A NNALI DELLA
S CUOLA N ORMALE S UPERIORE DI P ISA Classe di Scienze
G. B ELLETTINI G. D AL M ASO
M. P AOLINI
Semicontinuity and relaxation properties of a curvature depending functional in 2D
Annali della Scuola Normale Superiore di Pisa, Classe di Scienze 4
esérie, tome 20, n
o2 (1993), p. 247-297
<http://www.numdam.org/item?id=ASNSP_1993_4_20_2_247_0>
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of
aCurvature Depending Functional in 2D
G. BELLETTINI - G. DAL MASO - M. PAOLINI
1. - Introduction
The aim of this paper is to
study
the functionalwhere E C
R2
is a bounded open set of classC2,
p > 1 is a realnumber, x(z) = raE(Z)
is the curvature of 9E at thepoint
z, andJ/1
1 denotes the onedimensional Hausdorff measure in
We are interested in the
study
of the minimumproblem
where g
E is agiven
function. This can be considered as asimplified
version of a variational
problem proposed
in[15]
as a criterion forsegmentation
of
images
incomputer
vision.Moreover,
it wasrecently conjectured by
E. DeGiorgi [4]
thatproblem (1.2)
is connected with theasymptotic
behaviour ofsome
singular perturbations
of minimumproblems arising
in thetheory
ofphase
transitions.
Assume,
forsimplicity, that g
isnon-negative
forIzllarge enough.
If weapply
the direct method of the calculus of vatiations toproblem (1.2),
we areled to consider a
minimizing
sequence{ Eh } h,
whoseelements, according
to ourhypothesis
on g, are contained in a suitable ballindependent
of h andsatisfy
Pervenuto alla Redazione il 10 Marzo 1992.
By
a well known compactness theorem there exists asubsequence
whichconverges in to some bounded set E of finite
perimeter.
We shall prove that the functional 7 is lower semicontinuous withrespect
to the convergence in This allows us to showthat,
if the limit set E is of classC2,
thenE is a minimizer of
(1.2). Since,
ingeneral,
it is hard to provedirectly
thatthe limit of a
minimizing
sequence is of classC2,
we want to extend the functional 7 to the set .M of allLebesgue
measurable subsets ofJR2,
in sucha way that the extended
functional 7
is stilllower
semicontinuous. As usual in the relaxation method(see [2]),
we define7:.M --4 [0,+oo]
as the lower semicontinuousenvelope
of 7 with respect to thei.e.,
The main purpose of this
paper
is tostudy
thefunctional 7
and to determinethe
family
of sets E for which7(E)
+oo. Thestudy
of the minimumproblem (1.2)
led us to consider the functionalas a function of E rather than of aE.
Indeed,
the compactnessproperties
of a mi-nimizing
sequence of(1.2)
ensure agood
convergence for the setsEh,
but thecorresponding
weak convergence of the boundariesaEh
does not seem to be ap-propriate
for variational purposes. The main difficulties that we encountered weredue
essentially
to the lack ofgood contintuity properties
of the map E - BE.Lower
semicontinuity
results and existence of minimizers under suitableboundary
conditions are much easier to reach for the functionalrelated to
(1.3), where -1
varies now over all curves of classC2 satisfying
pre- scribedboundary
conditions and s is thecorresponding
arclenght
parameter. In thecase p = 2, this
problem
is classical and theminimizers,
discoveredby
Euler[8]
in1744,
are called elastica because of theirapplication
to thetheory
of flexi-ble inextensible rods. For a
complete
treatment of the elastica we refer to[14, 10]. Unfortunately,
these results cannot beapplied directly
to thestudy
of(1.2).
In this paper we have considered the
problem
in theplane.
The extensionof our results to the n-dimensional case is a difficult open
problem
and seemsto
require
the methods ofgeometric
measuretheory [9, 18].
We want to stress that all ourproofs
are obtainedby using only elementary
tools. We cannotexclude that some of these results could also be obtained in a more direct
(but
less
elementary)
wayby using
varifoldstheory [13, 6, 7].
We describe now in detail the content of the paper.
In Section 2 we fix the notation and introduce the
problem.
Section 3 is devoted to the
study
of the lowersemicontinuity
of 7.Precisely,
we prove(Theorem 3.2)
thatgiven
a sequencefehlh of
boundedopen sets
of
classC2 converging
in to a bounded open set Eof
classC2,
then 1* 11where K and Kh are the curvatures
of
aE andof aEh, respectively.
The definition
( 1.1 )
of 7 makes sense if E is a set whoseboundary
canbe
parametrized, locally, by
arcs ofregular
curves of classH 2,p .
Theprevious semicontinuity
theorem holds also in this case(Corollary 3.2).
_
We
emphasize
that thesemicontinuity
of 7 and the definition of ~ are considered with respect to the This means that the sequencelehlh approximates
E if andonly if
+oo,where ~ ~ ~ I
denotes the
Lebesgue
measure and A is thesymmetric
difference of sets. No further conditions arerequired
on8Eh
and 8E._
Simple examples
show that there exist sets E E N with +oo, whoseboundary
is not smooth. Inparticular,
let us consider the set E ofFigure
1.1 and suppose that 8E is
parametrized by
arcs of curves of class Coo, except for the cusppoints.
Fig.
1.1: A set E with-7(E)
+oo, and whoseboundary
is not smooth.Fig.
1.2: Thisapproximating
shows that7(E)
+oo.Fig.
1.3: Thisapproximating
sequence{Eh } h
is such thatSuPh +00.
Figure
1.2 shows that7(E)
+oo.Indeed,
the sequence{Eh}h
describedin
Figure
1.2 converges to the set E in the norm and SUPH +00.On the other
hand,
theapproximating
sequence ofFigure
1.3gives
rise to an infinite energy. Here the cusp
points
are smoothedby
circular arcs,and it is easy to prove
that, if p >
1, thenfaEh
- +oo as h - +oo.Note that
Figure
1.2 showsthat,
in thelimit,
the sequence(8Eh)h
creates ahidden arc
(with multiplicity two), given by
the segmentjoining
the two cusppoints.
The
main issue isobviously
to characterize those subsets E ofJR.2
such that7(E)
+oo. We find some necessary conditions and a number of different sufficientconditions,
but thecomplete
characterization of this class of sets still remains an openproblem.
In
Section 4 we present someregularity properties
of the sets E suchthat
7(E)
+oo. To beprecise,
we prove thefollowing
result(Theorem 4.1,
Proposition 4.3):
Let E CR2
be a measurable set such that7(E)
+oo.Then, up to a
modification of
E on a setof
measure zero, we have that E isbounded,
open, and)l1(BE)
+oo. Moreover, there exists a systemof
curvesr =
{~y 1, ... , ~ym } of
classH2,p
such that 9E is contained in the union(r) of
the traces
of
the curvesIi
and E =int(Ar
U(r)),
whereAr
is the setof
allpoints of JR.2B(r) of
index 1 with respect to r.Finally,
9E has a continuousunoriented tangent and can have at most a countable set
of
cusppoints.
At the end of Section 4 we
classify
thepoints
of 8Eaccording
to thelocal behaviour of the normal line with
respect
to aE.In Section 5 we show
examples
of sets E such that7(E)
+oo,despite
of their boundaries
being
veryirregular. Precisely,
inExample
1 a set Eis described whose
irregular boundary points
havepositive
one dimensional Hausdorff measure. InExample
2 a sethaving
an infinite number of cusppoints
is shown.In Section 6 we deal with the
following problem: which
conditions must be satisfiedby
theboundary
of a set E in order to have7(E)
+oo? To answerthis
question,
we firststudy
those systems of curves that can be obtained aslimits,
in theH2,p
norm, of a sequence of boundaries of smooth sets. To this aim we introduce the definition of system of curvessatisfying
the finitenessproperty
(Definition 6.1 ),
and thecompatibility
condition(Definition 6.6).
Thenthe
following
result holds(Theorem 6.2):
Let r be a systemof
curvesof
classH 2,p
withoutcrossings
andsatisfying
thefiniteness
property, anddefine
E asthe set
of all points of JR.2B(r) of
odd index with respect to r. Then aE C(r).
Moreover
7(E)
+oo, i.e., there exists a sequencelehlh of
bounded open setsof
class Coo such thatEh -->
E inL 1 (II~2 )
as h --> +oo and SUPH +00. Inaddition,
up to a suitable surgeryoperation
on the parameter spaceof
r, wehave that r in
H2,p
as h --+ +oo.Then, quite surprisingly, using only elementary properties
ofgraphs,
weprove one of the main results of the paper
(Theorem 6.5): Suppose
that 19E is smooth exceptfor
afinite
number nof
cusppoints.
ThenIn Section 7 we localize the definition of 7’ to all open subsets of
JR.2, i.e.,
we consider the functional
where Q C
R2
is an open set and E is a bounded open subset ofJR.2
such that Q n aE is of classC2.
We prove(Theorem 7.1 )
that:1(., Q)
isL1(Q)-lower semicontinuous.
Let
7( . , Q):
M 2013~[0,
+oo] I denote the lower semicontinuousenvelope
of:1( . , Q)
with respect to theL1(Q)-topology.
One of the main results of this section isthat,
asconjectured by
E. DeGiorgi in
aslightly
different context[4, Conjecture 5],
there are sets E such that7(E, - ),
if considered as a setfunction,
is not a measure.Precisely,
we construct anexample
of a set E whoseboundary
is smooth except for two cusppoints
and such thatwhere
Q2 C R2 are
two open sets, withS21
USZ2 - JR.2.
This shows that
Q) cannot
berepresented by
anintegral
of the form(1.3). Finally,
the value of iscomputed explicitly
in thisexample (Theorem 7.2).
Acknowledgements
We are indebted to Prof. E. De
Giorgi
forhaving suggested
us thestudy
of this
problem.
2. - Notations and
preliminaries
A
plane
p curve0 1
Y:[ 0,1
] 2013>of
classC is
said to beregular
if0
for2
g
dt ,every t
E[o, 1 ] .
Each closedregular
curve -1:[0,1]
2013>R2 will
beidentified,
in theusual way, with a map ~y:
S’ ~ ]R2,
whereS’
denotes the oriented unit circle.By (i)
=,([0,1]) = {-I (t):
t E[0, 111
we denote the traceof -1
andby l(,)
itslength;
s denotes the arc
length
parameter, and1, i
the first and the second derivativeof i
with respect to s. Let us fix a real number p > 1 and letp’
> 1 be itsconjugate
J g exponent, pi.e., 1 + I =
1. If the secondderivative y
in the sense ofp
p,
1’distributions
belongs
to LP, then the curvatureK(i) of y
isgiven by
andin this case we say
that -1
is a curve of classH2,p
and wewrite y
EH2,P.
Moreover, we put
If z E
I~2B(~y), 1(1,
z) is the index of z with respect to 7[3].
~’
denotes the 1-dimensional Hausdorff measure inR~ [9];
for any zo E o > 0,B,2(zo) = ol
is the ball centered at zo with radius o.We remark that if
f
is apositive
Borel function defined on(~y),
then(see
for instance[9,
Theorem3.2.6])
for any Borel set B C
[0, 1(-1)1.
Given a measurable set E C
R2,
xE denotes its characteristicfunction,
that is
xE(z)
= 1 if z E E,XE(Z)
= 0if z V
E;JEJ
is theLebesgue measure
ofE. For any subset C of
]R2,
we denoteby int(C)
the interior ofC, by
C theclosure of C, and
by
9C theboundary
of C. Let E CR2;
we say that E is of classH 2,p (respectively C2)
if E is open,and,
near eachpoint z
E aE, the setE is the
subgraph
of a function of classH 2,p (respectively C2)
with respect toa suitable
orthogonal
coordinate system. Notethat,
if 8E can beparametrized, locally, by
arcs ofregular
curves of classH 2,p (respectively C2),
and E lieslocally
on one side of itsboundary,
then E is of class(respectively C2).
If E is a bounded subset of
JR.2
of classH 2,p
then 9E can beviewed, locally,
as thegraph
of a function of classH2,p.
This allows us todefine, locally,
the notion of curvature of aE at everypoint
ofaE, by using
the classical formulas
involving
second derivatives. It is easy to see that the functionK(z)
does notdepend
on the choice of the coordinate system used to describe aE as agraph,
andbelongs
toLP( BE, ~( 1 ).
If M denotes the class of all
Lebesgue
measurable subsets of]R2,
we definethe
map 7: [0,
+oo]by
if E is a bounded open set of
class C2,
elsewhere on
M,
where K(z) is the curvature of aE at the
point
z.Let g
E be a function such that{z
C]R2: g(z) 01
CBR(O),
for a suitable R > 0. Let us consider the minimumproblem
where E varies over all bounded open subsets of
JR.2
withboundary
of classC2.
It is clear that
(2.3)
isequivalent
to theproblem
in the sense that
(2.3)
and(2.4)
have the same minimum values and the sameminimizers. If the
region where g
is less than aprescribed negative
number islarge enough,
then it is immediate toverify
that the minimum value ofproblem (2.3)
isnegative,
hence every solution of(2.3)
must be non-empty.We shall
identify
.M with a closed subset ofby
means of the map E H XE. The on .Mis, therefore,
thetopology
on M inducedby
the distanceEl AE2 1,
where E1,E2
E N and A is thesymmetric
difference of sets.
Let us prove that any
minimizing sequence {Eh}h of 9
isrelatively
compact in
Z~(R~).
Letfehlh
be such aminimizing
sequence;clearly,
we canassume that SUPH +oo, hence
Eh
is a bounded open set of classC2
for any h.Then,
sinceit follows that
Now we will show that there exists
ho c
N such that Eh CBR+H(O)
for any h >ho.
If this condition is notsatisfied,
then there exists asubsequence
ofstill denoted such that for every h E N. As the total
length
of is boundedby
H, each set E has a connected componentCh
which does not meetDenoting by Eh
the setEh
=EhBCh,
we getg(Eh) (recall that g >
0 on Since also is aminimizing
sequence, it follows that
necessarily
0 as h - +cxJ. Onthe other
hand,
one can show(see (3.5))
thatCh gives
apositive
contribution to theenergy .7 independent
of h, that is.7(Ch) 7Z+
0, whichgives
a contradiction.Hence there exists
ho E
N such thatWe deduce that
Using
the RellichCompactness
Theorem in BV(see [12,
Theorem1.19]),
itfollows that there exist a bounded set E of finite
perimeter
and asubsequence
such that
Ehk -~
E inLl(JR.2)
as k - +oo, and this shows that anyminimizing
sequenceof 9
isrelatively
compact inWe denote
by 7
the lower semicontinuousenvelope
of X with respect to thetopology
ofL1(JR.2).
It is knownthat,
for every E EM,
we haveFor the main
properties
of the relaxed functional we refer to[2].
Inparticular,
one can prove that
In
addition,
everyminimizing sequence
of +fE g(z)
dz has asubsequence
converging
to a minimumpoint
of7(E)
+fE g(z)
dz and every minimumpoint
of
7(E) + fE g(z)
dz is the limit of aminimizing
sequence ofl(E) + fE g(z)
dz._ The main purpose of this paper is to
study
theproperties
of the functional7.
From thedefinition,
it followsimmediately
that7(E)
+oo if andonly
if there exists a sequence
{Eh}h
of bounded open sets of classC2
such thatEh -
E inL 1 (JR.2)
as h - +oo andwhere Kh denotes the curvature of
aEh.
DEFINITION 2.1.
By
a system of curves we mean a finitefamily
r =~ 7 > .. > ’Y m }
of closedregular
curves of classC
1 such thatI
is constant on[o,1]
for any i =1,... ,
m.Denoting by
,S thedisjoint
union of m unit circlesSi, ... , Sm,
we shallidentify
r with the maph: ,S -~ I1~2
definedby
=,~,
for i =
1, ... ,
m. The trace(r)
of r is the union of the traces of the curvesY,
DEFINITION 2.2. If r =
~~y 1, ... , ~ym }
is a system of curves of classwe define
and
If z E
JR.2B(r)
we define the index of z with respect to r as is constant on[0, 1],
we have s(t) = hence3. -
Semicontinuity
of 7DEFINITION 3.1.
By
adisjoint
system of curves we mean a system ofcurves r =
{ 71, ... , ~ym }
such that(Y)
n = 0 for anyi, j
=1,...,
m,i ~ j .
Let
{Eh}h
be a sequence of setssatisfying (2.5).
Notethat,
for any h, there exists a suitableparametrization rh
ofaEh
such that the sequence{rh}h
satisfies
LEMMA 3.1. Let Then
be a system
of
curvesof
classH2,p.
PROOF.
Let -1
be asimple
closedregular
curve of classH2,p;
let us prove thatIf i
EC2,
then[5,
Theorem5.7.3]
By
a standardapproximation
argument,inequality (3.4)
holds for any curve -1 of classH2,p .
Then(3.3)
follows from the Holderinequality, since, if i
EHence
1(-i’)
> for any I =1,..., m.
Then,recalling
Definition2.2, it follows that
(1-’" 1 (1-"" 1
The
following generalization
ofinequality (3.3),
as well asCorollary 3.1,
willbe useful in Section 7.
LEMMA 3.2. Let 1:
[0,
1 ] - R2 be aregular
curveof
classH 2P
and let0:
[0,
1] ] - R be a continuousfunction
such that0(t)
is the orientedangle
between the x-axis and the tangent vector
of
I at t. ThenPROOF. Let us write with obvious notation
1(S) =
(cos0(s),
sin0(s)). Then,
using
the Holderinequality,
it follows thatCOROLLARY 3.1. 01 be as in Lemma 3.2. Then
PROOF. Lemma 3.2
implies
thatHence, since the minimum I
point
of the functionis reached at
DEFINITION 3.2. Let E C be a bounded open set of class
C 1.
We saythat a
disjoint
system of curves r is an orientedparametrization
of aE if eachcurve of the system is
simple, (r)
= aE,and,
inaddition,
PROPOSITION 3.1. Each bounded subset E
of JR.2 of class (respectively C2)
admits an orientedparametrization of
classH2,p (respectively C2).
PROOF. Since E is of class
H2,p
each connected component of BE can beparametrized by
aregular
closed curve of classH~>P.
As BE is compact andlocally connected,
it has a finite number of connected components. Thestatement about the index follows from Jordan’s Theorem. D DEFINITION 3.3. We say that a sequence of systems of curves of class converges
weakly
inH2,p
to a system of curves r =~~y 1, ... , ~m ~
of class
H2,p
if the number of curves of each system rhequals
the numberof curves of r for h
large enough, i.e.,
r h =~~yh, ... , ~yh ~, and,
inaddition,
~yh ~ 7i weakly
in as h - +00 for any i =1,...,
m.Note
that,
if(rh)h
converges to r ={ ~y 1, ... , ~ym ~ weakly
inH2,p,
thenfor any In
particular,
THEOREM 3.1. Let
{rh } h
be a sequenceof
systemsof
curvesof
classH 2,p satisfying (3.1)
such that the traces(rh)
are contained in a bounded subsetof
independent of
h. Then{rh }h
has asubsequence
which convergesweakly
in
H 2,p
to a systemof
curves r.PROOF.
By (3.2), using (3.1 )
it follows that the number mh of curves of the systemrh
is boundeduniformly
with respect to h.Hence,
for asubsequence (still
denotedby { rh } h ),
there exists aninteger
m such thatrh
=... iml
for any h. Fix i E
{l,...,m}; using (3.3)
and(3.1)
weget
that there exist twopositive
constants C1, C2 such thatThen, using (2.6)
we obtain that there exists apositive
constant c3 such thatI - If)
whence,
as(rh )
are bounded1 I
_
uniformly
with respect to hby
thehypothesis,
thefamily
isequibounded
in
H2,p. Then,
for asubsequence,
there exist m curves71, ... , 7m
of classH2~~
such that
1~
-i’ weakly
inH2,p
as h - +00, for any i =1,...,
m. This shows that{rh}h
converges to the system of curves r ={71, ... , ~ym} weakly
inH2,p.
lJ DEFINITION 3.4. We say that r is a limit system of curves of class
H2,p
if r is the weakH2,p
limit of a sequence{rh }h
of orientedparametrizations
ofbounded open sets of class
The
following
remark is an easy consequence of(3.6).
REMARK 3.1. If r is a limit system of curves of class
H2~~,
thenI(r, z)
E{0,1}
for any z EJR.2B(r).
In order to prove the
semicontinuity
Theorem3.2,
we need thefollowing
Lemma.
LEMMA 3.3. Let E C
R2
be a measurable set, let{ Eh } h be a
sequenceof
bounded open sets
of
classsatisfying (2.5),
and suppose that E in as h ~ +00. Let usdefine
Then E* and F* are open, E* is
bounded,
F* =int(JR.2BE*),
andMoreover there exists a limit system r
of
curvesof
classH2,p
with thefollowing
properties:
where where
PROOF. It follows
immediately
from the definition that E* and F* are open.For any h,
by (3.1)
andby
Lemma 3.1,aEh
admits an orientedparametrization
Ah of class and the number mh of curves of the system Ah is bounded
uniformly
with respect to h. Hence, for asubsequence (still
denotedby fahlh),
there exists an
integer
m such that Ah = for any h. Inorder
toapply
compactness arguments, we shall transform the sequence which is notnecessarily bounded, (see Figure 3.1),
into a sequence{rh } h
of orientedparametrizations
whose traces are contained in a bounded setindependent
of h.Fig.
3.1: The sequence not bounded: two connected componentsof
(Ah)approaches
+00 withlength
and curvatureuniformly
bounded with respect to h.Let us consider first the sequence of curves
{’yh ~ h .
If the sequenceI/A(t)1
converges to +00 as h - +00, then we eliminatelA,
i.e.,
wereplace
Ahby
the systemOh - ~ ~yh, ... , ~yh ~ .
Notethat,
as isuniformly
bounded with respect to h, the behaviour ofgives that,
forany r > 0, there exists
hr
E N such that = 0(hence
=for
any h
and for any z E If does not tend to +00, there exists asubsequence,
still denotedby
such that the traces(~yh)
arebounded
uniformly
with respect to h. In this case we defineOh
= Ah.Starting
from we repeat the same
procedure
forobtaining
a new sequence of systems of curves~Oh } h.
After m steps, we end up with a sequence ofsystems which we shall denote
by {rh}h. By construction,
for every h,rh
is adisjoint
system of curves of classH~,P,
and for every r > 0 there existshr
E N such thatfor
any h > hr
and for any z EBr(0))(Ah).
It is clear alsoby
construction that the traces(rh)
are boundeduniformly
with respect to h,i.e.,
there exists R > 0such that
From
(3.8)
and(3.9)
it followseasily that,
for hlarge enough,
for any z ER2B(rh)
we havez)
C{O, I}.
Let us defineAh
=fZ
EJR.2B(rh): I(rh,
z) =1}.
Since
Th
is adisjoint
system of classH2,p , Ah
is a bounded open set and(r). Using
Theorem3.1,
there exists asubsequence {Th}h
whichconverges
weakly
to a limitsystem
r of curves of classH 2,p . By (3.9)
we have(r)
CBy
Remark3.1, I(r, z)
E10, 1 )
for any z EJR2B(r).
Let
Ar
be the open set defined in(i).
It is clear thatAr c BR(O).
SincexAh(z)
=I(rh, z)
for any z EJR2B(rh),
andXAr(Z)
=I(r, z)
for any z ER2B(r),
by
thecontinuity
property of the index andby
the DominatedConvergence
Theorem we have that
Ah --~
Ar in as h - +oo. Let us prove thatBy (3.8)
and(3.9),
for any r > R, we have thatAh
=Eh n Br(0)
for hlarge enough; passing
to the limit as h -~ +oo, we obtainthat Ar0(E
n = 0.Since r is
arbitrary,
we get(3.10).
Let us prove
that
= 0.By (3.10),
it isenough
to prove thatSince
Ar
is openand
= 0, for any z EAr
there exists r > 0 such that Ar, sothat IB,(z)BArl
= 0, hence z E E*. ThereforeTo prove
(3.11), being I (r) = 0,
it is sufficient to show thatIf z g Ar
U(r),
thenI(h,
z) = 0. Since the index islocally
constant, there existsr > 0 such that = 0 for any w E
Br(z). As IBr(z)
nAr T
= 0,we
have z V
E*. This proves(3.13),
so we can conclude that 0.Moreover
(3.13)
shows that E* is bounded.Let us prove
(i).
The inclusionAr
follows from the fact thatAr
is open. The inclusion E* Cint(Ar
U(r))
follows from(3.13)
and from thefact that E* is open. To prove the
opposite inclusion,
let z Eint(Ar
U(r)).
Then there exists r > 0 such thatBr (z)
CAr U (r). Hence IBr(z)BEI = lBr(z)BArl
= 0by (3.10), and, therefore, z
E E*. This shows that E* =int(Ar
U(h)).
Let us prove
(ii).
The inclusionint(BrU(r)) D Br
follows from the fact thatBr
is open. Since E* and F* aredisjoint
open sets, we have F* Cint(JR.2BE*).
To prove the
opposite inclusion,
let z Eint(Br
U(r)).
Then there exists r > 0 such thatBr (z)
CBr
U(r), hence Br (z)
nE I =
nArl =
0(see (3.10)), and, therefore, z
E F*. This shows thatint(Br U (r))
C F*,which, together
with(3.14), gives
and concludes the
proof
of(ii).
Since E* and F* are
disjoint
open sets, we have E* Cint(JR.2BF*).
As Ar U
(r) = by (ii)
we have C Ar U(r),
henceE* C
int(JR.2BF*)
Cint(Ar
U(r)). By (i)
we conclude thatwhich, together
with(3.15), gives
and proves
(3.7).
We will now prove
(iii).
SinceAr
and Br aredisjoint
open sets, we haveaArU8Br
C(r).
Let us prove that BE* CAr.
For every z E aE*and for every
neighbourhood
U of z we selectint(Ar
U(r)).
Then there exists r > 0 such that and
Br(w) g
Ar U(r).
As(r)
hasLebesgue measure
zero, we have hence Thisimplies
that z
E A r .
Let us prove that aE* C Br. For every z E aE* and for every
neighbourhood
U of z we select w EUBE*
=UBint(Ar
U(r)).
Then thereexists r > 0 such that
Br(w)
CU and Br(w)
nBr
=Br(w)B(Ar
U(r))f=0.
Thisshows that U n
Brf=0, hence z
EBr.
Therefore
aE* CAr
nBr.
SinceAr
and Br aredisjoint
open sets, wehave
Ar
n Br =aAr n aBr,
hence aE* CaAr
naBr.
Let us prove that
aAr n o9Br C
aE*.By (i)
we haveC Ar
CE*,
by (ii)
we haveaBT
C Br 9F*,
andby (3.16)
we have E* =R2BF*,
henceF*
=II~2BE*.
It follows thathence aE* =
aAr n aBr.
From(3.7)
we obtain aE* - which concludesthe
proof
of(iii).
DLEMMA. 3.4. Let E C
JR.2
be a bounded open setof
classH2,p,
and let rbe a system
of
curvesof
classH2,p
such that aE C(F).
ThenHence
(see Definition 2.2).
PROOF. Let r =
~71, ... , 7m ~ .
For any measurable set T C 8E and anyi = 1,..., m let
Since BE C (T), we have T =
U:1
To prove the thesis it will beenough
toshow that
where T varies over a suitable finite measurable
partition P
of aE. Infact,
from(3.18)
it follows thathence
that proves the assertion. We choose the finite
partition P
of 8E as follows.Any
element of P must be contained in arectangle
in which 8E is a cartesiangraph.
This means that for any T E P we can choose a suitableorthogonal
coordinate system, two bounded open intervals I, J, and a function
f :
I - Jof class
H2e
such thatLet us fix i E
f 1, ... , ,m}
and leti’(s) = (li(s), -li (s)).
For any s EVi
we haveyi2(s)
= Since thefunctions -yi2
andare
of classH 2,p
on an openset
containing Y
and coincide onVi,
we have[ 11 ]
Then, since I
, we obtainand
It follows that
Hence, using (2.1 ),
that is
(3.17).
This concludes theproof.
DTHEOREM 3.2. The
functional 1(.)
is semicontinuous on the classof
all bounded open subsetsof JR.2 of
classC2,
i. e.,given
a sequencelehlh of
bounded open setsof
classC2 converging
inL 1 (I1~2 )
to a boundedopen set E
of
classC2,
thenwhere K and Kh are the curvatures
of
aE andof 8Eh, respectively.
PROOF. Let
{ Eh } h
be a sequence of bounded open sets of classC 2
such thatEh -
E inL1(JR.2)
as h - +oo. We can suppose that theright
hand side in(3.19)
isfinite,
otherwise the result is trivial. Let be asubsequence
ofwith the
property
thatFor
simplicity,
thissubsequence (and
any furthersubsequence)
will be denotedby
For any k, has a finite number of connected components.Let 0,~ be an oriented
parametrization
ofaEk.
Sincefeklk
satisfies(2.5),
thesequence satisfies
(3.1).
As in theproof
on Theorem3.1,
we can suppose that Ak= 1 -11, - - - , -,m I
with mindependent
of k. Let be theequibounded
sequence and let r be the limit system of curves of classH2,p
constructed in theproof
of Lemma 3.3.Then, by construction, n
m,r and for
any j
we have that-yk
-weakly
inH2,p
as k - +oo.Using (iii)
of Lemma 3.3 and the weak lowersemicontinuity
of the LP norm, we have
Since
by
Lemma 3.4 andwe conclude that
which is the assertion of the theorem. D
The
following
resultgeneralizes
Theorem 3.2.COROLLARY 3.2. Let E C
JR.2
be a bounded open setof
classH 2,p .
ThenIn
particular, 7(E)
+oo.PROOF. Theorem 3.2 holds with the same
proof
if E is of classH2,p ,
hence,passing
to the infimum withrespect
to theapproximating
sequencelehlh
in(3.19),
we infer thatLet us prove the
opposite inequality. Proposition
3.1implies
that there existsan oriented
parametrization
r =~7’ , ... , 7m }
of aE of classH2,p.
Hence inparticular
BE = where-1’ [0, 1]
2013~I~2
are closedsimple regular disjoint
curves of class
H 2,p .
For any i =1,
..., m, let us consider a sequence Ofcurves of class
C~([0,1])
such thati’ strongly
inH2,p
as h - +00. Itfollows
that,
for hlarge enough,
theapproximating
system rh = is adisjoint
system of curves. Moreover, the curvesig
aresimple
for hlarge enough.
If not, there exist a
subsequence,
still denotedby
and two sequences{th}h
ofpoints,
with 0 Sh th 1, such thatBy
compactness, we may also assume that sh- s and th
-~ t. Then~~(s) == YM.
If t 1, this
implies
s = t.By
the Mean ValueTheorem,
there existpoints ~h
and 1/h between sh
and th
such that the first component of the derivative ofig
vanisheson ~h
and the second component vanishes on 1/h.As ~h
-~ s = tand qh ~ s = t, we conclude that the derivative
of ~
vanishes at s, whichcontradicts our
assumption
on -1(see
Definition2.1 ).
If t = 1, then either s = 1or s = 0. The former case leads to the same contradiction obtained before. The latter case can be treated with obvious modifications. Hence the curves
ig
aresimple
for hlarge enough.
By construction,
the traces(rh)
areequibounded uniformly
with respect toh. For any h, let us define
Eh
=fZ
EI(rh, z)
=1 }.
ThenaEh
={ Eh } h
convergesto ~ z
E =11 =
E in as h - +oo, and+
f o~~h~[1
+ ds. It follows thatby
construction. This concludes theproof
of(3.20).
D4. - Some
properties
of the sets E with7(E)
+ooWe recall the definition of tangent cone for an
arbitrary
subset ofJR.2.
DEFINITION 4.1. Whenever C C E C, we define the
(unoriented) tangent
cone of C at z, denotedby Tc(z),
as the set of all v EJR.2
such that for every E > 0 there exist w E C andwith Iw - zl
Eand
E.Such vectors v will be called