Elliptic equations in divergence form, geometric critical points of solutions, and Stekloff eigenfunctions

SIAM J. MATH. ANAL.

Vol. 25, No. 5,pp. 1259-1268,September 1994

1994 Society for Industrial and Applied Mathematics 0O2

ELLIPTIC

EQUATIONS

IN DIVERGENCE

FORM,

GEOMETRIC

CRITICAL POINTS OF

SOLUTIONS,

AND STEKLOFF

EIGENFUNCTIONS

*

G.

ALESSANDRINI

AND R.

MAGNANINI:

Abstract. The Stekloff eigenvalue problem (1.1) has a’countable number of eigenvalues

(Pn }n=1,2

...

eachoffinitemultiplicity. Inthis paper the authors giveanupperestimate,in terms

ofthe integern, ofthemultiplicity ofPn, andthenumberof critical pointsandofnodaldomainsof theeigenfunctionscorresponding toPn.

Inview ofapossible applicationto inverse conductivity problems, theresult for thegeneral caseof elliptic equations with discontinuous coefficientsindivergenceformisprovenbyreplacing the classicalconceptof critical point with themoresuitablenotionof geometric critical point.

Key words, eigenvalue problems, geometric properties of elliptic equations, critical points, inverseconductivity problems

AMS subjectclassifications. 35J25,35P99, 35C62, 30C15

1. Introduction.

In

this paper we are concerned with weak solutions of the ellipticequation in divergence form:

(1.1a)

div(AVu)

0

in,

and especially with Stekloff eigenfunctions, that is, those nontrivial solutions

that,

forsome constant p, the Stekloffeigenvalue, satisfyin the weak sense the boundary condition

(1.1b)

AVu

v pu

on.

Here

is asimply connected boundeddomain intheplane, with Lipschitzboundary

0,

vistheexterior unit normal to

0,

and

A

(aij)

isa2 2 symmetric matrixof

L()

coefficients satisfying, for some constant

,

0

<

A <_

1, the uniform ellipticity condition

(1.2)

2 i,j=l

forevery z E

,

E /i

2.

Here

and inwhat

follows,

we usethecomplex coordinate z x

₊

iyintheplane. The study ofthiseigenvalue problemwas startedby Stekloff

[St]

in 1902.

In 3,

we recall the definitions ofStekloff eigenvalues and eigenfunctions; a review oftheir knownproperties canbefoundinBandle

[B].

Researchonthissubject has been mainly devotedto estimateson eigenvalues

(see,

for instance,

In-P-S]

andalso

[B]

for further

references).

Let

us mention in passing

that,

in connection with applied problems in fluidmechanics,mixedtypeproblemsalsohavebeen considered

(see

[F-K]).

Typically,

1993.

*Receivedbythe editorsFebruary 2, 1993; acceptedfor publication(inrevisedform)June 18,

Dipartimento di ScienzeMatematiche,Universith diTrieste, Italy.

:

Dipartimento diMatematica,Universith diFirenze, Italy. 1259

insuchproblems, we assumethat

(1.1b)

holds only ononeportionof

0t,

whereason

the restof

OFt, AVu.

u 0is assumed.

Our

main

results,

which aresummarized inTheorem 3.2, consistofestimates on the multiplicities of Stekloff eigenvalues and the numbers of nodal domains and of critical points ofStekloff eigenfunctions.

One

up-to-datemotivationofourstudy of the Stekloff eigenvalue problemcomes from the so-calledinverseconductivity problem: supposethatthe coefficientmatrix

A

(the

conductivity)

is

unknown;

wewishto determine it, or,more generally, torecover

partialinformationabout itfromtheknowledge ofthe so-called Dirichlet to

Neumann

map

_AA.

Here,

AA

H1/2(0t)

H-1/2(O)

is defined as the operator that maps the Dirichlet data

_ulo

for

_(1.1a)

into the corresponding

Neumann

data

(see,

for instance,

[Sy-V], [Sy]).

Then it is evident that the Stekloff eigenvalues and the boundary traces of the Stekloff eigenfunctions are exactly the eigenvalues and eigenfunctions of

_AA.

Such traces and eigenvalues can be approximately measured by experiments and couldbe effectively used as the data of the inversion procedure.

For

a discussion of a related spectral approachto the inverse conductivity problem,

see

Gisser,

Isaacson,

and Newell

[G-I-N].

A

deeper understanding of the geometrical features of Stekloff eigenfunctions would then behelpful in assessing uniqueness and continuous dependence questions for the inverse conductivity problem and perhaps alsoin the designofreconstruction algorithms.

In

addition tothepossible applicationsto inverseproblems, theauthorshavebeen inspired by the workof

Payne

and Philippin

[P-P]

onquestions of symmetry related to theStekloffeigenvalueproblem

(see

also

[A-M2]).

In

thisrespect,wementionthat

some ofour

results,

when restricted toLaplace’sequation and to the second Stekloff eigenvalue, have alreadybeenpresented in

[P-P].

The flavorofour resultsis similartothose ofCheng

[C]

for theeigenfunctionsof theLaplacianon

surfaces; however,

themethods in theproofs and thespecificresults are differentbecause oftheintrinsic differences between the two eigenvalueproblems.

To

mentionthe mostapparentpeculiarity ofStekloff eigenfunctions, observe

that,

bythemaximumprinciple, everysolutionof

(1.1a),

and thuseveryStekloff eigenfunc-tion, can have neither interior maxima nor minima, and each of itslevel sets must reach the

boundary. Due

to this observation, the geometric-topologic properties of levellinesand level setsofStekloff eigenfunctionsarequite different from those ofthe vibratingmembrane problems.

In fact,

in somerespects, thestudy ofsuch properties

isperhaps easierforequations like

(1.1a),

for which wehave theorems that permit us

to estimate the number ofinterior critical points in terms of the boundary data

[A],

[A-M1].

Theorems like this will beour basic

tool,

along withsimple variationsof the fundamental

Courant’s

nodal domain theorem.

Still, in view of the application to the inverse conductivity problem, we have chosen to treat the Stekloff eigenvalue proble.l when no moothness assumptio is

imposedon.the coefficients in

(1.1a).

In fact,

thereare practical caseswhen the

con-ductivitycoefficientsarediscontinuousandweareespeciallyinterested indetermining the discontinuities

(see,

for instance,

[B-F-I], [P],

[Su-U]).

This generMity causes ad-ditional technical diificultes: solutions of

_(1.1a)

need not to be differentiable i the classicalsenseand thus thenotionofcritical point .mst beadaptedto this nonsmooth setting.

For

thisreason, weintroducethe conceptof geometric critical point

(see

Def-inition

2.3).

Roughly speaking a point will be called a geometric critica| _point

,r

a olution u of

_(1.13)

if it is a critical point with repect to an. appropriate

(possi-bly

nonsmooth)

cha.nge ofvariables that makes u become smooth.

We

alamo give the definitionof geometric index, whichgenerMizcs the notionof multipli.cityofa critical

ELLIPTIC EQUATIONS AND GEOMETRIC CRITICAL POINTS 1261 point

(see

Definition

2.4).

These definitions are based onthe crucial remark that the unique continuation property andarepresentationtheorem also hold for solutions of

(1.1a).

We

have not been able to find in the literaturetheorems ofsuch a kind for equations like

(1.1a),

whereastheyare well known for equations withsmoother coefficients

_(see

_[Sch])

and for equations not indivergence form

(see

[B-N]).

Althoughthemethod ofproofof such results may sound familiarto the experts in quasi-conformal mappings and complex analytic

methods,

weincludeour ownproofsof the uniquecontinuationproperty and ofrepresentationformulas forsolutionsof

_{(1.1a) (Theorem}

2.1andCorollary

_2.2).

We

trust that these results mightbeofsome independentinterest and that theycan find useful applications, especially in thefield of inverseproblems.

We

conclude this introductionby pointing out thefollowingconsequence of such results.

In

[A], [A-M1],

when the coefficients in

(1.1a)

are

smooth,

an estimate on

the maximum number of interior critical points ofa solution u E

CI(Ft)

C?

C2(f)

of

(1.1a)

was given in termsofthe number oftimes some boundary data ofu changes

its signonOft

(see

[A,

Thm.

1.1], [A-M1,

Thms.

2.1,

2.2],

for

details).

Theorems 2.7 and 2.8 in this papergeneralize the aboveresults to equation

(1.1a)

with nonsmooth coefficients.

2. Geometric critical points. Throughout thissection,

BR(O)

willdenote the disk with radius

R

centered at z 0.

THEOREM

2.1

(representation

formula).

Let

u

W1,2(ft)

be a nonconstant

so-lution

_of

_(1.1a).

There exists a quasi-conformal mapping _X f ---+

BI

(O)

and a real-valued

har-monic

_function

h on

BI

_(O)

such that

(2.1)

u=hox

in

Ft.

The dilatation

_{coefficient Xe/Xz}

_of

_X is bounded by the constant

(1

Proof.

By

(1.1a),

the 1-form co

_-(a12ux

+

a22uy)dx

+

(allux

+

a12uy)dy

is

closed in f.

Therefore,

we can find v

W1,2(ft)

such that dv co. The function v

willbecalled thestream

_function

associated with

u,

inanalogywiththetheory ofgas dynamics

(see,

for instance,

[B-S]).

In

other

words,

u andvsatisfythefollowing elliptic system:

(2.2)

Vv

(

O1

-1)

almost everywhere in ].

Note

that

_(2.2)

isjust a specialcase of the elliptic systems studied by

Bers

and Nirenbergin

[B-N].

By

setting

_f

u

+

iv and using the standard notation for complex

derivatives,

(2.2)

takes the forn

where

a22 a.t

2ia2

#

1 -t-

a

+

a22

+

alia22

a212’

"

1 alla22-t-

a2

1

₊

a-

+

a22 zr-all.a22

a122

and thefollowingestima,te canbe ea,sily deducedfrom

_(1.2)

Consequently,since

_f

E

W1,2(’, C),

thenit is aquasi-regularmapping with dilatation bounded by

(1- A)/(1

+

A).

Since

_f

is

nonconstant,

it can befactored as

(2.3)

f

Fo X

ingt,

where

F

isa holomorphic function onthe disk

B1 (0)

and X

12

-+

BI(0)

is a quasi-conformal homeomorphism

(see

[L-V]).

Finally,

1-A

-I+A"

COROLLARY

2.2

(Unique

continuation

principle).

_If

there exists

zo

ft and

pos-itive constants

C1,

C2,...,

CN,

such that

f

(2.4)

]

[Vu[

2

dxdy

_

CNR

N

_VR

_>

_O,

_VN

_1,

_2,...,

B zo

thenu is

constant

in

.

Proof.

Let

us suppose

by

contradiction that u is nonconstant. Without loss of generality, wemay set z0 0 and

u(0)

v(0)

0.

Note

that,

by

(2.2),

thestreamfunctionv associated with usatisfies the following equation:

(2.5)

div

(delt

A

AVv)=0

in,t,

intheweaksense. Suchan equationsatisfies the ellipticitycondition

(1.2)

aswell. The local boundedness estimate

[G-T,

Thm.

8.17]

is applicable to both

(1.1a)

and

(2.5).

Thus,

bythisestimate, the Poincarinequality, and

(2.4),

we have max

[u--

uR[

2

<

_KCNR

N

BR/.(O)

max

Iv

vRI

2

<

_KCNR

N

BR/2(0)

/N=1,2,.-.

and

VR, 0<R<R0.

Here R0

dist(0,

OFt),

K

isapositiveconstant dependingon

A

only, andltR,VRdenote the mean values on the disk

Ba(0)

of

u,

v,

respectively. Since

u(0)

v(0)

0,

we also have

_U2R,

_v2R

<_

KCNRN,

andhence

max

(u

2+v

2)

_8KCNRN

_VR,

_O

_<

_R

_<

_Ro,

_N

_l,

_2,

Bn/2(O)

Now,

weclaimthat thereexistp, 0

<

p

<

R0,

aquasi-conformal homeomorphism

)

of

Bp(0)

in

itself,

and a positive integer

M

such that the quasi-regular mapping

f

-u

+

iv canbe factoredas

(2.7)

f

(z)=

(z)]

M

_Izl

<

p,

where has dilatation bounded by

(1

)/(1

+

A)

and

)(0)

0. This factorization

isreadily obtained from

(2.3),

first, by choosing

M

astheorder ofthe first nontrivial term in the Taylorseries for

F-

F(x(O))

at

X(0),

and,

second,

bynoticing thelocal invertibility ofthe branchesofthe multivaluedfunction

IF-

F(x(O))]I/M.

ELLIPTIC EQUATIONS AND GEOMETRIC CRITICAL POINTS 1263 Since

:

is quasi-conformal, we have that

_--1

isH61der continuous at zero with someexponent 0

<

a

_<

1.

From

(2.6), (2.7),

we have that

Q

_Izl

<-

KCN

Vz,

_Izl

<

p,

VN

1 2, forsome positiveconstant

Q,

andthis isimpossible.

DEFINITION

2.3.

We

say that

zo

E is a geometric critical point

_for

u,

_if

we have

Vh(x(zo))

O,

where _{h and X are, respectively, the} harmonic

_function

and the quasi-con.formalmapping appearing in

(2.1).

Remark.

It

is an obvious, but essential, consequence ofthis definitionthat

geo-metriccritical pointsof nonconstant solutions of

(1.1a)

are isolated.

We

nowrecall aclassicaldefinition ofthe indexofasmoothfunction

_{(see [Mi]).}

For

a

C

function h with isolated critical points in thedisk

_B1

₍₀₎

and a subdomain

DcC

B1 (0)

suchthat

OD

issmooth andcontains no critical point of

h,

theindexof hin

D

is

I

(D, h)

2rl

0D

d arg

(Vh).

With suchachoice of the sign, ifh is

harmonic,

I(D, h)

gives the number of critical points of h in

D,

when counted according to their multiplicities.

Moreover,

I(D,

h)

is constant under perturbations of

D

that contain the same critical points, and its definitioncanbe extended tothecasewhen

OD

isnonsmooth.

We

generalizethis notion to nonconstantsolutionsof

_(1.1a).

DEFINITION

2.4.

Let

D

C C be an open set.

_If

u has no geometric critical points on

OD,

we

_define

thegeometric indexofu in

D

as

I

_{(D, u)}

I

_{(X (D), h),}

where_{h and X} are as in Theorem 2.1.

Moreover,

we

_define

the geometric indexofu at

zo

as

I

(z0, u)

lim

I

(Br (z0) u).

Such a limit exists, since the geometric critical points

_of

a solution

_of

(1.1a)

are

iso-lated.

The next lemma gives a sort of justification for the term "geometric" in the previous definitions and shows that these do not depend onthe particular choice of the representation

(2.1).

LEMMA

2.5.

Let

u be a nonconstant solution

_of

_(1.1a).

_If

_zo

is ageometric critical point

for

u with geometric index

I

I(zo,

u),

then there exists aneighborhood

U

c

_of

zo

such that the level line

_{z

e U:

u(z)

u(z0)}

is made

_of

I

₊

1 simple arcs, whosepairwise intersection consists

_of

_{z0}

only.

Proof.

By

therepresentation

_(2.1),

since_Xis aquasi-conformal homeomorphism, it is enoughto look at the levelline

{

e

BI(0):

h

h(x(zo))}

near

_X(zo).

Since

_I(x(zo),

_h)

_{I(zo, u)}

I,

then

h-h(x(zo))

isasymptotictoahomogeneous harmonic polynomial of

degree I

+

1 near

_X(zo).

Remark. Observe

that,

if u is

C

in a neighborhood of z0

(which

happens, for

instance,when

A

isHhlder continuous; see

[Sch]),

thenz0 is ageometric critical point

with geometric index

I

ifand only if

_Vu(z0)

0 with standard index

I.

This is a

consequenceof

Lemma

2.5 above and

Lemma

3.1 in

_[A-M1].

Note

that in

[A-M1]

the opposite sign is chosen in the definition of theindex.

PROPOSITION2.6

(Continuity

ofthe geometric

index).

Let

{Am}m=1,2,...

be a

se-quence

_of

symmetricmatrices with

L

(t)

entriessatisfying

(1.2).

Let

Um

be weaksolutions

_of

div

_(AmVum)

0 in

which converge tou in

Wlo’

2

If

D

C

c

is such thatu has no geometric critical pointon

OD,

thenwe have

(2.8)

lim

I

(.D, u,)

I

(D, u).

m---,c

Pvof.

By

the proof of Theoreln 2.1, for each Um we may construct a stream function Vm such that

_fm

Um

+

iVm are quasi-regular mappings with dilatation coefficientsuniformly boundedby

(1 A) / (1

+

A).

By

(2.1),

wealso haveUm

hm

o

and the dilatation coefficients ofthe quasi-conformal mappings Xm arealso uniformly bounded. Since Um

--

Uin

Wllo’c2(),

by using the uniforminteriorboundsfor

_fm

and Xm in

C

a

_(see

_[G-T,

_Thm.

_8.24]),

_{we have}

_that,

_{possibly passing tosubsequences,}

_hm

and_{Xm converge, respectively,}in

_Clo(Bl(0))

and

_Coc(t)N

Wllo’c2

(Ft),

tothe functions h and_{X corresponding to}u intherepresentation

(2.1).

By

definition,

I(D,

u)

I(x(D), h)

and

I(D, urn)

I(Xm(D),

hm).

Furthermore,

byourhypothesis, Vh does notvanishon

_Ox(D),

sothat

_IVhml

isuniformly bounded away fromzero on

_Ox(D),

for mlarge enough.

Thus,

I

_{(X (D), h)}

lim

I

_{(X (D), hm),}

and

hence,

by the

_Cloc(f)

convergence ofXm, we arrive at

(2.8).

Observe now

that,

sincetheverybeginning ofour argmnent, wecouldhave replacedthe sequence

{urn}

with anyofitssubsequences.

Therefore,

the limit in

(2.8)

existsandthe statedequality holds.

THEOREM

2.7.

Let

g

H1/2(O)

be

_of

boundedvariation on

O

and such thatOgt canbe splitinto2Marcs onwhich altenativelygis anondecreasingand nonincreasing

function

of

the arclength parameter.

Let

u

W1,2(gt)

be the unique solution

_{of (1.1a)}

satisfying theDirichlet condition

u g on0.

Then the geometric critical points

_of

u in

,

when counted according to their indices, are at most

.M-

1.

.Proof.

In

view of

Lemma

2.5

above,

this isjust a rephrasing ofTheorem 1.1 in

[.A1].

We

omit the details.

THEOREM 2.8.

Let

g

H-1/2(O)

be such that

Ot

can be split into 2M closed arcs

F1,...,F2M

such that

(-1)Jg

0 on

_{Fy,j- 1,...,2M,}

in the sense

_of

distribu-tions.

.Let

u

W,2(t)

be a solution

_of

_(1.1a)

satisfying the

Neumann

condition

AVu.

v=g on Ogt.

Then,

the geometric critical points

of

u in

,

when counted according to their indices, are at most

M-

1.

Proof.

We

may suppose that

c9

is

C

a.

Ifit were not so, we could construct a

Lipschitz napping that transibrms

Ft

into adisk. Sucha mapping does not alter the natureofthe eq.uaionnor the sign conditionsonthe

Neumann

data g.

Let

us choose a sequence

{Am}

of

C()

symnetric matrices satisfying

(1.2)

and suchthat

.A

--.

A

in

LP(t),

for some 1.

_<

p

<

.

It

is astraightforwardexercise

ELLIPTIC EQUATIONS AND GEOMETRIC CRITICAL POINTS 1265 that

_fon

g’ ds 0 and

(--1)Jgm

>

0 intheinterior of each

F

j,j 1,...,

2M,

for all

m= 1,2,

For

any integer m, let Um E

C()

be the unique solution of the following problem:

div

(AmVum)

0

in,

AmVum.

=gm

on0/,

such that

_f

u, dx dy

_fa

u dx dy. Since Um is smooth on

OFt,

we can apply Theorem 2.2 in

[A-M1]

and obtain

I(D, urn)

_< M-

1 for every integer m.

Moreover,

we can easily see

that,

by possibly passing to subsequences, Um u

in

Wllo’2(t),

and hence Proposition 2.6 is applicable.

Therefore,

for any

D

CC

Ft

such that

OD

does not contain any geometric critical point of

u,

we have

I(D,

u)

limm-+

I(D, u,),

and hence

I(D,

u)

<_ M-

1.

By

the arbitrariness of

D

in

Ft,

we

obtainour thesis.

3. Multiplicity of Stekloff eigenvalues and geometric critical points of Stekloffeigenfunctions.

As

iswell

known,

by observingthat the trace imbedding

W1,2()

L2(0)

iscompact, the Stekloff eigenfunctionsandeigenvaluesin

W1,2(gt)

are characterized asthe critical points and critical values ofthe Rayleighquotient

(3.1)

R(u)

f

AVu

Vudxdy.

u2

d8

here ds denotes the arclength element on 0fl

_(see

_[St]).

The nth Stekloff eigenvalue Pn can berecursivelydefined asthe minimumof the quotient

(3.1)

over all functions of class

W1,2(gt)

thatare orthogonalin

_L2(0Ft)

to thesubspaces

Vk,

k 1,...,n- 1, where

(3.2)

Vk

{u

Wi,2

_(t)

_{uis aweaksolution of}

_(1.1)

_{with p}

_Pk}.

In

this way, we can form a divergent sequence 0 pl

<

p2

<

<

_Pn

<

Of

eigenvalues, each ofthem offinitemultiplicity.

DEFINITION

3.1.

Let

_Pn be thenth

Stekloff

eigenvalue; we denote by# its

mul-tiplicity, that is,

(3.3)

#n dim

V.

For

n

>

2, we will also set

(3.4)

n, max

_@-

{

geometric critical points

of

u in

counted according to theirindex

A

nodal domain

_of

u

V.,

is a connected component

_of

the set

{z

u(z)

0},

while a connected component

of

aset

Ok

0 will be

_rc.rred

to as aboundary nodal domain ofu.

We

then

_define

(3.5)

A,,,

max

v,,’\ 0

[nodal

domains

_of

THEOREM 3.2. Thefollowing inequalities hold" n

(3.7)

in+l

_<

1

+

#}, n---- 1, 2, k--1

(3.8)

gn

_<

An-

2,

n-2,

3,...,

(3.9)

n

_<

2

(an

-t-

1),

n-

2, 3,

COROLLARY

3.3.

For

every integer n

>_

2,

we have

(3.10)

#n

_<

2.3

n-2,

(3.11)

an _<

3n-2 _1,

(3.12)

A.

<

3n-.

₊

1.

Proof.

By

applying

(3.7)-(3.9)

we obtain the recurrence relation _#n+l

_

2 x

n

k=l

#k,n-

1,2,

Since #

I,

we obtain

(3.10);

(3.11)

and

(3.12)

then easily

follow from

_(3.8)and

_(3.7).

Remark. When n

2,

(3.11)

gives a2 0. This provides a different proof of

Lemma

3in

_[P-P].

The proof of Theorem 3.2 requiresthe following

lemma,

which will be provedat the end ofthis section.

LEMMA

3.4.

Let

An

and

n

be

_defined

as in

_Definition

3.1. Then

(3.13)

5n

_<

2

_(A

1),

n- 2, 3,

Proof of

Theorem 3.2.

Step

1.

We

prove

_(3.7)

bycontradiction. This argument has been used already in

[K-S]

for the case of theLaplace operator.

Suppose

there exists a nontrivial eigenfunction u E

_Vn+l

with A nodal domains

n

and A

>

2

₊

_k=l

#k

Let

us denote by

u

k),...,k(k) a basis of the vector space

Vk,

l <_k<_n.

Now

consider thefunction v

--J-=aj(ulnj);

here

lj

denotes the character-istic functionoftheset

j.

The real numbers a,...,aA_ canbechosen notallzero and such that

fo

k)

forallg=l, ..,#k,

k=1,2,

n;

(3.14)

vu ds

O,

in fact

(3.14)

provides

E=I

k

_<

A-2 linear homogeneous conditions on A- 1 parameters.

In

viewof

(3.14),

the functionv isadmissiblefor thevariational characterization

(3.1)

of_Pn+I.

From

thedefinition of

v,

wehave

f

AVv

Vv

dx dy

_]

_(AVv

₎

vds Jo

2/

(AVu. )

uds,

Oj J_O2

Hence,

(1.1b)

implies

AVv Vv

dxdy pn+l_aj fJ j=I,...,A-1. u2

_ds,

_{j 1, A 1}

ELLIPTIC EQUATIONS AND GEOMETRIC CRITICAL POINTS 1267

and,

by addingthe above A 1 relations, weobtain

AVv Vv

dxdy Pn+l

o

v2ds.

Therefore,

v is anontrivialStekloff eigenfunction correspondingtothe eigenvaluepn+1.

Now,

since v 0on

/x,

by the unique continuation property, we have v 0 on

whichis acontradiction.

Step

2.

Let

Un E

V;

by

(1.1b)

andby

Lemma 3.4, AVUn’p

satisfies thehypotheses of Theorem 2.7with

M

_<

_An

1;

thus,

(3.8)

follows easily.

Step

3.

By

contradiction,suppose

_n

2(an

q-

₁₎

₊

1.

Let

u(J),j

1,...,tin,beabasisof

V

and fix

an

+

1 distinct pointszl,...,Zn+

in

t. As

wedid in theproof ofTheorem

2.8,

weapproximate

A

bya sequence of ma-trices

{A,},=l,2,...

with

C()-coefficients

satisfying

(1.2).

For

eachj, 1

<_

j

<_

the weak solutions

U(m

j) ofthe

Drichlet

problem

div

(Am

Vu)

)

-0

in’t,

e

are

C(a)-functions

and forma sequence

U(m

j) that convergesto u(J) in

Wlo’c2(a).

By

our hypothesison ttn, for each j, 1

_<

j

_<

#n, we can findrealnumbers

a(m

j) suchthat

"n

Ej--1

1,m 1,

2,... and,

also

(3.15)

E

a(mY)VU(mJ)(ze)=

0, for all

t?

1,

an

q- 1, m 1, 2,

j=l

For

each j

1,...,

#n, thesequence

C(m

j) can be chosen to converge to some number

c(J) so thatwe have

E

(c(J))

2-

1.

j=l

Now,

let

D

be an open set with

D

c

,zl,...

,Ztn+l

D,

and such that

OD

does not contain any geometric critical point of thefunction v

_{-jn__=l}

o(J)u(J). The

m

_{c) u)}

sequence of functions Vm

y=l

is such that

I(Vm,

D)

+

1, by

(3.15).

Moreover,

by possibly passing to a subsequence, Vm v in

Wo’

(),

as m

,

so

that Proposition 2.6 implies that

I(v, D)

gn

₊

1, that is, v is a nontrivial eigen-function in

Vn

with at least gn

+

1 geometric critical points in

D

C

.

This is a

contradiction.

We

conclude by giving a sketch of the proof of

Lemma

3.4.

To

this

end,

we

introduce the following definitions.

DEFINITION

3.5.

We

say that a simply connected open subset

A

_of

is acp,

_if

O

OA

is connected and

nonempty.

Let

,...

,K

be open subsets of

;

we say that

{k}k=l

K is an admissible K

covering

_of

,

if

,...,

_K

are pairwise disjoint, C

=

k,

and

Ok

O

0,

for everyk-

1,...,K.

Proof

of

Lemma

3.4

(Sketch).

Let

u

Vn,

unontriviM, and let

,...,

g

be the nodal domains ofu in

;

these sets form an admissible coveringof

.

Now,

let

_N

be the number ofboundary nodM domainsofu in 0. Then

(3.13)

number

K

ofnodal domains andby using the following

facts,

theproofsof which are straightforward:

(i)

Every

nodal domain

_fk

issimplyconnected.

(ii)

The covering

{ftk}=l

_K

contains at least one cap.

(iii)

If ft/( is a cap, then f

_\

_fK

is a simply connected open set and

{ftk}k=l

K-1 is anadmissible covering of

Ft. Moreover,

Na

<_

Nfi +

2.

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