Structure of level sets and Sard-type properties of Lipschitz maps
GIOVANNIALBERTI, STEFANOBIANCHINI ANDGIANLUCACRIPPA
Abstract. We consider certain properties of maps of classC2fromRdtoRd 1 that are strictly related to Sard’s theorem, and we show that some of them can be extended to Lipschitz maps, while others require some additional regularity. We also give examples showing that, in terms of regularity, our results are optimal.
Mathematics Subject Classification (2010):26B35 (primary); 26B10, 26B05, 49Q15, 58C25 (secondary).
1. Introduction
In this paper we study three problems which are strictly interconnected and ulti- mately related to Sard’s theorem: structure of the level sets of maps fromRd to Rd 1, weak Sard property of functions from R2 to R, and locality of the diver- gence operator. Some of the questions we consider originated from a PDE problem studied in the companion paper [1]; they are however interesting in their own right, and this paper is in fact largely independent of [1].
Structure of level sets. In case of maps f : Rd ! Rd 1 of classC2, Sard’s theorem (see [17] and [12, Chapter 3, Theorem 1.3])1states that the set of critical values of f, namely the image according to f of the critical set
S:= x : rank(rf(x)) <d 1 , (1.1) has (Lebesgue) measure zero. By the Implicit Function Theorem, this property implies the following structure result: for a.e.y 2Rd 1the connected components of the level set Ey := f 1(y)are simple curves (of classC2).
This work has been partially supported by the italian Ministry of Education, University and Re- search (MIUR) through the 2006 PRIN Grant “Metodi variazionali nella teoria del trasporto ot- timo di massa e nella teoria geometrica della misura”, and by the European Research Council through the 2009 Starting Grant “Hyperbolic Systems of Conservation Laws: singular limits, properties of solutions and control problems”.
Received July 20, 2011; accepted December 12, 2011.
1For more general formulations see also [9, Theorem 3.4.3] and [2, 3, 6].
For Sard’s theorem to hold, the regularity assumption on f can be variously weakened, but in any case f must be at least twice differentiable in the Sobolev sense.2 However, a variant of the structure theorem holds even with lower regular- ity: in Theorem 2.5, statement (iv), we show that if f :R2 !Ris Lipschitz, then for a.e. y 2Rthe connected components ofEy are either simple curves or consist of single points, and in statement (v) we show that the same holds ifd 3 and
f :Rd !Rd 1is a map of classC1,1/2.
Note that ford 3 it is not enough to assume that f is Lipschitz: in Section 3 we construct examples of maps f : Rd ! Rd 1of classC1,↵ for everyd 3 and↵<1/(d 1)such that for an open set of valuesythe level setEycontains a Y-shaped subset, ortriod(see Subsection 2.3 for a precise definition), and therefore at least one connected component ofEy is neither a point nor a simple curve.
Weak Sard property. Given a function f : R2 ! Rof class C2, consider the measure µon Rgiven by the push-forward according to f of the restriction of Lebesgue measure to the critical setSdefined in (1.1). The measureµis supported on the set f(S), which is negligible by Sard’s theorem, and thereforeµis singular with respect to the Lebesgue measure onR:
µ?L1. (1.2)
Formula (1.2) can be viewed as a weak version of Sard’s theorem, and holds under the assumption that f is (locally) of classW2,1,cf.[1, Section 2.15(v)].
In Section 4 we prove that the last assumption is essentially optimal; more precisely we construct a function f :R2 !Rof classC1,↵for every↵<1 (and therefore also of classW ,pfor every <2 andp+1) such that (1.2) does not hold.
We actually show that this function does not satisfy an even weaker version of (1.2), calledweak Sard property, where in the definition ofµthe critical set S is replaced by S\E⇤, with E⇤the union of all connected components with positive length of all level sets.
The relevance of the weak Sard property lies in the following result [1, The- orem 4.7]: letb be a bounded divergence-free vector field on the plane, then the continuity equation@tu+div(bu)=0 admits a unique bounded solution for every bounded initial datumu0if and only if the potential f associated tob, namely the Lipschitz function that satisfiesb=r?f,3satisfies the weak Sard property.
Locality of the divergence operator. It is well-known that given a functionu on Rdwhich is (locally) of classW1,1, the (distributional) gradientruvanishes a.e. on
2For Sard’s Theorem for maps of classC1,1see [2], for maps in the Sobolev classW2,psee [4, 10]. The constructions in [11, 19] and [9, Section 3.4.4] give counterexamples to Sard’s Theorem of classC1,↵for every↵<1, and therefore also of classW ,pfor every <2 and 1 p 1.
3Given a vectorv=(v1, v2)we writev? :=( v2, v1), andv>:=(v2, v1). Thusr?:=
( @2,@1). If a vector fieldbis bounded and divergence-free thenb>is bounded and curl-free, and therefore there exists a Lipschitz function f such thatrf =b>, that is,r?f =b.
every Borel set Ewhereutakes a.e. a constant value. This property is summarized by saying that the gradient is strongly localfor Sobolev functions; it follows im- mediately that every first-order differential operator, including the divergence, is strongly local for (first-order) Sobolev functions.
It is then natural to ask whether first-order differential operators are strongly local even on larger spaces.
In Section 5 we show that, somewhat surprisingly, the answer for the diver- gence operator is negative,4and that in two dimensions this fact is strictly related to the (lack of) weak Sard property for Lipschitz functions (Remark 5.1). More pre- cisely, we construct a bounded vector fieldbon the plane whose (distributional) di- vergence belongs toL1, is non-trivial, and is supported in the set wherebvanishes;
we than usebto construct another example of Lipschitz function f :R2!Rwith- out the weak Sard property.
ACKNOWLEDGEMENTS. We thank Ludˇek Zaj´ıˇcek for pointing out reference [15]
and David Preiss for reference [16].
2. Structure of level sets of Lipschitz maps
We begin by recalling some basic notation and definitions used through the entire paper, more specific definitions will be introduced when needed.
2.1. Basic notation. Through the rest of this paper, sets and functions are tacitly assumed to be Borel measurable, and measures are always defined on the appropri- ate Borel -algebra.
We writea^banda_brespectively for the minimum and the maximum of the real numbersa,b.5
Given a subset E of a metric space X, we write 1E : X ! {0,1}for the characteristic function of E, Int(E)for the the interior ofE, and, for everyr > 0, IrEfor the closedr-neighbourhood ofE, that is,
IrE:= x 2X : dist(x,E)r .
The classF(X)of all non-empty, closed subsets ofX is endowed with theHaus- dorff distance
dH(C,C0):=inf r 2[0,+1] : C⇢IrC0, C0⇢IrC . (2.1)
4 This answers in the negative a question raised by L. Ambrosio; a simpler but less explicit example has been constructed by C. De Lellis and B. Kirchheim.
5 The symbol ^is sometimes used for the exterior product; the difference is clear from the context.
A function (or a map) defined on a closed set EinRdis of classCk if it admits an extension of classCk to some open neighbourhood of E, and is of classCk,↵ with 0 ↵ 1 if it is of classCk and the k-th derivative is H¨older continuous with exponent↵.
Given a measureµonXand a positive function⇢onXwe denote by⇢·µthe measure on Xdefined by[⇢·µ](A):=R
A⇢dµ. Hence 1E·µis the restriction of µto the setE.
Given a map f : X ! X0 and a measure µ on X, the push-forward of µ according to f is the measure f#µon X0 defined by[f#µ](A) :=µ(f 1(A))for every Borel set Acontained inX0.
As usual, Ld is the Lebesgue measure onRd whileHd the d-dimensional Hausdorff measure on every metric space – the usual d-dimensional volume for subsets ofd-dimensional surfaces of classC1in some Euclidean space. Thelength of a setEis just the 1-dimensional Hausdorff measureH1(E). When the measure is not specified, it is assumed to be the Lebesgue measure.
A setEinRd isk-rectifiableif it can be covered, except for anHk-negligible subset, by countably manyk-dimensional surfaces of classC1.
2.2. Curves.A curve in Rd is the image C of a continuous, non-constant path : [a,b]!Rd(the parametrization ofC); thusCis a compact connected set that contains infinitely many points. We say thatCissimpleif it admits a parametriza- tion that is injective,closedif satisfies (a)= (b); andclosed and simpleif
(a)= (b)and is injective on[a,b).
If is a parametrization ofC of classW1,1, thenH1(C) k˙k1 and the equality holds whenever is injective. Moreover it is always possible to find a strictly increasing function : [a0,b0] ! [a,b] such that is a Lipschitz parametrization which satisfies|( )0| =1 a.e.
For closed curves, it is sometimes convenient to identify the end points of the domain [a,b]. This quotient space is denoted by [a,b]⇤, and endowed with the distance
d(x,y):= |x y|^(b a |x y|). (2.2) A set E inRd ispath-connectedif every couple of pointsx,y 2 E can be joined by a curve contained in E(that is,x,yagree with the end points (a), (b)of the curve).
2.3. Triods. A (simple)triodinRd is any setY given by the union of three curves with only one end pointyin common, which we callcenterofY. More precisely
Y =C1[C2[C3
where eachCi is a curve inRdparametrized by i : [ai,bi]!Ci and i(ai)= y, and the sets i((ai,bi])are pairwise disjoint.
2.4. Lipschitz maps.Through this sectiond,kare positive integers such that 0<
k <d, and f is a Lipschitz map from (a subset of)Rd toRd k; we denote by fi, i =1, . . . ,d k, the components of f.
For every y 2 Rd k we denote by Ey := f 1(y) the corresponding level set of f, and by Cy the family of all connected components C of Ey such that H k(C) >0;6 E⇤y is the union of allC inCy, and E⇤ is the union of allE⇤y with y 2Rd k. BothE⇤y andE⇤are Borel sets (Proposition 6.1).
By Rademacher’s theorem f is differentiable at almost every point, and at such points we define the Jacobian
J := [det(rf ·rtf)]1/2.
We denote bySthe set of all pointsx 2Rd where either f is not differentiable or J(x)=0; note thatJ(x)6=0 if and only if the matrixrf(x)has rankd k.
If k = 1 and x 2/ S then there exists a unique unit vector ⌧ = ⌧(x) such that⌧ is orthogonal torfi for everyi and the sequence(rf1, . . . ,rfd 1,⌧)is a positively oriented basis ofRd.7
We can now state the main result of this section. Statements (i) and (ii) are immediate consequences of the coarea formula and have been included for the sake of clearness. Ford = 2 andk = 1, a variant of statement (iii) was proved in [16, Section 11] under the more general assumption that f is a continuousBVfunction.
Even though we could not find statement (iv) elsewhere, the key observation behind its proof – namely that for a continuous function on the plane only countably many level sets contain triods – is well-known.
Theorem 2.5. Let f : Rd ! Rd k be a Lipschitz map with compact support. In the notation of the previous subsections, the following statements hold for almost everyy2Rd k:
(i) the level setEy isk-rectifiable andHk(Ey) <+1;
(ii) Hk(Ey \S) = 0, which means that for Hk-a.e. x 2 Ey the map f is differentiable atx and the matrixrf(x)has rankd k; moreover the kernel of this matrix is the tangent space toEy atx;
(iii) the familyCy is countable andH k(Ey\E⇤y)=0;
(iv) fork=1andd =2every connected componentCofEyis either a point or a closed simple curve with a Lipschitz parametrization : [a,b]⇤ !Cwhich is injective and satisfies (t) /2Sand ˙(t)=⌧( (t))for a.e.t;
(v) the result in the previous statement can be extended to k = 1and d 3 provided that f is of classC1,1/2.
Remark 2.6. (i) The assumption that f is defined onRdand has compact support was made for the sake of simplicity. Under more general assumptions, the results on the local properties of generic level sets (rectifiability and so on) are clearly
6Since the length of a connected set is larger than its diameter, fork=1 the connected compo- nents inCyare just those that contain more than one point.
7Ifd=2 andk=1 thenJ= |rf|and⌧is the counter-clockwise rotation by 90 ofrf/|rf|.
the same, while the results concerning the global structure (notably statement (iv)) require some obvious modifications.
(ii) In statement (iv), the possibility that some connected components ofEyare points cannot be ruled out. Indeed, for every↵<1, there are functions f :R2! R of class C1,↵ such that the set Ey \E⇤y – namely the union of all connected components of Ey which consist of single points – is not empty for an interval of valuesy.8
(iii) In Section 3 we show that for everyd 3 and every↵<1/(d 1)there exist maps f : Rd ! Rd 1 of class C1,↵ such that the level set Ey contains a triod (cf. Subsection 2.3) for an open set of values y, and therefore its connected components cannot be just points or simple curves. This shows that the exponent 1/2 in the assumption f 2 C1,1/2in statement (v) is optimal ford = 3. However we believe that the exponent 1/2 is not optimal ford 4.
(iv) The fact that a connected component ofEyis a simple curve with Lipschitz parametrization (statement (iv)) does not imply that it can be locally represented as the graph of a Lipschitz function: there exist functions f on the plane of classC1,↵
with↵<1 such that every level set contains a cusp.
(v) Even knowing that a connected component of Ey is a simple curve, the existence of a parametrization whose velocity field agrees a.e. with ⌧ (cf. state- ment (iv)) is not as immediate as it may look, because ⌧ andEy lack almost any regularity. Our proof relies on the fact that generic level sets of Lipschitz maps can be endowed with the structure of rectifiable currents without boundary.
(vi) One might wonder if something similar to statement (iv) holds at least for functions f : Rd ! Rwithd > 2. As shown below, the key points in the proof of statement (iv) are that a family of pairwise disjoint triods in the plane is countable, and that a connected set in the plane with finite length which contains no triods is a simple curve. A natural generalization of the notion of triod could be the following: a connected, compact setEinRd is ad-triod with centeryif, for every open ballBwhich containsy, the setB\Ehas at least three connected components which intersect@B. However, even if it is still true that a family of pairwise disjoint d-triods in Rd is countable, very little can be said on the topological structure of d-triod-free connected sets with finiteHd 1measure.
The rest of this section is devoted to the proof of Theorem 2.5.
2.7. Coarea formula.The coarea formula (seee.g.[9, Section 3.2.11], [14, Corol- lary 5.2.6], or [18, Section 10]) states that for every Lipschitz map f :Rd !Rd k
8Simple examples can be obtained by modifying the construction in [11]. Moreover, since the level setsEyof any continuous function contain isolated points only for countably manyy, it turns out thatEy\E⇤yhas the cardinality of continuum for a set of positive measure ofy. An example of Lipschitz function on the plane such that the setEy\E⇤yhas the cardinality of continuum for a.e.yis also given in [13].
and every positive Borel function :Rd ![0,+1]there holds Z
Rd J dLd = Z
Rd k
Z
Ey
dHk dLd k(y) (2.3) (quite obviously, (2.3) holds even for real-valued functions provided that the in- tegral at the left-hand side makes sense,e.g., for 2L1(Rd)).
The coarea formula has two immediate consequences:
(i) Let be the constant function 1: if f has compact support then the integral at the left-hand side of (2.3) is finite, and therefore Hk(Ey), which is the value of the integral between square brackets at the right-hand side, is finite for a.e.y 2 Rd k.
(ii) If is the characteristic function of the set S defined in Subsection 2.4, then identity (2.3) implies that for a.e. y 2 Rd k there holdsHk(Ey \S) = 0.
This means that forHk-a.e.x2 Eythe map f is differentiable atxandJ(x)6=0, that is, rf(x) has rankd k. Hence Ey admits a k-dimensional tangent space at x, namely the kernel of the matrixrf(x). This implies that Ey isk-rectifiable, cf.[9, Theorem 3.2.15] or [18, Theorem 10.4].
Proof of statements(i)and(ii)of Theorem2.5. These statements are included in Subsection 2.7.
Statement (iii) of Theorem 2.5 will be obtained as a corollary of Lemmas 2.11, 2.12, and 2.13. In the next three subsections we recall some definitions and a few results that will be used in the proofs of these lemmas.
2.8. Connected components. We recall here some basic facts about the connected components of a setEinRd, or more generally in a metric spaceX; for more details see for instance [7, Chapter 6].
Aconnected componentofEis any element of the class of connected subsets of Ewhich is maximal with respect to inclusion. The connected components ofE are pairwise disjoint, closed in E, and coverE.
Assume now that E is compact. Then each connected componentC agrees with the intersection of all subsets of E that are closed and open inEand contain C[7, Theorem 6.1.23].
Every set which is open and closed inEcan be written asU\EwhereU is an open subset ofXsuch that@U\E =?.9HenceCis the intersection of the closures of all open setsU which containCand satisfy@U \E =?. Therefore, under the further assumption that Xis second countable (or, equivalently, separable), we can find a decreasing sequenceUn of such sets, the intersection of whose closures is stillC.10
9IfDis open and closed inE, thenDandE\Dare disjoint and closed inX, and sinceXis a normal space there exist disjoint open setsU,V such thatD⇢UandE\D⇢V; it is then easy to check thatUmeets all requirements.
10 Recall that in a second countable space every familyFof closed sets admits a countable subfamilyF0such that the intersection ofF0and the intersection ofFagree.
2.9. Rectifiable currents.We recall here some basic definitions about currents; for further details see for instance [14, Chapter 7] or [18, Chapter 6] .
Ak-dimensionalcurrentonRd is a linear functional on the space ofk-forms on Rd of class Cc1. The boundaryof a k-currentT is the (k 1)-dimensional current@T defined byh@T;!i :=hT;d!i, whered! is the exterior derivative of
!. The mass ofT, denoted byM(T), is the supremum ofhT;!iamong allk-forms
! that satisfy|!| 1 everywhere; Mis clearly a norm on the space of currents with finite mass.
LetEbe ak-rectifiable set inRd. AnorientationofEis a map⇣that associates toH k-a.e.xinEa unit, simplek-vector that spans the approximate tangent space to E at x; a multiplicity is any integer-valued functionm on E which is locally summable with respect toHk. To every choice ofE,⇣,mis canonically associated thek-dimensional current[E,⇣,m]defined by
h[E,⇣,m];!i:=
Z
Eh!;⇣im dH k
for everyk-form! of classCc1 onRd. Hence the mass of [E,⇣,m]is equal to R
E|m|dHk. Currents of this type are calledrectifiable.11
2.10. Current structure of level sets.Take S as in Subsection 2.4. For every x 2 Rd \S the kernel of the matrixrf has dimensionk, and therefore we can choose an orthonormal basis{⌧1, . . . ,⌧k}such that(rf1, . . . ,rfd k,⌧1, . . . ,⌧k)is a positively oriented basis ofRd; we denote by⌧the simplek-vector⌧1^· · ·^⌧k.12 Statement (ii) in Subsection 2.7 shows that for a.e.ythek-vector field⌧defines an orientation of Ey. We denote byTy thek-dimensional current associated with the setEy, the orientation⌧, and constant multiplicity 1, that is,Ty := [Ey,⌧,1].
The essential fact about the currentTy is that its boundary vanishes for a.e. y, that is
h@Ty;!i:=hTy;d!i:=
Z
Ey
hd!;⌧idHk =0 (2.4) for every(k 1)-form!of classCc1onRd.
The proof goes as follows: the currentsTy are the slices according to the map f of thed-dimensional rectifiable currentT := [Rd,⇣,1], where⇣is the canonical orientation ofRd. In general, the boundaries of the slicesTy agree with the slices of the boundary @T for a.e. y(see [9, Section 4.3.1]), and since in this particular case@T =0, then@Ty =0 for a.e.y.
11Examples ofk-dimensional rectifiable currents are obtained by taking ak-dimensional oriented surfaceE of classC1, endowed with constant multiplicity 1. In this case the mass agrees with thek-dimensional volume ofE, and the boundary ofEin the sense of currents agrees with the (k 1)-dimensional current associated to the usual boundary@E, endowed with the canonical orientation and constant multiplicity 1.
12Note that⌧is is uniquely determined by the assumptions on⌧1, . . . ,⌧k, and fork=1 it agrees with the vector defined in Subsection 2.4.
Lemma 2.11. LetT := [E,⇣,m] be a rectifiablek-current inRd, and let T0 :=
[E\A,⇣,m]whereAis a set inRd. If@T =0and the boundary of Adoes not intersect the closure ofE, then@T0=0.
Proof. Since @A\ E = ?, the sets E \ A and E \ A have disjoint closures, and we can find a smooth function : Rd ! [0,1] that is equal to 1 on some neighbourhood ofE\A, and to 0 on some neighbourhood ofE\A. Then for every form!there holds
h@T0;!i=hT0;d!i=hT; d!i=hT; d!+d ^!i
=hT;d( !)i=h@T; !i=0
(the second identity follows by the definition ofT0and the fact that =1 onE\A and =0 on E\A. The third identity follows by the fact thatd =0 onE).
Lemma 2.12. LetT := [E,⇣,m]be a rectifiablek-current inRdwithEbounded, and letT0:= [E\C,⇣,m]whereCis a connected component of the closure of E.
If@T =0then@T0 =0.
Proof. SinceCis a connected component ofE, we can find a decreasing sequence of open sets Un such that the intersection is C and @Un \E = ? for every n (see Subsection 2.8). We setTn := [E\Un,⇣,m]. Hence Lemma 2.11 implies
@Tn =0, and sinceTnconverge toT0in the mass norm, then@Tn converge to@T0 in the weak topology of currents, and therefore@T0=0.
Lemma 2.13. LetEy be a level set of f such thatHk(Ey) <+1and the asso- ciated currentTy is well-defined and has no boundary(cf. Subsection2.10). Then H k(Ey\Ey⇤)=0.
Proof. SetB:=Ey\E⇤y, fix >0, and take an open setA such thatB⇢ A and Hk(Ey\A )Hk(B)+ .
Recall that B is the union of the connected components of Ey which areH k- negligible. For every such connected component C, we can find an open neigh- bourhood U such thatC ⇢ U ⇢ A , Hk(Ey \U) , and@U \ Ey = ? (see Subsection 2.8). From such family of neighbourhoods we extract a countable subfamily{Un}that coversB, and setVn :=Un\(U1[· · ·[Un 1)for everyn. The setsVnare pairwise disjoint and coverB, and one easily checks that@Vn\Ey =? for everyn.
We then setTn := [Ey\Vn,⌧,1]. Thus the sumP
nTnagrees with the current T := [Ey\V,⌧,1]whereV is the union of the setsVn. Moreover the properties ofUn andVn yieldM(Tn)= Hk(Ey \Vn) Hk(Ey\Un) and@Tn = 0 (apply Lemma 2.11). Thus the isoperimetric theorem (see [14, Theorem 7.9.1], [18, Theorem 30.1]) yieldsTn=@Snfor some rectifiable(k+1)-currentSnthat satisfies
M(Sn)c[M(Tn)]1+1/k cM(Tn) 1/k =cHk(Ey\Vn) 1/k,
where c is a constant that depends only ond andk. Since the flat norm13 of Tn
satisfiesF(Tn)M(Sn), the previous inequality yields F(Tn)cHk(Ey\Vn) 1/k.
Taking the sum over all n we obtain F(T) cHk(Ey) 1/k, and therefore T tends to 0 with respect to the flat norm as !0.
On the other hand, the inclusions B ⇢ V ⇢ A and the choice of A imply H k((Ey\V )\B) . HenceT = [Ey\V,⌧,1]converge to the current[B,⌧,1] with respect to the normM, and therefore also with respect to the flat normFas
!0. Thus[B,⌧,1]must be equal to 0, which means thatHk(B)=0.
Proof of statement(iii)of Theorem2.5. Let A be the set of all y such that state- ments (i) and (ii) of Theorem 2.5 hold andTy := [Ey,⌧,1]is a well-defined current without boundary (cf.Subsection 2.10). ThenAhas full measure inRd k, and we claim that statement (iii) holds for every y 2 A. Indeed, the elements of Cy are pairwise disjoint subsets of Ey with positiveHk-measure, and sinceHk(Ey)is finite,Cy must be countable. MoreoverHk(Ey\E⇤y)=0 by Lemma 2.13.
Next we give some lemmas used in the proof of statements (iv) and (v) of Theorem 2.5.
Lemma 2.14. Let T := [C,⇣,1] be a 1-dimensional rectifiable current in Rd, whereCis a curve with Lipschitz parametrization : [a,b]!Csuch that| ˙ | = 1a.e. Assume that@T =0andCis simple. Then
(i) Cis closed;
(ii) either⇣ = ˙ a.e. in[a,b]or⇣ = ˙ a.e. in[a,b].
Proof. Step1.Since⇣( (t))and ˙(t)are parallel unit vectors for a.e.t, there exists : [a,b] !{±1}such that⇣( (t))= (t) ˙(t)for a.e.t. Since is injective at least on[a,b), the assumption@T =0 can be re-written as
0= Z
Chd ;⇣idH1= Z b
a hd ; ˙i dL1
for every function (0-form) onRd of classCc1. Sincehd ; ˙iis the (distri- butional) derivative of , we obtain that
0= Z b
a '˙ dL1 (2.5)
for every test function': [a,b]!Rof the form'= where is a function onRdof classCc1.
13Here the flat normF(T)of a currentTwith compact support is the infimum ofM(R)+M(S) over all possible currentsR,Ssuch thatT =R+@S,cf.[9, Section 4.1.12].
Step2:proof of statement(i).Assume by contradiction that is injective on[a,b], that is,C is simple but not closed. Then Corollary 7.4 implies that equality (2.5) holds for all Lipschitz functions': [a,b]!R, and this allows us to conclude that
=0 a.e., in contradiction with the fact that = ±1 a.e.
Step 3: proof of statement(ii). SinceC is simple, the parametrization is injec- tive on the interval with identified end points[a,b]⇤(cf.Subsection 2.2), and then Corollary 7.4 yields that (2.5) holds for all Lipschitz functions': [a,b]!Rsuch that'(a)='(b). This implies that the distributional derivative of vanishes, and therefore either =1 a.e. or = 1 a.e., and the proof is concluded.
Lemma 2.15. LetFbe a family of pairwise disjoint triods inR2. ThenFis count- able.
The above lemma has been proved in [15, Theorem 1]. For reader’s conve- nience, we give a self-contained proof of this result in Section 8.
Lemma 2.16. Given a map f :Rd !Rd 1, the following statements hold: (i) ifd =2, then the level setEycontains no triods for ally 2Rexcept countably
many;14
(ii) ifd 3and f is of classC1,1/2, then the level setEy contains no triods for a.e. y2Rd 1.
Proof. Since the level sets of any map are pairwise disjoint, statement (i) follows immediately from Lemma 2.15.
We prove statement (ii) by reduction to the cased=2. Consider the open set U := x 2Rd : rank(rf(x)) d 2 .
Since f is of classC1,1/2, a refined version of Sard theorem [2, Theorem 2] shows that the level setEyis contained inUfor a.e.y2Rd 1, and therefore it is sufficient to prove statement (ii) for the restriction of f toU.
Now, by applying the Implicit Function Theorem toy= f(x)we can coverU by open setsV whered 2 of the variablesxi can be written in terms ofd 2 of the variables yi; in other words, for every suchV there exists an open setW inRd, a diffeomorphism : W !V of classC1, and a mapg:W !Rsuch that, after a suitable re-numbering of the variables,
f (t)=(t1, . . . ,td 2,g(t)) for allt 2W. Then it suffices to prove statement (ii) for the map f˜:= f .
Let N be the set of all y 2 Rd 1 such that the level set E˜y of f˜contains a triod, and for every y0 = (y1, . . . ,yd 2)2 Rd 2 letNy0 be the set of all y00 2 R such that(y0,y00)2N. Then statement (i) shows thatNy0is countable for everyy0, and therefore Fubini’s Theorem implies thatNis negligible.
14Note that f does not need to be Lipschitz, and not even continuous.
Lemma 2.17. Let E be a closed, connected set inRd with finite, strictly positive length. If E contains no triods, then it is a simple curve, possibly closed. More precisely, there exists a Lipschitz parametrization : [a,b] ! E which is either injective on[a,b]or satisfies (a)= (b)and is injective on[a,b).
Proof. Step1.Recall the following well-known fact: a connected closed setEwith finite length is connected by simple curves. More precisely, for everyx,yinEthere exists an injectivemap : [a,b] ! E such that (a) = x, (b) = y, and is 1-Lipschitz, that is, it has Lipschitz constant at most 1 (see for instance [8, Lemma 3.12]).
Step2. LetFbe the family of all 1-Lipschitz maps : [a,b] ! E(the domain [a,b] may vary with ) that are injective on(a,b). We orderFby inclusion of graphs, that is, 1 2if[a1,b1]⇢ [a2,b2]and 1 = 2on[a1,b1]. One easily checks thatFadmits a maximal element : [a,b] ! E. In the next steps we show that this is the parametrization we are looking for.
Step3:either is injective on[a,b]or satisfies (a) = (b)and is injective on [a,b). Since is injective on(a,b), it suffices to show that for everyt 2 (a,b) there holds (t) 6= (a), (b). Assume by contradiction that (t) = (a)for somet, and take a positive such thata+2 <t < b . Then, contrary to the assumptions of the statement, E contains the triod with center y := (a) = (t) given by the union of the following three curves: C1 := ([a,a+ ]), C2 :=
([t ,t]),C3:= ([t,t + ])(see Figure 2.1(a)).
Figure 2.1.Construction of triods in the proof of Lemma 2.17.
Step 4:the image of is E. Assume by contradiction that there exists x 2 E \ ([a,b]). By Step 1 there exists an injective 1-Lipschitz map 0 : [a0,b0] ! E such that 0(b0) = x and 0(a0) is some point in ([a,b]). Now, lett0 be the largest of all t 2 [a0,b0] such that 0(t) 2 ([a,b])and taket1 2 [a,b]so that
(t1)= 0(t0).
If is injective on[a,b]– the other case is similar – we derive a contradiction in each of the following cases: i)t1 = b, ii)t1 = a, iii)a < t1 < b. Ift1 = b, we extend the map by setting (t):= 0(t b+t0)for allt 2[b,b+b0 t0]; one easily checks that the extended map belongs to F, in contradiction with the maximality of . A similar contradiction is obtained ift1 =a. Finally, ifa<t1<
bthen Econtains the triod with center y := (t1)= 0(t0)given by the union of
the following three curves: C1:= 0([t0,b0]),C2:= ([a,t1]),C3:= ([t1,b])– see Figure 2.1(b).
Proof of statements(iv)and(v)of Theorem2.5. Letd = 2 ord 3 and f be of classC1,1/2. Then for a.e. y 2 Rd 1 the level set Ey has finite length (Subsec- tion 2.7) and contains no triods (Lemma 2.16), and the associated current Ty :=
[Ey,⌧,1]is well-defined, rectifiable, and without boundary (Subsection 2.10). In particular the tangent vector⌧is defined atH1-a.e. point ofEy.
For every suchy, letCbe a connected component ofEy which is not a point.
Since C contains no triods, Lemma 2.17 implies thatC is a simple curve with a Lipschitz parametrization , and we can further assume that| ˙ | =1 a.e. The latter assumption implies that⌧ is defined at (t)for a.e.t.
By Lemma 2.12 the current T0 := [C,⌧,1] has no boundary, and therefore Lemma 2.14 implies that C is a closed curve such that either ⌧ = ˙ a.e. or
⌧ = ˙ a.e.; in the latter case we replace (t)by ( t) and the proof is concluded.
3. Examples of maps with no triod-free level sets
In this section we show that the assumption that f is of classC1,1/2in statement (v) of Theorem 2.5 cannot be dropped. More precisely, givend 3 and↵<1/(d 1), we construct a map f of classC1,↵ fromRd to a cubeQ0inRd 1such thatevery level set contains a triod,15and therefore at least one of its connected components is neither a point nor a simple curve (Proposition 3.7(iii)).
This example shows that the H¨older exponent 1/2 in Theorem 2.6(v) is optimal for d = 3. As pointed out in Remark 2.6(iii), we believe the the optimal H¨older exponent ford > 3 is the one suggested by this example, namely 1/(d 1), and not 1/2.
3.1. Idea of the construction. Assume for simplicity thatd = 3. The strategy for the construction of f is roughly the following: we divide the target square Q0 in a certain number N of sub-squaresQi with side-length⇢, then we define f on a cubeC0minus Ndisjoint sub-cubesCi with side-lengthr, so that the following key property holds: for every y 2 Q0the level set f 1(y)contains three disjoint curves connecting three points x1,x2,x3 on the boundary ofC0 with three points on the boundary of the cube Ci, where i is chosen so that y belongs to Qi, see Figure 3.1(a).16
15At first glance, this claim seems to contradict the fact that a generic level set of a Lipschitz map is a rectifiable current with multiplicity 1 and no boundary (cf.Subsection 2.10). It is not so, since we do not claim that the level setcoincideswith the triod in a neighbourhood of the center of the triod.
16This is the only part of this construction that requires more than two dimensions: as one can easily see in Figure 3.1, in two dimensions the joining curves for different values of ywould necessarily intersect, in contradiction with the fact that they are contained in different level sets.
x3 (a)
x2 x1
x3 (b)
x2 x1
x3 x~3 x~2 x~1
C0
Ci , i=1,2...
x~1 x~2
x~3 Ey Ey~
Figure 3.1.The components of the level setsEyandEy˜starting from the pointsxj and x˜j (j =1,2,3) after the first step of the construction (a), and at the end (b).
In the second step we replicate this construction within each Ci, so that f takes values in Qi, is defined outsideN sub-cubes with side-lengthr2, and now f 1(y) contains disjoint curves connectingx1,x2,x3to three points on the boundary of one of these smaller cubes. And so on . . .
As one can see from Figure 3.1(b), in the end each level set Ey will contain three disjoint curves connectingx1,x2,x3to the same point, which means that Ey
contains a triod (cf.Subsection 2.3).
Note that in the(n+1)-th step we define f on each one of theNncubes with side-lengthrn left over from the previous step minusNsub-cubes with side-length rn+1, so that each cube is mapped in a sub-square ofQ0with side-length⇢n. Thus the oscillation of f on this cube is⇢n, and this is enough to guarantee that in the end the map f is continuous. As we shall see, if things are carefully arranged, f turns out to be of classC1,↵.
3.2. Notation.We denote the points inRd byx=(x0,xd)withx0:=(x1, . . .,xd 1), and the points inRd 1 by the letter y. For every x0 2 Rd and every` > 0, we denote byC(x0,`)the closed cube inRd with centerx0and side-length`given by
C(x0,`):=x0+
` 2,`
2
d
.
We denote byQ(y0,`)the closed cube inRd 1, similarly defined.
Through this section we reserve the letterCford-dimensional closed cubes in Rd of the formC(x0,`), and the letterQfor the(d 1)-dimensional closed cubes inRd 1or inRd with axes parallel to the coordinate axes. With the cased =3 in mind, we often refer to the former ones simply as “cubes” and to the latter ones as
“squares”.
We set
C0:=C(0,1)=
1 2,1
2
d
, Q0:=Q(0,1)=
1 2,1
2
d 1
,
and for every cube C = C(x0,`) we denote bygC the homothety onRd which mapsC0intoC, that is,
gC(x):=x0+`x for allx 2Rd. Similarly, for every squareQ =Q(y0,`)we set
hQ(y):=y0+`y for ally2Rd 1.
3.3. A complete norm forCk,↵. Given a real or vector-valued map f defined on a subset EofRdand↵2(0,1], the homogeneous H¨older (semi-) norm of exponent
↵of f is
kfkC0,↵
hom := sup
x,y2E x6=y
|f(x) f(y)|
|x y|↵ . (3.1)
Let be given an open setAinRd, a pointx02 A, and an integerk 0. Among the many (equivalent) complete norms on the spaceCk,↵(A), the following is particu- larly convenient:
kfkCk,↵ :=krkfkC0,↵
hom+ Xk h=0
|rhf(x0)|. (3.2) The following interpolation inequality will be useful: if E is a convex set with non-empty interior and f is of classC1then17
kfkC0,↵
hom 2kfk11↵krfk↵1. (3.3) 3.4. Construction ofCi, Qi,gi,hi. For the rest of this section we fix a positive real number↵such that
↵< 1
d 1. (3.4)
We also fix an integerN >1 which is both ad-th and a(d 1)-th power of integers, e.g.,N = 2d(d 1). SinceN is a(d 1)-th power of an integer, we can cover the (d 1)-dimensional square Q0 by N squares Qi with pairwise disjoint interiors and side-length⇢:=N 1/(d 1). SinceNis thed-th power of an integer, for every positive real numberr such that
r < N 1/d (3.5)
we can findNpairwise disjoint cubesCi with side-lengthrcontained in the interior ofC0. For everyi =1, . . . ,N we set
gi :=gCi, hi :=hQi
(thusgi andhi are homotheties with scaling factorsr and⇢, respectively).
17It suffices to estimate the numerator|f(x) f(y)|at the right-hand side of (3.1) by 2kfk1 when|x y| kfk1krfk11, and bykrfk1|x y|otherwise.
For the rest of this section we fixrso that it satisfies (3.5) and
⇢<r1+↵. (3.6)
This assumption is compatible with (3.5) because the upper bound on ↵ and the definition of⇢imply⇢< (N 1/d)(1+↵).
Finally, we define the open set
A:=Int C0\(C1[· · ·[CN) .
3.5. Construction of f0.We construct a map f0 : Rd ! Q0 with the following properties:
(i) f0is of classC2and has compact support;
(ii) f0agrees withhi f0 gi 1on a neighbourhood of@Ci fori =1, . . . ,N;
(iii) there exists three pairwise disjoint setsGj contained in@C0such that the fol- lowing holds: for everyi = 1, . . . ,N and every y 2 Qi both f0 1(y)\Gj
and f0 1(y)\gi(Gj)consist of single points, denoted byxj andx0j respec- tively, and there exist pairwise disjoint curves joiningxj andx0j and contained in f0 1(y)\Aexcept for the end points,cf.Figure 3.1(a).
The construction of f0is divided in three steps.
Step1.Forj =1,2,3 we defineGj :=Q0j⇥{1/2}whereQ01,Q02,Q03are pairwise disjoint squares contained in Q0. Then we choose">0 so that
(a) the"-neighbourhoodsI"Gj with j =1,2,3 are pairwise disjoint;
(b) I"(@C0)andIr"(@Ci)are disjoint fori =1, . . . ,N(see Figure 3.2(a)).
Then we set
f0(x):=hQ10
j(x0) for everyx 2I"Gj, and take an arbitrary smooth extension of f0to the rest ofI"(@C0).
Step2. We define f0on the setsIr"(@Ci)withi = 1, . . . ,N so that property (ii) above is satisfied, that is, f0:=hi f0 gi 1.18
Step3.So far the map f0has been defined on a neighbourhood of the union of@Ci
withi = 0, . . . ,N. Now we extend it to the rest ofRd so that property (iii) above is satisfied.
Lety2 Qi be fixed: by Step 1 there exists a unique point inGj, denoted byxj(y), such that f(xj(y))= y, and by Step 2 there exists a unique pointx0j(y)ingi(Gj) such that f(x0j(y))= y.
Roughly speaking, the idea is to choose for all y 2 Q0 and all j = 1,2,3 pairwise disjoint curves joiningxj(y)andx0j(y)and contained inAexcept the end
18This definition is well-posed because the setsI"(@C0)andIr"(@Ci)are pairwise disjoint.
