology”, [K ∪ Γs] ∩ B(y, t)contains a connected piece F that separates Jj fromJk in B(y, t).

First H1(F ) ≤ H1(Γs) +H1(K ∩B(y, t)) ≤ 8t.Also H1(F ) ≥ 1

2Min|Jj |, |Jk|

≥ ρ/(2C1), and then

H1(F ∩ K) ≥ H1(G) − H1(Γs) ≥ ρ/(3C1) ≥ r/C(by (15) and if ε is small).

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Lots of C1 pieces in K

Here we assume that h(r) ≤ Crα for some α > 0.

Theorem [Ambrosio, Fusco, Pallara]. For almost ev-ery x ∈ K, we can find r > 0 such that K coincides inB(x, r) with a C1 and 10−2-Lipschitz graph.

Also see D. and Bonnet when n = 2, and thefollowing improvement∗ (when n > 2) from Rigot:

There exists C ≥ 1 such that whenever (x, r) ∈ ∆and r ≤ C−1, we can find y ∈ K ∩ B(x, r/2) andt ∈ [r/C, r/2], such that K coincides in B(y, t) with aC1 and 10−2-Lipschitz graph.

Main point: if K ∩ B(x, r) is flat enough, and∫B(x,r)

|∇u|2 is small enough∗, then K coincides witha C1 and 10−2-Lipschitz graph in B(x, r/2).

Recent improvement by A. Lemenant for n = 3:if K ∩ B(x, r) is close enough enough to a minimalcone, and

∫B(x,r)

|∇u|2 is small enough, then K is C1-equivalent to a minimal cone in B(x, r/2).

Comments:- List of minimal cones below- Connection with minimal sets and the Jean Taylortheorem- Proof: control of many constants, improvement fromB(x, r) to B(x, r/2), and iteration. Not today.

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3. Blow up limits and global minimizers

Important developments, following A. Bonnet.Again let (u,K) be a reduced local minimizer in

Ω ⊂ Rn, with gauge function h.Let xk be a sequence in K and rk a sequence

in (0,+∞), with limk→+∞ rk = 0. Also assume thatlimk→+∞ r−1

k dist(xk,Rn \ Ω) = +∞. Then Ωk =r−1k (Ω− xk) tends to Rn.

Often we just take xk = x (blow-up at a given point).

Set Kk = r−1k (K − xk) and uk(y) = r

−1/2k u(rky+ xk).

Then (uk, rk) is a local minimizer in Ωk, with gaugefunction h(r/rk).

We can easly∗ extract subsequences so that Kk

tends to a closed set K and uk tends to some u ∈W 1,2loc (Rn \K), in the sense that for every ρ > 0,

(16)Dρ(K,Kk) = supy∈K∩B(0,R) dist(y,Kk)

+ supy∈Kk∩B(0,R) dist(y,K)

tends to 0, and, for every connected component W ofRn \K, we can find constants ck = ck(W ) such that

(17)uk − ck converges to u uniformly

on compact subsets of W .

[In effect, ∇uk converges to ∇u and we integrate.]

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Theorem [Bonnet + . . .]: If (u,K) is a limit of the(uk,Kk) as above, then (u,K) is a global minimizer inRn.

Definition of global minimizers soon. The mainpoint of the proof is that Kk is uniformly concentrated,with uniform bounds, so

(18) Hn−1(K ∩ U) ≤ lim infk→+∞

Hn−1(Kk ∩ U)

for every open set U ⊂ Rn. Then we consider a com-petitor (u, K) for (u,K), use it to construct a competi-tor (uk, Kk) for (uk,Kk), use the almost minimality of(uk,Kk), and use (18) to get a useful comparizon with(u,K).

Comments: with this sort of argument, a limitof reduced almost minimizers in Ω is a “topologicalalmost minimizer” with the same gauge function h.This allows many compactness arguments.

Also, there is a proof by Dal Maso-Morel-Solimini(n = 2) and Maddalena-Solimini (n > 2) that Mumford-Shah minimizers exist. For instance, when n = 2, firstminimize under the constraint that K has at most Ncomponents, and then take a limit. Not so simple, butit works.

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Global minimizers

Denote by A the set of pairs (u,K) such that K ∈Rn is closed, and u ∈W 1,2

loc (Rn \K).A competitor for (u,K) ∈ A is a pair (u, K) ∈ A

such that for R large,

(19) u = u and K = K out of B(0, R)

and

(20)if x, y ∈ Rn \ [B(0, R) ∪K] and K separates

x from y, then K separates x from y.

By “K separates x from y”, we mean that x andy lie in different connected components of Rn \K.∗

Definition A global minimizer is a reduced pair (u,K) ∈A such that

Hn−1(K ∩B(0, R)) +∫B(0,R)\K

|∇u|2

≤ Hn−1(K ∩B(0, R)) +∫B(0,R)\K

|∇u|2

whenever (u, K) is a competitor for (u,K) and R is solarge that (19) and (20) hold. [A Dirichlet conditionat infinity]

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Expected: the study of global minimizers should besimple (no domain Ω, no image g or gauge functionh), to the point that we could even give the full list.

And once we have information on the global min-imizers, we shall return and get information on thelocal minimizers.

Examples of global minimizers when n = 2:- K = ∅, u is constant;- K is a line, u is constant on each component of R2\K;- K is a Y , u is constant on each component of R2 \K;- The cracktip: K = (−∞, 0] ⊂ R and

u(r cos θ, r sin θ) = C ±√

2πr1/2 sin

θ

2

for r ≥ 0 and |θ| < π.

The 120 angle in the Mumford-Shah conjecture comesfrom the Y (the only global minimizers for which u islocally constant are as above).The fact that Cracktip is a global minimizer is true,but non trivial [D., Bonnet]. But is it a blow-up limit?

The constant√

2rr is forced by balance betwen length

and energy (otherwise, make the crack longer or shorter).Strong Mumford-Shah conjecture: modulo rotations(for the cracktip), there is no other global minimizer.

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What is known in R2?

Theorem [Bonnet]. If (u,K) is a global minimizer inR2 and K is connected, then (u,K) is in the list above.

Main ingredient: prove that r → 1r

∫B(x,r)\K

|∇u|2 is

nondecreasing, and use limits.

Consequence: If (u,K) is a minimizer of the Mumford-Shah functional in Ω ⊂ R2 and K0 is an isolated com-ponent of K, then K0 is a finite union of C1 curves.Use blow-up limits and perturbation results near linesand sets Y .

Similarly, the strong Mumford-Shah conjecture wouldimply the standard one.

Leger’s formula: if (u,K) is a global minimizer in R2,

2∂u

∂z(z) = − 1

2π

∫K

dH1(w)(z − w)2

for z ∈ R2 \K ≈ C \K (Beurling transform of H1∣∣K).

In particular, u is essentially unique given K.

Theorem [D., Leger]. If (u,K) is a global minimizerin R2 and R2 \ K is not connected, then (u,K) is inthe list above.

Etc. . .

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What is known in R3?Less, but this makes more interesting questions.

Examples of global minimizers in R3 for which u islocally constant. That is, minimal sets K in R3, withthe topological constraint (20) for competitors:- ∅;- planes;- products Y of a Y with an orthogonal line: Y is theunion of three half planes bounded by a common lineL and making 120 angles along L;- cones T over the union of the edges of a regular tetra-hedron centered at the origin (six infinite triangularfaces bounded by four half lines).

Theorem [J. Taylor+D.] Every Mumford-Shah mini-mal set K in R3 is one of these cones.

Curiously recent, and no answer yet for Almgrenminimal sets, where we still minimize H2(K) locally,but competitors for K are sets K = ϕ(K), where ϕ :R3 → R3 is Lipschitz, with ϕ(x) = x out of some ball.

Recall that Mumford-Shah competitors for K aresets K such that K = K out of some big ball B, andK separates x and y ∈ R3 \ [K ∪B] whenever K sepa-rates them. Almgren competitors are Mumford-Shahcompetitors. so there may be more Almgren minimalsets.

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So we control the global minimizers for which u islocally constant.

Even locally: A. Lemenant’s result says that if(u,K) is a local minimizer in Ω ⊂ R3, with h(r) ≤ Cr,and if one of the blow-up limits of K at x is one of thecones above, then x has a neighborhood B where Kis C1-equivalent to this cone and u is smooth in eachcomponent of B \K.

Example where u is not constant: cracktip times aline, so K = (−∞, 0]× 0 × R (a vertical half plane)and

u(r cos θ, r sin θ, z) = C ±√

2πr1/2 sin

θ

2.

Comments:- Not too hard to check the minimality, by slicing;- Here u is essentially unique given K [Lemenant];- This is the only known (or suspected) global mini-mizer in R3 where u is not locally constant.

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Questions

- Are there other global minimizers? What happenswhen you cut cut a Y locally∗? I had suggestions, butB. Bourdin and B. Merlet don’t seem to like them.- Can we first describe (u,K) when K is a cone?[Lemenant: u is homogeneous of degree 1/2; henceconnections with the spectrum of ∆ on ∂B(0, 1) \K.]- Is u essentially unique given K? [True in the exam-ples above.]- Suppose K contains a small (flat) disk; is it one ofthe examples above?- Suppose u is constant somewhere?- Is every connected component of R3 \K a John do-main (true when n = 3); how many components?

And, even in dimension 2,- prove the strong Mumford-Shah conjecture- Does Cracktip really show up as the blow-up limitat x of (u,K) for some Mumford-Shah minimizer in adomain?- Suppose it does, can K spiral at x? [I think not,proof by Bonnet.]- Would the list of global minimizers change if we al-lowed u to be valued in Rk?

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