Strong minimizers of Blake & Zisserman functional

A NNALI DELLA

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

M ICHELE C ARRIERO

A NTONIO L EACI

F ^RANCO T ^OMARELLI

Strong minimizers of Blake & Zisserman functional

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

^e

série, tome 25, n

^o

1-2 (1997), p. 257-285

<http://www.numdam.org/item?id=ASNSP_1997_4_25_1-2_257_0>

© Scuola Normale Superiore, Pisa, 1997, tous droits réservés.

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Strong Minimizers

of

Blake &#x26; Zisserman Functional

MICHELE CARRIERO - ANTONIO LEACI - FRANCO TOMARELLI (1997), pp.

With deep ^{sorrow for the loss of our Teacher and Friend, we} dedicate this _paper to Ennio De Giorgi.

We have tried to pursue ^our research in the direction that he had pointed ^out.

Abstract. We prove the existence of strong minimizers for functionals depending

on free discontinuities, ^free gradient discontinuities and second derivatives, ^which

are related to image segmentation.

1. - Introduction ^’

The issue of

minimizing

functionals

depending

^{on both a} bulk energy and

a surfacic

(or

^{lineic in two}

dimensions) discontinuity

energy has attracted the interest of _many researchers

(see

for instance

[A 1 ], [A2], [AFP], [AV], [BZ], [C], [CL], [CLT1]-[CLT5], [Co], [DA], [DCL], [DG], [DMS], [DS], [Fo], [LS], [MS], [MSh], [Tl], [T2]).

Here we focus the functional

where Q c M" is an open set,

R,with

are

given;

^while

Ko, Ki

^{C are Borel sets}

(a priori unknown)

^with

Ko

^U

K1 closed,

^{and u E}

B ^(Ko

^U

^K1))

^is

approximately

continuous on S2

B Ko.

We notice that in case n ⁼ p ⁼ q - 2 the energy

( 1.1 )

^{reduces to the}

following

^one

Functional

(1.3)

^was introduced

by

^Blake &#x26; Zisserman

(thin plate

surface under tension

[BZ])

^{as an} energy ^to be minimized in order to achieve a

segmentation

of a monocromatic

picture.

^{In this} context g describes the

light intensity

^level

on the screen

0, A

^{is a scale}

parameter,

^{a is a contrast}

parameter

^{and a measure} of

immunity

^to

^noise, fl

^{is a}

gradient-contrast

parameter.

The elements of a

minimizing triplet ^{Ki, u)} play respectively

^{the role}

of

edges,

^{creases and}

smoothly varying intensity

^{in the}

region

^Q

B (Ko

^U

for the

segmented image.

The second-order model

(1-.3)

^was introduced to overcome the

over-segmentation

^of

steep gradients (ramp effect)

^{and other in-}

convenients which occur in lower order models as in case of Mumford &#x26; Shah functional

([MSh]).

We prove the

following

statements.

THEOREM 1.1. Under

assumptions (1.2)

^{with n =} p ⁼ 2 there is at least one

triplet

among

Ko, KI

i

C R 2

^{Borel sets with}

Ko U.K1

closed and u E

C2(QB(KoUKl)

approximately

continuous ^on Q

B Ko minimizing functional ( 1.1 )

^and

having finite

energy. Moreover the sets

Ko

^{f1 Q and}

Kl

^{fl Q are}

(?nC 1, 1 ) rectifiable.

THEOREM 1.2. Under

assumptions (1.2)

^{with n =} p ⁼ 2 and a ⁼

there

^is

at least one

pair

among K ^C closed set and u E

B K) minimizing

^the

functional

and

having finite

energy. Moreover the set K n S2 is

1) rectifiable.

We

proved

the existence of

strong

solutions for different free

discontinuity problems involving

second derivatives in

previous

papers: models of elastic-

plastic plates ^[CLT3]

^and

rigid-plastic

^slabs

^[CLT4],

^{but in those cases the}

continuity

^of

competing

^{functions was} built into the functional

(Ko*= 0),

^and

this

property

^{entailed an "a}

priori"

L°° estimate which is not true any ^more in the

present

^{case. Here the} finiteness of energy

(1.3)

entails neither essential boundedness of u or Vu nor local

summability

^{of Vu}

^(see example ^(1.4)

of

[CLT5]),

and this fact

develops

many substantial difficulties.

In the paper

[CLT5]

^we introduced a relaxed version of

(1.3) (weak

^for-

mulation of the Blake &#x26; Zisserman

functional)

^{which in case of}

(1.1)

^{leads to}

to be minimized over V E

GSBV(Q),

^{with Vv E and we}

proved

the existence of minimizers of

(1.4).

^{Here we} prove that for p = n = 2

a weak minimizer

actually provides

^a

minimizing triplet

^of

(1.3) by taking

^a

representative

of the function and the closure of

singular

^sets of the function itself and of its

gradient.

The

proofs rely

^{on a}

deep

^{use of the coarea} formula and of a new Poincare

type inequality

ⁱⁿ

GSBV(Q) (see

^Theorem

4.1 ).

These tools allow the

study

of a normalized energy

density (see (5.1 ))

^{for which we} prove ^a suitable

decay

outside the

singular

^set of weak solutions.

By

^{a careful}

application

^{of our Poincare}

inequality

^{we show that blown}

up sequences

(vh)

^{of a} weak minimizer are nice around

points

^with

vanishing

normalized _energy

density:

^{in the sense} that the functions vh,

though converging only

^{in measure, are}

strongly convergent

ⁱⁿ

togheter

with their

approxi-

mate

gradient,

^{if modified on sets of small area and}

perimeter

^{in a way preserv-}

ing

^{median and}

gradient

^median

(see

^Theorem

4.3).

^This

compactness property

is the most technical and difficult

point

^{of the whole} argument and allows us to show that the

pointwise

^{limit of}

(vh)

^{solves an}

elliptic

^{fourth order}

equation (see

^Theorem

5.1):

^the

decay

^{estimates of its solutions are} transferred

(by

^a

technical

joining lemma)

^to the sequence

(Vh) (see

^Theorem

5.4).

This entails that for a weak minimizer v of

(1.4),

^{we find U = 0}

and v E

B Sv

^U

5’vu).

We stress the fact that for the Mumford &#x26; Shah functional

([DCL]),

the

elastic-plastic plate

functional

([CLT3])

^{and the}

rigid-plastic

^{slab func-}

tional

([CLT4]),

^the minimizers u are

quasi-minimizers

of the main

part

^of the functional so that the

forcing

^{term in the} functional

plays

^{the role of a}

small

perturbation.

^{While for the} Blake &#x26; Zissermann

functional,

^{as like as in} the Mumford &#x26; Shah functional with unbounded datum

(see [L]),

^the

forcing

term has to be considered in the main

part

of the functional and not as a lower order

perturbation.

The

only point

^{where the}

assumption p

^{= 2} is needed is the

decay

^estimate

for solutions of a linear

elliptic equation

^of the fourth order

(see

^Theorem

5.2);

elsewhere in the

proofs

^the

only assumption

p &#x3E; n is

enough.

The

assumption

g ^E is

sharp

^{in the sense that for} every ^T nq there exist _g E such that functional

(1.1)

^{has no}

minimizing triplet.

^We

refer for a

counterexample

^{to a}

forthcoming

paper where additional

properties

of solutions are

given:

^lower and upper

density estimates,

necessary conditions satisfied

by

strong

minimizers,

^a

study

^{of the vector valued case and a more}

general

^case

for p

^{and n.}

The outline of the paper is ^the

following.

1. Introduction

2. Notation and functional _spaces 3.

Preliminary

^results

4.

Compactness

^{and lower}

semicontinuity

5.

Blow-up equation

6. Proof of the main results

2. - Notations and functional spaces

From now on we denote

by Q

^an open ^{set in} 2.

For a

given

^{set U C R" we denote}

by

^{8U its}

topological boundary, by

its k-dimensional Hausdorff measure and

by U ~ I

^its

Lebesgue

^outer

measure; Xu is the characteristic function of U. We indicate

by Bp (x)

^the

open ball

{y

^E

~y - x I p},

^{and we set}

Bp

⁼

⁼ If ^{Q, Q’}

are open subsets ⁱⁿ

R’, by

^{Q C C S2’ we mean that Q is} compact ^{and Q c} Q’.

We introduce the

following

notations: s A t ⁼

min{s, t}, s

^{V t =}

max{s, t}

^for

every s, t ^E R;

given

two vectors a,

b,

^{we set a.b =}

Ei ^aibi

^{and =}

In the

following

^{with the same letter c we} denote suitable constants which may

change

in different

inequalities.

For any Borel R the

approximate

^upper and lower limits of v are the Borel functions

v+,

v- : S2 -~ R ⁼ R U defined for _any xES2

by

The set

is a Borel set, of

negligible Lebesgue

^measure

(see

e.g.

[F], 2.9.13);

^we

say

that v is

approximately

continuous on Q

B Sv

^{and we denote}

by v :

^Q

B Sv

^R the function

-

Let ^x E Q

B Sv

^{be such}

that v(x)

^{E R; we} say that v is

approximately

differentiable at x if there exists a

vector VV(X)

^{E R"}

(the approximate gradient

of v at

x )

^{such that}

If v is a smooth function then Vv is the classical

gradient.

^{In the}

following

^with

the

notation lvvl

^{we mean} the euclidean norm of Vv and we set ⁼

denoting

the canonical basis of R". In the one dimensional case

(n

⁼

1)

we shall use the notation v’ in

place

We recall the definition of the space of functions of bounded variation in S2 with values in R:

where Dv ⁼

(D1 v, ... , Dn v)

denotes the distributional

gradient

^{of v and}

denotes the space ^of vector-valued Radon measure with finite total variation.

We denote

by fo ^IDvl

^the total variation of the measure D v in Q.

If v _{= XE} is the characteristic function of a set E, ^{then v E}

B V (Q)

^if

and

only

^{if E is a set with finite}

perimeter

ⁱⁿ

Q;

^the

perimeter

^{of E in Q is}

given by

r

For every v ^E

B V (S2)

^the

following properties

^hold:

1) v+(x), v-(x)

^{E R for all x E Q}

(see [Z], 5.9.6);

2) Sv

^is

countably (H ’- 1, n - 1)

rectifiable

(see [F], 4.5.9(16));

3)

^{Vv exists a.e.} in S2 and coincides with the

Radon-Nikodym

derivative of Dv with respect ^{to the}

Lebesgue

^measure

(see [F], 4.5.9(26));

4)

^{for almost all X} there exists a

unique V

^{= E}

^8Bi

^{1 such}

that, setting Bp

⁼

^{y

^E

^{Bp(x) : (y - x) ~}

^v &#x3E;

0}

^and

Bg = ^[y

^E

^{Bp(x) :}

(y - X) - v 0},

^then

(see [Z], 5.14.3)

and also

(see [F], 4.5.9(15))

Moreover is an

approximate

^{normal vector to}

Sv

^{at x}

(see [Z], 5.9.6).

We recall the definitions of some function spaces and ^we refer to

[DA]

and

[A 1 ]

^{for their}

properties.

DEFINITION 2. l.

SBV (Q)

^{denotes the} class of functions v E

B V (Q)

such that

11 p 11

SBVI,,,(Q)

denotes the class of functions v E

SBV(Q’)

^for every Q’ cc Q.

Moreover we define

_ ___ .__ r - - - . - ~ , ~_~ .~... , rt, 1

LEMMA 2. 2. Let v E

GSB V (0).

^Then

Sv is countably (~-~C n-1, n -1 ) rectifiable

and V v exists a. e. in Q.

PROOF. See

[A2], Proposition

^{1.3 and 1.4. El}

LEMMA 2.3. Let S2 C R nbe a bounded _open set with

Lipschitz boundary

^and

v E

GSB V (S2).

Assume that

Then for

every a, b E

R,

^{a b we have (}

and

(a

^{V v A}

b) XE

^E

SBV (Q) for

every ^set E

with finite perimeter

^{in Q.}

PROOF. Follows

easily by

Definition 2.1 and the coarea formula

(see

e.g.

[Z],

5.4.4).

^D

LEMMA 2.4.

Let a, b

^{E R and v E}

GSB V (a, b).

The

previous lemma,

^when

applied

^to any function in

GSB V 2 (a, b)

^and

to its

approximate derivative,

shows that in dimension one the

strong

^formula- tion

( 1.1 )

^{and the} weak formulation

(1.4)

^coincide.

Now we may define the

following

function spaces; ^even if we will use

only

^{in the}

following,

^we recall the definitions of various _spaces related to bounded second

derivatives,

^to avoid confusion with other authors’

notations.

DEFINITION 2.5. We set

Notice that Dv ⁼ Vv in

SBH(Q), Dv =A

^{Vv in and in}

GSB V2(Q);

moreover we set

-

Eventually

^we introduce the weak energy functional ^{and a} space for ^com-

peting

^functions.

DEFINITION 2.6. For S2 C R’ open set, ^under the

assumption (1.2), setting

= n

Lq (Q),

^{we define F:}

X(0)

^--~

[0, +00]

^as

We need a localization of the

previous

functional ,~’.

DEFINITION 2.7. For _every Borel subset A C S2 and V E we define

We shall use

0(v, A)

^if g, ~, a,

f3

^are

clearly

^defined

by

^{the context.}

The

following

^{two theorems have been}

proved

ⁱⁿ

^[CLT5]

^{with the} inessential restriction _{p = q} ⁼ 2. In the first statement we use the

following

sequence of smooth truncation functions:

for _every

integer k &#x3E;

^{2. The}

property

^{v E}

GSB V (Q) (see

Definition

2.1 )

^is

clearly equivalent

^{to the}

requirement

(see [A2],

^Section

1 ).

THEOREM 2.8

(Interpolation inequality).

^{Let n E l~ and let}

Q

^{C JRn be a cube}

with

edges of lenght

^I

and parallel

^{to the} coordinate axes, p &#x3E; 1 and v E GSB

V 2 ( Q).

Then

for

every i ⁼

1,

^{... , n the}

following inequality

^{holds true}

THEOREM 2.9

(Existence

^{of weak}

solutions).

^{Let Q c JRn be an} open ^set.

Assume

(1.2)

^with g ^E

Lq (0).

Then there is _Vo E

X (Q)

^{such that}

We recall that

assumption fl

^a

2~8

^is necessary for lower

semicontinuity (see [BZ], [Co]).

3. -

Preliminary

^results

’

In this section we prove ^some technical results.

LEMMA 3.1

(Density

upper

bound).

^{Let v E}

minimizer for the functional (2.1 ) with

^{g E}

(r 2: nq ).

^Then

for

every 0 p 1 and

for

every ^{x E} S2 such that

B p ^(x)

^{C S2 we have}

where (

PROOF.

By minimality

^{of v for .~’ we} get

.~’(w),

^{where w =}

VXQBB,(,).

^{· Since}

Sw

^U

Svw

^c

((Sv

^U

Svv) B B,(x))

^U

a Bp (x)

^and

taking

^into

account

f3

^a,

by

subtraction we obtain

hence, by

^the

assumptions

^{on T and} p, ^we achieve the

proof.

^C

LEMMA 3.2. Let Q C R" be an open ^set and let

Ko, K1

1 C be Borel sets with

Ko

^U

K,

^{closed. Let v E}

B (Ko

^U

K1 ))

^{and let v be}

approximately

^continuou6

in S2

B Ko.

^{Assume that}

then ^r

PROOF. The function v is

obviously

ⁱⁿ

Lq(0) and, by

Lemma 2.6 in

[CLT3],

we have ~v E

So, by

^{Lemma 2.3 in}

[DCL],

^the

proof

^{will be}

achieved as soon as we show that for _every cube

Q

c Q with

edges parallel

^to

the coordinate axes and for _every

integer k &#x3E;

^{2 we E}

(possibly

^not

uniformly

ⁱⁿ

^k, Q),

^{where satisfies}

(2.2).

Fix

Q

c Q ^as above and i E

{ 1, ... , n }, then, by

^Theorem

2.8,

^{we obtain}

the

following interpolation inequality

where 1 is the

edge length

^of

Q.

^{Hence v E}

X (Q)

^and the other assertions

follow

immediately.

⁰

REMARK 3.3.

By

^{Theorem 2.9 and Lemma 3.2 we obtain}

immediately

^that

min

.~’(v) inf f F(Ko,

"

Ki, v) : Ko, KI

^{C JRn Borel} sets,

Ko

^U

KI closed,

v E

c2(Q B

^U

Ki)), v approximately

continuous

on Q B Ko } .

LEMMA 3.4

(Scaling).

^{Let v E For À} &#x3E; 0 and

for

every

x E

BI

^{set v}

and,

Then Vr E

PROOF. Follows

immediately by change

of variables.

DEFINITION 3.5. A function v E

X (Q)

^{is a local minimizer of}

(2.1 ) if,

^for

every

compact

^{set T C}

Q,

Actually

^{the infimum in the} Definition 3.5 is a minimum due to Theorem 2.9 and to the fact that

{z

^e

X (S2) :

^{z = v}

T }

^{is closed in}

LEMMA

3.6

(Matching).

^{Let 2 : and}

Then, by setting

we have

PROOF. Follows

immediately by

^the definitions. 0 LEMMA 3.7

(Joining).

^{Let u, v E and let 0 s t 1 be such}

that

B t (x )

^{C Q. Then}

for

^{every 8 E}

(0, 1)

there exist c ⁼

c (8, p, n)

&#x3E; 0 and a

cut-o, ff function 1/1

^E

C2 0 (Bt (x)),

^with

^1/1

^{=- 1 in a}

neighborhoud of

^such

that, by setting

^{U =}

1/Iv

⁺

( 1 - ~ ) u,

^{we have}

PROOF. To

simplify

^{notation we consider x = 0. We fix 8 E}

^{(0, 1)}

^and

. Let

and *j ( j

^-

0,...,

^{N -}

1)

cut-off functions

between Bj

^{1 of class}

C2

such that and _! in

(where

and define

Then, by setting

^{a 0 b}

= 1 ^{(a ®}

^{b + b ®}

^a),

^{we get, for every}

^j,

Since

Cj

^f1

Cj+l = ^0, by adding

^with

respect to j

^{from 0 to N -}

1,

^we

get

min

By choosing

^the

index j

for which the above minimum is

achieved,

^{we set}

so that

hence the thesis follows with ^o

LEMMA 3.8. Let n &#x3E;

2,

_p &#x3E; 1 and let Q C JRn be open. Let ⁱ such

that, for

every compact ^{set T C}

Q,

Then

for

PROOF.

Apply

^{the same}

argument

of Lemma 2.6 in

[DCL].

4. -

Compactness

^{and lower}

semicontinuity

We prove ^a Poincare type

inequality

ⁱⁿ the class GSB V which extends Theorem 3.1 in

[DCL] allowing

^{a more}

general

truncation. We

enphasize

^that

v E does not even entail that either v or Vv

belongs

^to

Let B be an open ball in L~n . For every measurable function v : B --* R

we define the least median of v in B as

We remark that

med(., B )

^{is a non linear} _operator ^{and in}

general

^{it has no}

relationship

^with

f B dy . Obviously

^{we have}

med(vXB~E

⁺

med(v, B) XE, B) = med ( v , B )

^for every E ^C B. For every ^{v E}

GSBV(B)

^{and a E R with}

n -

, we set

where _yn is the

isoperimetric

^{constant relative to} the balls of

R",

^{i.e. for} every Borel set E

0 we define the truncation operator

We

get easily ^{T (T (v,}

^a,

1]),

^a,

q)

⁼

T (v, a, y?), med(T (v,

^a,

B)

⁼

med(v, B)

and a, ⁼

1])

^for every À &#x3E; 0 .

Moreover IVT(v,

^a,

1])1 I

a.e. on B and

In case v is vector-valued the

operators

^{med and T are defined}

componentwise.

THEOREM 4.1

(Poincaré

type

inequality).

^{Let B C JRn be an open}

^ball,

ⁿ &#x3E;

2,

p &#x3E; 1. Let ^{v E}

GSBV (B)

^{and a E R with}

let _r~ &#x3E; 0 and

T (v,

a,

q)

^{as in}

(4.1).

^Then

If

p n,

setting

^{, we have also}

If

p &#x3E; n,

for

eve ry ~ ^we have also

PROOF. We may ^assume that

med(v, B) =

^{0 and the}

right

^{hand sides are}

finite. If 0 and a ⁼ 0 then V E

T (v, 0, r~)

^{= v and}

the

inequalities

^are well-known.

By

^Lemma

^2.3,

^E

^SBV(B)

^{and we}

obtain

By

^{the coarea formula of Lemma 2.3 and the}

isoperimetric inequality

^{we obtain}

By

^the

assumption

^{on a and}

by comparison

^with

(4.7)

ⁱⁿ the case t7 ⁼ 0 we

have

hence

By

substitution in

(4.7),

^we obtain for _every

so

(4.4)

^follows

by

^Holder

inequality. By (4.8)

^and

(4.10)

^we

get

^also

By

^the classical Poincare

inequality (see [F],

pag.

504) applied

^to

we

get

We define

Then, by taking

^{into account}

(4.1), (4.2)

^and

(4.4),

^we have for every

Hence if _p ⁼

1, by (4.13 )

^{with s =}

^{1 * ,} (4.12)

^and

(4.10)

^for

T ( v ,

^a,

0)

^we get

so that

(4.5)

^is

proved

^{in the case p = 1. We focus now the case}

We set

-*

and we notice that w E

SBV(B), med(w, B)

⁼

med(v, B)

^and

By plugging

^{w in}

(4.12)

^and

(4.10) with 17

^{= 0 and}

by

^Holder

inequality

Dividing by

which is finite since

T (v,

^a,

0)

^is

bounded,

we

get

By (4.13)

^{with s =}

p *

^and

(4.14)

^{we obtain}

If p &#x3E; ^{n, fix and set}

r = 2013~.

^{Then r n and r * = s,}

hence

by (4.5)

and Holder

inequality,

^still

assuming med(v, B)

^{= 0 and}

setting

, we have

and the

proof

^is

completed.

We show

that,

^beside

(4.2),

^{also the}

perimeter

^{of the set}

can be estimated for _many suitable _1] E

(0, 1).

LEMMA 4.2. Let B C R" be an _open

ball,

ⁿ &#x3E;

2,

^s &#x3E; 1. Let ^{v E}

GSB V (B)

and a E R with

(

^Then there exists _q E

(0, 1 )

^such that

Actually

^{: both}

(4.15)

^and

(4.16) hold ) I

PROOF.

By

^coarea formula of Lemma 2.3 and

by

the definition of

SBV (B)

and

analogously

We

get

the thesis

by Chebyshev

^{and Holder}

inequalities

^and

by (4.2).

^D

For any

given

function in

GSB V,

^{we define an affine}

polynomial

^correction

such that both median and

gradient

median vanish.

Let C Q and V E for _{every y} E R’ we set

Since

med(v -

c,

Br (x ) )

⁼

med(v, B,. (x ) ) -

^{c for} every ^{c E} R and

then we have say

We notice that there are v such that

med(v, # med(Px,rv, B,(x)),

^take

e.g.

v(x)=

^{where H is the} Heaviside function.

In the

following

^{we denote}

by s’

^the

conjugate exponent

of s in the Holder

inequality.

THEOREM 4.3

(Compactness

and lower

semicontinuity).

^Let

^2,

Br (x )

^{C be an} open

ball, (Vh)

^C

GSBV2(Br(x)).

^Set

L h

Assume

Then there exist Z E

W 2’ p ( Br (x)),

^a sequence

(Zh)

^{C GSB}

V 2 ( Br (x))

^{and a}

positive

constant c

(depending

^{on the}

left

^{hand side}

of (4.19))

^such

that,

up ^to

a finite

^number

of ^indices,

Moreover there is a

subsequence

^such

that for

every

PROOF. We can assume x ⁼

0,

^{0 and we extract}

subsequences

without

relabeling.

^{We use}

properties

of functions that hold true

only

up ^{to a} finite number of

indices,

^hence

by (4.20)

^{we can assume that}

Arguing separately

^on

each component Of V Vh

^we deduce that V k E

{ 1, ... , n },

there exist

q§

^E

^{(0, 1)}

^{and c}

^depending

^{on the l.h. s. of}

^(4.19)

^such

that

Inequality (4.28)

^follows

by (4.2),

^while

(4.29)

^follows

by

^{Lemma 4.2 with}

s = p

applied

^to

Vkvh

^and

1) through (4.19).

We define small subsets

Eh

^C

Br

^{where we have to}

modify vh

^{in order}

to force boundedness

of Vvh,

^{and we}

perform

^{a former}

tuning

^of Vh. To this aim set

Since

med(vh, Br ) = 0

^then

med(wh,

^and

by

^the definition

med(Vwh , Br)

=

med(Vvh, Br)

^{= 0.}

By (4.29) Eh

^{has finite}

perimeter,

^hence

by

^{Lemma 2.3}

we have _wh E

GSB V2 (B,)

^and

V Wh

^E

SBV(Br,

ⁿ

L’(B,, JRn).

^Then

by summarizing

and

by (4.20), (4.28)

^and

(4.29)

p. Since 0

1]~ ^1, by (4.32), (4.6), (4.19)

^and

(4.20)

^we

By (4.32), (4.9)

^and

by

^the

assumptions (4.19), (4.20)

^and p &#x3E; n, we get

Now we want to force boundedness of

(wh). By (4.32)

^{we can assume}

and there are ?7h ^E

(0, 1)

^{and a} constant,

depending

^on the l.h.s. of

(4.19),

such that

Inequality (4.36)

^follows

by (4.2),

^while

(4.37)

^follows

by

^{Lemma 4.2 with}

s ⁼ p,

applied

^{to Wh,}

taking

^{into account}

(4.32)

^and

(4.34).

^{Then we define}

and we

perform

^the

following tuning

of the sequence

We notice that zh ^E and

Hence

(4.21 )

^and

(4.22)

^follow

by (4.28), (4.36),

^and

(4.29), (4.37) respectively,

taking

^{into account the}

assumption

p 2: ^n.

By (4.7), (4.9)

^and

(4.34)

^{we have}

Moreover, by (4.39)

^and

(4.34),

and

by (4.32), (4.39)

^we get

hence

by ^(4.11)

^{and the}

assumption p

^{we have}

Since 0 _17h

1, by (4.6)

^and

(4.41)

By (4.41), (4.42)

^{and the}

compactness

^{theorem in}

applied

^to

there exists

f

^{E such that}

By (4.39), (4.19), (4.20)

and Theorem 2.1 in

[All, f

^E

SBV(Br, JRn)

^{and also}

hence

f

^E

By [B],

Theorem 18 in

Appendix,

^and

by (4.34)

we get

By (4.40), (4.43)

^{and the} compactness theorem in

B V (Br ) applied

^to

(Zh) ,

there exists z E

B V (Br)

^{such that}

By (4.39), (4.41)

and Theorem 2.1 in

[Al]

^{we have z E}

SBV(Br)

^{and also}

hence z E and also

by (4.43)

By (4.45)

^and

(4.46) f

^{= Vz = Dz so that z E}

W2,p (B,). By

^the arbitrariness

of 6, (4.23), (4.24), (4.25)

^{hold true and}

(4.26), (4.27)

^follow

by (4.21).

⁰

5. -

Blow-up equation

The aim of this section is to prove for _any minimizer v of

(2.1 )

^{a faster}

decay

^estimate of the functional around

points X

^{such that}

THEOREM 5.1

(Blow

up

equation).

Let p &#x3E; n &#x3E; 2 and

B,. (x )

^{C S2 be an}

open

ball, (Vh)

^C

X (Q), (~ph )

^{C let}

(ah ),

^and

(,c,ch )

^three sequences

of positive

^{numbers with}

Ph

^{ah and let Z E Assume that}

Then, for

every p r, z minimizes ^the

functional

over all v E such that v ⁼ z in

Br (x ) B Bp (x)

^and

In

particular

^{z is a weak solution}

of

^the

fourth

^order

elliptic equation

PROOF. Assume x ⁼ 0 and set for

brevity’s

sake. We will extract

subsequences (without saying explicitly

and without

relabeling)

^{each time}

is needed and we will omit the

arguments

^ah,

Ph

^{in First we estimate}

and we have

by (iv), (v)

and even

by (v),

Hence

and

then, by

^Fatou’s

^lemma,

By (ii)

^and

(iii),

^the

assumptions

of Theorem 4.3 are

satisfied,

^{so that we can} choose _zh and z as like as in Theorem 4.3 and z turns out to be the same of

assumption (iv). By (4.25), (5.2)

^and

(iii)

^{we obtain}

To achieve the

proof

^{we want to show that} for almost _{every p} r , for every v ^E with v ⁼ z in

Br B Bp

By monotonicity (o

^is continuous for a.e. p ^E

(0, r).

^Assume

by

contradiction that there exist s r, v

E W2,P(Br)

^{and E} &#x3E; 0 such that cv is continuous at s,

v = z in ^and

As like as in Theorem

4.3,

^{we set}

Lh

⁼

1t n-l (SVh

^{U and}

^Ah

⁼

^Eh

^U

^Wh

⁼

(vh - PVh))-

^To

get

^a contradiction we

paste together zh

⁺

PVh and Vh along

^the

boundary

^{of a}

suitably

^choosen ball and then we

join

^{the new function}

with v

(which

reduces the energy in

Bs).

^{To this}

aim,

^we remark that for a.e.

p ^E

(0, r)

^{we have}

and, by (4.22),

Integrating

ⁱⁿ

polar

coordinates and

by

^the

isoperimetric inequality

^we get

Since

by (iii)

the sequence

~ah Lh ~

^is

^bounded,

^then

^{by (ii)}

Hence,

^{for a.e.} p ^E

(0, r),

^{we have}

Now, by

^the

continuity

^{of cv at s and}

by (5.4),

^{we choose t E}

(s, r)

^{such that}

and we

glue together vh

^{and Zh +}

~vh by defining

By

^Lemma

3.6,

u h ^E and

We join

^{u h and Let}

1frh

^{be the} cut-off functions

between BS and Bt provided by

^{Lemma 3.7 and set}

Then

Uh belongs

ⁱⁿ

Br B Bt

^{and we}

have, by

^the

minimality of vh

ⁱⁿ

Bt, by v

^{= z}

on B, B BS

^and

by (5.7),

By taking

^{into account}

(4.22)-(4.27), (4.31), (4.38), v

and assump- tions

(i), (iii)

^and

(v),

^we take the limit as h -~ +00 and we have

Then

by

^the arbitrariness of 3 and

by (5.3)

^and

(5.5)

^we get

contradicting

^the

monotonicity

^{of cv. 0}

We recall the

following

well-known result for the biharmonic operator. ^For the

proof

^{we refer to}

[G],

^Ch.

III,

^{sect. 2.}

THEOREM 5.2

(Estimate

^{on the solutions of the} biharmonic

equation).

Let B,

^{C R’ and let u E solution}

of

Then the

following inequality

^holds:

where _cn is an absolute constant

depending only

^{on the dimension.}

REMARK 5.3.

For q

&#x3E; 1 there exists ^{a constant c}

depending only

^{on n and}

q such that _{for every} affine

function ~

^and every 0 p ^{r we} have

hence there exists a constant cn,q such that for ^every biharmonic function ^u and for every ^affine

function ~

^{we have}

THEOREM 5.4

(Decay).

^Assume

(1.2)

^{with n =} p = 2. For every k &#x3E;

2,

ri, ^{a E}

(0, 1 )

^with

r

^{there is} 80 &#x3E; 0 such that

for

every ^{8 E}

(0, EO]

^and

.4

is a local minimizer it, a,

f3, Q)

^with

and

then

PROOF. Assume that the theorem is false. In such case, there exist k &#x3E;

2,

77, Or E

(0, 1)

^with

r0’ £,

^{a sequence}

^{of 8h}

&#x3E; 0 with

limh 8h =

^{0 and}

C2.a

local minimizers of

(.,

JL, a,

~, Q)

^{such that}

and

By translating

^{into the}

origin

^and

rescaling

^{we set}

where

The

functions vh

^are local minimizers in

B1

^{for , where}

By

^{Lemma 3.4 we} get

hence

By (5.9)

^and

by

^Theorem

^4.3,

up ^{to a}

subsequence

^{that we do not}

^relabel,

^there

exists z such that

Now we

verify

^{that the}

functions vh

^{and z}

satisfy

^{all the}

remaining assumptions

of Theorem 5.1.

Assumption (i)

^follows

by

^{Lemma 3.4 and}

(ii)

^follows

by (5.10).

We prove

(iii):

^let

(tj)

^{be a} sequence of radii dense in

(0, 1). By (5.9), by extracting

^for

any tj

^a

subsequence of vh

^and

by diagonalizing,

there exists for

any tj

Since tj

^{-~ is monotone}

nondecreasing,

^hence continuous _up to a countable set, then the above limit exists for almost _every

radius p

^E

(0, 1).

^To

verify (v)

of Theorem

5.1,

^we prove first that

By

^{Lemma 3.1}

^(hh)

^is

^bounded,

^{so our claim is obvious}

if q

&#x3E;

2,

and in the case q 2 it follows

by

^the

assumption

ph

8h k .

Moreover,

by changing

variables and

using

^Holder

inequality,

^we

get

hence

Finally

^we prove that is bounded in

Bl .

^{Since are affine}

functions,

it will be

enough proving

^{that ph is}

^bounded,

^{and the}

convergence follows

possibly extracting

^a

subsequence.

^In

^fact, referring

^{to the}

functions Zh

introduced in Theorem 4.3 and

setting Ah =

-~

for every h

big enough

^{we have}

due to

(5.9)

^and

(5.12). By

Theorem 5.1 we have

On the other

hand, by ^(5.8),

hence which contradicts the

assumption

^on 17.

6. - Proof of the main results

In this

concluding

^{section we} prove Theorems ^1.1 and 1.2.

Let v E

GSBV2(Q)

^{be a} minimizer of .~’

given by

Theorem 2.9. We remark first that if B is an open ball such that B C S2 ^{U =} 0 then v is smooth in

B,

^{since we have v E}

W2,2 (B)

^{so that}

by

the standard

regularity theory

^{we obtain D E}

C2(B) (see

e.g.

[B]

^{for the} case q &#x3E; 1 and use an

approximation argument for q

⁼

1).

^{Hence we have v E}

C2(Q B

^U

We now estimate ’ Set

We prove ^that

Qo

_{is open.} In order to

apply

^Theorem

5.4,

^{we fix k} &#x3E; 2 and r~, ^{o~ E}

(0, 1)

^with

r~~ ~2 ;

^{; let 80 be as in Theorem}

^5.4,

^{let x E}

^Qo

^{and let}

r &#x3E; 0 be such that

We show that If z then

hence we may

apply

Theorem 5.4

inductively

^{to obtain}

for _every h E N. For _every 0 t qr there

exists j

^{such that}

so that

~ , , ,

then

passing

^{to the limit as t - 0}

(so that j

^{- we} get ^{z E}

S2o.

^Since

Sv

^{U is}

countably 1) rectifiable, by

Theorem 3.2.19 in

[F]

^{we have}

H1«Sv

^U

Svv)

ⁿ

S2o)

^{= 0 so and}

(Sv

^U

SVv)

ⁿ

Qo

⁼ 0. Since

Qo

_{is open} ^{then U}

Sov

^{= 0.}

By

^{Lemma 3.8 we have}

Setting

we have that is

closed,

^{and ’}

hence are ( )

rectifiable,

^moreover

and, by

^Remark

3.3,

^{we conclude that}

(Ko, K1, C)

^{is a} minimizer for F in the

class of the admissible

triplets.

^D

Theorem 1.2 follows

immediately by

^{Theorem 1.1}

choosing

^{K =}

^Ko

^U

^Kl

and

taking

^{into account the}

assumption

^{a =}

~8.