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ChCAbstracWe cboundarfind the 2004RsumOn cla classdsingu 2004MSC: 49Keyword1. IntrIn rebesidesenergieE-m0294-144doi:10.10Ann. I. H. Poincar AN 21 (2004) 839880www.elsevier.com/locate/anihpcaracterization and representation of the lower semicontinuousenvelope of the elastica functionalaractrisation et reprsentation de lenveloppe semi-continueinfrieure de la fonctionnelle de lelasticaG. Bellettini a, L. Mugnai ba Dipartimento di Matematica, Universit di Roma Tor Vergata, via della Ricerca Scientifica, 00133 Roma, Italyb Dipartimento di Matematica, Universit di Pisa, via Buonarroti 2, 56127 Pisa, ItalyReceived 22 May 2003; received in revised form 18 December 2003; accepted 30 January 2004Available online 9 June 2004tharacterize the lower semicontinuous envelope F of the functional F(E) := E [1 + |E |p]dH1, defined onies of sets E R2, where E denotes the curvature of E and p > 1. Through a desingularization procedure, wedomain of F and its expression, by means of different representation formulas.Elsevier SAS. All rights reserved.aractrise lenveloppe semi-continue infrieure F de la fonctionnelle F(E) := E[1 + |E |p]dH1, dfinie sure des frontires des domaines E R2, o E dnote la courbure de E et p > 1. Grce une mthode delarisation, on trouve le domaine de F et son expression, laide de diffrentes formules de reprsentation.Elsevier SAS. All rights reserved.J45; 49Q20s: Semicontinuity; Curvature depending functionals; Elastica; Relaxationoductioncent years a growing attention has been devoted to integral energies depending on curvatures of a manifold;the geometric interest of functionals such as the Willmore functional [2,24,25], curvature dependings arise in models of elastic rods [11,15,17], and in image segmentation [8,1823]. In the case of planeail addresses: belletti@mat.uniroma2.it (G. Bellettini), mugnai@mail.dm.unipi.it (L. Mugnai).9/$ see front matter 2004 Elsevier SAS. All rights reserved.16/j.anihpc.2004.01.001840 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880curves the main example is the functional of the so-called elastic curves [11,13] which reads as [1 + | |2] ds.This fuFwhereH1 is tThecompuappearOnecomesbulk teEfor an athe clasa conjeis addrTheon g, s(especithe domTheF = FF(E) 1 is a real number, E(z) is the curvature of E at z andhe one-dimensional Hausdorff measure in R2.map F , considered as a function of the set E rather than of its boundary E, appears in problems ofter vision [8,22,23] and of image inpainting [3,18,19]. It is a simplified version of the building blocking in the model suggested in [23] to segment an image taking into account the relative depth of the objects.of the motivations of looking at F as a function of the sets E, which are endowed with the L1-topology,from the above mentioned applications, where one is typically interested in minimizing F coupled with arm; for instance, one looks for solutions of problems of the forminfM{F(E)+Eg(z) dz}, (2)ppropriate given bulk energy g, where F stands for the L1-lower semicontinuous envelope of F , defined ons M of all measurable subsets of R2. Another motivation for adopting this point of view is represented bycture in [14], where the approximation of the Willmore functional through elliptic second order functionalsessed.choice of the L1 topology quickly yields the existence of minimizers of (2) under rather mild assumptionsee the discussion in [4]; however, it is clear that, being the L1 topology of sets a very weak topologyally for functionals depending on second derivatives), several difficulties arise when trying to characterizeain of F and to find its value.study of the properties of F was initiated by Bellettini, Dal Maso and Paolini in [4]. After proving thaton regular sets [4, Theorem 3.2], the authors exhibited several examples of nonsmooth sets E having+, see for instance Fig. 1. However, some of these examples are rather pathological (for instance, setslocally around a point p have a qualitative shape as in Fig. 2) and show that the characterization of theof F is not an easy task.us briefly recall the partial characterization of F obtained in [4, Theorems 4.1, 6.2]. If E R2 is suchE) < +, then there exists a system of curves = {1, . . . , m} (that is, a finite family of constant speedions of the unit circle S1, see Definition 2.2) such that i H 2,p(S1), the union of the supportsmi=1(i) =:vers E and has no transversal crossings, and E coincides in L1(R2) with {z R2 \ ( ): I(, z) = 1} =:he set E is made by two connected components having one cusp point. The sequence {Eh} consists of smooth sets converging to E inwhose energy F is uniformly bounded with respect to h. Hence F(E) < +.G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 841A , where I(, ) is the index of (see Definition 2.7). As a partial converse of the previous result, given a systemof curvonly atthe hypset of in [4, E6.4] it iof simof cuspnot admdue toa reminput in robjectEvetwo cuconfiguThecan chaa d exhma desboWe remwhichLetto a chsubseqlet F(if theirthat ifcurvesis the sNThis approof,based oFig. 2. Tsingulares = {1, . . . , m} H 2,p(S11 S1m), if has no transversal crossings and self-intersects tangentiallya finite number of points, then F(Ao ) < +, where Ao := {z R2: I(, z) 1 (mod 2)}. We stress thatothesis of finiteness for the set of self-intersection points of (which in the sequel will be called the singularand denoted by Sing ) is an effective restriction since it may happen that H1(Sing ) > 0, as was shownxample 1, p. 271]. To conclude the list of the known results concerning the domain of F , in [4, Theorems proved that, if E can be locally represented as the graph of a function of class H 2,p up to a finite numberple cusp points (see Definition 2.33) then F(E) < + is equivalent to the condition that the total numbers is even. Finally, as far as the value of F is concerned, in [4, Theorem 7.3] it is proved that F(,) doesit an integral representation, where F(,) is the localization of F on an open set . This phenomenon isthe presence, in the computation of F(E,), of hidden curves (not in general contained in E) which areiscence of the limit of the boundaries Eh of a minimizing sequence {Eh}. Such hidden curves could beelation with the problem of reconstructing the contours of an object which is partially occluded by anothercloser to the observer [6].ntually, the computation of F(E) is carried on in [4, Theorem 7.2] in one case only, i.e., when E has onlysps which are positioned in a very special way (as in Fig. 1), the proof being not adaptable to more generalrations.aim of this paper is to answer the above discussed questions left open in the paper [4]. More precisely, weracterize the domain of F , thus removing the crucial finiteness assumption in Theorem 6.2 of [4], throughesingularization procedure on systems of curves having an infinite number of singularities;ibit different representation formulas for F (obviously not integral representation in the usual sense),king computable (at least in principle) the value of F(E) for nonsmooth sets E;cribe the structure of the boundaries of the sets E with F(E) < +, and extend [4, Theorem 6.4] toundaries with more general singular points rather than simple cusp points.ark that, in the discussion of the above items, we also characterize the structure of those systems of curvesare obtained as weak H 2,p limits of boundaries of smooth bounded open sets.us briefly describe the content of the paper. In Sections 2, 3 we prove some preliminary results, leadingaracterization of the singular set of systems of curves. To explain with some details our results in theuent sections, let us introduce some definitions. If = {1, . . . , m} is a system of curves of class H 2,p, we) :=mi=1 [1 + |i |p]ds. We say that two systems , of curves are equivalent (and we write )traces coincide, i.e., ( ) = ( ) and if { 1(p)} = { 1(p)} for any p ( ). It is not difficult to show , then F( ) = F( ). In Theorem 5.1 and Corollary 5.2 we show that, given an arbitrary system of = {1, . . . , m} of class H 2,p, without transversal crossings, there exists a system of curves whichtrong H 2,p-limit of a sequence {EN } of boundaries of smooth, open, bounded sets such thatlimF(EN)=F( ), limNEN = Ao in L1(R2).proximation result generalizes [4, Theorem 6.2], since no finiteness assumptions on Sing is required. Thewhich is quite involved, requires a desingularization of around the accumulation points of Sing , and isn several preliminary lemmata, see Section 4. Observe that in Theorem 5.1 we show that among all systemshe grey region denotes (possibly a part of) the set E. If E, locally around the singular point p (which is an accumulation point ofpoints of E), behaves as in the figure, it may happen that F(E) 842 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880of curves of class H 2,p without transversal crossings, those which have finite singular set are a dense subset inthe enein enerpointsIn SFwhereCorollaTheoryin Theo AMoreoFTheoreconstrafor proin PropIn SsectionsingulaIn Sof thisprove tFHere Rclass Hand doset E h2. NotA pregularorientedenotenumbebelongin thisFrgy norm. Hence we also have that every E with F(E) < + can be approximated both in L1(R2) andgy by a sequence of subsets {EN } such that SingEN consists only of a finite number of cusps and branch(see again Definition 2.33).ection 6 we give some representation formulas for F . In particular, in Proposition 6.1 we show that(E)= min{F( ): A(E)}, (3) A(E) if and only if ( ) E and E = A in L1(R2). This formula is much in the spirit of [10,ry 5.4], where a similar, but in some sense weaker, result is proved in the framework of Geometric Measure. Motivated by the density result of subsets with finite singular set given in Theorem 5.1 and Corollary 5.2,rem 6.3 we prove that if E has a finite number of singular points then the collectionQfin(E) of all systems(E) with finite singular set is dense in A(E) with respect to the H 2,p-weak convergence and in energy.ver(E)= inf{F( ): Qfin(E)}. (4)m 6.3 is stronger than Theorem 5.1, since the approximating sequence now must fulfill the additionalint of being made of elements of Qfin(E). Moreover Theorem 6.3 turns out to be the key technical toolving the results of Section 8. Note carefully that the minimum in (4) in general is not attained, as we showosition 8.8.ection 7 the regularity of minimizers for problem (3) is studied in the case p = 2. The main result of thisis Theorem 7.1 where we show that any solution of the minimum problem (3) has, out of E, a finiter set and consists of pieces of elastic curves.ection 8 we focus our attention on subsets E with finite singular set and with F(E) < +. The main resultsection is Theorem 8.6, where we give a (close to optimal) representation formula for F(E). Precisely, wehat(E)=RegE[1 + E(z)p]dH1(z)+ 2 min(E)F( ).egE denotes the regular part of the boundary of E, (E) is (roughly speaking) the class of all curves of2,p connecting the singular points of E in an appropriate way, which do not cross transversally each othernot cross transversally E. This result is a wide generalization of the example discussed in [4], where thead only two cusps and a very specific geometry.ation and preliminarieslane curve : [0, a] R2 of class C1 is said to be regular if d (t)dt= 0 for every t [0,1]. Each closedcurve : [0,1] R2 will be identified, in the usual way, with a map :S1 R2, where S1 denotes thed unit circle. By ( ) = ([0,1]) = { (t): t [0,1]} we denote the trace of and by l( ) its length; ss the arc length parameter and , the first and second derivative of with respect to s. Let us fix a realr p > 1 and let p be such that 1/p + 1/p = 1. If the second derivative in the sense of distributionss to Lp , then the curvature ( ) of is given by | |, and( )pLp=]0,l( )[| |p ds < +;case we say that is a curve of class H 2,p, and we write H 2,p. Moreover, we put( ) := l( )+ ( )pLp.G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 843If z R2 \ ( ), I(, z) is the index of with respect to z [7].ForboundaForfor anyDefinitthe setsuitablLetdefine,secondsystemGivE| | denE = FLetthroughby theNowFWe calthe topF2.1. SyIn thwe proDefinitsuch thof unit( ) ofByH 2,p(SDefinit for aany C R2 we denote by int(C) the interior of C, by C the closure of C, and by C the topologicalry of C. All sets we will consider are assumed to be measurable.every set E R2 let E denote its characteristic function, that is E(z) = 1 if z E, E(z) = 0 if z / E;z0 R2, > 0, B(z0) := {z R2: |z z0| < } is the ball centered at z0 with radius .ion 2.1. We say that E R2 is of class H 2,p (respectively Ck , k 1) if E is open and if, for every z E,E can be locally represented as the subgraph of a function of class H 2,p (respectively Ck) with respect to ae coordinate system.E R2 be a set of class H 2,p. Since E can be locally viewed as the graph of an H 2,p function, we canlocally, the curvature E of E at H1-almost every point of E using the classical formulas involving thederivatives. One can readily check that the definition of E does not depend on the choice of the coordinateused to represent E as a graph, and also that E Lp(E,H1).en a set E R2, we define := {z R2: r > 0: Br(z) \E= 0},oting the Lebesgue measure. If stands for the symmetric difference between sets and |EF | = 0, then.M be the collection of all measurable subsets of R2. We can identify M with a closed subset of L1(R2)the map E E . The L1(R2) topology induced by this map on M is the same topology induced on Mmetric (E1,E2) |E1E2|, where E1,E2 M.we define the map F :M [0,+] as follows:(E) :={E[1 + |E(z)|p]dH1(z) if E is a bounded open set of class C2,+ elsewhere onM.l L1-relaxed functional of F , and denote it by F , the lower semicontinuous envelope of F with respect toology of L1(R2). It is known that, for every E M, we have(E)= inf{lim infh F(Eh) :Eh E in L1(R2) as h }. (5)stems of curvesis subsection we list all definitions and known facts on systems of curves used throughout the paper, andve some preliminary results.ion 2.2. By a system of curves we mean a finite family = {1, . . . , m} of closed regular curves of class C1at | didt| is constant on [0,1] for any i = 1, . . . ,m. Denoting by S the disjoint union of m circles S11 , . . . , S1mary length, we shall identify with the map : S R2 defined by |S1i := i for i = 1, . . . ,m. The trace is defined as ( ) :=mi=1(i).a system of curves of class H 2,p(S) we mean a system = {1, . . . , m} such that each i is of class1i ). In this case we shall write H 2,p(S).ion 2.3. By a disjoint system of curves we mean a system of curves = {1, . . . , m} such that (i) (j )=ny i, j = 1, . . . ,m, i = j .844 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880Definition 2.4. We say that a system of curves = {1, . . . , m} is without crossings if di(t1)dt and dj (t2)dt areparalleDefinitlandFAs |]0GivennotatioFRemarfunctioDefinitto asDefinitorienteEIn [4orienteDefinitstronglh equweaklyIf {for anyl, whenever i(t1)= j (t2) for some i, j {1, . . . ,m} and t1, t2 [0,1].ion 2.5. If = {1, . . . , m} is a system of curves of class H 2,p, we define( ) :=mi=1l(i),( )pLp:=mi=1]0,l(i)[i(s)p ds,( )=mi=1F(i) :=mi=1l(i)+(i)pLp .didt| is constant on [0,1], we have s(t) = tl(i), hence,l(i)[|i|p ds = l(i)12p]0,1[d2idt2p dt.a set A = A1 Am S and a system of curves = {1, . . . , m} H 2,p(S), we fix the followingn:(,A) :=mi=1Ai{didtdt + l(i)12p Aid2idt2p}dt.k 2.6. With a small abuse of notation, with the same letter F we denote a functional defined on M and anal defined on regular H 2,p curves.ion 2.7. Let = {1, . . . , m} be a system of curves. If z R2 \ ( ) we define the index of z with respectI(, z) :=mi=1 I(i, z).ion 2.8. Let E R2 be a bounded open set of class C1. We say that a disjoint system of curves is and parametrization of E if each curve of the system is simple, ( )= E, and, in addition,= {z R2 \ E: I(, z) = 1}, R2 \E = {z R2 \ E: I(, z) = 0}., Proposition 3.1] it is proved that any bounded subset E of R2 of class H 2,p (respectively C2) admits and parametrization of class H 2,p (respectively C2).ion 2.9. We say that a sequence {h} of systems of curves of class H 2,p converges weakly (respectivelyy) in H 2,p to a system of curves = {1, . . . , m} of class H 2,p if the number of curves of each systemals the number of curves of for h large enough, i.e., h = { h1 , . . . , hm}, and, in addition, hi converges(respectively strongly) to i in H 2,p as h for any i = 1, . . . ,m.h} weakly converges to = {1, . . . , m} in H 2,p, thenhi i in C1 as h ,i = 1, . . . ,m. In particular, l( hi ) l(i) as h .G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 845The following result is proved in [4, Theorem 3.1]; it states the coercivity of the functional F with respect tothe weaTheoreboundeshThen {Definitof orienDefinitexists a2.2. NiDefinit (p) bcounteRwherewith redoes noRemarthat eanice reLetimplicivector2.3. DDefinit denoLemmon m aProof.Remar (z) ik H 2,p convergence of systems of curves.m 2.10. Let {h} be a sequence of systems of curves of class H 2,p such that all (h) are contained in ad subset of R2 independent of h andupNF(h) < +.h} has a subsequence which converges weakly in H 2,p to a system of curves .ion 2.11. We say that is a limit system of curves of class H 2,p if is the weak limit of a sequence {h}ted parametrizations of bounded open sets of class H 2,p.ion 2.12. We say that a system of curves verifies the finiteness property in an open set U R2 if therefinite set S U such that ( ) \ S is a one-dimensional embedded submanifold of R2 of class C1.ce rectanglesion 2.13. Let = {1, . . . , m} be a system of curves of class H 2,p without crossings. Let p ( ), lete a unit tangent vector to ( ) at p, and let (p) be the rotation of (p) of /2 around the origin inrclockwise order. We say that R(p) is a nice rectangle for ( ) at p if(p) = {z R2: z = p + l (p)+ d(p), |l| a, |d| b},a > 0 and b > 0 are selected in such a way that ( ) R(p) is given by the union of the cartesian graphs,spect to the tangent line Tp( ) to ( ) at p, of a finite number of functions {f1, . . . , fr } such that graph(fl)t intersect the two sides of R(p) which are parallel to Tp( ) for every l = {1, . . . , r}.k 2.14. By regularity properties of systems of curves of class H 2,p without crossings, one readily checksch point p ( ) admits a nice rectangle R(p) at p. Moreover, if H 2,p and () ( ), then R(p) is actangle at p also for .p ( ); when we write a nice rectangle R(p) at p for ( ) in the form R(p) = [a, a] [b, b], wetly assume that p is the origin of the coordinates, that Tp( ) is the x-axis, and that (p) agrees with the(0,1). In this case we also set R+(p) := [0, a] [b, b] and R(p) := [a,0] [b, b].ensity function of a system of curvesion 2.15. Let be a system of curves of class H 2,p. We define the density function of as : ( ) N {+}, (z) := { 1(z)},ting the counting measure.a 2.16. Let = {1, . . . , m} be a system of curves of class H 2,p. Then there exists M N depending onlynd on F( ) such that{1i (p)}M p (i), i = 1, . . . ,m.The statement is a consequence of step 1 in the proof of Theorem 9.1 in [6]. k 2.17. As a direct consequence of Lemma 2.16 we obtain that if is a system of curves of class H 2,p thens uniformly bounded with respect to z ( ).846 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 8398802.4. Definitions of A , Ao , A(E), Qfin(E), Ao(E)If AARemarparticuDefinitsatisfyi(We indby Ao((NotAo(E)it holdsRemar(respec2.5. ODefinitthe tanthe reaTheTheoreand EFhence iRemarEThen, aTheref1 Letis a system of curves of class H 2,p, in the following we set :={z R2 \ ( ): I(, z) = 1},o :={z R2 \ ( ): I(, z) 1 (mod 2)}. (6)k 2.18. If is a limit system of curves of class H 2,p then I(, z) {0,1} for any z R2 \ ( ), see [4]; inlar A = Ao .ion 2.19. Let E R2. We denote by A(E) the collection of all limit systems of curves of class H 2,png ) E, E = int(A ( )). (7)icate byQfin(E) the collection of all systems A(E) verifying the finiteness property in R2.1 We denoteE) the collection of all systems of curves of class H 2,p satisfying ) E, E = int(Ao ( )). (8)e that the elements of Ao(E) are not, in general, limit systems of curves. Moreover Qfin(E) A(E) . Finally, in view of Theorem 2.22 and (7) (respectively (8)), for every A(E) (respectively Ao(E)),|AE| = 0 (respectively |AoE| = 0), provided F(E) < +.k 2.20. If A(E) (respectively Ao(E)) and F R2 is such that |EF | = 0, then A(F )tively Ao(F )).n sets E with F(E) < +ion 2.21. Let C be a subset of R2. We say that C has a continuous unoriented tangent if at each point z Cgent cone TC(z) to C at z (see [4, Definition 4.1]) is a straight line and the map TC : z TC(z) from C intol projective space P1 is continuous.following results are proved in [4, Theorems 4.1, 6.2, 7.3].m 2.22. Let E R2 be such that F(E) < +. Then E is bounded, open, |EE| = 0,H1(E) < + has a continuous unoriented tangent. Moreover(E) inf{F( ): A(E)}, (9)n particularA(E) is nonempty.k 2.23. Let be a system of curves of class H 2,p without crossings and define:= Ao , F :={z R2 \ ( ): I(, z) 0 (mod 2)}. (10)s noticed in [4], E,E,F,F are open, E is bounded, |EE| = 0 andE = F = {z R2: 0 < Br(z)E< Br(z) r > 0}= E F ( ), E = int(Ao ( )). (11)ore Ao(E).us remark that in [4] the set Qfin(E) was denoted by Q(E).G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 847Theorem 2.24. Let be a system of curves of class H 2,p without crossings and satisfying the finiteness property.ThenFhence tsuphNspace SdefinedTheore2.6. ReDefinita regulclass HWe indof ( )Remarlet p RWesuch thevery xof r >H 2,p fJIf J =inductif1(x) /ByIf p B(z):Definitsufficiefor ( )Fig.Remarinvolveof R2 o(Ao ) < +,here exists a sequence {Eh} of bounded open sets of class C2 converging to Ao in L1(R2) and such thatF(Eh) < . In addition, there exist oriented parametrizations h of Eh defined on the same parameter, such that {h} converges strongly in H 2,p(S) to a system of curves equivalent to (see Definition 2.30),on S, and whose trace contains Ao .m 2.25. There exists a set E R2 such that F(E) < + and F(E, ) is not subadditive.gular and singular points of . Equivalent systemsion 2.26. Let be a system of curves of class H 2,p without crossings and let p ( ). We say that p isar point for ( ) if there exists a neighborhood Up of p such that ( ) Up is the graph of a function of2,p with respect to Tp( ). We say that p ( ) is a singular point of ( ) if p is not a regular point of ( ).icate by Reg the set of all regular points of ( ) and by Sing = ( ) \ Reg the set of all singular points.k 2.27. If H 2,p(S) is a system of curves, then Reg = . This is obvious if Sing = . If Sing = ,Sing and let R(p) = [a, a] [b, b] be a nice rectangle for ( ) at p, and write(p) ( )=rl=1graph(fl), fl H 2,p(]a, a[).proceed by induction over the number r of graphs. Suppose r = 2. As p Sing , there exists 1 ]a, a[at f1(1) = f2(1); hence we can find an open neighborhood U ]a, a[ of 1 such that f1(x) = f2(x) for U . Therefore Reg R(p) {(x, fl(x)): x U} = , l = 1,2. Assume that when R(p) ( ) consists2 graphs of H 2,p functions, then Reg R(p) = . Suppose that ( ) R(p) consists of r + 1 graphs ofunctions f1, . . . , fr+1. Define:= {x ]a, a[: f1(x) / {f2(x), . . . , fr+1(x)}}. then graph(f1) is contained in the union of the remaining r graphs and the thesis follows by theon hypothesis. Otherwise there is 1 J and an open neighborhood U ]a, a[ of 1 such that{f2(x), . . . , fr+1(x)} for every x U . Hence Reg R(p) {(x, f1(x)): x U} = .an arc of regular points we mean a connected component of ( ) consisting of regular points of ( ).( ) by B+ (p) (respectively B (p)) we mean {z B(p): (z p) (p) 0} (respectively {z (z p) (p) 0}), where (p) is a unit vector parallel to d/dt in p.ion 2.28. We say that p Sing is a node of ( ) if there exists Np N, Np > 1, such that for any > 0ntly small either B+ (p) ( ) \ {p} or B (p) ( ) \ {p} consists of the union of Np arcs of regular pointswhich do not intersect each other. We indicate by Nod the set of the nodes of ( ).3 explains the meaning of the definition of node.k 2.29. Since in the definition of regular point (respectively singular point and node) only the set ( ) isd and not the map , similar definitions can be given for every (immersed) one-dimensional submanifoldf class C1 without crossings.848 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880Fig. 4. TparticulaDefinitequivalIf of curvsystem2.7. SiDefinitWe saya functp ERemarthe setDefinitthat thkl=1 gFig. 3. The point p in (a) is a node of ( ), while the point q in (b) is not a node of ( ).wo equivalent systems of curves and ; observe that is a limit system of curves, while is not a limit system of curves. Inr Ao(E) \A(E) (the set E is the interior of the two drops).ion 2.30. Let H 2,p(S) and H 2,p(S) be two systems of curves without crossings. We say that isent to , and we write , if ( )= ( ) and = on ( ).A(E) and if , then does not necessarily belong toA(E), since in general is not a limit systemes. In Fig. 4 we show two equivalent systems of curves and , with A(E), such that is not a limitof curves. Eventually, observe that if then Ao = Ao .ngular points of E. Cusps and branch pointsion 2.31. Let E be an open subset of R2 such that E has continuous unoriented tangent, and let p E.that p is a regular point of E if there exists a neighborhood Up of p such that Up E is the subgraph ofion locally defined over Tp(E). We will indicate by RegE the set of all regular points of E. We say thatis a singular point of E (and write p SingE) if p / RegE .k 2.32. If E R2 is such that F(E) < +, by Theorem 2.22, near every regular point p, the boundary ofE can be represented as the graph of an H 2,p function with respect to Tp(E), see also Lemma 4.3 below.ion 2.33. Let E be an open subset of R2 with continuous unoriented tangent, and let p E. Supposeere are > 0 and an integer k 2 such that either B+ (p) E =kl=1 graph(fl) or B (p) E =raph(fl), where the fl are functions defined on Tp(E) whose graphs meet each other only at p. If k isG. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 849Fig. 5. Tcusp poieven wB (p)TheEacpoints,Remarat p. Apropert thethe if pand3. SomProposMoreovSandRRemarand SinProof.Nod Letcenterehe grey region locally stands for the set E. The point p of (a) is a branch point, but not a cusp point (k = 3). The point q of (b) is ant but not a simple cusp point, and the same happens for the point w in (c).e say that p is a cusp. If k is odd we say that p is a branch point. If k = 2 and either B+ (p) E = {p} or E = {p}, then we say that p is a simple cusp point.definition of the set NodE of the nodes of E is the same as Definition 2.28 where we replace by E.h connected component of the set E in Fig. 1 has a simple cusp; in Fig. 5 we show examples of branchand of cusp points which are not simple cusp points.k 2.34. Let E be such that F(E) < +, p NodE , Ao(E) and let R be a nice rectangle for ( )s noticed in [4, p. 269], the function is odd on the regular points of E. Since verifies the train tracksy in R (see Definition 3.6 below) we can conclude thatpoint p is always a cusp or a branch point, and cannot be a cusp and a branch point simultaneously, sinceconstants k corresponding to B+ (p) E and to B (p) E have the same parity;is a cusp (respectively a branch) point then (p) is even (respectively odd). Conversely if p NodE (p) is even (respectively odd) then p is a cusp (respectively a branch) point.e useful results on systems of curvesition 3.1. Let be a system of curves of class H 2,p without crossings. Then Nod is at most countable.er Sing has empty interior,ing = Nod , (12)eg = ( ). (13)k 3.2. It may happen thatH1(Sing ) > 0 (see [4, Example 1, p. 271]), therefore in this case Sing = Nodg is not countable.It is obvious that every node of ( ), or any accumulation point of nodes of ( ), is a singular point, so thatSing .us prove the opposite inclusion. Let p Sing . We can select a nice rectangle R = [a, a] [b, b]d at p = 0 where ( ) consists of a finite union of r > 1 graphs of H 2,p functions defined on R Tp( )850 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880and all passing through the point p. In particular, in [0, a ] [b, b] the set ( ) cannot be represented as thegraph oWegraph(fIAs f1compof1(0) =possibiaccumuAssp NoJThen Jthesis fand conWe divCasp Noofr+j=r+1j=1 gCasthroughthere a(1, f1R1 weNowk, j ah 0ph := (proof iLetof ( )of Nodof ( )To clet R bq intffor evepoint oof thisrectangf one function only for any [0, a[.now reason by induction on the number of graphs. Suppose first that [0, a] [b, b] ( ) = graph(f1) 2), and set 1(x) := (x, f1(x)), 2(x) := (x, f2(x)), x [0, a]. Define:= {x [0, a]: f1(x) = f2(x)}.and f2 are continuous, I is open, therefore it is the union of a possibly countable number of connectednents Ih. It is clear that, if x belongs to the boundary of one of the Ih, then (x, f1(x)) is a node. Sincef2(0), it follows that 11 (p) cannot belong to the interior of any of the Ih. Hence there are only twolities: either 11 (p) belongs to the boundary of one of the Ih, and then p Nod , or 11 (p) is anlation point of boundary points of the intervals Ih, and so p Nod .ume that when R ( ) consists of r > 2 graphs of functions of class H 2,p then p Sing impliesd . Suppose that R ( ) consists of r + 1 graphs of functions f1, . . . , fr+1 of class H 2,p. Define:= {x [0, a]: f1(x) / {f2(x), . . . , fr+1(x)}}.is open. If J is empty, then the graph of f1 is contained in the union of the remaining r graphs, and theollows by the induction hypothesis. Therefore we can suppose that J is nonempty. Define 2 := supJ > 0,sider the connected component ]1, 2[ of J having 2 as the right boundary point. Note that 0 1 < 2.ide the proof into two cases.e 1. 1 = 0. If p is a regular point for r+1j=2 graph(fj ), then it is a node for r+1j=1 graph(fj ), and henced . If p is a singular point forr+1j=2 graph(fj ), then p is a node (or an accumulation point of nodes)12 graph(fj ) by the induction assumption. Therefore p is a node (or an accumulation point of nodes) ofraph(fj ), and hence p Nod .e 2. 1 > 0. There are two subcases: either all functions fl , with l {2, . . . , r + 1}, whose graph passesthe point (1, f1(1)) coincide in an interval of the form ]1, 1 + [, or in any interval of this formre at least two functions fh, fk , with h, k {2, . . . , r + 1} that do not agree. In the first subcase we have(1)) Nod . In the second subcase we can select a nice rectangle R1 R centered at (1, f1(1)). Insidecan repeat the arguments of case 1 for the functions {f1, . . . , fr+1}. We conclude that (1, f1(1)) Nod ., using the C1-regularity of the fk and the fact that fk(p) = fj (p) and f k(p) = f j (p) for every 1 r + 1, we take a countable family of shrinking nice rectangles of the form [ah, ah] [bh, bh], withand bh 0 as h , and repeat the above arguments. In this way we obtain a sequence of pointsh1 , f1(h1 )) [ah, ah] [bh, bh] Nod , which converges to p and p Nod . This concludes then case 2, and the proof of (12).us now prove (13). Let p ( ); we have to prove that in each neighborhood of p there are regular points. This is immediate if p Reg . If p Sing , then by (12) either p Nod or p is an accumulation point ; in both cases, from the definition of node, we have that in each neighborhood of p there are regular points.onclude the proof of the proposition, it remains to show that Nod is at most countable. Let p ( ) ande a nice rectangle centered at p. Suppose that ( )R =hi=1 graph(fi), where fi H 2,p(Tp( )R). If(R) is a node of ( ), then there are k, l {1, . . . , h}, k = l, and 1 [a, a] such that (1, fk(1))= q andk(1)= fl(1) fk(1 + x) = fl(1 + x),ry x ],0[ or x ]0, [ (where > 0 is a number small enough). Therefore the point 1 is a boundaryf some connected component of the set {x [a, a]: fk(x) = fl(x)}, but this is an open set, and so pointskind can be at most countable. Now, since we can cover the whole of ( ) with a finite number of niceles, we have that Nod is at most countable. G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 851Remark 3.3. Since Sing = Nod by Proposition 3.1, it is clear that if ( ) verifies the finiteness property in Uthe setAs aCorollnice re(Proof.collectl=Clearly[b, b]Withouan opeevery xa2 I1are reg({ar} LemmrectangProof.wherespeed pIIand ithat (iand Sing U consists of a finite number of nodes.consequence of Proposition 3.1 we obtain the followingary 3.4. Let H 2,p(S) be a system of curves without crossings and let p Sing . Then there exists actangle R for ( ) at p such that ) R Reg . (14)Suppose p = 0 Sing and let [a, a] [b, b] be a nice rectangle for ( ) at p. Let f1, . . . , fr be aion of H 2,p(]a, a[) functions such thatr1graph(fl) =([a, a] [b, b]) ( ).for every ]a, a[, the set [,] [b, b] is still a nice rectangle for ( ) at p and ( ) ([,] ) is still represented by the graphs of f1, . . . , fr . By (13) we can find q1 ([a, a] [b, b]) Reg .t loss of generality we can suppose that q1 has coordinates (a1, f1(a1)) and a1 ]0, a[. As Reg isn subset of ( ), we can select an interval I1 ]0, a[ centered at a1 such that (x, f1(x)) Reg for I1. From Proposition 3.1 we know that Sing has empty interior in ( ). Therefore we can findsuch that (a2, f2(a2)) Reg and then select an interval I2 I1 such that (x, f1(x)) and (x, f2(x))ular points for every x I2. Repeating the same argument r times we find a point ar ]0, a] such thatR) ( ) = {z1, . . . , zh} Reg . Setting R := [ar, ar ] [b, b], we get (14). a 3.5. Let be a system of curves of class H 2,p. Let p = 0 ( ) and let R = [a, a] [b, b] be a nicele for ( ) at p. Then (p) =z( )({x}[b,b]) (z) x [a, a]. (15)Write = {1, . . . , m}. Let i {1, . . . ,m} be such that R i(S1i ) = , and write1i(int(R))= Mil=1Iil ,Mi and Iil are the connected components of 1i (int(R)). Using the fact that i is a constantarametrization we haveil Iik = l = k,i(Iil) int(R) ( ), i(Iil) ( ) R l,il = (s1, s2), i(s1) {a} ]b, b[ i(s2) {a} ]b, b[,is injective over each Iil , so that we can take Mi = {1i (p)}. Taking the union over all i {1, . . . ,m} such)R = , we get1(int(R))= mi=1{1i (p)}l=1Iil ,(p) =mi=1 {1i (p)}. Therefore (15) holds. 852 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880Definition 3.6. In the following, we will refer to property (15) as to the train tracks property of in R.Propos(a) for(b) Proof.class H( )Ron Up(a) Li thatIn thRemarLemmFProof.3.5, wewhereunder p andusing tffor eveFition 3.7. Let = {1, . . . , m} be a system of curves of class H 2,p. Thenany p ( ) and for any s0 S1j such that j (s0)= p there holds(j (s0)) lim supS1jss0(j (s)). (16)is constant on each connected component of Reg .(b) Let p Reg and let Up be a neighborhood of p such that ( ) Up is the graph of a function of2,p. Let us select a nice rectangle R Up for ( ) at p. From Lemma 3.5 we have that is constant on. Covering ( )Up with an appropriate countable family of nice rectangles, we obtain that is constant ( ).et p Sing and let R be a nice rectangle for ( ) at p. Then (a) follows from (15) and the fact that everyintersects R(p) passes through p. e following we will refer to property (a) of Proposition 3.7 as to the upper semicontinuity of .k 3.8. The boundedness and the upper semicontinuity of have been proved, in different contexts, in [16].a 3.9. Let H 2,p(S) and H 2,p(S) be two equivalent systems of curves without crossings. Then( )=F( ). (17)Let p ( ) and let R = [a, a] [b, b] be a nice rectangle at p for ( ) = ( ). As proved in Lemmahave1(R) = (p)i=1Ii , 1(R) = (p)i=1Ii ,the Ii S (respectively Ii S) are open connected pairwise disjoint arcs. Furthermore, the image of each Ii(respectively of Ii under ) is the graph of a function fi (respectively fi ) of class H 2,p passing throughthe restriction of over Ii (respectively of over Ii ) is injective. Since and are without crossings,he locality of the weak derivatives in Sobolev spaces (see for instance [1, Proposition 3.71]), we havei = fj , f i = fj a.e. on {fi = fj },ry i, j {1, . . . , (p) = (p)}. Hence, as by hypothesis ( )R = ( )R, we have(, 1(R))= (p)i=1]a,a[(1 + |fi |p(1 + (f i )2)3p/2)1 + (f )2 dx= (p)i=1]a,a[(1 + |fi |p(1 + (f i )2)3p/2)1 + (f )2 dx =F( , 1(R)). (18)G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 853Using Besicovitch Covering Theorem (see [12, Theorem 2.8.15]) we can find a countable family of nice rectangles{R(pn) ( )=FwhichRemarand ( )UFLemmReg ,Proof.(Theref = R = [(Using LwhichLemmp. LetProof. (p).either b1 (modUsin1 (modPropos{Then|} such that int(R(pn)) int(R(pm)) = if pn = pm and such that nNR(pn) covers H1-almost all( ). Using (18) we have( )=nNF(, 1(R(pn)))=nNF( , 1(R(pn)))=F( ),is (17). k 3.10. Using essentially the same proof, we can prove a local version of Lemma 3.9, that is: if H 2,p(S) H 2,p(S) are two systems of curves which verify the hypothesis of Lemma 3.9 in U R2, i.e.,= ( )U and = on ( )U , then(, 1(U))=F( , 1(U)).a 3.11. Let H 2,p(S) and H 2,p(S) be two systems of curves. If Reg = Reg and = onthen and are equivalent.From (13) we have )= Reg = Reg = ( ).ore to prove that and are equivalent it remains to check that = on ( ). Since by hypothesis on Reg = Reg it is enough to check that = on Sing = Sing . Let p = 0 Sing anda, a] [b, b] be a nice rectangle for ( )= ( ) at p, such that R verifies (14), that is ) R = ( ) R Reg = Reg .emma 3.5 and the hypothesis = on Reg we have (p) =hl=1 (zl)=hl=1 (zl) = (p),concludes the proof. a 3.12. Let be a system of curves without crossings, let p ( ) and be a unit vector normal to ( ) atz1 := p + t, z2 := p t. Then I(, z1)+ I(, z2) (p) (mod 2) for every t > 0 small enough.Using [4, Lemma 4.2] it follows that |I(, z1)I(, z2)| = |k d|, where k, d N are such that k + d =If I(, z1) + I(, z2) 0 (mod 2) then |I(, z1) I(, z2)| = |k d| is even. Therefore k and d areoth odd or even, hence (p) is even, and I(, z1)+ I(, z2) (p) (mod 2). If I(, z1)+ I(, z2)2) then |I(, z1) I(, z2)| is odd. Therefore (p) is odd, and I(, z1)+ I(, z2) (p) (mod 2).g Lemma 3.12 we prove that given a system of curves without crossings, the set {q ( ): (q) 2)} characterizes the set Ao in L1(R2).ition 3.13. Let H 2,p(S) and H 2,p(S) be two systems of curves without crossings. Assume thatq ( ): (q) 1 (mod 2)}= {q (): (q) 1 (mod 2)}. (19)AoAo| = 0. (20)854 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880Proof. Let C be the closure of the set {q ( ): (q) 1 (mod 2)}. We claim that C = (int(Ao ( ))).Using LFrom tit follohave thassumpand letare trannumbeare twoAo A4. PreIn thLemmto (0on [0,1Proof.arc lenSince C1 andparts anan Lpon ]l(Definitgraph(gthen gjany j LemmtangenThen Remarof [4].emma 3.12 it follows that C (int(Ao ( ))). Now let p (int(Ao ( ))) and suppose that p / C.he local constancy of the index it follows that (int(Ao ( ))) ( ), therefore p ( ). Since p / C,ws that (q) is even for every r > 0 small enough and every q Br(p) ( ). Using Lemma 3.12 weat I(, z) must be either always odd or always even for every z Br(p) \ ( ) which contradicts thetion p (int(Ao ( ))). Using (19) we have C = (int(Ao ( ))) = (int(Ao ())). Let z / ( ) be a continuous curve connecting z with such that all the intersections between ( ) () and ()sversal. Since I(, z) (mod 2) (respectively I(, z) (mod 2)) can be computed using the parity of ther of the intersections of ( ) (respectively of ()) with () and since int(Ao ( )) and int(Ao ())bounded open subsets of R2 with the same boundary we have int(Ao ( )) = int(Ao ()). Thereforeo ( ) (), so that |AoAo| |( ) ()| = 0. liminary lemmatais section we prove some lemmata needed in the proof of Theorems 5.1, 6.3, 7.1.a 4.1. Let , : [0,1] R2 be two regular curves of class H 2,p such that (1)= (0) and (1) is parallel). Then there is a regular curve : [0,1] R2 of class H 2,p such that ( )= () () and | | is constant].Let : [0, l()] R2 (respectively : [0, l()] R2) be the reparametrization of (respectively of ) bygth such that (l()) = (0) and (l()) = (0). Define : [0, l()+ l()] R2 by(s) :={(s) if s [0, l()],(s l()) if s ]l(), l() + l()].and are regular curves of class C1 and (l()) = (0), (l()) = (0), then is a regular curve of class = (respectively = ) on [0, l()] (respectively on [l(), l() + l()]). Using two integrations byd the assumptions on and one checks that the second distributional derivative of is represented byfunction and = (respectively = ) almost everywhere on ]0, l()[ (respectively almost everywhere), l() + l()[). Reparametrizing with t := s/(l() + l()) we obtain the thesis. ion 4.2. Let a > 0 and {g1, . . . , gr } be a finite family of functions in C1([0, a]). We say that1), . . . ,graph(gr ) meet tangentially in [0, a] if given j, l {1, . . . , r} and x [0, a] such that gj (x)= gl(x),(x) = gl (x). We say that graph(g1), . . . ,graph(gr) pass through zero horizontally if gj (0)= gj (0) = 0 for{1, . . . , r}.a 4.3. Let a > 0 and f1, . . . , fr be a family of distinct functions of class H 2,p(]0, a[) whose graphs meettially in [0, a] and pass through zero horizontally. Define:={g C0([0, a]): graph(g) rl=1graph(fl)}.is a bounded subset of H 2,p(]0, a[).k 4.4. The fact that is a bounded subset of C1([0, a]) was already observed in the proof of Theorem 6.4G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 855Proof. Let g . Assume first that [0, a] =di=1[i,i], with d > 0 a natural number and [i,i ] intervals (with0 =: 14.4 we]0wherethat g Using (whereAssg in theand eaFix n can alsh {0,Cason [xh,CasNoticein the uand fl1and fl(and setRep[xh, xhObsfl(x) =necessawe obtintervaNowReca subse< 1 < 2 < < d1 < d < d := a) where g is equal to some fli . Let Cc (]0, a[). By Remarkhave that g C1([0, a]), therefore,a[g dx =di=1]i,i [f li dx=di=1[]i,i [f li dx + (i)f li (i) (i)f li (i)]= di=1]i,i [f li dx, (21)we used the fact that the fj meet tangentially and the compactness of the support of . Using (21) it followsH 2,p(]0, a[) andgLp(]0,a[) rl=1f l Lp(]0,a[). (22)22) and the fact that is a bounded subset of C1([0, a]) (Remark 4.4) we deduce thatgH 2,p (]0,a[) C, (23)C > 0 is a constant independent of g.ume now that g is arbitrary. Fix a dense countable subset D = {xk} of [0, a]. We want to approximateweak topology of H 2,p(]0, a[) with a sequence {gn} such that gn(xk)= g(xk) for every k = 1, . . . , nch gn satisfies the hypothesis of the preceding step. To construct {gn} we proceed in the following way.N and relabel the first n elements of D in such a way that x0 := 0 < x1 < < xn < xn+1 := a. Weo assume thatnN{x0, . . . , xn+1} = D. We give the definition of gn over each interval [xh, xh+1]. Let. . . , n}. We have two cases.e 1. There exists l {1, . . . , r} such that fl(xh)= g(xh) and fl(xh+1)= g(xh+1). In this case we set gn := flxh+1].e 2. For every l {1, . . . , r} either fl(xh) = g(xh) or fl(xh+1) = g(xh+1). Define1 := inf{x ]xh, xh+1[: l {1, . . . , r}: fl(x)= g(x) and fl(xh+1) = g(xh+1)}.that xh < 1, otherwise we are in case 1; moreover, the fact that g is continuous and its graphs is containednion of the graphs of the fi imply that 1 < xh+1. Finally, there is l1 {1, . . . , r} such that fl1(1) = g(1)(xh+1) = g(xh+1). We set gn := fl1 on [1, xh+1]. Now, if there is l {1, . . . , r} such that fl(xh) = g(xh)1) = g(1) we set gn := fl on [xh, 1] and the algorithm stops. Otherwise we define2 := inf{x ]xh, 1[: l {1, . . . , r}: fl(x)= g(x) and fl(1)= g(1)},gn := fl2 on [2, 1], where l2 {1, . . . , r} is such that fl2(2) = g(2) and fl2(1)= g(1).eating the same argument i-times, i an arbitrary natural number, the function gn is defined on [ i, xh+1] +1] and gn agrees with one of the fl on each interval [j , j1], with j = 1, . . . , i .erve that, if for some j {1, . . . , i} and l {1, . . . , r} we have fl(j ) = g(j ) then, by definition of j ,g(x) for every x [xh, j ]. Since we deal with r distinct functions, after a finite number K r of steps,rily there is lK {1, . . . , r} such that flK (xh)= g(xh) and flK (K)= g(K). Setting gn := flK on [xh, K ],ain that there is a finite number of closed intervals (with pairwise disjoint interior), whose union is the wholel [xh, xh+1], where gn agrees with one of the fl ., repeating this construction for every h = 1, . . . , n, we obtain the desired function gn.alling (22), we have that the H 2,p norm of gn is uniformly bounded with respect to n. It follows that {gn} hasquence that converges weakly in H 2,p(]0, a[) to a certain g H 2,p(]0, a[). Since H 2,p weak convergence856 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880implies uniform convergence, we have that g and g coincide on the dense set D, hence g g on [0, a]. Thereforeg H 2GivPAs prolemmaLemm{(observThen thmadmitssuch thProof.p. 74].minimuFig. 6. Tminimum,p(]0, a[) and (23) holds. en an open interval I and a function g H 2,p(I ), we define(g) :=I[1 +( |g|(1 + (g)2)3/2)p]1 + (g)2 dx. (24)ved in [4], P(g) equals the energy F( ) of a simple curve whose support is the graph of g. The nextis concerned with P-minimal connections between the origin and a given point zj , see also Fig. 6.a 4.5. Let a, f1, . . . , fr and be as in Lemma 4.3. Setz1, . . . , zh} ={(a,f1(a)), . . . , (a, fr(a))}e that in general h r). Letj :={g : g(a)= zj}, j {1, . . . , h}.e problemin{P(g, ]0, a[): g j} (25)a solution. Moreover if j = l, there exist a minimizer gj of P over j and a minimizer gl of P over lat the following property holds: if for some c ]0, a[ we have gj (c)= gl(c), then gj gl on [0, c].The weak H 2,p sequential lower semicontinuity of the functionalP(, ]0, a[) follows from [9, Theorem 3.4,Using Lemma 4.3 it follows that j is H 2,p-weakly compact for every j {1, . . . , h}. Therefore them problem (25) admits a solution.hese two figures show the construction in the proof of Lemma 4.5. In the first figure we depict three solutions gj , j = 1,2,3 of theproblem (25), and g2(c) = g3(c). In the second figure we depict the resulting minimizers: in this case g2 g3 on [0, c].G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 857Let us fix g1 1 solution of (25) for j = 1. Take a function g2 2 solution of (25) for j = 2 and letsClearlyNote thPIndeedgUsing aof g1 oNowgBy (26a functand maReppropertRemarallows4.1. FiDefinitYbe a famgWe calthe genRemarzClearly:= sup{x [0, a]: g1(x)= g2(x)}.s < a. If s = 0 then the graphs of g1 and g2 meet only at 0 and in this case we set g2 := g2. Suppose s > 0.at(g1, ]0, s[)=P(g2, ]0, s[). (26), if not, assuming by contradiction for instance that P(g1, ]0, s[) >P(g2, ]0, s[), we can define the function1 :={g2 on [0, s],g1 on ]s, a].lso Lemma 4.1 we have g1 1; moreoverP(g1, ]0, a[) >P(g2, ]0, a[), thus contradicting the minimalityver 1.we define2 :={g1 on [0, s],g2 on ]s, a].) we have P(g2, ]0, a[)= P(g2, ]0, a[). Therefore g2 is still a minimizer of P(, ]0, a[) over 2. Now takeion g3 3 solution of (25) for j = 3, define:= sup{x [0, a]: either g3(x)= g2(x) or g3(x)= g1(x)},ke the same operation above to obtain the function g3.eating the same argument for each zj we obtain a family of minimizers of P satisfying the requiredies. k 4.6. The last assertion concerning gj and gl in Lemma 4.5 is crucial in the proof of Theorem 5.1, since itto locally modify an arbitrary H 2,p system of curves into a new system verifying the finiteness property.nite unions of graphs, generalized multiplicity, canonical familiesion 4.7. Let r N \ {0}, I R a closed interval and:= {(g1,1), . . . , (gr ,r)}ily of pairs where gl : I R is a continuous function and l N \ {0} for every l = 1, . . . , r . We setraph(Y ) :=ri=1graph(gi).l the functionY : graph(Y ) N \ {0}, Y (x, y) :=l:gl(x)=yl, (27)eralized multiplicity of Y .k 4.8. Let b be a real number with b > max1lr glL(]0,a[). Then({x}[b,b])graph(Y )Y (z)=rl=1l x [0, a].if all gi(x) have the same value at x = 0, then Y (0)=rl=1 l for any x [0, a].858 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880In the following lemma we do not assume that the union of the graphs of the functions gj (in R+) is containedin ( )Lemma nice{Let s Y := {( gra {(a if (If the gthen thproper((Proof.whereUsing (for anywe canthat (athat, foconstruDefinit[b, b]{1, . .(LemmProof.and (13that R+.a 4.9. Let H 2,p(S) be a system of curves without crossings. Let p = 0 ( ), R = [a, a] [b, b] berectangle for ( ) at p and setz1, . . . , zh} = ( )({a} [b, b]).N, s h, let {g1, . . . , gs} H 2,p(]0, a[) be a collection of distinct functions, {1, . . . ,s} N \ {0}, andg1,1), . . . , (gs,s)}. Assume thatph(g1), . . . ,graph(gs) meet tangentially in [0, a] and pass through zero horizontally;, g1(a)), . . . , (a, gs(a))} = {z1, . . . , zh};a,gl(a))= zj for some l {1, . . . , s} and j {1, . . . , h}, then the vector (1, gl (a)) is parallel to Tzj ( );sl=1 l = (p).eneralized multiplicity Y of Y satisfiesY (zj ) = (zj ) j {1, . . . , h}, (28)ere exists a system of curves H 2,p(S), having the same number of curves as , with the followingties:) R+ = graph(Y ) and = Y on graph(Y );) \R+ = ( ) \R+ and = out of R+.Let = {1, . . . , m}. As observed in the proof of Lemma 3.5, we have1(R+)=mi=11i (R+)=mi=1{1i (p)}k=1Iik, (p) =mi=1 {1i (p)} and Iik are closed, connected, pairwise disjoint arcs of S1i . Fix j {1, . . . , h}.28) and (27) we have {Iik : zj i(Iik)} = (zj ) = Y (zj ) =l:(a,gl(a))=zj l . Write Iik = (s1, s2). As,k = 1, . . . ,m, the first components of the two vectors i (s1), i (s2) are either both positive or both negative,apply Lemma 4.1 and obtain a new H 2,p curve whose image in R+ is given by graph(gl) for some l such, gl(a)) = zj . Fixed zj we repeat the same argument for every Iik such that zj i(Iik) in such a wayr every l {1, . . . , s} such that (a, gl(a)) = zj , graph(gl) is parametrized exactly l times. Repeating thisction for every j {1, . . . , h} we obtain the new system of curves . ion 4.10. Let H 2,p(S) be a system of curves without crossings. Let p = 0 ( ) and R = [a, a] be a nice rectangle for ( ) at p. Let {f1, . . . , fr } H 2,p(]a, a[) be a collection of distinct functions and. ,r } N \ {0}. We say that Y := {(f1,1), . . . , (fr ,r)} is a canonical family for ( ) in R if )R = graph(Y )R and = Y on ( )R. (29)a 4.11. Let and R be as in Definition 4.10. Then there exists a canonical family Y for ( ) in R.Since takes nonnegative integer values, we can consider 1 := min{ (q): q R} N \ {0}. From (16)), it follows that we can find q1 Reg R such that (q1) = 1. From (b) in Proposition 3.7, it follows 1 on a whole connected component C1 of Reg R containing q1. Now let f1 H 2,p(]a, a[) beG. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 859such that C1 graph(f1) ( ) R (the existence of f1 is ensured from the fact that R is a nice rectangle for( )). Tand dein int(Rsemicowith G1Reg after rsuch thfand satWerelated5. MaThemain reTheorespace Sclass Hand(Proof.Stepsuch th ( N F(Fix Nthan 2Shen consider the function1 : ( )R N, 1 :={ 1 on graph(f1), otherwise on R,fine G1 := {q int(R): 1(q) > 0}. As verifies (15) we have that 1 verifies the train tracks property). Now, observing that G1 is still a finite union of H 2,p graphs, that the train tracks and the upperntinuity properties still hold for G1 and that Sing SingG1 , we repeat the argument above replacing ( )and with 1. In this way we obtain 2 := min{1(q): q R} N \ {0}, a connected component C2 ofG1 and a function f2 H 2,p(]a, a[) such that C2 graph(f2) G1 ( ). Repeating this construction, (p) steps, we obtain that Gr+1 = . In this way we construct a family Y := {(f1,1), . . . , (fr ,r)}atl = fj for every l = j, since if l < j then graph(fj )Cl = and Cl graph(fl),isfying (29). conclude this section by observing that the definition of canonical family for a system of curves could bewith the notion of C1, multiple function appearing in varifolds theory, see for instance [16].in result on the approximation of systems of curvesfollowing theorem is the crucial approximation result for systems of curves of class H 2,p, and is one of thesults of the paper.m 5.1. Let be a system of curves of class H 2,p(S) without crossings. Then there exist a parameter, a limit system of curves H 2,p(S) equivalent to and a sequence {N } of limit systems of curves of2,p(S) satisfying the finiteness property, such thatN weakly in H 2,p(S), limNF(N )=F( ),N) ( ), F(N)F( ) N N.The proof is divided into three steps.1. We construct a sequence {N } H 2,p(S) of systems of curves (not necessarily limit systems of curves)at, for every N N, the following properties hold:N) ( );verifies the finiteness property;N)F( ). N. For any p Sing let R(p) be a nice rectangle for centered at p, with diameter strictly smallerN. By (12) the set Sing is compact, hence there are p1, . . . , pm(N) points of Sing such thating m(N)i=1R(pi). (30)860 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880Recalling (14), we can assume that ( ) R(pi) Reg for any i {1, . . . ,m(N)}. In order to construct thesystem1 i int(R(p(i) ((ii) (iii) F(iv) Letand tha(Ni1)(Ni1){f+1 , . .ForConsidmwhereminimu[0, c]. TNow cogeneraLetfamilieSincfind a ssame aand thepropertTo pcoincidUsinObservRecFNote all:N we use a recursive algorithm consisting of m(N) steps. We proceed as follows: let N0 := , letm(N), and suppose that Ni1 has been defined. Then Ni is obtained by modifying Ni1 only oni )), in particular (Ni ) \R(pi)= (Ni1) \R(pi), in such a way that:Ni ) (Ni1);Ni verifies the finiteness property in int(R(pi));(Ni )F(Ni1);Ni and Ni1 are defined on the same parameter space.us define Ni . To simplify the notation, we assume that pi = 0, that Tpi (Ni1) coincides with the x-axist R(pi) = [a, a] [b, b]. We shall work on (Ni1)R+(pi), since the modification of Ni1 on the set R(pi) is similar. Because of the assumptions on R(pi) and the inclusion (Ni1) ( ), we have that ({a} [b, b]) consists of a finite set of distinct points z+1 , . . . , z+h , labelled by their y-coordinate. Let. , f+r+} H 2,p(]0, a[) be the family of distinct functions such that (Ni1)R+(pi)=r+l=1 graph(f+l ).any j {1, . . . , h} let+j :={g C0([0, a]): graph(g) (Ni1)R+(pi), g(a)= z+j }.er the problemin{P(g, ]0, a[): g +j },P is defined in (24). According to Lemma 4.5, for every j = 1, . . . , h we can select a function g+j +j ,m of P over +j , such that, if j = l and if for some c ]0, a[ we have g+j (c)= g+l (c), then g+j g+l onhen we replace all the f+1 , . . . , f+r+ with the g+1 , . . . , g+h . Observe thathk=1 graph(g+k )r+l=1 graph(f+l ).nsider the family Y+ := {(g+1 , Ni1(z+1 )), . . . , (g+h , Ni1(z+h ))} and let Y+ : graph(Y+) N \ {0} be thelized multiplicity of Y+.fl , zj , j , gk , (zj ), Y, Y be the analog for the interval [a,0] of the spaces, functions, points,s and densities that we used in the construction on the interval [0, a].e Ni1(p) =hj=1 Ni1(zj ) and, by construction, Ni1(zj ) = Y(zj ) we can apply Lemma 4.9 andystem of curves in H 2,p(S), which will be our Ni , whose trace and density function outside R(pi) are thes Ni1, while on R+(pi) (respectively on R(pi)) the trace is given by graph(Y+) (respectively graph(Y))density function agrees with Y+ (respectively with Y ). By construction, and recalling Remark 4.6,ies (i), (ii) and (iv) hold (note that Ni verifies the finiteness property overji R(pj )).rove the validity of (iii) we need the concept of canonical family. Since the supports of the system of curvese outside R(pi) it is enough to verify inequality (iii) inside R(pi).g Lemma 4.11 we can choose a canonical family {(f+1 ,+1 ), . . . , (f+r+,+r+)} for (Ni1) in R+(pi).e that {z+1 , . . . , z+h } = {(a, f+1 (a)), . . . , (a, f+r (a))} and h r+ Ni1(pi).alling Definition 4.10 it follows(Ni1, (Ni1)1(R(pi)))= r+l=1+l P(f+l , ]0, a[)+ rl=1l P(fl , ]a,0[). (31)so thatfl (a)=zjl = Ni1(zj ) j {1, . . . , h}. (32)G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 861We now group the terms in the summationr+l=1 as follows:lWe obsPThereflSimilarlUsing (Fand (iiiWeWe hav(Conseqfiniteneon R2.Fand thiStepis not nSincsNwe canH 2,p(SWeLet p1/2N 0. So, for every N withdist(p,Sing ), the point p is outside the region where we made our modifications and therefore there862 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880is a whole neighborhood of p where the support of N and its density function are the same as the support and thedensity( )=By cStepLetSN andboundeof orieAN =any N .SinchaveFFurthersubseqsystemFinaFand theFig.TheCorollspaceparameEIn partFProof.converN,h :=can selorientewe havNIt remafor evethe Do of . Therefore p Reg and Reg Reg. Hence, recalling (13) and the inclusion () ( ), we getReg (). So ( )= () and therefore Reg = Reg.onstruction we have = on Reg = Reg. Hence by Lemma 3.11.3. Construction of the sequence {N }.us fix N N. As N verifies the finiteness property we can apply Theorem 2.24 and find a parameter spacea limit system of curves N H 2,p(SN) such that: SN has a number of connected components uniformlyd with respect to N ; N N and N is the strong H 2,p-limit of a sequence {N,h}h H 2,p(SN)nted parametrizations of bounded smooth open sets with equibounded energy and L1(R2)-converging toAoN . Passing to a suitable subsequence (still labelled by the index N ) we can suppose that SN = S fore N and N are equivalent, from step 1 we have SingN = NodN Sing and, using Lemma 3.9, we(N)=F(N)F( ). (39)more from step 1 we also have (N) = (N) ( ). Therefore we can apply Theorem 2.10 and find auence (still indicated by N ) whose elements are all defined on S, and weakly converging in H 2,p to aof curves H 2,p(S). Using the same arguments of step 2, one can prove that .lly( )=F( ) lim infN F(N ) lim supNF(N)F( ),refore F( ) = limNF(N). 7 illustrates the construction of the sequence {N } of Theorem 5.1 in a particular situation.following result is an improvement of Theorem 2.24.ary 5.2. Let be a system of curves of class H 2,p(S) without crossings. Then there exist a parameterS, a system of curves H 2,p(S) A(Ao ) equivalent to and a sequence {N } of orientedtrizations of bounded open smooth sets EN R2, such thatN Ao in L1(R2), N weakly in H 2,p(S), limNF(N) =F( ). (40)icular(Ao ) < +. (41)Let H 2,p(S) and {N } be as in Theorem 5.1. The convergence of the energies, together with the weakgence, implies that limN N2,p = 2,p, hence the strong H 2,p-convergence of {N } to . WriteEN,h, where N,h are introduced in the proof of step 3 in Theorem 5.1. Using a diagonal argument weect a subsequence {EN,hN }, which for simplicity we denote by {EN }, such that the sequence {N } of thed parametrizations of the elements of {EN } converges strongly in H 2,p(S) to . Therefore, since ,elimF(N)=F( ) =F( ).ins to prove that EN E in L1(R2) and that A(E). For every N N we have EN (z) = I(N, z)ry z R2 \ (N) and A (z)= I( , z) for every z R2 \ ( ). By the continuity property of the index andminated Convergence Theorem we have that EN = AN A in L1(R2) as N . Using the fact thatwe have A =Ao = E, so that EN E in L1(R2). Moreover, A(E). G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 863Fig. 7. Tcomponethat =Theoremsystemshave denresults oRemarfiniteneTheore6. RepAccrepresea functgraphsTheProposFProof.Fhe set E := E1 E2 E3 has smooth boundary except for the simple cusps of E1 and E2. The boundary of the smooth connectednt E3 oscillates and meets (from above) infinitely many times the horizontal line connecting the two cusps. Let A(E) be such1 on Reg E and = 2 on Reg (R2 \ E). The system N is obtained through the desingularization procedure described in5.1, while the system n is obtained through the desingularization procedure of Theorem 6.3. The main difference between the twois explained in Remark 6.5. These two systems of curves are equivalent to out of the two respective (dotted) nice rectangles, andsity constantly equal to 3 inside the rectangles. The energies of the systems n converge to F(E) (this will be a consequence of thef Section 8) whereas the sequence itself does not converge to an element of Qfin(E).k 5.3. Inequality (41) was proved in [4, Theorem 6.2], under the further assumption that satisfies thess property. Removing this assumption is one of the interesting and useful aspects of Corollary 5.2 (and ofm 5.1).resentation formulas for Fording to Theorem 2.25, the functional F(E, ) is not local. As a consequence,F does not admit an integralntation. In this section we study how to represent F as a minimum problem involving F , considered asional defined on systems of curves. Using tools of geometric measure theory (namely generalized Gauss) in [10] there are some partial results in this direction.following result is an improvement of (9).ition 6.1. Let E R2 be such that F(E) < +. Then(E)= min{F( ): A(E)}= min{F( ): Ao(E)}. (42)Thanks to (9), to show the first equality in (42) it is enough to prove that(E) inf{F( ): A(E)} (43)864 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880and that the infimum in (43) is attained. Given A(E), let {N } and {EN } be as in Corollary 5.2. Recalling thatA = AFand (43Nowlimh+ HEh h A(FwhereLetthe inccan beAo(E)DefinitF( )=Theorethere eIn partFRemar A(RemarapproxEThe diisolated(an infiof ( )Remaro , |EA | = 0, using (5) and (40) we have(E) lim infh+F(EN)= limN+F(N)=F( ),) follows.we select a sequence {Eh} of smooth bounded open sets converging to E in L1(R2) and such thatF(Eh) = F(E). As in the proof of [4, Lemma 3.3], we can find a parameter space S, a system2,p(S) and a sequence {h} H 2,p(S) of oriented parametrizations of smooth bounded open sets Eh Eh,Eh, such that (h) are all contained in a bounded subset of R2 independent of h, Eh E in L1(R2) andweakly in H 2,p(S) as h +. To show that the infimum in (43) is attained, it is enough to observe thatE) and(E)= limh+F(Eh) lim infh+F(Eh)= lim infh+F(h)F( ) F(E),we used the weak H 2,p lower semicontinuity of F on systems of curves and (43).us now prove that F(E) = min{F( ): Ao(E)}. As a direct consequence of the above arguments andlusionA(E)Ao(E) we have F(E) inf{F( ): Ao(E)}. On the other hand, the opposite inequalityproved as in the proof of (43), using the fact that |EAo | = 0. Eventually, the proof that the infimum inis attained follows from the inclusion Ao(E) A(E) and the above observations. ion 6.2. Let E R2 be such that F(E) < +. Any A(E) (respectively Ao(E)) satisfyingF(E) will be called a minimal system of curves in A(E) (respectively in Ao(E)).m 6.3. Let E R2 be such that F(E) < + and suppose that SingE is a finite set. Let A(E). Thenxist a sequence {n} Qfin(E) and a system of curves such thatn weakly in H 2,p, limn+F(n)=F( ).icular(E)= inf{F( ): Qfin(E)}. (44)k 6.4. The set Qfin(E) is not empty only if E has a finite number of singularities. Indeed, for everyE) we have Sing SingE ; therefore, if SingE is infinite, cannot verify the finiteness property.k 6.5. The main difference between Theorem 6.3 and Theorem 5.1 is that in Theorem 6.3 we are able toimate under the additional constraint that = int(An (n)) n N. (45)fficulty to keep (45) true is related to the following observation: even if the singular points of E are, it may happen that they are accumulation points of singularities of ( ), see Fig. 8; similarly, there may benite number of) regular points of E which are singular points (or accumulation points of singular points).k 6.6. We shall see in Section 8.1 that the infimum in (44) in general is not achieved.G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 865Proof oLet accnice re ( if p if pallNoeveSeleassumepThe coStep(a) E(b) |A(c) Sin(d) limIn the cthat coFig. 8. A cusp of E which is accumulation point of singular points of .f Theorem 6.3. Write = {1, . . . , m} : S R2. Recall that, as E ( ), we have SingE Sing .sing( ) be the set of the accumulation points of Sing . Fix n N. For every p accsing( ) let R(p) be actangle for ( ) at p with diameter less than 2n such that:) R(p) Reg (recall Corollary 3.4);/ E then R(p)R2 \ E; SingE then E is represented in R+(p) (or in R(p)) by a finite union of graphs of H 2,p functions,passing through p, that do not intersect each other at any point of R+(p) \ {p} (or of R(p) \ {p}) andF(, 1(R(p)))< 1MC2n, C := SingE , M := L(( ),H1). (46)te that these graphs coincide with all points of ( ) (R(p) \ {p}) with odd density (in general p may haven density, for example if it is a cusp point of E).ct a finite family {R(p1), . . . ,R(p(n))} covering the set accsing( ). Since SingE is finite, we can alsothat {R(p1), . . . ,R(p(n))} satisfies the following additional property: SingE accsing( ) R(p) {R(p1), . . . ,R(p(n))}.nstruction of {n} is divided into two steps.1. We construct a sequence {n} H 2,p(S) of systems of curves such that (n) ( );onE| = 0;gn E is a finite set;n+F(n)=F( ).onstruction of {n} we are not able to bound the energy of n with the energy of ; however, we can provendition (d) is valid, and at the same time the constraint in condition (b) is fulfilled.866 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880In order to construct n we use a recursive algorithm consisting of (n) steps. Let n0 := , let 1 i (n)and supni1 o(i) thod(ii) Si(iii) thWe supni :=LetRwith kif pi Defiand setpi Xof H 2,zero hoFinallyk > 1,ArgYfor (,XWe nowYWe havgIf R(pon wheOn thesingulapose that the system ni1 of curves of class H 2,p has been defined. Then ni is obtained by modifyingnly on int(R(pi)), in such a way that:e set of the points of (ni ) where ni is odd is the same as the set of the points of (ni1) where ni1 isd;ngni R(pi) E is a finite set;e following estimate holds:F(ni1, (ni1)1(R(pi)))F(ni , (ni )1(R(pi))){ 1C2n if pi SingE ,M(n)2n if pi RegE .pose pi = 0 and R(pi) = [a, a] [b, b]. If either pi R2 \ E or pi / accsing(ni1) then we setni1 in R(pi), and (i)(iii) are trivially satisfied.us now suppose that pi accsing(ni1) E. Write+(pi) E =kl=1graph(+l ), 1, +l H 2,p(]0, a[), +l (0) = 0 for every l = 1, . . . , k and +l < +j on ]0, a] for 1 l < j k (k = 1RegE ).ne: (ni1)R+(pi) 2N, :={ni1 1 onkl=1 graph(+l ),ni1 otherwise in (ni1)R+(pi),X := {q int(R+(pi)): (q) > 0}. As pi accsing(ni1), from (16) it follows that ni1(pi) > 1, hence. Since ni1 verifies the train tracks property in int(R+(pi)) and (ni1) int(R+(pi)) is a finite unionp graphs, we have that also X is a finite union of H 2,p graphs meeting tangentially and passing throughrizontally. Furthermore |X verifies the train tracks property and is upper semicontinuous in int(R+(pi))., we remark that, since the set of points where ni1 is odd coincides with kl=1 graph(+l ) (possibly, forwith the exclusion of pi ), then |X is everywhere even.uing as in the proof of Lemma 4.11 we construct a canonical family+ := {(f+1 ,2+1 ), . . . , (f+r+ ,2+r+)} H 2,p(]0, a[) (2N \ {0})X) in R+(pi), hence= graph(Y+), = Y+ on X.define+ := {(+1 ,1), . . . , (+k ,1), (f+1 ,2+1 ), . . . , (f+r+,2+r+)}.eraph(Y+)= (ni1) R+(pi), Y+ = ni1 on graph(Y+).i )E = we repeat the same construction in R(pi). We now proceed in two different ways dependingther pi SingE or pi RegE . The case pi SingE is easier, since by assumption SingE is finite.other hand, there may be an infinite number of regular points of E which are accumulation points ofr points of , and this makes case 2 more delicate.G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 867Case 1 of step 1. Suppose pi accsing(ni1) SingE (a situation like the one depicted in Fig. 8).In thWe repis a unigRoughneighbDefiZBy conwhereR+(piIf RAs(respeccurves(ni )=By csRSince (Fand (iiiWeCasWework oUsinRwhere1 l Iis case we have k > 1. Let l {1, . . . , r+} and definel := sup{x [0, a]: (x,f+l (x)) kj=1graph(+j )}.lace the function f+l with the function g+l defined as follows: if l = 0 then g+l := f+l . If l ]0, a] thereque j {1, . . . , k} such that f+l (l)= +j (l); in this case we set+l :={+j on [0, l],f+l on [l, a].ly speaking, the above definition means that the graph of g+l coincides with E in a small half-orhood of pi , thus leading by construction to the finiteness property of ni on R+(pi) E.ne+ := {(+1 ,1), . . . , (+k ,1), (g+1 ,2+1 ), . . . , (g+r+,2+r+)}. (47)struction R+(pi) E graph(Z+) (ni1) R+(pi). In addition the set of points of graph(Z+)Z+ is odd coincides with the set of points of R+(pi) where ni1 is odd (which coincides, in turn, with) E, possibly with the exclusion of pi ).(pi) E {pi} we repeat the same construction in R(pi).R(pi) Reg Regni1 and Y = ni1 , we have that Z+ = ni1 on ({a} [b, b]) (ni1)tively Z = ni1 on ({a} [b, b]) (ni1)), so we can apply Lemma 4.9 and find a system ofin H 2,p(S), which will be our ni , such that (ni ) R(pi) = graph(Z) and ni |R(pi ) = Z , while(ni1) and ni = ni1 outside of R(pi).onstruction we have that (i) and (ii) are satisfied. Furthermoreup(pi)ni= ni (pi)= ni1(pi)= supR(pi)ni1 = = supR(pi)n0= supR(pi) M.ni ) (ni1) ( ), we get, using (46),(ni , (ni )1(R(pi)))MF(, 1(R(pi))) 1C2n, (48)) follows.now consider the most difficult case.e 2 of step 1. Suppose pi accsing(ni1) RegE .keep the notation introduced at the beginning of step 1, but we omit the super/subscript , since we directlyn the whole of R(pi).g the assumption that pi is a regular point for E it follows that(pi) E = graph(), H 2,p(]a, a[), and (0) = (0) = 0. Let {z1 , . . . , zj} := ({a} [b, b]) (ni1). Fix fl withr . Letl :={x [a, a]: fl(x) = (x)}= kNIlk,868 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880where the Ilk are open pairwise disjoint intervals. We replace the function fl with the function gl defined as follows.If Il isgl, HgThe reunchanObslimsuch thRepeatH 2,p(]ZBy conZ = trace angraph(Z(i) is vaHenceWeBy conwhereobtainSincalso (iicomposed by a finite number of connected components then we let gl := fl . Otherwise, let N and define2,p(]a, a[) asl, :={ on Ilk with k and a / Ilk,fl otherwise in [a, a].quirement a / Ilk is needed to ensure that the conditions on the lateral boundary of R(pi) remainged.erve that gl, = f l on [a, a] \k Ilk and gl, = f l almost everywhere on [a, a] \k Ilk . Since+k H1(Ilk) = 0, using the absolute continuity of the Lebesgue integral, we can choose l(n) NatP(fl, ]a, a[)P(gl,l(n), ]a, a[) P(fl, kl(n)Ilk)+P(,kl (n)Ilk) 12n(n)r. (49)ing the same arguments for every l {1, . . . , r} we obtain a collection of functions {g1, . . . , gr } a, a[) defined as gi := gi,i (n). Let us consider the family:= {(,1), (g1,21), . . . , (gr ,2r)}. (50)struction we have R(pi) E graph(Z) (ni1) R(pi), graph(Z) R(pi) = {z1 , . . . , zj} andni1 on R(pi). Applying Lemma 4.9 we obtain a system of curves in H2,p(S), which will be our ni , whosed density function outside R(pi) are the same of ni1, while (ni ) R(pi) = graph(Z) and ni = Z on). By construction we have that (ii) is satisfied; moreover {q graph(Z): (q) is odd} = graph(). Hencelid. Using (49) we obtainF(ni , (ni )1(R(pi)))F(ni1, (ni1)1(R(pi)))=(P(, ]a, a[)+ rl=12lP(gl, ]a, a[))(P(, ]a, a[)+ rl=12lP(fl, ]a, a[))Mrl=1P(gl, ]a, a[)P(fl, ]a, a[) M2n(n) . (51)(iii) is valid and this concludes the proof in case 2.are now in a position to conclude the proof of step 1. Definen :=n(n).struction we have E (n) ( ) and (Singn E) < +. Furthermore since the set of all pointsn has odd density is the same as the set of all points where has odd density, from Proposition 3.13, wethat |AonE| = 0. Therefore (a), (b) and (c) hold.e by construction the support and density function of ni and ni1 agree outside of R(pi), we have, usingi),F(n)F( )= F(n(n))F(n0) (n)i=1F(ni )F(ni1)=(n)i=1F(ni , (ni )1(R(pi)))F(ni1, (ni1)1(R(pi)))G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 869= F(ni , (ni )1(R(pi)))F(ni1, (ni1)1(R(pi)))HenceStep(a) E(b) |A(c) Si(d) FIn ordesupposthat:(i) tho(ii) R(iii) FWe proCasRepGsuch thRRNoticeof stepWe(ii), (iigraph oTo tApplyiand if vHi: piSingE+i: piRegEF(ni , (ni )1(R(pi)))F(ni1, (ni1)1(R(pi))) 1 +M2n .also (d) is valid, and the proof of step 1 is concluded.2. We construct a sequence {n} H 2,p(S) of limit systems of curves such that (n) (n);nAon| = 0;ngn (R2 \ E) is a finite set for any n N;(n)F(n) for any n N.r to construct n we use a recursive algorithm consisting of (n) steps. Let n0 := n, 1 i (n) ande that ni1 has been defined. Then ni is obtained by modifying ni1 only on int(R(pi)), in such a waye set of the points of ( ni ) where ni is odd is the same as the set of the points of (ni1) where ni1 isdd;(pi) Sing ni (R2 \ E) is finite;( ni , (ni )1(R(pi)))F( ni1, ( ni1)1(R(pi))).ceed in two different ways depending on whether pi E, pi R2 \ E.e 1 of step 2. Suppose pi E.eating the construction at the beginning of step 1 we can find a family+ := {(+1 ,1), . . . , (+k ,1), (u+1 ,2+1 ), . . . , (u+r+,2+r+)} H 2,p(]0, a[) (N \ {0}),at the functions ui are all distinct and +i < +j on ]0, a] for i < j , and+(pi) E =kl=1graph(+l );+(pi) ( ni1) = graph(G+);G+ = ni1 in R+(pi).that if pi SingE (respectively pi RegE ) the function +l coincides with the function +l of case 11 (respectively k = 1 and +1 coincides with |R+(pi) where is the function of case 2 of step 1).want to modify ( ni1) R(pi) leaving the functions +l unchanged in order to fulfill (i), while, to obtaini), we want to replace every u+l with a function v+l whose graph has energy lower than the energy of thef u+l and the graphs of the v+l intersect each other tangentially and only a finite number of times.his aim we let+l :={v C0([0, a]): graph(v) graph(G+), v(a)= u+l (a)}, l {1, . . . , r+}.ng Lemma 4.5 we obtain a family {v+1 , . . . , v+r+} H 2,p(]0, a[) such that v+l is a minimizer for P in +l+l (c)= v+j (c) for some c ]0, a] then v+l v+j in [0, c]. Then we consider the family+ := {(+1 ,1), . . . , (+k ,1), (v+1 ,2+1 ), . . . , (v+r+,2+r+)} H 2,p(]0, a[) (N \ {0}).870 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880We remark that ( ni1) R+(pi) = graph(H+) R+(pi). In addition, if z graph(H+) R+(pi) thenH+(z)kl=1 gTheAs athat (Sinc(and theFwhichCasIn thsupposset of pwe havFand (WeApplyin thisPropos(n)FwhichNow|and henhave {{n} = G+(z) = ni1(z). Finally we note that, by construction, H+(q) is odd if and only if q raph(+l ) (possibly with the exclusion of pi ).n we repeat the same construction in R(pi).ll the hypotheses are fulfilled, we can apply Lemma 4.9 and obtain a system of curves ni H 2,p(S) suchni )R(pi) = graph(H) and ( ni )|R(pi) = H .e ni )1((R2 \ E)R(pi)) rl=1graph(vl )set of singular points ofrl=1 graph(vl ) is finite we have that ni satisfies (ii). Furthermore( ni , (ni )1(R(pi)))=kl=1P(l , ]a,0[)+ k+l=1P(+l , ]a,0[)+ rl=12l P(vl , ]a,0[)+ r+l=12+l P(v+l , ]0, a[)kl=1P(l , ]a,0[)+ k+l=1P(+l , ]a,0[)+ rl=12l P(ul , ]a,0[)+ r+l=12+l P(u+l , ]0, a[)=F( ni1, ( ni1)1(R(pi))),is (iii).e 2 of step 2. Suppose pi R2 \ E.is case we obtain ni simply repeating the construction used in step 1 in the proof of Theorem 5.1. Since weed that R(pi)R2 \ E, in R(pi) there are not points of ( ) or of ( ni1) with odd density. Therefore theoints of ( ni ) where ni is odd coincides with the set of points of (ni1) where ni1 is odd, by constructione( ni , (ni )1(R(pi)))F( ni1, ( ni1)1(R(pi))),ni ) verifies the finiteness property in R(pi). This concludes the proof of case 2 of step 2.are now in a position to conclude the proof of step 2. Definen := n(n).ng Theorem 5.1 we can find a limit system of curves which is equivalent to n. Let us still denote bynew limit system of curves. Since we did not modify the set of points with odd multiplicity, thanks toition 3.13 we have that |AnAon | = 0 which is the assertion of (b). By construction we have E (n) ( ), and (Singn (R2 \ E)) < + which are the assertions of (a) and (c). Furthermore,(n)=F( n(n))F( n(n)1) F( n1 )F( n0 ) =F(n),proves (d ) and this concludes the proof of step 2., using the properties of n and n, we can conclude the proof of the theorem. Thanks to (b), (b), we haveAnE| |A nAon | + |AonE| = 0,ce (45) holds. Since every n is a limit system of curves and (45) holds, using Remarks 2.23 and 2.20, wen} A(E). Given n N, from (c) and (c) it follows that (n) verifies the finiteness property, thereforeQfin(E).G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 871By construction we have nL L and from (a), (a) it follows that (n) ( ), hencesnSo wecertainrectangRThese rUsinFEveFig.7. RegIn ththe regthe optTheoreA(E) vFurequatio2Proof.StepLetand Cc[( ) \(and with odcurveslupNF(n) < +.can apply Theorem 2.10 to obtain a subsequence (still indicated by {n}) H 2,p-weakly converging to asystem of curves . Again as in the proof of Theorem 5.1, due to the fact that the diameters of the niceles used to cover the set accsing( ) uniformly decrease to 0, we obtaineg = Reg , ( )|Reg = ( )|Reg .elations, together with Lemma 3.11, imply that .g (d), (d ), Lemma 3.9 and the H 2,p weak lower semicontinuity of F , we have( )=F( ) lim infn+F(n) lim supn+ F(n) lim supn+F(n)= limn+F(n) =F( ).ntually, assertion (44) is a direct consequence of (42) and the assertions concerning {n}. 7 illustrates the construction of the sequence {n} of Theorem 6.3 in a particular situation.ularity of minimal systems in A(E)e following theorem we prove a regularity result for minimal systems of curves. We limit ourselves to studyularity of ( ) in R2 \ E and locally around regular points, since we know that is without crossings andimal regularity for ( ) in Reg E is given by the local representation with functions of class H 2,p.m 7.1. Let p = 2 and let E R2 be such that F(E) < +. Then every minimal system of curves inerifies the finiteness property in any open subset U R2 \ E.thermore, every connected component B of Reg U is an analytic curve and its curvature verifies thend2ds2 + 3 = 0, s [0,H1(B)]. (52)Let be a minimal system in A(E) and fix an open set U R2 \ E. The proof consists of two steps.1. Every connected component B of Reg U is an analytic curve and its curvature verifies (52).B be a connected component of Reg U ; B is a one-dimensional submanifold of R2 of class H 2,2is constant and even on B . Let : [0,H1(B)] R2 be a parametrization by arc length of B , let([0,H1(B)]) and consider the curve : [0,H1(B)] R2, := + . For || 1 we have () B] = . Using Lemma 4.1, we can find a system of curves A(E), whose trace is given by)=[( ) \B] (),= on ( ) \B , while on () assumes the same constant value of on B . Since the set of pointsd multiplicity of and coincides, using Corollary 5.2 and Proposition 3.13, we can find a system of A(E) which is equivalent to . Therefore, from the minimality of on A(E), we haveim0F( + )F()= 0.872 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880Using [13] and the regularity theory of ordinary differential equations, it follows that B is an analytic submanifoldof R2 aStepSupin U . H( ) atR((R(LetYbe a caINote thwe havPIndeedffrom Lgraph(YcontradUsinif Iij isa contrRemarU Rpropertliteratuholds,8. ChaAsapproxto givesectionthat Sinan H 2,nd (52) holds.2. verifies the finiteness property in U .pose by contradiction that there exists U R2 \ E such that does not verify the finiteness propertyence accSing( )U = . Let p accSing( )U and let R(p) = [a, a] [b, b] be a nice rectangle forp such thatp)R2 \ E;+(p) Sing )= +;) R(p) Reg .+ := {(f+1 ,+1 ), . . . , (f+r ,+r )} H 2,p(]0, a[) (2N \ {0})nonical family for in R+(p). For every i, j {1, . . . , r} letij :={x [0, a]: (x,f+i (x))= (x,f+j (x)) Sing }, ij := sup{x: x Iij }.at, since ( ) R(p) Reg , we have a / Iij . If 0 < ij < a then, due to the minimality of in A(E)e(f+i , ]0, ij [)=P(f+j , ]0, ij [). (53), suppose by contradiction that P(f+i , ]0, ij [) G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 873The following definition is contained in [4, p. 282] and is needed to count the number of singularities of Ewith anDefinitwhereBr (p)Remarp NoBr(p)small,TheTheorepFurtheFActuTheoreFTheneededWeLetcusp poSForAccordDefinitthat(i) i(ii) iffo(iii) i. .appropriate multiplicity.ion 8.1. For every p SingE we define the balanced multiplicity E(p) asE(p) := |+(p) (p)|2,+(p) (respectively (p)) is the number of distinct graphs necessary to cover B+r (p) E (respectively E), for r > 0 small enough.k 8.2. As observed in [4, p. 282], |+(p) (p)| is even for any p NodE = SingE . Indeed, givendE , there exists r > 0 such that Br(p) contains both points of E and of R2 \ E, the intersection betweenand E is transversal and hence the number of the elements of E Br(p) is even. If r is sufficientlythis number coincides with +(p)+ (p), which has the same parity of |+(p) (p)|.following result is contained in [4, Theorems 6.3, 6.4].m 8.3. We haveSingE(p) is even F(E) < +.rmore if SingE = {p1, . . . , pn} and pi is a simple cusp point for every i = 1, . . . , n, then(E) < + n is even. (54)ally, a more refined result can be proved. Indeed, the following theorem holds.m 8.4. We have(E) < + pSingEE(p) is even.orem 8.4 is based, among other tools, on formula (44) and on Theorem 8.3; since no new techniques are, we omit its proof, which can be found in [5].now want to prove Theorem 8.6, which is one of the main representation results for F of the paper.E R2 be such that F(E) < +. Suppose that SingE is not empty, finite and composed only by simpleints. Using Theorem 8.3 we haveingE = {p1,p2, . . . , p2M}, M N \ {0}.every pi SingE we choose a unit vector (pi) normal to Tpi (E) in such a way that +E(pi) E(pi).ingly, the half-nice rectangle R+(pi) corresponds to +(pi).ion 8.5. Let E R2 be as above. We define (E) as the set of all collections {1, . . . , M} of curves such H 2,p(0,1) and |di/dt| is constant for every i = 1, . . . ,M;i(t1) = j (t2) for some t1, t2 [0,1] then di(t1)/dt and dj (t2)/dt are parallel; moreover if i(t) Er some t [0,1], then di(t)/dt is parallel to Ti(t)(E);(0), i(1) SingE for every i {1, . . . ,M}, and there exists a bijective application between {1(0), 1(1),. , M(0), M(1)} and SingE ;874 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880(iv) for every i {1, . . . ,M}It isTheore{p1, . .FProof.StepFThanks(E)FWe wilLetRMoreoR+(q)Letwe wriConsome i(a)whpotha(b) the(c) if qmodidt(0) is parallel to Ti(0)(E) and points in the direction of R(i(0)),didt(1) is parallel to Ti(1)(E) and points in the direction of R+(i(1)).immediate to see that the set (E) is not empty.m 8.6. Assume that F(E) < + and that SingE consists of a finite number of simple cusp points. , p2M}. Then we have the following representation formula for F(E):(E)=RegE[1 + |E|p]dH1 + 2 min(E)F( ). (55)The proof is divided into three steps.1. We have(E)RegE[1 + |E|p]dH1 + 2 inf(E)F( ). (56)to (44), to obtain (56) it is enough to prove that for every Qfin(E) we can find = {1, . . . , M} such that( )RegE[1 + |E|p]dH1 + 2F( ). (57)l see that satisfies also ( ) ( ). Qfin(E). For every q Sing SingE , we denote by R(q) a nice rectangle for at q such that(q) Sing = {q}. (58)ver for every q Sing \SingE we make an arbitrary choice of a normal unit vector to ( ) at q so that,R(q) are defined.us construct 1. From now on with the symbol j we denote +1 or 1. Accordingly, for every q Singte Rj (q) in place of R(q) when j = 1.struction of 1. Set (G0,0, (q0, 0)) := (( ), , (p1,+1)). Suppose we have defined (Gi ,i, (qi, i)) for 1, with:i :Gi1 N, i :={i1 2 on Hi ,i1 on Gi1 \Hi, Gi :={z Gi1: i(z) > 0}, (59)ere Hi Gi1 is a connected component of RegGi1 such that: i1 2 is constant on Hi ; qi Sing is aint of the relative boundary of Hi (which is composed either by qi1 itself, and in this case we understandt qi1 = qi , or by two points {qi1, qi} Sing ); Hi crosses Ri1(qi1) and reaches qi crossing Ri (qi);function i verifies the train tracks property in the rectangle R(q) for every q Gi (Sing \{p1, qi});i = p1 we havezGiR+(p1)i(z)=zGiR(p1)i(z)+ 2;reoveri = 1 zGiR+(qi)i(z)=zGiR(qi)i(z) 2;G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 875(d) if q(e) if qLetpropertconstanLetRi ((e), we(Otherwcrossescontainwe havGi+1 asreplaciThealgorithqn = pFig. 9. The construction of 1 in step 1 in the proof of Theorem 8.6.i = p1 and i = 1 thenzGiR+(p1)i(z)=zGiR(p1)i(z)+ 4;i = p1 and i = +1 thenzGiR+(p1)i(z)=zGiR(p1)i(z).us construct the first step (i = 1). Since p1 := q0 is a simple cusp point and verifies the train tracksy, we can find a relative connected component of Reg which crosses R(p1) and over which 0 := ist 2. Hence we can define (G1,1, (q1, 1)) satisfying properties (a)(e), see Fig. 9.us now explain in which way the algorithm constructs the step i + 1 from the step i 1. If {z qi): i(z) 2} = then, from the hypothesis that SingE is composed only by simple cusps and (d),have that qi SingE \{p1}; in this case the algorithm stops and we set1) := H1 Hi.ise, in view of (c), (d), (e) and (58) we can find an arc of regular points of ( ) which is contained in Gi ,Ri (qi) and is such that i 2 is constant on this arc. Let Hi+1 be the connected component of RegGiing this arc and let {qi, qi+1} be the relative boundary of Hi+1 (possibly with qi = qi+1). Again from (b)(e)e that i 2 is constant on Hi+1 and Hi+1 reaches qi+1 crossing Ri+1(qi+1). In addition, setting i+1 andin (59) with i replaced by i + 1, we have, thanks to (58), that (Gi+1,i+1, (qi+1, i+1)) satisfies (b)(e)ng everywhere i with i + 1, see Fig. 9.refore we can iterate the algorithm as specified above. Since Qfin(E) we have Sing < +, hence them stops after a finite number n of steps. Furthermore qn = pj1 SingE \{p1}. Indeed, if by contradiction1, from (b)(e) we could iterate the algorithm also at step n+ 1.876 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880We now define(Since Hline to(1) =Conthan 2)with athe two(G10 ,0Since pRwheresupporpointsBy conat z. HSingEItera(i) proof oStepFWe staGwe hav{z G:connecForthat foR(pi1) := H1 Hn.i and Hi+1 have qi as a boundary point and belong to opposite half planes with respect to the normalGi at qi , using Lemma 4.1 we can find 1 H 2,p(0,1) parametrized with constant speed and such thatni=1 Hi ( ), 1(0)= p1, 1(1)= pj1 .struction of 2. In order to obtain (2) (which is meaningful in the case that the number of cusps is largerwe make a similar construction, but taking into account that parts of ( ) have already been deletedsuitable weight in the construction of 1. As we shall see, we will also modify the set E locally aroundpoints p1, pj1 in such a way that p1 and pj1 becomes regular points of the new set. We start from1, (pj2 ,+1)), where pj2 SingE \{p1,pj1}, and G10 , 10 are obtained as follows. Let10 (z) :={ (z) 2{11 (z)} if z (1), (z) if z ( ) \ (1), G10 :={z ( ) : 10 (z) > 0}.1,pj1 are simple cusp points, for m {1, j1} we have(pm) E = R+(pm) E = graph(m1 ) graph(m2 ),m1 , m2 are functions of class H2,p. Now, for m {1, j1}, we replace graph(m1 ) graph(m2 ) with thet of a curve m H 2,p(0,1) such that: (m) R(pm); (m) ( ) int(R(pm)) = {pm}; m joins the twograph(m1 ) R+(pm), graph(m2 ) R+(pm); m intersects ( ) tangentially. Then we define10 := 10 on G10 \m{1,j1}, l{1,2} graph(ml ), 10 1 onm{1,j1}, l{1,2} graph(ml ),1 on (1) (j1),G10 :={z G10 : 10 (z) > 0}.struction 10 verifies the train tracks property on G10 and every z G10 admits a nice rectangle for G10ence we can repeat the construction we used to obtain 1 to get 2 H 2,p(0,1) joining pj2 ,pj3 \{p1,pj1} with pj2 = pj3 . Note that (2) ( ).ting this argument exactly M times, we obtain the desired (E). Now we observe that, since( ), (z) 2Mi=1 {1i (z)} by construction and ( ) E, we also have (57). This concludes thef step 1.2. Given = {1, . . . , M } (E) we can find = {1, . . . , m} Ao(E) such that( )=RegE[1 + |E|p]dH1 + 2F( ). (60)rt noticing that if we set:=Mi=1(i) E, :G N \ {0}, (z) :={2Mi=1 {1i (z)} if z Mi=1(i) \ RegE ,1 + 2Mi=1 {1i (z)} if z RegE ,e that G admits a nice rectangle at any z G, verifies the train tracks property on G and E = (z) 1 (mod 2)}. Recall that since SingE is finite, then RegE consists of a finite number of (relative)ted components, whose (relative) boundary is composed by at most two distinct points of SingE .every pi SingE let R(pi) be a nice rectangle for G at pi such that R(pi) SingE = {pi}. Recallr every pi SingE the unit vector (pi) normal to G at pi is such that R+(pi) RegE = and) RegE = .G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 877Construction of 1. We will construct 1 by gluing together the parametrizations of the elements of a finiteordered . LetdistincstartingLetembedorientenecessahave alFsuch th(a) pl(b) if lpl,(c) if l(d) if lFirstlyreachesin theadditiohence Now(respeceven l Fwhere(respecSinccurves,that n iK of Rthere aelemenThisonly ancontradin the pboundafact thaAs aas pn+chain composed by oriented relative connected components of RegE and oriented supports of elements of() G be the support of a curve of class H 2,p connecting pi,pj SingE (with pi,pj not necessarilyt) such that has constant speed. From now on by writing (, (pi,pj )) we mean that we move along ()from pi and reaching pj .K1 be a relative connected component of RegE . If K1 does not have relative boundary then K1 is anded closed curve of class H 2,p and hence we can find a curve 1 H 2,p(S1) which is a constant speedd parametrization of K1 and we stop. Let us suppose that K1 = {p1,p2} with p1,p2 SingE notrily distinct. We set F1 = (1, (p1,p2)), where 1 is the arc length parametrization of K1. Suppose that weready constructed a chaini =(1, (p1,p2)),(2, (p2,p3)), . . . ,(i1, (pi1,pi)),(i, (pi,pi+1)),at SingE for every l {1, . . . , i + 1};is odd l is the arc length parametrization of one of the relative connected components of RegE havingpl+1 as boundary points;is even l is the arc length parametrization of the support of the unique il connecting pl,pl+1;= m are both odd then (l) (m)= ; if l is even there is at most an even m = l such that (l)= (m).we notice that, since if l is odd (respectively even) l starts crossing R+(pl) (respectively R(pl)) andpl+1 crossing R+(pl+1) (respectively R(pl+1)), thanks to Lemma 4.1, we can glue together all the lorder, and obtain a unique constant speed curve i H 2,p(0,1) whose support is given by il=1(l). Inn, thanks to (d) we have that i satisfies{1i (z)} (z) z (i), (61)i covers (i) at most twice and once each (relative) connected component of RegE (i).if i is odd (respectively even) and we can find a curve j having as starting or ending point pi+1tively a connected component K of RegE having pi+1 as boundary point) such that there is at most one{1, . . . , i} such that (l)= (j ) (respectively for every odd l {1, . . . , i} we have (l) = K), then we seti+1 := Fi ,(i+1, (pi+1,pi+2)),i+1 is the arc length parametrization of (j ) (respectively of K) and pi+2 is the other extreme of jtively {pi+1,pi+2} = K). Otherwise we stop.e RegE consists of a finite number of (relative) connected components and consists of a finite number ofthe above construction stops after a finite number n of steps. It is immediate to check that n > 3. We claims even and pn+1 = p1. Suppose by contradiction that n is odd. Then n parametrizes a connected componentegE such that {pn,pn+1} is the relative boundary of K and n 5. Furthermore as our construction stops,re two even numbers l,m {1, . . . , n 1}, l < m such that (l) = (m) = (j ), where j is the uniquet of having pn+1 as starting or ending point.means that we crossed twice R(pn+1) at the step m n 1. If l m < n 1, since there is at mostother relative connected component of RegE having pn+1 as a boundary point, in view of (d) we have aiction. If m = n 1 then (n1,pn+1,pn) and l n 3 (notice that n 3 2 as n 5). Therefore asrevious case since there is at most only another relative connected component of RegE having pn+1 as ary point, in view of (d) we have a contradiction. Hence n is even. With a similar argument and using thet n is even, one can prove that pn+1 = p1.lready noticed, we can find a constant speed curve n H 2,p(0,1) such that (n) =nl=1(l). In addition,1 = p1 and n reaches p1 crossing R(p1), while 1 moves from p1 crossing R+(p1), we can find a878 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880constant speed curve 1 H 2,p(S1) such that (1) = (n) and also by (61) {11 (z)} = {1n (z)} (z) forevery zConfrom G{12 (Sincat a bo{1, . . .{z (As aFHenceFThe proStepRLetconverreplaceFDue to{n} wEvehypoth12pmultipl{whereih=1 Definitof Defi(iii) i1 (1).struction of 2. To obtain 2 we repeat the same construction used to obtain 1, but this time we start1 := {z G: 1(z) > 0}, where 1(z) := (z) {11 (z)} and taking into account that 2 verifiesz)} 1(z).e the number of connected components of RegE is finite and each component of starts and endsundary point of a relative connected component of RegE , iterating this procedure we obtain :=, m} such that ( ) =Mi=1(i) E and = . Hence (60) holds. Moreover, since by construction ): (z) 1(mod 2)} = E, by Proposition 3.13 we also have Ao(E).consequence of step 2 and (42) we get(E)RegE[1 + |E|p]dH1 + 2 inf(E)F( ).by step 1 we deduce(E)=RegE[1 + |E|p]dH1 + 2 inf(E)F( ).of of the theorem then follows from the following final step.3. There exists (E) such thategE[1 + |E|p]dH1 + 2F()=RegE[1 + |E|p]dH1 + 2 inf(E)F( ). (62) be a minimal element inA(E). Thanks to Theorem 6.3 we can pick a sequence {n} H 2,p(S)Qfin(E)ging to in H 2,p. Let n be the elements of (E) constructed in step 1, i.e., such that (57) holds with d by n. Since (n) (n) we get(E) lim supnRegE[1 + |E|p]dH1 + 2F(n) limnF(n)=F(E).the strong convergence of the sequence {n} and the finiteness of SingE , we can find a subsequence ofhich converges to a certain (E). Using the lower semicontinuity of F on (E) we have (55). ntually let us sketch very briefly how the representation formula (55) can be proved removing theesis that every element of SingE is a simple cusp point (see [5] for a more detailed proof). Let M :=SingE E(p). Recall that M N by Theorem 8.4. In order to consider each singular point with the correcticity, let us represent the set {q SingE: E(q) = 0} = {p1, . . . , pd} as follows:q SingE : E(q) = 0}= {p1,p2, . . . , p2M},d 2M , pj := p1 for every 1 j E(p1), and pj := pi for every j with i1h=1 E(ph) j E(ph) and every i = 2, . . . , d .ion 8.7. We define (E) as the set of all collections {1, . . . , M} of curves such that properties (i) and (ii)nition 8.5 hold, and(0), i(1) SingE for every i {1, . . . ,M}, and there exists a bijective application between {1(0),(1), . . . , M(0), M(1)} and {q SingE : E(q) = 0};G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880 879Fig. 10.RegE [1that (E(iv) foorwNotthe poiThe(a) suiofsm(b) suito8.1. AUsinProposclass QProof.The sea simp(a) shows the easiest example of formula (55): if E is the set consisting of two drops (as in Fig. 1), then F(E) equals+ |E |p]dH1 plus twice the distance between the two simple cusp points. (b) and (c) (where E consists of four drops) show) does not necessarily reduce to a unique possible collection {1, . . . , M }.r every i {1, . . . ,M} eitherdidt(0) is parallel to Rc(i(0))anddidt(1) is parallel to R(i(1))didt(0) is parallel to R(i(0))anddidt(1) is parallel to Rc(i(1)),here Rc (respectively R) denotes the rotation of /2 in clockwise (respectively counterclockwise) order.ice that (iii) implies that for i {1, . . . ,m} we have i(0)= pi0 , i(1)= pi1 SingE and i0 = i1 (howevernts pi0 and pi1 may coincide).proof of (55) then follows bytably approximating E with a sequence {En} of sets obtained by modifying E locally around each pointSingE in such a way that En = E outside the union of a family of nice rectangles of diameter strictlyaller than 1/2n covering SingE , and in addition SingEn is composed only by simple cusp points;tably passing to the limit as n + in the formula (55) where E is replaced by En, which is valid thanksTheorem 8.6, see [5].counterexampleg Theorem 8.6 we show an example of a set E for which the minimum in (44) is not attained.ition 8.8. There exists a set E with SingE = 2 such that F(E) < + and the minimum of F over thefin(E) in (44) is not achieved.Let Ej with j = 1,2,3 be as in Fig. 7: they are three connected sets, whose closure are pairwise disjoint.t E3 is smooth and contained in {y 0}, while Ei , i = 1,2, are smooth except for the point pi , which isle cusp point, p1 = (1,0), p2 = (0,1). The unoriented tangent to Ei at pi is the x-axis, i = 1,2. We880 G. Bellettini, L. Mugnai / Ann. I. H. Poincar AN 21 (2004) 839880suppose that the oscillating part of E3 touches the segment ]1,1[{0} an infinite number of times. Since thesegment joining p1 and p2 is an absolute minimizer for F in (E) the thesis follows from Theorem 8.6. Corollary 8.9. There exists a set E R2 such that F(E) < +, with only two simple cusp points and such thatthe minimal system in A(E) has multiplicity equal to 3 on a set of positive H1 measure.Proof.of posiRefere[1] L. A[2] L. A[3] L. Asub[4] G. BNor[5] G.Uni[6] G.App[7] H. C[8] A. C[9] B. D[10] S. DMa[11] L.Om[12] H. F[13] M.199[14] E. DTrie[15] O.Cal[16] J.E.Sym[17] A.E[18] S. M[19] S. M[20] J.MApp[21] D. M[22] D. M199[23] M.[24] L. S[25] T.J.It is enough to choose, in Proposition 8.8, the set E3 in such a way that E3 intersects ]1,1[{0} on a settive H1 measure. ncesmbrosio, N. Fusco, D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Science Publication, 2000.mbrosio, C. Mantegazza, Curvature and distance function from a manifold, J. Geom. Anal. 5 (1998) 723748.mbrosio, S. Masnou, A direct variational approach to a problem arising in image reconstruction, Interfaces and Free Boundaries,mitted for publication.ellettini, G. Dal Maso, M. Paolini, Semicontinuity and relaxation properties of a curvature depending functional in 2D, Ann. Scuolam. Sup. Pisa Cl. Sci. (4) 20 (1993) 247299.Bellettini, L. Mugnai, Characterization and representation of the lower semicontinuous envelope of the elastica functional, Preprintv. Pisa, May 2003.Bellettini, M. Paolini, Variational properties of an image segmentation functional depending on contours curvature, Adv. Math. Sci.l. 5 (1995) 681715.artan, Theorie Elementaire des Fonctions Analytiques dUne ou Pluiseurs Variables Complexes, Hermann, 1961.oscia, On curvature sensitive image segmentation, Nonlin. Anal. 39 (2000) 711730.acorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, 1989.elladio, Special generalized gauss graphs and their application to minimization of functionals involving curvatures, J. Reine Angew.th. 486 (1997) 1743.Euler, Additamentum I de curvis elasticis, methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, Operania I (24) (1744) 231297.ederer, Geometric Measure Theory, Springer-Verlag, 1969.Giaquinta, S. Hildebrandt, Calculus of Variations I, in: Grundleheren der Mathematischen Wissenschaften, vol. 310, Springer-Verlag,6.e Giorgi, Some remarks on -convergence and least squares method, in: Proc. Composite Media and Homogeneization Theory,ste, 1991, pp. 135142.Gonzales, J.H. Maddocks, F. Schuricht, H. von der Mosel, Global curvature and self-contact of nonlinearly elastic curves and rods,c. Var. Partial Differential Equations 14 (2002) 2968.Hutchinson, C1, -multiple functions regularity and tangent cone behaviour for varifolds with second fundamental form in Lp , Proc.p. Pure Math. 44 (1986) 281306..H. Love, A Treatise on the Mathematical Theory of Elasticity, Dover, 1944.asnou, Disocclusion: a variational approach using level lines, IEEE Trans. Image Process. 11 (2002) 6876.asnou, J.M. Morel, Level lines based disocclusion, in: Proc. ICIP98 IEEE Internat. Conf. on Image Processing, 1998, pp. 259263.. Morel, S. Solimini, Variational Methods in Image Segmentation, in: Progress in Nonlinear Differential Equations and Theirlications, vol. 14, Birkhuser, 1995.umford, Elastica and computer vision, in: Algebraic Geometry and its Applications, 1994, pp. 491506.umford, M. Nitzberg, T. Shiota, Filtering, Segmentation and Depth, in: Lecture Notes in Computer Science, vol. 662, Springer-Verlag,3.Nitzberg, D. Mumford, The 2.1-D sketch, in: Proc. of the Third Internat. Conf. on Computer Vision, Osaka, 1990.imon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 2 (1993) 281326.Willmore, An introduction to Riemannian Geometry, Clarendon Press, 1993.

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