Numerical optimization of Dirichlet-Laplace eigenvalues on domains in surfaces

Diri hlet-Lapla e Eigenvalues on

Domains in Surfa es

Thèse

présentée à laFa ulté des s ien es

pour obtenirle gradede do teur èss ien es par

Régis Straubhaar

soutenue ave su ès le 31mai2013

et a eptée sur proposition du jury

Prof. OlivierBesson o-dire teurde thèse, rapporteur

Prof. BrunoColbois o-dire teurde thèse, rapporteur

Prof. Pedro Freitas rapporteur (Université de Lisbonne)

Prof. Alexandre Girouard rapporteur (Université Laval)

•

U11

1e

UNIVERSITÉ DE

NEUCHÂTEL

Faculté des sciences

Secrétariat-décanat de Faculté

Rue Emile-Argand 11

2000 Neuchâtel - Suisse

Tél:+ 41 (0)32 718 2100

E-mail: [email protected]

IMPRIMATUR POUR THESE DE DOCTORAT

La Faculté des sciences de l'Université de Neuchâtel

autorise l'impression de la présente thèse soutenue par

Monsieur Régis STRAUBHAAR

Titre:

"Numerical Optimization of Dirichlet-Laplace

Eigenvalues on Demains in Surfaces"

sur le rapport des membres du jury:

•

Prof. Bruno Colbois, Université de Neuchâtel, co-directeur de thèse

•

Prof. Olivier Besson, Université de Neuchâtel, co-directeur de thèse

•

Prof. Pedro Freitas, Université de Lisbonne, Portugal

•

Prof. Alexandre Girouard, Université Laval, Québec

Neuchâtel, le 20 juin 2013

Le Doyen, Prof. P. Kropf

The spe trum of the Diri hlet-Lapla e operator dened on a bounded domain in

a smooth and omplete surfa e onsists in a stri tly positive sequen e, in reasing to

innity. Theaimof thisthesisistoapproximate numeri ally thersteigenvalues ofthis

operatorusinganiteelementbasedmethod,thentoaddressthefollowing optimization

problem: what is the domain whi h minimizes the

k

-th eigenvalue among all domains of a given area, and what is this eigenvalue equal to? This latter has its roots in the

Faber-KrahnandKrahn-Szeg®theorems,whi hanswerthequestionfortherstandthe

se ond eigenvalue ofa domain inthe Eu lidean spa e. For higher eigenvalues and other

underlying surfa es like the sphere and hyperboli spa e, shape optimization has been

performed to provide domains whi h are andidates to be solutions. This gives rise to

some observations about the omparison of eigenvalues of domains in various surfa es.

Theproblemoflo atinga ir ularobsta leinsideaballtomaximizethersteigenvalues

isalso addressedinthis do ument.

Keywords: Spe tral geometry; Diri hlet-Lapla e operator; Eigenvalues; Numeri al

Le spe tre de l'opérateur de Lapla e-Diri hlet déni sur un domaine borné d'une

surfa elisseet omplèteestunesuitestri tementpositive, roissante,tendantversl'inni.

Le but de ette thèse est d'appro her les premières valeurs propres de et opérateur

de manière numérique à l'aide d'une méthode d'éléments nis, puis de onsidérer le

problèmed'optimisationsuivant:quelestledomainequiminimisela

k

-èmevaleurpropre parmi tous les domaines d'aire donnée, et que vaut ette valeur propre? Ce dernier

trouve son origine dans les théorèmes de Faber-Krahn et Krahn-Szeg®, qui règlent le

as de lapremière et de la deuxième valeur propre d'un domaine de l'espa e eu lidien.

Des méthodes en optimisation de forme ont été élaborées pour proposer des domaines

andidats à être solution pour des valeurs propres plus élevées ainsi que pour d'autres

surfa es sous-ja entes omme la sphère et l'espa e hyperbolique. Cela a donné lieu à

des observations sur la omparaison de valeurs propres asso iées à des domaines sur

diérentessurfa es.Leproblèmedupla ement d'unobsta le ir ulaireàl'intérieurd'une

boulean demaximiser lespremières valeurspropres est aussiabordédans ette thèse.

Mots lés : Géométrie spe trale; Opérateur de Diri hlet-Lapla e; Valeurs propres;

Approximations numériques;Optimisationdeforme;Méthodedesélémentsnis;

Cetravailaétéenpartienan éparlasubventionn o

20-137696/1duFondsNational

Suissede lare her he s ientique (FNS).

Enpremierlieu,mesremer iementslesplus haleureuxvontàmesdire teursdethèse,

OlivierBessonetBrunoColbois.Ilnem'estpaspossibled'énumérervosnombreuses

qua-litésdontj'aibéné iépendant esquatreans,aussin'ensoulignerai-jequ'une,donttout

do torant n'a pas la han e de pouvoir proter. Je n'aurais sans doute pas eu

l'oppor-tunité de ren ontrer autant de her heurs sans tes nombreuses relations, Bruno. Cela

m'apermisd'élargirmes onnaissan esetderemettre avantageusementen questionmon

travail.Et lorsqueledoute s'insinuait tropvivement, laporte dubureaud'enfa e, elui

d'Olivier, était toujours ouverte. J'ai énormément appré ié ta apa ité à transmettre,

en plus de tes onnaissan es indéniables, ta motivation etta onan e. Cha un à votre

manière, vous avez guidé mes premiers pas dans la re her he. Mer i pour tout à vous

deux.

Mes remer iements vont aussiaux ProfesseursPedroFreitas et Alexandre Girouard

pourm'avoirfaitleprivilègedeprendrepartàmonjuryetpouravoir onsa rédutempsà

liremathèse.Vosquestionspertinentes,vossuggestionsderéféren esbibliographiqueset

vos ommentairesinstru tifsm'ont permisd'améliorer edo ument. Desur roît,malgré

le ara tèresolennel delasoutenan e,vousavezsu yfairerégnerun limatdé ontra té.

Qu'ilestagréabled'imagineretd'é rire esquelqueslignesdontjemesuislongtemps

refusé àesquisserlesmots.Il esttemps deremer iermes ollègues,quiresterontpourla

plupart,j'ensuissûr,mesamis.Etmêmesansné essairementavoirélu idélesproblèmes

que j'airen ontrés dansmon travail, leurprésen em'a étébénéque.

Celuiquim'aleplusapporté,notammentsurleplanmathématique,est ertainement

Alex.Tuastoujoursprisletempsdem'é outeretderépondrepré isémentauxquestions

plus ou moins pertinentes que je me posais. Mer i de m'avoir fait proter de ta vaste

expérien e. Par ailleurs, l'anglaisemployé dans ette thèse,s'il n'est deloin pasparfait,

auraitétédebienplusmauvaisequalitésansl'aidedemes o-bureaux,AnaetPN,quise

sontprisaujeudelatradu tion.Disonsquevousaveztouslesdeuxgagnéla ompétition.

Je penseaussiàFabienqui m'afourni son odeau débutde mathèse.

Mer i David d'avoir partagé plus d'une fois un souper, si propi e à es pré ieuses

dis ussions qui nous ont animés. Ces quatre ans de thèse m'auront également permis

Olivier, Kola (et ta petite famille), Greg, Dennis (tu reviens quand tu veux), vous

quim'avez pré édé à l'Institut, 'était un plaisir de vous toyer etde partager nos

im-pressionssur epériplequereprésenteunethèse.Quel bonheurdepouvoirvous ompter

parmi mesamis.

Je ne t'oublie pas Bastien,toi qui a veillé à ne pasme faire manquer es désormais

mythiques pauses au Saloon, ni tous leurs protagonistes qui se re onnaîtront. Danke

Raphaelfür deine Freundli hkeit unddeinen Humor.

Je tiens également à adresser un mot à Christine, la se rétaire, toujours de bonne

humeur etprête à rendreservi e, ainsiqu'aux étudiants qui,même s'ils ne s'en rendent

pas ompte,m'ont donnéàmaintesreprises unballond'oxygènefortappré iable.Enn,

je pense à tous eux ave qui j'ai bu un verre,pris un repas, regardé un mat h de foot,

bref,simplement passéun bon moment.

Finalement, mes remer iements vont à mes parents et à mon frère. Qu'il est bon

et rassurant de pouvoir ompter sur votre soutien et votre é oute indéfe tibles. Votre

générosité touten simpli itéestinestimable.

Mumu,tuasvumonmoralos illerdurant ettethèsefaitedehautsetdebas.Maistu

astoujourssu mefaire es aladerlesommet suivant.Ave toi, je meréjouisdedé ouvrir

Abstra t iii

Résumé v

Remer iements vii

1 Introdu tion 1

2 Fundamental tools 11

2.1 Geometry and al ulus onmanifolds . . . 11

2.2 Somenotions about the Finite Element Method . . . 23

2.3 The Lan zosmethod . . . 33

3 Computation of eigenvalues of the Diri hlet-Lapla ian 41

3.1 Theoreti al statement of theproblem . . . 42

3.2 Numeri alpro essing oftheproblem . . . 46

3.3 Estimation ofthe error

|λ

h,k

− λ

k

|

. . . 48 3.4 Estimation oftheerror

ku

h,k

− u

k

k

H

1

0

(Ω)

. . . 53

3.5 Numeri alexperimentson surfa es . . . 57

4 Preliminariestooptimizationofeigenvalueswithrespe ttothedomain 69

4.1 Detailsofthe shapeoptimization step . . . 70

4.2 TheUzawaalgorithm. . . 83

4.3 Te hni al aspe ts about the displa ement . . . 87

5 Optimization of eigenvalues of the Diri hlet-Lapla ian with respe t to

the domain 91

5.1 Theoreti al statement of the optimization problem . . . 92

5.2 Numeri al omputations . . . 94

A Some notions on fun tional analysis, distributions theory and Sobolev

spa es 103

A.1 General notions andresults about Hilbert spa es . . . 103

B A omplete example: the optimization of

λ

7

in the Poin aré dis 111 B.1 Starting fromvarious initial domains . . . 111

B.2 Taking themultipli ity into onsideration . . . 118

C Some additional numeri al values 121

C.1 Computation of the rst fortyeigenvalues of a ball of volume 1 in

R

2

, in

the sphere

S

2

and inthe Poin aré dis

D

2

. . . 121

C.2 Pla ement ofa ir ular obsta leinsidea ball. . . 123

C.3 First fteeneigenfun tions ona ball in

R

2

Introdu tion

Thisthesisismainly on ernedwithanoptimizationproblemfromtheeldofspe tral

geometry. The notions involved in its denition are addressed within this framework.

However, the approa h hosen to deal with this problem omes mostly from numeri al

analysis. This ontext madeoftwo dierent areasof mathemati sispresent throughout

this do ument. In order to be understandable for people whoare lessfamiliarwith one

of them, some relatively elementary notions from both are re alled. As an illustration,

spe ial areistakentodevelopexpli itly geometri notionsaswellastooutlinethepart

of the FiniteElement Method required for thiswork.

To get qui kly to the heart of the matter in this introdu tion, some notions are

postponed to the next se tions where they are properly dened. However when this

happens, the orresponding laim is arefully indi ated. After settingthe framework of

thetopi withafewmotivations,thisintrodu tiondealswiththeissuesaddressedinthis

thesis,throughtheoreti alstatements,state-of-the-artresultsandpersonal ontributions.

Context and motivations

Let

(M, g)

be a smooth, omplete Riemannian manifold 1

and let

Ω

M

⊂ M

be a domain,namelyaboundedopensetin

M

. Moreoverassumethat

g

issmooth. Although this introdu tion takespla e inanydimension, onlytwo-dimensional manifoldsare

on-sidered intherest ofthis thesis. Let

∆

g

denote theLapla e operator 2 . The underlying problemis this:

(

P)







Finda map

u := u

Ω

M

: Ω

M

→ R, u 6≡ 0,

anda s alar

λ := λ

Ω

M

su h that

−∆

g

u = λu

in

Ω

M

,

u = 0

on

∂Ω

M

.

1. ThefundamentaldenitionofaRiemannianmanifoldisnotrepeatedinthisdo ument.See[dC76,

Denition5-10.5a℄foradenition.

Thespe traltheorem 3

ensures thatthereexist astri tly positivesequen e

0 < λ

1,Ω

M

≤ λ

2,Ω

M

≤ ... ր +∞,

tending to

+

∞

and a sequen e of fun tions

(u

n,Ω

M

)

n∈N\{0}

, forminga Hilbert basis of

L

2

(Ω

M

)

,su hthatforall

n

∈ N\{0}

,

(λ

n,Ω

M

, u

n,Ω

M

)

isasolutionof

(

P)

. Of ourse,these eigenvalues

λ

n,Ω

M

and eigenfun tions

u

n,Ω

highly depend 4

on the underlying domain

Ω

M

. Inthis ase, thesetofalltheeigenvaluesformsthespe trumoftheLapla e operator on

Ω

M

itisalso alledthespe trumof

Ω

M

for onvenien e. Inthisthesis,anumeri al study of the spe trum is proposed, from the approximation of eigenvalues of ertain

domains in various manifolds to the optimization of eigenvalues with respe t to the

domain.

Twodomains whi hhave thesamespe trum are alledisospe tral. Oneofthe

inter-estingproperties of the spe trumis its invarian e under isometries 5

,meaning that two

isometri domains are isospe tral. The onverse statement does not hold as proved by

J.Milnor whoexhibited apair of 16-dimensional isospe tralat tori whi h arenot

iso-metri [Mil64℄. Withregard to the parti ular aseof two-dimensional domains, M. Ka

asked his famous question Can one hear the shape of a drum? 6

[Ka 66℄. A negative

answer in the form of two isospe tral planar domains was later given by C. Gordon,

D.Webb and S.Wolpert in[GWW92℄ and followed thereafter by familiesof isospe tral

planar domains[BCDS94℄. See Figure1.1. Bothrely onqualitative argumentsavoiding

expli it omputations ofthespe trumof thedomainsinvolved. A tually, thebehaviour

ofthespe trumofa domainsubje ttosmallperturbationshasbeen intensively studied,

resulting innumeroustheorems. The lassi al referen es [BGM71℄, [Bér86℄ and [Cha84℄

present a qualitative study in spe tralgeometry. It shows that thespe trumis auseful

toolfor omparing several domains, whi h is observed using numeri al experiments for

two and three-dimensional domains[Reu06℄. Thisobservation gave rise to the

develop-ment ofappli ations for shapere ognition.

As M. Ka already knew[Ka 66℄, geometri and topologi al properties of a smooth

andbounded planardomain an be derived fromits spe trum. Moregenerallyfor a

do-main

Ω

M

inamanifoldofdimension

N

∈ N\{0}

withboundary

∂Ω

M

regularenough,the asymptoti behaviour of large eigenvalues give information about some of thedomain's

features. An illustrationof thisis the famous Weyl asymptoti formula 7 ,

λ

N/2

_k,Ω

M

∼

(2π)

N

ω

N

k

vol

g

(Ω

M

)

,

as

k

→ ∞,

3. SeeTheorem3.1.2.

4. Thisdependen eisalsoindi atedusingthenotation

λ

n

(Ω

M

)

.

5. Thedenitionofanisometryimpliesdire tlythatthereexisttwo hartswithinthe oe ientsof

themetri oftwoisometri Riemannianmanifoldsareequal. Hen e,theexpressions(3.1)oftheLapla e

operatorinbothsurfa esarethesame.

6. Asstatedin[CH53,Se tionV.5℄,everyeigenfrequen y

f

k

ofadrum orrespondsto

pλ

k,Ω

,where

λ

k,Ω

isthe

k

-theigenvalueasso iatedtothedomain

Ω

representingthedrumhead. 7. See[Cha84,Se tionVII.3℄where

∂Ω

M

issupposedtobepie ewisesmooth.

Figure1.1: Theisospe tral domainsgivenin[GWW92℄.

where

ω

N

denotes the volume of the unit ball of

R

N

and

vol

g

the volume measured using the metri

g

. Another general formula involving the spe trum and properties of theunderlying domain isgiven in[MS67℄,namely

(4πt)

N

2

X

k≥1

e

−λ

k,ΩM

t

_{= vol}

_g

_(Ω

_M

_{) +}

(4πt)

1

2

4

vol(∂Ω

M

) +

t

3

Z

Ω

M

K

₋

t

6

Z

∂Ω

M

J + o(t

3

2

),

where

J

is themean urvature oftheboundary

∂Ω

M

and

K

isthes alar urvature. Individual eigenvalues an also deliver information. As an illustration, onsider the

hara terization of the eigenvalues given by the Rayleigh quotient and the Min-max

theorem 8 ,namely

λ

k,Ω

M

= min

E

k

∈V

k

max

v∈E

k

\{0}

Z

Ω

M

|∇v|

2

dV

g

Z

Ω

M

v

2

dV

g

,

where

V

k

denotes the set of all subspa es

E

k

of

H

1

0

(Ω

M

)

of dimension

k

. In parti ular for

k = 1

,

λ

1,Ω

M

=

min

u∈H

1

0

(Ω

M

)\{0}

Z

Ω

M

|∇u|

2

dV

g

Z

Ω

M

v

2

dV

g

.

Together withsymmetri de reasingrearrangementsoffun tions 9

,itleads tothe

Faber-Krahn inequality, onje tured byLord Rayleigh [Ray45℄.

Theorem1.1.1(Faber-Krahninequality,[Fab23℄,[Kra25℄). Let

Ω

R

N

⊂ R

N

beabounded

open set of volume

V > 0

and

Ω

∗

1,R

N

⊂ R

N

be the openball of samevolume. Then,

λ

1

(Ω

∗

1,R

N

)

≤ λ

1

(Ω

R

N

),

withequality if and onlyif

Ω

R

N

= Ω

∗

1,R

N

.

Thisresultalso holdsinthe sphere andinhyperboli spa e asmentioned in[Cha84,

Se tion IV.2℄. Note it will be re overed numeri ally for these three surfa es in

Se -tion 5.2. About arbitrary Riemannian manifolds, a theorem from [PS09℄ asserts that

in the neighbourhood of ea h non-degenerated riti al point

p

of the s alar urvature, thereexist small extremaldomainsfor thersteigenvaluewhi h are lose to ageodesi

ball entredat

p

. Byextremaldomains, theauthorsmeanthatthederivativeoftherst eigenvalueseenasarealvaluedfun tionofavolumepreservingdeformationvanishes.

Withregardtothese ondeigenvalue, theanalogousresulttotheFaber-Krahninequality

istheKrahn-Szeg® inequality.

Theorem1.1.2(Krahn-Szeg®inequality,[Kra26℄ 10

). Let

Ω

R

N

⊂ R

N

be aboundedopen

set ofvolume

V > 0

and

Ω

∗

2,R

N

⊂ R

N

be the domain onsistingof two disjointopenballs of volume

V /2

. Then,

λ

2

(Ω

∗

_2,R

N

)

≤ λ

2

(Ω

R

N

),

withequality if and onlyif

Ω

R

N

= Ω

∗

2,R

N

.

Itsprooffollows dire tlyfromtheFaber-KrahninequalitytogetherwiththeCourant

nodaltheorem 11

andares alingargument. Thelattermakesuseoftheinvarian eunder

homothetyof thefun tional 12

Ω

_{7−→ λ}

_k,R

N

(Ω) vol(Ω)

2/N

,

k

∈ N \ {0},

(1.1)

dened onregularbounded domainsin

R

N

.

For a volume

V = 1

,the minimalvaluerea hed bytheball

Ω

∗

1,R

N

⊂ R

N

is given by

λ

1

(Ω

∗

_1,R

N

) = ω

2/N

N

j

N/2−1,1

2

,

where

j

N/2−1,1

denotes therstzeroof the

N/2

− 1

Besselfun tion

J

N/2−1

,whereas

λ

2

(Ω

∗

_2,R

N

) = 2

2/N

λ

1

(Ω

∗

_1,R

N

),

see [Cha84, Se tion IV.2, Remark 4℄. The denition of the Bessel fun tions and their

detailedstudymake expli itthe eigenvaluesfor theballof

R

N

. However,ex eptforvery

10 . G. Pólya attributed the result to G. Szeg® in his paper [Pól55℄, but this inequality was also

pusblishedindependentlybyI.Hong[Hon54℄oneyearearlier.

11 . A nodal domain of a fun tion

u

dened ona domain

Ω

M

is a onne ted omponent of the set

Ω

M

\ {x ∈ Ω

M

| u(x) = 0}

. Thistheoremassertsthatthenumberofnodaldomainsofaneigenfun tion

spe i domainssu hasare tangle intheplane, getting expli itvalues isunfeasible for

generaldomains. Thisisarstreasontodealwiththespe trumoftheDiri hlet-Lapla e

operator usinga numeri al approa h.

Another argument to onsider omputational approximations is related to the

op-timization problem generalizing the Faber-Krahn and Krahn-Szeg® inequalities to any

eigenvalue

λ

k

,namely

(

P

opt

)



















Findan open set

Ω

∗

k,M

⊂ M

ofvolume

V > 0

,su hthat

λ

k

(Ω

∗

k,M

)

≤ λ

k

(Ω

M

)

,

forall opensets

Ω

M

⊂ M

of volume

V

.

Indeed,for anyinteger

k

≥ 3

,noanalogous results tothe Faber-Krahnor to the Krahn-Szeg®inequalitiesexist. Nevertheless,legitima yoftheproblem

(

P

opt

)

hasbeenenhan ed byare ent resultbyD.Bu ur[Bu 12℄for quasi-opensetsinsteadofopensetsin

R

N

,

N

_{≥ 1}

,also rea hed independently byD.Mazzoleniand A.Pratelli [MP13℄. Thisresult laimsthatasolutionexistsinsu ha lassofdomainsforany

k

. Furthermore,itensures theoptimizer tobebounded and to have nite perimeter.

Several numeri al experiments have been performed to nd a andidate to be the

optimizer in

(

P

opt

)

. E. Oudet is a pre ursor in this eld with his work [Oud04℄. It is restri ted to the domains in

R

2

minimizing the rst ten eigenvalues. It suggestsas

expe ted by the mathemati al ommunitythat the andidate asso iated to the third

and fourtheigenvalues are adis and twodis of dierent arearespe tively. Thereafter,

his results were improved by P. R. S. Antunes and P. Freitas in their paper [AF12℄,

wheretheyfoundadierentshapeforthe andidateasso iatedtotheseventheigenvalue.

They also extendedthe results to therst fteen eigenvalues, see Figure 1.2, aswell as

to Neumann and Robin boundary onditions with J.B. Kennedy in [AFK13℄. With

regard to Neumann-Lapla e eigenvalues whi h form a positive sequen e

0 = µ

0

(Ω

M

) <

µ

1

(Ω

M

)

≤ µ

2

(Ω

M

)

≤ ... ր +∞

,therelevantoptimization problemistomaximizethe

k

-theigenvalue

µ

k

amongall domainsof agiven volume. The ounterpart toFaber-Krahn inequality is the Szeg®-Weinberger inequality [Wei56℄. It laims, for domains ofvolume

1,that

µ

1

(Ω)

≤ µ

1

(Ω

∗∗

_1,R

N

) = ω

2/N

N

p

2

N/2,1

,

where

Ω

∗∗

1,R

N

isaballofvolume1in

R

N

and

p

N/2,1

denotestherstzeroofthederivative of the fun tion

t

7→ t

1−N/2

_J

N/2

. M. S. Ashbaugh and R. D. Benguria extended this inequalityto domains ontainedina hemisphereof

S

N

,aswell astosmooth domainsin

D

N

[AB01℄.

Moreover,a theorembyA.Girouard,N.Nadirashvili,andI. Polterovi h[GNP09℄ is

theanalogous resultto theKrahn-Szeg®inequalityindimension2. Itasserts,for simply

onne ted planar domainsofvolume 1,that

µ

2

(Ω) < 2µ

1

(Ω

∗∗

_1,R

N

) vol(Ω

∗∗

_1,R

N

),

Figure 1.2: Planar domains minimizing the fth (left) up to thefteenth (right)

eigen-value amongdomains ofa given volume from [AF12℄.

Issues addressed in this thesis and ensuing results

The two main topi s of this thesis are the numeri al approximation of eigenvalues

of the Lapla e operator on domains in surfa es, and their numeri al optimization with

respe t to the domain. Thenumeri al method usedto perform approximationof

eigen-values isbased onthe FiniteElement Method. To takeinto onsiderationthe urvature

ofa general manifold, the omputationstake pla e intheopen setof a hart. Contrary

to lassi al surfa e dis retization whi h interpolates a surfa e in

R

3

see [JU08℄, the

advantage is that surfa es whi h are non-embeddable in

R

3

, su h as hyperboli spa e,

an be onsidered. The metri ontribution is then introdu ed into the omponents of

the matri esinvolved inthe approximated problem. Indeed, the eigenvalues appearing

ina nite linear systemareused to approximate thedesiredeigenvalues of theLapla e

operator. For this purpose, a Lan zos method is performed. The surfa es hosen for

numeri al approximationsofeigenvaluesare

R

2

,thesphere

S

2

,hyperboli spa ethrough

thePoin aré dis model

D

2

anda family ofsurfa eswithnon- onstant urvature.

The optimization problem

(

P

opt

)

is addressed numeri ally by minimizing the rst fteeneigenvalues for adomain in

R

2

to re overtheresults in[AF12℄. Then, the

analo-gousoptimizationprobleminthesphere

S

2

andinthePoin arédis model

D

2

are arried

out numeri ally. It leads to Table 1.1 13

repeated from themain part of this do ument.

Notethevaluereportedforthethirteenth eigenvalueisslightlylarger inthespherethan

hyperboli spa e. Moreover, the optimizers

Ω

∗

k,D

2

,

k = 1, . . . , 15

, in the Poin aré dis aredisplayed inFigure1.3.

The shape optimization pro ess to deal with

(

P

opt

)

relies ona des ent method algo-rithm, whi h takes advantage of the Hadamard Variational Formula (4.6) given

subse-quently. Thevolume onstraint 14

appearingin

(

P

opt

)

ishandledwithaUzawaAlgorithm, whi h is a new approa h. The optimal domain is obtained by ndinga saddle point of

a Lagrangian

L

of theform

L(Ω, µ) = λ

k

(Ω) + µ(vol(Ω)

− V )

, where the notations are pre isely dened in Chapter 4. Up to now, optimization on surfa es dierent from the

planedoesnotseem tohavebeenstudiednumeri ally. Asa omparisonin

R

2

,theFinite

ElementMethodisalsousedin[Oud04℄to omputeeigenvaluesapproximation, whereas

an algorithm based on the Method of Fundamental Solutions 15

is employed in [AF12℄.

13 . Thevaluesdisplayedinthistableare omputedwithmasslumping(thenotionofmasslumpingis

re alledinDenition2.2.9).

14 . The invarian e underhomothety of the fun tional given by (1.1) allows to bypassthis volume

onstraintfor optimization in

R

2

. Indeed, dierentvaluesinthe volume onstraintslead to the same

Table 1.1: Numeri al approximationof

λ

k

(Ω

∗

k,M

)

,for

Ω

∗

k,M

the optimizerof volume 0.1 in

M = S

2

,

R

2

,

D

2

for the

k

-theigenvalue,

k = 1, . . . , 15

.

k

λ

k

(Ω

∗

_k,S

2

)

⊂ S

2

λ

k

(Ω

_k,R

∗

2

)

⊂ R

2

λ

k

(Ω

∗

_k,D

2

)

⊂ D

2

1 180.855 181.7 182.639 2 364.356 363.9 364.827 3 460.927 463.0 464.068 4 639.377 647.8 653.612 5 784.251 785.3 789.829 6 888.975 890.5 894.214 7 1063.127 1065.1 1089.251 8 1199.235 1200.1 1207.212 9 1330.355 1340.6 1341.360 10 1439.525 1448.2 1445.205 11 1583.765 1605.5 1632.550 12 1738.957 1743.7 1757.700 13 1890.493 1888.4 1887.360 14 1999.437 2022.2 2026.394 15 2125.772 2111.6 2148.878

Figure1.3: Optimizers ofvolume 0.1in

D

2

for the

k

-theigenvalue,

k = 1, . . . , 15

,leftto right, thendownwards. Thepoint inthe domainsdenotes theorigin of

D

2

.

Contrary to the former, this is a meshless method, representing omputer memory and

omputationaltime savings.

Another optimization problemis about thepla ement of a ir ular obsta leinside a

domain. Thisissueisdis ussed in[Hen06,Se tion 3.5℄. Inthis thesis,the maximization

of an eigenvalue of a ball with respe t to the lo ation of a ir ular obsta le inside is

addressednumeri ally. Aftervalidationfor therstandse ondeigenvalues intheplane,

in the sphere and in the Poin aré dis whi h are theoreti ally known 16

, investigations

are arried out for the third, fourth and fth eigenvalues in these three surfa es. Note

theoptimal pla ement seems to be relatedto theextremal points of the orresponding

eigenfun tion dened onthe ball without obsta le.

Asregardwiththenumeri aloptimizationofeigenvalueswithrespe ttothedomain,

this work does not laim to deliver a proof of the optimality of a domain but only

approximate andidates to be an optimizer. Indeed, on the one hand the algorithm

rea hes lo aland not globalminima, and on the other hand it is in general almost

unfeasible to prove theoreti ally that a domain is in fa t a solution. Even for

k = 3

in

R

2

, it is not yet proven that the dis is the optimizer, despite numeri al onrmations and agreement

17

inthe mathemati al ommunity. Noti ethat not only an an expli it

expression for theboundary of the optimal domain not be guessed for

k

≥ 5

, but it is 16 . Therst and se ond eigenvaluesare maximalwhenthe obsta le is lo ated at the entre of the

ball. For

λ

1

,thisresultisstatedin[Her63℄for

R

2

andin[AA05℄for

S

2

andfor

D

2

. Itisextendedto

λ

2

in[ESK08℄.

17 . As stated in Open Problem 8 from [Hen06℄. This referen e also states that S. A. Wolf and

J. B.Keller proved that the dis in

R

2

is a lo al minimizer for

λ

3

[WK94℄. However, omputations togetherwithnumeri alexperimentsbyA.Bergerseemtoinvalidatethattheballin

R

3

also very likely that no su h expli it des ription is possible. However, some interesting

properties anbederived,su hasthe fa tthatthe andidateobtainednumeri allytobe

the domain minimizing the thirteenth eigenvalue inthe plane is not symmetri [AF12℄.

Another uriosity raised by numeri al investigations is given by the omparison of the

valueof thersteigenvalue

λ

1

(Ω

∗

1,M

)

asso iated to theoptimal domain

Ω

∗

1,M

invarious urvedsurfa es

M

. Forthisnumeri alexperiment, thesphere

S

2

whose urvatureequals

1, the uppersheet

H

of a hyperboloid whose urvature lies between 0 and 1, theplane andthePoin arédis whose urvatureequals-1areaddressed. Theoptimizerisaball

entred at the point of maximal urvature for

H

in ea h of these surfa es and for a volume of 0.01, the following inequalities hold

λ

1

(Ω

∗

_1,S

2

) < λ

1

(Ω

∗

_1,R

2

) < λ

1

(Ω

∗

_1,D

2

) < λ

1

(Ω

∗

1,H

)

and the orrespondingvalues are

1816.57 < 1816.80 < 1817.6 < 1819.10.

It ould have been expe tedat least for small volumesthat there is a ranking of

su h eigenvalues with respe t to the urvature. But the eigenvalue resulting from the

experiment in the upper sheet of a hyperboloid is not between those oming from the

plane and the sphere. It is even higher than theeigenvalue of the ball in the Poin aré

dis . Asame rankingappears alsofor these ondandfor highereigenvalues. Su h kinds

of observations werepossiblethanksto numeri al investigations.

Organisation of the thesis

This do ument is organized inve hapters and three appendi es. After this

intro-du tion, Chapter 2 presentsbasi notions and tools from geometry, the Finite Element

Method and the Lan zos method to nd eigenvalues asso iated to a nite linear

sys-tem. The third hapter deals with the underlying problem, that is, the omputation

of eigenvalues of the Diri hlet-Lapla e operator. Afterits theoreti al statement and its

numeri alapproximation,theestimationoftheerrorbetweenanexa teigenvalueandits

approximationisperformedandalso illustratedusingnumeri alexamples. Itisfollowed

by numeri al validations for spe i domains su h as a ball in

R

2

and by experiments in the plane

R

2

, the sphere

S

2

,the Poin aré dis

D

2

and a manifold withnon- onstant

urvature. Theproblemoflo ating anobsta leinaballto minimizeitsrsteigenvalues

is alsoaddressed, aswell asa omparison of thersteigenvalues of a ballin

R

2

,

S

2

and

D

2

. It is based on the prepubli ation [Str12a℄. Then, Chapters 4 and 5 are devoted to the optimization problem

(

P

opt

)

. The former introdu es shape optimization required to establish the main formula to deform a domain and also the Uzawa algorithm to

extend the problem on domains in manifolds. The latter states the problem

theoret-i ally and displays numeri al results: some validations to re over optimal andidates

already obtained in [AF12℄ and some investigations in

S

2

and

D

2

are those in [Str12b℄. Finally, the do ument ends with three appendi es: the rst one

deals with some notions of fun tional analysis, espe ially Sobolev spa es whi h are the

suitableframeworkfortheunderlying andoptimization problems. AppendixBprovides

a detailed example of the optimization of an eigenvalue with respe t to a domain in a

manifoldnamely

λ

7

(Ω

M

)

for

Ω

M

in the Poin aré dis

D

2

, whereas some additional

Fundamental tools

This hapter isdevotedto re allsome lassi al notions and tools about various

on- eptsinvolved inthefollowing hapters. Thisthesistakespla ebetween two mainelds

ofmathemati s,namelygeometryandnumeri alanalysis. Tobeasa essibleaspossible,

a hoi ehasbeenmadetopresentsomebasi notionsrequiredfromboth. Notationsvery

ommon inmathemati s are usedinthis hapter inorder to avoid needless omplexity,

even if they do not oin ide with the usual ones whi h exist in these parti ular elds.

However, spe ial are has been taken to mention the more frequent notations when it

happens.

This hapter is divided into three parts: the rst one is about Riemannian

geome-try and al ulus on a manifold. In this part, some lassi al tools are introdu ed in the

framework of a manifoldand the expressionof the volume element, of thegradient of a

fun tionand ofthe Lapla eoperatorareexpli itlyestablishedinlo al oordinatesusing

a hart

(U, α)

. Pre isely,performingthe omputationsusinga hartisaspe i ityofour approa h and itisparti ularly helpful for thenumeri al implementation ofour method.

The se ond se tion is a short introdu tion to the Finite Element Method, restri ted to

theaspe ts and results thatareuseful inthesequel. It leads to onsiderapproximation

of fun tionalspa es bydis retized spa es,and to approa h the omputation ofintegrals

using quadrature rules. The notion of masslumping is also addressed. Finally, the last

se tion is on erned with the Lan zos method, used to solve nite dimensional

eigen-problems. In ea hpart, some lassi alreferen es about thesetopi saregiven.

Throughoutthis hapter,

N

∈ N \ {0}

standsforthedimension oftheambient spa e.

2.1 About dierential geometry and al ulus on manifolds

In this se tion,

M

denotes a dierentiable manifold of dimension

N

. Assume that

M

is smooth, that is,

M

is of lass

C

∞

, although less regularity would be su ient.

2.1.1 Dierential forms,volumeelementandintegrationonamanifold

Thissubse tion beginswiththedenitionofseveralnotionsandtoolsusefulto

intro-du e operators and integration on a manifold, and with some of them properties. The

referen e book they are derived from is [Boo75℄, espe ially its hapter V. Refer to it

for theproofsof theresultspresentedbelow aswell asfor some omplements

intention-ally skipped here. With regard to the development of the notion of integration over a

manifold, see [dC94, Chapter 4℄. Thereafter, a more te hni al part is dedi ated to the

expressionof the volume element inlo al oordinates.

In thissubse tion,

V

denotes ave torspa e over

R

of dimension

N

and

V

∗

its dual

spa e. Althoughonly dimension

N = 2

isneeded forour purpose, thistopi sis exposed inanydimension, be ause itdoesnot add anyextradi ulties.

Denition 2.1.1 ( Derived from [Boo75, Denition V-5.1℄ ). A tensor

φ

on

V

is a multilinearmap

φ : V

_{× · · · × V}

|

{z

}

r

times

× V

∗

× · · · × V

∗

|

{z

}

s

times

−→ R,

where

r

∈ N

denotes the ovariant order of

φ

and

s

∈ N

its ontravariant order.

Notation 2.1.2. Fromnowon,

r

∈ N

and

s

∈ N

always standfor non-negative integer. Thesetofalltensorsof ovariantorder

r

and ontravariantorder

s

isdenotedby

T

r

s

(V )

. It isa ve tor spa eover

R

of dimension

N

r+s

,see [Boo75, Theorem 5.2℄ for aproof. In

thefollowing,only ovariant tensorsareused, that is

s = 0

,and thesetof all ovariant tensorsoforder

r

isdenotedby

T

r

_{(V )}

insteadof

T

r

0

(V )

. Set

T

0

_{(V ) = R}

by onvention.

Notation2.1.3. Throughoutthisdo ument,ve torsandve toreldsaredenotedusing

boldfont. Moreover,thetransposeofamatrix

A

isdenotedby

A

T

andthetransposeof

theinverse

A

−1

of amatrix

A

by

A

−T

.

Denition2.1.4([Boo75,DenitionV-5.3℄). A

C

∞

(M )

- ovarianttensoreldof order

r

∈ N

on amanifold

M

isa fun tion

φ : M

→ ∪

p∈M

T

r

(T

p

M )

,

p

7→ φ

p

,su hthat for all

C

∞

(M )

-ve tor elds

X

1

, . . . , X

r

,the map

φ(X

1

, . . . , X

r

) : M

→ R

p

_{7→ φ}

p

(X

1

(p), . . . , X

r

(p)),

isa

C

∞

(M )

-fun tion. The set ofall

C

∞

_{(M )}

- ovariant tensor elds of order

r

on

M

is denoted by

T

r

_{(M )}

.

Set

T

0

_{(M ) =}

_C

∞

_{(M )}

by onvention.

Denition 2.1.5 ( [Boo75, Denitions V-5.4 and V-6.12℄ ). A ovariant tensor

φ

∈

T

r

_{(V )}

, oforder

r

∈ N

issymmetri ,respe tively alternating,iffor ea h

v

1

, . . . , v

r

∈ V

,

respe tively

φ(v

1

, . . . , v

i

, . . . , v

j

, . . . , v

r

) =

−φ(v

1

, . . . , v

j

, . . . , v

i

, . . . , v

r

),

∀ 1 ≤ i, j ≤ r.

Byextension, atensor eldissymmetri ,respe tively alternating,ifithasthisproperty

at ea h point. Moreover, an alternating ovariant tensor eld of order

r

on a manifold

M

is alledanexterior dierential form of degree

r

,or simplya

r

-form.

Remark 2.1.1. Thesetofallsymmetri tensors, respe tively alternatingtensors, formsa

subspa e of

T

r

_{(V )}

,denoted by

Σ

r

_{(V )}

,respe tively

Λ

r

_{(V )}

. Moreover,

Σ

r

_{(V )}

_{∩ Λ}

r

_{(V ) =}

{0}

. The same remarkholds for amanifold

M

instead of

V

,withthenotations

Σ

r

_{(M )}

,

respe tively

Λ

r

_{(M )}

.

Remark 2.1.2. Let

{e

1

, . . . , e

N

}

be a basisof

V

and

φ

6= 0

be a alternating ovariant tensor of order

r = N

on

V

. Then, a dire t omputation involving themultilinearity of

φ

gives, for all

v

1

, . . . , v

N

∈ V

,with

v

i

=

P

N

j=1

C

i,j

e

j

,

i = 1, . . . N

,

φ(v

1

, . . . , v

N

) = det C φ(e

1

, . . . , e

N

),

where

C

is thematrix with omponent

C

i,j

onthe

i

-throwand

j

-th olumn.

Denition - Proposition 2.1.6 ([Boo75,Denition V-6.1and TheoremV-6.2℄ ). Let

φ

∈ T

r

_{(V )}

and

ψ

∈ T

s

_{(V )}

be two ovariant tensors. The produ t of

φ

and

ψ

, denoted

φ

_{⊗ ψ}

isatensor of order

r + s

dened by

φ

_{⊗ ψ(v}

1

, . . . , v

r

, v

r+1

. . . . , v

r+s

) = φ(v

1

, . . . , v

r

)ψ(v

r+1

, . . . , v

r+s

),

for all

v

1

, . . . , v

r+s

∈ V

. The produ t denes a mapping

(φ, ψ)

7→ φ ⊗ ψ

of

T

r

_{(V )}

_×

T

s

_{(V )}

_{→ T}

r+s

_{(V )}

whi h isbilinear and asso iative.

By extension, theprodu t of two ovariant tensor elds on a manifold

M

isdened at ea h point

p

∈ M

,using the previousdenition onthe ve tor spa e

T

p

M

:

Denition - Proposition 2.1.7 ( [Boo75, Theorem V-6.3℄ ). Let

φ

∈ T

r

_{(M )}

, and

ψ

_{∈ T}

s

(M )

be two ovariant tensor elds over amanifold

M

. The produ t of

φ

and

ψ

, denoted

φ

⊗ ψ

,isa ovariant tensor eld oforder

r + s

on

M

dened by

φ

_{⊗ ψ(p) = φ}

p

⊗ ψ

p

,

∀p ∈ M.

Theprodu tdenesanappli ation

T

r

_{(M )}

_{× T}

s

_{(M )}

_{→ T}

r+s

_{(M )}

,

(φ, ψ)

7→ φ ⊗ ψ

,whi h isbilinear and asso iative.

Remark 2.1.3. The tensor produ t of alternating tensors on

V

is not, in general, an alternating tensor on

V

. It leads to introdu e another notion ofprodu t, whi h veries

Denition 2.1.8 ( [Boo75, Denition V-6.5 and Lemma V-6.6℄ ). The mapping

∧ :

Λ

r

_{(V )}

_{× Λ}

s

_{(V )}

_{→ Λ}

r+s

_{(V )}

,

(φ, ψ)

7→ φ ∧ ψ

, dened by

φ

_{∧ ψ(v}

1

, . . . , v

r+s

) =

1

r!s!

X

σ∈S(r+s)

sgn(σ)φ

_{⊗ ψ(v}

_σ(1)

, . . . , v

_σ(r+s)

)

for all

v

1

, . . . , v

r+s

∈ V

,where

S

(N )

denotes theset ofall permutations of

{1, . . . , N}

and

sgn(σ)

denotes the signature of

σ

, is alled exterior produ t or wedge produ t of

φ

and

ψ

. This produ t isbilinear andasso iative.

Remark 2.1.4 ([Boo75,CorollaryV-6.7℄). Itisastraightforward al ulation toseethat

if

φ

i

∈ Λ

r

i

_{(V )}

,

r

i

∈ N

,

i = 1, . . . , k

,thenfor all

v

1

, . . . , v

r

1

+···+r

k

∈ V

φ

1

∧ · · · ∧ φ

k

(v

1

, . . . , v

r

1

+···+r

k

)

=

1

r

1

! . . . r

k

!

X

σ∈S(r

1

+···+r

k

)

sgn(σ)φ

1

⊗ · · · ⊗ φ

k

(v

σ(1)

, . . . , v

σ(r

1

+···+r

k

)

).

Denition2.1.9([Boo75,DenitionV-7.5℄). Amanifold

M

isorientableifitispossible to dene a

C

∞

_{(M )}

-

N

-form

φ

on

M

whi h is not zero at any point. In this ase,

M

is saidto beoriented by

φ

.

Theorem 2.1.10 ( [Boo75,Theorem V-7.7℄ ). Let

(M, g)

be an orientable Riemannian manifold. Corresponding to an orientation of

M

there is a uniquely determined

N

-form

Φ

whi hgivesthe orientationandwhi h hasthe value+1on everyoriented orthonormal frame.

Denition2.1.11. The

N

-form

Φ

oftheprevioustheoremis alledvolumeelement and isdenoted

1

by

dV

g

.

Notation 2.1.12. Let

T (V ) = ⊕

∞

i=0

T

r

(V )

and

Λ(V ) =

⊕

∞

i=0

Λ

r

(V )

. These two dire t sums are a tually asso iative algebra, see [Boo75, Corollary V-6.8℄. Moreover it holds

that

Λ(V ) =

⊕

N

i=0

Λ

r

(V )

,see [Boo75,Theorem V-6.10℄.

Theorem2.1.13 ([Boo75,DenitionV-6.11℄). Let

V

and

W

be two nitedimensional ve tor spa es and

F

∗

: W

→ V

be a linear mapping. Then, the mapping

F

∗

_:

_{T (V ) →}

T (W )

dened by,

F

∗

_{(φ)(w) = φ(F}

∗

(w))

for all

φ

∈ T (V )

and

w

∈ W

, takes

Λ(V )

into

Λ(W )

and isa homomorphism of these (exterior)algebras.

In parti ular, if

α

denotes themap of a hart

(U, α)

in a neighbourhood of a point

p

∈ M

,the derivative

T

·

α

−1

: R

N

→ ∪

q∈U

T

q

M

of

α

−1

isa linear mapping. Thus it an

be employed to transport the volume element from

M

to

R

N

using

T

·

α

−1∗

.

Therequiredtoolsarenowatourdisposaltogivetheexpressionofthevolumeelement

inlo al oordinates. Thisexpressionisintensivelyusedinthesequelandespe iallyinthe

expli itly,although itis a parti ular aseof Remark2.1.2. Let

(M, g)

be a Riemannian manifold oriented by the volume element

dV

g

, and let

(α, U )

be a hart of

M

in the neighbourhoodofapoint

p

∈ M

. Let

{E

1

(p), . . . , E

N

(p)

}

p∈U

denote

2 thebasisof

T

p

M

su h that

T

p

α E

i

(p) = ∂x

i α(p)

,

∀i = 1, . . . , N,

(2.1) where

{∂x

i

}

N

i=1

denotes the usual lo al oordinates 3

. However, there is no parti ular

reason for

{E

1

(p), . . . , E

N

(p)

}

p∈U

to be orthonormal with respe t to the Riemannian metri

g(p)

,soingeneral

dV

g

(E

1

(p), . . . , E

N

(p))

6= 1

. Thus,we onsideranorthonormal (withrespe tto

g(p)

) basis

{F

1

(p), . . . , F

N

(p)

}

p∈U

of

T

p

M

,thatis

g(p)(F

i

(p), F

j

(p)) =

δ

ij

. So,itallowsustoexpresstheve tors

E

i

(p)

,

i = 1, . . . , N

,usingthebasis

{F

k

(p)

}

N

k=1

:

E

i

(p) =

N

X

k=1

A

i,k

(p)F

k

(p),

A

i,k

∈ R, 1 ≤ i, k ≤ N,

or equivalently,





|

|

E

1

(p)

· · · E

N

(p)

|

|





|

{z

}

=: E(p)

=







A

1,1

(p)

· · · A

1,N

(p)

. . . . . . . . .

A

N,1

(p)

· · · A

N,N

(p)







|

{z

}

=: A(p)







− F

1

(p)

−

. . .

− F

N

(p)

−







|

{z

}

=: F (p)

.

At ea h point

p

∈ M

, the metri

g

an be represented by a matrix

G(α(p))

using the lo al oordinates, thatis

G

i,j

(α(p)) = g(p)(E

i

(p), E

j

(p)),

1

≤ i, j ≤ N.

(2.2) Then,

G

i,j

(α(p)) = g(p)

N

X

k=1

A

i,k

(p)F

k

(p),

N

X

k=1

A

j,k

(p)F

k

(p)

!

=

N

X

k=1

A

i,k

(p)A

j,k

(p) = (A(p)A

T

(p))

i,j

.

(2.3)

2. Although

α

doesnotappearinthenotationof

E

i

(p)

,

i = 1, . . . , N

,theseve torsa tuallydepend onthe hart.

3. A tually,thelo al oordinates donotdependonthepoint

α(p)

wheretheyare estimated. They aresometimesdenotedby

n

∂

∂x

i

o

N

i=1

.

Besides,

dV

g

(E

1

(p), . . . , E

N

(p))

=

dV

g





N

X

k

1

=1

A

1,k

1

(p)F

k

1

(p), . . . ,

N

X

k

N

=1

A

N,k

N

(p)F

k

N

(p)





=

N

X

k

1

,...,k

N

=1

A

1,k

1

(p)

· · · A

N,k

N

(p)dV

g

(F

k

1

(p), . . . , F

k

N

(p))

σ(i):=k

i

=

X

σ∈S(N)

sgn(σ)A

_1,σ(1)

(p)

_{· · · A}

_{N,σ(N )}

(p)

=

det A(p) =

p

det G(α(p)),

(2.4) where the last equality omes fromequality(2.3) .

There isstill a manipulationremaining, onsisting inexpressingthevolume element

in

α(U )

⊂ R

N

, where

U

is the open set of the hart. Sin e

p

7→ T

α(p)

α

−1∗

_dV

_g

is an

N

-form 4 on

R

N

byTheorem 2.1.13,it an bewritten at a point

p

∈ α(U)

as

T

α(p)

α

−1∗

dV

g

= f (p) dx

1

(p)

∧ · · · ∧ dx

N

(p),

(2.5)

where

f : α(U )

→ R

isa

C

∞

(α(U ))

-fun tion to be determined and

{dx

j

}

N

j=1

isthedual basisof

{∂x

k

}

N

k=1

,that is,

dx

j

(∂x

k

) = δ

j,k

, where

δ

jk

denotes the Krone ker symbol. First,noti e that, for

p

∈ U

,

T

_α(p)

α

−1∗

dV

g

(∂x

1

, . . . , ∂x

N

) := dV

g

T

α(p)

α

−1

∂x

1

, . . . , T

α(p)

α

−1

∂x

N

=

dV

g

(E

1

(p), . . . , E

N

(p))

=

p

det G(α(p))

(2.6)

where the last equality omes from (2.4) . Then, for any

v

i

=

P

N

k

i

=1

v

i,k

i

∂x

k

i

∈ R

N

,

i = 1, . . . , N

,itholdsthat

T

α(p)

α

−1∗

dV

g

(v

1

, . . . , v

n

)

=

T

α(p)

α

−1∗





N

X

k

1

=1

v

1,k

1

∂x

k

1

, . . . ,

N

X

k

N

=1

v

N,k

N

∂x

k

N





=

N

X

k

1

,...,k

N

=1

v

1,k

1

· · · v

N,k

N

T

α(p)

α

−1∗

dV

g

(∂x

k

1

, . . . , ∂x

k

N

)

=

X

σ∈S(N)

sgn(σ)v

_1,σ(1)

_{· · · v}

_{N,σ(N )}

T

_α(p)

α

−1∗

dV

g

(∂x

1

, . . . , ∂x

N

)

(2.6)

=

p

det G(α(p))

X

σ∈S(N)

sgn(σ)v

_1,σ(1)

_{· · · v}

_{N,σ(N )}

.

4. This

N

-formis alledpullba k of

dV

g

by

α

−

1

.Itismoreoftendenotedby

α

−

1∗

_dV

g

.

Ontheother hand,

dx

1

∧ · · · ∧ dx

N

(v

1

, . . . , v

N

)

= dx

1

∧ · · · ∧ dx

N





N

X

k

1

=1

v

1,k

1

∂x

k

1

, . . . ,

N

X

k

N

=1

v

N,k

N

∂x

k

N





=

N

X

k

1

,...,k

N

=1

v

1,k

1

· · · v

N,k

N

dx

1

∧ · · · ∧ dx

N

(∂x

k

1

, . . . , ∂x

k

N

)

=

X

σ∈S(N)

sgn(σ)v

_1,σ(1)

_{· · · v}

_{N,σ(N )}

dx

1

∧ · · · ∧ dx

N

(∂x

1

, . . . , ∂x

N

)

|

{z

}

=1

.

Finally,thevolume element an be expressedinlo al oordinates by

T

_α(p)

α

−1∗

dV

g

(v

1

, . . . , v

n

) =

p

det G(α(p)) dx

1

∧ · · · ∧ dx

N

(v

1

, . . . , v

N

),

(2.7) for all

v

1

, . . . , v

N

∈ R

N

. Hen e,

f =

√

det G

inequality(2.5) .

Following [dC94, Chapter 4℄, the notion of integrals over a Riemannian manifold

(M, g)

an be nowaddressed. Theaim is to ompute integrals of a fun tionover

α(U )

, for agiven hart

(α, U )

using theexpressionof thevolumeelement inlo al oordinates. Indeed, it will be useful again for the numeri al omputations involved in the sequel.

Before dealing with integrals of a fun tion, let us rst dene the integral of an

N

-form overa bounded subset of

R

N

.

Denition 2.1.14. Let

φ

be an

N

-form in an open subset

D

⊂ R

N

with ompa t

support

K

ontained in

U

. If

φ

iswritten as

φ = f dx

1

∧ · · · ∧ dx

N

,fora

C

∞

_(D)

-fun tion

f

,thentheintegral of

φ

over

D

isdened by

Z

D

φ =

Z

K

f dx

1

. . . dx

N

,

where

dx

1

. . . dx

N

denotes the Lebesguemeasure on

R

N

.

Themap

α

ofa hartallowstoextendthisdenitiontoanorientedmanifold. Toavoid onvergen e problems,itis onvenientalthough generallynotrequiredto assumethe

supportof the

N

-form to be ompa t. Itholds for instan e if

M

is ompa t. Moreover, makerst theassumptionthatthe supportof the

N

-form is ontained inan open setof a hart.

Denition 2.1.15. Let

(M, g)

beanorientedRiemannianmanifold 5

,

φ

bea

N

-formon

M

having ompa t supportinthe open set

U

of a hart

(α, U )

. The integral of

φ

over

M

is dened by

Z

M

φ =

Z

α(U )

T

_·

α

−1∗

φ.

Theorientability ofthe manifoldensuresthatthisdenitiondoesnot depend onthe

hoi e of the map. The hoi e of an orientation for

M

xes the sign of the integral. Finally, if

φ

has ompa t support, but not ompletely inside the open set of a hart, then to integrate

φ

over the entire manifold, a partition of unity

{ψ

i

}

ompatible with the overing by the open sets of the harts is required. Indeed, it allows to apply the

previousdenitiontoea h

ψ

i

φ

. Theintegralof

φ

over

M

isthenthesumoftheintegrals ofthe

ψ

i

φ

over

M

.

Now, the denition of an integral of a fun tion

f : M

→ R

over a Riemannian manifold

(M, g)

follows naturally.

Denition 2.1.16. Let

(M, g)

beanorientedRiemannianmanifold, andlet

dV

g

bethe asso iated volumeelement. Afun tion

f : M

→ R

isintegrable over

M

if

f

has ompa t support in

M

. Furthermore, the integral of

f

over

M

is the integral of the

N

-form

6

f dV

g

.

Remark 2.1.5. Assume

f

to be as in the previous denition. If the support of

f

is ompa t andin luded inthe open set

U

ofa hart

(α, U )

,then,

Z

M

f dV

g

=

Z

α(U )

T

_·

α

−1∗

(f dV

g

) =

Z

α(U )

f

_{◦ α}

−1

√

det G dx

1

. . . dx

N

,

byequation (2.7). Thisformulawill beintensively usedfortheeigenvalueproblemon a

Riemannianmanifold.

The lassi al and expe ted properties of the integral dened above are proved in

[Boo75, Se tion VI-2℄.

2.1.2 Expression of the gradient in lo al oordinates

Thissubse tion isbased onthe se ond hapterof [GHL04℄.

Denition 2.1.17 ( [Cha84, Denition I-1℄ ). Let

f : M

→ R

be a fun tion of lass

C

∞

_{(M )}

. Thegradient of

f

,denoted

∇

f

is the ve tor eldon

M

dened by

g(p)(∇ f (p), Z(p)) = Z(f )(p),

∀Z ∈ χ(M), ∀p ∈ M,

where

χ(M )

denotes the ve tor spa eofall ve tor elds of lass

C

∞

_{(M )}

.

Remark 2.1.6. TheRieszrepresentation Theorem(Theorem A.1.3)ensuresthatthe

gra-dient ofa fun tion

f

iswell dened. Proposition 2.1.18. Let

f, h

∈ C

∞

_{(M )}

. Then,

∇

_{(f + h) = ∇ f + ∇ h,}

∇

_{(f h) = f ∇ h + h∇ f .}

Remark 2.1.7. Let

(α, U )

bea hartof

M

inthe neighbourhood ofapoint

p

∈ M

. Asin theprevioussubse tionwithequation(2.2) , onsiderthefamilyofmatri es

{G(α(p))}

p∈U

representing the metri

g

at ea h point

p

∈ U

in the usual lo al oordinates

{∂x

i

}

N

i=1

. Let us arry out an analogous development as the one that gave the expression of the

volume element in lo al oordinates. The gradient of a fun tion

f

∈ C

∞

(M )

an be expressed at apoint

p

∈ U

inthe basis

{E

i

(p)

}

N

i=1

dened inequation(2.1) , thatis

∇

_f

_{(p) =}

N

X

k=1

β

k

(p)E

k

(p),

thus,for all

i = 1, . . . , N

,

g(p)(∇ f (p), E

i

(p)) =

N

X

k=1

β

k

(p)G

k,i

(α(p)).

For

p

∈ U

,thedenition ofthe gradient applied to

Z(p) = E

i

(p)

,

i = 1, . . . , N

,and the denitionof

{E

i

(p)

}

N

i=1

,yield

g(p)(∇ f (p), E

i

(p)) = E

i

(f )(p) = T

α(p)

α

−1

∂x

i

(f )(p) = ∂x

i

(f

◦ α

−1

)(α(p)).

Hen e, for all

i = 1, . . . , N

,

∂x

i

(f

◦ α

−1

)(α(p)) =

N

X

k=1

β

k

(p)G

k,i

(α(p)),

thatis,

∇

_us

_{(f ◦ α}

−1

_{)(α(p)) = G}

T

_{(α(p))∇ f (p),}

where

∇

us

denotes the usual gradient operator a ting on fun tions dened on an open setof

R

N

. Finally,the gradient of

f

inlo al oordinates isgiven,for all

p

∈ U

,by 7

∇

f

= G

−T

∇

_us

_{(f ◦ α}

−1

₎

_{◦ α,}

or by

∇

_f

_{= G}

−1

∇

_us

_{(f ◦ α}

−1

₎

_{◦ α,}

(2.8)

thanksto thesymmetryof

G

sin e itrepresents ametri .

Inordertodenethedivergen eandtheLapla eoperatorsonaRiemannianmanifold

(M, g)

,several toolsneed to beintrodu ed.

Denition 2.1.19 ( [GHL04, Denition 1.52 bis℄ ). Let

U

⊂ M

be an open set of

M

. The Liebra ket isthe mapping

[

·, ·] : χ(U) × χ(U) → χ(U)

dened by

[X, Y ] = XY

− Y X, X, Y ∈ χ(U).

7. Asmentionedbefore,

G

−

T

Remark 2.1.8. The Lie bra ketis a

R

-bilinear, anti ommutative mapping, and satises theJa obi identity,that is,

[X, [Y , Z]] + [Y , [Z, X]] + [Z, [X, Y ]] = 0.

_{∀X, Y , Z ∈ χ(U).}

Denition 2.1.20 ( [GHL04, Denitions 2.49 and 2.50℄ ). A onne tion on

M

is a mapping

∇ : χ(M) × χ(M) → χ(M)

, denoted by

(X, Y )

7→ ∇

X

Y

, su h that for all

X, Y , ξ, ζ

_{∈ χ(M)}

andfor all

f

∈ C

∞

_{(M )}

:

(i)

∇

ξ

(f X + Y )(p) = ξ(f )(p)X(p) + f (p)

∇

ξ

X

(p) +

∇

ξ

Y

(p)

; (ii)

∇

f ξ+ζ

(X)(p) = f (p)

∇

ξ

X

(p) +

∇

ζ

X

(p)

.

Moreover,

∇

X

Y

issaidto be torsion-free ifitalsosatises (iii)

[X, Y ] (p) = (

∇

X

Y

− ∇

Y

X

)(p)

.

Theorem 2.1.21 ( [GHL04, Theorem 2.51℄ ). Let

(M, g)

be a Riemannian manifold. Then,there existsa unique torsion-free onne tion

∇

satisfyingfor all

X

, Y , ξ

∈ χ(M)

,

ξ(g(X, Y )(p)) = g(

_∇

ξ

X

(p), Y (p)) + g(X(p),

∇

ξ

Y

(p)).

Denition 2.1.22 ( [GHL04, Denition 2.53℄ ). The onne tion dened in the above

theoremis alled theLevi-Civita onne tion.

Remark 2.1.9. It an benshown 8

that theLevi-Civita onne tionis hara terized by

g(

_∇

X

Y

, Z) =

1

2

(X(g(Y , Z)) + Y (g(Z, X))

− Z(g(X, Y ))

+ g(Z, [X, Y ])

− g(X, [Y , Z]) + g(Y , [X, Z])) .

for all

X

, Y , Z

∈ χ(M)

. Hen eforth,

∇

denotes theLevi-Civita onne tionon

(M, g)

. Denition 2.1.23 ( [GHL04, Denition 2.67℄ ). Let

c : I

→ M

be a smooth urve. A ve tor eld along

c

is a urve

X : I

→ T M

,su hthat

X(t)

∈ T

c(t)

M

,for any

t

∈ I

.

The ve torspa e ofall ve tor elds along

c

isdenotedby

χ

c

(M )

.

Denition - Proposition 2.1.24 ( [GHL04, Theorem 2.68℄ ). Let

c : I

→ M

be a smooth urve. There exists a unique operator, denoted by

D

dt

and alled ovariant derivative, dened on the ve tor spa e of all ve tor elds along

c

, whi h satisesto the following onditions:

(i) forall

X

∈ χ

c

(M )

,

f

∈ C

∞

_(I)

,

D

dt

f X(t) = f (t)

D

dt

X(t) + f

′

(t)X(t);