1
2
Modica–Mortola Functional
-Convergence
Let (X; d) be a metric space and consider a sequence fFn g of functionals Fn :
X ! [ 1; 1]. We say that fFn g -converges to a functional F : X ! [ 1; 1]
if the following properties hold:
(i) (Liminf Inequality) For every x 2 X and every sequence fxn g
such that xn ! x,
F (x) lim inf Fn (xn ):
(1)
(ii) (Limsup Inequality) For every x 2 X there exists fxn g
xn ! x and
lim sup Fn (xn ) F (x):
(2)
n!+1
X
X such that
n!1
Remark 1 An important property of -convergence is that if each Fn is bounded
from below and admits a minimizer xn , that is,
Fn (xn ) = min Fn (y) ;
y2X
if xn ! x for some x 2 X, and if fFn g -converges to F , then x is a minimizer
of F and
min F (y) = F (x) = lim Fn (xn ) = lim min Fn (y) :
n!+1
y2X
n!+1 y2X
To see this, let y 2 Y and apply property (ii) to …nd a sequence fyn g
that yn ! y and
lim sup Fn (yn ) F (y):
X such
n!1
Using the fact that Fn (xn ) = miny2X Fn (y) and property (i) for the sequence
fxn g, we have that
F (x)
lim inf Fn (xn )
n!+1
lim sup Fn (xn )
n!1
lim sup Fn (yn )
F (y);
n!1
which shows that x is a minimizer of F . Taking y = x, gives that there exists
limn!+1 Fn (xn ) = F (x).
3
Compactness
For " > 0 consider the functional
F" : W 1;2
; Rd ! [0; 1]
1
de…ned by
F" (u) :=
Z
1
W (u) + "jruj2
"
dx;
where the double well potential W : Rd ! [0; 1) satis…es the following hypotheses:
(H1 ) W is continuous, W (z) = 0 if and only if z 2 f ; g for some
with 6= .
;
2 Rd
(H2 ) There exist L > 0 and R > 0 such that
W (z)
for all z 2 Rd with jzj
L jzj :
R.
Theorem 2 Let
RN be an open bounded set with Lipschitz boundary.
Assume that the double-well potential W satis…es conditions (H1 ) and (H2 ).
Let "n ! 0+ and let fun g W 1;2 ; Rd be such that
M := sup F"n (un ) < 1:
(3)
n
Then there exist a subsequence funk g of fun g and u 2 BV ( ; f ; g) such that
unk ! u for in L1
; Rd :
Proof. We begin by showing that fun g is bounded in L1 ; Rd and equiintegrable. By (3) and (H2 ), for every t R,
Z
Z
1
W (un (x)) dx M "n ;
jun j dx
L fjun j tg
fjun j tg
which implies that fun g is equi-integrable. Moreover, since has …nite measure,
Z
Z
Z
L jun j dx = L
jun j dx + L
jun j dx
fjun j Rg
fjun j<Rg
Z
L
W (un (x)) dx + LR j j M "n + LR j j :
fjun j Rg
In view of the Vitali’s convergence theorem, to obtain strong convergence of a
subsequence, it su¢ ces to prove convergence in measure or pointwise LN a.e.
in . We divide the proof in two steps.
Step 1: Assume …rst that d = 1. De…ne
W1 (z) := min fW (z) ; 1g ;
Since 0
W1
z 2 R:
W , for every n 2 N, we have
Z p
1
F"n (un )
W1 (un (x))jrun (x) j dx
2
Z
=
(jr ( 1 un ) (x) j) dx;
2
where
1
2
1 (t) :=
Then by (3),
sup
n
Moreover, since
p
W1
Z tp
W1 (s) ds;
0
Z
(jr (
j
1
un ) j) dx
1
t 2 R:
(4)
M:
(5)
1,
(un (x))j
jun (x)j
for LN a.e. x 2 and for all n 2 N. Since fun g is bounded in L1 ( ), it follows
that the sequence f 1 un g is bounded in L1 ( ). By the Rellich–Kondrachov
theorem, there exist a subsequence funk g and a function w 2 BV ( ) such that
wk :=
1
unk ! w in L1loc ( ) :
By taking a further subsequence, if necessary, without loss of generality, we may
assume that wk (x) ! w (x) and that W (unk (x)) ! 0 for LN a.e. x 2 . Since
the function W1 (t) > 0 for all t 6= ; , it follows from (4) that the function 1
is strictly increasing and continuous. Thus, its inverse 1 1 is continuous and
unk (x) =
1
1
1
(wk (x)) !
1
(w (x)) := u (x)
for LN a.e. x 2 . It follows by (H1 ) and the fact that W (unk (x)) ! 0 for
LN a.e. x 2 , that u (x) 2 f ; g for LN a.e. x 2 . In turn, w (x) 2
f 1 ( ) ; 1 ( )g for LN a.e. x 2 , and since w 2 BV ( ), we may write
w=
for a set E
1
( )
E
+
1
( ) (1
E)
(6)
of …nite perimeter. Hence,
u=
belongs to BV ( ).
Step 2: Assume that d
E
+
(1
E)
(7)
2 and that j j =
6 j j. For every t
0 de…ne
V (t) := min W (z) :
jzj=t
Then V is upper semicontinuous, V (t) > 0 for t 6= j j, j j, V (j j) = V (j j) = 0,
and V (t) Lt for t R. For every u 2 W 1;2 ; Rd de…ne
G" (u) :=
Z
1
V (juj) + "jr juj j2
"
dx
F" (u) :
Then by (3),
sup G"n (un )
n
sup F"n (un ; ) < 1;
n
3
and so by the compactness in the scalar case d = 1, there exist a subsequence
funk g and w 2 BV ( ) such that
wk :=
2
where
1
2 (t) :=
2
and
junk j ! w in L1loc ( ) ;
Z tp
V1 (s) ds;
t2R
0
V1 (z) := min fV (z) ; 1g ;
z 2 R:
Hence,
junk j ! v :=
1
w in L1 ( ) :
2
By taking a further subsequence, if necessary, without loss of generality, we may
assume that junk (x)j ! v (x) and that W (unk (x)) ! 0 for LN a.e. x 2 .
This implies that v 2 BV ( ; f ; g). De…ne
if v (x) = j j ;
if v (x) = j j :
u (x) :=
We claim that
unk ! u in L1
; Rd :
To see this, …x x 2
such that junk (x)j ! v (x) and that W (unk (x)) ! 0.
Then, by (H1 ), necessarily, unk (x) ! u (x).
Step 3: If d 2 and j j = j j, let ei be a vector of the canonical basis of
Rd such that
ei 6=
ei . Then j + ei j =
6 j + ei j. It su¢ ces to apply the
previous step with W replaced by
^ (z) := W (z
W
ei );
z 2 Rd ;
and un by un + ei .
4
Gamma Convergence of the Modica–Mortola
Functional
In view of the previous theorem, the metric convergence in the de…nition of convergence should be L1 ; Rd . Thus, we extend F" to L1 ; Rd by setting
8 Z
1
<
W (u) + "jruj2 dx if u 2 W 1;2 ; Rd ;
F" (u) :=
"
:
1
if u 2 L1 ; Rd n W 1;2 ; Rd :
Let "n ! 0+ . Under appropriate hypotheses on W and , we will show that
the sequence of functionals fF"n g -converges to the functional
F (u) :=
cW jDuj ( )
1
if u 2 BV ( ; f ; g) ;
if u 2 L1 ; Rd n BV ( ; f ; g) :
4
We begin with the liminf inequality. Consider a sequence fun g
such that un ! u in L1 ; Rd for some u 2 L1 ; Rd . If
L1
; Rd
lim inf F"n (un ) = 1;
n!+1
then there is nothing to prove, thus we assume that
lim inf F"n (un ) < 1:
n!+1
(8)
Let f"nk g be a subsequence of f"n g such that
lim inf F"n (un ) = lim F"nk (unk ) < 1:
n!+1
k!+1
Then F"nk (unk ) < 1 for all k su¢ ciently large. Hence, unk 2 W 1;2 ; Rd
for all k su¢ ciently large. Moreover, if hypotheses (H1 ) and (H2 ), then by
Theorem 2, u 2 BV ( ; f ; g). Finally, by extracting a further subsequence,
not relabelled, we can assume that un (x) ! u (x) for LN a.e. x 2 .
Hence, in what follow, without loss of generality, we will assume that (8)
holds, that fun g W 1;2 ; Rd , that u 2 BV ( ; f ; g), that lim inf n!+1 F"n (un )
is actually a limit, and that fun g converges to u in L1 ; Rd and pointwise
LN a.e. in .
5
Liminf Inequality, N = 1, d = 1
We begin with the case N = d = 1 and assume that (H1 ) and (H2 ) hold with
<
and that
= (a; b). Consider u 2 BV ( ; f ; g). Without loss of
generality we may assume that there exists a partition
a = t0 < t1 <
< tm = b
such that u (x) = in (t2i 2 ; t2i 1 ) and u (x) = in (t2i 1 ; t2i ). Let "n ! 0+
and let fun g W 1;2 ( ) be such that un ! u in L1 ( ) and pointwise L1 a.e.
in . Fix > 0 small. Then
Z
m Z ti +
X
1
1
W (un ) + "n ju0n j2 dx
W (un ) + "n ju0n j2 dx:
"n
"
n
t
i
i=1
Consider one term
Z
ti +
1
W (un ) + "n ju0n j2 dx:
"
ti
For simplicity we can assume that ti = 0 and, by taking
smaller, that
un ( ) ! and un ( ) ! . Then
Z ="n
Z
1
0
2
W (un (x)) + "n jun (x) j
dx =
W (un ("n y)) + "2n ju0n ("n y) j2 dy
"n
="n
(9)
Z ="n
=
W (vn (y)) + jvn0 (y) j2 dy;
="n
5
where we have made the change of variables x = "n y and vn (y) := un ("n y). It
follows that
Z
Z ="n
1
2
W (vn (y)) + (vn0 (y)) dy
W (un (x)) + "n ju0n (x) j2 dx
"n
="n
Z ="n p
2 W (vn )vn0 (y) dy
=
Z
="n
un ( )
un (
)
Z
p
2 W (s) ds !
p
2 W (s) ds
as n ! 1, where we have made the change of variables s = wn (y). In turn,
lim inf
n!1
6
Z
1
W (un ) + "n ju0n j2
"n
dx
Z
p
2 W (s) ds (number of jumps of u).
Limsup Inequality, N = 1, d = 1
Assume in addition to (H1 ) and (H2 ) that W is of class C 1 . Let g be the
solution of the Cauchy problem
g0 =
p
W (g);
Then g is globally de…ned, strictly increasing,
lim g (t) =
cW :=
Z
R
< g (t) <
= 2 lim
n!1
Hence, if
n
Z g(n)
W (g (t)) + jg 0 (t)j
2
g( n)
n!1
p
W (s) ds = 2
for all t, and
t!1
Note that by (10) we have
Z n
Z
2
0
cW = lim
W (g (t)) + jg (t)j dt = 2 lim
n!1
(10)
lim g (t) = :
t! 1
De…ne
+
:
2
g (0) =
Z
p
W (s) ds:
= R and
if y < 0;
if y 0;
u (y) =
then the right sequence would be
un (x) := g
6
x
"n
;
dt:
n
n
(11)
p
W (g (t))g 0 (t) dt
since
Z
R
1
W (un (x)) + "n ju0n (x) j2
"n
dx =
Z
1
W (g
"n
R
=
Z
1
x
) + "n 2 g 0
"n
"n
x
"n
2
u0n
(x)
W (g (y)) + jg 0 (y) j2 dy = cW :
R
In the case of a general u 2 BV ( ; f ; g), we need to glue g to and . Let
u be as in the previous section. Fix > 0 and let b be such that g (b ) =
and let a be such that g (a ) = + . De…ne
8
if y b + 1;
>
>
>
>
if b < y < b + 1;
< (y b ) +
g (y)
if a
y b ;
g (y) :=
>
>
(y
a
)
+
+
if
a
1
<y<a ;
>
>
:
if y a
1;
and
un; (x) :=
8
>
>
< g
x t2i
"n
t2i+1 x
"n
if t2i
if t2i+1 r < x < t2i+1 + r;
otherwise,
g
>
>
:
u (x)
where 2r < min fti+1 ti : i = 0; : : : ; m
x
ables x "nt2i = y or t2i+1
= y we get
"n
Z
1
W (un; ) + "n ju0n; j2
"n
dx =
=
1g. Then using the change of vari-
m Z
X
ti +r
1
W (un; ) + "n ju0n; j2
"n
i=1 ti r
m Z r="n
X
r=an
i=1
=
r < x < t2i + r;
m Z
X
i=1
dx
W (g (y)) + jg 0 (y) j2 dx
b +1
a
W (g (y)) + jg 0 (y) j2 dx
1
for all n su¢ ciently large.
Since W ( ) = 0, by the mean value theorem
W (t) = W ( ) + W 0 ( ) (t
for some
W (g (y))
) = 0 + W 0 ( ) (t
)
between t and . Hence, in [b ; b + 1],
(
g (y)) max jW 0 j = ( ( (y
[ ; ]
b )
)) max jW 0 j
[ ; ]
0
g (y) = :
It follows that
Z
a
b +1
1
W (g (y)) + jg 0 (y) j2 dx
7
C ;
2 max jW 0 j ;
[ ; ]
!
dx
where we have made the change of variables s = wn (y). On the other hand,
Z b
Z b
W (g (y)) + jg 0 (y) j2 dx =
W (g (y)) + jg 0 (y) j2 dx
a
a
Z
W (g (y)) + jg 0 (y) j2 dy = cW :
R
Hence, we get
Z
1
W (un; ) + "n ju0n; j2
"n
dx
cW m + C m:
1
W (un; ) + "n ju0n; j2
"n
dx
cW (number of jumps of u).
In turn,
lim sup lim sup
!0+
n!1
Z
x t2i
"n
Moreover, using the change of variables
Z
jun;
uj dx =
m Z
X
i=1
=
m
X
i=1
ti +r
jun; (x)
ti r
"n
= y or
Z
u (x) j dx =
t2i+1 x
"n
m
X
i=1
"n
Z
= y,
r="n
r=an
jg (y)
v (y)j dy
(12)
b +1
a
1
jg (y)
v (y)j dy
as n ! 1, where
C "n ! 0
if y > 0;
if y 0:
v (y) :=
We now diagonalize the sequence fun; g to obtaining a sequence fun; n g converging to u in L1 ( ) and such that
Z
1
lim
W (un; n ) + "n ju0n; n j2 dx cW (number of jumps of u).
n!1
"n
7
Liminf Inequality, N = 1, d
1
In this case the constant cW should be replaced by
Z
cW := inf
W (g (t)) + jg 0 (t) j2 dt :
(13)
R
1
g 2 Hloc
R; Rd
such that
lim g (t) =
t! 1
We proceed as in the case d = 1 up to (9). Fix
and un ( ) ! , we have that
jvn (
="n )
j = jun (
)
j< ;
8
lim g (t) =
t!1
> 0 small. Since un (
jvn ( ="n )
j = jun ( )
:
)!
j<
for all n large.
We now extend vn to R by setting
8
>
>
>
>
vn ( ="n )) (y
="n ) + vn ( ="n )
< (
vn (y)
wn (y) :=
>
>
) (y + ="n ) + vn ( ="n )
> (vn ( ="n )
>
:
if y
="n + 1;
if ="n < y < ="n + 1;
if
="n y
="n ;
if
="n 1 < y <
="n ;
if y
="n 1;
Since W ( ) = 0 by the mean value theorem
W (z) = W ( ) + rW ( ) (z
for some
) = 0 + rW ( ) (z
)
in the segment between z and . Hence, in [ ="n ; ="n + 1],
W (wn (y))
jvn (y)
j max jrW j
B( ;2)
C ;
wn0 (y) = :
A similar estimate can be made in the interval [ ="n 1; ="n ]. It follows
that
Z ="n
Z
W (vn (y)) + jvn0 (y) j2 dy
W (wn ) + jwn0 j2 dy C ;
="n
R
cW
C :
In turn,
lim inf
n!1
Z
1
W (un ) + "n ju0n j2
"n
dx
! 0 gives
Z
1
lim inf
W (un ) + "n ju0n j2
n!1
"n
cW (number of jumps of u)
C :
Letting
8
dx
cW (number of jumps of u):
Limsup Inequality, N = 1, d
1
1
Fix
> 0 and by (13) …nd g 2 Hloc
R; Rd such that limt!
limt!1 g (t) = and
Z
2
W (g (t)) + jg 0 (t)j dt cW + :
R
9
1
g (t) =
,
(14)
Let u be as in the previous section. Fix > 0 and let b >> 1 be such that
jg (b )
j < and let a << 1 be such that jg (a )
j < . De…ne
8
if y b + 1;
>
>
>
>
g (b )) (y b ) + g (b ) if b < y < b + 1;
< (
g (y)
if a
y b ;
g (y) :=
(15)
>
>
(g
(a
)
)
(y
a
)
+
g
(a
)
if
a
1
<
y
<
a
;
>
>
:
if y a
1;
and
un; (x) :=
8
>
>
< g
x t2i
"n
t2i+1 x
"n
if t2i
g
>
>
:
u (x)
r < x < t2i + r;
if t2i+1 r < x < t2i+1 + r;
otherwise,
where 2r < min fti+1 ti : i = 0; : : : ; m 1g. Reasoning as in the previous
sections we get
Z
m Z ti +r
X
1
1
0
2
W (un; ) + "n jun; j
dx =
W (un; ) + "n ju0n; j2 dx
"n
"
n
i=1 ti r
m Z r="n
X
W (g (y)) + jg 0 (y) j2 dx
=
=
r=an
i=1
Z
m
b
+1
X
i=1
m
Z
R
a
1
W (g (y)) + jg 0 (y) j2 dx
2
W (g (t)) + jg 0 (t)j
dt + C
(cW + ) m + C
for all n su¢ ciently large. Note that we can take = . It follows that
Z
1
lim sup lim sup
W (un; ) + "n ju0n; j2 dx cW m
"
+
n!1
n
!0
Z
and as in (12), jun;
uj dx ! 0 as n ! 1. Again we can diagonalize to get
a sequence fun; n g.
9
Liminf Inequality, N
1, d = 1
10
Liminf Inequality, N
11
Limsup Inequality, N
For N
1, d
1, d
1, given u 2 BV ( ; f ; g), we write
u=
E
+
10
nE :
1
1
Assume …rst that E is a regular set, that is, that E is an open set with \
@E of class C 2 , and that E meets the boundary of
transversally, that is,
HN 1 (@E \ @ ) = 0. Let g and g be as in (14) and (15). In this case we take
8
if dist (x; @E) < "n L
<
g (dist (x; @E) ="n ) if jdist (x; @E)j "n L ;
un; (x) :=
:
if dist (x; @E) > "n L ;
where dist is the signed distance, and L > 0 is such that g (L ) =
and
g ( L ) = . Note that the function dist (x; @E) is of class C 2 in the set
x 2 RN : jdist (x; @E)j < r
for r small, with jr dist (x; @E)j = 1. Moreover,
lim HN
r!0+
1
x 2 RN : dist (x; @E) = r
= HN
1
(@E) :
Hence, by the coarea formula
Z
1
W (un; ) + "n jrun; j2 dx
"n
Z
1
=
W (g (dist (x; @E) ="n )) + jg 0 (dist (x; @E) ="n ) j2 dx
"n jdist(x;@E)j "n L
Z "n L
1
=
W (g (r="n )) + jg 0 (r="n ) j2 HN 1 x 2 RN : dist (x; @E) = r
"n
"n L
Z L
=
W (g (r)) + jg 0 (r) j2 HN 1 x 2 RN : dist (x; @E) = "n r dr
Z
!
L
L
In turn,
Z
lim sup
n!1
dr
L
W (g (r)) + jg 0 (r) j2 drHN
1
W (un; ) + "n jrun; j2
"n
and so taking
dx
1
(@E) :
Z
R
W (g (t)) + jg 0 (t)j
+ C HN
and using (14),
Z
1
W (un; ) + "n jrun; j2
lim sup lim sup
"n
n!1
!0+
1
2
dt HN
(@E) ;
=
11
dx
cW HN
1
(@E) :
1
(@E)