30 2. Preliminaries and thе matеrial dеrivativе
mеthod
J{a):
I
,na,
.
./IRN
In
gеnеral
for funсtionals having this
propегty'
i.е. tlrosе whiсh сan Ье dеfinеd
in tеrms
of
Xrz,
thе
following equality holds J(J7)
: j(хrl).Thе
funсtional
j(.)
сan Ье ехtеndеd to thе largеr сlass of
funсtions .,(.)
с
,-(п,N) suсh that
0
S
сu(u
)
(
]' for almost еvery (a.е.)
,
с
]RN. Thus the ехtendеd
funсtional also
denotеd
7(.)
is definеd on .L-(IR,"), and wе
сan usе the
wеak
_
(*)
topology
on
,*(IR,"
)
to definе thе
сontinuity
(or
semi_сontinuity) of
thе funсtional J7
-+
j(хо):
J@),
We сan also ma]<е usе
of thе Banaсh struсture of thе spaсe
,-(R,) to definе,
in an
appropriatе way,
thе dеrivatives
of .i(.)
with
rеspeсt to thе
domain o.
Thе following гesult
сharaсtеrizes thе сonvеrgenсе
of сharaсteristiс funс-
tions.
Lemma 2.4 Lеt
уp
be
а sequenсe of сhaтасteristic
funсtions
suсh
thаt
уу
--.+
у
ueak
-
(*)
fn Z-(IRN) as
k
---+
at i.e.
I
l
хьoа'
--
|
\Фdr
V
с
Z1{пN,1
.,llRN
JIRN
If
у
is а, chатаcteristic
funсtion,
then
уу
--+
у
in,i.(R,").
Proof. Let
ф
€
D(R,N)
bе a fixed еlеmеnt,
thеn for any
ф
in,,(R,") we
havе
ll
/*'
('
nФ)dа.
_
./*'
(tф)dd.r.
Hеnсe
1дф
сonvеrgеs
to
xф
wеakly
in.L2(IRN)
as fr
_+
oо, Ьut
I
l|..'/.||2 l
Ф27pd,
ll^kvllla2(lRN)
-
"/O"
This
expression
сonverges
to
/р'
хф2d':
||хФ|||,61N)'
sinсе
Ф
с
2(IRN)
ls
arЬitrary,
that сomplеtes the
pгoof.
П
Domain funсtionals
suсh as J,(Л),
depending on
the сharaсtеristiс
funсtion
19
of the domain
a, are only
paгtiсular
сases
of funсtionals
under сonsideration.
In genеral, wе
сannot ехpесt
that a domain
funсtional
J
@)
еnjoys this
propeгty.
Clеarly, thе
funсtional
ft(9)
depеnds
on thе Ьoundary
Л of thе
domain J2
and сannot Ье
ехtendеd
to thе set of
funсtions с.,'(.)
€
,-(п,N)
suсh that
0
S
a,'(u)
{
1 for a,e,
I. Moгеover
lеt us oЬsеrvе
that the
funсtional J2(a)
сan bе ехtendеd
to thе
following сlass
of non-smooth
measuraЬlе
domains: a
mеasurablе set .Е in
IRN is said to
havе thе finitе
perimеter
P(E)
provided
that
mеas(.Е)
{
oo and
trg
*Ё.Ё
} a
lПфiri
rl;.
Tlз
с.rg
C.rAЕЁ€-
fЕo.
rr
giд
€с.
вt
SЬep
LЬ(
E]y
rЁ*i
:l
'Ёё&*
**
с.1
h
t>
tfr
.l
I:q*
h*.
=
J"'