In Copyright - Non-Commercial Use Permitted Rights ...29852/eth-29852-02.pdfContents Abstract vii Zusammenfassung ix Résumé xi 1Weaklybiharmonic maps 1.1 Definitions and basic properties

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Doctoral Thesis

Biharmonic maps

Author(s): Angelsberg, Gilles

Publication Date: 2007

Permanent Link: https://doi.org/10.3929/ethz-a-005463226

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Diss. ETH No. 17187

Biharmonic Maps

A dissertation submitted to the

Swiss Federal Institute of Technology

Zurich

for the degree of

Doctor of Science

presented by

Gilles Angelsberg

Dipl. Math. Université de Fribourg

born June 3, 1980

citizen of the

Grand Duchy of Luxembourg

accepted on the recommendation of

Prof. Dr. Michael Struwe, examiner

Prof. Dr. Tristan Rivière, co-examiner

Prof. Dr. Norbert Hungerbühler, co-examiner

2007

r

i

e

BlankLeer/leaf

Von einem gewissen Punkt an gibt es keine Rückkehr mehr.

Dieser Punkt ist zu erreichen. "

Franz Kafka

Seite Leer /Blank leaf

Fir main Papp

Seite Leer /

Blank leaf

Abstract

This work investigates existence and regularity properties of extrinsic weakly biharmonic

maps from an Euclidean domain Q into a smooth compact Riemannian manifold J\f.

The first part deals with the existence of large solutions for biharmonic maps in four

dimensions. Introducing the notion of topological degree for Sobolev maps from R4 to S4,we show that there exists locally minimizing extrinsic biharmonic maps u* of topological

degree —1 and 1. The proof is based upon P.L. Lions' concentration compactness princi¬

ple. This allows us to exclude the phenomena of concentration and vanishing at infinity,for minimizing sequences for the Hessian energy with prescribed topological degree —1 or

1, up to rescalings and translations. We infer that the degree is preserved in the limit.

Then, for fl = Bi unit ball in R4 andM = S4, we show the existence of two non homotopicbiharmonic maps for certain Dirichlet boundary data. The key step is a

"

sphere attachinglemma" stating the existence of a map u, non homotopic to the absolute minimizer u of

the Dirichlet problem, having less energy than the sum of the energies of u and u*. Thus,we can exclude bubbling of minimizing sequences in the considered homotopy class in

order to conclude compactness.

In the second part, we consider the regularity problem for extrinsic weakly biharmonic

maps. We first give a rigorous proof of the monotonicity formula of S.-Y.A. Chang,

L.Wang and P.Yang for extrinsic stationary biharmonic maps of class W2,2, allowingto show partial regularity in higher dimensions.

Then, we show regularity results for polyharmonic maps in higher dimensions. Our

method uses a sharp interpolation inequality by Adams-Frazier and avoids the use of

moving frames. In view of the monotonicity formulae for stationary harmonic and bihar¬

monic maps we conclude, in theses cases, partial regularity.

Seite Leer /

Blank leaf

Zusammenfassung

Diese Arbeit befasst sich mit Existenz- und Regularitätseigenschaften von extrinsisch

schwach biharmonischen Abbildungen zwischen einem Euklidischen Gebiet ü und einer

glatten kompakten Riemannschen Mannigfaltigkeit N.

Im ersten Teil untersuchen wir die Existenz von grossen Lösungen der biharmonischen

Gleichung in vier Dimensionen. Nach Einführung des topologischen Grades für Sobolev-

Abbildungen von R4 nach S4, zeigen wir die Existenz von lokal minimierenden extrinsisch

biharmonischen Abbildungen u* vom Grad —1 und 1. Der Beweis beruht auf P.L. Lions'

"concentration compactness principle". Dies erlaubt uns ein mögliches Verschwinden der

Energie gegen unendlich, sowie die Konzentration an Punkten für minimierende Folgender Hesseschen Energie (bis auf Reskalierungen und Translationen) auszuschliessen. Somit

bleibt der Grad im Limes erhalten.

Des weiteren zeigen wir für Q = B\ C R4 und M — S4 die Existenz von zwei nicht ho-

motopen biharmonischen Abbildungen für gewisse Randdaten. Der entscheidende Schritt

ist das "sphere attaching lemma". Es liefert die Existenz einer Abbildung u, die nicht

homotop zur absolut minimierenden Abbildung u des Dirichlet-Problems ist, und deren

Energie kleiner ist als die Summe der Energien von u und u*. Diese Aprioriabschätzungerlaubt uns, das Ablösen einer Sphäre für minimierende Folgen in der betrachteten Ho-

motopieklasse auszuschliessen.

Im zweiten Teil betrachten wir Regularitätsfragen für extrinsisch schwach biharmonische

Abbildungen. Zunächst beweisen wir die Monotonieformel von S.-Y.A.Chang, L.Wangund P.Yang für extrinsisch stationär biharmonische Abbildungen in W2'2. Diese erlaubt

die Herleitung der partiellen Regularität in höheren Dimensionen.

Weiter zeigen wir Regularitätsresultate für polyharmonische Abbildungen. Unsere Me¬

thode verwendet eine Interpolationsungleichung von Adams-Frazier, und vermeidet die

Anwendung von "moving frames". Angesichts der Monotonieformeln für stationär har¬

monische und biharmonische Abbildungen, erhalten wir in diesen Fällen partielle Regu¬larität.

! Se/te Leer /Blank leaf

Résumé

Nous étudions dans cette thèse quelques propriétés d'existence et de régularité des appli¬cations faiblement biharmoniques (au sens extrinsèque), définies sur un domaine Euclidien

et à valeurs dans une variété Riemannienne compacte de classe C°°.

Dans la première partie, nous analysons l'existence de grandes solutions pour l'équation

biharmonique. Introduisant le degré topologique pour les applications de Sobolev de R4 à

valeurs dans S4, nous démontrons l'existence de minimiseurs locaux u* de l'énergie Hessi¬

enne ayant un degré topologique prescrit -1 ou 1. La preuve est fondée sur le principede concentration-compacité de P.L. Lions. Par suite, les phénomènes de concentration et

d'annihilation à l'infini s'excluent pour des suites minimisantes de l'énergie Hessienne (àtranslations et dilatations près). Le degré est donc préservé à la limite.

De plus, pour Ü — B\ boule d'unité dans R4 et M — S4, nous considérons des problèmesde Dirichlet pour l'équation biharmonique admettant (au moins) deux solutions non-

homotopes. L'outil principal est un lemme, appelé "sphere attaching lemma". Celui-ci

fournit une application u, étant non-homotope au minimiseur global u du problème de

Dirichlet, et ayant une énergie inférieure à la somme des énergies de u et u*. Nous en

déduisons la convergence forte des suites minimisantes dans les classes d'homotopie con¬

sidérées.

La deuxième partie est consacrée au problème de régularité des applications faiblement

biharmoniques. Nous démontrons d'abord la formule de monotonie de S.-Y.A. Chang,

L.Wang et P.Yang pour les applications biharmoniques stationaires de classe W2,2, per¬

mettant d'établir la régularité partielle en dimensions supérieures à quatre.En dernier lieu, nous établissons des résultats de régularité pour les applications polyhar-

moniques. Notre méthode utilise une inégalité d'interpolation d'Adams-Frazier. De plus,nous évitons des repères mobiles. En vue des formules de monotonie pour les applications

harmoniques et biharmoniques stationaires, nous obtenons la régularité partielle dans ces

cas particuliers.

fe'fe Leer/°iank leaf

Contents

Abstract vii

Zusammenfassung ix

Résumé xi

1 Weakly biharmonic maps 1

1.1 Definitions and basic properties 1

1.2 Euler-Lagrange equation 3

1.3 Noether equation 5

1.4 Extensions of biharmonic maps, Paneitz functional 6

1.5 Weakly intrinsic biharmonic maps 7

2 Large solutions in four dimensions 9

2.1 Introduction 9

2.2 The Dirichlet problem for harmonic maps 11

2.3 Topological degree of Sobolev maps 12

2.4 Weak continuity properties of the volume functional 14

2.5 Concentration compactness lemma 17

2.6 Gluing lemmas 18

2.7 Large solutions on M4 23

2.8 Large solutions on the unit disk I : Energy gain 26

2.9 Large solutions on the unit disk II : Proof of Theorem 2.2 34

2.10 Outlook. .

36

3 Partial regularity in higher dimensions 37

3.1 Introduction 37

3.2 Proof of the monotonicity formula 40

3.3 Polyharmonic maps 44

3.4 Morrey decay estimates 47

3.5 Proof of Theorem 3.2 55

3.6 The harmonic and biharmonic cases 56

3.7 Outlook 57

A Sobolev spaces and embedding theorems 59

B Linear Estimates 63

xiv CONTENTS

C Miscellaneous 73

Bibliography 75

Acknowledgements 81

Curriculum Vitae 83

Chapter 1

Weakly biharmonic maps

1.1 Definitions and basic properties

Let £1 C Km be a smooth domain and Mn a smooth compact Riemannian manifold without

boundary of dimension n. According to Nash's embedding theorem in [51], we may assume

that M is isometrically embedded in some Euclidean space RN for N sufficiently large.For k G N and 1 < p < oo, we define the Sobolev spaces

Wk*(Cl,M) := {u G Wk'p(n,MN) : u(x) G Af for a.e. x G Ü},

equipped with the topology inherited from the topology of the linear Sobolev space

wk*(n,uN).The Hessian energy (or extrinsic biharmonic energy) for maps u in W2,2(Q,Af) is defined

as

En(u) ~ [ \Au\2dx,Jo.

where A is the standard Laplace operator on Em.

The Hessian energy functional now raises three classes of maps.

Definition 1.1 A map u in W2'2(Q,Af) is called (extrinsically) weakly biharmonic if it

is a critical point of the Hessian energy Eq with respect to compactly supported variations

on the target manifold, i.e. if

dt

for all £ G C(Ü,RN), where Tlfj- denotes the nearest point projection onto M

£n(ILv o (u + #)) = 0 (1.1)£=0

Remark 1.1 Equation (1.1) is the Euler-Lagrange equation corresponding to the Hes¬

sian energy functional, or biharmonic map equation.

Definition 1.2 A weakly biharmonic map u in W2,2(Q, A/") is called (extrinsically) sta¬

tionary biharmonic if, in addition, it is a critical point of the Hessian energy Eq, with

respect to compactly supported variations on the domain manifold, i.e. if

£dt

£?n(uo(id + ^)) = 0 (1.2)£=0

2 Weakly biharmonic maps

for all £ G C£°(fï,lRm), where id denotes the identity map.

Remark 1.2 Equation (1.2) is the Noether equation to the Hessian energy func¬tional. Maps in W2,2(Vt,M) satisfying the Noether equation, and not necessarily the

Euler-Lagrange equation, are called Noether biharmonic. See for example Hélein [27,p.42] in the harmonic case.

Definition 1.3 A map u in W2'2(Cl,Af) is called (extrinsically) minimizing biharmonic

if, for all maps v G W2'2(Q,Af) satisfying u — v G W0' ,it holds

En(u) < En{v).

Remark 1.3 Related to extrinsic weakly biharmonic maps are the intrinsic weakly bihar¬

monic maps (mentioned in section 1.5). However, we do not consider the latter in the

present work. Therefore, we omit henceforth the adjective "extrinsic" without any possible

confusion.

The constant maps are clearly minimizing biharmonic. The existence of non-trivial mini¬

mizing biharmonic maps follows from the direct method of the calculus of variations. For

7 G W2'2(£l,Af) we consider the complete metric space

W2'2(ü,Af) := {u G W2'2{ü,Af) : u - 7 G W*>2}.As En is weakly lower semi-continuous with respect to the iy2,2-topology, there exists

u G W2>2{Ü,N) such that En(u) < En(v) for all v G W2'2{Ü,Af). Explicit (non-trivial)weakly biharmonic maps are given in section 1.2.

Minimizing biharmonic maps are clearly stationary biharmonic. We thus have the follow¬

ing relation between these maps

{Minimizing biharmonic maps}

C {Stationary biharmonic maps}

C {Weakly biharmonic maps}.

For sufficiently regular maps, the Euler-Lagrange equation (1.1) implies the Noether equa¬

tion (1.2), and consequently every weakly biharmonic map is stationary biharmonic. See

section 1.3.

An important property of the Hessian energy functional is the invariance under transla¬

tions t, and rescalings r in four dimensions (m — 4). More precisely, for t0 G Rm and

r0 > 0, we consider the translation t : x 1—> x — t0 and the rescaling r : x 1— r0x. For

u G W2,2(Vl,N), we define v(x) :— u(t(x)) and w(x) :— u(r(x)). In view of the formula

for changes of coordinates, we then compute

En(y) - / \Av\2dx = f \Au\2dx = Em(u)JQ Jt(Q)

and

En(w) - / \Aw\2dx = r4~m [ \Au\2dx = r4~mEr(çi){u).Jü Jr(Q)

Thus, weakly biharmonic maps are preserved under translations, and rescalings in four

dimensions. However, contrary to the Dirichlet energy, Eq(-) is not invariant under con-

formal change of metrics. We refer to section 1.4.

1.2 Euler-Lagrange equation 3

1.2 Euler-Lagrange equation

We deduce the geometric form of the Euler-Lagrange equation for weakly biharmonic

maps (1.1). We consider, for S > 0, the tubular neighborhood Vs of A/" in MN defined as

Vs := {p eRN : dist(p,Af) < 5}.

As N is smooth and compact, there exists S > 0 sufficiently small s.t. for every p G Vsthere exists a unique point Ujv(p) G M s.t.

|p-ILv(p)|=dist(p,A/).

n^ : Vs — M is the nearest point projection. For p G A/", P(p) :— Vn(p) is the

orthonormal projection onto the tangent space TpM (see for example Moser [48, Lemma

3.1.]). The orthonormal projection onto the normal space will be denoted by P1. Observe

that P + PL = id. Then, we have

Lemma 1.1 (Euler-Lagrange) If u G W2,2(fl,Af) is weakly biharmonic, then it satis¬

fies

P(u)(A2u) = 0 (1.3)

in the sense of distributions.

Proof. For £ G Cg>(fi,RN), we compute

d0 =

,

dt[\A(Ilô(u + tO)\2dx

2 [ AuA(P(u)(Ç))dx.In.

Remark 1.4 For a weakly harmonic map u inC°°(Ü,N)f\W2a!(SÎ,N) andÇ G C^(Q,RN),with £(x)//TU(X)N', we have P(u)(Ç) = £. Thus, u satisfies

A2u ± TUN


Example 1.1 ForM = WN, the Euler-Lagrange equation reduces to

A2u = 0 (1.4)

in the sense of distributions. Due to the Weyl Lemma (see for example Dacorogna [17]),solutions to (1-4) are smooth, and consequently (1-4) holds pointwise. If we prescribeboundary data, the biharmonic functions are even unique. For sufficiently regular do¬

mains, the set of solutions can be written explicitly. For example on £1 — BT \ Bp C R4,the biharmonic functions are of the form

u(r,v) = Y. Yl (.°>k,nrk+2 + bk,nrk + cKnr~k + dKnr-k-2)SKn(n)feeN* l<n<(fc+l)2

+e In r S0(n),


where r, n denote the radial, resp. the angular coordinates, and {Sk,n}k,n forms an L2(S3)-orthonormal basis of the spherical harmonics of order k in four dimensions. The vectorial

coefficients ak,n,bk,niCk,n and e are uniquely determined by the boundary data. We referto Müller [50] for further details on spherical harmonics.

Example 1.2 For A/" — Sn C Mn+1, a biharmonic map u satisfies

A2u = -(|Au|2 + A(|Vw|2) + 2VuVAu)u (1.5)

in the sense of distributions. Equation (1.5) follows by differentiating four times \u\2 — 1

and observing that A2u // u inV.

The simplest weakly biharmonic maps into the sphere are the constant maps. For Ct — Em

and n — m, the inverse of the stereographic projection is a non-trivial weakly biharmonic

map, as shown in the appendix C .

For Q. = Br and n = m — \, the radial projection

u{x) =R

is even minimizing biharmonic. Indeed, for v G W2,2{BT,Sm~l) we have

Avava = -\Vv\2.

As v is a unit vector, it follows

f \Av\2dx > f \Avava\2dx = / \Vv\4dx.JBT JBr JBr

Observe that equality holds for u. Now, as shown by Coron-Gulliver [16], the radial

projection is a minimizing 4'harmonic map. Hence, it is minimizing biharmonic. This

argument is taken from Hong- Wang [28],

Consider now the second fundamental form A(-)(-, ) of Af in RN, given by

A(p)(X, Y) := P1(VX(Y)) for p G Af and X, Y G I^V,

where X, Y have been extended to tangent vector fields of A/" in a neighborhood of p.

Then, we can reformulate Lemma 1.1 as follows.

Lemma 1.2 (Euler-Lagrange 2) If u G W2,2(Q,J\f) is weakly biharmonic, thenu sat¬

isfiesA2u =< A(P(u)),Au > -A(A(u)(Vu, Vu)) + 2 < V(P(u)), VAw > (1.6)


Proof. Applying the product rule, we compute in the sense of distributions

P{u)(A2u) = A(P(u)(Au))- < A(P(u)),Au > -2 < V(P(u)), VAu >.

Moreover, as P{-)(-) is the orthonormal projection onto the tangent space, we have

P(«)(V«) - Vu.

1.3 Noether equation 5

Differentiating gives

Hess ILv(u)(Vu, Vu) + P(u)(Au) = Au.

As A(u)(Vu)(Vu) — —Hess Ilj^(u)(Vu, Vu) (see for example Moser [48, Lemma 3.2.]),we get

P(u)(A2u) - A2u + A(A(u)(Vu, Vu))- < A(P(u)), Au > -2 < V(P(u)), VAu >.

The claim follows now from (1.3). D

Thus, weakly biharmonic maps satisfy a fourth order elliptic nonlinear partial differential

equation in a weak sense. If u satisfies (1.6) pointwise, u is called biharmonic. However,it is customary not to distinguish between both notions. Therefore, we omit the adverb

"weakly" in the sequel.

Remark 1.5 The Euler-Lagrange equation can also be expressed, in terms of moving

frames. However, we do not need this formulation and refer the interested reader to

Lamm [35] and Wang [75], [76].

1.3 Noether equation

First of all, we derive the explicit partial differential equation expressing the stationaritywith respect to inner variations. We infer from (1.2) the following

Lemma 1.3 If u is a stationary biharmonic map in W2'2(B2r,J\f), then we have:

/ (Aukkui3H + 2ukkUjiJü - | Au|2V £) dx = 0 (1.7)

for every test function £ G C$°(B2r, Em).

Proof. We compute

Au, = (Au), + t (2(uy)t# + (u^é) + t2{ujk)tikïi,

where for any / we denote ft(y) := f(y + tÇ(y)) for £ G C^(52r,Rm). The change of

variables x~

y + t£ for |t| sufficiently small induces a C°°-diffeomorphism of B2r onto

itself. Observe that the Jacobian determinant of the transformation satisfies

-lox1

det( —dyJ

det

and

£dt

t=o

det

dxi

dx1

IMP

= 1 - iV • £ + 0(t2)

= -V.£.

The stationarity assumption (1.2) then implies:

f \Aut\2(x-tÇ)(l-tV-Ç + 0(t2))dx

= / {AukkUij$,{ + 2ukkuj$i - |Au|2V f) dx.

JBlr

0 = T

dt


D

For sufficiently regular maps u : f2 —> A/", we derive equation (1.7) from the Euler-Lagrange

equation. Thus, we have

Lemma 1.4 Every biharmonic map u G W3'2(B2r,J\f) is stationary biharmonic.

Proof. A biharmonic map u G W3,2(B2r,J\f) satisfies, in accordance to Lemma 1.1, for

every test function £ G C^(B2r,Rm)

0= f A2uau^jdx.J Bir

Repeated integration by parts gives

0 —

— / (AujiUij^dx — / (AujiUjÇjdxJb2t Jb2t

- - / (\Au\2)jtfdx + 2 / Auuij£{dx+ I Auu^dx^JBlr J Bir JB2t

= 2 AuUijÇJdx + / Auuj^dx - - / |Au|2V • £dx.JB2r JB2r

2 Jb2t

D

Remark 1.6 The statement of Lemma 1-4 is not sharp in the sense that the regularity

assumption on u can be weakened. However, the integrations by part in the proof arc not

valid for maps in W2,2 anymore.

1.4 Extensions of biharmonic maps, Paneitz func¬

tional

There are two possible extensions of biharmonic maps to arbitrary smooth domain mani¬

folds (M, g).

The more obvious one is to consider the Hessian energy functional

EM(u) = / \Agu\2dvolgJm

for maps in W2,2(Ai,Af), where Ag denotes the Laplace-Beltrami operator associated to

(A4,g). The critical points with respect to compactly supported variations on the tar¬

get manifold are called extrinsically weakly ^-biharmonic. Similar to the Euclidean case,

we define extrinsically stationary ^-biharmonic and extrinsically minimizing ^-biharmonic

maps.

However, as in the present work, the domain manifold plays no decisive role, we restrict

ourselves to the Euclidean case. The proof of partial regularity in chapter 3 remains es¬

sentially the same, but technically a little more complicated.

1.5 Weakly intrinsic biharmonic maps 7

Moreover, in four dimensions, the Hessian energy functional E^ is not invariant under

conformai change of metrics. Therefore, biharmonic maps are not preserved under con-

formal transformations.

The related conformally invariant fourth order energy functional in four dimensions is the

Paneitz functional

P4(u) := / (\Au\2 + ~Rg\Du\2 - 2Ricg(Du, Du))dvolg,

where Rg and Ricg are the scalar and Ricci curvature of (M4,g) and Ricg(Du, Du) :=

Yl%=\ Ricg{Dul, Du1). Critical points with respect to compactly supported variations in

the target manifold M are solutions to

P4u JL TUM

in the sense of distributions, where P4 is the Paneitz operator in four dimensions defined

as

P4 := (-Ag)2 + Si-Rggn - 2Ri^)d.

8 is the negative divergence and d is the differential (acting on vector-valued functions).For M — R4 equipped with the Euclidean metric, the scalar and the Ricci curvature

vanish. Hence, the Paneitz functional describes the behaviour of Emi under conformai

transformations. Thus, criticality of the Paneitz functional seems to be an adequatealternative generalization of the biharmonicity in four dimensions from the point of view

of conformai geometry.The Paneitz operator was introduced by Paneitz [54] in 1984. We refer to Chang [13]and Chang-Yang [14] for further information on the Paneitz operator and its conformai

invariance property in four dimensions.

1.5 Weakly intrinsic biharmonic maps

Closely related to (extrinsic) biharmonic maps are the weakly intrinsic biharmonic maps.

They are defined as the critical points with respect to compactly supported variations in

the target manifold of the tension energy functional

M«) := / \{Agu)T\2dvolg.JM

The tension field (Agu)T is the component of Agu tangent to TUN'. Stationary and min¬

imizing intrinsic biharmonic maps are defined in the same way as in the extrinsic case.

As weakly harmonic maps have vanishing tension field, they are minimizing intrinsic

biharmonic, and therefore they are weakly intrinsic biharmonic. Thus, weakly intrin¬

sic biharmonic maps are a direct generalization of harmonic maps. Conversely, for A/"

with sectional curvature k < 0, Jiang [29] proved that every intrinsic biharmonic map is

harmonic. In the case of target manifolds with positive sectional curvature, there exist


non-harmonic intrinsic biharmonic maps. See for example Ou [53].

Contrary to the Hessian energy, the tension energy does not depend on the embeddingM w R^. Therefore, from the geometric point of view, the tension energy seems more

natural. However, the Hessian energy bounds the iy2'2-norm. This is no longer true for

the tension energy as can be seen by considering geodesies from S1 to S1 which transverse

the circle k times. As k — oo, the W2'2-nomi tends to infinity whereas the tension energy

still vanishes. Moreover, the tension energy functional is not coercive (contrary to the

Hessian energy). Thus, from the analytic viewpoint, the latter seems more adequate in

order to apply standard methods from the calculus of variations and partial differential

equations.

Existence and regularity results of weakly intrinsic biharmonic maps were studied in Ku

[32], Lamm [36], Moser [49] and Wang [75], [76]. A survey of the intrinsic biharmonic

map scene from the point of view of differential geometry (construction of examples and

classification results) can be found in Montaldo-Oniciuc[45].

Chapter 2

Large solutions in four dimensions

2.1 Introduction

Let Ü C R4 be a smooth domain, S4 the unit sphere in M5 and consider

Vl'p(tt, S4) := {u G L°°(ß, S4) : V'u G Lp(fi,R54')}

equipped with the semi-norm ||u||p,iP,n q4):— îai=i/n if^l7"^- Observe that the spaces

Vl'p locally coincide with the usual Sobolev spaces Wl,p. Furthermore, the Hessian energy

is finite and the notion of biharmonic maps is well defined on V2,2{Q, S4).S.-Y.A. Chang, L.Wang and P.Yang proved in [15] the smoothness of biharmonic maps

defined from a 4-dimensional domain to a sphere. For recent improvements, we refer to

Strzelecki [73], Wang [75] and Lamm-Rivière [37]. Thus, biharmonic maps verify .

A2u - -(|Au|2 + A(|Vu|2) + 2VuVAu)u (2.1)

pointwise.

The simplest examples of biharmonic maps R4 — S4 are the constant maps, that are even

the absolute minimizers of the Hessian energy. Our first aim here is to show the existence

of local ndnimizers u : R4 —> SA having a non-trivial topological degree. As in [10] we

introduce

Qn(u) = —— J J4(Vu)dx

for u G £>2'2(ft,S4) C Vl>4(Çl,S4), where J4(Vu) :- det(u, g-,..., g) is the Jacobian

determinant in 4 dimensions. When Q — R4, we observe that due to Corollary 2.5

Qr*(u) G Z. Thus, for u G V2>2(R4,S4) the topological degree deg(u) := Qî{u) is well

defined. For k G Z we then consider the non-empty homotopy class

Ek := {u G 2?2'2(R4, S4) : deg(u) = k}.

The existence of locally minimizing maps R4 —> S4 now follows from

Theorem 2.1 For k G { — 1,1}, there exists a smooth map u G Sfc such that

16H4(S4) < ERi(u) =1- inf ERi(v) < 24H4(Sâ).VEk

10 Large solutions in four dimensions

By symmetry of the Jacobian determinant and of the Hessian energy, it is sufficient to

prove Theorem 2.1 in the case k — 1. Moreover, we have

2 = inf E(u)= inf E(u).

Our second problem focuses on the existence of non-minimizing biharmonic maps from

the unit ball B C R4 into S4. We consider 7 G W2'2(B, S4) and define

E1 - {u e W2'2{B, S4) : u - 7 G W02'2}.

As £7 is a complete metric space and EB is weakly lower semi-continuous with respect to

the T>2'2-topology, there exists some u G E1 such that

EB(u) = inf EB(u).

In view of Corollary 2.6, for u G £7 we have

Qb(u) - QB(u) G Z.

For k G Z, define the homotopy class

S7 := {m£H7: Qb(u) - QB(u) = k}.

It is not hard to see that Ek is non-empty for any k.

Fix a smooth map u* G H1 with Era(u*) — X as given, for instance, by Theorem 2.1. For

R > 0, let 7(2) :— u*(Rx), x E B. For i? > 0 sufficiently small, the above minimizer

« £ ST of Eß coincides with 7. We fix such an R > 0. We show

Theorem 2.2 inf=i EB is achieved.

This gives the following

Corollary 2.3 There exist (at least) two distinct critical points in H7 of EB-

In view of the results of Brezis-Coron [10] and Jost [30] for harmonic maps, the specialchoice of 7 may seem unnecessary. However, the present problem is of fourth order.

Carrying over the method of sphere-attaching from the second order case therefore be¬

comes very delicate. In particular, geometric considerations require that the gradientsof the absolute minimizer u of EB in S7 and u* are almost identical at some point. If

7(x) :— u*(Rx) (for R sufficiently small), this condition is trivially satisfied. However, at

this moment, we have no general criteria to guarantee this condition. See Lemma 2.17

and Remark 2.3 for further details.

In the proof of both theorems, we have to deal with the problem that the classes Ek,respectively Ek, are not closed in the weak P2,2-topology. If we take arbitrary minimizing

sequences for EM4 (resp. EB) in Ek (resp. Ek), we may encounter the phenomena of con¬

centration and vanishing at infinity as introduced by P.L. Lions in [40] and [41]. Therefore,we have to choose our minimizing sequences carefully in order to assure compactness in

the limit.

2.2 The Dirichlet problem for harmonic maps 11

Following the scheme presented in [40] and [41], we show that every minimizing sequence

(ufc)fceN in E1 for £k4(-) converges, up to translations and rescalings, to a map u in E1. Our

proof strongly relies on uniform estimates for fR4 | J4(Vufc)|dx for a minimizing sequence

(wfc)fceN of prescribed topological degree d — 1, that allow us to show that the degree is

conserved in the limit k — oo.

In order to prove Theorem 2.2 we show the existence of a map u £ El^, verifying

EB(u) < EB(u) + I — 5 for some S > 0. For a minimizing sequence (uk)kew C H*

s.t. EB(uk) < EB(u) +X— S we then can exclude bubbling and conclude that the limiting

map u belongs to S7.Similar constructions first appear in Wente's "sphere attaching lemman in [79, Theorem

3.5] in the context of surfaces of prescribed constant mean curvature. Further results

relying on this technique can be found in Steffen [65], Steffen [66], Wente [80], Struwe

[69], Steffen [67], Brezis-Coron [9], Brezis-Coron [11], Struwe [70] and Struwe [71] in the

context of surfaces of prescribed constant mean curvature, and in Brezis-Coron [10], Jost

[30], Giaquinta-Modica-Soucek [22], Soyeur [64], Hardt-Lin-Poon [25], Qing [57], Kuwert

[34], Rivière [58] and Weitkamp [77] in the case of harmonic maps, and in Kusner [33],Bauer-Kuwert [7] for Willmore surfaces.

The chapter is organized as follows. In section 2.2 we compare our results with the respec¬

tive results in the theory of harmonic maps. In section 2.3 we assign a topological degreeto the Sobolev maps in 2?2'2(M4, SA). In 2.4 we describe the behaviour of the volume func¬

tional under weak Z>2'2-convergence. In 2.5 we prove a concentration compactness lemma

stating that the minimizing sequences for the biharmonic energy locally converge in V2,2

except on a countable set of points. Subsection 2.6 is devoted to some gluing construc¬

tions allowing us to isolate possible concentration points of the minimizing sequences. In

section 2.7 we prove Theorem 2.1; in section 2.8 we prove the sphere-attaching lemma,and in 2.9 we finally present the proof of Theorem 2.2.

This chapter is based upon the paper [4].

In what follows we denote u — (ul,u2,u3,u4,u5) and we tacitly sum over repeated indices

(unless otherwise stated). We let Br(x) be the ball of radius r > 0 centered at x G R4

and define BT •— Br(0), B := B\ and A(r, q) :— P(i+9)r \ Br. Moreover, V^ is the usual

gradient on the Riemannian manifold M. We let V := VK4 and E(-) := Era(-).

2.2 Short review of the Dirichlet problem for har¬

monic maps

For a smooth domain Ü C R2 consider for u G Wl'2(ü, S2) the Dirichlet energy Dn(u) =

fn \Vu\2dx. The (weak) harmonic maps from fl into S2 are the critical points of D^(u)with respect to compactly supported variations on S2. The resulting Euler-Lagrangeequation states that u G Wl'2(fl, S2) is harmonic iff u satisfies

Au = -|Vu|2u (2.2)


in the sense of distributions. We define u to be D-minimizing if u minimizes the Dirichlet

energy among all maps v G W1,2(fl,S2) satisfying u — v G W0' . Thus, D-minimizing

maps are harmonic. F. Hélein [26] proved that harmonic maps defined on a 2-dimensional

domain are smooth. Therefore, harmonic maps verify (2.2) pointwise.Here again, the simplest examples of harmonic maps are the constant maps. A non-trivial

harmonic map is given by the inverse of the stereographic projection k. Indeed, observe

that

|Vu|2 > 2J2(Vu), (2.3)

where J2(Vu) :— det(u, J^-, £r) is the Jacobian determinant in 2 dimensions, and con¬

sider X := {u G Wl'2(R2\s2) : /R3 J2{Vu)dx = H2(S2)}. It follows from (2.3) that

Dmi(v) > 2TC2(S2) for v X, and equality holds for k G X (and all maps being confor-

mally equivalent to k). Thus, k minimizes the Dirichlet energy in the homotopy class X

of W1'2- maps of topological degree 1. Hence, k is harmonic.

Consider now the space W^2{B,S2) := {v G Wh2(B,S2) : v = 7 on dB}, where B de¬

notes the unit disk in R2 and 7 G Wl'2(B,S2). It is generally impossible to determine

explicit harmonic maps in Wl,2{B, S2). However, as W*'2(B, S2) is a complete metric

space and DB(-) is weakly lower semi-continuous with respect to the W^1,2-topology, there

exists a D-minimizing map u G W*,2(B,S2). H.Brezis and J.-M.Coron [10], and J. Jost

[30] proved independently that Dn also attains its infimum in (at least one) homotopyclass different from [u] when 7 has non-constant boundary data. Conversely, if 7 has

constant boundary data, L. Lemaire [38] proved that the constant maps are the only har¬

monic maps. See Giaquinta-Modica-Soucek [22], Kuwert [34], Qing [57], Soyeur [64] and

Weitkamp [77] for further results on the Dirichlet problem for harmonic maps in two di¬

mensions.

2.3 Topological degree of Sobolev maps

In this section, we show that we can define a topological degree for maps in the Sobolev

class V2'2(R4,S4).In view of the embedding X>2'2(R4, S4) w Î?1-4(R4, S4) it is sufficient to show the following

density result for maps in Z>1,4(R4, S4). This is a direct generalization of the density result

of Schoen-Uhlenbeck [62] for maps in Hl. See also Brezis-Coron [10].

Theorem 2.4 (Density) Foru G I?1,4(1R4, S4) there exists a sequence of maps (uk)ke® CC(R4, S4) D 2?1,4(R4, S4) and a sequence of radii rk —> 00, as k —» oo; such that

Uklw^B^ = etc.

and

Ufc — u in V1'4.

Proof. Let a : S4 — R4 be the stereographic projection, which maps the south pole into

0. We set v(p) = u(o-(p)) for p G S4. We define

Vl'4(S4,S4) := {/ e L4(S4,S4) : \\f\\v^(S^) < 00},

2.3 Topological degree of Sobolev maps 13

with

N p\\A I in ,m4 t

[54oi,4(54)S4):= / IV54/I dvols

Observe that ||v||x,i,4(Ä4)i94) — ||u||2ji,4(R4jS4) < 00. Hence, u G Vl,4(S4, S4). Consider

Mi(p):={qeS4:\q-P\<l}and

Thus, we have

and

vk(p) := f v(q)dvolsi(q).JB1 (p)

vkC(S4,R5)DV1A{S4,R5)

Vk^vinV1'4^4^5).

Now we define, for 4>k G C^B^/V)) (N north pole), with 0 < (f>k < 1 and ^fc|Bl(jv) = 1,

Vk(p) := Vk(p) + MP) (vk(N) - vk(p)).

We estimate using Poincaré's inequality

dist(vk(p), S4) < f \v(q) - vk(p)\dvolS4{q) + \vk(p) - vk(p)\Jm 1 (p)

< Ckl f \VS4V\2) +\<Pk(p)\{ \vk(N) - v(q)\dq\ J® 1 (p) / Jv 1 (p)

< Ckl f \VS4v\2) +C-f \vk(N)-v(q)\dqWBi(p) I JMi(N)

/ \ 4/

< c

14

/ \ 4

/ |V5H4 ) +C\ f |V<H4 ] . (2.4)Ju^(p) I

wb|(jv) y

As the right hand side decreases monotonically to zero as k —> 00, Dini's theorem impliesuniform convergence. Thus, for k sufficiently large, we can project vk(p) onto S4 and

assume that vk G C(S4, S4). Moreover, vk —> v in V1'4 and vk is constant in a neighborhoodof the north pole TV.

The sequence uk :— vk(a~1(x)) has now all the required properties. O

Remark 2.1 The approximating sequences uk can be chosen in the class of smooth maps

after mollification with a smooth kernel.

It follows immediately


Corollary 2.5 For u G X>1>4(R4,54), we have

I J4(Vu)dx = fcft4(S4)

with k G Z.

Moreover, we get

Corollary 2.6 .Fbr u,v 2?1,4(5, S4) wuVi u — von dB, we have

/(J4(Vu) - J4(Vw))dz = fcft4(S4)

wrf/i fceZ.

Proof. We define w : R4 - S4 as

J u(.x) for x Ç. B

W{x)'"\v(w) for -xeR4\R

As u, u G r>1-4(JB, S4), we verify that w G X>1-4(R4, S4) and

/ (J4(Vu) - J4(Vv))dx - / J4(Vu;)dx.Jb jr4

Thus, the result follows from Corollary 2.5. D

2.4 Weak continuity properties of the volume func¬

tional

In this section we prove a generalization of Wente's weak continuity property of the volume

functional

V(u) = / JAÇVu)dxJq

in [78, Sect. III]. More precisely, the following properties for the weak Z?2,2-topology hold.

Proposition 2.7 1. Let uk—^ u in V2'2(BR, S4). Then, we have for a.e. 0 < r < R

I J4(Vuk)dx= ,h{V{uk-u))d,x+ I J4(Vu)rf.x + o(l), (2.5)J Br J Bf J Br

where o(l) —> 0 as k —> oo.

We call r a good radius if r satisfies (2.5).

2. For ukûin X»2-2(R4, S4) it holds

/ </4(Vufc)dz= / J4 (V(ufc - «)) dx + / Ji(Vu)dx + o(l).7r4 7k4 yffi4

2.4 Weak continuity properties of the volume functional 15

3. For uk->> u in £>2-2(R4, S4) and Ç G C%°(R4, R>0) it holds

dvk dvk dvk dvk[ t\J4(Vuk) - J4(Vu)\dx 0) or VI — R4 let ek-*

e, fk —>- f, gk—^

g, hk —^ 0 in

V2'2(Q, S4) as k -> 00, and w G V2>2(Vt, S4). Then, as k ^ 00 we have that

Lr(det I ek,

dfk dfjk dhk dw

dxi'

dx2'

dx3'

d,x4rfx

Proof. We consider, for e > 0 and fi — £?r (compact case), we, smooth such that

\\w — wc\\V2a — 0 as e -+ 0. It follows from the multilinearity of the determinant,Holder's inequality and the continuity of the embedding W2,2 ^-> W1*4 (Sobolev's embed¬

ding theorem)

/J Br

det ek, dx

<

lKe*#

dfk dgk dhk dw

dx\'

9x2'

dx3'

<9x4

dfk dgk dhk d(w - wc)

-4-

fA r(

9f*

+ / det[ ek,—-,

Jbt V 0x1

dx2'

<9x3'

dx^dfk

dgk dhk dw,

dx

dxdx2

'

dx$'

dx4

< ||efc||L~(Br)||V/fc||Li(flr)||V^fc||L4(Br)||V/ifc||L4(Br)l|V(u; - we)\\LHBr)+ ||efc||L»(Br)||V/fc||i4(flr)||Vfl'fc||li4(flr)||V/lfc||ta(flr)||Vlü£||L»(Br)

< C\\w - Wt\\V2^BT) + #11 V7lfc||L:»(Br),

where C is independent of k and e, and K is independent of k.

The embedding V2'2(Br,S4) = W2>2(Br,S4) --> H/1'2^,^4) (Br c R4) is compact byRellich's theorem. Thus, the second term converges to zero as k tends to infinity. Then,we let e tend to zero and the claim follows for SI = BT (compact case).For Q, — R4, we choose ri —> 00 (/ —+ 00) and estimate with Holder's inequality

det 1 e.ËA ËEa ?h ËE.) dxOX\ OX2 ÖX3 0x4/

dfk dg± dhk dw_dx2

'

<9x3'

<9x4

/ deti

Jr* V

-in \det{Ck^ ) dx + C\ f \Vw\4dx

where C is independent of k and I. First, we let & — 00 and the first term converges to

zero (as in the compact case Cl — Br). Then, we let / tend to infinity and the second term

vanishes due to the absolute continuity property of the Lebesgue integral. D


Now we are able to prove Proposition 2.7. Define vk :— uk — u. Using the multilinearityof the determinant we have

4

J4{Vuk) - J4{Vu) = J] Aj + J4 (Vvk),j=0

(e.g. det(u, g, g, ft, !£) appears in 42; det(«fc) £, £, g, £) appears in ^.)

where A,- is a sum of terms where j derivatives of vk (and 4 — j derivatives of u) appear.

For j = 1, 2,3, and Ü = BT (r > 0) or ft - M4, it holds

IIÎIl1^)

as k —» oo. This follows directly from Lemma 2.8. Moreover, we observe that

du du du du\M = det Vk,

dxi'

dx2'

öx3'

öx4< C|Vu|4 G L\

Thus, applying Lebesgue's dominate convergence theorem implies that A0 —> 0 in L1^).Since A4 + J4(Vvfc) = det fUfc, f^, g£, §j£, fjj) this completes the proof of the third

affirmation.

It remains to consider A4 in order to complete the proof of the two first affirmations. We

rewrite using integrations by parts

/ AAx - f detlu^ ^ ^ ^\ dxJ Br J Br dx\

'

ÔX2'

dx3'

dx4

idvl dvt dvi

„M,.JBis, 9xi dxs dxm

+ E / (-dx

(2.6)

ou'3 c^f dvk dvk

Using Fubini's theorem we have for a.e. r that (vk)k^ is bounded in W2'2(dBr, SA).By the compactness of the embedding W2>2(dBr,S4) ^ W1<3(dBr,S4) the first term

converges to zero as k tends to infinity. The second term may be treated as in the proofof Lemma 2.8. This completes the proof of the first affirmation.

It remains to consider the case Q = R4. Using Theorem 2.4, we may assume that uk = cte.

and u = cte. + 0^2,2 on R4 \ Bnk for some sequence of radii Rk — 00 as k — 00, and

Ox»2,2 —> 0 in D2,2. Thus, equation (2.6) implies

. dvk dvk dvk dvk, ,

det I u, -—, -—, -—,-—

1 dxI det (dx\

'

öx2'

dx3'

<9x4

< E fyiyv^^H^-dx*, r «, x *

»/ Ire. ox, dx,- öxm dxn+ o(l),

where o(l) — 0 as A; —> 00. The second affirmation follows now as in the proof of Lemma

2.8 and we are done.

2.5 Concentration compactness lemma 17

2.5 Concentration compactness lemma

Lemma 2.9 (Concentration compactness) Suppose uk-± u G V2,2(R4,Sâ) and ßk =

IV2uk\2dx —^

ß, vk = J4(Vuk)dx —^ f weakly in the sense of measures, where u. and u are

bounded signed measures on R4.

Then, we have:

There exists some at most countable set J, a family {xß\j G J} of distinct points on R4,and a family {v^\j G J} of non-zero real numbers such that

v = J4{Vu)dx + J2 v(j)öxb),

where 5X is the Dirac-mass of mass 1 concentrated at x G R4.

Proof. We follow the scheme of [41]. Let vk := uk-u£ X>2'2(M4,R5). Then we have that

vk—^ 0 weakly in Z>2,2. Moreover, we define Xk :— |V2wfc|2dx, u)k '•= vk — J4(Vu)dx, uk

the positive part of ujk, u^ the negative part of uok and ü~k :— uk +uk the total variation.

We may assume that Xk —^ A, while uk—^ u — v — J4(Vu)dx, u^ —^ u+, u^ —^ u~ and

Uk —^ U — u+ + io~ weakly in the sense of measures, where X,u+,uj~,uj > 0.

For £ G C£°(R4,R) with Proposition 2.7 we have

/ Ç4du — lim / £4dÜkJR4 k^°° JR4

= lim / iA\J4{Vuk)- J4{Vu)\dxk^°° JR4

< lim inf / f4kâo 7]R4/

det I uk,dvk dvk dvk dvk

dx.dxi

'

c?x2'

dx3'

ôx4

As £ has compact support Rellich's theorem implies that lower order terms vanish as

oo, and with Nirenberg's interpolation inequality in [52, p. 11] we get

d{Çvk) d(Çvk) (Çdvk) d(Çvk)'

k

/JM.4

£ dû < lim inf / det ( ufc,Jr4 \ dxi

f \V(Çvk)\4dxJr4

C lim inf f / \V2(£vk)\2dxfc->00 \JR4

dxo dx* ÖXadx (2.7)

C lim inf

< C lim infk—»oo

(/\Jr4

e\v2vk\2dx

< c ed\m.4

.

(2-8)

.jej/*U)ArU>>Let {x^;j G J} be the atoms of the measure u> and decompose u — u0 + Ylwith p^) > 0 and ü70 free of atoms.

Since JR4 dû < oo, J is an at most countable set. Now, for any open set fl C R4, by (2.7)with £ — £k G C£°(f2) converging to the characteristic function of $7 as k —> oo, we have

dw < C ( I dX) . (2.9)/Jn


In particular, uj is absolutely continuous with respect to A and by the Radon-Nikodymtheorem there exists / G LÏ(R4, X) such that dû — fdX, A-almost everywhere.

Moreover, for A-alraost every x G R4 we have

fix) = limp{)

J[x) Rr.rfA-JBp{x)

But then, using (2.9), if x is not an atom of A,

/(*) < lim

CVB>{X)J

= cum I dX = 0,

^ $BMdX j P^]Bp(x)

A-almost everywhere. Since A has only countably many atoms and ö70 has no atoms, this

implies that uJq — 0.

We conclude that

v = J4(Vu)dx + Y,v{J)öxU),

with v^ ^ 0. This completes the proof. D

2.6 Gluing lemmas

Lemma 2.10 Suppose u,v G W2>2(S3, S4). Fore>0 there exists w W2'2(S3x [0, e], R5)such that w agrees with u in a neighborhood of S3 x {0}, w agrees with v in a neighborhood

ofS*x{e},

I |V!sxMu;|2 < Cef (|V|3u|2 + |V|3*f)

-fCe"1 f |VS3(u-u)|2 + Ce-3 / \u-v\2,JS* JS3

and

dist(w, S4) < \\u — u||i<=ü(53)almost everywhere on S3 x [0,e].

Proof. Choose ip G C°°([0, e]) with 0<^<l,V = lina neighborhood of 0, ip = 0 in a

neighborhood of e, |V>'| < Ce"1 and \ip"\ < Ce~2. We define

w(x, s) :— v(x) + y:)(s)(u(x) — v(x))

for (x,s) G S3 x [0,e]. We estimate

|V|axMu;|2 < C (|V2u|2 + |V2u|2 + 6"2|V(u - v)\2 + e"4|u - v\2) .

Integration over S3 x [0, e] yields the energy estimate.

As v(S3) C S4, we have for a.e. (x,s) G S3 x [0,e]

dist(w(x, s), S4) < \\u — ü||£,00(53).

n


Lemma 2.11 Let uk—^ u in T>2,2(R4,S4) as k —> oo. After passing to a subsequence,

there exists, for a.e. p > 0, a sequence of positive real numbers ek —» 0 (k —> oo) and a

sequence of maps (vk)k^ C V2,2(R4,S4) such that

J ufe for x G BpVk~\u for xeR4\B{l+k)p

and

|VV|2dx — 0,JAi

as k —> oo.

Proof. As ufc is bounded in P2'2 Lemma 2.12 below applied to A(r) :— jQB |V2Ufc|2cteimplies that, after passing to a subsequence, we have for a.e. p > 0 and every sequence

of positive real numbers jk — 0

f |V2ufc|2dx^0, (2.10)

Mprik)

as k — oo. Henceforth, we consider a fixed p > 0 satisfying (2.10) and ek — 0 {k — oo)to be fixed later. Fubini's theorem implies that there is a set of ak G (p, (1 + ^-)p) of

positive measure, such that for p G {2, |} we have

CkO~k f (|V2ufc|2 + |V2u|2) < cf (|V2ufc|2 + |V2u|2)JdBak JA{p,ek)

Wk f |V(ufc-u)|p < cf |V(ufc-u)|p (2.11)

Cfcö-fc / |ufc-u|2 < C |Ufc-u|2.JdBak JA(p,ek)

Now we can apply Lemma 2.10 (with ^ instead of e.k) to the functions uk(x) :— uk(o-kx)and u~(x) :— u(crkx), thus giving vk on S3 x [0,^] with vk — vLk in a neighborhood of

S3 x {0}, vk — ü in a neighborhood of S3 x {^} and satisfying

lV!*x[o,^]u*r <S3x[0,^]

r n r n r

< Cek j (|V|3üfc|2 + |V|3ü|2) + - f |Vs-3(üfc - ü)|2 + ~ I |üfc

S3 Ä3 s3

<Cekak f (IWI2 + |V2u|2) + — / |V(ufc-u)|2

dB"k dB"k

C f+7

T7/ \uk-u\2 (2.12)

[ekO-k)3 J

<C f (|V2Ufc|2 + |V2u|2) +-^ /|V(ufc-u)|2 +^ /K-u|2J akek J ak(k J

A(p,ek) B2 ni,


by (2.11), and almost everywhere on S3 x [0, %]4

2

dist(vk,S4) < C\\uk-u\\Tjoo(S3)<C ( I |V(ufc-u)|s

2

I l f 7 \< C[4f |V(ufc-u)|H

~ CPC*1X |v(wfc~w)|lJ ' (2-13)

by Sobolev's inequality and (2.11). As Ufc —> u in W/o'c2 (up to a subsequence), we may

choose ffc tending sufficiently slowly to zero such that the two last terms in (2.12) and the

right hand side in (2.13) converge to zero as k —> oo. The first term on the right hand

side in (2.12) converges to zero due to (2.10). This implies that

L'53x[0,^l

' 4

and almost everywhere on S3 x [0, ^]

dist(vk,S4)— o(l),

V'sô^fkl2 = o(l) (2.14)

4

?4ï

where o(l) —> 0 as A; —> oo. Let II be the nearest point projection onto S4. Then, for k

sufficiently large, II o vk ' S3 x [0, ^] —» S4 is well-defined and smooth. So finally, after

passing to a subsequence, we can define a suitable sequence of maps vk by

Ufc(x) for |x| < Ofc

Vk(r)-=\ "Nll'S-1)) f°r "k<\x\<(l + ^)ak^

uM\x\)ft) for (1 + *)<rk < |x| < (1 + fKu(x) for |x| > (1 + ^)crfe,

where ip(t) is a C2(R) function with the properties ip((l + c-f)crk) — ak, i/j((1 + (-%)ok) —(1 + f )akl -0'((1 + ^)o-k) = 0 and ^'((1 + f)o~k) = 1

.In view of (2.14), it is now easy

to conclude. D

In order to complete the proof of Lemma 2.11, we have to show the following

Lemma 2.12 Let fk > 0 be a bounded sequence in L1(R). After passing to a subsequence,we have for a.e. p > 0 and every sequence of positive real numbers ~/k —> 0

rp+ik

I fkdx —+ 0,Jp

as k — oo.


Proof. Define Fk(y) := Jl^fkdx. Observe that Fk G WM(R) w C°(M) increases

monotonically and \Fk\ < C. Thus, after passing to a subsequence, Fk —» F a.e. and

F increases monotonically. It follows that the set sing(F), where F is discontinuous, is

countable. Hence, for any p G" sing(F) and any sequence on — 0, as / —> oo, we have

F(p + cfy) — F(p), as Z —» oo.

Now fix p G1 sing(F) and consider sequences of positive real numbers ßm —*-* 0 and

7A; —^ 0 such that for any fixed m G N, as k —» oo, we have Fk(p ± /3m) -^? F(p ± ßm).Then, for k > ko(m) we have

0 < Fk{P + 7fc) - Fk(p) < Fk(p + ßm) - Fk(p - ßm)= F(p + ^m)-F(p-/?m)+o(l),

where o(l) —» 0 as k — oo. The claim follows as we first let k --* oo and then also pass

to the limit m — oo. D

A direct consequence of Lemma 2.11 is the following

Lemma 2.13 Let uk—^ u m X>2'2(R4, S4) as A; —» oo. After passing to a subsequence,

there exists a sequence of positive real numbers ek —» 0 (k —» oo) ared a sequence of maps

(wfc)fceN C £>2'2(R4,S4) sucä tfiaf

Ûfe /or x G BekWk

\u for xeR4\ B(1+^k

andr

\V2wk\2dxjJAt'A(f.k,^k)

as k —* oo.

Remark 2.2 Lemma 2.11 and 2.13 may be viewed as a variant of Luckhaus' lemma (see[42] and [43]) for second derivatives in the critical Sobolev dimension.

Lemma 2.14 Let u G V2'2(M.4,S4). For every sequence of positive real numbers ek —> 0

(k —-> 00), there exists a sequence of maps (wk)k^N C V2'2(R4,S4) such that

{u for x G Btk

<uBtk) for xeR4\B2<k

and

f \V2wk\2dx —> 0,

as k —> 00, 7T is the nearest point projection onto S4 and uBt :— + udx.

JBck

Proof. Choose ip G C£°([0,2efc]) with 0 < ip < 1, if} = 1 on [0, ek], ip= 0 in a neighborhood

of 2ek, IV'I < Cql and |<| < Cef. We define

wk(x) := uß +V(|x|)(u(x) -u«),


and estimate with Poincaré's and Sobolev's inequality

f |V2u;*|2dx < cf (|V2u|2 + efc2|Vu|2 + efc4|u-ußJ2)^Blzk •* B2ek

< C f |V2u|2dx + C\ f |Vu|4rfxJB2lk \JB2ek

< C f |V2u|2dx = o(l),JB2lk

where o(l) —> 0 as A: —> oo. Moreover, we have for a.e. x G B2tk

dist(Wk(x),S4) < I \u(y)-uBek\dy+m\x\)\\u(x)-uBek\. . (2.15)JBth{x)

As in (2.4) Poincaré's and Sobolev's inequality imply that the first term in (2.15) is

bounded byi

/ \u(y)-uBJdy<C\ f \Vu\4dy) <C\f |V2u|2du). (2.16)

JBk(x) \JB3ck ) \JB3ck JFor the second term in (2.15) we define

Ufc(x) := f u(y)dy.JBik(x)

As Ufc — u in L2 it follows, after passing to a subsequence, that uk converges to u pointwise

a.e. Thus, we estimate

\^(\x\)\\u{x) - uBik\ < I \u(y) - uBJdy + \uk(x) - u(x)\. (2.17)JB,k(x)

The last term in (2.17) converges to zero a.e., as k —» oo. Therefore, the inequalities

(2.15), (2.16) and (2.17) imply that dist(wk(x),S4) converges to zero, as k — oo, for

a.e. x G B2ek. Thus, for k sufficiently large, we can project wk onto S4 and assume that

wk G W2'2(R4,S4) and wk — 7r(uße ) in a neighborhood of dB2tk. This completes the

proof. D

A direct consequence of Lemma 2.13 and Lemma 2.14 is

Lemma 2.15 Let uk-^ u in V2'2(R4,S4) as k —> oo. After passing to a subsequence,

there exists a sequence of positive real numbers ek —» 0 (k — oo) and a sequence of maps(wk)kEN C £>2'2(R4, S4) defined as

_

/ Ufc for x £ Bkk'

I *(«0 for x£R4\B4tkand

2.7 Large solutions on R' 23

2.7 Large solutions on R4

In this section, we give a proof of Theorem 2.1. As mentioned in the introduction to the

present chapter, it is sufficient to show that infgi E is achieved. Therefore, let (uk)ke^ CS1 be a minimizing sequence for E(-). We deduce from |ufc|2 = 1 that Auku^ — —|Vufc|2.Combining this with Lemma C.2 in the appendix and integrating by parts we obtain

16H4(S4) < 16 f |J4(Vufc)|dx < / |Vufc|4dx (2.18)Jm4 Jr4

= / |<A<|2dx < / |Aufe|2dx = / |V2ufc|2dx<C, .

Jr4 Jr4 Jm4

where C < oo is independent of k.

Now, as E(-) is invariant under rescalings and translations, there are sequences (rfc)fc^N CR and (xk)k^n C R4 such that ufc, defined as

üfc(x) :=ufe ( — JGH1,is a minimizing sequence for E(-) verifying (2.18) and

/ J4(Vük)dx = sup / J4{Vük)dx = -H4(S4), (2.19)JBi(0) x0eR4JBi(xo) 4

for all k G N. Replacing uk by uk, if necessary, henceforth we may assume xfc = 0 and

rfc = 1.

Consider now the families of measures pk :— \V2uk\2dx, vk :— J4ÇVuk)dx, vk the positive

part of uk, vk the negative part of vk and vk :— v^ + vk — \J4{Vuk)\dx. We have

/ duk = H\S4),JR.4

f dVk = (l+ëk)H4(S4), (2.20)Jr4

f dvt = (l + ek)H4(S4),JR4

f dvk = ekH4(S4),JR4

with

ëfc = 2efc > 0.

Let a denote the stereographic projection from S4 to R4. From the appendix C we getthat er-1 G S1 is biharmonic with E(o~~l) — 24H4(S4). If a~l is minimizing, we are done

and minueSi E(u) — 247i4(S4). Otherwise, it follows from (2.18), that there exists 5 > 0

independent of k such that

0 < Hfc < - - 25

for k sufficiently large. Thus, we get with (2.20), that there exists 5 > 0 independent of

k such that

0 < / dvk < -n\S4) - 6 (2.21)Jr4 4


for k sufficiently large. This implies with (2.19), that for all r > 1 and A; sufficiently large

0 < 5 < f dvk< ~H4(S4).r(0)

Hence, we get, after passing to a subsequence, that

uk^ u weakly in £>2'2(R4,S4),

pk—^

p weakly in the sense of measures,

vk—^ v weakly in the sense of measures,

with

0 < / dv< lim inf / dv+ < ^H4(S4) < 2H4(S4). (2.22)Jr4 k-+°° Jr4 4

From Lemma 2.9, we deduce that

v = J4{Vu)dx + Y^ vU)8xu) (2.23)

for certain points x^ G R4 (j G .7), v^ G R \ {0} and J a countable set.

We prove in Lemma 2.16 below that J — 0. This implies with (2.22), (2.23) and Corollary2.5 that

/ J4(Vu)dx = H4(S4).Ju4

Thus, u eE1 and E(u) = min2i E. The smoothness of u follows from [15].Now we want to show

16H4(S4) < f \Au\2dx < 247i4(S4). . (2.24)Jr4

Let v be the minimizer of E in S1. Then,

\m4{S4) < 16 f |J4(Vv)|da:< / |Vu|4dx < f \Av\2dx < 24H4(S4). (2.25)JR4 JR4 JR4

The second inequality is achieved iff v is conformai, i.e. v — <r_1 o c, where c : R4 —> R4 is

conformai. Thus, c belongs to the Möbius group, generated by the translations, rescalingsand the inversion at the unit sphere. Since a'1 is not harmonic, then also no conformai

map v G S1 is harmonic. However, the third inequality is achieved iff v is harmonic.

Thus, (2.24) has to hold.

In order to complete the proof of Theorem 2.1, we need to show

Lemma 2.16 J is empty.

Proof. Suppose J ^ 0. Choose / G J and set x® = 0 (translation). Applying Proposition2.7 and Lemma 2.12, we may choose a good radius 0 0

as k — oo, we have

/ |V2ufc|2dx^0 (2.27)

A(p,1k)

as A: —> oo. Moreover, we may assume

Jbdu)dx < f \J4(Vu)\dx < -j- f \Vu\4dx

Jbp 16 Jbp

< ^j \Au\2dx<\n\S4), (2.28)

and that Lemma 2.11 is valid for p, uk and u. Hence, we get ek — 0 (k —> oo) and maps

K)fceN C £>2'2(R4,S4) such that

VkUfc for x G Bpu for x G R4 \ B{}+fk)p

and

f \Vvk\4dx<C f |V2Ufe|2da:—>0 (2.29)

Mptk) A{p,ek)

as k —> oo. As Vk—*- u in X>2,2, we get from Proposition 2.7

/ J4 (V(ufc - «)) dx - / J4(Vvk)dx- j J4(Vu)dx + o(l)JR4 JM4 JR4

= (dk-d)H4(S4)+o(l),'

(2.30)

where dfc is the topological degree of vk and d the degree of u. Notice that, according to

Corollary 2.5, dk — d G Z.

We get from (2.19), (2.21), (2.23), (2.28) and (2.29) that

/ J4 (V(vk - u)) dx\Jm4

lim / J4 (V(ufc — u)) dxk^°° Jbp

lim / J4(Vuk)dx — / J4(Vu)rfxfc--<*> Jbb Jb0

+ o(l)

}£J(P)

+ o(l) (2.31)

+ o(l)/ dv — \ J4(Vu)dxJbp Jb„

< f dük + ^H4(S4)+o(l)J Bp ^

< / dvk + 2 f dvk + \n4(S4) + o(l)./Bi 7r4 4

< H4(#4),


for A; sufficiently large. Thus,

/ J4(V{vk-u))dx = 0.

JR4

This implies with (2.31) that

J2 »U) = o.

ieJ(p)

This is a contradiction to (2.26). Thus, J = 0. D

2.8 Large solutions on the unit disk I : Energy gain

Let tz be the inversion at the unit sphere centered in z given by tz : M4U{oo} —» R4U{oo},tz(x) :— i^Tgii + z for x $ {z, oo}, tz(z) := oo and r^(oo) := z. We abbreviate r :— t0.

We consider for / G C°°(B, S4) and y G B the sets

S := X»2'2(R4,54) n C°°(R4, S4)

and

•^7,» := iu e f : w o ry G 5, (u o Ty)(y) = f(y)

and V(u o Ty)(y) — AV/(u) for some A > 0}.

In general, for u G £, we do not have that u o ry G £. However, Lemma 2.22 below states

that for 7, as defined in the introduction, TltV is non-empty.

We have the following energy comparison lemma.

Lemma 2.17 Let v G W2>2(B, 54)nC°°(B, S4) be such that Vv(x0) ^ 0 for some x0 G R4,and consider w G Fyô- There exists u G W2'2(B, S4) such that u — v G W02,2,

/ (J4(Vu) - J4(Vu)) dx = / J4(Vw)dx (2.32)

and

/ |Au|2dx < / |Av|2dx + / |Aw|2dx. (2.33)jb Jb Jr4

Proof. Let Br(x0) C B. We may assume, after performing a translation and dilation

that x0 — 0 and r — 1. Since w G TVtQ, the inversion at the unit sphere r gives the map

y :— w or G S with Vu(0) = AVu(O) for some A > 0. Thus, w — y o r. We may assume,

due to Lemma 2.18 below, that u(0) = v(0) = N = (0,0,0,0,1), u;(0) is orthogonal to

Uj(0) and Vi(0) is orthogonal to Vj(0) for i ^ j. Indeed, we replace v by R\ o v o R2 and

w by Ri o w o R2, with suitable fii G 50(5) and R2 G 50(4). Furthermore, we have

yf(0) = 0 = ^(0) for 1 < i < 4.


We consider on R4 \ {0} the spherical coordinates 0\, 82 and 03, which are related to the

usual Euclidean coordinates by xl — rrf (1 < i < 4), where

r/1 — cos 0i

n2 — sin 0i cos 82

if — sin #i sin 82 cos #3

rf — sin 6\ sin 02 sin 03.

We define for e sufficiently small

(v(x)

for x G B \ B2e

u(x) for x G ^2e \ #£

w(x) for ieSf]

where ü;(x) :— u;(/_1e_2x), / is a constant to be fixed later, ü is defined as

û°(r, r?) - aV + &V + car + da for 1 < a < 4, (2.34)

^5m = J1- E («aM)2>y i<a<4

where aa, fea, ca and da depend only on 8, <j>, ip, and are such that

ü? = w°, f^ = Ç ond£?t,

ü" = < ^ = ^ on052e. (2.35)

We clearly have u G W2,2(B,S4). Now we define z(x) :— w(l~lr(x)) G £, i.e. w(x) —

z(l~1r(x)), w(x) — z(e2r(x)) and z(x) — y(lx). Thus, w agrees with z on dB( and the

boundary conditions on dBt read

FfTïa f)?a

ua(x) = za(x) and —(a:) = -^-(x). (2.36)or or

We consider the Taylor expansions for 2 and v for any fixed /

za(x) = za(0) + zf(Q)x{ + o(x) and va(x) - va(0) + ^(0)^ + o(x), (2.37)

their partial derivatives

z?(x) = zf(0) + o(l) and vf(x) = v?(0) + o(l),.

(2.38)

their radial derivatives

ßza ßva

—(x) = z?(0W + o(l) and —(x) = <(0)t/ + o(l), (2.39)

and their Laplacians

Aza(x) = Aza(0) + o(l) and Ava(x) = AvQ(0) + o(l). (2.40)


Combining (2.34), (2.35), (2.36), (2.37) and (2.39) with u(0) - v(0) = N = (0,0,0,0,1)gives, for 1 < a < 4, the following system of linear equations for aa, ba, c° and da

aae3 + bae2 +cae + da = zf(0)nie + o(e)3aae2 + 2bae + ca - -zf(0)rf + o(l) (

.

8a°e3 + 4bae2 + 2cQe + da = 2vf(0)nie + o(e) ^ '

12aae2 + 4bae + ca = <(0)t/ + o(l).

We verify that the solution to (2.41) satisfies

aa = afrf + o(e"2), af := (zf(0) - 3<(0))e"2ba = bavi + 0(e-i); h<* .= (_4zf (0) + 14<(0))e"1c<* = cfrf + o(l), cf := Azf (0) - 19uf (0)d" = df^ + o(e), df := 8<(0)e.

Exactly in the same way, we verify, for 1 < j < 3, that

3flj u'i ddj^ u\t )i

80, Ui de3 ^°\t )i

(2.42)

d0j cî ÖÖj^ Uv-U'

(2.43)

^ =^ + 0(0.

It holds for the spherical harmonics rf (1 < i < 4)

(ASa + 3)r/ = 0.

Thus, we verify as before

AS30a - afAs^rf + o(t~2) = -3<t/ + o(e"2),AS3Öa - ofAW + ofe-1) - -3&JV + o(e"1),AS3C° - cfAsstj' + o(l) - -3cfr/ + o(l),A5sda = d?AW + o(e) = Sdfrf + o(e).

(2.44)

We are now able to compute the biharmonic energy expense Ea of u on the annulus

B2e\Bt, given by

jEa = E / iAî2ffa:-

We consider the Laplacian in spherical coordinates

For 1 < a < 4, the identities (2.34) and (2.44) give on B2e \ B

r-iAs*v? - -3(aar + ba + car~l + dar~2) + o(e"1). (2.46)

Furthermore, we compute

r-s^-r3d ,düa

Or dr15aQr + 8ba + S^r'1. (2.47)


Thus, inserting (2.46) and (2.47) into (2.45) yields

Aü° - (12aJV + bbf - Zdfr'2)^ + o(e~l).

Moreover, we have

JS$

7T

rfdvolS3 = —Sij.

Hence, we compute for 1 < a < 4 (here we do not sum over a)

4 / \Av?\2dvolsa = 144|<|2r2 + 25|fef|2+9|df|2r-4

+120af6fr - 72afdfr~l - 30bfdfr~2 + o(e~2),

and with (2.42)

4 / |Aü°fda; =TT2 J

B-it\Bc

(2.48)

(2.49)

375,.ai.0,5

- 1512|<|2e6 + ^\bf \2e4 + 9\n2\df\2 + 744afbfe5

-mafdfe3 - Abbfdfe2 + o(e2)= (36(af (0))2 - 132<(0)zf (0) + (576 In2 - 273)«(0))2) e2 + o(e2).

Now we show that

/ |Aü5|2dx = o(e2).

As«5- Jl- Yla=i (ü01) iwe compute

(2.50)

(2.51)

ü\ =

Aü5 -

Vî - £l=i i^f

ELidv^l2 + Âü") (Eti^ff)2

'i-E4a=i(«a)2 (i-£Li(^)2)3

From (2.34), (2.42), (2.43) and (2.48), we infer that

\aar3 + bar2 + car + da\ < C(e~2r3 + e"V + r + e) < Ce,

|3aV + 2bar + ca\ < C(r2r2 + e~V + 1) < C,

(2.52)

auQ

ôr

MiII

II

II

oua

daa,

56a „ öcQ ddQ

d82

ôua

de«

dÔi ddi d6i Ô6i

<... <CesinÖi,

<...

< Ce sin öi sinö2,

< C(e"V + e"V + r + e) < Ce,


and

lAû"! - \(12afr + 56? - Zdfr~2)rt\ + o(e~l) < C(t~2r + e"1 + er"2) < Ce~l

on B2 \ B. Consequently, we have

and

|Vü°|2 =öua

dr+ r

düa

dOi

a=l

+sin2 0i

düa

d0.+

1

sin2 0i sin 02

&üa

de*<c.

Introducing these estimates in (2.52) gives

|Aü5|2 < C,

and

/JB2e\B,

Aü5|2dx<Oe4.

This completes the proof of (2.51). Combining (2.50) and (2.51) gives

Ea = TT2 (l8(zf(0))2 - 66<(0)zf(0) + (2881n2 - HZ?)(v<*(0))2) ? + o(e2). (2.53)

(Since y\(0) — 0 — vf (0) for 1 < i < 4, we sum over all possible values of a.)The biharmonic energy expense E0 of u on B \ B2 is

dx.E0< I \Av\Jb

On B(, the biharmonic energy of u equals

Et = f |AüJ|2dx.JBt

Recalling w(x) — z(e2r(x)), we compute with z(x) := w(e2x) = z(t(x))

f \Aw\2dx= fJR4\Be JR4

Azrdx = / \AzaJBe

4z?---\2dx.

lR4\Be JR4\Bt

From the Taylor expansion (2.38) and (2.40) on Bt: we deduce

xl

1 Ixl21

\Aza - 4zf--,Y = lGzf(Q)zU0)vWr~2 + o(e~2).x

It follows with (2.49)

/ |AüJ|2dx - 4tt2(z?(0))2e2 + o(e2),JR4\B,

(2.54)


and consequently

Ei= f \Aw\2dx-4n2(zf(0))2e2 + o(e2). (2-55)JR4

Combining (2.53), (2.54) and (2.55) with z*(0) = lyt(0) and yt(0) = Av((0) gives the total

energy EB(u) = E0 + Ea + Eioiu

EB(u) < f \Av\2dx + f \Aw\2dxJb Jr4

+7T2 (l4(uf(0))3/2 - 66uf (0)uf (0)/ + (288In2 - ^)(,;?(0))2) e2

+o(62)

- f \Av\2dx + f \Aw\2dxJb Jm.4

+7T2 (UX212 - 66X1 + (288 In 2 - )\ |Vu(0)|V + o(e2).

As Vu(0) t^ 0, we may choose an adequate I and e0 > 0 sufficiently small such that (2.33)holds for 0 < e < e0. We take for example I — |.

We show now that u satisfies (2.32). We have

/ J4(Vu)dx- f J4(Vv)dx + o(l). (2.56)Jß\B2r Jb

From (2.53), we infer

/ J4(Vu)dx< f \Au\2dx = o(l). (2.57)

B2c\Bf B2c\Be

As the volume functional is conformally invariant, we get

/ J4(Vu)dx = / J4(Vw)dx- / J4(Vw)dxJbc Jr4 Jr4\b£

= / J4(Vw)dx- f J4(Vz)dx (2.58)Jr4 Jbc

= / J4(Vw)dx + o(l).Jr*

Combining (2.56), (2.57), (2.58) gives

/ (J4(Vu) - J4(Vv)) dx = f J4(Vw)dx + o(l).Jb Jr4

For 0 < e < e0 sufficiently small, Corollary 2.6 gives (2.32). D


Remark 2.3 At first sight the assumptions on the gradient of v and w in Lemma 2.17

may seem to be unnecessarily restrictive conditions. Moreover, it would seem that the

choice of the interpolant could be improved by replacing ua for 1 < a < 4 in (2.34)with the biharmonic function satisfying A2vP = 0 and the boundary conditions (2:35), or

that the ratio of the inner and outer radii of the interpolating annulus could be modified.

However, neither the insertion of the biharmonic functions nor a different choice of the

radii of the annulus lead to a significantly better energy gain that would allow to weaken the

assumptions. Therefore, in order to circumvent more ponderous notations and involved

computations, we have chosen the present approach.

Furthermore, from a geometric point of view, it seems quite natural, that the gradients ofv and w should be almost identical (up to rescalings) in order to control the biharmonic

energy of the interpolant.

In order to complete the proof of Lemma 2.17, we need to show

Lemma 2.18 Let f G C°°(Br,S4), 0 < r < oo. There exists rotations Ri G 50(5) and

R2 G 50(4), such that j:=Äio/ofl2e C°°(Br,S4) satisfies

1. 0(0) = N,

2. |£(0) is orthogonal to |£(0) for i ^ j,

3. EBr(f) = EBT(g)-

Proof. It is clear that there exists a rotation Ri G 50(5) s.t. h = Ri of verifies condition

1. dh(Q) may be viewed as a 4x5-matrix. If we define g(p) :— h(R2p), we verify

dg(p) = dh(p) R2,

where is the usual matrix multiplication. Hence,

(d9(0))T • dg(0) = Rl (dh(Q))T dh(0) R2.

As (dh(0)) -dh(0) G Symm(4), the theorem of principal axes gives a rotation R2 G 50(4),s.t. (dg(0)) dg(0) G Diag(4). Thus, g verifies condition 2. The last condition follows

from the invariance of the biharmonic energy under rotations. D

We have furthermore the following

Theorem 2.19 (Unique Continuation) Let u, ü be two biharmonic m,aps in C°°(f2,54).// they agree to infinitely high order at some point, then u — u~ everywhere on Cl.

Proof. Any smooth biharmonic map u G C°°(f2, S4) verifies the Euler-Lagrange equationassociated to the Hessian energy functional

A2u - - (|Au|2 + 2|V2u|2 + 4VuVAu) u.

Defining the new variables v — Vu, w — Au, it follows that any biharmonic map satisfies

the elliptic second order equation

Ay = F(u, Vu, Vv, w, Vw), (2.59)


with

(wVw | . (2.60)

-(u;2 + 2|Vt;|2 + 4VuVw)u

Consider now two biharmonic maps u,ü~ C°°(Ù, S4) with the corresponding new variables

v,v,w,w,y and y. Moreover, we define z :— y — y. As u, ü and all their derivatives are

locally bounded in U, it follows from (2.59) and (2.60)

iAî<o{^>n+EM}i,a

on every open and bounded U C £2- As zfi vanish to infinitely high order at some point,

Aronszajn's generalization of Carleman's unique continuation theorem in [6] implies that

z — 0 on U. The result follows from the connectedness of fi.

Corollary 2.20 If a biharmonic map u G C°°(£2, S4) is constant on some open set U C £1,then u is constant on Q.

Corollary 2.21 The minimizers of E(-) in E1 have non-vanishing gradient almost ev¬

erywhere.

Now let u* and 7 be defined as in the introduction to the present chapter. Then, we have

the following

Lemma 2.22 For y G B, there exists u G Flfy n S1 s.t. E(u) — 1.

Proof. We assume, after possible translations, that y — 0. Define

u := u* o aN o «j"1 : W \ {0} — 54,

where R4 :— R4 U 00, <Jjv and as are the stereographic projections s.t. o~m(N) = 00,

o-n(S) - 0, as(S) - 00 and as(N) = 0. Observe that u - u* o r, (u o r)(0) - u*(0),V(u o r)(0) — Vu*(0) and deg(u)= 1. We show now that u has a removable singularityat 0. Indeed, we infer from [14] (or direct computation)

E(u) = / \Au\2dx= \ASi(u*oaN)\2dvolS4+2 \Vs*(u* o aN)\2dvolS4Jm4 .Is4 Js4

= f |Au*|2dx=J.Jr4

Thus, u verifies the biharmonic equation on R4 \ {0}. We show that u verifies the bihar¬

monic equation on R4. Consider ip G C£°(R4), and smooth cut-off functions 0 < rje < 1

satisfying r]e(x) = 0 for |x| < e, ^(x) = 1 for \x\ > 2e, \Vne\ < Ce~l and |V27/| < Ce~2.

Inserting <j> — rîß G C(R4 \ {0}) into the biharmonic equation (2.1) gives

/ (Au + IVu|2u)A(r)tip)dx + 2 (|Vu|2Vu- < Au, Vu > u)V(n(iß)dxJr4 Jm4

- (\Au\2u+\Vu\2Au)nÊipdx = 0 (2.61)Jr4


We compute with a :— Au + |Vu|2u and b := |Vu|2Vu— < Au, Vu > u that

/ aA(nip)dx = / aAipnedx + 2 aS7ipVntdx + / aipAnedx (2.62)Jr4 Jr4 Jr4 Jr4

and

/ bV(nip)dx = / bVipn£dx + / bipVrj.dx. (2.63)Jr4 Jr4 Jr4

Recalling that rjÊ is constant outside the annulus Aç :— B2c\Bc, |u| 1 and | Vu|4 < | Au|2,we estimate using Holder's inequality

2 / aVipVndx <c( f \Au\2dx J*

( f \Vr],\2dx J*

< Of ^ 0, (2.64)

f atpAnedx<c(f \Au\2dxY ( f \Anf\2dxY <C ( f |Au|2dxY ^ 0 (2.65)

and

/ bipSJn,dx < (f \Au\2dxY ( f \Vu\4dxY ( f |V77£|4dxY

< c(f \Au\2dx J4

^ 0. (2.66)

Combining (2.61)-(2.66) gives, as e —» 0, that

/ (Au + |Vu|2u)A^dx + 2 (|Vu|2Vu- < Au, Vu > u)VipdxJr4 Jr4

- / (|Au|2u+|Vu|2Au)V>dx = 0

Jr4

for every test function iß G C£°(R4). Hence, u is a weak extrinsic biharmonic map on

R4. Thus, we may assume that u is smooth on R4. This completes the proof. Similar

arguments can be found in the proof of [36, Theorem 2.2.]. D

2.9 Large solutions on the unit disk II : Proof of The¬

orem 2.2

Now we are able to prove Theorem 2.2. Let u*, 7 and R be defined as in the introduction

to the present chapter. Recall that u — 7. From Corollary 2.21, we infer the existence of

a point x0 G B, s.t. Vu(x0) ^ 0. Furthermore, Lemma 2.22 gives u* G Ty,^ H H1 such

that E(u*) — J. Applying Lemma 2.17 to u* and u yields 5 > 0 and / G S* such that

EB(f)<EB(u)+l-ö.

Thus, let (ufc)fc6N be a minimizing sequence for EB in El such that

EB(uk) <EB(u)+l-6. (2.67)

2.9 Large solutions on the unit disk II : Proof of Theorem 2.2 35

Consider pk :— |V2Ufc|dx, vk :— J4(\?Uk)dx, v^ the positive part of vk and vk the negative

part of z/fc. We may extract a subsequence, such that

uk^ u weakly in V2'2(B,S4),

Pk—^

p weakly in the sense of measures,

z/fc-a y weakly in the sense of measures,

and

f du- f J4(Vu)dx = H4(S4). (2.68)Jb Jb

We deduce from Lemma 2.9 (after extension of uk,u,pk,p,uk and v to R4), that

v = J4(Vu)dx + J2 v(j)sx^ (2-69)jeJ

for certain points x^ G B (j G J), v^"1 G R \ {0} and J a countable set.

Suppose J t^ 0. Choose / G J. After performing a translation t, we may assume x^ — 0.

Henceforth, we set B :— t(B). Let 7r be the nearest point projection onto 54 and define

for r > 0

wjjr :— t udx.

Jbt

Applying Lemma 2.15, we get maps (wk)k^ C T>2'2(R4,S4) s.t.

_

Ufc for x G 5^W"-l

tt(ub) for ieR4\ß4,

and

as k -^ oo. Thus,

/ |V2wfc|2dx — 0,

B<t(k\Bek

E(wk) = EBtk(uk)+o(l),with o(l) — 0 as k —» oo. For A C R4 open set, we consider the characteristic-

function xa' R* —* {0,1}- Then, Xb\b< Auk converges weakly to Au in L2 and

Eb(u) < EB\Bck(uk). Hence, we infer from (2.67) and EB(u) < EB(u)

E(wk) < EB(uk)-EB(u)+o(l)< 1-5 + o(l) < 24H4(S4). (2.70)

On the other hand, as (wk)k^ C X>2,2(R4,54), we have

/ J4(Vwk)dx = dkH\S4),Jr4

with dfc G Z. Moreover,

/ J4(Vu;fc)dx - / J4(Vuk)dx + o(l) - v(l) + o(l) 7^ 0.


Thus, |dfc| > 0 for k sufficiently large. Furthermore, as \wk\2 = 1 (i.e. Awkwk = -\X7wk\2),we have with Lemma C.2

24H4(S4) > E(wk) = / |Au;fc|2dx > / \Vwk\4dxJr4 Jr4

> 16 f |J4(Vu;fc)|dx > 16|dfc|H4(54).Jr4

In particular, if |deg(u>fc)| = 1, Theorem 2.1 implies E(wk) > 1. This is a contradiction to

(2.70). Hence, J = 0. This implies with (2.68) and (2.69) that u G H*, and we are done.

2.10 Outlook

In view of the Dirichlet problem for harmonic maps (see Section 2.2), the results of the

present chapter seem not yet optimal. As the inverse of the stereographic projection min¬

imizes the Dirichlet energy among the maps u G X>1,2(IR2, 52) of topological degree 1, the

natural question arises if the corresponding minimizers u* G S1, given by Theorem 2.1,

can be identified.

Moreover, the conditions on the boundary data in Theorem 2.2 are suboptimal. Even-

though a general result, as in Brezis-Coron [10] for harmonic maps, seems unlikely to the

present fourth order problem (see Remark 2.3), there should be weaker assumptions on

the boundary data, allowing to apply the"

sphere-attaching lemma" Lemma 2.17. Con¬

ceivably, the identification of the minimal bubble u* will considerably ease the analysis.

Furthermore, we mention that the present method is not applicable to intrinsic biharmonic

maps. We are unable to bound the volume functional by the tension energy (as in the

fundamental estimate (2.18)). In addition, the tension energy gives no bound for the

W/2,2-norm. Thus, we can not infer weak compactness for minimizing sequences.

Chapter 3

Partial regularity in higherdimensions

3.1 Introduction

For several decades regularity properties of weakly harmonic maps have been intenselystudied. For a manifold M of dimension m = 2, C.B. Morrey [46] showed in 1948 that

every minimizing map u G W1,2(M,J\f) belongs to C°°(M,N). The regularity problemfor general critical points of the harmonic energy functional had remained open for a

long time. In 1981, still for the case m — 2, M. Grüter [24] proved smoothness of confor¬

mai weakly harmonic maps. R. Schoen [61] introduced the notion of stationary harmonic

maps and extended Grüter's result to this class. Finally, F. Hélein [26] showed that every

weakly harmonic map in the case m — 2 is smooth. For m > 3, more complex phe¬nomena show up. R.Schoen and K.Uhlenbeck [62] showed that if u G W1,2(M,N) is

energy minimizing, then u is smooth except on a closed subset S with Hausdorff dimen¬

sion dimî(S) < m — 3. In particular, for m = 3, they show that 5 is reduced to at most

isolated points. This result is optimal since the radial projection from Bm into 5m_1 is

a minimizing map for m > 3, as shown by H.Brezis, J.-M. Coron and E. Lieb [12] for

m — 3 and F.H. Lin [39] for m > 3. On the other hand, T. Rivière [58] proved existence of

everywhere discontinuous weakly harmonic maps. For the intermediate class of stationaryharmonic maps, L.C. Evans [18] showed partial regularity for maps into the sphere and

F. Bethuel [8] generalized this result for arbitrary target manifolds. Their proofs rely on

a monotonicity formula for stationary harmonic maps adapted from P.Price [55].

Similar questions for (extrinsic) weakly biharmonic maps were studied by S.-Y.A. Chang,

L.Wang and P.Yang in [15]. They showed smoothness for weakly biharmonic maps into

the sphere and m < 4 (see also Strzelecki [74]), and asserted partial regularity for station¬

ary biharmonic maps into the sphere and m > 5. C. Wang generalized these results for

arbitrary target manifolds in [75] and [76]. Once again, a monotonicity formula derived

from the stationarity assumption is crucial in the proof of partial regularity for m > 5

whose derivation in [15, Proposition 3.2] is only given for sufficiently regular maps.

Here, we give a rigorous proof of this monotonicity formula in the case of stationary

38 Partial regularity in higher dimensions

biharmonic maps of class W2'2(Br,N), concluding the regularity results in the above-

mentioned papers.

Theorem 3.1 (Monotonicity formula) For K > 0 and u G W2'2(BK, N) (extrinsi¬cally) stationary biharmonic, it holds for a.e. 0 < p < r < ~

r4-m / |Au|2dx - pA-m f \Au\2dx = P + R,J Br */ Bp

where

= *J (JBr\Bp V

(uj + x'u.j)2|(m - 2)(xiui)2

I im-2+

|_imI dX

\x\ \X\

*%*,+ 2 (x^Ç _

2\Vuf^R, — 2 1

i,,_ii. , i „

JdBT\dBp \ \x\ \x\ \x\

Here and henceforth, we use the following notations. We denote u — (u1,..., un), Uj — ^Letc. We tacitly sum over repeated indices and abbreviate

f fda := f fda - f fda.JaBr\dB„ JdBr JdBnldBr\dBp JdBr

Conceivably, this monotonicity formula may allow to study the singular behaviour of sta¬

tionary biharmonic maps and especially minimizing biharmonic maps as suggested by

M.-C.Hong and C.Wang in [28].

In the case of target manifolds without symmetry, another important tool for proving

(partial) regularity for harmonic and biharmonic maps is the technique of moving frames.

This was introduced for harmonic maps in two dimensions by F.Hélein [26], applied to

stationary harmonic maps by F. Bethuel [8] and later to (stationary) biharmonic maps byC. Wang in [75] and [76]. Only very recently, T. Rivière, in [59], succeeded to rephrase the

harmonic map system as a conservation law when m = 2, allowing him (amongst other

results) to give a direct proof of regularity of weakly harmonic maps in two dimensions

avoiding the use of moving frames. Prompted by [60], T. Rivière and M. Struwe devel¬

oped a related gauge-theoretic approach to prove partial regularity in higher dimensions.

Moreover, this new approach allows the authors to reduce Hélein's C5-assumption on the

target manifold to C2, which seems to be the natural assumption in order to render the

second fundamental form well defined. Finally, T. Lamm and T. Rivière [37] could show

smoothness for weakly biharmonic maps in four dimensions avoiding moving frames, and

M. Struwe [72] proves partial regularity for stationary biharmonic maps in higher dimen¬

sions via gauge theory.

Strengthening the natural hypotheses for the regularity of a stationary biharmonic map

u slightly by assuming some higher integrability of the leading order derivative, we show

here similar partial regularity results for biharmonic maps without using moving frames.

Our method is not restricted to this fourth order problem and provides regularity results

for polyharmonic maps. Therefore, we now switch to the general setting of polyharmonic

3.1 Introduction 39

maps.

More precisely, for k G N and u G Wk,2(fl,Af), we consider the fc-harmonic energy

functional

E(u)= f \Vku\2dx.Jn

Define the BMO space and the Morrey spaces Mp'x as

BMO(Cl) - {u G Ll(Ü) : [u]BMO(Q) := sup {r~m / \u- üBrnn|dx} < 00}BrcRm Jßrnn

M?>x(n) := {u G L"(SÎ) : [u]pM„X{n) := sup {rA"TO / |u^dx} < 00},BrcRm JBrnn

where uBr :— + udx.

Jbt

Definition 3.1 A map u G Wk'2(fl,Af) is called weakly k-harmonic, if u is a critical

point of the k-harm,onic energy functional with respect to compactly supported variations

on Af, that is, if for all £ G C(Q,RN) we have

and

d_Jt E(n(u +1£)) - 0,

f=0

where it denotes the nearest point projection onto Af.

Definition 3.2 A weakly k-harmonic map in Wk'2(Q, Af) is called stationary k-harmonic

if, in addition, u is a critical point of the k-harmonic energy E(-) with respect to compactly

supported variations on the domain manifold, i.e. if

d_dt

E(uo (id +1£)) = 0 for all Ç G C?(SÎ, Rm), (3.1)*=o

where id denotes the identity map.

Remark 3.1 (Stationary) 1-harmonic maps are (stationary) harmonic maps. Observingthat |V2u|2 and \Au\2 only differ by a divergence term,, we conclude that the (stationary)2-harmonic maps are precisely the (stationary) biharmonic maps.

Our main result then reads

Theorem 3.2 For p > 1 and 2kp < m, let u G Wk>2p(Q,Af) be weakly k-harmonic.

There exists e > 0, such that for each point x0 G Cl for which there exists some r0 > 0

withk

E[VÛ] 2i 2k <,1JMM'2fe(B (Xo))

-

ß=l

we have

ueCco(Br_£,(Xo)1Af).


Remark 3.2 For 2kp > m, Sobolev embedding implies that every map u in Wk'2p(fl,Af)is Holder-continuous. Smoothness then follows at once from elliptic bootstrapping argu¬

ments.

In view of the monotonicity formulae for stationary harmonic and biharmonic maps, the

condition on the Morrey estimate is, in these cases, satisfied almost everywhere. More

precisely, we deduce the following

Corollary 3.3 For k G {1,2} and p > 1, let u G Wk>2p(fl,Af) be stationary k-harmonic.

Then, u is smooth outside a closed set S with Hm~2kp(S) — 0.

Conceivably, a monotonicity formula allowing to guarantee the Morrey norm condition in

Theorem 3.2 will hold for k > 3.

The proof of Theorem 3.2 is based on Morrey decay estimates for the rcscaled poly¬harmonic energy. We employ an interpolation inequality by Adams-Frazier [2] (see also

Meyer-Rivière [44], Strzelecki [74] and Pumberger [56]) in order to bound the Wk,2p-novm

by the BMO- and W2k<p-norm.

The idea of proving e-regularity results with interpolation inequalities first appeared in

Meyer-Rivière [44] in the context of Yang-Mills fields.

Of course, the critical case p — 1 is the natural exponent for the present problem. More¬

over, Corollary 3.3 directly follows from Bethuel [8] resp. Wang [76] and Poincaré's

inequality. However, our proof is more direct and avoids the moving frame technique.

We want to remark that polyharmonic maps have already been studied by A. Gastel in

[19], where he considered the polyharmonic map heat flow in the critical dimension.

The chapter is organized as follows. In section 3.2, we derive the monotonicity formula

for stationary biharmonic maps. Section 3.3 gives the Euler-Lagrange equation for poly¬harmonic maps. In 3.4 we prove the Morrey decay estimates. In 3.5 we deduce Theorem

3.2 and finally, in 3.6, we conclude Corollary 3.3.

Section 3.2 is published in the paper [3] and sections 3.3, 3.4, 3.5 and 3.6 are based on

the joint work [5] with David Pumberger.

3.2 Proof of the monotonicity formula

Here, we want to prove Theorem 3.1. Choose in Lemma 1.3 the test functions £(x) :—

ipe ( ^ j x, where iß — ißc : R+ —> [0,1] is smooth with compact support on [0,1] and iß* = 1

on [0,1 — e]. Then, it follows with £/ — ißiX^ -f ißöij and £;? — ißuxi + 2ißj

0 = / ((4 - m)|Au|2V' - \Au\2xßiX% + 4ukkUij4)ix:> + 4ukkujißj + 2ukkujißHxj) dx.JR


We have iß, (M) = & (H) g and ißn (M) = ^ (fei) + te=V (tel) ^.Thus,

f lf 4 f x'xJ0 = (4-m) \Au\2ißdx / |Au|V|x|dx + - / ukkUijip' -r-r-dx

A"> t Jr t Jr \x\

2(m + l) f „a?*,

2 fH / ukkUjiß —rdx +

— I UkkUjiß xPdx.t Jr \x\ t Jftm

This implies for F(t) := r4"m /Rm | Au|Ve f^f\ dx

rm-3-^-/f(r) - (4 -m) f \Au\2i()dx - - f |Au|V|x|dxûT Jm. t Jr

4 f xixi 2(m +1) /" xJ= / UkkUijlß'-r-r-dx — / UkkUjiß'—dx

t Jr \x\ t Jm \x\

—2 / UkkUjiß"xjdx. (3.2)T Jr

Furthermore, we have:

r J \x\ \ p J \x

+3-m ,

{i±i\ T2-mdT_

Thus, applying Fubini's theorem twice gives:

/ r1_m / UkkUjiß" ( — J xjdx drJp Jr \ t J

f i<,>{\XW r3~m;

f1,,f\x\\ Pd~m

j= - / UkkUjx'ij) I — ——dx + / ukkUjXJiß I — —j—j—arc

Jm \ r / \x\ Jr \ P J \x\

+(3-m) f T2~m f Ufcfc7^xV ( — ) Adx dr..

(3.3)Jp Jr V T J \x\

Multiplying both sides of (3.2) with r3~m, integrating over r from p to r and inserting

(3.3) yield

T(r)-T(p) = -aTt2- f UkkUijii/ (W)*¥-dxdrJP Jr \ r J \x\

_8 f r2"m / ukkUjiß' (^] ^jdx dr

JP Jr V T J \x\

+2 f ukkUjxjiß' (M ) r*-m^-dxJr V r J \x\

-2 f UkkUjXJiß' (^) p3~m~dx.Jr \ P J \x\


r

For all Lebesgue points p and r of the function g(s) :— JaB ^V^ do G Ljoc(]0, K]), as e —>

0, by applying Lemma 3.4 below to the first two terms and the Lebesgue differentiation

theorem to the last two terms, we obtain

A'm f \Au\2dx - p4~m f \Au\2dxJBr JBp

=

/ (^kkUijxy ukku^\ dx_f ukku^d(j1

. \ \rr to—2 Tra-2 / /,, „

U. m—3

Jbt\bp \ \x\ \x\ j JdBT\dBp \x\

= 2f ÛkkUijXiXj

_

UkkjUjXiXj UkkUjXj \Jbt\bp

V \x\~~

m—2 \rr\m—2 |t.|to—2

p\R \ \X\—

- X '"_iX

Here and henceforth, all third derivatives are interpreted in the sense of distributions.

The monotonicity formula now follows by several integrations by parts. We proceed as in

[15] and compute for a.e. p and r

f UkkUjjXlXJ_

f UkUjjXlT?Xkj

f UfcUjjfcXV ^/ i i__ o

ax — ié i i

ou ii i r»

ax

l„ , ~ \rr\m—2 I, „

«.to— 1 / ,Lm-2

JBr\Bp \X\ JdBr\dBp \x\ Jbt\bp \x\

r / ukuikxi (m - 2)ukuijXixj2Jbabp V \x\m'2 \x\m

__

f fukUijXlX:!Xk_

UkUjkXl \

JdBr\ÖBp V \x\m~l \x\m~3 J

i / (ukUikXi UikUjkXiXj \

JBr\Bp\\x\m-2 \Am~2 )

+(m -2) f UkUf^Jxkdx,Jbt\bp \x\m

f _Ufcfc,UjXV= _

f UkiUjXiX3XkJbt\bp \x\m~2

X ~

Jaws,, \x\m~l

f /ukjUjXk u3ukkxJ\

JbabAm-2 \x\m~2 )+

>BT\BP

t

JBr\Bp V Ixh \x\m'2'

(2 — m)ujUikxtx^xk UikUjkX%x?

f ^^dx=

/ p^do-

f j^-dxI

, tto—2 /„

,-v. w—1 / Lra-2

Jßr\Bp\X\ JdBT\dBp

\x\

JBr\B0 \x\

JJbt\bp

Combining these equations leads to

r\Bp \X\JdBr\ÖBp

FlJßr\Bp

i I (m-2)(ufcXfc)2_

UkUjkX3 '.

'x|m~2

m / |Au|2dx - p4~m f \Au\2dxJBr .JBp

= 2JJ8BT

I iXjJb } CLjLLjjjL .

\dBr V \X\ \X


+2 /.'ujUjjX*

| 2(y')2 |UjUjjX1 |Vu|2

|(m-2)(u^);

Jbt\bp \x\m~^ lxlm—2 lxlm—^ lxlm—Îxl

Moreover, it holds

0=_4/ i^+s/ (î+m4)dxI U. to—3 / \ U. to—2 Ly. rn—2 /

JdBT\dBpFl

./bab,, VFI Fl /ldBT\dBpFl

v/ßj-Xi

and

(UjX')2= 2/

JdBT

-2/J BT

TO—1da

\öör ^

u^x*+

|Vu|2+u^x*

+(2-m)(uixi)2 .

^,

,,,.,7n—2

ItIto—2 ItItti—2\sp

\q.lltl~4 \rp ini—

i

Adding the last three equations yields the monotonicity formula. It remains to prove the

following

Lemma 3.4 For /, 0 G L^R), define

{<j>*f)(x):= f <ß(l)f(y)dy= f <ß(z) f (xz)xdz.Jr xx/ Jr

Suppose /, (cßk)keN G L*(R, R), with

4>k>0, / 0fc = 1 and supp^fc C [1 - -, 1], k G N.

TTien, as k — oo /or 0 k * f-+id • f in V-{\p,r]),

where (id f) (x) = xf(x).

Proof. Choose /, G C0(M,R) s.t. ||/j - /||li(r,r) -> 0 as ; -> oo. It follows

WW) - f\\Li < \\f(h-)-fj(h.)\\Ll + \\f-f^ + \\f,(h-)-fj\\Ll< 2||/ - fj\\» + Cl(supp(fJ(h-) - mifÄh-) - fj\\suP.

As fj are uniformly continuous, the second term on the right hand side tends to zero as

h tends to 1. Thus, we infer that

\\f(h-)-f\\v(R,R)ÔÎoTh^l. (3.4)

Moreover, we have

((pk * f)(x) - xf(x) - x / <j>k(z) (f(xz) - f(x)) dz.Jr


Thus, we have using Fubini's theorem and Holder's inequality:

Hk * f - xf\\Ll(M) < / |x| / \4>k(z)\\f(xz) - f(x)\dzdxJp Jr

- rLlMz)l iflf{xz) ~f{x)idx)dz< rWMv sup \\f(h.)-f\\L*.

ftesupp 4>k

As supp^fc C [1 — |, 1] and H^Hl1 = 1> assumption (3.4) implies that the right hand side

converges to zero for k —> oo.

The monotonicity formula implies the following

Proposition 3.5 Let u G W2,2(B2,Af) be stationary biharmonic and 0 < R < 1. There

exists e > 0 and po > 0 sufficiently small such that if

R4~m f (|Vu|4 + |V2u|2)dx < ,Jb*

'BR

we have

[V7x]M4,4(Bpo) + [V2u]M2,4(jBpo) < Ce.

Proof. See C.Wang [76] and Struwe [72]. D

3.3 Euler-Lagrange equation for polyharmonic maps

We deduce the geometric form of the Euler-Lagrange equation for weakly polyharmonic

maps. We consider, for S > 0 sufficiently small, the tubular neighborhood Vs of Af in RN

and the smooth nearest point projection Uj^ : Vs —> Af. For p G Af, P(p) := VU(p)is the orthonormal projection onto the tangent space TpAf. The orthonormal projectiononto the normal space will be denoted by P1. Recall that P + PL — id. Then, we have

Lemma 3.6 (Euler-Lagrange) If u G Wk'2(Çl, Af) is weakly k-harmonic, then it satis¬

fies

P(u)(Aku)=0 (3.5)


Proof. For f G C^(Q,RN), we compute

0 = ^1 f\Vk(IlMo(u + tt;))\2dx

= 2 f VfcuVfc(P(u)(0)dx.Jn

3.3 Polyharmonic maps 45

Remark 3.3 For a weakly polyharmonic map u in C°°(0,,Af) D Wk,2(fl,Af) and £ G

C^(Ü,RN), with £(x) parallel to Tu{x)Af for all x G tt, we have P(u)(£) = £. The proof

of Lemma 3.6 then shows that

Aku JL TuAf


In order to formulate the following lemma, we introduce the /-divergence V^- as follows.

We define V^ • u :- V • u and Vw • u := V (V*'"1* • u) for I > 2.

Lemma 3.7 If u G Wk'2(fl,Af) is weakly k-harmonic, then there exist f and gji with

Aku = f+ ]T V^-Aigju (3.6)1,1 >o

1 < 2j + I < k

where

k

l/l < C ^2 u I v/iwl7A'" with J] Pl\n = 2k for everV A G A>AeA ß=0 ß

k

\gji\ < C Y^ IIIVul7*'" with J2 Pix,» = 2k- (2j + I) for every A G A,AeA ß=0 ß

with A consisting of finitely many indices and 7A,M > 0 for every A G A and 0 < p < k.

Remark 3.4 Note that the representations in Lemma 3.7 are not unique. See for exampleRemark 3.5.

Proof. We observe that

Ak(a • 6) = Yl 4AV9a • AjWqb

0 < i, }, q < k

i + j + q— k

where ckjq are positive integers. In particular, we have ck00 = c^ — 1. Combining this

with equation (3.5) shows that u satisfies

0 - P(u)(Afeu)= Afc-1(P(u)(Au))- J2 ($j-q1AiVl(P(u))A3+1V'u ,

0 < i,j,q < k - 1

M J i- q = k - 1

(i, }.<!)? (0,*-l,O)

in the sense of distributions. Now we consider the second fundamental form A(-)(-, ) of

jV in RN. The property P(u)(Au) = Au + A(u)ÇVu, Vu) implies

A*M= Jl 4r1A'V9(P(u))A^1V'?u-Afc-1(J4(u)(Vu,Vu))."

(3.7)

i + 3 -\- q — k — 1

(i,3, q) * (0, k- 1,0)


First we consider the case when k is even and analyze the Euler-Lagrange equation (3.7)term by term. It is sufficient to show that every term in (3.7) can be written in the desired

form.

For i, q such that i + \ = §, we have that cfr1 A*V9(P(u))Aj+1V% is of the form /. Here

we used the fact that

ß

\vßp(u) i < cyn ivn7a,m with y ßix>»=&for every ^ -k and a e a- ^3-8ÂeA ß—0 ß

Indeed, the chain rule gives V(P(u)) = VP(u)Vu and V2(P(u)) = V2P(u)VuWu +

WP(u)V2u. We infer estimate (3.8) by iterating this computation and taking the smooth¬

ness of the nearest point projection into account.

For i, q such that i + | > ^,we compute

A'V9(P(u))AJ+1V%= V • (AI-1V9+1(P(u))AJ+1V(?u) - A'-1V,+1(P(«))AJ'+1V«+1u

and/or

A'V9(P(u))AJ+1V<?u= V • (AiV<7-1(P(u))AJ+1V9u) - AiV?-1(P(«))Aj+2V9-1u.

Iterating these computations, we get with estimate (3.8) that c*~1A'V9(P(u))AJ'+1V%is of the form

/ + y v(0 • "su3,1 > 0

1 < 2) f ( < k

whenever i + | > |. The terms for i,q such that i + | < | are estimated similarly.

Moreover, we have

Afc"1(^(7i)(Vu, Vu)) = V • A^7l,with 7=^-1,completing the case when k is even.

For k odd we distinguish between the three cases i+ | = ^, i +1 > ^ and i + ^ < ^,and proceed similarly to the case when k is even. Moreover, we get

A*-'(/lM(Vu, V«)) = { ^ ^ 7 = ¥ £ J=

J odd

This completes the proof. D

Remark 3.5 Observe that harm,onic maps (k = 1) satisfy

Au - -A(u)(Vu, Vu) in V.


Thus, the harmonic map equation is of the form Au = / with

/ = -,4(u)(Vu,Vu)<C|Vu|2.

Weakly biharmonic maps (k — 2) satisfy

A2u = AP(u)Au + 2VP(u)AVu - A(A(u)(Vu, Vu))= -AP(u)Au + V • (2VP(u)Au - V(A(u)(Vu, Vu))) in V,

i.e. the biharmonic map equation is of the form A2u = / + V • goi with

f = -AP(u)Au < <7|V2u|2

and

goi =2VP(u)Au- V(A(u)(Vu,Vu)) < C|V2u||Vu|.

However, we could also write the biharmonic map equation as

A2u = -AP(u)Au + V (2VP(u)Au) + A(-A(u)(Vu, Vu)) in V,

i. e. it is also of the form A2u — f + V • #oi + Agw with

f = -AP(u)Au < C|V2u|2,

g01 = 2VP(u)Au < C|V2u||Vu|

and

gw = -A(u)(Vu,Vu) < C\Vu\2.

This illustrates the nonuniqueness of the representation mentioned in Remark 3.4-

3.4 Morrey decay estimates

We will deduce Theorem 3.2 from the following

Proposition 3.8 For p > 1, let u G Wk,2p(Çl,Af) be weakly k-harmonic. There exist

e. > 0 and r G (0,1) such that for each point y0 G ft for which there exists a radius ro > 0

withk

V[V"u] 2k2k <e,

ß=i

we have

k k

(TT)2*p-m£ / |v^|^dx < K2kp~mY f |V«u|^dx, (3.9)/i=1

JBTr(x0) 4^=l

Jb,.(x0)

for all x0 G Bro(y0), 0 < 4r < dist(x0,dBro(y0)).


Proof. We consider the fe-harmonic extension v of u, defined as the unique solution to

the following Dirichlet problem:

Akv = 0 in Br(x0){ 0 = 0 ondBr(x0),

for 0 < / < k — 1, where v denotes the unit normal vector to dBr(xo). According to

Lemma B.4, we have

it

/. IVvl^dx < C

BpOo)

(P-YY f |VA^dx (3.10)

for 0 < p < j and 1 < p < k. It follows that

,2k],m dx

ß^l Jbp(xq)

<Cj2f \Vv\i?dx + C^2 f |V^(u-7;)|^dx/j=l Jbp(xo) ^=1 Jbp(x0)

k k

<cß)mS2f IVÎ^dx + cV / |V(u-v)|^da:

k k

<cf^)mV;/ IVûl^dx + cV / \Vß(u-v)\2J?dx. (3.11)

In view of Lemma 3.7, we introduce the auxiliary maps Uf and u3l for all j,I > 0 such

that 1 < 2j + I < k as the solutions to the Dirichlet problems

A'

Afcuf

k,2p/

U,

9jl

f with Uf-UE W^p(Br(xo)),

V^-A^jt with ug3leWk^(Br(x0)),where / and gj\ satisfy (3.6). Observe that the uniqueness of the Dirichlet problem implies

(3.12)u = uf + E %

93r

3,1 >01 < 2j + I < k

k,2p.Moreover, U/

— v G W0' p(Br(x0)) satisfies

Afc(U/-«) = /.

Lemma 3.7, Holder's inequality and Nirenberg's interpolation inequality (A.l) give

k

Lv(Br(x0))< C EIIlV/J

AeA /j=0

k

7^|7A,M

LP(Br(xü))

< cEnii^««7»A<=A ß=0

* cEYlAeA m=o

|2

2fcp

L ** (Br(xo))

, ||Tx,M(l-f) iiiiÎ "Nz,»(Br(x0)) II ullWfc.2p(^(ieo))

<CIMIwM^bô)) < OO,


and

lft'llLrJ'(flr(xo)) ^ C yjniv^rAeAM-0 LT3i(Br(x0))

k

< cEniiv"<A4AeA /j=0

fc

CL 11 \\U\yL(BT{xQ))\\U\\Wk,2p(Br(xo))AeA m=0

<

2fcp

h » (Br(xo))

,•

*> !U,I! —*

^ ClMlvvMpfB,(Br(*0» < oo, (3.13)

where

r,/ =

2kp 2k - (2j + 1)n

and 1 < n-,1 :— ; < 2.

kJl

2k - (2j + I)

Thus, Corollary B.2 and Lemma B.7 imply that

uf-ve W2k*(Br(x0)), u9jl G W2fc-^+/)^(£r(xo)),

and

l|V2*(«/-f)||LP(fl,(x0)) < C\\f\\mBr{ro)),

l|V2fc-(2j+/)uffJLV(Ma:o)) < C\\g3l\\Lr,l(Brixo)). (3.14)

We remark that here is the only place where we need that p > 1 in order to ensure the

first estimate for uj— v.

We have

/. ...2kp

|V(u—v)\ a dx

Bj(io)

<C f \^(uf-v)\^dx+C E /JBr(xo) 3j>0 JBt{x0)

.2kp

|V%,| * dx

1 < 2j -| 1 < fc

(3.15)for 1 < /i < k. We apply the Gagliardo-Nirenberg interpolation inequality (A.2), Lemma

B.3, estimates (3.14) and Lemma 3.7 to obtain

1_Ji_L IL

V(«/ - v)\\L& < C[uf- v]b0{b^(xq)) \\uf - v\\%2k<p{B (xo))

< C[uf -v]B0{Br{oro))

< C[uf-v]^à

V2k(uf-v)\\%{Br{Br{Xo))JL2k

BMO(Br(x0)) IM \\Lp(Br(r0))4"

< C[u,-v]B** (lo))AeA /j=0

u

JL

2%

iP(Br(*0))

wi*h Su^7a,/* = 2fc for every A G A. Next we use Holder's inequality and Young's

inequality to deduce

AeA m=o

u ITA,/

£^(flr(a;o))

< EniivAeA m=o

|7A,;I 2fcpL M (BT(*o))


< CEllV"WHM:ß=l

2kp 1

L >J (Br(xo))

where we remark that VJ^=1 ^^ — 1 and the L°°-norm of u enters into the constant C.

Combining the above estimates we obtain

/ |V>, - v)ffdx < C [uf - ^mSSL)Ê /JßtlXo) X ß=\jBr

"èhz

|V"u|~^dx. (3.16)'BjOeo)

For the second term in (3.15), as before, the Gagliardo-Nirenberg interpolation inequality

(A.2) gives

I Vuwllr2ifiL » (B^(x0))

< C\u11-^'

-

U lU93llBMO(j \Un\&«

where

< On :=

BMO(B%(x0)) "

"fclII v^w-^+O.r,,(B (a.o))

»

^=

„-1< 1

2k - (2j + 1) ljl -

(3.17)

for all j, I > 0 such that 1 < 2j +1 < k. Furthermore, we again apply Lemma B.3 with

p — 2k — (2j + l), estimates (3.14), Lemma 3.7, Holder's inequality and Young's inequalityto get

Ugil\\w2k-{.23+l),T < liv2k-(2j+l)QJl(Br(x0)) - " V U9,l\\LT3'.(Br{x0))

< C\\3jl\\Lrß(Br(x0))i

k

< C ÎJlVulW"AeA ß=0

k_

A£Am=0 Lr^^(BT(x0))

k

Ll(Br(x0))

l

2kp

< cEIIiiv» ITA,,

AeA ß=0

k krjji

2kpL M (Br(xo))

< CEllVûlß=l

2kpL m (Br(z0))

(3.18)

where

Vji =

Combining (3.17) and (3.18) gives

2^ ~ (2j + Q ^

k

f |VXJ^dx < C [u9/BMt^Z)) E / |Vû|^dx. (3.19)


Thus, we conclude with (3.15), (3.16) and (3.19) that

I'B^(xo)L V~ß(u-v)f?dx (3.20)

k

<c([uf- v]BlMf{B%Q)) + E ["JbJoQZ)) E / iVN^dx.

From Lemma 3.10 below we infer

[«/ - -IbmIIL))+e kj ilMoàZ)) ^ c"ß <3-21)hl

for some ß > 0, all 1 0 such that 1 < 2j + / < k. Then we can

combine this with inequalities (3.11) and (3.20) into

p2^mJ2 f \ûf?dx < Cx ((P-)2kP + e? (P)2kp-m) rKP-mJ2 f \V^dx.ilîJbp(x0) \\î / \r' ) îJbAxo)

We conclude the proof of this proposition by setting r :—£ equal to (2Ci)~^ and

choosing e > 0 sufficiently small so that CießT2kp~m <\.Now it remains to show (3.21). In a first step, we prove the subsequent lemma.

Lemma 3.9 There exists a constant C > 1 such that

JL- 2k *L 2k *L f 2k

E[vx.]>,fc,fl, „^CY,^%^(B. +^2fe_mE / iv%,f^.^ MM' (Br(*o)) ^ M" (Bir(xo)) ^TiJBr(xo)

(3.22)

Proof. For Bs(x) C -E?4r(xo), we consider the fc-harmonic extension Vji of ugß on Bs(x),where uffj, is set to zero outside Br(x0). Then u^ := u^

—

Vji G iyofc,2(P>5(x)) satisfies

AS, = V<'> • A?~g3l,

where <?_,/ :— XBr(x0)9ji- Analogous to (3.13) we conclude that w3i G W2k'^Jrl'i,r'1 (Bs(x)),and similar to (3.18) we get

krht

hhi\\w^-^^3i{Bs{x)) ^ cY,WVßuWT%m,„_1

'> ^ (Ms(Xß—

1

Here and henceforth in the proof of Lemma 3.9, we set p — 1. Observe that the second

estimate in (3.14) is still valid in this case.

Applying the Gagliardo-Nirenberg inequality (A.2) and the preceding estimate gives

Jb*(x) ^" J=S JBs(x)fc

< CHÄÊ/ \û\fdx (3.23)«

ß=1 Jbb(x)


for 1 < A < k. As Vji is the fc-harmonic extension of ug t,we obtain with Holder's

inequality, Poincaré's inequality and Lemma B.6

2k

Mmo(b,(i)) ^ SUP \t \w3i - wjlB\dx1 BcBs(x) Ub't

< CN^Àm^b^x))l2* , „rr-, l2fc

^ C[Vu*«Lw(B. (x))

+ C [VVjAm^b {x))

(k\>l

E/ |V%j"dx . (3.24)ß=i JB'(X) /

Arguing with the help of Poincaré's inequality and (3.13) we show that

\\ua,i\\w^(Br(x0)) < C||V ugJ\L2{Br(xo)) < C\\gji\\L2{Br(xo)) < C\\u\\Wk,2{BT{xo)) < Ce < 1,

provided e > 0 is sufficiently small. Using this we can omit the exponent p in the last

term of (3.24) and estimate

h"*

2k

[^\2BMO{B&)) < Ê [*%]>.»,fl (x))(3-25)

ß— 1

Combining estimates (3.23) and (3.25) with Young's inequality yields

r. ( k2k \ (.Ä~2fe-(23to) k

ç

/ ivvi2^ < cEfvxJ;wBf J E/ ivN^rfx./B|W \~[ MM' {Bs(x))J ß^lJßs(x)

k2k

k2fcC(A)

< ^ [^J;»,WW) + CM£ ||V«|| „• (3.26)ß=l

v v "

//=1

with 7 > 0 and C(X) > 1 for 1 < 2j +1, A < &. With a rescaling argument this gives

/ |vs«l"dx < c,-2SE tvX«]>.„,„, „

+ CWE / |Vû|ïdx,

(3.27)where the constants are now independent of s and A. Here we also used the fact that

||Vu|| & < 1 for 1 0 is sufficiently small. Combining

estimate (3.27) with Lemma B.4, we estimate for 0 < p < f as in (3.11)

E/ \û9ßdx < c(9mE/ \û9jl\fdx + C±f iV^^dxß=lJBP(x) V«/ ß=lJB%{x) ß^JB^x)

< Cßyf^f \Vßugi\fdx + C(-f)Y] f |Vu|tdxV,S/ ß^lJMx)

'

ß=\JBs{x)k

2k

+cs-2fc7E[vxJ%2fc • (3-28)


The proof of the lemma is completed by the following iteration argument. To simplify, 2k

notation, we define T(p) := P2k~mJ2ß=i Ib <x) I^^-iI"^ so tnat tne above estimate

becomes

fc

T(p) < c(^)2kT(s) + C(1)(^)2k~ms2k-my f |V«u|^dx

2k 2k.A /• '2k(Bs(x))

**(;) E[v%J;¥

ß=i

Next we choose r := ^ sufficiently small such that CV2fc < |, which implies that

fc

T(ts) < -T(s) + C(i)r2k-ms2k-mf] f |Vû|^dx2

/1=17bs(x)

___ a*

+C7x—|:[V%I];f,21(aw). (3.29)

Now we consider Ba(x) C B2r(xo). There exists JêFJ (simply set i — [log ^/logrj)with ri+l2r <<t< Tl2r and note that therefore (ri2r)2k~m < o2k~m < (T(Hi)2r)(2fc-m)We estimate

TV) = ^-e/ ivx#^M=l JB<r(x)

< (r<"»2r)<2'-"£/ |VX,I*^,U=1 jBT'2r(X)

< CT^r). (3.30)

Furthermore, estimate (3.29) gives

fc

T(r'2r) < ^(r^r) + C(7) (T2r)2k-m T^-^2k~m) E / |V"u|*<£e2

^=i ^T*-i2»

2fc

M+C7r2^^[v%r^ M » (Brl_l2rW)

fc

2fc1K

,

< -Tfr*-12r) + C7(7)r2fe- E iVN "

* a

£i—1

+C7r2fc- E IVXJ2*

_,MM'ZK(Br(xo))

^—1

where in the last term we used the fact that u9il is supported in Br(xo) allowing us to

reduce to the Morrey norm on Br(x0). Setting

k2k

k

s := c(7)r2k-e \?"%**>IB,n+C^2k~mE [v%J ;*.»,„

,,1-pv M f- (Bir(x0)} J M ** (B,.(xo))


the above becomes

T(r{2r) < \T(Tl'l2r) + S(r)it

which after iteration yields

T(r*2r) < T(2r) + E ^5-

,2M

m=i

< T(2r) + S.

Combining this with (3.30) gives

2k r

T(o) < C(1)J2lVßu]ir2_k2k +Cr2Ê/ lVX#^

k2k

+o» £[*%.];•»,/i=1

'M^Bô))

The desired result now follows by taking the supremum over all such balls Ba(x), and

choosing 7 > 0 sufficiently small to absorb the last term on the right-hand side. D

Now we are able to complete the proof of Proposition 3.8 with the following

Lemma 3.10 We have

iUf ~ V]ßMO(Br(xo)) + E [W»JBMO(ß|(zo))

~

Ct

3,1

for some ß > 0 and all j, I > 0 such that 1 < 2j + I < k.

Proof. Similar to (3.23) and (3.26) we estimate

kP 2k

lM û'ßdx -

Cl ["»"]-°«m«o»+°w E »v'»«iV(Br(I0)) <3-31)

for 1 < A < A: and every 7 > 0. Lemma 3.9, estimate (3.31) and Poincaré's inequalitygive

[Vu,J M2k,2k{Br(xo))< C7 [u9jt] BMO(BAxo)) + C(7) E ^^-(b^))

/i—1

ß—1

/j—1

3.5 Proof of Theorem 3.2 55

which for 7 > 0 sufficiently small implies

k"*

2k

ß—1

Applying Holder's inequality and Poincaré's inequality together with the above estimate

we infer

\U93l\BMO{Br(x0)) -

C V*U93i\M2k>2k{BT(xo))

k

< cE[VN%2,^ M f 'ik(BiT(xo))ß—1

< Ceß

for some ß > 0. From (3.12), we deduce

lUf ~ V]ßMO(Bj(x0)) -

C E [U9,H BMO(Br (x0))+ C lU ~ V\ßMO(Br(xü)) • (3-32)

3,l>0*

Holder's inequality and Poincaré's inequality imply

iU~ ^lBMO(Br (xo)) -

C ^(U ~V)\m'^(B^ (xo))

< C (n;2,2(Bc(io)) + [Vv]*^^),) . (3.33)

As v is the k-harmonic extension of u, Lemma B.6 and Holder's inequality imply

[V«]m*.»(BÇ(*o)) + Nm2^(Bj(ï0)) - CZ^^tJ"U^M2,^(B2r(x0))ß=l

k

Zsl JM /x '3*(Bar(*o))

< Ce (3.34)

for some /? > 0. Estimate (3.21) now follows from (3.32)-(3.34). This completes the

proof.

3.5 Proof of Theorem 3.2

Theorem 3.2 now follows from Proposition 3.8. Consider x0 G Bm(yo)- An iteration of4

estimate (3.9) implies

2kP-m k

^ (riYkP-m± f \V^dx

(3.35)


for all i For 0 0.

Observe that (~)1 < (t1)1'. Thus, inequality (3.35) implies

p2kP-mJ2 |V"u|^dx < r2*"-!^) E/ iVûl^dxM=1-/Bp(xo) V 4 /

î^^ô)

< (rji)2kp-^y f |v-u|^dx^-< C(P.

for all Bp(x0) C B^(yo). Hence, by Morrey's Dirichlet growth theorem in [47, Theorem

3.5.2], we conclude that u G C0'» in a neighborhood of y0. The smoothness of u near y0

follows now from elliptic bootstrapping arguments. This completes the proof of Theorem

3.2.

3.6 The harmonic and biharmonic cases

Here we give a derivation of Corollary 3.3. For stationary harmonic maps, i.e. k — 1,

Corollary 3.3 follows from Theorem 3.2 and

Proposition 3.11 (Monotonicity formula [55]) Foru G Wx'2(B2,Af) stationary har¬

monic and 0 7"),

I r^° Jbt(x0) )

with 7 > 0 small. We have by Ziemer [81, Corollary 3.2.3.] that rim~2p(S) = 0. ApplyingHolder's inequality, we get that for any y0 G fl \ S, there exists R > 0 s.t.

R2~m f |Vu|2dx < C (R2p~m f \Vu\2pdx) "

< 7.JBR(y0) V JBR(y0) J

3.7 Outlook 57

Hence, the monotonicity formula (3.36) implies for x0 G BR(y0) that

dxp2~m f |Vu|2dx < CR2~m f |Vu|Jbp(x0) Jbr(x0)

~2~

< C f |Vu|2dxJbr(Vo)

< C2y.

Fix 7 :—£-,

where e is given by Theorem 3.2, and the claim follows.

For stationary biharmonic maps, i.e. k = 2, we replace Proposition 3.11 by Theorem 3.1.

We observe that Nirenberg's interpolation inequality (A.l) implies that Vu G LAp(Q), and

define

S := (x0 G n : limsupr4p-m / (|V2u|2p + |Vu|4p) dx > VP\I r^° JbAxq) )

with 77 > 0 small. We have by Ziemer [81, Corollary 3.2.3.] that Hm-4p(S) = 0. For any

yo £ ^ \ S, there exists R > 0 s.t.

R4-m f(|Vu|4 + |V2W|)2c/x < (7 ( H4p~m / (|Vu|4p + IV2u|2p)dx )

"

< Tj.JBR(yo) V JBR(y0) /

As we mentioned in Proposition 3.5, the monotonicity formula implies the existence of

p0 > 0 and ë0 > 0 such that if R4'm /Bä( }(| Vu|4 + | V2u|2)dx < ë < ë0, we have

p4-m j (|Vu|4 + |V2W|2)dx < C3ë

for all Bp(x0) C Bpo(yo). Fix ë :— min(,ë0) > 0 and 77 :— ë > 0, where e > 0 is given byTheorem 3.2. This completes the proof.

3.7 Outlook

Our approach to regularity for polyharmonic maps can easily be extended to intrinsic

polyharmonic maps. In fact, the nonlinearity of the Euler-Lagrange equation is not es¬

sential in the proof. It is merely required to be of the form as stated in Lemma 3.7. So,Theorem 3.2 is even extendable to a more general class of elliptic partial differential equa¬

tions. However, we have restricted our attention to the case of extrinsic polyharmonic

maps, as the monotonicity formulae for stationary harmonic and extrinsic stationary bi¬

harmonic maps allow to conclude partial regularity in these cases.


Appendix A

Sobolev spaces and embeddingtheorems

Here we collect some results about Sobolev spaces.

Let il be a domain in Rm. We define a = (an,..., am) G Nm, |a| = X^j=i aji and define

the distributional derivative VQu — ;- • • • JpsrU via

fvau<pdx= /(-l)HuV>dx,Jn Jn

for all 0 G C£°(S1). By definition, Vu belongs to 1^(0,) if there is a function g G //(O)s.t.

f g<j)dx= /(-l)lQluVa0dx.We then identify Vau with # G Lp(ty.For & G N and 1 p(tl) = {ue Lp(Q) : V"u G LP(Q) for all a : \a\ < k},

with norm

IMI{U»(n) := E IIVauWl*(n) for 1 < p < oo

\a\<k

and

l|w||w*.»(n) :=max||Vau||L«(n).|a|<fc

Moreover, we set

Wk'p(Q,RN) - {u G tf(Çl,RN) : u7 G Ty*'p(ft) for all 1 < 7 < N}.

Wk,p(Çl) is a Banach space. For 1 p(n) is even separable, and Wk'p(ty nC°°(Q) is dense in Wk*(i1).Furthermore, we have the following theorems.

60 Sobolev spaces and embedding theorems

Theorem A.l (Sobolev, Rellich-Kondrakov [1]) Let il C Rm be a bounded domain

with Lipschitz boundary and consider fc G N, 1 < p < oo. Then the following holds:

1. For kp < m and 1 < q < -£Fk ,we have

Wk'p(il) ^ Lq(il).

The embedding is compact for q < -^-.

2. ForOKk-^Kl + l andO<a<k-l-^, we havep

— —

p

wk'p(n) ^cl'a(U).

The embedding is compact for a < k — I — —.

Theorem A.2 (Extension operator [1]) Let il be a smooth domain tut"1. Fork G N

and 1 

Wk'p(Rm) s.t. for every u G Wk'p(il) the following holds:

1. Eu(x) — u(x) a.e. in il

2. ||£û||Wfc,p(Em) < C||u||VK*,P(n).

The following interpolation inequality has been proven by L. Nirenberg in [52],

Theorem A.3 (Gagliardo-Nirenberg type inequality 1) For k G N and 1 < q,r <

oo, let u G Vk'r(Rm) n Lq(Rm). Then, for 0 < j < k, we have

||VJu||LP(Em) <C||Vfcu||2,(Mm)||u||^(affim), (A.l)

where1 J (l k\^n ï1

= a +(1- a)-,P m \r mj q

for all

{<a<l.k

The constant C is independent of u.

Remark A.l For a bounded domain il with smooth boundary, the result remains true ifwe add to the right side of (A.l) the term C\\u\\Lq(ty for any q > 0. The constants then

depend also on the domain.

In particular, we infer from Theorem A.3 that for a—\ and q = oo

II * Wlll,P(Rm) — k||V U||Lr(]Itm)||u||z,°°(]Rm),

where

p m 2r 2m

However, for our purposes to prove e-regularity for polyharmonic maps in section 3.4,

this is insufficient. Therefore, we need to employ an improvement of the preceding in¬

equality, where the L^-norm is substituted by the £?M0-seminorm. Such an inequalityfirst appeared in Adams-Frazier [2]. See also Meyer-Rivière [44], Strzelecki [74] and Pum¬

berger [56]. The following version of the Gagliardo-Nirenberg type inequality is stated in

Pumberger [56].

61

Theorem A.4 (Gagliardo-Nirenberg type inequality 2) Assume that u G Wk'r(il)for some r > 1 and 1 < j < k, with j,k G N. If u G BMO(il), then Vju G Lp(il) for

p :— ~r and

||Vû||L,<C[u]^0||uCfc,r, (A.2)

where 6 :— £, for some constant C — C(k,j, r).

62 Sobolev spaces and embedding theorems

and

Appendix B

Linear Estimates

Singular Integral Operators

We recall the following singular integral theorem in Stein [68, Theorem II.3.2].

Theorem B.l Let K : Rm —> R be a measurable function such that

\K(x)\<C\x\~m for\x\ >0, (B.l)

f \K(x-y)-K(x)\dx<C for\y\>0 (B.2)

/ K(x)dx = 0 for all 0 < Ri < R2 < oo. (B.3)Jri<\x\<R2

For e > 0, 1<-

Then, we have

\\TJ\\rjP<C\\f\\LP,

where C is independent of e and f. Moreover, there exists Tf G Lp(Rm) such that

TJ — Tf in Lp,

aseÔ, for all f G Lp(Rm).

For m > 2k + 1, the fundamental solution of Ak on Rm is

rfe(x-u) = c|x-u|2fc-,

i.e.

A*Tfc(x -y) = 5{x - y) for x, y G Em.

The kernel K :— V2fcTfc verifies the hypotheses of Theorem B.l. Thus, we conclude

64 Linear Estimates

Corollary B.2 Let f G Lp(il), 1 < p < oo, K = V2ferfc and u = Tf. Then, u G

W2k'p(il),Aku = f a.e.

and

||V2fcu||Lp(n) < C||/||Lp(n)

where C depends only on n and p.

Remark B.l Corollary B.2 also follows from the proof of Gilbarg-Trudinger [23, Theo¬

rem 9.9]. The arguments can be carried over line by line to the case of general k.

Furthermore, we have

Lemma B.3 For 1 (fi) for p even,

and

\\u\\ww(n) < C'IIVAûUlp^) for p odd.

Proof. We follow the scheme of Gilbarg-Trudinger [23, Lemma 9.17]. We consider the

case p even. If Lemma B.3 is not true, there exists {u/}/epj C WfJ"p(il)DW0 'P(H) satisfying

IKIIw^.p(n) = 1 and ||A2^||LP(r!) —> 0,

as I — oo. After passing to a subsequence, we may assume the existence of u G Wß'p(il) D

Wq'p(ü) s.t. ||u||w^.p(n) = 1 and

utû in Wß>p(il),

as I — oo. Since

j gVauidx^ f gVaudxJn Jn

p

for all |a| < p and g G Lp-1 (il), we must have

gA2 udx — 0J:Jn

for all g G Lp-1 (il). Thus, Aû — 0 and u — 0 by the uniqueness assertion, contradictingthe condition ||u||H^,P(n) — 1- For p odd, the proof is similar.

65

Decay Lemmas

Here we prove the following decay lemmas used in the proofs of partial regularity.

Lemma B.4 Let u G Wk'2(il) be a weakly k-harmonic function with \\u\\Wk,P^) < 1- F°r

x0 G U, 0 < p < r < dist(x0, OU) and 2 < p < oo we have

P~mlb I IVÎ^dx < cr~mE / |Vzu|^dx,/_! Jbp(x0) /„! Jbt(xq)

where c is independent of u and p.

Proof. Due to the Weyl Lemma we know that u is smooth. Moreover, we have the

following Cacciopoli type estimate. For all Bu C il and for all 7 G N, it holds

IMIw^b«) <c(-f,R)\\u\\Wk-i,2{Bli). (B.4)

To see this choose n G C(BR) satisfying 0 < 77 < 1, 77= 1 on Br. For ip := 77fe+1u, we

obtain with

/ VfcuVfc-0dx = O

Jbr

that

cu| dx

'br'

k

f nk+1\VkJbr

= -2V / Va(rik+1)VkuVk-audx

= 2E / (Va+1(r?fc+1)Vfc-Qu + VQ(77fc+1)Vfc+1-au)Vfc-1

\ f V277/:+1|Vfc-1u|2dx2 Jbr

< c(R) E / |Vau|2dx.-.*su 1

J Br

a=2J ß«

Vnk+1Vk-1uWkudxBR

k

udx

~2JBH

a<k~l

Thus we have shown estimate (B.4) for 7 = k. For 7 > k -f 1, we observe that Vu is

fc-harmonic. Repeating the preceding argument for Vu we deduce the case 7 — k + 1 and

by iteration we conclude (B.4) for all 7.

Now let p < |. Applying equation (B.4) to V'u (which is fc-harmonic) and Sobolev

embedding theorem, we infer with s sufficiently large

p~m f |V'u|^dx < c sup IV'ul^x)Jbp(x0) xBp(x0)

66 Linear Estimates

< c(r)||V'u||^8i2(ß5)*

21

< c(r)(E / |V°u|2dx)k-l

r

< c(r)E(/ |VQu|2dx)

Using the fact that ||u||Wfc,p(n) < 1 this can be estimated by

fc-1p k£

c(r)E(/ |Vau|2dx)2aa=l JBt

from which by Jensen's inequality we deduce that

fc-i

p~m f IV^dx^c^J^ f \Vau\%dx.Jbp(x0) a=l

JBr

To get the desired we apply a rescaling argument showing that c(r) — cr'm and sum over

all 1 < I < k.

Lemma B.5 Let u G Wk'2(il) be a weakly k-harmonic function. For x0 G il, 0 < p <

r < dist(xQ, Oil) and 2 < p < oo we have

p~m f |Vfeu|pdx < cr-m f |Vfcu|pdx,Jbp(xq) J Br(xo)

where c is independent of u and p.

Proof. We begin by observing that we can add any polynomial of order k — 1 to a k-

harmonic function and still obtain a fc-harmonic function. Therefore the estimate in (B.4)remains valid if one subtracts the average of u on Br from u, i.e. together with Poincaré's

inequality we get

fc—l

jg \Vu\Hx < C]L jju-ü^/dx+CY.x^ JjV'-ufdx

l—^ ]-£ { I J

*

%*^LruHx-

For Vu we define the polynomial pi(x) = YîL\ xiûBR(x0) wnere we noê that ^Pi ~

Vmbä(Io). Since the previous inequality still remains valid for u — pi we apply Poincaré's

inequality again which yields

/ |Vfcu|2dx < Cj^ f |Vu-VuBR(lo)|2dx + CEö2(L) / \Vau\2dxJBR j*' JBr a=2

J BR

* cEné^)f ivî2^-

67

Iterating this procedure for Va for 2 < a < fc - 1 in the same way we note that (B.4) can

actually be sharpened to only involve the term with Vfc_1u on the right-hand side. Thus

we deduce a better Cacciopoli-type inequality for Vfcu

llV7fc7/IM„ < HNyfc-S/lr,,Hv uWl1(br) -cR2»y u\\L2(BRp

and iterating again as in the previous lemma

||Vfcu||^,2(Bf) < c(rR)\\Vfcu||L2(ßR). (B.5)

To deduce the decay estimate we proceed similarly to the above proof. Letting p < ~ we

apply Sobolev embedding, inequality (B.5) and Jensen's inequality (noting that p > 2)to get

p-mf

\vku\pdx < c sup |Vfcu|p(x)Jbp(x0) x&Bp(x0)

< c(r)||Vfcu||^,2(ß5)< c(r)( f |Vfcu|2dxV

< c(r) f |Vfc«|p<te./ Br

The same scaling argument as before shows that c(r) = cr~m and the lemma is proved.D

Lemma B.6 For r > 0 we consider u G Wk'2(Br) and its k-harmonic extension on Br,i.e. v solves the Dirichlet problem

Akv = 0

u-veW^'2(Br).

We have

E[v^m^(b,)<Ê/ iv"ui2^/i=l //=1

-'Br

Proof. Observe that v satisfies the Cacciopoli type estimate (B.4). For all p < |, x G Bl

and s > 0 sufficiently large, Sobolev embedding theorem and the Cacciopoli type estimate

imply

p~m f |Vu|2dx < C sup |Vu|2JBp(x) XB„(x)

s

<

fc

< CE / |V\f dx,

CE / |VAu|2dx\=ojBiW

'E / ivA'üi\-nJßt(x)

68 Linear Estimates

It follows that

fc

p2ß~m j |v"v|2rfa: < C'Y" / \Vxv\2dx (B.6)

fc

A=0

Applying Poincaré's inequality yields

< Cj2(f |VAu|2dx+/ IV^v-u)!2^\_n \JBr JBT /

f \Vx(v-u)\2dx<C f |Vfc(u-u)|2dx. (B.7)J BT J BT

As v is fc-harmonic, we have

/ VkvVk(u - v)dx = 0,JBT

i.e.

f \Vkv\2dx < f |V*u|2dx. (B.8)»/ Br J Br

Combining (B.6) - (B.8) completes the proof. D

Divergence form

This section is devoted to the proof of the following lemma.

Lemma B.7 Consider a ball B C Rm, g G Lr(B), k>landj,l>0 with 1 < 21+j < fc.

There exists a unique weak solution u G WQ'2(B,RN) of

Aku = Vw Ajg

satisfying

||V2fc-(2û||Lr(B) < C\\g\\Lr{B)

for 1 < r < oo.

The proof is based on Gilbarg-Trudinger [23] and Giaquinta [20], [21]. In a first step we

prove existence of a unique solution.

Lemma B.8 Consider a ball B C Rm and g G L2(B). Then there exists a unique weak

solution u G w£'2(B,RN) of

Aku = Vw • A3g

for j, I > 0 with l<2l+j<k.

69

Proof. This follows from a direct application of the Lax-Milgram theorem. OnW0' (B,we define the functional F(v) - JBgV(l) A'vdx for v G Wk'2(B,RN). Then by Holder's

inequality this functional is bounded on W0 ' (B, RN). Also note that the bilinear form de¬

fined by C(u, v) = fB VkuVkvdx is again bounded by Holder's inequality but also coercive

since

7fc„.!l2 -^1

||„.||2C(U,U) = ||V U\\L2(B)>

ç\\u\\wk,2{B)

by Poincaré's inequality. Thus we apply the Lax-Milgram theorem [23, Theorem. 5.8] to

deduce existence of a unique weak solution. D

In a second step, using the method of difference quotient, we show

Lemma B.9 Consider a ball B C Rm and g G L2(B). If u G Wk,2(B,RN) is a weak

solution ofAku = V(,) • Ajg

for j, I > 0 with 1 < 21 + j < k, then

ueW2k~(-2j+l)'2(B,RN).

Proof. We define (Aû) (x) :— $u(x + hev) — u(x)) for 1 < u < m. Choose the test

function Aîp with ip G w£'2(B,RN). It follows

(-1)' / gVlAjA-hi^dx = / VkuVkA-hipdx = / VûA^V^dxJb Jb Jb

= / AhuVkuVkil)dx = / VkAû\7kipdx. (B.9)Jb Jb

Now we choose ip = r/A:+1Aû for arbitrary n G C(B) with 0 < n < 1. We compute

fc

/ VkAûVkipdx = / 77fc+1|VfcAû|2dx + ci E / Va(nk+1)VkAh„uVk-aAh„udxJb Jb

a=]Jb

k

= f nk+l\VkAû\2dx + c2 V / â(vk+1)^k~1ûVk+1~aAtudxJb

a=1Jb

+c3V f Va+l(rik+l)Vk-lAh„uVk-aAhuudx, (B.10)1

Jb

o=lJB

where —oo < Cj < oo for 1 (v%A-„h(Ahwu) + A;h(nk+1)Ahuu)\dxJb

< f \9Vk_+h]yA3(A-h(Ahvu))\dxJb

70 Linear Estimates

+E / l^llVX^HV^ÂÂSldxa=ijB

+E / M|VQA;V+1||V2j+'-QA£u|dx. (B.ll)t^Jb

Combining (B.9)-(B.ll) with Young's inequality for some e > 0 to be chosen later gives

(using also that 0 < r/(x) < 1 and that |Vr/fc+1|2 < C(\Wn$nk+l)

fc-i

f nk+l\VkAhvu\2dx < - /"îV^+'AÂû^dx + cE / |VaA£u|2dxJb 2 JB

a=o yspt77

2j'+/ „ „

+cE / \V2j+l~aA:hAh„u\2dx + c g2dxa=lJsptr,h,v Jb

+Ce f î)k+l\VkAhwu\2dx, (B.12)Jb

where we can therefore absorb the last term in the left-hand side provided e > 0 is chosen

sufficiently small. For h > 0 sufficiently small, we infer that the second and the third

term on the right-hand side are uniformly bounded. For example, applying [31, Lemma

8.2.1] to Vau, we estimate

/ |V"A£u|2dx < / |A£Vau|2dx < / |Va+1u|2dx < CJsptrj Jsptt] Jb

for 0< a < fc-1.

If 2j +1 + 1 < fc, we similarly show that the first term on the right-hand side is uniformlybounded. Otherwise 2j +1 + 1 = fc, and we again apply [31, Lemma 8.2.1] twice (notingthat the presence of a cut-off function does not affect the proof) to obtain

1

J^\V2Â-hAtu\2dx <\j^\AhuVku?dx.Using the continuity of 77 and choosing h > 0, e > 0 sufficiently small we can absorb the

terms involving t^ÎA^VûI2 in the left-hand side of (B.12). Combining the above steps

shows that from (B.12) we can deduce

f |VfcA£|2dx < C.J{xB:rj(x)=l}

Now we observe that for all il CC B there exists r/ G C(B) such that 0 < 77 < 1 and

77= 1 on il. We conclude

/Jn

|VfeA£|2dx < Cn

for all il CC B. Hence, from [31, Lemma 8.2.2.] we infer that u G Wk+l>2(B). The

existence of higher derivatives now follows by induction. This completes the proof. D

71

Thus, we look at solutions u G W2k~{2j+l)'2(B,RN) f) Wk'2(B,RN) satisfying

(_1)* f V2fc-(2j+07iV2j+Vdx - / gWlAjijjdx (B.13)Jb Jb

for every test function ip G W^j+l(B,RN).Set T(#) :— V2fc~^2:'+û. Lemma B.IO below states that T is a continuous linear operator

T : L2(B) — L2(£),

respectivelyT : L°°(j5) — BMO(B).

Thus, Stampacchia's interpolation theorem (see [21, Theorem 4.6]) then implies that

T : Lq(B) —> Lq(B) is also continuous for 2 < q < oo.

For 1 < ç < 2 we deduce continuity from a duality argument as follows. Let / G LQ,

g G LP with i + -7 — 1, where by density we can assume that /, g G C(B). Thus

noting that then the fundamental solution of Afcu = V^ • A^g is given by ug(x) —

C(~l)1 fB VlyAirk(x - y)g(y)dy we compute

I (Tf)(x)g(x)dx = f V2xk-(2j+l)uf(x)g(x)dxJb Jb

= (_l)2fc-^+0 / u}(x)Vf-{2^g(x)dxJb

= (_1)2/=-(2J+/)c f f ViytiyTk(x-y)f(y)dyV2k-(2j+l)9(x)dxJb Jb

= (_1)2fc-(2,+oc f f vlyAiTk(x - y)f(y)V2k-^g(x)dxdyJb Jb

= C f f(y) f Vf-{2mVlyA{Tk(x - y)g(x)dxdyJb Jb

= C f f(y) f V2yk-M+lWlxAirk(x - y)g(x)dxdyJb Jb

= C f /(y)Vf-(W / V< A^x - y)g(x)dxdyJb Jb

= C f f(y)Vf-{2j+l)ug(y)dyJb

= C Jj(Tg)< C\\f\\Lq\\Tg\\Lq,

Since q' > 2 we can apply the above continuity result to g to obtain

f(Tf)g<C\\f\\Lq\\g\\Lq,Jb

After taking the supremum over all \\g\\Lq> — 1 by the dual characterisation of the norm

this enables us to conclude that

||T/||W < C\\f\\Lq

72 Linear Estimates

for q > 1. The proof of Lemma B.7 is thus complete.

It remains to show

Lemma B.IO Consider a ball B C Rm, g G L2(B), k>landj,l>0 with 1 < 21+j < fc.

Lfu G Wq'2(B,Rn) is a weak solution of

Aku = V(/) Ajg,

then g G C?{X(B) implies that sjU-W+Du e c}x(B) for 0 < A < m + 2, where C2'X(B) :=

\u L2oc(B) M^(B) := suPbp(x0)cb P"A /Bp(a.o) I« - üBp(x0)?dx < ooj ts the Campanato

space.

Proof. The proof is similar to Giaquinta [21, Theorem 3.3]. Set a — 2fc — (2j + I).Consider Br(xo) C B and let v be the a-harmonic extension of u on Bt(xq), i.e. w :—

u — v G W^'2(ßr(a:o)) and Aav = 0 on Br(xo). From Poincaré's inequality and the

application of Lemma B.5 we can estimate

f \Vav-V"vBp{xo)\2dx < Cp2 f \Va+lv\JBo(x0)

Jbo(x0)

I

'Bç(xo)

2dx(xo) Jbp(x0)

Cp2 (?)m f |Vö+17j|2dx

C (P-\m+2 f \vav - VavBAxo)\2dx. (B.14)Vf/

JBr(xn)

<^r'

JBt(x0)

for p < \. On the other hand, w satisfies (5.13) and ip - y^-2(23+i)w £ W*i+W [s an

admissible test function. Thus we estimate

/ |V%fdx = (-1)*/ (g-gBr(X0))VlAiwdxJBr(x0) JBt(x0)

< If \9-9Br(xo)\2dx + l f |Vaufdx,Z JbAxo) L JbAx0)

i.e.

/ |V"u>|2dx < C f \g- gBr(xo)\2dx < C[g]2c2,HB)rx. (B.15)JBT(xo)

JBr(x0)

r(xo) JBr(x0)

Combining (B.14) and (B.15) yields

/ |V"u-VauBp(:rû)|2dxJbp(x0)

< f \Vav - Wv~Bp{x!))\2dx + / \Vaw - Wv^Bp(xo)\2dxJbp(x0) Jbp(xq)

< c ß)m+2 f \Vau ~ WTiBr{xo)\2dx + C f \Vaw\2dxVr/ Jßr(x0) JBr(xo)

* ° (rT+2 I Û ~ V^Brtotfdx + C[g?C2,HB)rx.Vr/ Jb,(xo)

The conclusion now follows as in the proof of Giaquinta [20, Theorem III.2.2]. We applyLemma III.2.1, Theorem III. 1.2 and Theorem III. 1.3 in [20] to conclude. D

Appendix C

Miscellaneous

Young's inequality

Lemma Cl Let a,b,p,q be positive real numbers with - + - = 1. Then, we have

,ap bq

ab< h—.p q

Proof. Considering the convexity of the exponential function, we have

ab = e>g(a*)+>g(6*) < IeiogK) + Ielog(W) = ^ +hl

p q p q

An inequality

Lemma C.2 Consider a, b,c,d G R. Then, it holds

16abcd<(a2+b2 + c2 + d2)2.

Proof. Applying 2xy < x2 + y2 gives

4abcd < 2a2b2 + 2c2d2, 4abcd < 2a2c2 + 2ft2d2,4abcd < 2a2d2 + 262c2, 4a6cd < a4 + b4 + c4 + d4.

Adding these inequalities establishes the desired inequality. D

Inverse of the stereographic projection

Here, we give some explicit computations concerning the inverse of the stereographic

projection k :— <r_1. k, is explicitly given (for example) as follows. For 1 < i < 4, we

define

k: R4 -h- Sa

74 Miscellaneous

We compute for 1 < i,j < 4

^ -

^7(l+kl2)-4x'^ 5_

4x3

K3~

(1 + M2)2 ' ^3~

(1 + kl2)2'

^K _

(1 +M2)2+

(1+M2)3'^ ~~

(l+k|2)3'A2,,j

„

768s' A2„5_

384(1-|a|2)^ K -

ü+iW^ K ~~

(i+M2)6

We deduce that

2384

A"~(l + |x|2)4AC'i.e. k is biharmonic.

Moreover, we compute

|A/t|2 =2_

64(4+ |x|2)

(l + |x|2)4'

It follows with H3(S3) = 2tt and HA(S4) = |tt2, that

E(k) = / \An\2dx = 64H3(S3) / —L--r3dr

= 647T2/ 1 l

(1 + r2)3 (1 + r2)2 l + r/

= 64tt2 = 247t4(54).

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Acknowledgements

I want to express my gratitude to my supervisor Michael Struwe for the constant support

during the last years. I have benefited much from the many valuable discussions and

careful explanations. I appreciate his vast knowledge and skills, and his advice that en¬

abled me to accomplish my thesis. Moreover, I want to thank the co-examiners Norbert

Hungerbühler and Tristan Rivière for very stimulating discussions.

I thank Ruben for his careful proofreading of the entire thesis, and David, Ivo, Joachim,

Reto and Tobias for proofreading several parts. I thank the participants of the Séminaire

hexagonale d'analyse and the Oberseminar Analysis at the University of Fribourg for help¬ful remarks and suggestions.

Let me also mention the Swiss National Science Foundation that partially supported the

research carried out in this thesis, as well as the National Research Fund Luxembourg,that funded the printing of the present thesis.

Furthermore, I owe many thanks to my actual and former office colleagues Ivo, Joachim,

Lydia, Michael, Stefan and Thierry, as well as the assistants of the analysis group for

the pleasant and serene working atmosphere. I thank the AS Campanato team-mates

for thrilling soccer matches and amusing after-soccer-activities, Läszlo for the enjoyable

evenings at Jozsi's, Johannes for the magnificent flight along the Swiss Alps, and Liver¬

pool FC for winning the Champions League in 2005, and almost in 2007.

Moreover, I thank all my friends outside ETH giving me additional energy, vitality and

motivation provided through the enjoyable time we spent together. In particular, I appre¬

ciated the many visitors from Luxembourg, and my visits in Aix- en- Provence (François),Copenhague (Marc), Edinburgh (Ben) and Berlin (François).

Last but not least, I thank my brother Steve, my mother Anne and my whole familyfor the constant support, and Lara for showing me that life outside mathematics exists.

Thank you Sheila for all your love and support!

I thank my father for teaching me what really matters in life!

82 Acknowledgements

Curriculum Vitae

Personal data

Name

Date of birth

Place of birth

CitizenshipCivil Status

Gilles AngelsbergJune 3, 1980

Luxembourg-Ville, LuxembourgGrand Duchy of Luxembourgunmarried

Education

since October 2003

Oct 1999 - Jul 2003

Sep 1992 - Jul 1999

Sep 1986 - Jul 1992

Doctoral Studies in Mathematics under the

supervision of Prof. M. Struwe,ETH Zürich, Switzerland

Diploma Studies in Mathematics

Université de Fribourg, Switzerland

Lycée Classique de Diekirch, Diekirch, Luxembourg

Ecole primaire, Erpeldange, Luxembourg

Professional Experience

since October 2003

Oct 2001 - Sep 2003

Teaching assistant, Department of Mathematics,ETH Zürich, Switzerland

Tutor and Underassistant,

Department of Mathematics,Université de Fribourg, Switzerland

Documents

In Copyright - Non-Commercial Use Permitted Rights ...29852/eth-29852-02.pdfContents Abstract vii Zusammenfassung ix Résumé xi 1Weaklybiharmonic maps 1.1 Definitions and basic properties