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Approximation of non-convex functionals in GBV Roberto Alicandro, Andrea Braides Jayant Shah

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Approximation of non-convex functionals in GBV Roberto Alicandro, Andrea Braides Jayant Shah
Approximation of non-convex functionals in GBV
Roberto Alicandro, Andrea Braides
SISSA, via Beirut 4, 34014 Trieste, Italy
Jayant Shah
Northeastern University, Department of Mathematics
360 Huntington Avenue, Boston, MA 02115, USA
1
Introduction
In many problems of Computer Vision, the unknown is a pair (u; K ) with K
varying in a class of (suciently smooth) closed hypersurfaces contained in a
xed open set R2 and u : n K ! R belonging to a class of (suciently
smooth) functions. A variational formulation of some of these problems was given
by Mumford and Shah [12] introducing the functional
Z
Z
2
1
F (u; K ) =
jruj dx + c1H (K ) + c2
ju 0 gj2 dx :
(1)
n
K
n
K
In this case g is interpreted as the input picture taken from a camera, u is the
`cleaned' image, and K is the relevant contour of the objects in the picture; c1 and
c2 are contrast parameters, and H1 (K ) denotes the total length of K (which is
a union of curves). Problems involving functionals of this form are usually called
free-discontinuity problems (see [9], [2], [5]).
The presence of the unknown surface K leads to numerical problems, to
solve which some kind of approximation of this functional is needed to obtain
approximate smooth solutions. The Ambrosio and Tortorelli approach [3] provides
a variational approximation of the Mumford and Shah functional (1) via elliptic
functionals. The lack of convexity of the limiting functional is overcome by the
introduction of an additional function variable which approaches the characteristic
of the complement of the jump set. The approximating functionals have the form
Z
Z Z
1
F" (u; v) = v2 jruj2 dx + c1
"jrvj2 + (1 0 v)2 dx + c2 ju 0 gj2 dx ; (2)
4"
dened on functions u; v such that u; v 2 H 1 (
) and 0 v 1. The interaction
of the terms in the second integral provide an approximate interfacial energy. The
adaptation of the Ambrosio and Tortorelli approximation to obtain as limits more
complex surface energies does not seem to follow easily from their approach.
In this paper we study a variant of the Ambrosio Tortorelli construction by
considering functionals of the form
Z
Z Z
1
2
2
2
G"(u; v) = v jruj dx +
"jrvj + (1 0 v) dx + c2 ju 0 gj2 dx: (3)
4"
1
Even though the form of these functionals is quite similar to the previous one, the
Rdomain of the limiting functional will be dierent. In fact, as we have G"(u; 1) jruj dx, it is clear that the limit of these functionals will be nite if u 2 BV (
).
In fact we prove (Theorem 4.1 and Example 4.6) that G" converge to functionals
related to the function-surface energy
Z
Z
j
u+ 0 u0 j
1
G(u; K ) = jDuj(
n K ) +
ju 0 gj2 dx ; (4)
dH + c2
K 1 + ju+ 0 u0 j
nK
where jDuj(A) denotes the total variation on A of the distributional derivative
Du, and u6 are the traces of u on both sides of K . We push this approach further,
constructing a variational approximation for a wide class of non-convex functionals
dened on spaces of functions of bounded variation.
The paper is divided as follows. In Section 2 we introduce the spaces of
generalized functions of bounded variation GBV and GSBV , which are needed for
a weak formulation of the functionals in (1){(4), and the notion of 0-convergence,
which precises in which sense the convergence of these functionals is understood.
In Section 3 we state the many preliminaries which are needed in the course of
the proof. Section 4 is devoted to the statement and proof of the main result, in
a slightly more general form than above. The proof of the result lies on a lower
bound which is obtained by a new denition of the limit interfacial energy density,
taking into account the interaction of the rst two integrals of the approximating
energies G", and on an upper bound which is obtained by direct construction
and a density result of pairs function-polyhedral surface. Section 5 contains the
statement and proof of the approximation result for general isotropic functionals
with convex bulk energy density and concave surface energy density dened on
GBV .
2
Notation
We use standard notation for Sobolev and Lebesgue spaces. Ln will denote the
Lebesgue measure in Rn and Hk will denote the k-dimensional Hausdor measure.
A(
)
and
B(
)
will be the families of open and Borel sets, respectively. If
is a Borel measure and
E
is a Borel set, then the measure
B
is dened as
B (A) = (A \ B ). Let A A be open sets. By a cut-o function between
A and A we mean a function 2 C0 (A) with 0 1 and = 1 on A .
0
0
2.1
Let
1
0
Generalized functions of bounded variation
u 2 L1 (
). We say that u is a function of bounded variation
on if its
distributional derivative is a measure; i.e., there exist signed measures
that
Z
ui Dj dx = 0
2
Z
dij
ij
such
2 Cc1(
). The vector measure = (ij ) will be denoted by Du. The space
of all functions of bounded variation on will be denoted by BV (
).
It can be proven that if u 2 BV (
) then the complement of the set of
Lebesgue points Su , that will be called the jump set of u, is rectiable, i.e. there
exists a countable family (0i ) of graphs of Lipschitz functions of (n 0 1) variables
S
such that H n 1 (Su n i=1 0i ) = 0. Hence, a normal u can be dened Hn 1 -a.e.
on Su , as well as the traces u of u on both sides of Su as
for all
1
0
0
6
u (x) =
Z
+ 0fy2B (x):6hy0x;u (x)i>0g u(y) dy ;
6
R
0B u dy = jB j 1
If u 2 BV (
) we
where
0
R
B
lim
!0
u dyg.
D a u, Dj u and Dc u as follows.
By the Radon Nikodym Theorem we set Du = D u + D s u where D a u << Ln and
Ds u is the singular part of Du with respect to Ln . Da u is the absolutely continuous
Su is the jump
part of Du with respect to the Lebesgue measure, D j u = Du
part of Du, and Dc u = Ds u (
n Su ) is the Cantor part of Du. We can write
dene the three measures
a
then
Du = Da u + Dj u + Dc u :
It can be seen that D j u = (u+ 0 u )u Hn 1
Su , and that the Radon Nikodym
derivative of Du with respect of Ln is the approximate gradient ru of u.
A function u 2 L1 (
) is a special function of bounded variation on if
Dc u = 0, or, equivalently, if its distributional derivative can be written as
0
0
Du = ru Ln + (u+ 0 u )u Hn
0
0
1
Su :
The space of special functions of bounded variation on is denoted
SBV (
). We
will also use the auxiliary spaces
SBV p (
) = fu 2 SBV (
) : jruj 2 Lp (
); Hn
0
1
(Su )
< +1g:
GBV (
) of generalized functions of bounded variation
u 2 L1 (
) whose truncations uT = (0T )S_ (u ^ T )
are in BV (
) for any T > 0. For such functions we can dene Su =
S ,
T > 0 uT
and the approximate gradient and the traces u as the limits of the corresponding
quantities dened for uT . Moreover, we dene the measure jD c uj : B (
) ! [0; +1]
We dene the space
as the space of all functions
6
as
If
jDc uj(B ) = sup jDc uT j(B ) = T lim
jDc uT j(B ):
+
u 2 BV (
) jD uj
c
T >0
!
1
coincides with the usual notion of total variation of
Dc u.
Finally, we set
GSBV (
= fu 2 GBV (
) : jDc uj = 0g = fu 2 L1 (
) : uT 2 SBV (
) for all T g:
For a detailed study of the properties of
BV -functions
we refer to [2], [10]
and [11]. For an introduction to the study of free-discontinuity problems in the
BV
setting we refer to [2].
3
2.2 Relaxation and 0-convergence
Let (X;d) be a metric space. We rst recall the notion of relaxed functional. Let
F : X R + . Then the relaxed functional F of F , or relaxation of F , is
the greatest d-lower semicontinuous functional less than or equal to F .
We say that a sequence Fj : X
[ ; + ] 0-converges to F : X
[ ; + ] (as j + ) if for all u X we have
(i) (lower limit inequality) for every sequence (uj ) converging to u
! [ f 1g
01 1
! 1
! 01 1
2
!
F (u) limjinf Fj (uj );
(5)
(ii) (existence of a recovery sequence) there exists a sequence (uj ) converging to u
such that
(6)
F (u) lim sup Fj (uj );
j
or, equivalently by (5),
F (u) = lim
Fj (uj ):
(7)
j
The function F is called the 0-limit of (Fj ) (with respect to d), and we write
F = 0-limj Fj . If (F") is a family of functionals indexed by " > 0 then we say that
F" 0-converges to F as " ! 0+ if F = 0-limj!+1 F" for all ("j ) converging to 0.
j
The reason for the introduction of this notion is explained by the following
fundamental theorem.
Theorem 2.1 Let F = 0-limj Fj , and let a compact set K X
inf X Fj = inf K Fj for all j . Then
F = lim
inf Fj :
9 min
j X
X
u
Moreover, if ( j ) is a converging sequence such that
then its limit is a minimum point for .
F
exist such that
(8)
limj Fj (uj ) = limj inf X Fj
The denition of 0-convergence can be given pointwise on X . It is convenient
to introduce also the notion of 0-lower and upper limit, as follows: let F" : X
[ ; + ] and u X . We dene
01 1
!
2
f
u ! ug;
0- lim sup F (u) = inf flim sup F (u ) : u ! ug:
!0+
!0+
0- lim inf
F"(u) = inf lim
inf F" (u") :
"! 0 +
"! 0 +
"
"
"
"
"
"
"
(9)
(10)
If 0- lim inf"!0+ F"(u) = 0- lim sup"!0+ F"(u) then the common value is called the
0-limit of (F") at u, and is denoted by 0- lim"!0+ F" (u). Note that this denition
is in accord with the previous one, and that F" 0-converges to F if and only if
F (u) = 0- lim"!0+ F"(u) at all points u X .
2
4
We recall that:
(i) if F = 0-limj Fj and G is a continuous function then F + G = 0- limj (Fj + G);
(ii) the 0-lower and upper limits dene lower semicontinuous functions.
From (i) we get that in the computation of our 0-limits we can drop all
d-continuous terms. Remark (ii) will be used in the proofs combined with approximation arguments.
For an introduction to 0-convergence we refer to [8]. For an overview of 0convergence techniques for the approximation of free-discontinuity problems see
[5].
3
Preliminaries
In the following will denote a bounded open set in Rn with Lipschitz boundary.
We denote by (
) the space of all functions w SBV (
) satisfying the
following properties:
(i) n01(S w Sw ) = 0;
(ii) S w is the intersection of with the union of a nite number of pairwise
disjoint (n 1)-dimensional simplexes;
(iii) w W k;1 (
S w ) for every k N.
The following result is due to Cortesani [7] (see also [6]).
Theorem 3.1 (Strong approximation in SBV 2) Let u SBV 2 (
) L1 (
).
Then there exists a sequence (wj ) in
(
) such that wj u strongly in L1 (
),
2
n
wj u strongly in L (
; R ), lim suph!+1 wj 1 u 1 and
H
0
2
r !r
n
W
2
n
2
W
Z
(wj+ ; wj0 ; wj ) dHn01 lim sup
j !+1 S w
j
Z
2
!
k k k k
Su
\
(u+ ; u0; u ) dHn01
: R 2 R 2 S n0 1
(a; b; ) = (b; a; 0 ), for every a; b 2 R and 2 S n01 .
for every upper semicontinuous function
! [0; +1)
such that
The next result is a particular case of a theorem by Bouchitt
e , Braides and
Buttazzo [4], and deals with relaxation in
Theorem 3.2
(Relaxation in
BV
BV ) Let g : R ! [0; +1] be a lower semicontinuos
function with
g(0) = 0;
t ! g(jtj)
BV (
) ! [0; +1] be dened by
and such that the map
of isotropic functionals.
lim
t!0+
g(t)
t
= 1;
is subadditive and locally bounded. Let
8
Z
Z
>
< jruj dx + g(ju+ 0 u0j) dHn01
Su
F (u) :=
>
: +1
5
if
F
u 2 SBV 2 (
) \ L1 (
)
otherwise in
BV (
)
:
F
Then the relaxation of
the functional
F (u) =
Z
with respect to the
jruj dx +
Z
Su
L1 (
)-topology is given on BV (
) by
g(ju+ 0 u0 j) dHn01 + jDc uj(
):
The following lemma is a commonly used tool (see [5]).
Lemma 3.3
R
(Supremum of measures) Let
superadditive function, let
2 M+ (
), let
i
(A) A i d for all A 2 A(
) and let
for all A 2 A(
).
:
A(
) ! [0; +1)
be an open-set
R
be positive Borel functions such that
(x) = supi
i (x).
Then
(A) A
d
We nally include a `slicing' result by Ambrosio (see [1]). We introduce rst
2 Sn01, and let 5 := fy 2 Rnn : hy; i = 0g be the linear
hyperplane orthogonal to . If y 2 5 and E R we dene E;y = ft 2 R :
y + t 2 E g. Moreover, if u : ! R we set u;y : ;y ! R by u;y (t) = u(y + t ).
Theorem 3.4 (a) u 2 BV (
)
2 S n01
u;y
n
0
1
BV (
;y )
H
y 2 5
y
t 2 ;y
(11)
u0;y (t) = hru(y + t ); i
Su;y = ft 2 R : y + t 2 Su g;
(12)
6
7
v(t6) = u (y + t );
(13)
v(t6) = u (y + t )
hu; i > 0 hu; i < 0
hu; i = 0
some notation. Let
Let
belongs to
. Then, for all
for
-a.a.
. For such
the function
we have
for a.a.
or
according to the cases
or
(the case
being negligi-
ble). Moreover, we have
Z
jDcu;yj(A;y )dHn01(y) = jhDcu; ij(A)
5
for all
A 2 A(
), and
Z
for all Borel functions
X
5 t2Su;y
(b) Conversely, if
u;y 2 BV (
;y ) and
then
u 2 BV (
).
g(t) dHn01(y) =
Z
(14)
g
Su
g(x)jhu; ijdHn01 :
(15)
u 2 L1 (
) and for all 2 fe1 ; . . . ; eng and for a.a. y 2 5
Z
5
jDu;y j(
;y) dHn01(y) < +1 ;
6
(16)
4
The main result
Using the space GBV dened in the previous section, it is possible to give a weak
formulation for problems as in (1) and (4), which has been successfully used to
obtain solutions of free-discontinuity
problems (see [2]). In what follows we drop
R
the term containing u g 2 dx, which is of lower order, and does not aect the
form of the 0-limit, and we generalize the form of the functional (3).
Theorem 4.1 Let W : [0; 1] [0; + ) be a continuous function such that
W (x) = 0 if and only if x = 1, and let : [0; 1] + be an increasing lower
semicontinuous function with (0) = 0, (1) = 1, and (t) > 0 if t = 0. Let
G" : L1(
) L1 (
) [0; + ) be dened by
8Z >
>
(v) u + 1 W (v) + " v 2 dx if u; v H 1(
)
j
0
j
!
1
!
1
6
2
G" (u;v)
!
1
>
< =>
>
>
:
"
jr j
2
jr j
and 0 v 1 a.e.
+
otherwise.
Then there exists the 0- lim"!0+ G"(u; v) = G(u; v) with respect to the
L1(
)-convergence, where
1
8Z
>
jr
>
>
< u dx +
G(u; v) = >
>
>
:
j
Z
Su
g( u+ u0 ) d n01 + Dc u (
)
j
j
0
H
j
j
+
2
if u 2 GBV (
)
and v = 1 a.e.
otherwise,
1
and
L1 (
)
g(z ) := min (x)z + 2cW (x) : 0 x 1 ;
with cW (x) := 2 x W (s) ds.
f
R1p
(17)
g
The proof of the theorem above will be a consequence of the propositions in
the rest of the section . Before entering into the details of the proof, we dene also
a `localized version' of our functionals as follows:
8Z 1 W (v) + " v 2 dx if u; v H 1(
)
>
>
(
v
)
u
+
>
<
"
G" (u; v;A) = > A
and 0 v 1 a.e.
jr j
>
>
:
and
2
jr j
+
otherwise.
1
8Z
>
jr
>
>
< A
u dx +
G(u; v; A) = >
>
>
:
j
Z
g( u+ u0 ) d n01 + Dc u (A)
Su \A
if u GBV (
) and v = 1 a.e.
j
0
j
H
j
2
+
otherwise,
1
7
j
for any A bounded open set.
Remark 4.2 By the assumptions on
satises the following properties
(i) g is increasing, g(0) = 0 and
2
z
and W , it can be easily proved that g
g(z ) = 2cW (0) = 4
!lim
+1
Z 1p
W (s) ds;
0
(ii) g is subadditive, i.e.
g(z1 + z2 ) g(z1 ) + g(z2 )
z1 ; z2 R+ ;
8
2
(iii) g is Lipschitz-continuous with Lipschitz constant 1;
(iv) g(z ) z for all z R+ and
lim g(z ) = 1;
2
! 0+ z
(v) for any T > 0 there exists a constant cT > 0 such that z cT g(z ) for all
z 2 [0; T ].
Proposition 4.3 Let n = 1. Then G(u; v) 0- lim inf "!0+ G" (u; v) for all u; v 2
L1(
).
z
Proof. It suces to consider the case in which the right-hand side is nite.
Let "j ! 0+ , uj ! u and vj ! v in L1 (
) be such that limj !+1 G"j (uj ; vj ) =
0- lim inf"!0+ G"(u; v). Up to passing to subsequences we may suppose
uj ! u;
We have
Z
and vj
! v a.e.
(18)
W (vj ) dx < c"j ;
08
91
hence, by the
continuity of W , for 91
any > 0 L1 x 2 : W (v(x)) > =
08
1
limj !+1 L x 2 : W (vj (x)) > = 0. We conclude that W (v) = 0 a.e., i.e.
v = 1 a.e.
By simplicity, suppose that = (a;b) (otherwise we split into its connected
components). We now use a discretization argument similar to the one used in the
proof of [3]. Let N 2 N and consider the intervals
INk = a + (k N0 1) (b 0 a); a + Nk (b 0 a) ;
8
k 2 f1;::; Ng:
Up to passing to subsequences we may suppose that
vj
!lim1 inf
INk
exists for all N 2 N and k 2 f1; ::; Ng. Let z 2 (0; 1) be xed and consider the set
j +
o
n
JNz = k 2 f1; ::;Ng : j !lim
inf vj z :
+1 I k
N
Note that for any (; ) interval in
Young's inequality,
Z 1
W
(w) + "jw0 j2 dx 2
"
R and for any w 2 H 1(; ) we
Z w ( )
p
Z p
W (w)jw0j dx 2
have, by
w ( )
W (s) ds:
From this inequality we deduce, arguing as in [3], that
Z 1p
2
z
W (s) ds #JNz j !lim
G"j (uj ; vj ) < +1:
+1
Then
z
#JN
C
C independent of N . Hence, up to a subsequence, we may suppose
JNz = fk1N ; ::; kLN g
with L independent of N , and up to a further subsequence that there exist S =
ft1 ; ::; tLg [a; b] such that
kiN = t
lim
i
N !+1 N
for any i 2 f1; ::;Lg. For every > 0 we have
INk S := S + [0; ]
z and for N large enough. Then
for all k 2 JN
lim inf G"j (uj ; vj ) lim inf G"j (uj ; vj ; n S )
j !+1
j !+1
with
+ lim inf
j !+1
L
X
Z =1
G (u ;v ; (t 0 ; t + ))
"j
j
j
i
i
i
0
lim
!+inf
1 (z ) n ju j dt
j
X
S
L
+
i
=1 j !+1
9
lim inf
j
G j (u ; v ; (t 0 ; t + )):
"
j
j
i
i
(19)
With xed i 2 f1; . . . ;Lg, we focus our attention on the term G j (u ; v ; (t 0
; t + )). By denition and by (18), we have that for any > 0 there exist
x1; x2 2 (t 0 ; t + ) such that
inf u + ;
u (x1) = u(x1) < ess!lim
+1
( i0 i+ )
u (x2) = u(x2) > ess- sup u 0 ;
!lim
+1
( i 0 i+ )
lim v (x1 ) =
lim v (x2 ) = 1:
(20)
!+1
!+1
"
j
j
i
i
i
i
j
j
j
j
i
j
i
j
j
estimate:
G j (u ; v ;I
j
t
;t
Z
x1 ;x2
] vj . Then we obtain the following
Z qW v jv0 j dx
Z ( ) pW s ds Z ( ) pW s ds
G j (u ; v ; (x1; x2))
2
(v (x ))
u0 dx + 2
1
(v (x ))ju (x2) 0 u (x1 )j
ki
)
N
j
j
j
x 2 [x1; x2] be such that v (x ) = inf [
"
;t
j
j
Let
t
"
j
j
i
j
j
i
j
j
n
t
;
j
x
j
x2
(
vj x 2
( )
( )
vj xij
+2
t ju (x2) 0 u (x1 )j
Z ( ) pW s ds Z
( )
j
j
( )
( )
vj x 2
+
t
( )
( )
vj xij
vj x 1
t
pW s dso:
( )
(21)
j ! +1 and taking into account (20), we get
lim inf G j (u ; v ; I i
!+1
t
Z
1p
o
W s ds :
u 0 ( 0 + )u 0 +)
>
u ; v ; t 0 ; t g
u0
u
n t 2[0 1]
( 0
"
j
j
k
N
j
( ) ess- sup
inf
;
ti
;ti
Thus, by the arbitrariness of
!+1 G j (
lim inf
j
"
j
j
(
ti
;ti
2
get the equiboundness of
(22),
i
+ ))
t
S
"
j
j
(ti0;ti +)
ess- sup
ess- inf
(ti 0;ti+)
(22)
G (u ; v ) < +1, by (19) we
0 j dt. Hence u 2 BVj (
n S ) and, by (19) and
j
u
"
j
j
j
X g
L
!+1 G j (u ; v ) (z )jDuj(
n S
lim inf
( )
+4
0,
i
R
n
ess- inf
Now we turn back to the estimate (19). Since supj
j
j
j
+2
Letting
j)
x1
vj x 1
+2
2inf
[0 1]
x
) +
i
10
=1
ess- sup
u 0 essinf u :
( i0 i+ )
(ti0;ti +)
t
;t
(23)
, we deduce that u 2 BV (
n S ), i.e., since S is nite,
u 2 BV (
). Then, letting ! 0 in (23), we get
By the arbitrariness of
!+1 G j (u ; v ) (z )jDuj(
n S ) +
lim inf
j
"
j
(z )jDuj(
n S
u)
j
+
X g ju+ 0 u0j t
(
2
( ))
X g ju+ 0 u0j t
L
i
=1
(
( i ))
^ (z )ju+ 0 u0 j(t) :
(24)
t Su
z ! 1 in (24) we obtain the required inequality, since g(t) t.
We recover, now, the n-dimensional analogue of the previous inequality, by
Finally, letting
using Theorem 3.4.
Proposition 4.4
L1(
).
Let n 2 N. Then G(u; v) 0- lim inf"!0+ G" (u; v) for all u; v 2
Proof. In the following we will use the notation G0 = 0- lim inf !0+ G .
Let 2 S 01 be xed and let be the hyperplane through 0 orthogonal to
. For any u 2 L1 (
), A 2 A(
), y 2 we set
A := ft 2 R y t 2 Ag; uy t u y t :
"
n
"
:
y
In particular, if
u 2 H 1 (
),
+
u; v
(
+
)
we get
u0y (t)
For any
( ) :=
:=
hru(y + t ); i:
2 H 1(
), 0 v 1, we have, by Fubini's Theorem,
G (u; v; A)
Z" Z
v y + t ))jru(y + t )j
( (
=
Ay
+
W (v(y + t )) + "jrv(y + t )j2 dt dHn01 (y)
Z Z "
Z
0
2
vy (t))ju0y j + W (vy (t)) + "jvy
dt dHn01(y)
(t)j
"
1
(
=
where
1
Ay
G"(uy ; vy ; Ay ) dHn
0
1
y ;
( )
(25)
G" is dened by
8Z 1
2
>
>
(
v
)
j
u
dt
W
(
v
)
+
"
j
v
j
+
j
>
< I
"
G"(u; v; I ) = >
>
>
:
0
00
1
+
11
if
u; v 2 H 1 (I )
and 0
v1
otherwise,
u; v
Let " j
for any
2 L1 (I ) and I R open and bounded.
! 0 and let uj ! u, vj ! v in L1 (
) be such that
lim inf
j ! +1
v
2 H 1 (
);
uj ; vj
Then
0
vj
G"j (uj ; vj ) +1:
1 a.e. and, as in the proof of Proposition 4.3,
uj )y
= 1 a.e. Moreover, by Fubini's Theorem, (
for
H
n01 -a.a.
2 .
y
(26)
! uy ; (vj )y ! 1 in L1 (
y )
Thus by Proposition 4.3 and by Fatou's Lemma we get
lim inf
j !+1
Z
G"j (uj ; vj ; A)
lim inf
j !+1
Z Z
G"j ((uj )y ; (vj )y ; Ay ) dHn
juy j dt +
Z
0
Ay
Suy \Ay
g(ju+
y
0
1
y
( )
0 uy j) d# + jD cuy j(Ay ) dHn
0
0
1
y :
( )
(27)
T >0
Let
Since
u
by
and set
0T ) _ (u ^ T ):
uT
= (
g is increasing, it is clear that we decrease the last term in (27) if we substitute
uT . Moreover, since uT 2 L1 (
), with kuT k1 T , by Remark 4.2(v), we
have
ju+T 0 uT j cT g(ju+T 0 uT j)
0
0
for a suitable constant
cT
depending only on
Z
jDuT j(Ay ) dHn
Thus, applying Theorem 3.4, we get that
uj )
of (
vj ),
uT
1
Then, by (26) and (27), we have
y < +1:
( )
2 BV (
) and, by the arbitrariness
and(
G0(u; 1; A) for all
0
T.
Z
A
A 2 A(
)
jhruT ; ij dx +
and
2 Sn
0
Z
Su \A
n0 1
0
c
g(ju+
+ jhD uT ; ij(A)
T 0 uT j)jhu ; ij dH
(28)
1.
Consider the superadditive increasing function dened on
(A)
:=
A(
) by
G0(u; 1; A)
and the Radon measure
:= Ln
+
0
n0 1
g(ju+
T 0 uT j)H
12
SuT
+
jDc uT j:
i )i2
Fixed a sequence (
N
, dense in
S n0 1 ,
(A)
for all
i2
N
we have, by (28),
Z
A
i d
, where
8 jhru (x); ij Ln a.e. on T
i
>
>
>
<
jhu(x); i ij jDc uT j a.e. on n SuT
i (x) =
>
>
>
:
n 1
jhu(x); i ij
H
0
a.e. on
SuT .
Hence, applying Lemma 3.3, we get
G (u; 1; A) Z
0
for all
A 2 A(
).
A
jruT j dx +
SuT \A
g(ju+
T
0 uT j) dHn
0
0
1
+
jDc uT j(A)
(29)
In particular
G (u; 1; ) Z
0
jruT j dx +
Finally, by the arbitrariness of
T
Z
! +1 in (30).
Proposition 4.5 We have -
Z
SuT
g(ju+
T
0 uT j) dHn
0
T > 0, u 2 GBV (
)
0
1
+
jDc uT j(
):
(30)
and the thesis follows letting
+ G"(u; v) G(u; v)
0 lim sup"!0
for all u; v 2 L1 .
(
)
Proof.
It suces to prove the inequality for v = 1 a.e. Since we will use
density and relaxation arguments, we divide the proof into ve steps, passing from
a particular choice of u to the general one. In the following we will use the notation
G00 = 0- lim sup"!0+ G" .
Step 1. Suppose that u
2 W (
) and
Su = \ K
with K a (n 0 1)-dimensional simplex. Up to a translation and rotation argument,
we can suppose that K is contained in the hyperplane := fxn = 0g. Set
h(y) := u+ (y) 0 u0 (y);
y
2 S u:
By our hypotheses on u, h is regular on S u ; hence, xed > 0, we can nd a
triangulation fTi gN
i=1 of S u such that
jh(y1) 0 h(y2)j < 13
if y1 ; y2
2 Ti :
Let h : S u
! R be dened as
h (y) := zi
where zi := min fh(y) : y
have that
Z
Su
y
2 Ti ;
2 T ig. Since kh 0 h k
1
g(h (y)) dHn01
Z
Su
< , by Remark 4.2 (iii), we
g(h(y)) dHn01 + Hn01(S u ):
Let xzi realize the minimum in (17) for z = zi . Fixed > 0, there exists T () > 0
such that
min
nZ T jv j2 + W (v) dt : v 2 H 1 (0; T );
0
0
o
v(0) = xzi ; v(T ) = 1
cW (xz ) + i
(31)
for all T T () and for any i = 1; ::; N . Let v(zi ; 1) realize the minimum in (31).
For r > 0; " > 0 and i 2 f1; ::; N g, set
n
Br := (y; t) 2 : y
R(1n
and let i" :
kr k
i"
1
< C"
0
", and dene
1)
2 S u ; jtj < r
o
and
n
Ti" := y
2 Ti :
o
d(y; @Ti ) > " ;
R be a cut-o function between
Ti"
!
and Ti such that
. Fix a sequence (" ) such that lim"!0+ "" = 0, set T" := T ()" +
0
8
<1
v" (y; t) :=
: i (y)vi (t) + (1 0 i (y))
"
"
"
8x
< z
v"i (t) :=
: v zi ;
where
i
jtj0"
"
if
(y; t)
if
y
2 n BT
"
2 Ti ; jtj < T" ,
if
jtj < "
if
" < jtj < T" .
2 H 1(
) and v" ! 1 in L1 (
) as " ! 0+. Hence, we get
Z 1
"jrv"j2 + W (v" ) dx
(32)
"
Z T N Z
X
jtj 0 " 2 + W vz ; jtj 0 " dt dHn 1(y)
1 v zi ;
2
i
"
"
"
We have that (v" )
"
=
i=1
Ti"
+
+
0
"
N Z Z
X
Ti
i=1
Ti nTi"
XZ
"
i=1
N
0 "
0
Z
1
"jri"(y)j2 jzi 0 1j2 + W (v" (y; t)) dt dHn01(y)
"
T"
"
"jri"(y)j2 v zi ;
14
jtj 0 " 0 12
"
jtj 0 " 2 j
i"(y)j2 v zi ;
dt dHn 1(y)
"
"
ZT
N Z
X
1
+
W (v" (y; t)) dt dHn 1 (y)
"
+
1
0
0
"
i=1
XZ
N
0
"
Z T
jv (zi ; t)j2 + W (v(zi ; t)) dt dHn
2
0
Ti"
i=1
Ti nTi"
0
1
0
(y)
X
" n 0 1
H (Su ) + c() Hn01 (Ti n Ti" )
"
i=1
N
+c
Z
N
X
2
i=1
Ti
cW (xzi ) dHn01 (y) + 2Hn01 (Su ) + O("):
We now construct a recovery sequence u" . Let
8z
0T" < t < 0"
1
>
>
>
<
z2 z1
u
~"(z1 ; z2 ; t) =
2 (t + " ) + z1 jtj < "
>
>
>
:
0
"
z2
" < t < T"
8 u(y; t)
jtj > T"
<
u"(y; t) =
: u~" u(y; 0T" ); u(y; T" ); t jtj < T" .
and set
2
It can be easily veried that u"
Moreover, we have
Z
(v" )jru"j dx
Z
N Z
X
Ti"
i=1
Z
"
H 1(
) and u"
1
!
u in L1 (
) as "
!
0+ .
(xzi )ju(y; T" ) 0 u(y; 0T" )j dt dHn01(y)
0" 2"
jruj dx + cHn 1(Ti n Ti" ) + O(")
Z
N Z
X
jruj dx +
(xz )ju+ 0 u j(y) dHn 1 (y) + O("):
+
=
0
nBt"
i=1
Ti
i
0
0
(33)
Letting, now, " tend to 0+ , we obtain, by (32) and (33),
G00(u; 1) lim sup G"(u" ; v")
Z
"! 0 +
N
jruj dx +
XZ
i=1
Ti
(ju+
0 u j(y)
0
15
(xzi ) + 2cW (xzi )) dHn01 (y) + c
Z
=
Z
Z
jruj dx +
jruj dx +
jruj dx +
N Z
X
Zi=1
Z
Su
Su
Ti
(zi (xzi ) + 2cW (xzi )) dHn01 (y) + c( + )
g(h (y)) dHn01 (y) + c( + )
g(ju+ 0 u0j(y)) dHn01 (y) + c( + ):
Letting and tend to 0+ , we obtain the required inequality.
In order to use the same costruction as above in the case S u = \
SM
i=1 Ki ,
with M > 1, we now show that we can replace (u" ) by a new sequence (^
u") such
that u
^" 6= u only in a small neighbourhood of K . To this end we again use a cut-o
argument. Set
K" := fy 2 : d(y; K ) < "g
and let " :
c"01. Dene
Rn
0
1
! R be a cut-o function between K
and K" with
u
^" (y; t) := " (y)u" (y; t) + (1 0 " (y))u(y; t)
(y; t)
jr"j 1
2 :
We have
u
^" (y; t) = u"(y; t)
if (y; t)
u
^" (y; t) = u(y; t)
if (y; t)
Then
Z
nBT"
jru^"j dx Z
"
(34)
jruj dx
ZT jr"(y)jju" (y; t) 0 u(y; t)j dt dHn 1(y)
K)
T
ZT "(y)jru" (y; t)j
K)
T
+(1 0 "(y))jru(y; t)j dt dHn 1(y)
nK" 2(0T";T" )
Z
+
"
\ (K " n
Z
+
2 BT ;
2 n K" 2 (0T" ; T"):
Z
0
0
"
"
\ (K " n
0
"
0
jruj dx + c T"" Hn
Z
Thus
lim sup
"! 0 +
nBT"
0
1
(K"
jru^"j dx =
n K ) + O("):
Z
jruj dx;
and, by (34), we still have
lim sup G" (^
u"; v" )
"! 0 +
G(u; 1) + c( + ):
16
Step 2. If u
2 W (
) with S u = \
SM
i=1 Ki , we can generalize in a very
natural way the construction of the recovery sequences u
^" and v" in Step 1, since
this construction modies u and v only in a small neighbourhood of each sets Ki .
Step 3. Let u 2 SBV 2 (
) \ L1 (
). Then, applying Theorem 3.1 with
(a; b; ) = g(ja 0 bj), there exists a sequence (wj ) 2 W (
) such that
wj
! u in L1 (
);
and lim sup G(wj ; 1)
j !+1
G(u; 1):
Then, by the previous steps and by the lower semicontinuity of G00
G00(u; 1) lim inf G00(wj ; 1) lim inf G(wj ; 1) G(u; 1):
j !+1
j !+1
Step 4. Since g satises the hypotheses of Theorem 3.2, the relaxation with
respect to L1 (
)-topology of the functional
8
< G(u; 1)
F (u) :=
: +1
is given by
if
u 2 SBV 2(
) \ L1 (
)
otherwise in BV (
)
F (u) = G(u; 1)
for all u 2 BV (
). Then by the previous steps and by the lower semicontinuity of
G00 we get
G00(u; 1) F (u) = G(u; 1)
for any u 2 BV (
).
Step 5. We recover the general case by a truncation argument. Let u
GBV (
) and let uj = (0hj ) _ (u ^ j ). Then
2
lim G(uj ; 1) = G(u; 1):
j !+1
Since uj
! u in L1 (
) we get the thesis by the lower semicontinuity of G
00
Example 4.6
.
Let W (v) = (1 0 v)2 =4, so that cW (z ) = (1 0 z )2 =2. We then have
(a) if (v) = v2 then g(z ) =jz j=(1 + jz j);
jz j 0 (z 2 =4) if jz j 2
(b) if (v) = v then g(z ) =
1
if jz j > 2;
0 if v = 0
(c) if (v) =
then g(z ) = minfjz j; 1g.
1 otherwise,
Note that in the rst case we always have interaction between the bulk term
and the `surface term' of G" (i.e. x 6= 1 in the denition of g) contrary to what
happens in the Ambrosio Tortorelli approach. The interaction also occurs in the
second case for jz j < 2. Note moreover that in the third case the minimal x in the
denition of g(z ) does not vary with continuity at z = 0.
17
Approximation of general functionals
5
In this section we show how Theorem 4.1 can be used to obtain an approximation
of general (isotropic) energies dened on
GSBV .
Let be a bounded open subset of Rn, let W and be dened
as in Theorem 4.1, let f : [0; + ) [0; + ) be a convex function with minimum
in 0 satisfying
Proposition 5.1
1!
1
f (t) = 1;
t!+1 t
1
1
and let G" : L (
) 2 L (
) ! [0; +1) be dened by
lim
(35)
Z
8
1
2 dx
>
(v )f ( jruj) + W (v ) + "jrv j
>
<
"
G"(u;v) = > >
:
1
+
Then there exists the 0- lim"!0+ G"(u; v)
L1(
)-convergence, where
=
if u; v
and 0
2 H1
v
(
)
1
a.e.
otherwise.
G(u; v) with respect to the L1 (
) 2
Z
Z
8
+
0 n0 1
c
>
>
< f (jruj)dx + Su g(ju 0 u j)dH + jD uj(
)
G(u; v) = >
>
:
if u
and
1
2 GBV
(
)
v = 1 a.e.
otherwise,
+
and g is dened in (17).
Proof. The estimate for the 0-lim inf can be performed as in Proposition
4.3, noting that in (21) we obtain, by Jensen's inequality,
G" (uj ; vj ; INk0i ) (vj (xij ))jx2 0 x1 jf u(xx2) 00 ux(x1) + 2
2
j
Zx q
1
2
x1
W (vj ) vj dx;
j
0
j
from which the lower bound can be easily obtained taking into account (35). The
rest of the proof can be obtained following Propositions 4.4 and 4.5.
Remark 5.2 Let K > 0 and N
2, let
a0 < a1 < < aN = 1; 0 = bN < bN 1 < b0 = K;
and let f and W be as in the previous proposition. Then there exists
satisfying
L1(
) [0; + ) is
the hypotheses in Theorem 4.1 such that, if G" : L1 (
)
0=
111
0
111
2
18
!
1
8> Z >< v f u K" W v "K v 2 dx
G" u;v >
>:
dened by
( ) (jr
(
j)
+
( )+
jr j
)=
+1
then the thesis of the previous proposition holds with
by
u;v H 1(
)
and 0
v 1 a.e.
if
2
otherwise,
g : [0; +
1) !
;
[0 +1) given
g(z) = min aiz + bi :
In fact, in this case the formula for g can be easily inverted, obtaining
f
piecewise constant function given by
ai
g
as the
(0) = 0 and
c 1 b = < cW1l(bi=2);
0
( )=
if W ( i01 2)
0
cW is dened in Theorem 4.1.
Proposition 5.3 Let W be as in Theorem 4.1. Let ';# : [0; + )
[0; + ) be
functions satisfying
(i) ' is convex and even, limt + '(t)=t = + ;
(ii) # is concave and even, limt 0 #(t)=t = + .
Then there exist two sequences of functions ('j ) and ( j ), and two sequences
of positive real numbers (kj ) and ("j ), converging to sup # and 0, respectively, such
where
1
!
1
1
1
!
!
+
1
that if we dene
8> Z >< j v 'j u k"jj W v kj "j v 2 dx if u;v H 1
Gj u;v >
and
v
>:
otherwise,
( )
(
(jr
j) +
( )+
)=
0
+1
then there exists the 0- limj
L1(
)-convergence, where
!
+1
(jr
(
j)
+
(
0
0
)
Proof. Let #j : [0; +
1) !
1
111
j
j =
2
if u 2 GSBV (
)
and v = 1 a.e.
otherwise.
;
#j (z) = min Aji z + Bij ;
g
A0 < < A j converging increasingly to #, and let 'j : [0; +
j
a.e.
[0 +1) be functions of the form
f
with 0 =
n01
H
)=
+1
(
)
Gj (u;v) = G(u;v) with respect to the L1(
)
Z +
8> Z
'
u
dx
#u u d
>< Su
G u;v >
>:
;
2
jr j
1) !
[0 +1) be convex even functions with
lim
t! + 1
'j (t) = j;
t
19
'. Let kj = max 'j .
gj = #j =j, Kj = kj =j and fj = 'j =j. By the previous remark, applied
converging increasingly to
Set
with g = gj , f = fj and K = Kj , we can nd
G"j : L1(
) L1 (
)
[0; + ) be dened by
2
Z
8
>
>
<
G (u; v) =
>
>
:
!
=:
1
jruj) + k"j W (v) + "kj jrvj2
j (v )'j (
j
"
j
dx
+1
such that if we let
if u; v 2 H 1 (
)
and 0 v 1 a.e.
otherwise,
then there exists the 0- lim"!0+ Gj"(u; v) = Gj (u; v) with respect to the L1(
) 2
L1(
)-convergence, where
Z
Z
8
+
0
01
>
>
< ' (jruj)dx + u # (ju 0 u j)dH + jjD uj(
)
G (u; v) =
if u 2 GBV (
) and v = 1 a.e.
>
>
:
j
j
n
j
S
+1
c
otherwise.
Since the functionals Gj converge increasingly to G, they also 0-converge to G
as j ! +1. By the metrizable character of 0-convergence, we can then nd a
sequence ("j ) of real numbers converging to 0 such that Gj"j 0-converges to G,
that is, the thesis.
Remark 5.4 If ' is convex and even, # is concave and even, and
t
!+lim
inf ty
'(t)
#(t)
= lim+
= M;
t
t
t o
!
then there exist ('j ), ( j ) (kj ) and ("j ) such that the functionals Gj dened above
0-converge with respect to the L1 (
) 2 L1 (
)-convergence to
Z
8Z
>
'
(
jr
u
j
)
dx
+
>
<
G(u; v) =
>
>
:
Su
j 0 u0 j)dHn01 + M jDc uj(
)
if u 2 GBV (
) and v = 1 a.e.
#( u+
+1
otherwise.
The proof can be obtained directly from Remark 5.2, using the approximation
argument of Proposition 5.3.
Acknowledgements This paper was written while the second author was visiting
the Max-Planck-Institute for Mathematics in the Sciences at Leipzig, on MarieCurie fellowship ERBFMBICT972023 of the European Union program \Training
and Mobility of Researchers".
20
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Comm. Pure Appl. Math. 17
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