Improved estimate of the singular set of Dir-minimizing Q-valued functions via an abstract regularity result

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Journal

of

Functional

Analysis

www.elsevier.com/locate/jfa

Improved

estimate

of

the

singular

set

of

Dir-minimizing

Q-valued

functions

via

an

abstract

regularity

result

Matteo Focardia,∗_{, Andrea Marchese}b_{, Emanuele Spadaro}b

a

DiMaI,UniversitàdegliStudidiFirenze,VialeMorgagni67/A,50134Firenze, Italy

b

Max-Planck-InstitutfürMathematikindenNaturwissenschaften,Inselstrasse22, 04103Leipzig,Germany

a r t i c l e i n f o a bs t r a c t

Article history:

Received 8 September 2014 Accepted 16 February 2015 Available online 3 March 2015 Communicated by C. De Lellis MSC: 49Q20 54E40 Keywords: Quantitative stratification Q-valued functions

Area minimizing currents

In this note we prove an abstract version of a recent quantitative stratification principle introduced by Cheeger and Naber(2013) [6,7].Using thisgeneralregularity result pairedwithanε-regularitytheoremweprovideanewestimate of theMinkowskidimensionofthesetofhighermultiplicity points of a Dir-minimizingQ-valued function.The abstract principle isapplicable toseveralotherproblems:werecover recent results in the literature and we obtain also some improvementsinmoreclassicalcontexts.

© 2015ElsevierInc.All rights reserved.

* Corresponding author.

E-mailaddresses:focardi@math.unifi.it(M. Focardi), marchese@mis.mpg.de(A. Marchese), spadaro@mis.mpg.de(E. Spadaro).

http://dx.doi.org/10.1016/j.jfa.2015.02.011

1. Introduction

Anabstractregularityresult. Weproposeanabstractionofaquantitativestratification principle introduced and developed in a series of papers by Cheeger and Naber [6,7], Cheeger,Haslhoferand Naber[4,5]andCheeger,NaberandValtorta[8].

Theinterestinfindinggeneralformulationsofthiskindofregularityresultsisdriven byanumberofimportantapplicationsingeometricanalysis.Apartfromthosecontained in the papers quoted above, we mention the cases of Dir-minimizing Q-valued maps accordingtoAlmgren,ofvarifoldwithboundedmeancurvatureandofalmostminimizers oftheperimeter.Theformeristreatedindetail inSection4andSection5,thelatter in Section6.Weexplicitlyremark thatthepapers[4,5]dealalsowithparabolicexamples, acasethatisnotcoveredbyourresults.

To our knowledge the first example in this direction of abstraction is the general regularitytheoremprovenbySimon[18,AppendixA]basedontheso calleddimension reduction argument introduced by Federer in his pioneering work [15]. Similarly, the paperbyWhite[22]generalizestherefinementofFederer’sreductionargumentmadeby Almgreninhisbigregularity paper[2].

Thebasicprincipleand themain ingredients of ourabstract formulationcanbe ex-plainedroughlyasfollows.

Abstractstratification:thesetofpointswhereasolutiontoageometricproblem is faraway at every scale from being homogeneous with k + 1 independent invariant directionshas Minkowskidimensionlessthanor equaltok.

Themainsetsof quantitiesweconsiderare:

(a) afamilyofdensityfunctionsΘs,increasing w.r.t. s≥ 0;

(b) a family of distance functions dk, k ∈ {0,. . . ,m}, measuring the distance from

k-invariant homogeneoussolutions.

Inaddition,weassumesuitablecompatibilityconditions,namely

(i) aquantitative differentiationprinciplethatallowsto quantify thenumberof those scalesforwhichclosenesstohomogeneoussolutionsfails,andthattypically follows intheapplicationsfrom monotonicitytypeformulas;

(ii) aconsistencyrelationbetweenthedistancesdk:ifasolutionisclosetoak-invariant

one and additionally is 0-invariant with respect to another point away from the invariantk-dimensionalspace,thenitisactuallyclosetoa(k + 1)-invariantsolution (seeSection2.2forthedetailedformulation).

This set of hypotheses is common to many problems in geometric analysis such as theDirichletminimizingmultiple valuedfunctions,harmonic maps, almostminimizing

currents and several others (see [6–8] forother applications). Indeed,the stratification resultandtheestimateontheMinkowskidimensiononlydependonassumptions(i)and (ii), thusmakingthecommonaspects ofallpreviousresultsclear.

ItturnsoutthatthereisasimpleconnectionbetweenWhite’sapproachtoAlmgren’s stratificationandtheoneoutlinedabove.InSection3.1weshowhowtorecasttheresult by White inourframework. Inthis respect, westress thatthe stratification in[7]and inourTheorem 2.3canbe appliedtosomecasesnotcoveredbytheideasin[22], such as stationaryharmonicmaps(cp.[7,Corollary 2.6],Section2.6.2and [22,Section 6]).

Ourmainapplicationoftheabstractstratificationprincipleisoutlinedinthefollowing paragraph.

Applicationto Q-valued functions. Intheregularitytheoryforhighercodimension min-imalsurfaces(inthesenseofmassminimizingintegerrectifiablecurrents)afundamental roleisplayedbythemultiplevaluedfunctionsintroducedbyAlmgrenin[2],whichturn outtobe thecorrectblowuplimitsfortheanalysis ofsingularities(seealso[9,13,12,10, 11] forasimplifiednewproofoftheresultin[2]).

Following[9],aQ-valuedfunctionu isameasurablemapfromaboundedopensubset Ω ⊂ Rn _{(for simplicity we always assume that the boundary of Ω is smooth) taking}

valuesinthespace ofpositiveatomicmeasures inRm_{with massQ,namely}

Ω x → u(x) ∈ AQ(Rm) :=  _Q  i=1 JpiK : pi∈ Rm  ,

where JpK denotes the Dirac delta at p. Almgren proves in [2] (cp. also [11]) that the blowupsofhighercodimensionmassminimizingintegralcurrentsareactuallygraphsof Q-valued functionsu in asuitableSobolevclassW1,2_(Ω,A

Q(Rm)) minimizing a

gener-alizedDirichletenergy(cp.[9,Definition 0.5]): ˆ Ω |Du|2_≤ ˆ Ω |Dv|2 _{∀ v ∈ W}1,2_(Ω,A Q(Rm)), v|∂Ω= u|∂Ω,

(explicitexamplesofDir-minimizingQ-valuedfunctionsaregivenin[20]).

Inordertoestimatethesizeofthesingularsetofaminimizingcurrentitisessentialto bound thedimensionofthesetofpointswherethegraphofaDir-minimizingQ-valued functionhashighermultiplicity.Almgren’smainresultintheanalysisofmultiplevalued functionsisinfactanestimateoftheHausdorffdimensionofthesetΔQ ofmultiplicity

Q pointsofaDirichletminimizingQ-valuedfunctionu,i.e. theset ofpointsx∈ Ω such thatu(x)= QJpK forsomep∈ Rm_{,which turnsoutnotto exceedn− 2 inthecaseit}

doesnotcoincidewithΩ (cp.[9,Proposition 3.22]).

Inthis paperweimprove Almgren’sresultbyshowing anestimateoftheMinkowski dimension of ΔQ. To this aim we denote by Tr(E) := {z ∈ Rn : dist(z,E) < r} the

Theorem 1.1. Let u: Ω → AQ(Rm) be a Dir-minimizing function, where Ω⊂ Rn is a

boundedopensetwith smoothboundary.TheneitherΔQ = Ω,orforevery Ω ⊂⊂ Ω the

Minkowskidimensionof ΔQ∩ Ω is lessthan orequal ton− 2, i.e. forevery Ω ⊂⊂ Ω

andforevery κ0> 0 thereexistsaconstant C > 0 such that

|Tr(ΔQ∩ Ω)| ≤ C r2−κ0 ∀ 0 < r < dist(Ω, ∂Ω). (1.1)

Wealso obtainastratification resultforthewholeset of singularpointsof multiple valued functions that, even if known to the experts, we were not able to find in the literature. Tothis aim we introducethe following notation.Given aQ-valuedfunction u: Ω→ AQ(Rm),wedenotebySingu⊂ Ω itssingularset,i.e. x0∈ Sing/ u ifandonlyif

thereexists r > 0 suchthat

graph(u|Br(x0)) :={(x, y) ∈ R

n×m _{: |x − x}

0| < r, y ∈ supp(u(x))}

isasmoothn-dimensionalembeddedsubmanifold(notnecessarilyconnected).Forevery k∈ {0,. . . ,n},wedefinethesubsetSing_uk ofthesingularsetSing_u madeofthosepoints having all tangent functions with at most k independent directions of invariance (we refertoSection5.3 fortheprecise definition).

Theorem 1.2. Let u: Ω → AQ(Rm) be a Dir-minimizing function, where Ω⊂ Rn is a

boundedopen set with smooth boundary,and let Singk_u be thesingular strata defined in Section5.3.Then, Sing_u= Singn_u−2 and

Sing0_u is countable (1.2)

dim_H(Singk_u)≤ k ∀ k ∈ {1, . . . , n − 2}. (1.3) In the case Q = 2 a more refined analysis by Krummel and Wickramasekera [16]

showstherectifiabilityofthesingularset,remarkably improvingAlmgren’s work. WeproveTheorems 1.1 and1.2asaconsequenceofourabstractstratification princi-ple.Moreprecisely,Theorem 1.2isadirectconsequenceofit,whileTheorem 1.1requires afurtherstabilityproperty deducedbyanε-regularityresult(seeProposition 5.4). Applications to generalizedsubmanifolds. Inthe final Section6we apply theabstract stratificationprincipletovarifoldswithboundedmeancurvatureandalmostminimizers oftheperimeter,two relevantcasesforapplicationsthatarenotcoveredbytheresults in[7].Alsointhese caseswederive someimprovementsofwell-knownestimatesforthe singularset.Stratificationforthesingularsetofstationaryvarifoldswithboundedmean curvatureisaddressedinSection6.1.Eventually,inTheorem 6.7wegiveaboundonthe Minkowskidimensionofthesingularset ofanalmostminimizeroftheperimeterrather than the classical Hausdorff dimension estimate, and in Theorem 6.8 we show higher integrabilityforitsgeneralizedsecondfundamentalform.

Ontheorganizationofthepaper. Afewwordsareworthwhileconcerningthestructure of the paper. The first two sections of the paper are devoted to the abstract regular-ity results. Inparticular, Section 2contains theestimate of the volumeof the tubular neighborhood ofthe singular strata given inTheorem 2.2 (whichis proved inthe first partof Section3)and theabstract stratification inTheorem 2.3. Inorder tomake our statementsandhypothesesrecognizableand“natural”tothereaders,weillustratethem inSection2.6for themodelexamplesof areaminimizingcurrents andharmonicmaps. The last part of Section 3 is devoted to the comparison with the results by White in

[22].Then, we specializeourresults tothe caseofQ-valued functionsinSection5, the needed preliminaries are collectedinSection4. Wefinally focus on varifolds with suit-ablehypotheses ontheir meancurvatureand onalmost minimizersof theperimeterin Section6.

2. Abstractstratification

The general abstract approach we proposeis based ontwo main sets of quantities: namely,afamilyofdensityfunctionsΘsandanincreasingfamilyofdistancefunctions dk.

2.1. Densitiesanddistancefunctions

LetΩ⊂ Rn_{beopenandbounded,andforeverys≥ 0 setΩ}s_{:={x∈ Ω: dist(x,∂Ω)≥}

2s}.Weassumethefollowing.

(a) Foreverys suchthatΩs_{= ∅, thereexist functionsΘ}

s∈ L∞(Ωs) suchthat

0≤ Θs(x)≤ Θs(x),

for all 0 ≤ s < s and for all x ∈ Ωs_{. Moreover, for every s}

0 > 0 there exists

Λ0= Λ0(s0)> 0 suchthat

Θs(x)≤ Λ0,

forevery0≤ s≤ s0 andforeveryx∈ Ωs0.

(b) Setting U :={(x,s): x∈ Ωs, Θ0(x)> 0},there exist apositiveintegerm≤ n and

controlfunctionsdk : U→ [0,+∞) fork∈ {0,. . . ,m} such that

d0≤ d1≤ · · · ≤ dm.

2.2. Structuralhypotheses

(i) For every s0 > 0, ε1 > 0 there exist 0 < λ1(s0,ε1),η1(s0,ε1) < 14 such that if (x,s)∈ U, withx∈ Ωs0 _{ands< s}

0,then

Θs(x)− Θλ1s(x)≤ η1 =⇒ d0(x, s)≤ ε1.

(ii) Foreverys0> 0,foreveryε2,τ∈ (0,1) thereexists0< η2(s0,ε2,τ )≤ ε2suchthat

if(x,5s)∈ U,withx∈ Ωs0 _{and5s< s}

0,satisfiesforsomek∈ {0,. . . ,m− 1}

dk(x, 4s)≤ η2 and dk+1(x, 4s)≥ ε2,

then thereexistsak-dimensionallinearsubspaceV forwhich d0(y, 4s) > η2 ∀ y ∈ Bs(x)\ Tτ s(x + V ),

where Tτ s(x+ V ):={z : dist(z,x+ V )< τ s} isthetubularneighborhoodofx+ V

ofradius τ s.

2.3. Volumeof theneighborhoodsof singularstrata Thesetsweconsiderinourestimatesarethefollowing.

Definition 2.1 (Singular strata). For every 0 < δ < 1, 0 < r ≤ r0 and for every k ∈

{0,. . . ,m− 1} weset Sk r,r0,δ :=  x∈ Ωr0_{: Θ} 0(x) > 0 and dk+1(x, s)≥ δ ∀ r ≤ s ≤ r0  (2.1) and Sk r0,δ:=  0<r≤r0 Sk r,r0,δ and S k r0 := 0<δ<1 Sk r0,δ. (2.2)

Note that,by the monotonicityof the controlfunctions, Sk r,δ ⊂ Sk



r,δ if δ ≤ δ,r ≤ r

andk≤ k.

Ourabstractstratificationresultreliesonthefollowingestimateforthetubular neigh-borhoodsofthesingularstrata.ItsproofispostponedtoSection3.

Theorem2.2. UnderthestructuralhypothesesinSection2.2,foreveryκ0,δ∈ (0,1) and

r0> 0 there existsC = C(κ0,δ,r0,n,Ω)> 0 suchthat

|Tr(Sr,rk 0,δ)| ≤ C r

n−k−κ0 ∀ 0 < r < r

0 ∀ k ∈ {1, . . . , m − 1} (2.3)

S0

2.4. Hausdorffdimensionof thesingularstrata

ItisnowanimmediateconsequenceofTheorem 2.2thefollowingstratificationresult. Theorem2.3.UnderthestructuralhypothesesinSection2.2foreveryr0> 0 theestimate

dimH(S_rk₀)≤ k holds fork∈ {1,. . . ,m− 1}.Moreover, S_r0₀ iscountable. Proof. IndeedTheorem 2.2impliesthatdim_M(Sk

r0,δ)≤ k,wheredimMistheMinkowski

dimension. Since the Hausdorff dimension of aset is always lessthan or equal to the Minkowskidimension,wealsoinferthat

dimH(Srk0)≤ dimH δ>0 Srk0,δ  ≤ k

because,being theunionmonotone, itisenoughto consideracountablesetof parame-ters. 2

2.5. Minkowskidimensionofthesingular strata

ThedependenceoftheconstantC in(2.3)onδ preventsthederivationofanestimate ontheMinkowskidimensionofthesingularstrataSk

r0.Nevertheless,ifsuchdependence

drops,thenTheorem 2.2turnsactuallyintoanestimateontheMinkowskidimensionof thesingularstrata whichisnotimplied byAlmgren’s stratificationprinciple.

Theorem 2.4. Under the hypotheses of Theorem 2.2, if for some δ0 > 0 and k ∈

{0,. . . ,m− 1}

Sk r0,δ =S

k

r0 ∀ δ ∈ (0, δ0), (2.5)

then forevery 0< κ0< 1 andr0> 0 thereexistsC = C(κ0,δ0,r0,n,Ω)> 0 such that

|Tr(Srk0)| ≤ C r

n−k−κ0 ∀ 0 < r < r

0. (2.6)

In particular dim_M(S_rk₀)≤ k. 2.6. Examples

The meaning of the structural hypotheses in Section 2.2 is very well illustrated by the two familiar examples of area minimizing currents and stationary harmonic maps treatedin[7]forwhichTheorem 2.2 and2.3hold.Moreoverforareaminimizingcurrents of codimensiononeinRn Theorem 2.4canbe alsoappliedfork = n− 8.

2.6.1. Areaminimizing currents

LetT beanm-dimensionalareaminimizingintegralcurrentinΩ.Thenwecanset Θs(x) := T (Bs

(x)) wmsm

for s > 0 and Θ0(x) := lim

r↓0+Θr(x) andfork∈ {0,. . . ,m} dk(x, s) := inf  F(Tx,s− C) B1

: C is k-conical & area minimizing, where

• Tx,r istherescalingofthecurrentaroundanypointx∈ Rn atscaler > 0:

Tx,r:=

 ηx,r

#T (2.7)

andthepush-forwardisdoneviathepropermapηx,rgiven byy→(y− x)r; • F is theflatnorm(see[18, § 31]);

• an m-dimensionalcurrent C inRn _{is k-conical fork∈ {0,. . . ,m},ifthere exists a}

linearsubspaceV ⊂ Rn _{ofdimensionbiggerthanorequaltok suchthat}

Tx,r= T for all r > 0 and x∈ V.

Note thata0-conicalcurrentissimplyaconewithrespect totheorigin.

One canchoose Λ0(r0) := M(T )ωmr0m. Then (a) is a consequence of the Monotonicity

Formula(see[18,Theorem 17.6])and(b)followsfromtheinclusionofk-conicalcurrents inthek-conicalones when k ≤ k. Withthis choice,thestructural hypotheses in Sec-tion 2.2 are satisfied, indeed (i) is an other consequenceof the Monotonicity Formula and(ii)follows fromarigiditypropertyof conessometimescalled “cylindricalblowup” (see[18,Lemma35.5]).

ThenthequantitativestratificationprincipleinTheorem 2.2recoversthe correspond-ingresultin[7]:

the set of points that are faraway from (k + 1)-conical area minimizing currents, at everyscale in[r,r0],has Minkowskidimensionlessthanor equalto k.

2.6.2. Stationary harmonic maps

Similarlyletu∈ W1,2_{(Ω,N ) beastationaryharmonicmapfromanopensetΩ⊂ R}n

toaRiemannianmanifold(N m_{,h) isometricallyembeddedinsomeEuclideanspaceR}p

(see,e.g.,[19]).Wecanset Θs(x) := s2−n

ˆ

Bs(x)

and foreveryk∈ {0,. . . ,n} dk(x, r) := inf v∈Ck B1 dist2_N ux,r, v dy, with

• ux,r(y):= u(x+ ry) forx∈ Ω andr∈

0,dist(x,∂Ω) ;

• a measurable map v is said to be k-conical if there exists a vector space V with dim V ≥ k thatleavesv invariant,i.e.

v(x) = v(y + x) ∀ x ∈ Rn, y∈ V, (2.8) andsuchthatv is0-homogeneous withrespecttothepointsinV ,i.e.

v(y + x) = v(y + λ x) ∀ x ∈ Rn, y∈ V and λ > 0; (2.9) • Ck:={v : B1→N k-conical}.

Assumption (a)inSection2.1iseasilyverifiedandthemonotonicityformula

Θr(x)− Θs(x) = r ˆ s ˆ ∂Bt(x) t2−n∂u ∂t  2dHn−1dt

togetherwithanelementarycontradictionargumentshowthattheStructuralHypothesis (i)inSection2.2issatisfied.MoreovertheStructuralHypothesis(ii)followssimilarlyto theoneforminimizingcurrents(cp.[7]formoredetails),thusleadingtothestratification of Theorem 2.2.

InSection6wegiveotherapplicationsofthisabstractregularity resulttothecaseof varifoldswithboundedvariationandalmostminimizersofthemassincodimensionone. 3. Proofofthe abstractstratificationandcomparisonwithAlmgren’sstratification

To begin with, we stateasimple consequence ofthe Structural Hypothesis (ii) (cp. Section2.2)inthefollowing

Lemma 3.1. Forevery s0> 0, foreveryε,τ ∈ (0,1) there exists0< γ0 ≤ ε suchthat if

(x,5s)∈ U,with x∈ Ωs0 _{and5s< s}

0,satisfies forsomek∈ {0,. . . ,m− 1}

d0(x, 4s)≤ γ0 and dk+1(x, 4s)≥ ε,

then thereexistsalinear subspaceV withdim(V )≤ k such that

Proof. Let γ0 ≤ γ1 ≤ . . . ≤ γk+1 be set as γk+1 = ε andγj−1 = η2(s0,γj,τ ) with η2

theconstantintheStructural Hypothesis (ii).Leti ∈ {0,. . . ,k} be thesmallestindex such that di+1(x,4s) ≥ γi+1 (which exists because of the assumption dk+1(x,4s) ≥

ε= γk+1).Then,applyingtheStructuralHypothesis(ii)wededucethatthereexistsan

i-dimensionallinearsubspaceV suchthateverypointy∈ Bs(x) withd0(y,4s)≤ γ0≤ γi

belongstothetubularneighborhood Tτ s(x+ V ). 2

IntheproofofTheorem 2.2weshallrepeatedlyusethefollowingelementarycovering argument.

Lemma3.2. For every measurable setE ⊂ Rn with finite measureand forevery ρ> 0, thereexistsafinite covering{Bρ(xi)}i∈I of Tρ/5(E) with xi ∈ E and

H0_(I)≤5

n_|T ρ/5(E)|

ωnρn

. (3.2)

Proof. Considerthefamilyofballs{B_ρ/5(x)}x∈E.BytheVitali5r-coveringlemma,there

existsafinite subfamily{B_ρ/5(xi)}i∈I ofdisjointballssuchthatTρ/5(E)⊂ ∪i∈IBρ(xi).

Byasimplevolumecomparison weconclude(3.2). 2 Wearenowreadyto proveTheorem 2.2.

ProofofTheorem 2.2. Westartfixing aparameterτ = τ (n,κ0)> 0 suchthat

ωnτ

κ0

2 ≤ 20−n. (3.3)

We choose the other constants involved in the structural hypotheses in the following way:

1. let γ0 ≤ γ1 ≤ . . . ≤ γk be such that γk = δ and γj−1 = η2(r0,γj,τ ) for every

j∈ {1,. . . ,k} asintheStructuralHypothesis (ii);

2. letλ1= λ1(r0,γ0) andη1= η1(r0,γ0) beas intheStructuralHypothesis(i);

3. fix q∈ N suchthatτq_{≤ λ}

1.

Wedividetheproofinto foursteps.

Step 1: reduction to dyadic radii. Let Λ0 = Λ0(r0) given in Section 2.1. It suffices to

prove(2.3)foreveryr oftheform r = r0τp

5 with p∈ N suchthatp≥ p0 := q + M + 1

andM :=q Λ0/η1.Indeedfor r0τ

p0 5 < s< r0 wesimplyhave |Ts(Ss,rk 0,δ)| ≤ |Ω| ≤ |Ω| _r0τp0 5 n−k−κ0 s n−k−κ0 = C2(κ0, δ, r0, n, Ω) sn−k−κ0.

On theotherhand,ifweassumethat(2.3)holdswithaconstantC1> 0 foreveryr of

theform r = r0τp

5 withp≥ p0,weconcludethatforrτ < s< r itholds

|Ts(Ss,rk 0,δ)| ≤ |Tr(S

k

r,r0,δ)| ≤ C1r

n−k−κ0≤ C

1τk+κ0−nsn−k−κ0.

Therefore setting C := max{τk+κ0−n_C

1,C2} we deducethat(2.3) holds foreveryr∈

(0,r0).

Step2:selectionofgoodscales. Fixavaluep∈ N withp≥ p0asaboveandsetr =r0τp₅.

Forall(x,r0)∈ U wehave

p  l=q Θ_4τl_r0(x)− Θ_4τl+q_r0(x) = p  l=q l+q−1_ i=l Θ4τi_r 0(x)− Θ4τi+1r0(x) ≤ q p+q_−1 h=q  Θ_4τh_r0(x)− Θ_4τh+1_r0(x) = qΘ4τq_r 0(x)− Θ4τp+qr0(x) ≤ q Λ0.

Therefore, thereexist atmostM indicesl∈ {q,. . . ,p} forwhichitdoesnotholdthat Θ4τl_r

0(x)− Θ4τl+qr0(x)≤ η1. (3.4)

Forany subsetA⊂ {q,. . . ,p} withcardinalityM we consider SA:=



x∈ S_r,rk _0,δ :(3.4)holds ∀ l /∈ A. Weproveinthenextstepthat

|Tr(SA)| ≤ C rn−k−

κ0

2 (3.5)

for some C = C(κ0,δ,r0,n,Ω) > 0. From (3.5) one concludes becausethe number of

subsetsA asaboveisestimatedby 

p− q + 1 M



≤ (p − q + 1)M _{≤ C | log r|}M

forsomeC(κ0,δ,r0,n)> 0,and

|Tr(Sr,rk 0,δ)| ≤  A |Tr(SA)| ≤ C | log r|Mrn−k− κ0 2 ≤ C rn−k−κ0 forsomeC(κ0,δ,r0,n,Ω)> 0.

Step3: proof of(3.5). We estimatethevolumeofTr(SA) bycoveringititeratively with

familiesofballscenteredinSAandwithradiiτjr0forj∈ {q,. . . ,p}.Wecanthenproceed

asfollows.Firstlyweconsider acoverofTτq_r

0_/5(SA) madeofballs{Bτqr0(xi)}i∈Iq with

xi ∈ SA andbyastraightforwarduseofLemma 3.2

H0_(I

q)≤ 5nτ−nqr0−n

diam(Ω) + 1 n.

Iteratively, for every j ∈ {q + 1,. . . ,p}, we assume to be given the cover {Bτj−1r0(xi)}i∈Ij−1 of Tτj−1r0_/5(SA), and we select a new cover of Tτjr0_/5(SA) which

ismadeofballsofradiiτj_r

0centeredinSAaccordingtothefollowingtwo cases:

(a) j− 1∈ A, (b) j− 1∈ A./

Case (a). For everyxi inthe familyat level j− 1, using Lemma 3.2 we coverSA∩

Bτj−1r0(xi) with finitely many balls Bτjr0_/2(yl) with yl ∈ SA ∩ Bτj−1r0(xi) and the

cardinalityofthecoverisboundedby 5n|B (τj−1+τj_{/10) r}0(xi)| ωn _τj_r 0₂ n ≤ 20 n_τ−n (note that T_τj_r

0_/10(SA∩ Bτj−1r0(xi)) ⊂ B_(τj−1+τj_{/10) r}0(xi)). We claim next that the

unionofBτj_r

0(yl) coversthetubularneighborhood

Tτ j r0 5

(SA∩ Bτj−1r0(xi)).

Indeedfor everyz ∈ T_τj

r0_/5(SA∩ Bτj−1r0(xi)) there exists z

 _{∈ S}

A∩ Bτj−1r0(xi) such

that|z − z|<τj_r

0₅.Sincez∈ B

τjr0_/2(yl) forsomeyl,thenz∈ Bτjr0(yl).

Therefore, collecting allsuch balls, the cardinality of the new coveringis estimated by

H0_(I

j)≤ 20nτ−nH0(Ij−1). (3.6)

Case (b). Ifj− 1∈ A,/ then(3.4)holdswithl = j− 1.BytheStructuralHypothesis (i) and the choiceof λ1,η1 in (2) and τ in (3) at the beginning of the proof, we have

that d0(x,4τj−1r0) ≤ γ0 for every x ∈ SA. Since xi ∈ SA ⊂ Sr,rk 0,δ we have also

dk+1(xi,4τj−1r0)≥ δ.Wecanthenapply Lemma 3.1and concludethat

SA∩ Bτj−1r0(xi)⊂ Tτjr0(xi+ V ) (3.7)

forsomelinearsubspaceV ofdimensionlessthanorequaltok.Note that

Thus applying Lemma 3.2 we find acovering of T_τj_r

0_/5(SA) with balls Br0τj(yl) such

thatyl∈ SA andusing (3.8)thecardinalityof thecoveringisbounded by

H0_(I

j)≤ 10nωnH0(Ij−1) τ−k. (3.9)

Inanycasetheprocedureendsatj = p withacoveringofTτp_r

0_/5(SA) whichismade

of balls{Bτp_r

0(xi)}i∈Ip suchthatxi∈ SAand

H0_(I p)≤ 5nτ−nqr−n0  diam(Ω) + 1 n20nτ−n M10nωnτ−k p−q−M ≤ C τ−kp₍₂₀n_ω n)p≤ C τ−p  k+κ02 (3.10) with C = C(κ0,δ,r0,n,Ω)> 0 andwherewe used(3.3)inthelastinequality.Estimate

(3.5)follows atonce |Tr(SA)| ≤ H0(Ip)|Bτp_r 0| (3.10) ≤ C rn−k−κ0 2 , forsomeC = C(κ0,δ,r0,n,Ω)> 0.

Step4:proofof(2.4). Letjxbethesmallestindexsuchthat(3.4)holdsforeveryj ≥ jx,

and foreveryi∈ N set

Ai:={x ∈ Sr00,δ : jx= i}.

We will prove that Ai is discrete, and hence Sr00,δ is at most countable. Fix x ∈ Ai.

By the choiceof the parametersapplying the Structural Hypothesis (i)it follows that d0(x,4r0τj)≤ γ0foreveryj≥ i.Sincex∈ Sr00,δ,wecanapplyLemma 3.1andinferthat

thepointsy ∈ Br0τj(x) satisfyingd0(y,4r0τ

j_{)≤ γ}

0arecontainedinBr0τj+1(x).

There-foreAi∩ Br0τj(x)⊂ Br0τj+1(x) foreveryj≥ i,whichimpliesthatAi isdiscrete. 2

3.1. Almgren’sstratificationprinciple

In this section we make the connection to the approach to Almgren’s stratification principlebyWhite[22].IndeedunderverynaturalassumptionstheresultsbyWhitefor thetimeindependent casefollow fromours.

White’s stratificationcriterion initssimplestformulationisbasedon:

(a) an uppersemi-continuous function f : Ω→ [0,∞) defined onabounded open set Ω⊂ Rn_;

(b) foreveryx∈ Ω acompactclassofconicalfunctionsG(x) accordingtothefollowing definition.

Definition 3.3. (1)An uppersemi-continuous map g :Rn → [0,∞) isconical if g(z)= g(0) impliesthatg ispositively0-homogeneous withrespectto z,i.e.,

g(z + λx) = g(z + x) for all x∈ Rn and λ > 0.

(2)AclassG ofconicalfunctionsiscompact if forallsequences(gi)i∈N⊆G thereexist

asubsequence(gij)j∈Nand anelementg∈G suchthat

lim sup

j→∞ gij(yij)≤ g(y) ∀ y ∈ R n_{, (y}

i)i∈N⊂ Rn with yi→ y.

Inparticularaconicalfunctionis0-homogeneous withrespectto0.

The stratification theorem by White is then based on the following two structural hypotheses:

(i) g(0)= f (x) forallg∈G(x);

(ii) for allri↓ 0 thereexistasubsequencerij ↓ 0 andg∈G(x) suchthat

lim sup

j→+∞ f (x + rijyj)≤ g(y) for all y, yj ∈ B1 with yj→ y.

Bytheuppersemi-continuityofany conicalfunctiong,theclosedset Sg:={z ∈ Rn: g(z) = g(0)}

isinfactthesetofthemaximumpointsofg.Sg iscalledthespine ofg.MoreoverSg is

thelargestvectorspacethatleavesg invariant,i.e.,

Sg={z ∈ Rn: g(y) = g(z + y) for all y∈ Rn} (3.11)

(cp.[22,Theorem 3.1]).Wesetd(x):= sup{dim Sg: g ∈G(x)},and

Σ:={x ∈ Ω : f(x) > 0, d(x) ≤ }.

Thestratificationcriterionin[22,Theorem3.2] isthefollowing. Theorem3.4 (White).Under thestructuralhypotheses (i), (ii),

Σ0 is countable; (3.12)

dim_H(Σ)≤  ∀  ∈ {1, . . . , n}, (3.13)

wheredim_H denotestheHausdorffdimension.

Thereaderwhoisinterestedintheapplicationofthiscriteriontothemodelcasesof areaminimizingcurrentsandharmonicmapsisreferredto[22].

3.1.1. Relation betweenthestructuralhypotheses

Theorem 3.4 canberecovered fromourTheorem 2.3ifweassumethefollowing rela-tionsbetweenthestructuralhypotheses(i),(ii)inSection2.2and(i),(ii)inSection3.1:

(1) f = Θ0;

(2) foreveryx∈ Ω,if lim

j dk(x, rj) = 0 for some (rj)j∈N⊂ (0, dist(x, ∂Ω)),

thenx∈ Σ/ k−1.

Note that (1) and (2) are always satisfied in the relevant examples considered in the literature.

ToprovethattheconclusionsofTheorem 3.4areimpliedbyTheorem 2.3itisenough to showthat Σ⊂ r0>0 S r0. (3.14)

Thismeansthatforeveryr0> 0 andforeveryx∈ Σ∩ Ωr0 thereexistsδ > 0 suchthat

d+1(x, r)≥ δ ∀ 0 < r ≤ r0. (3.15)

Assumebycontradictionthat(3.15)doesnothold,wefindr0andx asabovesuchthat

for asequence rj ∈ (0,r0] we haved+1(x,rj)↓ 0. Then bySection 3.1.1(2) x cannot

belongto Σ.

4. PreliminaryresultsonDir-minimizingQ-valued functions

We follow [9] for the notation and the terminology, which we briefly recall in the following subsections.

ThespaceofQ-pointsofRm_{isthesubspaceofpositiveatomicmeasuresinR}m_with

massQ,i.e. AQ(Rm) :=  _Q  i=1 JpiK : pi∈ Rm 

where JpiK denotes the Dirac delta at pi. AQ is endowed with the complete metric G

given by:foreveryT =_iJpiK andS =iJpiK ∈ AQ(Rm)

G(T, S) := min σ∈PQ  _Q  i=1 |pi− pσ(i)|2 1/2

A Q-valued function is ameasurable map u : Ω → AQ(Rm) froma bounded open

set Ω ⊂ Rn _{(with smooth boundary ∂Ω for simplicity). It is always possible to find}

measurable functions ui : Ω → Rm for i∈ {1,. . . ,Q} such thatu(x) =



iJui(x)K for

a.e. x∈ Ω. Note that theui’s arenot uniquely determined:nevertheless,we often use

the notation u=_iJuiK meaning an admissiblechoice of the functionsui’s hasbeen

fixed.Weset |u|(x) := G(u(x), Q J0K) =   i |ui(x)|2 _1/2 .

ThedefinitionoftheSobolevspaceW1,2(Ω,AQ) isgivenin[9,Definition 0.5]andleads

tothenotionofapproximatedifferentialDu=_iJDuiK (cf.[9,Definitions 1.9&2.6]).

Weset |Du|(x) :=   i |Dui(x)|2 _1/2

andsaythatafunctionu∈ W1,2(Ω,AQ(Rm)) isDir-minimizing if

ˆ Ω |Du|2_≤ ˆ Ω |Dv|2 _{∀ v ∈ W}1,2_{(Ω), v|} ∂Ω= u|∂Ω

where the last equality is meant inthe sense of traces (cp. [9, Definition 0.7]). By [9, Theorem 0.9]Dir-minimizingQ-valuedfunctionsarelocallyHöldercontinuouswith ex-ponentβ = β(n,Q)> 0.

In what follows we shall always assume that u ∈ W1,2_(Ω,A

Q(Rm)) is a nontrivial

Dir-minimizingfunction,i.e. u≡ QJ0K,with

η◦ u := 1 Q Q  i=1 ui≡ 0. (4.1)

Asexplainedin[9,Lemma 3.23]themeanvalueconditionin(4.1)doesnotintroduceany substantial restrictiononthe spaceof Dir-minimizingfunctions. Moreover,inthis case ΔQ reducestotheset{x∈ Ω: u(x)= QJ0K}.Notethat,ifu≡ QJ0K,thenΔQ⊂ Singu

by[9,Theorem 0.11]. 4.1. Frequencyfunction

Westartbyintroducingthefollowingquantities:foreveryx∈ Ω ands> 0 suchthat Bs(x)⊂ Ω weset

Du(x, s) := ˆ Bs(x) |Du|2 Hu(x, s) := ˆ ∂Bs(x) |u|2 Iu(x, s) := s Du(x, s) Hu(x, s) .

Iuiscalledthefrequencyfunction ofu.Sinceu isDir-minimizingandnontrivial,itholds

thatHu(x,s)> 0 foreverys∈ (0,dist(x,∂Ω)) (cp.[9,Remark 3.14]),from whichIu is

well-defined.

We recall that the functions s → Du(x,s), s → Hu(x,s), and s → Iu(x,s) are

absolutely continuouson(0,dist(x,∂Ω)). Similarlyforfixeds∈ (0,dist(x,∂Ω)) onecan prove the continuity of x → Du(x,s), x → Hu(x,s) and x → Iu(x,s) for x ∈ {y :

dist(y,∂Ω) > s}. The former follows by the absolute continuity of Lebesgue integral; while fortheremaining twoitsufficesthefollowingestimate:

 Hu(x, s)−  Hu(y, s) ≤ ⎛ ⎜ ⎝ ˆ ∂Bs(y)

||u|(z) − |u|(z + x − y)|2_dz

⎞ ⎟ ⎠ 1 2 ≤ |x − y| ⎛ ⎜ ⎝ ˆ ∂Bs(y) 1 ˆ 0 |∇|u|(z + t (x − y))|2_{dt dz} ⎞ ⎟ ⎠ 1 2 ≤ |x − y| ⎛ ⎜ ⎝ ˆ

Bs+|x−y|(y)\Bs−|x−y|(y)

|Du|2 ⎞ ⎟ ⎠ 1 2 (4.2)

where weusethefactthat|u|∈ W1,2_{(Ω) with|∇|u||≤ |Du| (cp.[9,Definition 0.5]).}

ThefollowingmonotonicityformuladiscoveredbyAlmgrenin[2]isthemainestimate about Dir-minimizingfunctions(cp. [9,Theorem 3.15 &(3.48)]): forall 0≤ r1 ≤ r2 <

dist(x,∂Ω) itholds Iu(x, r2)− Iu(x, r1) = r2 ˆ r1 t Hu(t) ˆ ∂Bt(x) |∂νu|2 ˆ ∂Bt(x) |u|2₋ ˆ ∂Bt(x) ∂νu, u 2 dt. (4.3)

Wefinally recallthatfrom [9,Corollary 3.18]wealsodeducethat Hu(z, r) = O(rn+2 Iu(z,0

+₎₋₁

) (4.4)

4.2. Compactness

From[9,Proposition 2.11&Theorem 3.20],if(uj)j∈NisasequenceofDir-minimizing

functionsinΩ suchthat sup j ujL 2_(Ω)+ sup j ˆ Ω |Duj|2< +∞,

thenthereexistsu∈ W1,2(Ω,AQ) suchthatu isDir-minimizing,anduptopassingtoa

subsequence(notrelabeledinthesequel)G(uj,u)→ 0 inL2(Ω),andforeveryΩ⊂⊂ Ω

G(uj, u)L∞(Ω)→ 0 and ˆ Ω |Duj|2→ ˆ Ω |Du|2_.

Inparticularthisimpliesthat(|Duj|2)j∈Nareequi-integrableinΩ,and

lim

j→+∞Iuj(x, s) = Iu(x, s) ∀ x ∈ Ω, ∀ 0 < 2s < dist(x, ∂Ω). (4.5)

4.3. Homogeneous Q-valuedfunctions

We discuss next some properties of the class of homogeneous Q-valued functions: w∈ W_loc1,2(Rn_,A

Q(Rm)) satisfying

(1) w islocallyDir-minimizingwith η◦ w ≡ 0,

(2) w isα-homogeneous,inthesensethat

w(x) =|x|αw  x |x|  ∀ x ∈ Rn_{\ {0},}

forsomeα∈ (0,Λ0],where Λ0 isaconstanttobe specifiedlater.

We denote this class by HΛ0. Note that Iw(x,0

+_{) = 0 if w(x) = QJ0K. Thefollowing}

lemmaisanelementaryconsequenceofthedefinitions. Lemma4.1. Letw∈ HΛ0.ThenIw(·,0

+_{) isconicalin thesense ofDefinition 3.3(1).}

Proof. FirstlyIw(·,0+) is uppersemi-continuous.Indeedsincew isDir-minimizing, we

canuse (4.3)and deduce thatIw(x,0+)= infs>0Iw(x,s), i.e. Iw(·,0+) isthe infimum

ofcontinuous(by(4.2))functionsx→ Iw(x,s) andhenceuppersemi-continuous.

We need only to show that Iw(·,0+) is 0-homogeneous at every point z such that

Iw(z,0+) = Iw(0,0+). We can assume without loss of generality that w is nontrivial,

Iw(z, 0+) = Iw(0, 0+) = Iw(0, 1) > 0

whereinthelastequalityweusedthehomogeneityofw.Thereforeinparticularw(z)= QJ0K.NextweshowthatIw(z,r)= Iw(0,0+) forallr > 0.Byasimpleestimateweget

Iw(z, r) = r Dw(z, r) Hw(z, r) ≤ I w(0, r +|z|) Hw(0, r +|z|) Hw(0, r) Hw(0, r) Hw(z, r) . (4.6)

Sincew ishomogeneouswithrespecttotheoriginandthefrequencyofw at0 isexactly α (cp.[9,Corollary 3.16]),wehavealso

Hw(0, r) = Hw(0, 1) rn+2 α−1 Dw(0, r) = Dw(0, 1) rn+2 α−2. Inparticular Iw(0, r +|z|) = α = Iw(0, 0+) = Iw(z, 0+) Hw(0, r +|z|) Hw(0, r) → 1 as r ↑ +∞.

Forwhatconcernsthethirdfactorin(4.6)

Hw(0, r)

Hw(z, r)

= 1 +Hw(0, r)− Hw(z, r) Hw(z, r)

(4.7) and from(4.4)and(4.2)weinferthat

|Hw(0, r)− Hw(z, r)| =  Hw(0, r) +  Hw(z, r) |Hw(0, r)−  Hw(z, r)| ≤ C rn+2 Iu(0,02 + )−1|z|D w(0, r +|z|) − Dw(0, r− |z|) 1 2 ≤ C |z| rn+2 Iu(0,0+ )−1 2 (r +|z|)n+2 α−2− (r − |z|)n+2 α−2 1 2 ≤ C |z|3 2rn+2 α−2. (4.8)

This inturn implies

Hw(0, r)

Hw(z, r) → 1 as r ↑ +∞

and from(4.6)

lim

i.e. by Almgren’s monotonicity estimate (4.3) we infer that Iw(z,r) = Iw(z,0+) for

all r > 0. As a consequence (cp. [9, Corollary 3.16]) w is α-homogeneous at z which straightforwardlyimpliesthatIw(·,0+) is0-homogeneous atz. 2

Wecanthendefine thespine ofahomogeneousQ-valuedfunctionw∈ HΛ0:

Sw:={x ∈ Rn : Iw(x, 0+) = Iw(0, 0+)}.

Bytheproof ofLemma 4.1 it followsthatw is α-homogeneousat everypoint x∈ Sw.

SimilarlyitissimpletoverifythatSwisthelargestvectorspacewhichleavesw invariant,

aswell asIw(·,0+):

Sw=



z∈ Rn : w(y) = w(z + y) ∀ y ∈ Rn. (4.9) Indeeditisenoughtoprovethateveryz∈ Swleavesw invariant(theotherinclusionis

obvious).To showthis, notethatbytheα-homogeneity ofw atz and 0 itfollowsthat foreveryy∈ Rn w(y) = w (z + y− z) = 2αw  z +y− z 2  = 2αw  y + z 2  = w (z + y) .

WedenotebyCk fork∈ {0,. . . ,n} theset ofk-invarianthomogeneousQ-functions

Ck :={w ∈ HΛ0 : dim(Sw)≥ k}. (4.10)

Note thatCn =Cn−1 ={QJ0K},i.e. these sets are singleton consisting of theconstant

functionw≡ QJ0K.ForCnthisisfollowsstraightforwardlyfromthedefinitionand(4.9).

While for Cn−1 onecan argue via the cylindrical blowup in[9, Lemma 3.24]. Assume withoutlossofgeneralitythat

w∈ Cn−1, w≡ Q J0K and Sw=Rn−1× {0}.

Thenbytheinvarianceofw alongSwitfollowsthatw isafunctionofonevariable.By [9,Lemma 3.24]itfollowsthatw :˜ R→ AQ(Rm) islocallyDir-minimizingand

˜

w≡ Q J0K , η ◦ ˜w≡ 0.

ThisisclearlyacontradictionbecausetheonlyDir-minimizingfunctionofonevariable arenon-intersectinglinearfunctions(cp.[9,3.6.2]).

Finally, asimple consequence of (4.9)is that{w|B1 : w ∈ Ck} is aclosed subset of

Lemma 4.2. Let (wj)j∈N ⊂ Ck and w ∈ Wloc1,2(Ω,AQ(Rm)) be such that wj → w in

L2_loc(Rn,AQ(Rm)). Thenw∈ Ck.

Proof. Let αj be the homogeneity exponent of wj. Since for Dir-minimizing

α-homo-geneous Q-valuedfunctionsw it holdsthatDw(1)= α Hw(1),wededucefrom αj≤ Λ0

andwj→ w thatthefunctionswjhaveequi-boundedenergiesinanycompactsetofRn.

Therefore by the compactness inSection4.2 itfollows thatwj → w locally uniformly

and w∈ HΛ0.

Foreveryj∈ N letnowVjbeak-dimensionallinearsubspaceofRncontainedinSwj.

By the compactness of theGrassmannian Gr(k,n), we can assumethat up to passing toasubsequence(notrelabeled)Vj convergestoak-dimensionalsubspaceV .Usingthe

uniform convergenceofwj tow wethen concludethatforeveryz∈ V and y∈ Rn

w(z + y) = lim

j wj(zj+ y) = limj wj(y) = w(y)

wherezj ∈ Vj isanysequencesuchthatzj→ z.ThisshowsthatV ⊂ Sw,thusimplying

thatdim(Sw)≥ k. 2

4.4. Blowups

Letu beaDir-minimizingQ-valuedfunction, η◦u≡ 0 andu≡ QJ0K.Fixanyr0> 0.

For everyy ∈ ΔQ∩ Ωr0, i.e. foreveryy such thatu(y)= QJ0K and dist(y,∂Ω)≥ 2r0,

we definetherescaledfunctionsofu aty as

uy,s(x) :=

sm−22 u(y + sx)

D1/2u (y, s)

∀ 0 < s < r0, ∀ x ∈ Br0 s (0).

From[9,Theorem 3.20]wededucethatforeverysk↓ 0 thereexistsasubsequencesk↓ 0

suchthatuy,s_k convergeslocallyuniformly inRn to afunctionw :Rn→ AQ(Rm) such

thatw∈ HΛ0 with Λ0= Λ0(r0) := r0 ´ Ω|Du| 2 min_x∈Ωr0H_u(x, r₀). (4.11)

Note thatminx∈Ωr0Hu(x,r0)> 0. Indeed,bythe continuityof x→ Hu(x,r0) and the

closureofΩr0_{,theminimumisachievedandcannotbe0 becauseoftheconditionu}≡ 0.

Inparticular, Λ0∈ R.

5. StratificationforDir-minimizing Q-valued functions

In this section we apply Theorems 2.2, 2.3 and 2.4 to the case of Almgren’s Dir-minimizing Q-valued functions. Keeping the notation Ωs and U as in Section 2.1, we set

(1) Θs: Ωs→ [0,+∞) givenby

Θ0(x) := lim

r↓0+Iu(x, r) and Θs(x) := Iu(x, s) for s > 0, x∈ Ω

s_,

(2) foreveryk∈ {0,. . . ,n},dk : U → [0,+∞) isgivenby

dk(x, s) := min



G(ux,s, w)L2_(∂B1): w∈ C_k

 . Note thatsince{w|B1 : w ∈ Ck} is aclosed subset of L

2_(B

1) the minimum inthe

definitionofdk isachieved.

It follows from Almgren’s monotonicityformula (4.3)thatconditions (a)and (b) of Section2.1 aresatisfied.

WeverifynextthatthestructuralhypothesesinSection2.2arefulfilled.Forsimplicity wewritethecorrespondingstatementsforfixedr0.ThecorrespondingΛ0> 0 isdefined

asin(4.11)above.Therefore,thesetsHΛ0 andCk,introducedrespectivelyinSection4.3

and(4.10), aredefinedintermsofΛ0= Λ0(r0).

Lemma 5.1. For every ε1 > 0 there exist 0 < λ1(ε1),η1(ε1) < 14 such that, for all

(x,s)∈ U withx∈ Ωr0 _{ands< r}

0,itholds

Iu(x, s)− Iu(x, λ1s)≤ η1 =⇒ ∃ w ∈ C0 : G(ux,s, w)L2_(∂B 1)≤ ε1.

Proof. Weargue bycontradictionand assumethere exist points(xj,sj) withxj ∈ Ωr0

andsj < r0 suchthat

Iu(xj, sj)− Iu(xj,s₂jj)≤ 2−j and G(uxj,sj, w)L2(∂B1)≥ ε1 ∀ w ∈ C0

orequivalently,setting uj := uxj,sj,

Iuj(0, 1)− Iuj(0, 2−j)≤ 2−j and G(uj, w)L2(∂B1)≥ ε1 ∀ w ∈ C0. (5.1)

From[9,Corollary 3.18]itfollowsthat sup j Duj(0, 2)≤ 2 n−2+2 Iuj(0,2)Iuj(0, 2) Iuj(0, 1) ≤ C (5.2)

where C = C(Λ0) because Iuj(0,2) ≤ Λ0 by definition of Λ0. We can then use the

compactnessforDir-minimizingfunctionsinSection4.3 toinfertheexistenceofa Dir-minimizingw suchthat(upto subsequences)uj → w locallystronglyinW1,2(B2) and

uniformly.Wethencanpassinto thelimitin(4.3)andusing(5.1)weobtain

1 ˆ 0 t Hw(t) ⎛ ⎜ ⎝ ˆ ∂Bt |∂νw|2 ˆ ∂Bt |w|2₋ ⎛ ⎝ ˆ ∂Bt ∂νw, w ⎞ ⎠ 2⎞ ⎟ ⎠ dt = 0.

This impliesthatw isα-homogeneous (cp.[9,Corollary 3.16])with α = lim

j Iuj(0, 1)≤ Λ0

because of Section4.2. This contradicts G(uj,w)L2_(∂B1) ≥ ε₁ for all w∈ C₀ in (5.1)

and provesthelemma. 2

Remark 5.2.Using theregularity theory of Dir-minimizingfunctionsproven in[9]itis infactpossibletoproveastrongerclaimthenLemma 5.1,namelythatforeveryε1> 0

there exists0< η1(ε1)<14suchthatforall(x,s)∈ U withx∈ Ωr0 and s< r0

Iu(x, s)− Iu(x,s2)≤ η1 =⇒ ∃ w ∈ C0 : G(ux,s, w)L2_(∂B

1)≤ ε1. (5.3)

Since (5.3)is notneededinthesequel,we leavethedetailsoftheproofto thereader. For what concerns (ii) we argue similarly using a rigidityproperty of homogeneous Dir-minimizingfunctions.

Lemma 5.3. Forevery0< ε2,τ < 1 there exists0< η2(ε2,τ )≤ ε2 suchthat if (x,5s)∈

U , with x ∈ Ωr0 _{and 5s < r}

0, dk(x,4s) ≤ η2 and dk+1(x,4s) ≥ ε2 for some k ∈

{0,. . . ,n− 1} then thereexistsak-dimensional affinespaceV such that d0(y, 4s) > η2 ∀ y ∈ Bs(x)\ Tτ s(V ).

Proof. We prove the statement for V = Sw with w ∈ Ck such that dk(x,4s) =

G(u,w)L2_(∂B

4s(x)). We argue by contradiction. Reasoning as above with the

rescal-ings ofu (eventually composingwith arotationofthedomain to achieve(4) belowfor afixedspaceV ),wefind asequenceoffunctionsuj∈ W1,2(B5,AQ(Rk) suchthat

(1) sup_jDuj(0,5)< +∞;

(2) there existswj∈ Ck suchthatG(uj,wj)L∞(B4)↓ 0;

(3) G(uj,w)L2_(B

4)≥ ε2 foreveryw∈ Ck+1;

(4) thereexistsyj ∈ B1\Tτ(V ) suchthatd0(yj,4)↓ 0 andV = Swj isthek-dimensional

spineofwj(notethatby(2)&(3)thedimensionofthespineofwj cannotbehigher

thank).

Possiblypassingto subsequences(as usualnotrelabeled),wecanassumethatuj → w,

wj → w locallyinL2(Rn,AQ(Rm)) andyj → y for somew∈ Wloc1,2(Rn,AQ(Rm)) and

y ∈ ¯B1\ Tτ(V ). By Lemma 4.2 we deduce that w ∈ Ck with Sw ⊃ V ; since by (3)

w /∈ Ck+1, weconcludeSw= V .

Itfollowsfrom(4)thatwy,s= wy,1foreverys∈ (0,1].Indeedthereexistzj ∈ C0such

w(y) = 0 and by the upper semi-continuity of x → Iw(x,0+) we deduce also that

Iw(y,0+)= Iw(0,0+),i.e. y∈ Sw whichisthedesiredcontradiction. 2

WecantheninferthatTheorem 2.2 holdsforQ-valuedfunctions.

Theorem5.4.Letu: Ω→ AQ(Rm) beanontrivialDir-minimizingfunctionwithaverage

η◦ u≡ 0.

Forevery0< κ0,δ < 1 andr0> 0,there existsC = C(κ0,δ,r0,n)> 0 such that

|Tr(ΔQ∩ Sr,rk 0,δ)| ≤ C r

n−k−κ0 ∀ k ∈ {1, . . . , n − 1},

and S_r0_0,δ is countable.

Inparticular,Theorem 2.3appliesandweconcludethatdim_H(Sk

r0)≤ k andthatS 0

r0

isatmostcountable.WeshallimproveuponthelatterestimateonthestratumS_rn−1₀ in thenextsection.

5.1. Minkowskidimension

Wecanactually giveanestimateontheMinkowskidimensionofthesetof maximal multiplicity pointsΔQ bymeans ofTheorem 2.4.Anε-regularity resultis thekeytool

toprovethis.

Proposition5.5. Thereexistsaconstant δ0= δ0(r0)> 0 such that

Sn−1

r =Srn−2=Sr,δn−20 ∀ r ∈ (0, r0). (5.4)

Proof. The first equality is an easy consequence of Cn = Cn−1 = {QJ0K} that gives

dn≡ dn−1.

Setδ0:= (Λ0+ 1)−1/2, weshow thatSr,δn−2 ⊂ S n−2

r,δ0 foreveryδ∈ (0,δ0). Assumeby

contradictionthatthereexistsx∈ S_r,δn−2\S_r,δn−2

0 forsomeδ asabove.FromCn−1={QJ0K}

wededucetheexistenceofs∈ (0,r) suchthat

0 < δ≤ ux,sL2_(∂B1)< δ₀.

Inparticular,thecondition´_B

1|Dux,s| 2_{= 1 gives} Iux,s(0, 1) = ´ B1|Dux,s| 2 ´ ∂B1|ux,s| 2 ≥ 1 δ2 0 > Λ0.

ByrecallingthatIu(x,s)= Iux,s(0,1),thedesiredcontradictionfollowsfromAlmgren’s

monotonicityformula(4.3)andtheverydefinitionofΛ0in(4.11). 2

Proof of Theorem 1.1. It is adirect consequenceof Proposition 5.5 and Theorem 2.4. Given u : Ω → AQ(Rm) a nontrivial Dir-minimizing function (i.e. ΔQ = Ω), we can

consider thefunction

v(x) :=

i

Jui(x)− η ◦ u(x)K .

Then by [9, Lemma 3.23] v is Dir-minimizing with η◦ v ≡ 0. Moreover, the set of Q-multiplicity pointsofu inΩr0 _{correspondstotheset}Sn−2

r0 forthefunctionv andthe

conclusionfollowsstraightforwardly. 2 5.2. Almgren’sstratification

In this sectionwe show thatTheorem 3.4 appliesinthe caseof Q-valued functions, as well.Inparticular, this impliesthatthesingularstrata forDir-minimizing Q-valued functions canalso be characterized by the spines of theblowup maps, thus leadingto theproof ofTheorem 1.2intheintroduction.

ByfollowingthenotationinSection3.1.1(1),weset f (x) := Iu(x, 0+) ∀ x ∈ Ω.

For everyx∈ Ω such thatf (x)= 0 (or, equivalently, u(x)= QJ0K) we define G(x) to be thesingletonmadeoftheconstantfunction0,i.e.G(x)={QJ0K};otherwise

G(x) :=Iw(·, 0+) : w∈ Wloc1,2(Rn,AQ(Rm)) blowup of u at x



. (5.5) As explained in Section 4.3 G(x) is never empty because there always exist (possibly non-unique)blowupofu atany multiplicityQ point.

Sinceeveryblowupofu isanontrivialhomogeneousDir-minimizingfunction,itfollows from Lemma 4.1 thatevery functiong ∈G(x) is conicalin thesense of Definition 3.3

(1). Weneedthentoshowthefollowing.

Lemma5.6.Foreveryx∈ Ω theclassG(x) iscompactinthesense ofDefinition 3.3(2). Proof. If x is not a multiplicity Q point, then there is nothing to prove. Otherwise consider a sequence of maps gj = Iwj(·,0

+_{) ∈ G(x), with w}

j blowup of u at x. By

Section 4.3 wj is Dir-minimizing α-homogeneous with α = Iu(x,0+) and Dwj(1) = 1.

Then bythecompactnessinSection4.2,there existsw such thatwj → w locallyinL2

up to subsequences(notrelabeled)with Dw(1)= 1. By asimplediagonal argumentit

follows thatw isaswell ablowupofu atx,i.e. g = Iw(·,0+)∈ G(x).Foreveryyj∈ B1

lim sup j↑+∞ gj(yj)≤ lim supj↑+∞ Iwj(yj, s) = lim sup j↑+∞  s Dwj(y, s) Hwj(y, s) Dwj(yj, s) Dwj(y, s) Hwj(y, s) Hwj(yj, s)  = Iw(y, s) whereweused

- themonotonicityofIwj(yj,·) inthefirstline,

- thecontinuityofx→ Dwj(x,s) andx→ Hwj(x,s),

- theconvergenceofthefrequencyfunctionsIwj(y,s)→ Iw(y,s) (cp.4.2).

Sendings to0 providestheconclusion. 2

Finally we prove that the Structural Hypothesis (ii) of White’s theorem (cp. Sec-tion3.1)holdsaswell:

lim sup j↑+∞ f (x + rijyj) = lim supj↑+∞ Iu(x + rijyj, 0 +₎ ≤ lim sup j↑+∞ Iu(x + rijyj, rijs) = lim sup j↑+∞ Iux,rij(yj, s) = Iw(y, s)

whereweused thestrongconvergenceof thefrequencyofSection4.2.

Inparticular,Theorem 3.4holdstrue,whichinturnleadstotheproofofTheorem 1.2

byasimpleinductionargument. 5.3. Stratification Theorem 1.2

WedefinenowthesingularstrataSingk_uforaDir-minimizingmultiplevaluedfunction u: Ω→ AQ(Rm).Consideranypointx0∈ Singu, andlet

u(x0) =

J



i=1

κi JpiK

with κi ∈ N\ {0} such that Ji=1κi = Q andpi = pj fori = j. Thenby theuniform

continuity of u there exist r > 0 and Dir-minimizing multiple valued functions ui :

Br(x0)→ Aκi(R

m_{) fori∈ {1,. . . ,J} suchthat}

u|Br(x0)=

J



i=1

wherebyalittleabuseofnotationthelastequalityismeantinthesenseu(x)=_iui(x)

as measures.Foreveryi∈ {1,. . . ,J} letvi: Br(x0)→ Aκi(R

m_{) be givenby} vi(x) := κi  l=1 J(ui(x))l− η ◦ ui(x)K .

Then we say thatapoint x0 ∈ Singu belongs to Singku, k ∈ {0,. . . ,n},if the spine of

everyblowupofvi atx0,foreveryi∈ {1,. . . ,J},isat mostk-dimensional.

We can then proveTheorem 1.2 bya simpleinduction argument on the numberof valuesQ.

ProofofTheorem 1.2. ClearlyifQ= 1 thereisnothingtoprovebecauseeveryharmonic function is regular and Sing_u =∅.Now assume we haveproven the theorem for every Q∗< Q andweproveitforQ.

We can assume without loss of generality that ΔQ = Ω. Then, as noticed, ΔQ =

Sing_u∩ΔQ by [9,Theorem 0.11]. MoreoverSingku∩ΔQ = Σk, whereΣk isthatof The-orem 3.4. Indeed x0 ∈ Σk if and only if the maximal dimension of the spine of any

g ∈ G(x0) isat mostk. By(5.5)g ∈ G(x0) ifandonly ifg = Iw(·,0+) for someblowup

w of u atx0.Hence by(4.9) x0 ∈ Σk if and onlyif thedimension ofthe spines of the

blowupsofu atx0isatmostk.NotethatSingun−2∩ΔQ= ΔQsinceCn=Cn−1 ={QJ0K}

(weuseherethenotationinSection4.3)andu isnottrivial.Thereforewededucethat Sing0_u∩ΔQ is countable

dim_H(Singk_u∩ΔQ)≤ k ∀ k ∈ {1, . . . , n − 2}.

Nextwe considertherelatively open setΩ\ ΔQ (recallthatboth Singu andΔQ are

relativelyclosedsets).Thankstothecontinuityofu wecanfindacoverofΩ\(Sing_u∩ΔQ)

madeofcountablymanyopenballsBi⊂ Ω\ (Singu∩ΔQ) suchthatu|Bi =

q u1 i y +qu2 i y with u1

i and u2i Dir-minimizing multiple valued functions taking strictly less than Q

values. Since Singk_u∩Bi = Singku1 i∪Sing

k u2

i by the very definition, using the inductive

hypotheses foru1_i andu2_i wededucethat

Sing0_u∩Bi is countable

Singn_u−2∩Bi= Singu∩Bi

dim_H(Singk_u∩Bi)≤ k ∀ k ∈ {1, . . . , n − 2},

thus leadingto (1.2)and (1.3). 2

6. Applicationstogeneralizedsubmanifolds

InthepresentsectionweapplytheabstractstratificationresultsinSection2to inte-gral varifoldswithmeancurvatureinL∞andto almostminimizersincodimensionone

(bothframeworksarenotcoveredbytheresultsin[7]althoughtheycanbeconsidered asslightvariantsofthose).Thiscaseisrelevantinseveralvariationalproblems(seethe examplesin[22,§ 4])mostremarkablythecaseofstationary varifoldsorarea minimiz-ing currentsina Riemannianmanifold. Fora morecomplete account on thetheory of varifoldsandalmost minimizingcurrentswereferto[1,3] andthelecturenotes[18]. 6.1. Tubular neighborhoodestimate

In what follows we consider integer rectifiable varifolds V = (Γ,f ), where Γ is an m-dimensionalrectifiableset inthe boundedopensubset Ω⊂ Rn,and f : Γ→ N\ {0} is locally Hm_{-integrable. Weassume thatV hasbounded generalized mean curvature,}

i.e. there exists a vector field H_V : Ω → Rn _{such that H}

VL∞(Ω,Rn₎ ≤ H₀ for some H0> 0 and ˆ Γ divTyΓX dμV =− ˆ X· H_V dμ_V ∀ X ∈ C_c1(Ω,Rn)

whereμ_V := fHm _{Γ.Itisthenwell-known(cp.,forexample,[18,Theorem 17.6])that}

thequantity

Θ_V(x, ρ) := eH0ρμV(Bρ(x))

ωmρm

ismonotoneandthefollowinginequalityholdsforall0< σ < ρ< dist(x,∂Ω)

Θ_V(x, ρ)− Θ_V(x, σ)≥ ˆ

Bρ\Bσ(x)

|(y − x)⊥_|2

|y − x|m+2dμV(y) (6.1)

where (y− x)⊥ is the orthogonal projection of y− x on the orthogonal complement (TyΓ)⊥. In particular the family (Θ(·,s))s∈[0,r0] (with the obvious extended notation

Θ(·,0+_{):= lim}

r↓0Θ(·,r)) satisfies assumption (a) inSection2.1 forevery fixed r0 > 0

with

Λ0(r0) := eH0diam(Ω)

μ_V(Ω) ωmrm0

. (6.2)

Inordertointroducethecontrolfunctionsdk we recallnextthedefinitionofcone.

Definition 6.1. An integer rectifiable m-varifold C = (R,g) in Rn _{is a cone if the}

m-dimensionalrectifiablesetR isinvariantunderdilationsi.e. λ y∈ R ∀ y ∈ R, ∀ λ > 0

and g is0-homogeneous, i.e.

g(λ y) = g(y) ∀ y ∈ R, ∀λ > 0.

Anintegerrectifiablem-varifoldC = (R,g) inBρ,ρ> 0 isaconeifitistherestriction

to Bρ ofaconeinRn.

The spine of a cone C = (R,g) in Rn _{is the biggest subspace V ⊂ R}n _{such that}

R = R× V uptoHm_-nullsets.

The class of cones whose spine is at least k-dimensional is denoted by Ck and its

elements arecalledk-conical.

If d_∗ is adistance inducing the weak ∗ topology of varifolds with bounded mass in B1(cp.,forinstance,[17,Theorem 3.16]forthegeneralcaseofdualspaces),thecontrol

functiondk isthendefinedas

dk(x, s) := inf  d_∗Vx,s,C : C ∈ Ck, HCL∞(Ω,Rn₎≤ H₀  (6.3) where Vx,s:= (ηx,s(Γ),f◦ ηx,s−1) withηx,s(y):=(y− x)s.

By very definition, then (b) in Section 2.1 is satisfied. We are now ready to check that the conditions in the structural hypotheses are satisfied. As usual, we write the corresponding statementsforfixedr0andΛ0:= Λ0(r0),forsimplicity.

Lemma 6.2. For every ε1 > 0 there exist 0 < λ1(ε1),η1(ε1) < 14 such that for all

(x,ρ)∈ U,with x∈ Ωr0 _{andρ< r} 0,

Θ_V(x, ρ)− Θ_V(x, λ1ρ)≤ η1 =⇒ d0(x, ρ)≤ ε1.

Proof. Assume by contradiction that for some ε1 > 0 there exists (xj,ρj) ∈ U, with

xj ∈ Ωr0 andρj< r0, suchthat

Θ_V(xj, ρj)− ΘV(xj, j−1ρj)≤ j−1 and d0(xj, ρj)≥ ε1. (6.4)

Weconsiderthesequence(Vj)j∈NwithVj:=Vxj,ρj,andnotethatforallpositivet > 0

there isanindex¯j suchthattρj < r0 ifj≥ ¯j,sothat

μVj(Bt)≤ ωmt

m_Θ

Vj(xj, t ρj)≤ ωmt

m_Λ

0 ∀ j ≥ ¯j.

Therefore, upto theextractionof subsequencesandadiagonalargument, Allard’s rec-tifiabilitycriterion(cp.,forinstance,[18, Theorem 42.7,Remark42.8])yieldsalimiting m-dimensionalintegervarifoldVj→C = (R,g) withtheboundHCL∞(Ω ≤ H0.Since

Θ_V(xj,sρj)= ΘVj(0,s)→ ΘC(0,s) exceptat mostforcountablevaluesofs,by

mono-tonicityand(6.4)forallj−1< r < s< 1 wehaveΘ_C(0,s)= Θ_C(0,0+_{) foreverys≥ 0.}

The monotonicity formula (6.1) applied to C implies that C is actually a cone, thus contradicting d0(xj,ρj)≤ ε1. 2