on singularities of energy minimizing maps · the theory of regularity of energy minimizing maps is...

Master's thesis

On singularities of energy minimizing maps

Marc Pegon

supervised byVincent Millot

December 19, 2016

CONTENTS CONTENTS

Contents

1 Introduction 2

1.1 Energy minimizing maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Regular and singular sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Minimal surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Variations 4

2.1 Outer variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2 Inner variations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 The monotonicity formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 The ε-regularity theorem 103.1 Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 A lemma of Luckhaus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.3 A reverse Poincaré inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.4 A technical regularity lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.5 Proof of the ε-regularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4 Partial regularity 34

4.1 Compactness theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.2 The density function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Tangent maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.3.1 De�nition and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.3.2 Properties of homogeneous minimizing maps of degree zero . . . . . . . . . . . . 41

4.4 Strati�cation of the singular set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Case of study: maps into S3 47

6 Conclusion 51

A Analytic preliminaries 52

A.1 Hausdor� measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.2 Hölder continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52A.3 Markov's inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.4 Poincaré inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53A.5 Absolute continuity of H1 functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.6 Elliptic regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54A.7 An abstract lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

B Harmonic functions and harmonic maps 55

B.1 Harmonic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55B.2 Harmonic maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

1

1 INTRODUCTION

1 Introduction

The theory of regularity of energy minimizing maps is a vast subject with a wide variety of applicationsin mathematical physics and geometry. As we will see, energy minimizing maps are in particular weaklyharmonic maps, but with added constraints that allow for �ner regularity properties. Even more sothan weakly harmonic maps, we will see that energy minimizing maps are linked with minimal surfaces,which is not surprising as both can be seen as generalizations of geodesics in larger dimensions. Letus give some generalities on energy minimizing maps, regularity, and minimal surfaces before statingthe purpose of this thesis.

1.1 Energy minimizing maps

In the whole thesis we will conduct our study in �nite dimension real vector spaces, we do not considerRiemannian manifolds of even general manifolds but only submanifolds of some Rd, which is enoughbecause the main goal of this thesis is to give partial regularity results for energy minimizing mapsinto a sphere. Di�erential geometry prerequisites are kept to a minimal.

Let us de�ne energy minimizing maps.

De�nition 1.1 (Energy minimizing maps). Let u ∈ H1loc(Ω,N ) where Ω is an open subset of Rn andN is a compact smooth submanifold of Rp. Then we say that u is energy minimizing if for any ballBr(x0) such that Br(x0) ⊂ Ω, and any v ∈ H1(Br(x0),N ) that agrees with u in a neighborhood or∂Br(x0), we have

ˆBr(x0)

|∇u|2 dx ≤ˆBr(x0)

|∇v|2 dx,

i.e. u is a minimizer of the Dirichlet energy on each ball in Ω among all maps that agree with it onthe boundary.

Remarks 1.1.

(i) Note that since energy minimizing maps are minimizers of the Dirichlet energy, they are ofcourse critical point of this energy with respect to outer variations (see Section 2.1), whichmeans exactly that they are weakly harmonic maps. See Appendix B.2 for basic de�nitions ofsmooth harmonic/weakly harmonic maps and some more references.

(ii) Notice that if n = 1, a critical point of the Dirichlet is exactly a geodesic. This means that inthat case harmonic maps, and thus energy minimizing maps, are geodesics.

1.2 Regular and singular sets

Let us de�ne properly the regular and singular sets of a map.

De�nition 1.2. If u ∈ H1loc(Ω,N ), where Ω is an open subset of Rn and N is a smooth compactsubmanifold of Rp, then

reg u := {x ∈ Ω : u is smooth in a neighborhood of x}

is the regular set of u, and

sing u := Ω \ reg u

is the singular set of u.

2

1.3 Minimal surfaces 1 INTRODUCTION

Remarks 1.2.

(i) Note that by de�nition reg u is an open set, while sing u is a relatively closed set of Ω.

(ii) We will see later that if u is an energy minimizing maps, then u is smooth in a neighborhood ofx if and only if it is continuous at x.

As we have seen, if n = 1, then energy minimizing maps are geodesics, thus their regularity followsby solving the geodesic equation. In the case where n = 2, it was shown by Morrey and Charles [8] thatenergy minimizing maps are smooth as well. Beginning from dimension 3, there exist energy minimizingmaps which have singularities. Schoen and Uhlenbeck [12] have shown that the map u(x) = x|x| from

Rn → Sn is energy minimizing if and only if n ≥ 7, which gives an example of non regular minimizingmaps. Note that in the case n = 3, this map is not energy minimizing, although it is weakly harmonic,which shows that the two notions are indeed distinct. Schoen and Uhlenbeck [11] have also establishedthat if u :M→N is an energy minimizing map between smooth Riemannian manifolds of dimensionn and k, then dimH(sing u) ≤ n − 3. In this thesis, we give an almost self-contained proof of thisassertion in the case where M is an open subset of Rn and N is a smooth compact submanifold ofRp. We also introduce the required theoretical tools with full proofs, most of which have already beencompiled by Simon [13].

1.3 Minimal surfaces

Some notions in this paragraph come from di�erential geometry and are used here without beingproperly introduced. Their full understanding is not necessary to appreciate the link with harmonicmaps and energy minimizing maps.

Let us now state equivalent de�nitions of minimal surfaces to see how they are tied to harmonicand energy minimizing maps. We do not prove the equivalence of those de�nitions.

De�nition 1.3 (Minimal surfaces). We call surface a smooth 2-dimensional submanifold of some Rn.A surface S is said to be minimal if it satis�es one of the two equivalent properties:

1. its mean curvature is equal to zero at all points

2. it is a critical point of the area functional for all compactly supported variations.

Remark 1.3. While the �rst de�nition gives a �geometric� point of view of minimal surfaces, the secondone ties it to calculus of variations.

Now as here we deal with maps rather than curves and surfaces, we would like to express thisminimality condition in terms of maps.

De�nition 1.4 (Immersions). Let M and N be submanifolds of some Rd of respective dimensionsm and n, with m ≤ n. We say that a smooth map u : M → N is an immersion if its di�erential isinjective at each point.

Remark 1.4. Note that if u is an immersion and m = 2, u(M) is not necessarily a surface as de�nedabove, e.g. if u is not injective. But it is locally a surface, in the sense that for every x ∈ M thereexists a neighborhood N of x such that u(N) is a surface. Thus we can de�ne the mean curvature ofthe map u at x as the curvature of u(N) at u(x), where N is such a neighborhood. We can also saythat u(M) is locally a surface around points y where the preimage is a singleton.

De�nition 1.5. Let M and N be submanifolds of some Rd of respective dimensions 2 and n, with3 ≤ n. We say that a smooth conformal map u :M→ N , i.e. a map that preserves angles locally, is

3

2 VARIATIONS

a (branched) minimal immersion if its di�erential is injective and its mean curvature is equal to zeroeverywhere. In particular u(M) has mean curvature zero everywhere except at isolated points calledbranch points.

The image of a minimal immersion is sometimes called a branched minimal surface, although it isnot a surface as de�ned above. Essentially a branched minimal surface is smooth except in the setof branch points, and is a �classical� minimal surface (its mean curvature vanishes identically) in theneighborhood of any regular point.

We can now establish a link between branched minimal surfaces and smooth harmonic maps whenthe domain and target manifolds are speci�c [4, 3, 6].

Proposition 1.1. Let u :M→N be a smooth conformal map, whereM and N are smooth compactsubmanifolds of respective dimensions 2 and n ≥ 3. Then u is harmonic if and only if it is a minimalimmersion.

In particular we have the following result for harmonic maps from the euclidian 2− sphere [10, 4].

Proposition 1.2. Any smooth harmonic map from S2 into a smooth submanifold N of dimensionn ≥ 3 is conformal. In particular, it is a minimal immersion.

Recall that a minimal surface is a critical point of the area functional. In our study we ratherconsider the energy functional. Those two functionals are actually tightly linked. In the case whereu : Ω ⊂ R→ N , we know that critical points of the energy functionals are the same as critical pointsof the �area� functional (length), i.e. geodesics. For conformal maps between submanifolds, the areaand the energy functions are the same up to a constant factor [6].

Proposition 1.3. Let u :M→N be a smooth conformal map, whereM and N are smooth compactsubmanifolds of Rd of respective dimensions 2 and n ≥ 3. Let A be the area functional

A(u) =ˆM|det∇u| dx

and E the Dirichlet energy functional

E(u) =ˆM|∇u|2 dx,

then

2A(u) = E(u).

In view of the previous de�nitions and properties, we see how minimal surfaces are closely tied toharmonic maps and energy minimizing maps, especially in the case of maps from S2. In our case ofstudy � minimizing maps from some Ω ∈ Rn into the sphere S3 � we will see that smooth harmonicmaps from S2 → S3 appear naturally using tangent maps, and that the existence of singularities canbe reduced to the study of minimal immersions into S3.

2 Variations

In this section Ω will always denote an open subset of Rn. Let us consider a minimizing map u ∈H1loc(Ω,Sd−1), and R, x0 such that BR(x0) ∈ Ω. Then u is a critical point of the Dirichlet energy E(u) =´BR(x0)

|∇u|2 dx for any variation that agrees with u in a neighborhood of ∂BR(x0). Considering two

4

2.1 Outer variations 2 VARIATIONS

types of variations one can deduce nice properties of minimizing maps. From �outer� variations we getthat minimizing maps are actually weakly harmonic � we will derive the equation in our particular casewhere the target manifold is the sphere �, and from �inner� (or �domain�) variations we can deducethe so-called monotonicity formula for minimizing maps, which is a key tool for the study or theirregularity.

2.1 Outer variations

Using outer variations, we are going to establish the following proposition:

Proposition 2.1. An energy minimizing map u ∈ H1loc(Ω,Sd−1) is weakly harmonic and satis�es

−∆u = |∇u|2u

in the sense of distributions.

Proof. Let us consider variations of the form ut =u+tϕ|u+tϕ| , where ϕ ∈ C

∞c (BR(x0),Rd). Since ϕ is

smooth, and |u(x)| = 1, a.e., it is clear that |u+ tϕ| > 0 for t su�ciently small, a.e. in BR(x0). It isalso trivial that |ut(x)| = 1, u0 = u, and ut agrees with u in a neighborhood of ∂BR(x0). We still needto check that ∇ut belongs to L2(BR(x0),Rd×n), so that ut belongs to H1(BR(x0),Sd−1). We have, inthe weak sense,

∇ut =∇(u+ tϕ)|u+ tϕ|

− (u+ tϕ)∇(u+ tϕ)>(u+ tϕ)

|u+ tϕ|3

thus

|∇ut| ≤ 2|∇(u+ tϕ)||u+ tϕ|

. (2.1)

Since ϕ is smooth and has compact support, for t su�ciently small, |u+ tϕ| ≥ 12 , hence

|∇ut| ≤ 4|∇(u+ tϕ)|,

thus ut belongs to H1(BR(x0),Sd−1). By minimality of u, we have

E(u) ≤ E(ut),

for any t small enough. Then we will show that t 7→ E(ut) is di�erentiable at zero, which implies byminimality that

d

dt

∣∣∣∣t=0

E(ut) = 0.

This means exactly that u is weakly harmonic in BR(x0). A nice trick to make computations easier isto de�ne ψ(x) = ϕ(x)− 〈ϕ(x), u(x)〉u(x), and vt = u+tψ|u+tψ| . Now since ϕ has compact support, so doesψ, so vt agrees with u in a neighborhood of ∂BR(x0). We clearly have v0 = u, and vt still belongs toH1(BR(x0),Sd−1), even though ψ is not smooth. Using inequality (2.1) with vt, ψ instead of ut, ϕ,the result is clear provided that t is small enough. Variations with ψ are actually easier to computebecause we have

|u+ tψ|2 = |u|2 + t2|ψ|2 + 2t〈u, ψ〉

= 1 + t2|ψ|2.

5

2.2 Inner variations 2 VARIATIONS

For t small enough, we get

E(vt) = E(u) +ˆBR(x0)

〈∇u,∇ψ〉 dx+O(t2),

then by minimality of uˆBR(x0)

〈∇u,∇ψ〉 dx = 0.

This is true for any ψ = ϕ− 〈ϕ, u〉u, where ϕ ∈ C∞c (BR(x0),Rd). Now we want to express it in termsof ϕ instead of ψ. Let us compute ∇ψ. We have

∂jψ = ∂jϕ− (〈∂jϕ, u〉+ 〈ϕ, ∂ju〉)u− 〈ϕ, u〉∂ju, ∀j ∈ {1, . . . , n} ,

thus

〈∇u,∇ψ〉 = 〈∇u,∇ϕ〉 −n∑j=1

(〈∂jϕ, u〉+ 〈ϕ, ∂ju〉) 〈u, ∂ju〉 − 〈ϕ, u〉 |∇u|2.

Noticing that 〈u, ∂ju〉 =∑dk=1 uk∂juk =

12∂j(|u|

2) = 0, one gets

ˆBR(x0)

〈∇u,∇ϕ〉 − |∇u|2〈u, ϕ〉 dx = 0.

This being true for any ϕ ∈ C∞c (BR(x0),Rd), it means that in the weak sense, u satis�es

−∆u = |∇u|2u.

Remark 2.1. In the more general case where the target space is any smooth compact submanifoldN embedded in Rd, we can still make outer variations by projecting onto the manifold to get theharmonicity of energy minimizing map as well as a formula involving the second fundamental form ofN (see Appendix B.2).

2.2 Inner variations

Using domain variations we are again to establish the following proposition:

Proposition 2.2. An energy minimizing map u ∈ H1loc(Ω,Sd−1) satis�es, for all BR(x0) ∈ Ω,ˆBR(x0)

|∇u|2 div ~X − 2 Tr(∇u>∇u∇ ~X) dx = 0, ∀ ~X ∈ C∞c (BR(x0),Rn), (2.2)

or equivalently,

ˆBR(x0)

n∑i,j=1

(|∇u|2δij − 2〈∂iu, ∂ju〉

)∂iξ

j dx = 0, ∀(ξ1, . . . , ξn) ∈ C∞c (BR(x0),Rn). (2.3)

Remark 2.2. Setting T = (|∇u|2δij − 2〈∂iu, ∂ju〉)ij , this means div T = 0 in the sense of distributions.T is called the stress energy tensor.

6

2.2 Inner variations 2 VARIATIONS

Proof. Let us consider variations of the form ut = u ◦ φt where φt is a C1-di�eomorphism of BR(x0),such that φ0 = idBR(x0) and ut agrees with u in a neighborhood of ∂BR(x0) for t small enough. Inparticular, given x ∈ BR(x0), de�ne φ(·, x) the integral curve of the ordinary di�erential equation{

∂tφ(t, x) = ~X(φ(t, x))φ(0, x) = x

,

where ~X ∈ C∞c (BR(x0),Rd), and let φt(·) := φ(t, ·). Since ~X is smooth, the Cauchy-Lipschitz theoremgives us that φ(·, x) is smooth and uniquely de�ned for t in some ] − T−x , T+x [. In fact, since ~Xhas compact support, φ(·, x) is de�ned for all t ∈ R, and again the smoothness of ~X implies thesmoothness of φ(t, x) with respect to the initial condition x. Furthermore, the applications φt de�nea one-parameter group of smooth di�eomorphisms, i.e. they satisfy the following properties, which aretrivial to check:

1. φt : BR(x0)→ BR(x0) is a smooth di�eomorphism, ∀t ∈ R

2. φ0 = idBR(x0)

3. φs ◦ φd = φs+t, ∀t ∈ R

Trivially ut belongs to H1(BR(x0),Sd−1), clearly we have u0 = u ◦ φ0 = u, and ∀x ∈ BR(x0) \

supp ~X, φt(x) = x by the Cauchy-Lipschitz theorem, thus ut agrees with u in a neighborhood of∂BR(x0). By minimality, we then have

E(u) ≤ E(ut), ∀t ∈ R

Let us compute E(ut) =´BR(x0)

|∇ut|2 dx. We have

∇ut(x) = ∇u(φt(x)) ◦ ∇φt(x), a.e.,

and

|∇ut(x)|2 = Tr(∇φt(x)>∇u(φt(x))>∇u(φt(x))∇φt(x)>

).

By changing variables, we get

ˆBR(x0)

|∇ut|2 dx =ˆBR(x0)

Tr(∇φt(φ−t(x))>∇u(x)>∇u(x)∇φt(φ−t(x))

)|det∇φ−t(x)| dx,

and knowing that φt ◦ φ−t = φ0 = id, we have

∇φt ◦ φ−t = ∇φ−1−t ,

thus

E(ut) =ˆBR(x0)

Tr(∇φ−t(x)−1>∇u(x)>∇u(x)∇φ−t(x)−1

)|∇φ−t(x)| dx. (∗)

Now let us compute ∇φ−t(x) = ∇x φ(−t, x), where ∇x is the di�erential of φ w.r.t. the space variablex. Since φ is smooth, we have according to Schwarz theorem

∂t∇xφ(−t, x) = ∇x ∂tφ(−t, x),

7

2.3 The monotonicity formula 2 VARIATIONS

and then using the fact that φ is the solution of an ODE, and the chain rule

∂t∇xφ(−t, x) = −∇x(~X(φ(−t, x))

)= −(∇ ~X)(φ(−t, x)) ◦ ∇x φ(−t, x)

= −(∇ ~X)(φ−t(x)) ◦ ∇φ−t(x).

Thus we have the Taylor expansion when t goes to zero

∇φ−t(x) = ∇φ0(x)− t(∇ ~X)(φ0(x)) ◦ ∇φ0(x) + o(t)

= id− t∇ ~X(x) + o(t),

from which we derive

∇φ−t(x)−1 = id+ t∇ ~X(x) + o(t)

|det ∇φ−t(x)| = 1− tTr(∇ ~X(x)) + o(t).

Plugging this into (∗) we get

E(ut) =ˆBR(x0)

|∇u|2 dx+ t

(ˆBR(x0)

Tr(∇ ~X>∇u>∇u+∇u>∇u∇ ~X)

− |∇u|2 Tr(∇ ~X) dx

)+ o(t)

= E(u) + t

(ˆBR(x0)

2 Tr(∇u>∇u∇ ~X)− |∇u|2 Tr(∇ ~X) dx

)+ o(t),

hence (2.2) by minimality, and (2.3) by setting ~X = (ξ1, . . . , ξn).

Remark 2.3. Notice that the target space does not play a particular role for domain variations. Wecould have chosen a minimizing map u ∈ H1loc(Ω,N ), where N is any smooth compact submanifoldembedded in Rd, and would have found the same stress energy tensor satisfying div T = 0.

2.3 The monotonicity formula

The aim of this section is to establish the following proposition

Proposition 2.3 (Monotonicity formula). Let u ∈ H1loc(Ω,N ) be an energy minimizing map, whereN is a smooth compact submanifold of Rd, and BR(x0) ⊂ Ω. Then for any 0 < ρ < τ < R, we have

ρ2−nˆBρ(x0)

|∇u|2 dx− τ2−nˆBτ (x0)

|∇u|2 dx = 2ˆBτ (x0)\Bρ(x0)

r2−n∣∣∣∣∂u∂r

∣∣∣∣2 dσ,where r = |x− x0|, and ∂u∂r := ∇u

(x−x0r

)is the derivative of u in the radial direction.

First we are going to state and prove the following standard integration lemma.

Lemma 2.4. If a ∈ L1(BR(x0),Rn) satis�es div a = 0 in D′(BR(x0)), then for almost all ρ ∈ (0, R),ˆBρ(x0)

〈a,∇ξ〉 dx =ˆ∂Bρ(x0)

〈ξa, ~n〉 dσ, ∀ξ ∈ C∞(Bρ(x0)),

where ~n is the outward pointing unit normal vector of ∂Bρ(x0).

8

2.3 The monotonicity formula 2 VARIATIONS

Proof of the Lemma. Let us consider ηε a standard molli�er, and set aε = ηε ∗ a, so that aε ∈C∞(BR(x0),Rn), aε

L1loc−−−→ a, and still div aε = 0. By the Stokes formula we haveˆBρ(x0)

〈aε,∇ξ〉 dx =ˆ∂Bρ(x0)

〈ξaε, ~n〉 dσ. (2.4)

First recall that by Fubini's theorem, ρ 7→´∂Bρ(x0)

〈ξa, ~n〉 dσ is de�ned almost everywhere and inte-grable on (0, R). Since aε converges to a in L

1loc(BR(x0)), we have for any r ∈ (0, R)ˆ r

0

ˆ∂Bτ (x0)

|〈(aε − a)ξ, ~n〉| dσ dτ ≤ |ξ|0,BR(x0)ˆBρ(x0)

|aε − a| dxε→0−−−→ 0,

thus we can �nd εk → 0 such that for almost all ρ ∈ (0, r)ˆ∂Bρ(x0)

〈aεkξ, ~n〉 dσk→+∞−−−−−→

ˆ∂Bρ(x0)

〈aξ, ~n〉 dσ.

Plugging εk into (2.4) and passing to the limit, using the dominated convergence theorem on the leftterm, we get

ˆBρ(x0)

〈a,∇ξ〉 dx =ˆ∂Bρ(x0)

〈ξa, ~n〉 dσ,

for almost all ρ ∈ (0, r) and any ξ ∈ C∞(Bρ(x0)). We conclude using the arbitrariness of r.

Now we can establish the monotonicity formula.

Proof of the monotonicity formula. Let us consider an energy minimizing map u ∈ H1loc(Ω,N ), withBR(x0) ⊂ Ω. Now according to Remark 2.3, (2.3) stands true, and setting

aj = (tj1, . . . , tjn),

where T = (tij) is the stress energy tensor, it is equivalent to

ˆBR(x0)

〈aj ,∇ξ〉 dx = 0, ∀ξ ∈ C∞c (BR(x0)),∀j ∈ {1, . . . , n}

i.e. div aj = 0, ∀j ∈ {1, . . . , n}, in the sense of distributions D′(BR(x0)). Since aj is integrable, wecan apply the previous lemma, which yields

ˆBρ(x0)

n∑i=1

(|∇u|2δij − 2〈∂iu, ∂ju〉∂iξj dx =ˆ∂Bρ(x0)

n∑i=1

(|∇u|2δij − 2〈∂iu, ∂ju〉)ρ−1(xi − xi0)ξj dσ

∀j ∈ {1, . . . , n},∀ξj ∈ C∞(Bρ(x0)),

where x = (x1, . . . , xn) and x0 = (x10, . . . , x

n0 ). Now taking ξ

j(x) = xj − xj0 for each j, we haveξj ∈ C∞(Bρ(x0)) and ∂iξj = δij , thus

ˆBρ(x0)

|∇u|2 − 2|∂ju|2 dx =ˆ∂Bρ(x0)

|∇u|2ρ−1(xj − xj0)2

− 2n∑i=1

〈∂iu, ∂ju〉ρ−1(xi − xi0)(xj − xj0) dσ.

9

3 THE ε-REGULARITY THEOREM

Summing over j, noticing that

n∑j=1

〈∂iu, ∂ju〉ρ−1(xi − xi0)(xj − xj0) = ρ

∣∣∣∣∣n∑i=1

ri∂iu

∣∣∣∣∣2

= ρ

∣∣∣∣∂u∂r∣∣∣∣2,

we obtain the equality

(n− 2)ˆBρ(x0)

|∇u|2 dx = ρˆ∂Bρ(x0)

(|∇u|2 − 2

∣∣∣∣∂u∂r∣∣∣∣2)

dσ. (∗)

Recall that if f is integrable on Br(x0), then ρ 7→´Bρ(x0)

f dx is di�erentiable for almost every

ρ ∈ (0, r), and

d

dρ

ˆBρ(x0)

f dx =

ˆ∂Bρ(x0)

f dσ.

Now since |∇u|2 is integrable, let us compute

d

dρ

(ρ2−n

ˆBρ(x0)

|∇u|2 dx

)= (2− n)ρ1−n

ˆBρ(x0)

|∇u|2 dx+ ρ2−nˆ∂Bρ(x0)

|∇u|2 dσ,

then, using (∗)

d

dρ

(ρ2−n

ˆBρ(x0)

|∇u|2 dx

)= 2ρ2−n

ˆ∂Bρ(x0)

∣∣∣∣∂u∂r∣∣∣∣2 dσ

= 2

ˆ∂Bρ(x0)

r2−n∣∣∣∣∂u∂r

∣∣∣∣2 dσ= 2

d

dρ

ˆBρ(x0)

r2−n∣∣∣∣∂u∂r

∣∣∣∣2 dσ,for almost all ρ ∈ (0, r). Since the functions di�erentiated are absolutely continuous on (0, r), inte-grating over [τ, ρ] ⊂ (0, r) concludes the proof.

This formula is referred to as the monotonicity formula because it trivially implies the monotonicityof the function ρ 7→ ρ2−n

´Bρ(x0)

|∇u|2 dx, which is a very strong result. Indeed it allows to de�ne theso-called density function of u (see Section 4.2), a very useful tool to study energy minimizing maps.In particular, with the ε-regularity theorem, it will give us a simple characterization of their regularpoints.

3 The ε-regularity theorem

The purpose of this section is to establish the following theorem, originally proven by Schoen andUhlenbeck [11]. Unless stated otherwise, Ω denotes an open subset of Rn with n ≥ 2, and N a smoothcompact submanifold of Rp.

10

3.1 Statements 3 THE ε-REGULARITY THEOREM

3.1 Statements

Theorem 3.1 (ε-regularity). Let Λ > 0, θ ∈ (0, 1). There exists ε = ε(n,N ,Λ, θ) > 0 such that,if u ∈ H1loc(Ω,N ) is an energy minimizing map with BR(x0) ⊂ Ω, and if R2−n

´BR(x0)

|∇u|2 dx ≤ Λand R−n

´BR(x0)

|u − λx0,R|2

dx < ε, then u ∈ C∞(BR(x0)) and for small values of R we have theestimates:

Rj |∇ju|0,BθR(x0) ≤ C

(R−n

ˆBR(x0)

|u− λx0,R|2

dx

)1/2,

for each j ∈ {1, . . . , n}, where C depends only on j, Λ, N , θ and n.

Remarks 3.1.

(i) Naturally BR(x0) ⊂ Rn.

(ii) λx0,R is de�ned by λx0,R := R−n ´

BR(x0)u dx, and is also the value that minimizesR−n

´BR(x0)

|u−λ|2 dx.

(iii) It is enough to prove that for θ = 1/8 �xed, there is some ε(n,N ,Λ) such that, under theassumptions of the theorem, u ∈ C∞(BR/4(x0)) and we have the required estimates on BR/8(x0).We then deduce the theorem by coveringBθR(x0) with ballsB(1−θ)R/8(yj), yj ∈ BθR(x0). Indeed,since B(1−θ)R(yj) ⊂ BR(x0), we have

((1− θ)R)−nˆB(1−θ)R(yj)

|u− λyj ,(1−θ)R|2

dx ≤ ((1− θ)R)−nˆB(1−θ)R(yj)

|u− λx0,R|2

dx

≤ (1− θ)−nR−nˆBR(x0)

|u− λx0,R|2

dx

≤ (1− θ)−nε

and

((1− θ)R)2−nˆB(1−θ)R(yj)

|∇u|2 dx ≤ (1− θ)2−nΛ,

so we can apply the result with (1 − θ)R in place of R, (1 − θ)2−nΛ in place of Λ, to getan ε which also depends on θ such that we have u ∈ C∞(B(1−θ)R/4(yj)) and the estimates of((1−θ)R)j |∇ju|0,B(1−θ)R/8(yj) . This shows that u is actually smooth inBθR(x0), and summing overall j we obtain the needed estimates, with C depending also on θ. By virtue of the arbitrarinessof θ ∈ (0, 1), u is smooth on BR(x0).

An immediate corollary of this theorem is the following.

Corollary 3.2. Let u ∈ H1loc(Ω,N ) be an energy minimizing map, and x0 ∈ Ω. Then u is smooth ina neighborhood of x0 if and only if it is continuous at x0.

Proof. Assume that u is continuous at x0, then for any ε > 0 there is R small such that |x0 − y| <R =⇒ |u(x0)− u(y)| < ε. Thus

R−nˆBR(x0)

|u− λx0,R|2

dx ≤ R−nˆBR(x0)

|u− u(x0)|2 dx ≤ Cε2,

where C depends only on n. Then setting Λ := R2−n´BR(x0)

|∇u|2 dx, and taking ε such that Cε2 ≤ δ20 ,where δ0 = δ0(n,N ,Λ) given by the ε-regularity theorem, we get that u is smooth in BR(x0).

11

3.2 A lemma of Luckhaus 3 THE ε-REGULARITY THEOREM

We now state another direct corollary of the ε-regularity theorem, which is commonly used andoften also referred to as the ε-regularity theorem.

Corollary 3.3 (ε-regularity). There exists ε = ε(n,N ) > 0, such that, if u ∈ H1loc(Ω,N ) is an energyminimizing map with BR(x0) ⊂ Ω satisfying R2−n

´BR(x0)

|∇u|2 dx < ε, then u is smooth in BR(x0)and Rj |∇ju|0,BR/2(x0) ≤ C for each j ∈ N where C depends on j, n, and N .

Remark 3.2. In the case where n = 2, the regularity of u follows immediately.

Proof of the corollary. In view of the Poincaré inequality and Remark A.2, we have

R−nˆBR(x0)

|u− λ|2 dx ≤ CR2−nˆBR(x0)

|∇u|2 dx,

where C depends only on n. Then let us take the ε given by the ε-regularity theorem with Λ = 1,θ = 1/2, and rename it η for the sake of clarity. Now setting ε = min(1/C, η) yields the result. Indeed,

in view of the Poincaré inequality and Remark A.2, if R2−n´BR(x0)

|∇u|2 dx < ε, we have,

R−nˆBR(x0)

|u− λ|2 dx ≤ CR2−nˆBR(x0)

|∇u|2 dx < Cε < 1,

R−nˆBR(x0)

|∇u|2 dx < η,

thus u is smooth in BR(x0) and we have the required estimates thanks to the choice of η.

We postpone the proof of the ε-regularity theorem to Section 3.5, as we �rst need to establish somepreliminary results.

3.2 A lemma of Luckhaus

In this section we prove a lemma of Luckhaus, which will be needed to prove that, under some condi-tions, minimizing maps satisfy a reverse Poincaré inequality. First, let us give some useful de�nitions.

De�nition 3.1. Let v ∈ L2(Sn−1,Rp), then we say that v ∈ H1(Sn−1,Rp) is the homogeneous degreezero extension ṽ(rw) ≡ v(w), w ∈ Sn−1, r > 0 belongs to H1(V,Rp) for some V neighborhood of Sn−1.We say that v ∈ H1(Sn−1,N ) if v ∈ H1(Sn−1,Rp) and v(x) ∈ N for σ − a.e. x.

Remark 3.3. Note that if n ≥ 3, ṽ ∈ H1(V,Rp) if and only if ṽ ∈ H1(B1(0),Rp).

De�nition 3.2. Let v ∈ L2(Sn−1 × [a, b],Rp), then we say that v ∈ H1(Sn−1 × [a, b],Rp) if thehomogeneous degree zero extension ṽ(w, t) w.r.t the �rst variable belongs to H1(V × [a, b],Rp) forsome V neighborhood of Sn−1.

We can now state the lemma.

Lemma 3.4 (Luckhaus). Let N be an arbitrary compact subset of Rp, n ≥ 2, and u, v ∈ H1(Sn−1,N ).Then for each ε ∈ (0, 1), there is a function w ∈ H1(Sn−1 × [0, ε]) such that w agrees with u ina neighborhood of Sn−1 × {0}, w agrees with v in a neighborhood of Sn−1 × {ε}, and we have theestimates ˆ

Sn−1×[0,ε]|∇w|2 dx ≤ Cε

ˆSn−1

(|∇u|2 + |∇v|2) dσ + Cε−1ˆSn−1|u− v|2 dσ,

12


and

dist(w(x, s),N ) ≤

Cε1−n(ˆ

Sn−1|∇u|2 + |∇v|2 dσ

)1/2(ˆSn−1|u− v|2 dσ

)1/2+ Cε−n

ˆSn−1|u− v|2 dσ

for a.e. (x, s) ∈ Sn−1 × [0, ε], where C depends only on n, ∇ is the gradient on Sn−1 and ∇ is thegradient on the product space Sn−1 × [0, ε].

Proof. We are going to give di�erent proofs for the cases n = 2 and n ≥ 3.

n = 2 Here u and v are functions from S1 into N . According to Proposition A.7, we can assume thatu and v are absolutely continuous representatives of their L2 representatives, such that their classicalgradient is equal almost everywhere to their weak L2 gradient, and for all x ∈ S1, u(x), v(x) ∈ N . Letus de�ne ϕ(t) = |u− v|2(cos(2πt), sin(2πt)), t ∈ R, then by one-dimensional calculus we have, for anyt, s ∈ R

|ϕ(t)− ϕ(s)| =∣∣∣∣ˆ ts

ϕ′(τ) dτ

∣∣∣∣≤ 2π

ˆ 10

|∇(|u− v|2)|(− sin(2πt), cos(2πt)) dt

=

ˆS1|∇(|u− v|2)| dλ,

so

|ϕ(t)| ≤ˆS1|∇(|u− v|2)| dλ+ |ϕ(s)|,

and by integrating on [0, 1] we get

|ϕ(t)| ≤ˆS1|∇(|u− v|2)| dλ+

ˆ 10

|ϕ(s)| ds.

Replacing ϕ with its expression, taking the supremum over t ∈ [0, 1] and using Cauchy-Schwarzinequality, we get

||u− v|2|0,S1 ≤ˆS1|∇(|u− v|2)| dλ+ (2π)−1

ˆS1|u− v|2 dλ (1)

≤ 2ˆS1|∇(u− v)||u− v|+ (2π)−1

ˆS1|u− v|2 dλ

≤ C(ˆ

S1|∇(u− v)|2 dλ

)1/2(ˆS1|u− v|2 dλ

)1/2+ C

ˆS1|u− v|2 dλ.

Now let us de�ne w : Sn−1 × [0, ε] by

w(ω, s) = u(ω) +ψ(s)

ε(v(ω)− u(ω)), (2)

where ψ ∈ C∞c ((0, ε)) is nonnegative, |ψ|0,[0,ε] ≤ ε and |ψ|0,[0,ε] ≤ 2. Then, denoting by ∇w thegradient of w in S1 × (0, ε), we have

|∇w| ≤ |∇u|(

1− ψ(s)ε

)+ |∇v|

∣∣∣∣ψ(s)ε∣∣∣∣+ |v − u|∣∣∣∣ψ′(s)ε

∣∣∣∣13


≤ |∇u|+ |∇v|+ 2ε−1|v − u|.

Using the squared triangle inequality, it follows

|∇w|2 ≤ 2(|∇u|+ |∇v|)2 + 4ε−2|v − u|2

≤ 4(|∇u|2 + |∇v|2 + ε−2|v − u|2),

and by integrating over S1 × [0, ε]ˆS1×[0,ε]

|∇w|2 dλ ds ≤ CεˆS1|∇u|2 + |∇v|2 dλ+ Cε−1

ˆS1|v − u|2 dλ,

which is the �rst inequality of the lemma. Then using (1), (2), and the fact that u(x) ∈ N , we havefor any (ω, s) ∈ S1 × [0, ε]

dist2(w(ω, s),N ) ≤ |w(ω, s)− u(ω)|2

≤(sε

)2|(u− v)(ω)|2

≤ ||u− v|2|0,S1

≤ C(ˆ

S1|∇(u− v)|2 dλ

)1/2(ˆS1|u− v|2 dλ

)1/2+ C

(ˆS1|u− v|2 dλ

)1/2≤ C

(ˆS1|∇u|2 + |∇v|2 dλ

)1/2(ˆS1|u− v|2 dλ

)1/2+ C

(ˆS1|u− v|2 dλ

)1/2,

which is the second inequality of the lemma and concludes the proof in the case n = 2.

n ≥ 3 Here again u and v are to be understood as representatives of their L2 classes. We extendthem by homogeneity of degree zero to Rn, and note their respective extensions u and v as well. Notethat for almost every r ∈ (0,+∞), u|∂Br(0) (resp. v|∂Br(0)) de�nes a (unique) class of L2 functions inH1(∂Br(0),N ), whose weak L2 gradient coincides with the tangential part of the gradient of u (resp.v) almost everywhere. Note that since u(x) = u(x|x|−1) for almost every x, we have

∇u(x) = 1|x|∇u(x

|x|

)so, since n ≥ 3

ˆ[−1,1]n

|∇u|2 + |∇v|2 dx ≤ˆB√n(0)

|∇u|2 + |∇v|2 dx

≤ˆ √n

0

rn−2ˆSn−1|∇u|2 + |∇v|2 dσ dr

≤ CˆSn−1|∇u|2 + |∇v|2 dσ,

where C depends only on n. And similarly we also haveˆ

[−1,1]n|u− v|2 dx ≤ C

ˆSn−1|u− v|2 dσ.

14


Let us now slice [−1, 1]n into small cubes of side ε ∈ (0, 1/8), i.e. consider Qi,ε = [i1ε, (i1 + 1)ε]× . . .×[inε, (in + 1)ε] for i = (i1, . . . , in) ∈ Zn). Then for a measurable function f : [−1, 1]n → [0,+∞), wede�ne f̃ on any Qi,ε by

f̃(x) =∑

i|Qi,ε⊂[−1/2,1/2]nf(x+ εi).

We have ˆQ0,ε

f̃(x) dx(x) =

ˆQ0,ε

∑i|qi,ε⊂[−1/2,1/2]n

f(x+ εi) dx(x)

=∑

i|Qi,ε⊂[−1/2,1/2]n

ˆQi,ε

f(x) dx(x)

≤ˆ

[−1,1]nf dx,

thus by Markov's inequality (Proposition A.5) it follows

λ

x ∈ Q0,ε : εn ∑i|Qi,ε⊂[−1/2,1/2]n

f(x+ εi) > K

ˆ[−1,1]n

f dx

≤ K−1εn,

where λ is the one dimensional Lebesgue/Hausdor� measure. Let us de�ne

Q̃0,ε := ∪|i|≤2Qj,ε,

and note Q(n) the number of j ∈ Z such that |j| ≤ 2. For any vertex c of Q0,ε, and any vector ej ofthe canonical orthonormal basis of Rn, we have, using Fubini's theorem

ˆQ0,ε

ˆ ε0

f̃(x+ c+ tej) dt dx(x) =

ˆ ε0

ˆQ0,ε

f̃(x+ c+ tej) dx(x) dt

≤ˆ ε

0

ˆQ̃0,ε

f̃(x) dx(x) dt.

Noticing that´Qi,ε

f̃(x) dx(x) ≤´

[−1,1]n f dx it follows from the previous inequality

ˆQ0,ε

ˆ ε0

f̃(x+ c+ tej) dt dx(x) ≤ Q(n)ˆ ε

0

ˆ[−1,1]n

f dxdt ≤ εQ(n)ˆ

[−1,1]nf dx,

which can be rewrittenˆQ0,ε

∑i|Qi,ε⊂[−1/2,1/2]n

ˆ ε0

f(x+ c+ tej + εi) dt dx(x) ≤ Q(n)εˆ

[−1,1]nf dx,

Then by Markov's inequality this implies (shortening∑i|Qi,ε⊂[−1/2,1/2]n by

∑Qi,ε

)

λ

x ∈ Q0,ε : εn−1∑Qi,ε

ˆ ε0

f(x+ c+ tej + εi) dt > K

ˆ[−1,1]n

f dx

≤ Q(n)−1K−1εn.

15


By considering the multiples integrals´ ε

0. . .´ ε

0f̃(x+ c+ t1ej1 + . . .+ tlejl dt1 . . . dtl where ej1 , . . . , ejl

are pairwise distinct vectors of the orthonormal basis of Rn, we get similarly

λ

x ∈ Q0,ε : εn−l∑Qi,ε

ˆF (l)

f(x+ y + εi) dHl(y) > Kˆ

[−1,1]nf dx

≤ CK−1εn,

for any l-face F (l) of Q0,ε, where C depends only on n, any l ∈ {1, . . . , n}. Then noting that the sumover l of the number of l-faces depends only on n, this even implies

λ

x ∈ Q0,ε : εn−l

∑Qi,ε

∑all l-faces F (l)

of x+Qi,ε

ˆF (l)

f(y) dHl(y) > Kˆ

[−1,1]nf dx

≤ CK−1εn,

where C depends only on n. We want to apply this result to the functions |u− v|2 and |∇u|2 + |∇v|2.Note that for almost every x, for every l ∈ {1, . . . , n}, the restriction of u (resp. v) to each l-faceF (l) for x+Qi,ε is measurable and de�nes a unique class of functions in H

1(F (l),N ) whose weak L2gradient coincides with the tangential part of ∇u (resp. ∇v). Thus choosing K−1 small enough �depending only on n, since C depends only on n � there exists a ∈ Q0,ε such that

εn−l∑

i|Qi,ε⊂[−1/2,1/2]n

∑all l-faces F (l)

of x+Qi,ε

ˆF (l)

f(y) dHl(y) ≤Mεnˆ

[−1,1]nf dx, (3)

for f = |u − v|2 and f = |∇u|2 + |∇v|2, and any l ∈ {1, . . . , n}, where M depends only on n. Let Qbe some a+Qi,ε. We are going to de�ne a function w = w

(i,ε) in H1(Q× [0, ε],Rp) such that w(ω, t)agrees with u(ω) (resp. v(ω)) when t lies in a �xed neighborhood of 0 (resp. ε). To do that, we aregoing to de�ne it �rst on the 1-faces (i.e. edges) of Q, and then de�ne it recursively on all l-faces ofQ, for any l ∈ {0, . . . , n}. Let E be an edge of Q. Then by one-dimensional calculus along lines, wehave for any x, y in E

||u− v|2(x)− |u− v|2(y)| ≤Ê

|∇(|u− v|2)| dλ

so

||u− v|2(x)|0,E ≤Ê

|∇(|u− v|2)| dλ+ |u− v|2(y)

and by integrating over E and using Cauchy-Schwarz inequality

||u− v|2(x)|0,E ≤Ê

|∇(|u− v|2)| dλ+ ε−1Ê

|u− v|2 dλ (4)

≤ 2(ˆ

E

|∇(u− v)|2 dλ)1/2(ˆ

E

|u− v|2 dλ)1/2

+ ε−1Ê

|u− v|2 dλ

≤ C(ˆ

E

|∇u|2 + |∇v|2 dλ)1/2(ˆ

E

|u− v|2 dλ)1/2

+ ε−1Ê

|u− v|2 dλ.

Now we de�ne w on Q × {0} (resp. Q × {ε}) by w(·, 0) = u (resp. w(·, ε) = v), and on F (1)×[0,ε] forany edge F (1) of Q by

w(ω, s) =

(1− ψ(s)

ε

)u(ω) +

ψ(s)

εv(ω),

16


where ψ ∈ C∞c ((0, ε)) is nonnegative, with |ψ|0,[0,ε] ≤ ε and |ψ′|0,[0,ε] ≤ 2. Then in F (1) × [0, ε] wehave

|∇w| ≤(

1− ψε

)|∇u|+

∣∣∣∣ψε∣∣∣∣|∇v|

≤ |∇u|+ |∇v|+ ε−1|v − u|,

where ∇w denotes the gradient of w in F (1) × [0, ε]. Using the squared triangle inequality, it follows

sups∈[0,ε]

|∇w(ω, s)|2 ≤ 4(|∇u(ω)|2 + |∇v(ω)|2) + 2ε−2|u− v|2(ω), ∀ω ∈ F (1).

Thus by integrating over F (1)

ˆF (1)|∇w|2 dλ⊗ ds =

ˆF (1)

ˆ[0,ε]

|∇w(ω, s)|2 ds dλ (5)

≤ εˆF (1)

sups∈[0,ε]

|∇w(ω, s)|2 dλ

≤ CεˆF (1)|∇u|2 + |∇v|2 dλ+ Cε−1

ˆF (1)|u− v|2 dλ.

For 2 ≤ l ≤ n − 1, we proceed inductively as follows. Let us assume that that w is already de�nedon Q × {0}, Q × {ε} and on F (l−1) × [0, ε] for every (l − 1)-face of Q such that w(ω, t) agrees withu (resp. v) in a �xed neighborhood of 0 (resp. ε) for every ω ∈ F (l−1). Let us notice that for anyl-face F (l) of Q, ∂(F (l) × [0, ε]) is the union of F (l) × {0}, F (l) × {ε} and the sets F (l−1) × [0, ε] for all(l − 1)-faces F (l−1) of F (l). Thus by the induction hypothesis w is already de�ned of ∂(F (l) × [0, ε]),so we can de�ne w on F (l) × [0, ε] by degree zero extension with origin point (q, ε/2) where q is thecenter of F (l). Then, denoting by Ts the homothety of center (q, ε/2) and ratio s, we have

ˆF (l)×[0,ε]

|∇(l)w|2 dHl ⊗ dλ ≤ˆ ε/2

0

ˆTs(∂(F (l)×[0,ε]))

|∇(l−1)w|2 (dHl−1 ⊗ dλ)ds

≤ˆ ε/2

0

sl−2ˆ∂(F (l)×[0,ε])

|∇(l−1)w|2(dHl−1 ⊗ dλ)ds

≤ Cεl−1ˆ∂(F (l)×[0,ε])

|∇(l−1)w|2 dHl−1 ⊗ dλ

= Cε

ˆF (l)|∇u|2 + |∇v|2 dHl

+ Cε∑

all (l−1)-facesF (l−1) of F (l)

ˆF (l−1)×[0,ε]

|∇(l−1)w|2 dHl−1 ⊗ dλ,

where ∇(l−1) denotes the gradient on F (l−1)× [0, ε] for some (l− 1)-face F (l−1), and ∇(l) the gradienton F (l) × [0, ε]. We can expand this last inequality with the inequalities obtained for l− 1, l− 2, . . . , 2to reach

ˆF (l)×[0,ε]

|∇(l)w|2 dHl ⊗ dλ ≤ Cl∑

j=1

εl−j+1∑

all j-faces

F (j) of Q

ˆF (j)|∇u|2 + |∇v|2 dHj ⊗ dλ (6)

17


+ Cεl−1∑

all 1-facesF (1) of Q

ˆF (1)×[0,ε]

|∇(1)w|2 dλ⊗ dλ.

By induction w is de�ned on all l-faces F (l) of Q, and (6) holds, for all l ∈ {1, . . . , n}. With l = n,this implies that w is de�ned on all Q (which is its unique n-face). Moreover, we have

dist2(w(ω, s),N ) ≤ |(w(ω, s), u(ω)|2

≤∣∣∣∣ψ(s)ε

∣∣∣∣|u− v|2(ω),thus, since w was built from the edges of Q and then extended by zero homogeneity, (4) still holds onthe whole cube, which implies

dist2(w(ω, s),N ) ≤ C maxall 1-facesF (1) of Q

[(ˆF (1)|∇u|2 + |∇v|2 dλ

)1/2(ˆF (1)|u− v|2 dλ

)1/2(7)

+ Cε−1ˆF (1)|u− v|2 dλ

].

Taking l = n in (6), and then using (5), we get

ˆQ×[0,ε]

|∇w|2 dHn ⊗ dλ ≤ Cεn−1∑


ˆF (1)×[0,ε]

|∇(1)w|2 dλ⊗ dλ (8)

+ C

n∑j=1

εn−j+1∑

all j-faces

F (j) of Q

ˆF (j)|∇u|2 + |∇v|2 dHj

≤ Cεn−2∑


ˆF (1)|u− v|2 dλ

+ C

n∑j=1

εn−j+1∑

all j-faces

F (j) of Q

ˆF (j)|∇u|2 + |∇v|2 dHj

Let us recall that in fact w = w(i,ε) is de�ned on a+Qi,ε, and notice that by construction, for any i, j,and any common faces of a+Qi,ε and a+Qj,ε coincide. Thus we can de�ne w ∈ H1([−1/4, 1/4],Rp)by

w(ω, s) = w(i,ε) when x ∈ a+Qi,ε.

It well de�ned because for a ∈ Q0,ε, ε ∈ (0, 1/8), any x ∈ [1/4, 1/4]n belongs to some Qi,ε ⊂[−1/2, 1/2]n. Now by summing over i in (8), using (3) with f = |u − v|2 and f = |∇u|2 + |∇v|2,and shortening

∑i|Qi,ε⊂[−1/2,1/2],

∑all j-faces

F (j) of Qi,ε

by∑Qi,ε

and∑F (j) respectively, it follows

ˆ[−1/4,1/4]n×[0,ε]

|∇w|2 dHn dλ ≤∑i

ˆ(a+Qi,ε)×[0,ε]

|∇w(i,ε)|2 dHn dλ (9)

18


≤ Cε−1εn−1∑Qi,ε

∑F (1)

ˆF (1)|u− v|2 dλ

+ Cε∑Qi,ε

∑F (j)

εn−jˆF (j)|∇u|2 + |∇v|2 dHj

≤ Cε−1ˆ

[−1,1]n|u− v|2 dx+ Cε

ˆ[−1,1]n

|∇u|2 + |∇v|2 dx.

We also have using (7) and (3) again

dist2(w(ω, s),N ) ≤ C maxall 1-faces F (1) of

all Qi,ε

[(ˆF (1)|∇u|2 + |∇v|2 dλ

)1/2(ˆF (1)|u− v|2 dλ

)1/2(10)

+ Cε−1ˆF (1)|u− v|2 dλ

]

≤

(∑F (1)

ˆF (1)|∇u|2 + |∇v|2 dλ

)1/2(∑F (1)

ˆF (1)|u− v|2 dλ

)1/2

+ Cε−1∑F (1)

ˆF (1)|u− v|2 dλ

≤ Cε1−n(ˆ

[−1,1]n|∇u|2 + |∇v|2 dx

)1/2(ˆ[−1,1]n

|u− v|2 dx

)1/2

+ Cε−nˆ

[−1,1]n|u− v|2 dx

Eventually to complete the proof we want to de�ne a function inH1(Sn−1×[0, ε],Rp). To do that, let usnotice that by Fubini's theorem for almost every ρ ∈ [1/8, 1/4], w de�nes a function in H1(∂([−ρ, ρ]n)×[0, ε],Rp), and

ˆ 1/41/8

ˆ∂([−ρ,ρ]n)×[0,ε]

|∇w|2 dHn−1 ⊗ dλ ≤ˆ

[−1/4,1/4]n×[0,ε]|∇w|2,

so by Markov's inequality we have for any C > 0

λ

({ρ :

ˆ∂([−ρ,ρ]n)×[0,ε]

|∇w|2 dHn−1 ⊗ dλ > Cˆ

[−1/4,1/4]n×[0,ε]|∇w|2 dHn ⊗ dλ

})≤ C−1.

Thus taking e.g. C = 1/16, one can �nd ρ ∈ (1/8, 1/4) such that w de�nes a function inH1(∂([−ρ, ρ]n)×[0, ε],Rp) and ˆ

∂([−ρ,ρ]n)×[0,ε]|∇w|2 dHn−1 ⊗ dλ ≤ C

ˆ[−1/4,1/4]n×[0,ε

|∇w|2 dHn ⊗ dλ, (11)

where C is a constant. Now let ϕ be the radial function that maps Sn−1 to ∂([−ρ, ρ]n), which is aLipschitz, C1 piecewise function �H1 in particular� with Lipschitz, C1 piecewise inverse. De�ningŵ ∈ H1(Sn−1 × [0, ε],Rp) by

ŵ(ω, s) = w(ϕ(ω), s),

19


we have, using (9), (11), and the fact that |Jacϕ|0 is bounded from belowˆSn−1×[0,ε]

|∇ŵ|2 dHn−1 ⊗ dλ ≤ Cˆ∂([−ρ,ρ]n)×[0,ε]

|∇w|2 dHn−1 ⊗ dλ

≤ Cˆ

[−1/4,1/4]n×[0,ε]|∇w|2 dHn−1 ⊗ dλ

≤ Cεˆ

[−1,1]n|∇u|2 + |∇v|2 dHn + Cε−1

ˆ[−1,1]n

|u− v|2 dHn.

The estimate on dist2(ŵ,N ) follows directly from (10). Moreover, since w(x, t) = u(x) (resp. v(x))when t lies in a �xed neighborhood of 0 (resp. ε), for any x ∈ ∂([−ρ, ρ]n), and since u and v are degreezero homogeneous maps, we have when t lies in a neighborhood of 0

ŵ(ω, t) = w(ϕ(ω), t)

= u(ϕ(ω))

= u(ω),

and similarly ŵ(ω, t) = v(ω) when t lies in a neighborhood of ε. Hence ŵ satis�es the conclusions ofthe lemma.

We can then state and prove a very useful corollary.

Corollary 3.5 (Luckhaus). Let N be a smooth compact manifold embedded in Rp and Λ > 0. Thenthere exist δ0 = δ0(n,N ,Λ) and C = C(n,N ,Λ) such that the following hold:

1. If ε ∈ (0, 1), u ∈ H1(Bρ(y),N ) with ρ2−n´Bρ(y)

|∇u|2 dx ≤ Λ, and ρ−n´Bρ(y)

|u − λy,ρ|2 dx ≤ε2nδ20, then there is σ ∈ (3ρ/4, ρ) and a function w = wε ∈ H1(Bρ(y),N ) which agrees with u in aneighborhood of ∂Bσ(y) and satis�es

σ2−nˆBσ(y)

|∇w|2 dx ≤ ερ2−nˆBρ(y)

|∇u|2 dx+ ε−1Cρ−nˆBρ(y)

|u− λy,ρ|2 dx.

2. If ε ∈ (0, δ0], if u, v ∈ H1(B(1+ε)ρ(y)\Bρ(y),N ) satisfy the inequalities ρ2−n´B(1+ε)ρ(y)\Bρ(y)

|∇u|2+|∇v|2 dx ≤ Λ and ρ−n

´B(1+ε)ρ(y)\Bρ(y)

|u − v|2 dx ≤ δ20, then there is w ∈ H1(B(1+ε)ρ(y) \ Bρ(y),N )such that w = u in a neighborhood of ∂Bρ(y), w = v in a neighborhood of ∂B(1+ε)ρ(y) and

ρ2−nˆB(1+ε)ρ(y)\Bρ(y)

|∇w|2 dx

≤ Cρ2−nˆB(1+ε)ρ(y)\Bρ(y)

|∇u|2 + |∇v|2 dx+ Cε−2ρ−nˆB(1+ε)ρ(y)\Bρ(y)

|u− v|2 dx.

Proof. Here again u and v are to be understood as representatives of their L2 classes such that for allx, u(x), v(x) ∈ N .

Proof of (1) First notice that for almost every σ ∈ (ρ/2, ρ), u de�nes a function in H1(∂Bσ(y),N ),and we have, since σ/ρ < 1,

λ

({σ ∈ (ρ/2, ρ) : σ3−n

ˆ∂Bσ(y)

|∇u|2 dHn−1 > θ−1ρ2−nˆBρ(y)\Bρ(y)/2(y)

|∇u|2 dx

})

20


≤ λ

({σ ∈ (ρ/2, ρ) :

ˆ∂Bσ(y)

|∇u|2 dHn−1 > (θρ)−1ˆBρ(y)\Bρ/2(y)

|∇u|2 dx

}),

for any θ > 0, so by Markov's inequality (Proposition A.5) we have

λ

({σ ∈ (ρ/2, ρ) : σ3−n

ˆ∂Bσ(y)

|∇u|2 dHn−1 > θ−1ρ2−nˆBρ(y)\Bρ/2(y)

|∇u|2 dx

})≤ θρ,

and similarly

λ

({σ ∈ (ρ/2, ρ) : σ1−n

ˆ∂Bσ(y)

|u− λy,ρ|2 dHn−1 > θ−1ρ−nˆBρ(y)\Bρ/2(y)

|u− λy,ρ|2 dx

})≤ θρ.

Thus taking e.g. θ = 1/8, there exists a σ ∈ (ρ/2, ρ) such that u ∈ H1(∂Bσ(y),N ) and

σ3−nˆ∂Bσ(y)

|∇u|2 dHn−1 ≤ Cρ2−nˆBρ(y)\Bρ/2(y)

|∇u|2 dx (1)

σ1−nˆ∂Bσ(y)

|u− λy,ρ|2 dHn−1 ≤ Cρ−nˆBρ(y)\Bρ/2(y)

|u− λy,ρ|2 dx.

Since u(x) ∈ N for every x, we have dist2(λy,ρ,N ) ≤ |u(x)−λy,ρ|, so integrating over Bρ(y), it follows

dist2(λy,ρ,N ) ≤ Cρ−nˆBρ(y)

|u− λy,ρ|2 dx

≤ Cδ20ε2n,

assuming that ε−2nρ−n´Bρ(y)

|u−λy,ρ|2 dx ≤ δ20 for some δ0. Now as N is compact, there exists someλ ∈ N such that |λ − λy,ρ|2 ≤ Cδ20ε2n. Let us then apply the lemma of Luckhaus to the functionsũ(ω) ≡ u(y + σω) and v(ω) ≡ λ. Thus there exists a function w ∈ H1(Sn−1 × [0, ε]) such that w(ω, t)agrees with ũ (resp. λ) when t lies in a �xed neighborhood of 0 (resp. ε), and such that

ˆSn−1×[0,ε]

|∇w|2 dHn−1 dλ ≤ CεˆSn−1|∇ũ|2 dHn−1 + Cε−1

ˆSn−1|ũ− λ|2 dHn−1

≤ Cεσ3−nˆ∂Bσ(y)

|∇u|2 dHn−1 + Cε−1σ1−nˆ∂Bσ(y)

|u− λ|2,

where C depends only on n. Using (1) and the fact that |u−λ|2 ≤ 2|u−λy,ρ|2 + 2|λy,ρ−λ|2, it followsˆSn−1×[0,ε]

|∇w|2 dHn−1 dλ ≤ Cερ2−nˆBρ(y)

|∇u|2 dx+ Cε−1ρ−nˆBρ(y)

|u− λy,ρ|2 dx. (2)

The lemma of Luckhaus also gives the estimate

dist2(w(x, s),N ) ≤ Cε1−n(ˆ

Sn−1|∇ũ|2 dHn−1

)1/2(ˆSn−1|ũ− λ|2 dHn−1

)1/2+ Cε−n

ˆSn−1|ũ− λ|2 dHn−1

21


≤ Cε1−n(σ3−n

ˆ∂Bσ(y)

|∇u|2 dHn−1)1/2(

σ1−nˆ∂Bσ(y)

|u− λ|2 dHn−1)1/2

+ Cε−nσ1−nˆ∂Bσ(y)

|u− λ|2 dHn−1

≤ Cε1−nΛ1/2δ0εn + Cε−nδ20ε2n

≤ Cδ0,

where C depends only on n and Λ. Thus we can choose δ0 small enough, depending only on n and Λ,such that dist(w,N ) ≤ α, where α is such that the nearest point projection on N denoted ΠN is wellde�ned. Now de�ning w̃ on Bρ(y) as follows

w̃(x) =

λ for |x| ≤ σ(1− ε)ΠN

(w(x|x| , 1−

|x|σ

))for |x| ∈ [σ(1− ε), σ]

u(x) for |x| ∈ [σ, ρ],

we have w̃ = u in a neighborhood of ∂Bσ(y), and

σ2−nˆBσ(y)

|∇w̃|2 dx = σ2−nˆBσ(y)\Bσ(1−ε)(y)

∣∣∣∣∇(ΠN ◦ w( x|x| , 1− |x|σ))∣∣∣∣2 dx

≤ Cσ2−nˆ σσ(1−ε)

ˆ∂Bs(y)

∣∣∣∣∇x(w( x|x| , 1− |x|σ))∣∣∣∣2 dHn−1 ⊗ ds,

when C depends on N , n and Λ. A direct computation gives∣∣∣∣∇x(w( x|x| , 1− |x|σ))∣∣∣∣ ≤ 1|x| |∇ωw|+ 1σ |∂tw| ≤ 1|x| |∇w|,

thus

σ2−nˆBσ(y)

|∇w̃|2 dx ≤ Cσ2−nˆ σσ(1−ε)

sn−3ˆSn−1

∣∣∣∇w (ω, 1− sσ

)∣∣∣2 dHn−1(ω)ds≤ Cσ−1

ˆ σσ(1−ε)

ˆSn−1

∣∣∣∇w (ω, 1− sσ

)∣∣∣2 dHn−1ds= C

ˆSn−1×[0,ε]

|∇w|2 dHn−1 ⊗ dλ,

hence the result, in view of (2).

Proof of (2) Proceeding as in the proof of (1), we have for θ > 0

λ

({σ ∈ [ρ, ρ(1 + ε)] :

ˆ∂Bσ(y)

|∇u|2 dHn−1 > θ−1ε−1ρ2−nˆB(1+ε)ρ(y)\Bρ(y)

|∇u|2 dx

})≤ εθρ,

and the same for v. Now taking e.g. θ = 1/8 we �nd that there exists a σ ∈ (ρ, (1 + ε)ρ) such thatu, v ∈ H1(∂Bσ(y),N ) and

σ3−nˆ∂Bσ(y)

|∇u|2 + |∇v|2Hn−1 ≤ Cε−1ρ2−nˆB(1+ε)ρ(y)\Bρ(y)

|∇u|2 + |∇v|2 dx. (3)

22


Proceeding similarly with |u− v|2, σ can also be chosen such that

σ1−nˆ∂Bσ(y)

|u− v|2 dHn−1 ≤ Cρ−nε−1ˆB(1+ε)ρ(y)\Bρ(y)

|u− v|2 dx ≤ Cε2n−1δ20 . (4)

Let us apply the lemma of Luckhaus to the function ũ(ω) ≡ u(σω) and ṽ(ω) ≡ v(σω), which ε/4.Then we get w ∈ H1(Sn−1 × [0, ε/4],Rp) such that w(ω, t) agrees with ũ (resp. ṽ) when t lies in a�xed neighborhood of 0 (resp. ε/4), with estimates which give, using (3), (4)ˆSn−1×[0,ε/4]

|∇w|2 dHn−1 ⊗ dλ ≤ CεˆSn−1|∇ũ|2 + |∇ṽ|2 dHn−1 + Cε−1

ˆSn−1|ũ− ṽ|2 dHn−1 (5)

= Cεσ3−nˆ∂Bσ(y)

|∇u|2 + |∇v|2 dHn−1

+ Cε−1σ1−nˆ∂Bσ(y)

|u− v|2 dHn−1

≤ Cρ2−nˆB(1+ε)ρ(y)\Bρ(y)

|∇u|2 + |∇v|2 dx

+ Cε−2ρ−nˆB(1+ε)ρ(y)\Bρ(y)

|u− v|2 dx,

and

dist2(w̃,N ) ≤ Cε1−n(σ3−n

ˆ∂Bσ(y)

|∇u|2 + |∇v|2 dHn−1)1/2(

σ1−nˆ∂Bσ(y)

|u− v|2 dHn−1)1/2

+ Cε−nσ1−nˆ∂Bσ(y)

|u− v|2 dHn−1

≤ Cε

(ρ2−n

ˆB(1+ε)ρ(y)\Bρ(y)

|∇u|2 + |∇v|2 dx

)1/2(ε−2n

ˆB(1+ε)ρ(y)\Bρ(y)

|u− v|2 dx

)1/2+ Cδ20

≤ CεΛ1/2δ0 + Cδ20≤ Cδ0.

Thus for δ0 small enough, the nearest point projection on N is well de�ned. Let us de�ne w̃ ∈H1(B(1+ε)ρ(y) \Bρ(y),Rp) by

w̃(x) =

u(x) for |x| ∈ [ρ, σ]ΠN ◦ w

(x|x| ,

|x|σ − 1

)for |x| ∈ [σ, σ(1 + ε/4)]

v(ϕ(|x|) x|x|

)for |x| ∈ [σ(1 + ε/4), σ(1 + ε/2)]

v(x) for |x| ∈ [σ(1 + ε/2), (1 + ε)ρ]

,

where ϕ is an increasing smooth function on [(1 + ε/4)σ, (1 + ε/2)σ] such that ϕ((1 + ε/4)σ) = σ,ϕ((1 + ε/2)σ) = σ, and 1 ≤ |ψ′|0 ≤ 4. Then w̃ agrees with u in a neighborhood of ∂Bρ(y), with v in aneighborhood of ∂B(1+ε)ρ(y), and it is easy to check that it satis�es

ρ2−nˆB(1+ε)ρ(y)\Bρ(y)

|∇w̃|2 dx ≤Cρ2−nˆB(1+ε)ρ(y)\Bρ(y)

|∇u|2 + |∇v|2 dx

23

3.3 A reverse Poincaré inequality 3 THE ε-REGULARITY THEOREM

+ Cε−2ρ−nˆB(1+ε)ρ(y)\Bρ(y)

|u− v|2 dx.

Hence the result.

Remark 3.4. This corollary is particularly useful with minimizing maps to give an estimate of theenergy of u on the smaller ball B3ρ/4(y). Indeed, imagine that in (1), u is minimizing, then by themonotonicity formula and by minimality we have for any ε ∈ (0, 1)

ρ2−nˆB3ρ/4(y)

|∇u|2 dx ≤ σ2−nˆBσ(y)

|∇u|2 dx

≤ σ2−nˆBσ(y)

|∇w|2 dx

≤ ερ2−nˆBρ(y)

|∇u|2 dx+ ε2n−1Cδ20 .

3.3 A reverse Poincaré inequality

In this section we shall prove that energy minimizing maps satisfy a �reverse� Poincaré inequality,which will be useful to prove the ε-regularity theorem.

Lemma 3.6 (Reverse Poincaré inequality). Let u ∈ H1loc(Ω,N ) be an energy minimizing map, and Λ bea constant. Then there is C = C(n,N ,Λ) such that, if BR(x0) ⊂ Rn and R2−n

´BR(x0)

|∇u|2 dx ≤ Λ,we have

ρ2−nˆBρ/2(y)

|∇u|2 dx ≤ Cρ−nˆBρ(y)

|u− λy,ρ|2 dx,

for each y ∈ BR/2(x0), ρ < R/2.

Proof. From now on we will implicitly only consider functions u such that R−n´BR(x0)

|∇u|2 dx ≤ Λfor some ball BR(x0) ⊂ Ω. Let ε0 be a small positive number depending only on n, N , Λ, to be chosenlater. For those u, ρ ∈ (0, R/2], y ∈ BR/2(y) such that

ρ−nˆBρ(y)

|u− λy,ρ|2 dx > ε20,

we have the required inequality

ρ2−nˆBρ/2(y)

|∇u|2 dx ≤ Cρ−nˆBρ(y)

|u− λy,ρ|2 dx

with C = 2n−2Λ/ε20, which depends only on n and Λ. Indeed by the monotonicity formula, sinceρ ≤ R/2 and BR/4(y) ⊂ BR(x0), we have

ρ2−nˆBρ/2(y)

|∇u|2 dx ≤ 22−n(R/4)2−nˆBR/4(y)

|∇u|2 dx

≤ 2n−2R2−nˆBR(x0)

|∇u|2 dx

≤ 2n−2Λ = Cε20

24


< Cρ−nˆBρ(xy)

|u− λy,ρ|2 dx.

Thus we shall now focus on those u, ρ ∈ (0, R/2], y ∈ BR/2(y) such that

ρ−nˆBρ(y)

|u− λy,ρ|2 dx ≤ ε20, (1)

and show that they satisfy the same inequality for some other C depending only on n, N , Λ. Let ustake δ0 given by Corollary 3.5 (1), then δ ∈ (0, 1), and �nally ε0 ∈ (0, 1) such that ε0 < δnδ0 (thus ε0depends only in n, N , Λ and δ). Then using (1), we have

ρ−nˆBρ(y)

|u− λy,ρ|2 dx ≤ δ2nδ20 ,

so we can apply Corollary 3.5 (1) with δ in place of ε. Thus there exists σ ∈ ( 3ρ4 , ρ), C = C(n,N ,Λ),and w ∈ H1(Bρ(y),N ) which agrees with u in a neighborhood of ∂Bσ(y) such that

σ2−nˆBσ(y)

|∇w|2 dx ≤ δρ2−nˆBρ(y)

|∇u|2 dx+ δ−1Cρ−nˆBρ(y)

|u− λy,ρ|2 dx. (2)

Now since 3ρ/4 < σ < ρ, by the monotonicity formula, we have

ρ2−nˆB3ρ/4(y)


|∇u|2 dx, (3)

and by minimality of u,

σ2−nˆBσ(y)


|∇w|2 dx. (4)

Using (3) and (2), it follows

ρ2−nˆB3ρ/4(y)

|∇u|2 dx ≤ δρ2−nˆBρ(y)

|∇u|2 dx+ δ−1Cρ−nˆBρ(y)

|u− λy,ρ|2 dx (5)

≤ δΛ + Cδ2n−1δ20 ≤ Cδ

for some other C that depends only on n, N , Λ, since δ0 = δ0(n,N ,Λ) and δ2n−1 ≤ δ.Remark 3.5. Notice that this is not true for any ρ, y such that ρ ∈ (0, R/2], y ∈ BR/2(y). Indeed wealso ask that (1) is satis�ed. This prevents us from stating that this is true for any subball Bσ(a) suchthat B2σ(a) ⊂ Bρ(y). If this were the case, we could then use Lemma A.9 to conclude that the reversePoincaré inequality is in fact true in Bρ(y) directly. Instead we are going to consider any subball Br(z)with z ∈ Bρ/2(y) and r ∈ (0, ρ/4], and show that an equality similar to (5) stands true. Then this beingtrue for any such ball, it will also be true for any subball Bσ(a) such that B2σ(a) ⊂ Br(z), thus we canapply Lemma A.9 to show that the reverse Poincaré inequality is satis�ed on Br(z). We conclude bycovering Bρ(y) with balls Bρ/8(yj) and applying this result.

Now consider any ball Br(z), such that z ∈ Bρ/2(y) and r ≤ ρ/4, then we have

r2−nˆBr(z)

|∇u|2 dx ≤ 42−nρ2−nˆBρ/4(z)

|∇u|2 dx

25


≤ 42−nρ2−nˆB3ρ/4(y)

|∇u|2 dx

≤ Cδ,

where C = C(n,N ,Λ), thanks to the monotonicity formula and the fact that Bρ/4(z) ⊂ B3ρ/4(y). Wecan then apply the Poincaré inequality and change δ (which can be chosen arbitrarily) so that we have

r−nˆBr(z)

|u− λz,r|2 dx, r2−nˆBr(z)

|∇u|2 dx ≤ δ,

for any such balls Br(z). Recall that this is true whenever (1) holds, with ε0 small enough dependingonly on n, N , Λ, δ. Now let ε ∈ (0, 1) �xed. We want to apply again Corollary 3.5 with z, r, ε, inplace of ρ, y, δ. So ε and δ0 being �xed, we want

r−nˆBr(z)

|u− λz,r|2 dx ≤ ε2nδ20 .

In view of the arbitrariness of δ in the previous inequality, we can indeed chose ε0 depending only onn, N , Λ, ε, such that this condition holds. Repeating the steps (3), (4), (5) with Br(z), ε, r/2 insteadof Bρ(y), δ, 3ρ/4 yields

r2−nˆBr/2(z)

|∇u|2 dx ≤ Cεr2−nˆBr(z)

|∇u|2 dx+ Cε−1r−nˆBr(z)

|u− λz,r|2 dx.

This being true for any ball Br(z) such that r ≤ ρ/4 and z ∈ Bρ/2(y), it is also true for any subballBσ(a) where B2σ(a) ⊂ Br(z). Thus

σ2−nˆBσ/2(a)

|∇u|2 dx ≤ Cεσ2−nˆBσ(a)

|∇u|2 dx+ Cε−1σ−nˆBσ(a)

|u− λa,σ|2 dx.

But then notice thatˆBσ(a)

|u− λa,σ|2 dx ≤ˆBσ(a)

|u− λz,r|2 dx ≤ˆBr(z)

|u− λz,r|2 dx,

hence

σ2ˆBσ/2(a)

|∇u|2 dx ≤ Cεσ2ˆBσ(a)

|∇u|2 dx+ Cε−1I1,

where I1 :=´Br(z)

|u−λz,r|2 dx for any z ∈ Bρ/2(y), r ≤ ρ/4, and any Bσ(a) such that B2σ(a) ⊂ Br(z).The important fact here is that I1 does not depend on σ, but on ρ instead. We can then use Lemma A.9,taking ε such that Cε = ε0 (ε0 given by the lemma) with γ = Cε

−1I1 , k = 2 and ϕ(A) =´A|∇u|2 dx

to get

r2−nˆBr/2(z)

|∇u|2 dx ≤ Cr−nˆBr(z)

|u− λz,r|2 dx, (6)

where C depends only on n, which is the reverse Poincaré inequality for the ball Br(z). Eventuallywe obtain the inequality for the ball Bρ(y) by covering Bρ/2(y) with balls Bρ/8(yj), j ∈ {1, . . . , Q},

26

3.4 A technical regularity lemma 3 THE ε-REGULARITY THEOREM

where yj ∈ Bρ/2(y) and Q depends only on n. Indeed we can use (6) with ρ/4, yj instead of r, z, andnoticing that

ˆBρ/4(yj)

|u− λyj ,ρ/4|2

dx ≤ˆBρ/4(yj)

|u− λy,ρ|2 dx ≤ˆBρ(y)

|u− λy,ρ|2 dx,

and summing over j we reach the expected inequality. Note that in the cases where (1) stands true,the constant C depends only on n, but in the other case, it depends on n and ε0, which itself dependson n, N , and Λ, so eventually the constant C depends on all three (only).

3.4 A technical regularity lemma

We are going to establish the following technical regularity lemma.

Lemma 3.7. Let α ∈ (0, 1), and β ≥ 1. Then there exists δ0 = δ0(n, α, β) > 0 such that, ifu ∈ H1(BR(x0),Rp) satis�es

∆u = F (3.1)

in the weak sense D′(BR(x0)), where BR(x0) ⊂ Rn, F ∈ L1(BR(x0)) and if

(i) |F (x)| ≤ β|∇u(x)|2, for a.e. x ∈ BR(x0),

(ii) for any ball Bρ(y) such that y ∈ BR/2(x0) and ρ ≤ R/2, we have(ρ2

)2−n ˆBρ/2(y)

|∇u|2 dx ≤ βρ−nˆBρ(y)

|u− λy,ρ|2 dx,

(iii) R−n´BR(x0)

|u− λx0,R|2

dx ≤ δ20,

then u ∈ C1,α(BR/4(x0)) and for small values of R (e.g. R ≤ 1) we have the estimate

|u− u(x0)|1,BR/4(x0) + [∇u]α,BR/4(x0) ≤ CI1/20 , (3.2)

where I0 := R−n´BR(x0)

|u− λx0,R|2

dx, and C depends only on n, α and β.

Proof. The proof relies heavily on Campanato's lemma (Lemma A.3) and on the harmonic approxi-mation lemma (Lemma B.3). First we are going to show that u ∈ C0,α(BR/2(x0)) and give an upperbound of its norm.

Holder continuity of u According to (3.1), we have

ˆBR(x0)

〈∇u,∇ϕ〉Rnp dx = −ˆBR(x0)

〈F,ϕ〉Rp dx, ∀ϕ ∈ C∞c (BR(x0)),

thus by taking ϕ ∈ C∞c (Bρ/2(y)) with y ∈ BR/2(x0) and ρ ∈ (0, R/2], and using (i) and (ii), it follows∣∣∣∣∣(ρ2)2−nˆBρ/2(y)

〈∇u,∇ϕ〉Rnp dx

∣∣∣∣∣ ≤ |ϕ|0,Bρ/2(y)β2ρ−nˆBρ(y)

|u− λy,ρ|2 dx.

27


and then, using Lemma B.1 on ∇w, with j = 0, and Jensen's inequality

|∇w|20,Bθρ(y) ≤

(Cρ−n

ˆBρ(y)

|∇w| dx

)2(6)

≤ Cρ−nˆBρ(y)

|∇w|2 dx,

where the constants C depend only on n (see Remark B.1). Thus, putting together (5), (6) and (2a),it follows


|w − w(y)|2 dx ≤ Cθ2ρ2−nˆBρ(y)

|∇w|2 dx

≤ Cθ2,

where C depends only on n. Now plugging the upper bounds of the terms of the right side of (3), wereach


|v − w(y)|2 dx ≤ θ−nε2 + Cθ2.

In terms of u instead of v, setting λ := lw(y), this means


|u− λ|2 dx ≤ l2(θ−nε2 + Cθ2) (7)

= (β2θ−nε2 + β2Cθ2)ρ−nˆBρ(y)

|u− λy,ρ|2 dx.

Let us now choose θ small enough such that β2Cθ2 ≤ 12θ2α, which is possible since α ∈ (0, 1). Then let

us choose ε such that β2θ−nε2 ≤ 12θ2α. Note that θ and ε depends only on n, α and β. By de�nition

of λy,ρ, with θ and ε so chosen, (7) implies


|u− λy,θρ|2 dx ≤ θ2αˆBρ(y)

|u− λy,ρ|2 dx. (8)

This is true for any ball Bρ(y) with y ∈ BR/2(y), ρ ∈ (0, R/2], provided that (1) stands true. Letus note that, if this true for Bρ(y), then according to (8) it is also true for Bθρ(y). Thus by a trivialrecursion argument, we get

(θjρ)−nˆBθjρ(y)

|u− λy,θjρ|2

dx ≤ θ2jαρ−nˆBρ(y)

|u− λy,ρ|2 dx, ∀j ∈ N. (9)

Now setting I0 = R−n ´

BR(y)|u− λy,R|2 dx, for any y ∈ BR/2(x0), ρ = R/2, we have

ρ−nˆBρ(y)

|u− λy,ρ|2 dx ≤ 2nR−nˆBR(x0)

|u− λx0,R|2

dx = 2nI0,

thus choosing δ0 ≤ δ2−n/2, δ0 depends only on n, α, β, and if I0 ≤ δ20 (i.e. (iii) stands true), then thestarting hypothesis (1) is true for Bρ(y). Hence according to (9)

(θjR/2)−nˆBθjR/2(y)

|u− λy,θjR/2|2

dx ≤ θ2jα2nI0,

29


for any y ∈ BR/2(x0). Now taking σ ∈ (0, R/2), there exists j ∈ N such that θj+1R/2 ≤ σ ≤ θjR/2,and we have

σ−nˆBσ(y)

|u− λσ,y|2 dx ≤ θ−n(θjR/2)−nˆBθjR/2(y)

|u− λy,θjR/2|2

dx

≤ θ−nθ2jα2nI0= θ2α−n(θj+1)2α2nI0

≤ θ2α−n2n+2αI0( σR

)2α,

hence, by Campanato's lemma (Lemma A.3), u ∈ C0,α(BR/2(x0)),

[u]α,BR/2(x0) ≤ CR−αI

1/20 , (10)

where C is a constant depending only on n, α, β since θ depends only on those quantities. This endsthe �rst part of the proof.

Hölder continuity of ∇u Let y ∈ B3R/8(x0) and ρ ∈ (0, R/8). Then Bρ(y) ⊂ BR/2(x0), thusu ∈ C0,α(Bρ(y)). It is standard using Lax-Milgram theorem that there exists v ∈ H1(Bρ(y),Rp) suchthat {

∆v = 0 in Bρ(y) (weakly)v = u on ∂Bρ(y)

.

Since v is harmonic, we haveˆBρ(y)

〈∇v,∇ϕ〉Rnp dx = 0, ∀ϕ ∈ C∞0 (Bρ(y),Rp),

so with (3.1), it followsˆBρ(y)

〈∇(v − u),∇ϕ〉Rnp dx =ˆBρ(y)

〈F,ϕ〉 dx. (11)

Now notice that v−u ∈ H10 (Bρ(y),Rp) by de�nition of v, so by density of C∞c (Bρ(y)) inH10 (Bρ(y),Rp),we can take ϕ = u− v in (11), which yields, using (i)

ˆBρ(y)

|∇(u− v)|2 dx =ˆBρ(y)

〈u− v, F 〉Rp dx (12)

≤ β|u− v|0,Bρ(y)ˆBρ(y)

|∇u|2 dx.

Now let y0 ∈ ∂Bρ(y), then by the triangle inequality and the fact that u = v on ∂Bρ(y), we have|u− v|0,Bρ(y) ≤ |u−u(y0)|0,Bρ(y) + |v(y0)− v|0,Bρ(y). By the Hölder continuity of u and (10), providedthat δ0 was chosen smaller than 1, it follows

|u− u(y0)|0,Bρ(y) ≤ CI1/20

( ρR

)α≤ Cδ0

( ρR

)α≤ C

( ρR

)α,

where C depends only on n, α, β. And by virtue of the maximum principle applied to the harmonicfunctions vi, where v = (v1, . . . , vp) , we have

|v(y0)− v|0,Bρ(y) ≤√n maxi∈{1,...,p}

|vi(y0)− vi|0,Bρ

30


=√n maxi∈{1,...,p}

|ui(y0)− ui|0,∂Bρ(y)

≤ C( ρR

)α,

Plugging these two inequalities in (12), we get

ˆBρ(y)

|∇(u− v)|2 dx ≤ C( ρR

)α ˆBρ(y)

|∇u|2 dx,

where C depends only on n, α and β. Now since y ∈ B3R/8(x0) and 2ρ ≤ R/2, we can use the reversePoincaré inequality (ii) with 2ρ instead of ρ to obtain

ˆBρ(y)

|∇u|2 dx ≤ 2−nρ−2ˆB2ρ(y)

|u− λy,2ρ|2 dx,

thus, back to the previous inequality, by de�nition of λy,2ρ and the Hölder continuity of u, we get

ˆBρ(y)

|∇(u− v)|2 dx ≤ CR2( ρR

)α−2 ˆB2ρ(y)

|u− λy,2ρ|2 dx (13)

≤ CR2( ρR

)α−2 ˆB2ρ(y)

|u− u(y)|2 dx

≤ CR2( ρR

)3α+n−2I0.

Now let us take σ < ρ/2, and de�ne Λσ,y := |Bσ(y)|−1´Bσ(y)

∇u. We want to apply again Campanato'slemma on ∇u, so we want upper bounds for σ−n

´Bσ(y)

|∇u−Λy,σ|2 dx. Let us use the squared triangleinequality and (13) to compute

σ−nˆBσ(y)

|∇u−∇v(y)|2 dx ≤ 2σ−nˆBρ(y)

|∇(u− v)|2 dx+ 2σ−nˆBσ(y)

|∇v −∇v(y)|2 dx (14)

≤ CR2( ρσ

)n ( ρR

)3α−2I0 + 2σ

2|∇2v|20,Bσ(y).

Since v is harmonic, so is ∇v − Λ for any constant Λ, thus we can use Lemma B.1 with ∇v − Λ,R = 3ρ/4, θ = 2/3 and j = 1, which yields

|∇2v|20,Bσ(y) ≤ |∇2v|20,Bρ/2(y) ≤ Cρ

−2

(ρ−nˆB3ρ/4(y)

|∇v − Λ| dx

)2(15)

≤ Cρ−2ρ−nˆBρ(y)

|∇v − Λ|2 dx.

Now using the squared triangle inequality, the de�nition of Λy,ρ, and the inequalities above withΛ = Λy,ρ, we have

σ2|∇2v|20,Bσ(y) ≤(σ

ρ

)2ρ−nˆBρ(y)

|∇v − Λy,ρ|2 dx

≤ 2(σ

ρ

)2ρ−nˆBρ(y)

|∇(v − u)|2 dx+ 2(σ

ρ

)2ρ−nˆBρ(y)

|∇u− Λy,ρ|2 dx

31


≤ CR2(σ

ρ

)2 ( ρR

)3α−2I0 + 2

(σ

ρ

)2ρ−nˆBρ(y)

|∇u|2 dx,

thus with (14)

σ−nˆBσ(y)

|∇u− Λy,σ|2 dx ≤ σ−nˆBσ(y)

|∇u−∇v(y)|2 dx

≤ CR2( ρσ

)n ( ρR

)3α−2I0 + CR

2

(σ

ρ

)2 ( ρR

)3α−2I0

+ C

(σ

ρ

)2ρ−nˆBρ(y)

|∇u|2 dx.

Using the reverse Poincaré inequality (ii), we have

ρ−nˆBρ(y)

|∇u|2 dx ≤ Cρ−2ρ−nˆB2ρ(y)

|u− λy,2ρ|2 dx (16)

≤ CR2( ρR

)2α−2I0,

hence, since (ρ/R)3α ≤ (ρ/R)2α,

σ−nˆBσ(y)

|∇u− Λy,σ|2 dx ≤ CR2( ρσ

)n ( ρR

)3α−2I0 + CR

2

(σ

ρ

)2 ( ρR

)2α−2I0. (17)

Note that this is true for any 0 < 2σ ≤ ρ ≤ 1/8 and y ∈ B3R/8(x0). Now choose σ such that 2σ = ρκ,with κ := 1 + αn+2 , then clearly 2σ < ρ, and a simple computation shows that

R2( ρσ

)n ( ρR

)3α−2= C1R

2( σR

)2γR2(σ

ρ

)2 ( ρR

)2α−2= C2R

2( σR

)2γ,

where γ :=[α(

1 + 1n+2

)− 1]/κ, and C1, C2 depend only on n, α, β. Now we can choose an α

′ ∈ (0, 1)such that α < α′ < 1 and the corresponding γ is positive, and consider that δ0 was chosen so thatu ∈ C0,α′(B1/2(x0)) as well as C0,α(BR/2(x0)) whenever (iii) stands true. Then we can do all thecomputations above with α′ rather than α, and with γ so chosen, we can rewrite (17) in the form

σ−nˆBσ(y)

|∇u− Λy,σ|2 dx ≤ CR2I0( σR

)2γ, ∀σ < (1/8)

κ

2, ∀y ∈ B3R/8(x0).

Hence, by Campanato's lemma, ∇u ∈ C0,γ(B3R/8(x0)) and [∇u]γ,B3R/8(x0) ≤ CI1/20 R

1−γ , where C

depends only on n, α and β. Now we can show that |∇u|0,B3R/8(x0) ≤ CRI0. Indeed let us considerx, y ∈ BR/8(x0), then by the squared triangle inequality

|∇u(x)|2 ≤ 2|∇u(x)−∇u(y)|2 + 2|∇u(y)|2

≤ CR2I0 + 2|∇u(y)|2

32

3.5 Proof of the ε-regularity theorem 3 THE ε-REGULARITY THEOREM

Then taking the average integral over the ball BR/8(x0) and using (16) with y = x0 and ρ = R/8, weget

|∇u(x)|2 = −ˆBR/8(x0)

|∇u(x)|2 dx(y)

≤ CR2I0 + C(R/8)−nˆBR/8(x0)

|∇u(y)|2 dx(y)

≤ CR2I0,

thus

|∇u(x)| ≤ CRI1/20 . (18)

Hence u satis�es ∆u = F weakly in B3R/8(x0), where F ≤ β|∇u|2is bounded, so applying the elliptic

regularity lemma (Lemma A.8) to u− u(x0), we get u ∈ C1,α(B3R/8(x0)), and

[∇u]α,BR/4(x0) ≤ CR−1−α|u− u(x0)|0,B3R/8(x0) + CR

1−α|∇u|20,B3R/8(x0)≤ CR−α|∇u|0,B3R/8(x0) + CR

1−α|∇u|20,B3R/8(x0)≤ CR1−αI1/20 + CR3−αI0 ≤ CI

1/20 ,

where C depends only on n, α, β, because I0, R ≤ 1 and α ∈ (0, 1). Together with (18), and thefact that |u− u(x0)|0,BR/4(x0) ≤ R|∇u|0,BR/4(x0) ≤ CI

1/20 , this gives estimate (3.2) and concludes the

proof.

Remark 3.6. Note that the hypothesis �R small� was needed only to get estimate (3.2). Anyways weare almost always interested in the behaviour of u close to some point x0.

3.5 Proof of the ε-regularity theorem

Let us recall that we want to prove the theorem for θ = 1/4, and get an estimate on Rj |∇ju|0,BR/8(x0).

Proof. According to Lemma 3.6, u satis�es a reverse Poincaré inequality such as in Lemma 3.7 (ii),where β depends only on n, N and Λ. Also, according to Section B.2, u satis�es

∆u = −n∑j=1

Au(∇ju,∇ju), in D′(BR(x0)), (1)

where Au(x) is bilinear continuous, and y 7→ Ay is bounded by virtue of the smoothness and compact-ness of N , thus ∣∣∣∣∣∣

n∑j=1

Au(∇ju,∇ju)

∣∣∣∣∣∣ ≤ C|∇u|2,hence ∆u = F where F satis�es inequality (i) of the technical lemma (Lemma 3.7). Then considering

δ0 given by this technical lemma as ε, we get that if R−n ´

BR(x0)|u − λx0,R|

2dx ≤ ε2, then u ∈

C1,α(BR/4x0) for some α ∈ (0, 1), and for R small we have

|∇u|0,BR/4(x0) + [∇u]α,BR/4(x0) ≤ C

(R−n

ˆBR(x0)

|u− λx0,R|2

dx

)1/2,

33

4 PARTIAL REGULARITY

where C depends only on n, α, N , Λ. But then∑nj=1Au(∇ju,∇ju) ∈ C0,α(BR/4(x0)), which is the

right side of (1), so by the elliptic regularity lemma (Lemma A.8), u ∈ C2,α(BR/4(x0)) and we havethe estimate

R2|∇2u|0,B( 18+2−3)R

(x0)≤ C

(R−n

ˆBR(x0)

|u− λx0,R|2

dx

)1/2,

where C = C(n,N , α,Λ). By applying the elliptic regularity lemma recursively, we get that u issmooth on BR/8(x0) and the estimates

Rj |∇ju|0,BθjR(x0) ≤ C

(R−n

ˆBR(x0)

|u− λx0,R|2

dx

)1/2, ∀j ∈ N \ {0},

where 1/8 < θj :=(

18 + 2

−(j+1)) < 1/4. Taking e.g. α = 1/2, then ε and C depend only on n, N andΛ, hence the result.

4 Partial regularity

The aim of this section is to establish partial regularity results of energy minimizing maps, as well asintroduce standard tools needed for this study: the introduction of the density function will allow tostate a simple criterion for points to be singular or regular, while the compactness theorem is essentialto construct the so-called tangent maps. These will enable us to �slice� the singular set into particularincreasing subsets and eventually to prove that the Hausdor� dimension (see Appendix A.1) of thesingular set of an energy minimizing map is at most n− 3 for n ≥ 3.

Unless stated otherwise, Ω denotes an open subset of Rn with n ≥ 2, and N a compact submanifoldof Rp with dimensions m ≥ 2. The j-dimensional Hausdor� measure is denoted by Hj , and theHausdor� dimension by dimH.

4.1 Compactness theorem

In this section we show a compactness theorem for minimizing maps. Not only this is a very nice resultin itself, but it is also a precious tool to study the singular set of minimizing maps, especially throughthe de�nition of tangent maps which it allows us to de�ne.

Theorem 4.1 (Compactness for minimizing maps). Let (uj) be a sequence of energy minimizing mapsin H1loc(Ω,N ) such that supj

´Bρ(y)

|∇uj |2 dx < +∞ for all Bρ(y) ⊂ Ω, then there is a subsequence(uj′) and an energy minimizing map u ∈ H1loc(Ω,N ) such that uj′ → u in H1loc(Ω,Rp).

Remarks 4.1.

(i) Note that uj′ → u in H1loc(Ω,Rp) is equivalent to uj′ → u in H1(Bρ(y),Rp) for all Bρ(y) ⊂ Ω.

(ii) Note also that the boundedness of the energy on each ball Bρ(y) and the Rellich compactnesstheorem already give us that there exists u ∈ H1loc(Ω,Rp) and a subsequence (uj′) such thatuj′ converges to u strongly in L

2loc(Ω,Rp) and weakly in H1loc(Ω,Rp). Note that up to another

sequence extraction, uj′ → u, a.e., thus for almost every x ∈ Ω, u(x) ∈ N , i.e. u ∈ H1loc(Ω,N ).The interesting result of this theorem is that u is energy minimizing and that ∇uj′ convergesstrongly to ∇u in L2loc(Ω,Rnp).

34

4.1 Compactness theorem 4 PARTIAL REGULARITY

(iii) The main di�culty lies in the fact that, if Bρ(y) is a ball such that Bρ(y) ⊂ Ω, the maps uj donot agree with u in a neighborhood of ∂Bρ(y). If that were the case, then by weak lower semi-continuity of the energy, we would trivially get the minimality of u, and the strong convergence of∇uj . The key is to use Luckhaus lemma (Lemma 3.4) to glue uj and u and then the minimalityof the maps uj .

Proof. Let us consider the subsequence discussed in the above remark, which we note note (uj) for thesake of simplicity, and u such that uj → u, a.e., strongly in L2loc(Ω,N ) and weakly in H1loc(Ω,N ). LetBρ0(y) ⊂ Ω, and δ > 0, θ ∈ (0, 1) �xed. Then let M be any odd integer such that

lim supj

ρ02−nˆBρ0 (y)

|∇uj |2 dx < Mδ/2, (1)

and ε ∈ (0, (1− θ)/M). To get the idea, consider the annulus Bρ0(y) \Bθρ0(y), from which we cut Mslices Bρ0(θ+lε)(y) \ Bρ0(θ+(l−2)ε)(y), l ∈ {2, 4, . . . ,M + 1}, of size 2ερ0. Then by virtue of (1), for atleast one of those slices we have

ρ02−nˆBρ0(θ+lε)(y)\Bρ0(θ+(l−2)ε)(y)

|∇uj |2 dx < δ, (2)

for countably many j. Indeed, otherwise by summing over l we would have

ρ02−nˆBρ0(θ+Mε)(y)\Bρ0θ(y)

|∇uj |2 dx ≥Mδ/2

for all j except a �nite number of them, which contradicts (1) since Bρ0(θ+Mε)(y) \Bρ0θ(y) ⊂ Bρ0(y).Thus let l be such that (2) holds for countably many j. Now let ρ := ρ0(θ + (l − 2)ε). Then we cancheck that ρ(1 + ε) = ρ0(θ + (l − 1)ε+ (l − 2)ε2) < ρ0(θ + lε), so

ρ0θ < ρ0(θ + (l − 2)ε) = ρ < ρ(1 + ε) < ρ0(θ + lε),

which implies that Bρ(1+ε)(y) \ Bρ(y) ⊂ Bρ0(θ+lε)(y) \ Bρ0(θ+(l−2)ε)(y), hence by (2) there exists asubsequence uj′ such that

ρ02−nˆBρ(1+ε)(y)\Bρ(y)

|∇uj′ |2 dx < δ, ∀j. (3)

Note that since uj′ → u weakly in Bρ(y), by lower semi-continuity of the energy, this implies

ρ02−nˆBρ(y)

|∇u|2 dx ≤ δ. (4)

Then since ρ0 ≤ ρ, using (3) and (4) we get


|∇u|2 + |∇uj′ |2 dx ≤ 2δ, ∀j (5)

We can now apply the corollary of Luckhaus lemma (Corollary 3.5) with Λ = 1, to get δ0 and Cdepending only on n and N , such that for any δ < 1/2, any j large enough, there exists wj′ ∈H1(Bρ(1+ε)(y)(y) \ Bρ(y),N ) with w = u in a neighborhood of ∂Bρ(y), w = uj′ in a neighborhood of∂Bρ(1+ε)(y) and


|∇wj′ |2 dx (6)

35

4.1 Compactness theorem 4 PARTIAL REGULARITY

≤ Cρ2−nˆBρ(1+ε)(y)\Bρ(y)

|∇u|2 + |∇uj′ |2 dx+ Cε−2ρ−nˆBρ(1+ε)(y)\Bρ(y)

|u− uj′ |2 dx.

This holds by virtue of (5), provided that ρ−n´Bρ(1+ε)(y)\Bρ(y)

|u − uj′ |2 dx < δ20 for any j largeenough, which is obvious because uj′ → u strongly in Bρ(y). Now we are going to show that u isenergy minimizing on Bθρ0(y). Let us consider some v ∈ H1(Bθρ0(y),N ) which agrees with u in aneighborhood of ∂Bθρ0(y). Let us recall the chain of inequalities

0 < ρ0θ < ρ < ρ(1 + ε) < ρ0,

and extend v by ṽ ∈ H1(Bρ(y),N ) de�ned by

ṽ =

{v in Bθρ0(y)u in Bρ(y) \Bθρ0(y).

Also de�ne ũj′ ∈ H1(Bρ0(y),N ) by

ũj′ =

ṽ in Bρ(y)wj′ in Bρ(1+ε)(y) \Bρ(y)uj′ in Bρ0(y) \Bρ(1+ε)(y)

,

then uj′ = ũj′ in a neighborhood of ∂Bρ0(y), thus by minimality of uj′ we haveˆBρ0

|∇uj′ |2 dx ≤ˆBρ0

|∇ũj′ |2 dx,

which means, since uj′ = ũj′ in Bρ0(y) \Bρ(1+ε)(y),ˆBρ(1+ε)(y)

|∇uj′ |2 dx ≤ˆBρ(1+ε)(y)

|∇ũj′ |2 dx

=

ˆBρ(y)

|∇ṽ|2 dx+ˆBρ(1+ε)(y)\Bρ(y)

|∇wj′ |2 dx.

Plugging (5) and (6) in this inequality and passing to the lim infj we get

ρ2−nˆBρ(y)

|∇u|2 dx ≤ lim infj

ρ2−nˆBρ(y)

|∇uj′ |2 dx ≤ ρ2−nˆBρ(y)

|∇ṽ|2 dx+ Cρ2−nδ, (7)

for any δ small, where C depends only on n and N . Noting that u = ṽ in Bρ(y) \Bθρ0(y), this meansˆBθρ0 (y)

|∇u|2 dx ≤ˆBθρ0 (y)

|∇v|2 dx+ Cδ,

which implies the minimality of u in Bθρ0(y) since δ is arbitrary small. Now recalling the arbitrarinessof θ and ρ0, u is actually energy minimizing in Ω. There remains to prove that up to the extraction ofa subsequence, ∇uj converges strongly locally to ∇u in Ω. Choosing ṽ = u in (7), we have

lim infj

ˆBρ(y)

|∇uj′ |2 dx ≤ˆBρ(y)

|∇u|2 dx+ Cρn−2δ

≤ˆBρ0 (y)

|∇u|2 dx+ Cρ0n−2δ.

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4.2 The density function 4 PARTIAL REGULARITY

Now δ being �xed, take any ρ1 < ρ0, then since ρ = ρ0(θ+ (l− 2)ε) where (l− 2)ε ∈ (0, 1− θ), takingθ close to 1 gives a ρ such that ρ1 < ρ < ρ0, thus

lim infj

ˆBρ1 (y)

|∇uj′ |2 dx ≤ˆBρ0 (y)

|∇u|2 dx+ Cρ0n−2δ,

so by the arbitrariness of δ, and the de�nition of the lim inf the following inequality holds for the wholesequence (uj), and any ρ1 < ρ0:

lim infj

ˆBρ1 (y)

|∇uj |2 dx ≤ˆBρ0 (y)

|∇u|2 dx.

Hence by the arbitrariness of ρ0, letting ρ0 → ρ1 yields

lim infj→+∞

ˆBρ(y)

|∇uj |2 dx ≤ˆBρ(y)

|∇u|2 dx, (8)

for any ball Bρ(y) ⊂ Ω. Then by passing to the lim infj inˆBρ(y)

|∇u−∇uj |2 dx =ˆBρ(y)

|∇u|2 dx+ˆBρ(y)

|∇uj |2 dx− 2ˆBρ(y)

〈∇u,∇uj〉 dx

we get that lim infj´Bρ(y)

|∇u−∇uj |2 dx = 0, thus we can extract a subsequence (uj′) such that ∇uj′converges strongly to ∇u in Bρ(y). Then by covering Ω with countably many balls Bρj (yj) such thatBρj (yj) ⊂ Ω and a diagonal extraction we get the result.

4.2 The density function

Let us recall that by virtue of the monotonicity formula (Proposition 2.3), the function ρ 7→ ρ2−n´Bρ(y)

|∇u|2 dxis increasing, so the limit when ρ goes to zero exists.

De�nition 4.1 (Density function). Given an energy minimizing map u ∈ H1loc(Ω,N ), then we de�neΘu on Ω by

Θu(y) = limρ→0

ρ2−nˆBρ(y)

|∇u|2 dx.

The interesting property of the density function is its upper semi-continuity w.r.t to y and u.

Proposition 4.2 (Upper semi-continuity of the density). The function (u, y) 7→ Θu(y) is uppersemi-continuous in the sense that, if (uj)j∈N is a sequence of energy minimizing maps in H1loc(Ω,N )that converges locally in L2 to a minimizing map u ∈ L2loc(Ω,N ), with locally bounded energy (i.e.supj

´Bρ(y)

|∇u|2 dx < +∞ for any ball Bρ(y) ⊂ Ω), and if yj → y ∈ Ω, then u is energy minimizingand Θu(y) ≥ lim supj→+∞Θuj (yj).

Proof. According to the compactness theorem (Theorem 4.1), u is actually an energy minimizing mapin H1loc(Ω,N ), and uj converges to u locally in H1. Now let ε > 0 and ρ > 0 such that Bρ+ε(y) ⊂ Ω.Up to the extraction of a subsequence, we can assume Bρ(yj) ⊂ Bρ+ε(y), ∀j ∈ N. Let δ > 0, thensince uj converges locally to u in H

1, there exists J = J(ρ, ε, δ) such that

ρ2−nˆBρ+ε(y)

|∇uj |2 dx ≤ ρ2−nˆBρ+ε(y)

|∇u|2 dx+ δ, ∀j ≥ J,

37

4.2 The density function 4 PARTIAL REGULARITY

then we have by the monotonicity formula, for j ≥ J

Θuj (yj) ≤ ρ2−nˆBρ(yj)

|∇uj |2 dx

≤ ρ2−nˆBρ+ε(y)

|∇uj |2 dx

≤ ρ2−nˆBρ+ε(y)

|∇u|2 dx+ δ,

thus

lim supj→+∞

Θuj (yj) ≤ ρ2−nˆBρ+ε(y)

|∇u|2 dx+ δ.

Then, letting δ and ε go to zero, we get

lim supj→+∞

Θuj (yj) ≤ ρ2−nˆBρ(y)

|∇u|2 dx,

hence the result when ρ goes to zero.

Now let us state two direct corollaries of the ε-regularity theorem, which respectively give a lowerbound on the density of singular points, and characterize regular points in terms of density.

Proposition 4.3. The density of the singular points of any u ∈ H1loc(Ω,N ) is bounded from below bysome ε0 which depends only on n and N .

Proof. Trivial using Corollary 3.3.

Proposition 4.4 (Equivalence regularity/zero density). Let u ∈ H1loc(Ω,N ) be an energy minimizingmap. Then we have

y ∈ reg u ⇐⇒ Θu(y) = 0

Proof. The direct implication is trivial since u smooth in a neighborhood of y guarantees that |∇u| isbounded near y, and the reciprocal is an immediate consequence of Corollary 3.3.

It is not that di�cult to show that the Hausdor� measure of the singular set is at most n − 2, asstated in the next corollary. We will show later that it is even smaller, at most n− 3.

Proposition 4.5 (Partial regularity). Let u ∈ H1loc(Ω,N ) be an energy minimizing map. ThenHn−2(sing u) = 0.

Remark 4.2. In particular this shows that dimH(sing u) ≤ n− 2.

Proof. By virtue of Proposition A.1, it is enough to show that, for any compact subset K of Ω, forevery ε, δ > 0, there exists a countable covering of sing u ∩ K by balls Bρj (yj) such that ρj ≤ δand

∑j ρ

n

on singularities of energy minimizing maps · the theory of regularity of energy minimizing maps is...

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