Big pieces of <span class="mathjax-formula">$C^{1, \alpha }$</span>-graphs for minimizers of the Mumford-Shah functional

A NNALI DELLA

S ^CUOLA N ^ORMALE S UPERIORE DI P ^ISA Classe di Scienze

S ^ÉVERINE R ^IGOT

Big pieces of C

^1,α

-graphs for minimizers of the Mumford-Shah functional

Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4

^e

série, tome 29, n

^o

2 (2000), p. 329-349

<http://www.numdam.org/item?id=ASNSP_2000_4_29_2_329_0>

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Big ^{Pieces of} C1,03B1-Graphs ^for

Minimizers of the Mumford-Shah Functional

SÉVERINE

RIGOT

Abstract. We consider the

generalization

of the Mumford-Shah functional defined

by

where S2 is a bounded domain in ?" (n &#x3E; 2), g ^a bounded measurable function on

Q, K ^a relatively closed subset of Q,

Hn - (K)

denotes the (n - 1)-dimensional Hausdorff measure of K and ^u E

W 1 ° 2 (S2 B I~ ) .

We prove here that there exist

a E (0, 1) ^{and C} &#x3E; 1 such that if (u, K) ^{is an} irreducible minimizer for J and B (x , r) ^a ball centered on K, contained in Q, with radius r s 1, then there is a

ball B centered on K, contained in B(x, r), ^with radius &#x3E;

C - 1 r,

such that K n B is a

C l,a -hypersurface.

^{Moreover the} constants a, C and the Cl,’-constant for K n B depend only ^{on n and In} particular the Hausdorff dimension of the set of points in K around which K is not a C

l," -hypersurface

^is strictly ^{less than}

n - l.

Mathematics Subject Classification (2000) : ^49N60 (primary), 49Q20 (secondary).

(2000),

1. - Introduction

Let S2 be a bounded domain in M"

(n &#x3E; 2) and g

^{be a} bounded measurable function on S2. We consider the

following

functional defined

by

where the

integrals

^are taken with respect ^{to the}

Lebesgue

^{measure in}

^{R’, HI-1}

denotes the

(n - I)-Hausdorff

^{measure and the}

competitors (u, x) satisfy

^the

following

conditions:

Pervenuto alla Redazione 1’8 novembre 1999 ^e in forma definitiva il 2 maggio ^2000.

This is a

generalization

of the Mumford-Shah functional introduced in

[13]

which

corresponds

^{to n =} 2. The existence of minimizers for J was

proved

in

[11]: using

^a

larger

^{class of}

competitors,

^{E. De}

Giorgi,

^M. Carriero and A.

Leaci prove the existence of

"generalized"

minimizers and then

they

^{show that}

these

"generalized"

minimizers are

equivalent

^to

competitors

^which

satisfy (1.1) (see

^also

[2]).

Here we are interested in the

regularity properties

^{of the set K when}

^{(u, K)}

minimizes the functional J. Note that if

(u, X )

^{is a}

minimizer, adding

^{to K}

any closed set of

Hn-1-Hausdorff

measure 0 and

keeping

^{the same function u}

doesn’t

change

the value of the functional. This

gives

^{a new minimizer}

^(u, K)

for which the set

K

may look much

uglier

^{than the}

original

^{set K.} This is the

reason

why

^{we shall}

always

restrict ourselves to "irreducible minimizers".

DEFINITION 1. 2. Let

(u, K)

^{be a}

competitor

^which

satisfies ( 1.1 ).

We say that

(u, K) is an

irreducible

minimizer for

^J

if

J (u, K)

⁼

inf{ J (u’, K’) : ^{K’ is} relatively

^{closed in S2 and}

^u’

and

if

^{there is no} proper

relatively

^{closed set}

K

C K such that u has an extension in

Note also

that,

for each minimizer

(u, K),

^{there is an} irreducible minimizer

(u, k)

^with

^K

^{c K. From now on, we} will restrict ourselves to the case where 1 which is not

really

restrictive

by

^a

scaling argument.

^We will prove the

following

THEOREM 1.3. There exist ^{a E}

(0, 1)

^{and C} &#x3E; 1 such that

if (u, K)

^{is an} irreducible

minimizer for

^{J and}

B(x, r)

^a ball centered on K, ^{contained in}

Q,

^with radius r

1,

then there is ^a ball B centered on K, ^{contained in}

B(x, r),

with radius

~

C-1 r,

such that K n B is a Moreover the constants a, C and the

C1,a-constantfor K

^{n B}

depend only

^{on n.}

This is the n-dimensional version of the main result in

[7]

^which

gives

^this

regularity property

^{when n = 2. To} prove

this,

^{we will need a} uniform control

on two

quantities.

^{The first one} is the function

defined for all x and t E

(0,1]

^{such that}

B(x, t)

c Q. The second one is the number

f3 (x, t)

^{which measures} the flatness of K inside a ball

B(x, t)

^and

is defined for all x E K and t &#x3E; 0

by

where the infimum is taken on the set of all affine

hyperplanes

^{in JRn. See}

Lemma 3.1 below. Then we will use a criterion

proved

ⁱⁿ

^[3]

which involves

these two

quantities

^{to conclude. Let us}

point

^out

^that, using

^the

equivalence

between our irreducible minimizers and minimizers in the SBV

setting,

^{it can}

be shown that irreducible minimizers are

quasi

minimizers as in Definition 2.2 in

[3].

^More

precisely, following

the notations of

[3],

^{it can} be shown that if

(u, K)

^{is an} irreducible minimizer for J, ^{then u E with = Ct} for some absolute constant C and K ⁼

K (u) (see

Definition 2.2 and page ⁴² in

[3]

for the definitions of and

K (u)

and Remark 2.3 in

[3]

^{for the}

arguments

^{of a}

proof,

^{see also}

[ 11 ]).

^{Thus a direct}

application

^{of Theorem}

3.1 in

[3] gives

^the

Cl,’-regularity

^{of the set K in a small}

neighborhood

^of

Hn-1-almost

_every

points

^{in K but} without the kind of uniform estimates that

we want. This is the reason

why

^we shall first obtain a uniform control on the numbers W2

and f3

defined above. The ideas used here are a

generalization

ⁱⁿ

dimension

greater

^{than 2 of ideas}

developed

ⁱⁿ

[7].

ACKNOWLEDGEMENTS. I am very

grateful

^to

Guy

^{David for}

introducing

^me

to these

problems

^{and for so} many

helpful

discussions and remarks.

2. - Some known

properties

of irreducible minimizers

From now on,

(u, K)

^{will denote a} fixed irreducible minimizer for J.

We will first recall some known

properties

^{of u and K. First u is}

^bounded,

and u E

C1(QBK) (because,

^{when K is}

^fixed,

^{u is a solution}

of a Neumann

problem

ⁱⁿ

LEMMA 2. l. There is an absolute constant

Co

&#x3E; 1 such that

The upper estimate follows from ^a

simple

truncation argument. ^{The lower} estimate was

proved

ⁱⁿ

^[5]

^{when n = 2} and follows from Remark 3.13 in

[4]

when n &#x3E;

^{2. Set}

For each ₁ p

2,

^we will need a LP version of the function úJ2 defined above.

So,

^{for each}

(x, t)

^E

A,

^set

LEMMA 2.3. There is an absolute constant C &#x3E; 0 such that

For each 1 p

2,

^{there is a constant}

Cp

&#x3E; 0 such that

The estimate

(2.4)

follows from the same truncation argument ^{as for the} upper estimate of

(2.2).

The "Carleson measure estimate"

(2.5)

follows from

(2.4), Holder,

Fubini and the size estimate

(2.2).

^The

proof

^{is the same as the}

proof

^of

Proposition

^{4.5 in}

^[9],

modulo minor modifications due to the dimension n &#x3E; 2. We will also need ^some consequences of the uniform

rectifiability property

^{of the set K. It} is proven in

[10]

^{that the set K} is contained in a

uniformly

rectifiable set

(at

^least

locally).

See Theorem 5.4 and Theorem 5.48 there. From the

properties

^of

uniformly

rectifiable sets

(see

e.g.

[8])

^{we deduce}

the

LEMMA 2.6

(Weak

^Geometric

Lemma).

^{For each E} &#x3E;

0,

^{there is a constant}

C(e)

&#x3E; 0 such that

where

DE

⁼

{ ( y, s )

^{E A :}

fJ (y, s) &#x3E; E }

and X v£ denotes the

characteristic function of D’.

We will

only

^{need here the}

following

consequence of the "Carleson ^measure estimates"

(2.5)

^and

(2.7):

COROLLARY 2. 8. For each E &#x3E; 0 and 1 p

2, there is a constant C (s, p)

&#x3E; 1 such that

for

^each

(x, t)

^E A, ^{there is a}

pair (y, s)

^{E A with}

B(y, s)

^C

B(x, t),

s &#x3E;

C (E,

^and

PROOF. Let E &#x3E;

0,

1 p 2 be fixed and

(x, t)

^{E A. We} argue

by

contradiction and assume that we can not find

(y, s)

^{as in the} statement. Then

all pairs (y, S)

^{~ A with} y ^E

1)

^and

^{C - 1 t}

^s

~,

^{for some C} &#x3E; 2 to be fixed

later,

would be such that or

~.

^{Then we would}

have

s)

⁺ XVe 2:

(y, s) ? min( § , I )

^{and hence}

by (2.2).

^{The constant C’}

depends only

^{on the constant}

^Co

ⁱⁿ

^(2.2)

^{and on E.}

Then,

^{if C is}

large enough,

^this contradicts

(2.5)

^and

(2.7).

⁰

3. - Uniform estimate on

c~2 (x , t)

^and

t)

As announced before Theorem 1.3 will follow from the ^same kind of estimate as

(2.9)

^{but with} _top

replaced by

and from Theorem 3.1 in

[3].

More

precisely

^we will prove the

following

LEMMA 3.1. For each T &#x3E;

0,

^{there is a constant C} &#x3E; 1 and a radius to ^E

(0, 1)

such

that for

^each

(x, t)

^{E A with t} to, there ^{is a}

pair (y, s)

^{E A with}

B(y, s)

^C

B (x , t),

s &#x3E;

C -1 t,

^and

Before

giving

^the

proof

^{of this}

^Lemma,

^{let us}

explain

how Theorem 1.3 follows. We recall Theorem 3.1 of

[3].

THEOREM 3.3

([3]).

^{There exist a E}

(0, 1)

^{and some absolute constants} t &#x3E; 0 and C &#x3E; 1 such that

if

^the

pair (x, t) satisfies

then K fl

B(x,

Clearly

Theorem 1.3 follows from Lemma 3.1 and Theorem 3.3. To be

more

precise

Theorem 3.1 in

[3]

^{is more}

general

^and

P(x, t)

^is

replaced by

^a

~2

version of

it,

where the infimum is taken on the set of all affine

hyperplanes

in M".

Clearly

from

(2.2)

^and the definitions of the numbers

f3

^and

P2

^{we have that}

f32(X, t)

^:5

Cf3(x, t )

^{with a constant C which}

depends only

^{on the constant}

^Co

ⁱⁿ

^(2.2).

On the other hand we also have that

t) :5 (see

e.g. the relation

(1.73)

^op page 27 ⁱⁿ

[8]).

And hence condition

(3.4)

here and condition

(3.2)

ⁱⁿ

[3]

^are

equivalent.

Thus it remains to prove ^Lemma 3.1.

Taking

^{into account}

Corollary

^2.8

and the

fact that f3(y, Cf3(x, t )

^whenever

B ( y , s )

^C

B (x , t ) and s &#x3E; C - 1 t,

we

only

^{need to} find for each

(x, t )

^{E A a ball}

B ( y , s )

^{centered on}

K,

^contained in

B(x, t),

^{of size}

comparable

to t, where

c~ (y, s )

^{is as small as we want.}

Let us first

give

^{a few more} notations. For 1 p ^{2 and}

(x, t)

^E A, ^we define the

analogue

^of

^{t )}

where balls are

replaced by spheres,

where &#x3E;

⁼

Hn-1.

From now on we will

always

set &#x3E; ^{= when}

looking

at

integrals

^over

(n - I)-dimensional

^{sets. It will} be also useful to introduce

some subsets of 0. For

each f3

^E

(0, 1),

^{we define as the set of}

pairs (x , t )

^{E A} for which there is a

hyperplane

^P

through x

^{such that}

If we want to

specify

^a

hyperplane

^{P for which this inclusion}

^holds,

^we

will say ^that

(x , t )

^{E with}

hyperplane

^{P. Note that if}

then

(x , t )

^E

A(2p)

^with

hyperplane

^a

hyperplane through x parallel

^{to some}

hyperplane

that realizes If

(x , t )

^{E A with}

hyperplane

^{P, the set U =}

{y

^E

B(x, t) : dist(y, P)

&#x3E; has two connected

components (because fJ 1).

Call them

U+ (t)

^and

U- (t)

^{and set = n U = n}

a B(x, t).

Because u E the values of u on

8+(t)

^are well defined. We will denote

by

^{its mean value on}

Clearly

^{the sets}

U~ (t),

^{and the}

quantities m~ (t)

^don’t

depend only

^{on t}

but also on

fl,

^{x and P. To}

specify

^these

dependances

^when

^needed,

^{we will}

say ^that

U:L(t), a-~’(t)

^and

ml(t)

^are defined with respect ^to

(x, t)

^{E with}

hyperplane

^P.

First we want to

get

^a lower bound on

the jump t - ~ ~ m + (t ) - m - (t ) ~ ^I

^when

(x, t)

^{E for}

~8

^{small and with}

hp(x, ^t)

^{small. We} will need the

following

Lemma to fix which _p we consider.

LEMMA 3.5. Let

B,

^{be the unit ball in R n and} p ^E

(2(n-1) ^2).

^{There is}

a constant

C(p, n)

&#x3E; 0 such that

if

^{h E}

CO(ABI)

^{n and}

if

^{v is the}

harmonic extension

of h

^inside

Bl,

^then

The Lemma follows from well known Sobolev

imbeddings ^(see

e.g.

[ 1 ]).

Now we fix once and’ for all some p ^E

( 2~ n 1~ , ²⁾

^{so that the}

^previous

^Lemma

holds.

LEMMA 3.6. There exist some absolute constants 8 &#x3E;

0,

to ^E

(0, 1 )

^{and C} &#x3E; 0 such that

if (x, t)

^{E with t to and}

~8

^-E-

h p (x, ^t)

^{e, then}

PROOF. Let

C,

&#x3E; 1 be a

(large)

constant to be chosen later. Its value will

depend only

^{on n and on the constant}

^Co

ⁱⁿ

^(2.2).

^{Let 8} &#x3E; 0 be a small c-i

constant to be fixed

--

for the time

being.

^Consider

^{(x, t)}

^E

with

hyperplane

^{P and assume that}

fJ

⁺

hp(x, t)

^{E . Consider a function}

cp ^E

C °° (Il~n )

such that 0

1,

and

C t -’ for y

^E

9~(jc,~),

^{where C}

depends only

^{on n and}

^Ci.

We define a function h on

by

We have that h E

t)) because 8

c-1

w.

^{We now define a new}

function v. Inside

B(x, t),

^{we take v to} be the harmonic extension of h. On

we set v ⁼ u. We also set

K

⁼

(KBB(x,t))

^{U V where}

Then

(v, K)

^{is a}

competitor

which satisfies

(1.1).

^Because

1,

we have - -

and we get ^{from the}

minimality

^of

^{(u, K)}

^that

We have K n

t )

C J and hence

On the other

hand, by (2.2),

^{we also have}

Thus if we fix

C1

1

large enough

^and if t is small

enough, t

to for some to

depending only n

^and

^Co,

^we

get

Lemma 3.5

gives

and hence we have

Thus

(3.7)

will follow if we show that

Indeed if this

inequality

^{holds we}

get

^that

and if E is small

enough

^we get

(3.7).

^{Let us} prove

(3.8).

^{We have}

and hence

We have

Ctn-1

for some absolute constant C &#x3E; 0 because

. ,, ^..

e :5

1-õo

^{and we have}

^already

^fixed

^C1,

^and Poincaré’s

inequality applied

^to

the function u on

a+(t) gives

^us

Similarly

^{we have}

Hence we

get

and

finally

^we

get (3.8).

And this achieves the

proof

^{of Lemma 3.6. 0}

Next we want to say that if

(x, r)

^then

m + (t ) (and ~’(~))

^{does not}

vary ^too much when t describes

( 2 , r ) .

^{Note that if}

^~8 4

^and

^{(x , r )}

^E

with

hyperplane

^{P then}

(x, t)

^{E with the same}

hyperplane

^{P for all}

~~(~r).

LEMMA 3.9.

Let fl 4

^and

^{(x, r)}

^{e with}

hyperplane

^{P. Then}

and

for

all tl, t2 E

( 2, ^r),

^where

^{m~ (ti )}

^are

^defined

^with respect to

(x, ti)

^e

~(2~8)

^with

hyperplane

^{P. The constant C in}

(3.10)

^and

(3.11 ) depends only

^{on n.}

PROOF. This does not come from the

minimality

^of

^{(u, K )}

but from the fact that u E and that the

segment joining

any

point

^{in to} any

point

in

a + (t2 ) stays

^{in Let}

f3 §

^and

^{(x, r)}

^{e with}

hyperplane

^{P. In}

the

following proof U ~ (t ) , a ~ (t )

^and

m ~ (t )

^will

always

be defined with respect

to

(x , t )

^{E with}

hyperplane

^{P when t E}

( 2 , r ) .

^{tl t2 r.}

The set

is convex and U E

t2)).

Thus for each _y E

a+(tl)

^{and Z E}

a+(t2)

^we

have

_

Integrating

^{over y E}

a+(tl)

^{and .z E}

a+(t2)

^and

using

^Fubini’s

Theorem,

^we

get

We have

and

similarly

Hence we

get

and then

by

^Holder’s

inequality.

^{This is}

exactly (3.10).

^We _get

(3.11) by

^{the same}

argument.

And this achieves the

proof

^{of Lemma 3.9. 0}

Combining

^{Lemma 3.6 and Lemma 3.9 we} get ^the

following

LEMMA 3.12. There exist e &#x3E;

0,

ro ^E

(0, 1)

^{and C} &#x3E; 0 such that

if (x, r)

^E

with

hyperplane

^{P, r} ro and

{3

^-f-

top (x, ^r)

^{E, then}

where

Here is

defined

^with respect ^to

(x, t)

^E

A(2p)

^with

hyperplane

^{P as in}

Lemma 3.9.

PROOF. First

by

Fubini and

Tchebychev

^{we can find to E}

(~l, ^{r )}

^{such that}

i.e. such that Since

(x, to) E

^with

hyperplane

P,

^{if E and} ro ^are small

enough,

^we

get by

^{Lemma 3.6 that}

We may ^assume without loss of

generality

^that

m+(to)

&#x3E;

m-(to). By

^Lemma

3.9 we have

1 1 1 1

and

for all t E

(~, ^r).

^Hence if 8 is small

enough

^we get

for all t E

(~, ^r)

^{and for some absolute constant C} &#x3E; 0. On the other hand we

have

-~ I - - I

because

Hn-1 (a+(t)) - Hn-1 (a-(t)) &#x3E; ^Ctn-1

^{for some absolute C} &#x3E; 0

(recall

that

8+(t)

is defined with

respect

^to

(x, t )

^E for all t E

(~, ^r)

^{and we}

consider

only

small values of

P). Combining

^this

inequality

^and

(3.14)

^we get

which is

exactly

^{what we want. D}

1

The fact that

the jump [

^is

large

^{will tell us that K}

separates

well a

large portion

^of

U + (r )

^{from a}

large portion

^of

U - (r ) .

^Therefore

replacing K n B (x , t ) by

^{for some t}

comparable

^{to r} won’t increase too much the surface of the

singularity

^set.

LEMMA 3.15. There exist _Eo &#x3E;

0,

ro ^E

(o, 1 )

^{and C} &#x3E; 0 such

that for

^each

e :5 eo and

(x , r ) E

^with

hyperplane P, r :5

ro

and f3

⁺

c~p (x , r )

^{e, there}

exists t E

( 20 , r )

^{such that}

PROOF. We want to construct a function u * E

C 1 ( B (x , r ) B K )

^{such that its}

values on a

large portion

^of

U+ (r)

^are far from its values on a

large portion

^of

U- (r).

^The

point

^{is to} deduce from

(3.13) pointwise

^{estimates on the}

jump

^of

u*. Then if we also have that

fB(x,r)BK ^{IVu*1 [}

^is

^small,

^{we can}

^easily

^conclude.

The function u * will be a mollified version of u. First we can assume that E and

r are small

enough

^{so that}

(3.13) holds,

and that E

loo .

^Denote

^by

^{xl and x2}

the intersections of the line

through

^x

perpendicular

^{to P with}

respectively a+ ( 2 )

and

9’(~).

^Here

a + ( 2 )

^and

a - ( 2 )

^are defined with

respect

^to

(x, 2 )

^E

with

hyperplane

^{P. Set}

^B

^{1 =}

215)

^and

^{B2 =} ~(~2. ~).

If 8 is small

enough

^then

^{2B, =} B(xl, io)

^C

^U+(r)

^and

^{2B2 = B(x2,} j)

^C

U-(r) (with U + (r )

^and

U - (r )

defined with

respect

^to

(x, r)

^E

A(2fl)

^with

hyperplane ^P).

Set M _{= 17 *} u

were 17

^E is such that supp q ^C

r ), f

^{r~ = 1 and}

0

17 We want to

replace

^u

by u

^inside

^B

^U

^B2.

^Let cp ^E be such that

0

cp

1,

and

Cr-1.

Set

Then u * E

r)BK)

and u * - u on

Bi

^U

B2.

^{Let us show that}

for all YI ^E

B,

and Y2 E

B2

^{and for some absolute constant C} &#x3E; 0.

Keeping

the same notations as in Lemma

3.12,

^we have for each _y E

B 1,

by

Poincar6’s

inequality.

^Thus

for all _y E

Bl. Similarly

^{we have}

for all _y E

B2.

^Hence

combining

^these

inequalities

^and

^(3.13),

^we

^{get (3.17)}

^if

s is small

enough.

^{Set S =}

r ) )

^where

^rip

^denotes

the

orthogonal projection

^{onto P. The}

inequality (3.16)

will follow for some t a little

larger

^than

2o

^{if we show that}

Indeed if

t2

⁼

(fõ)2

^{+ we have}

if 8 1 is small

enough,

^and

Hence we get ^that

which is

exactly

^{what we want. To} prove

(3.18)

^set

a,

⁼

B1

^{n and}

a2

⁼

B2 n a B (x, ~).

^{If z E} S then the

segment ki, z2 ] perpendicular

^{to P and}

joining ^al

^to

^a2 through

^{z does not meet K.} Since u* is

C1

outside of K we

have that

Hence

by (3.17)

^{we have}

and

integrating

^{over S we}

get

If we show that

we will have

which is

exactly (3.18).

^{Thus let us} prove

(3.19).

^{We have}

hence

with D ⁼

f y

^E

B(x, ~) : ^dist(y, P) &#x3E; -LI.

^{For y E D we have}

with D’ -

{y

^E

B(x, r) : dist(y, P) &#x3E; 20}.

^{Then if 8 is small}

^enough

^we

can

apply

Poincar6’s

inequality

^{to the}

^C

^{1 function u on the convex domains}

D’ n and we

get

It remains to

estimate

For y ^E D we have

hence

and

finally

^we get

(3.19).

This achieves the

proof

of Lemma 3.15. D Next we claim that if

(3.16)

holds then

!P(17)BK

^{is as small as}

we want

provided 17

^{is small}

enough (and E

^even

smaller),

^where

LEMMA 3.20. For each i &#x3E;

0,

^{there exist} 17 &#x3E;

0,

e &#x3E; 0 and to ^E

(0, 1)

^such

that

if (x, t)

^{E with}

hyperplane

^{P, t to,}

~8

⁺

wp (x, ^t)

^{8 and}

^{if (3.16)}

holds,

^then

PROOF. We want to

replace K n B(x, t) by

the union of a thin torus of size

a little

larger than i7t

^near

a B (x , t )

^{and P n}

B(x, t)

^{and build a new function}

u ⁼

M(~), where 1/1

^{is a}

piecewise C -diffeomorphism

^whose

image

^{is contained}

in the domain of u and also not

meeting

^{And we} may

choose 1/1

^such

that

D1/I -

Id is small. Thus let T &#x3E; 0 be fixed. Let _q

l~

^{and E} &#x3E; 0 be two

positive

^{constants to be chosen small later and}

^{(x, t)}

^E

d(f3)

^with

hyperplane

P be as in the statement. If E

is

^small

enough (depending

on 17 but ^we will first

fix 17

^and

then E)

^{then K}

n B (x , t )

^C

P(77).

^Set

We define a new

singularity

^set

K by

We have

by (3.16)

and the definition of T. Thus

if il

^{and E are small}

enough,

^we

get

We now want to construct the function u . To

simplify

the notations we may

identify

^{RI with}

^{x R}

^{and denote}

by

y ⁼

(y’, yn )

^E

x R

a

generic point

in R’. We may ^assume that x ⁼ 0 and P ⁼

{y

⁼

(y’,

^E Yn ⁼

0}.

Let D+ denote the connected

component

^of

B(x, ~)B(T

^U

P)

that lies above P

(i.e.

yn &#x3E; 0 for each y ⁼

(y’, yn)

^E

D+).

^See

Figure

^{1. We will define M on}

D+

by

where 1/1

^{is a}

C1 diffeomorphism

^{from D+ onto a subset}

~(D+)

^of

B(x, t)

contained in the

region

This

mapping

^will be continuous up ^to the

boundary ^{a D+}

^{and we will have}

To define

1/1,

^we first choose a non

negative

C~ function h defined on R+ such that

and 4. We also consider a non

negative

^C~

function w

^{defined on}

R+such

that I I ’" ~ ,

and

5-1.

We first

define 1/1 = (*,, *2)

^on

(see Figure 2) by

and

On

D+BW+

^{we set}

~(y)

⁼ y.

Clearly (3.23)

holds. If _y E D+ 17t,

we

have Yn :5 9

^{otherwise y would be in}

^T,

^{and t}

^{- lOi7t}

^{for the}

same reason, ^so that

p2(y/, yn)

&#x3E; qt ^C V+ as announced before.

The

mapping 1/1

^{is a} continuous and

piecewise ^C1

^{function on D+. Let us}

estimate its derivatives.

By

definition

of 1/1

^we

only

^{need to} worry about

*2-

For _y E W+ set

y’ = (y 1,

^...,

yn _ 1 ) .

^{For all 1 i n - 1, we have}

The first term can

only

^{exist when} yn because of h and

5r¡t

10qt

because of

cp’.

^{And so} it vanishes on

W~B7B

^{The second term can}

only

exist when

4

^{Y, *} because of h’ and t

- Iy’l

&#x3E; because of _cp, and is dominated

by

Thus for all _y E

D+

and all 1 i

s n - I ,

^,

we have

Similarly

^{we have}

and the same

computations give

In

particular, when y’

^is

^fixed, p2

^{is a}

strictly increasing

function of _yn and so

1/1

^{is one to one from D+ onto}

~(D~)

and is in fact a

C 1-diffeomorphism

^on

D+.

To estimate

fW+ ^IVuI2,

^{we cut W+ into}

^pieces

k = 1, 2,...,

^{where =}

By (3.24)

^and

(3.25)

^{we have}

for some absolute constant C. We also have that

This can be viewed

using

^a

simple

truncation

argument.

^{We set v = u outside} of

1/!(Ek) ^{U K, v}

^{= 0}

inside 1/!(Ek)

^{and K’ = U}

a ~ ( Ek ) .

^Then

(v, X’)

^{is a}

competitor

which satisfies

( 1.1 ),

^{such that and we}

get

^{from the}

minimality

^of

^{(u, K )}

^that

We have

Hn-1 (a’~l~’ (Ek)) C2krJtn-l

^and

^{I ClEkl (}

^:5

C2krJtn-l 1) and

^we _get ^the

_required

estimate. Then

we have - -

A summation over k

gives

and thus

Let us now define M on the whole Let D- be the

image

^{of D+}

by

the

orthogonal symmetry

^with

respect

^{to P. We define M on D- in a} similar way. More

precisely,

^{we set}

M(y) = u(1fr-(y»)

^for y ^E

D-,

^where

1fr - (y) = a (* (o, (y))).

^The restriction of u to D - has the same sort of

properties

as its

restriction

^to

^D+.

^In

particular, u

^{coincide with u on a}

neighborhood aB(x, t)BT,

^and

And

hence,

^{we have}

On the

complement

^of

B (x , t),

^{we =}

u ( y )

^{and on T n}

B (x , t),

^{we set}

u(y)

^{= 0. The}

pair (u, k)

satisfies the conditions

(1.1)

^{and is a}

good

^candidate

for the functionnal J. Because

II u II 00’

^{we have}

And

by (3.22), (3.26)

^and

(3.27),

^{we have}

If t

and 17

^{are small}

enough (depending

^on

^t),

^{we then deduce}

^(3.21)

^{from the}

minimality

^of

^{(u, K).}

⁰

Now, using

the size estimate

(2.2)

^and

(3.21),

^{we will find a ball}

B ( y, 4 )

centered on K such that

We

keep

^{the same notations as above and}

(x, t)

^{is as in the statement of Lemma} 3.20. If E is small

enough

^then

B(y, 4 )

^c

^P(q)

^for all y E x ⁿ

B(x, ~).

^We

consider a maximal

family

^{,,4 of}

points

ⁱⁿ

K n B (x, ~)

^such

^that

^for

all _y, .z E

A, y #

^z. Then there exists such that the

previous inequality

holds.

Otherwise,

^{because the balls}

B (y, 4 )

^are

disjoints,

^{we would have}

Then the size estimate

(2.2)

and the fact that the balls

B(y, 2 ),

^cover

K n

B (x , 2 )

^would

^give

which contradicts

(3.21)

^{if C is}

large enough.

^{Let us resume all} these Lemmas in the

following

LEMMA 3.28. For each T &#x3E;

0,

^{there exist 8} &#x3E;

0,

ro ^E

(0, 1 )

^{and C} &#x3E; 1 such that

if (x, r)

^E

A (p)

^{with r} ro and

fJ

⁺

cvp(x, ^r)

^e, then there are y ^E K f1

B(x, r)

and s &#x3E;

C-lr

such that

B(y, s)

^C

B(x, r)

^and

PROOF.

Indeed,

^{if 8 and} ro ^are small

enough

^{and if}

(x, r)

^{E with}

hyperplane

^{P is as} in the statement, Lemma 3.15

gives

^{us t E}

(-L, r)

^such

that

(3.16)

holds. Because

(x, t)

^{E with}

hyperplane

^{P and}

CúJp(x, r),

^{we may}

^apply

^Lemma 3.20 and the

argument following

^{it to} get

the

required

conclusion. D

The

proof

^{of Lemma 3.1 is now almost} finished. Let i &#x3E; 0 be

given

^and

(x, t )

^{E A}

with t to,

^{for some} to ^to be fixed small later. First

by Corollary

2.8 we may find for each 8 &#x3E; 0 a

pair (z, r)

^{E A with}

B (z, r )

^C

B (x , t ),

r &#x3E; and

+ cp (z, r)

^E. Then if E and to ^{are small}

enough,

^we

may

apply

^{Lemma 3.28 to}

(z, r)

^E

r)).

^This

gives

^us y E K and s &#x3E; such that

B ( y , s )

^C

B (z , r )

^and

~. ^Finally

^{we note that}

and if 8 is small

enough,

^we also have that And Lemma 3.1 follows.

REMARK 3.29. A consequence of Theorem 1.3 is that the Hausdorff dimen- sion of the set E of

points

in K around which K is not a

Cl,"-hypersurface

(see

^a few lines below for a

precise definition)

^is

strictly

less than n - 1. Note that it was

only

known before that ⁼ 0

(see [3]).

^More

precisely

^we

set for all x ~

K,

and we define

It follows from Theorem 1.3 and from

[7,

pages

809-810]

^{that = 0 for}

some d n - 1.

Since,

^{once we} have Theorem

1.3,

^the

arguments

^to prove that n

B(x, r))

^{= 0 for all x} and r ¹ whenever d is close

enough

to n - 1 are not

really

new, ^we

just

sketch them here. The

key point

^{is the}

existence for every

Ahlfors-regular

^{set of a}

decomposition

^into

"dyadic

^cubes"

(see [6]). Using

^this

decomposition,

Theorem 3.1 and

arguing

^{as in}

[7,

pages

809-810],

^{it can} be shown that for all k ^E

N*,

^{E n}

B(x, r)

is contained in

a set

Ek.

^{This set}

Ek

is the union of cubes

Q

of diameter

comparable

^to

2-kN (where

^{N is a fixed}

constant)

and such that is

comparable

^to

2-k(n-l)N. Furthermore,

^{we have}

where il

^{is a fixed}

(small)

^constant

(we

tried here to

keep

^{the same notations}

as in

[7,

pages

809-810]

^{and the set}

Ek

^has

nothing

^to do with the set

Ek

of the

proof

^{of Lemma}

3.20).

^{Hence we can cover E n}

B(x, r)

^{with at most}

C(l -

^sets of diameter less than

C2-kN.

Thus

which tends to 0 when k ~ +00 if d is close

enough

^{to n - 1. D}

REMARK 3.30. After the

present

article had been

accepted

^for

publication,

the author was told that an

analogous

^{result was obtained}

independently by

F. Maddalena and S. Solimini

([12]).

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[3] L. AMBROSIO - N. Fusco - D. PALLARA, ^Partial regularity of free discontinuity ^sets II, Ann. Scuola Norm. Sup. ^{Pisa Cl. Sci.} (4) ²⁴ (1997), ^39-62.

[4] M. CARRIERO - A. LEACI, ^Existence theorem for ^a Dirichlet problem with free discontinuity

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[6] ^G. DAVID, "Wavelets and singular integrals ^{on curves and} surfaces", Lecture Notes in Math, Vol. 1465, Springer-Verlag, ^{Berlin, 1991.}

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[9] G. DAVID - S. SEMMES, ^{On the} singular ^sets of minimizers of the Mumford-Shahfunctional,

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[10] G. DAVID - S. SEMMES, Uniform rectifiability ^and singular ^{sets, Ann. Inst. H. Poincaré}

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[11] E. DE GIORGI - M. CARRIERO - A. LEACI, ^Existence theorem fora ^minimum problem ^with free discontinuity ^set, Arch. Rational Mech. Anal. 108 (1989), 195-218.

[12] ^F. MADDALENA - S. SOLIMINI, Regularity properties of free discontinuity ^sets, preprint.

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Université de Paris-Sud

D6partement ^de Math6matique

Bdtiment 425

91405 Orsay Cedex, ^France

[email protected]