Asymptotics of the eigenvalues of the Dirichlet-Laplace problem in a domain with thin tube excluded

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Asymptotics of the eigenvalues of the Dirichlet-Laplace

problem in a domain with thin tube excluded

Xavier Claeys

To cite this version:

Xavier Claeys. Asymptotics of the eigenvalues of the Dirichlet-Laplace problem in a domain with

thin tube excluded. Quarterly of Applied Mathematics, American Mathematical Society, 2016, 74 (4),

pp.595-605. �hal-01120422v2�

Lapla e problem in a domain with thin tube ex luded

X.Claeys

∗†

Abstra t

We onsider aLapla eproblem withDiri hlet boundary ondition inathree dimen-sionaldomain ontaininganin lusiontakingtheformofathintubewithsmallthi kness

δ

. Weprove onvergen einoperatornormoftheresolventofthisproblemas

δ → 0

, estab-lishingthattheperturbationindu edbythein lusionontheresolventisnotgreaterthan

O(| ln δ|

−γ

)

for some

γ > 0

. We dedu e onvergen eof theeigenvaluesoftheperturbed operatortowardthelimitoperator.

1 Introdu tion

Analysis of singular perturbations of ellipti boundary value problems by in lusions of small volume has re eived a lot of attention in the past de ades due to its numerous appli ationstomodelingandnumeri alsimulationofmulti-s aleproblems,anditspossible usein e ientinverseproblem[1,2℄orshapeoptimization[20℄ strategies.

For Lapla e equation, many works of theexisting literature havebeendedi ated to asymptoti analysis involving thepresen e of an in lusion that has small size in all di-re tionsofspa e(typi allysmallballs),inthe aseofeither impenetrable(homogeneous Diri hlet,Neumann,orRobin)orpenetrable(transmissionproblem)boundary onditions. Su hasymptoti shavebeenprovidedforboth2-Dand3-Dproblems,inthe aseofeither a singlein lusion ormany, see [19, 18, 17, 16, 10℄. Thebooks [14, 9℄are landmarkson this typeofproblem.

As regards Lapla e equation perturbed by a small in lusion that takes the form of a thin tube, mu h less work is available in the literature. Some works an be found in the ase of a tube entered around a smooth losed ontour without self-interse ting point. The analysis of Lapla e equation in 3-Dwith Neumann and Diri hlet boundary ondition was provided in [5, 6℄ where the author onsiders either a tube entered at a non-self-interse ting smooth losed ontour, oratube entered at astraightsegment. Thisanalysisisrenedandgeneralizedtoarbitrarydimensionsin[15℄and[14,Chap.12℄. Similarresultswereestablishedfora ousti s andHelmholtzequationin [3, 7,22℄.

It shouldbepointedoutthat,in the aseofa3-D Lapla eequation perturbedby a thin elongatedin lusion,the asesofthehomogeneousDiri hletorRobinboundary on-dition leadtomu hmore hallenginganalysisthanthe aseofNeumannortransmission onditions: while Neumann/transmission onditionslead to ratherstandardasymptoti onstru tions, mat hingof asymptoti sin theDiri hlet/Robin aseleadstoan ill-posed 1-Dintegralequation.

∗

LaboratoireJa ques-LouisLions,UPMCUniv. Paris6andCNRSUMR7598,75005,Paris,Fran e

†

ary onditionhasmanyimportantappli ationsinparti ularin ele tri alengineeringand antenna based ele tri /ele troni devi es, as it is ommonly taken as a rough model for ele tromagneti dira tion by perfe tly ondu ting thin wires, see for example [11, Chap.6℄. Inthis ontext,thestudy ofeigenvaluesis ofparti ularimportan easitis re-latedtoresonan ephenomena. Yet,inthe aseofthinelongatedin lusionswithDiri hlet boundary ondition, onlyPlanida [21℄ hasaddressedthis issueso far. Inthis referen e, the authorestablishes onvergen eof eigenvaluesin the asewhere thein lusion is en-teredaroundasmooth losed urvethatdoesnotself-interse t. Howevernoexpli itrate of onvergen eisprovided.

Inthepresentarti le, westudyLapla eequation withhomogeneousDiri hlet ondition, and theasso iated eigenvalueproblem, perturbedby the presen e of a small elongated in lusion that takesthe form of a thin tube. Weestablish onvergen e in the operator norm,oftheresolventoftheperturbedproblemtowardtheresolventofthelimitproblem. A rstnovelingredienthere omparedtopreviouslyestablishedresults on ern the geo-metri al ongurationsunder onsiderationthatonlyrequirethein lusionto on entrate around aparametrized urve. Contrary to pre-existingworks, thepresent analysisdoes not require the in lusion to be onne ted, or admit any symmetry of revolution. The limit urve onsideredheremayadmit self- rossingpointswhi h, to ourknowledge,has neverbeeninvestigatedin theexistingliterature(atleastin the aseofanhomogeneous Diri hletboundary ondition). Inaddition,weestablisha onvergen eresult(witherror estimates) for eigenvalueswith expli it error estimate showing that the presen e of an elongatedin lusionindu esashiftoforder

O(| ln δ|

−γ

₎

oftheeigenvalues,forsome

γ > 0

. Theoutlineofthepresentarti le isasfollows. InSe tion 2wedes ribein detailthe geometry under onsideration. In Se tion 3 we introdu e weighted

δ

-dependent norms adapted to our analysis, and establish a stability property for the Lapla e operator in terms ofthese norms. The nextse tion isdedi ated to provingHardytype inequalities thatwillbene essaryforthesubsequenterrorestimates. Inthelastse tionwederivean upperboundforthedieren e,inoperatornorm,betweentheresolventsoftheperturbed problem andthelimitproblem.

2 Geometry under onsideration

Westartbydes ribingindetailthegeometri alsettingunder onsideration. Let

Ω ⊂ R

3

refer to aLips hitz domain. Consider a

C

1

-fun tion

γ : R 7→ Ω

that is

L

-periodi for some

L > 0

, andsu h that

0 < α

−

≤ |

dγ

dz

(z)| ≤ α

+

, ∀z ∈ R

for xed onstants

α

±

> 0

. Thenweset

Γ := γ(R)

,whi hisaLips hitz urve. Weshallneedtorefertothedistan e fun tion

d(x) := inf

y∈Γ

|x − y|.

(1) Then we onsider

(Ξ

δ

)

δ>0

asany family of Lips hitz domains su h that there exists a xed onstant

C > 0

independentof

δ

forwhi h

sup

x∈Ξ

δ

d(x, Γ) ≤ C δ.

(2)

This onditionimposesthat,as

δ → 0

,thesets

Ξ

δ

aremoreandmore on entratedaround

Γ

. Changing the index parameter

δ

throughmultipli ation by somefa torif ne essary, wemayassumethat

C < 1

in(2)withoutrestri tinggenerality. Thenwedenote

lems in avities ontainingin lusions whose geometry is similar to wires. Note that we do not require

γ

to be one-to-one overaperiod so that, under the assumptions above, the urve

Γ

mayself-interse t. Note alsothat

Ξ

δ

donotneedto be onne tedand may degenerate,as

δ → 0

,toonlyapartof

Γ

withtips. Asa onsequen e,thepresentsetting overssu hgeometri al ongurationsastheonespresentedin pi turesbelow.

Figure1Threeexamplesofgeometri al ongurations overedbytheanalysispresentedhere.

3 Weighted Sobolev spa es GivenanyboundedopenLips hitzdomain

ω ⊂ R

3

,theset

L

2

_(ω)

shallrefertothespa eof square integrablefun tions over

ω

equipped withthepairing

u, v 7→ (u, v)

ω

:=

R

ω

u v dx

and the norm

kuk

L

2

(ω)

=

p

(u, u)

ω

. As usual,

H

1

0

(ω)

shall refer to the ompletion of

C

₀

∞

_{(ω) := {ϕ ∈ C}

∞

_(R

3

_{), ϕ = 0}

on

R

3

_{\ ω, supp(ϕ)}

bounded

}

forthenorm

kϕk

H

1

0

(ω)

:=

kϕk

L

2

_(ω)

+ k∇ϕk

_L

2

_(ω)

. Finally

H

−1

_(ω)

will refertothetopologi aldualto

H

1

0

(ω)

.

Foragivenparameter

δ > 0

smallenough,weareinterestedinstudying theeigenvalues oftheoperator

A

δ

: H

1

0

(Ω

δ

) → H

−1

(Ω

δ

)

denedby

hA

δ

(u), vi := (∇u, ∇v)

Ω

∀u, v ∈ H

1

0

(Ω

δ

).

(3)

Weshall ompareitseigenvalueswiththoseof thelimitoperator

A

0

: H

1

0

(Ω) → H

−1

(Ω)

that is dened similarly but in a domain without any perturbation i.e.

hA

0

(u), vi :=

(∇u, ∇v)

Ω

∀u, v ∈ H

1

0

(Ω)

.

Tostudythesetwooperators,weneedtointrodu eadapted

δ

-dependentweightedSobolev norms. Thesearenot lassi alweightednormssu hasthose hara terizingKondratiev's spa es,seee.g. [13℄. Thenormswe onsiderhereinvolve

δ

-dependentlogarithmi weights: givena

β ∈ R

,andaLips hitzopenset

ω ⊂ R

2

,wedene

kvk

2

V

1

β,δ

(ω)

:= |v|

2

V

1

β,δ

(ω)

+ |v|

2

V

0

β,δ

(ω)

where

|v|

2

V

0

β

(ω)

:=

Z

ω

| ln d

δ

(x)|

−2β

|v(x)|

2

|d

δ

(x) ln d

δ

(x)|

2

dx

|v|

2

_V

1

β

(ω)

:=

Z

ω

| ln d

δ

(x)|

−2β

|∇v|

2

dx

with

d

δ

(x) := d

⋆

(x) + δ

where

d

⋆

(x) := min(d(x), e

−1

_/2).

(4)

For

δ > 0, β ∈ R

xed, the norm above is obviously equivalent to

k k

H

1

(ω)

. However thesenorms"behave"dierentlyas

δ → 0

. Thenorms(4)arespe i allytailoredforour analysis.

The ondition

d

⋆

(x) ≤ e

−1

_{/2 ∀x ∈ Ω}

guarantees that

|d

⋆

(x) ln d

⋆

(x)| ≤ |d

δ

(x) ln d

δ

(x)|

for all

x

∈ Ω

and for

δ < e

−1

_/2

. Ontheother hand,re allthat

d

isadistan efun tion to a ompa t setsoit isLips hitzwith

k∇dk

L

∞

(Ω)

= 1

, see[4, Prop.1D.4℄forexample. Hen e

d

⋆

is aLips hitz fun tion,and

d

δ

is non-vanishing with

d

δ

(x) ≥ δ, ∀x ∈ Ω

. We dedu e that

Υ

δ

β

(v) := v(x) ln

β

_d

δ

(x)

isomorphi ally maps

H

1

0

(Ω)

onto itself, and there exist onstants

c

±

, δ

0

> 0

independentof

δ

su hthat

0 < c

−

≤

kΥ

δ

β

(v)k

V

1

β,δ

(Ω)

kvk

V

1

0

,δ

(Ω)

≤ c

+

∀v ∈ H

1

0

(Ω),

∀δ ∈ (0, δ

0

).

(5)

Weshallalso onsiderthe orrespondingdualnorms,ea hofwhi hprovidinganormfor

H

−1

_(Ω)

,and arryingatthesametimesomedependen ywithrespe tto

δ

,

kf k

_V

−1

β,δ

(ω)

:=

_v∈H

sup

1

0

(ω)\{0}

|hf, vi|

kvk

V

1

−

β,δ

(ω)

.

Let us examine the ontinuity properties of the operators

A

0

, A

δ

with respe t to these norms. Firstofall, both

A

0

and

A

δ

easily appear asuniformly ontinuouswith respe t to thesenormsi.e. adire t al ulusshowsthat

lim sup

δ→0



sup

v∈H

1

0

(Ω)\{0}

kA

0

(v)k

V

−1

β,δ

(Ω)

kvk

V

1

β,δ

(Ω)

+

sup

v∈H

1

0

(Ω

δ

)\{0}

kA

δ

(v)k

V

−1

β,δ

(Ω

δ

)

kvk

V

1

β,δ

(Ω

δ

)



< +∞.

Uniforminvertibilityof theseoperatorsalsoholdunder ertainhypothesisontheweight exponent

β

.

Proposition3.1.

There exists

β

⋆

> 0

su h that, for

|β| < β

⋆

, the inverses of both

A

0

and

A

δ

remain uniformly boundedwith respe t to

δ

intermsof the weighted norms

lim sup

δ→0



sup

f∈H

−1

_(Ω)\{0}

kA

−1

0

(f )k

V

1

β,δ

(Ω)

kf k

_V

−1

β,δ

(Ω)

+

sup

f∈H

−1

_(Ω

δ

)\{0}

kA

−1

_δ

(f )k

V

1

β,δ

(Ω

δ

)

kf k

_V

−1

β,δ

(Ω

δ

)



< +∞ .

Proof:

Weneedtoprovideanupperboundforboth

A

−1

δ

and

A

−1

0

. Weprovesu habound only for

A

−1

δ

, asthe derivation for

A

−1

0

is very similar and slightly easier. First of all observethat,a ordingto theuniformbounds (5),thisisequivalentto provingthat,for

β ∈ R

xedsu hthat

|β| < β

⋆

,thereexistsa onstant

C

β

> 0

independentof

δ

su hthat

C

β

≤

sup

v∈H

1

0

(Ω

δ

)\{0}

|hA

δ

· Υ

δ

β

(u), Υ

δ

−β

(v)i|

kuk

V

1

0,δ

(Ω)

kvk

V

1

0,δ

(Ω)

.

(6) Wehave

hA

δ

· Υ

δ

β

(u), Υ

δ

−β

(v)i = (∇Υ

δ

β

(u), ∇Υ

δ

−β

(v))

Ω

δ

,a ordingtothedenitionof

A

δ

. Expanding thisexpressionyields

(∇Υ

+β

(u), ∇Υ

−β

(v))

Ω

δ

= (∇u, ∇v)

Ω

δ

+β

Z

Ω

∇d

⋆

(x) ·

u∇v − v∇u

d

δ

(x) ln d

δ

(x)

dx − β

2

Z

Ω

u v

|d

δ

(x) ln d

δ

(x)|

2

dx.

(7)

Observe that, taking

v = u

, the se ond term in the right hand side above is purely imaginary. In addition, sin e

|d

⋆

(x) ln d

⋆

(x)| ≤ |d

δ

(x) ln d

δ

(x)|

for all

x

∈ Ω

, using Proposition4.1belowandthefa tthat

v ∈ H

1

0

(Ω

δ

) ⊂ H

1

0

(Ω)

,weobtain

ℜe{ (∇Υ

+β

(u), ∇Υ

−β

(v))

Ω

δ

} ≥ (1 − |β/β

⋆

|

2

_{) k∇uk}

2

L

2

_(Ω

δ

)

≥

1 − |β/β

⋆

|

2

1 + 1/β

2

⋆

kuk

2

_V

1

0

,δ

(Ω

δ

)

.

(8)

witha onstantindependentof

δ

.



4 Hardy type inequalities

This se tionisdedi ated toprovingtwoinequalitiesthat takeaform similartoHardy's inequality,butadapted toourgeometri alsetting. Therstresultbelowmaybe under-stoodasa ylindri alversionofHardy'sinequality. Theproof,though,ismadetri kyby the rather generalgeometry under onsiderationhereand, in parti ular, the possibility for

Γ

toadmitself- rossingpoints.

Proposition4.1.

Thereexistsa onstant

β

⋆

> 0

su hthat

1

β

2

⋆

:=

sup

v∈H

1

0

(Ω)\{0}

n

₁

k∇vk

2

L

2

_(Ω)

Z

Ω

|v(x)|

2

_dx

|d

⋆

(x) ln d

⋆

(x)|

2

o

< +∞.

(9) Proof:

Duringthisproof,weshallreferto

e

3

(t) := ∂

t

γ(t)/|∂

t

γ(t)|

, onsidera

C

0

-ve toreld

t 7→ e

1

(t) ∈ R

3

su hthat

e

1

(t) · e

3

(t) = 0, ∀t ∈ R

,andset

e

2

(t) := e

3

(t) × e

1

(t)

. Denoting

I

ǫ

:= (−ǫ, +ǫ)

and

D

ǫ

⊂ R

2

thediskof enter

0

andradius

ǫ

, forea h

t ∈ R

,there exists asmall ylinder

Q

ˆ

t

= D

ǫ

t

× I

ǫ

t

with

ǫ

t

> 0

su h thatthemap

φ

t

: ˆ

Q

t

→ R

3

denedby

φ

t

(x, y, z) := γ(t + z) + xe

1

(t) + ye

2

(t)

isanimmersion. Pi knitelymany

t

1

, t

2

, . . . t

n

su hthat

[0, L] ⊂ ∪

n

j=1

(t

j

− ǫ

t

j

, t

j

+ ǫ

t

j

)

. Denote

Q

ˆ

j

:= ˆ

Q

t

j

,

Q

j

:= φ

t

j

( ˆ

Q

j

)

and

Γ

j

:= Γ∩Q

j

sothat,inparti ular,

Γ ⊂ Q

1

∪· · ·∪Q

n

. Let

ω

η

= {x ∈ Ω | d(x) < η}

. Letusprovethat,if

η

is hosensmallenough,thenforany

x

∈ ω

η

thereisone

j

su hthat

x

∈ Q

j

and

d(x) = inf

y∈Γ

j

|x − y|

. Toshowthis,pro eed by ontradi tion,assumingforamomentthat su hisnotthe ase.

This means that there exists a sequen e

x

p

∈ Ω

with

lim

p→∞

d(x

p

) = 0

and su h that, for any

x

∗

p

∈ Γ

satisfying

|x

p

− x

∗

p

| = d(x

p

)

, none of the

Q

j

's both ontain

x

p

and

x

∗

p

. Sin e

Γ

is ompa t,extra tingasub-sequen eifne essary,wemayassumethat

lim

p→∞

x

p

= lim

p→∞

x

∗

_p

= x

∞

∈ Γ

. Takea

Q

m

ontaining

x

∞

. Then, sin e

Q

m

isan openneighborhood of

x

∞

, forsu ientlylarge

p

wehaveboth

x

p

∈ Q

m

and

x

∗

p

∈ Q

m

whi h ontradi tsourinitialassumption.

Fromnow on, weassume that

η > 0

is hosensmall enough to guaranteethe property dis ussedinthepreviousparagraph. Denote

d

j

(x) = inf

y∈Γ

j

|x − y|

,andlet

1

Q

j

referto the hara teristi fun tion of

Q

j

. Whatpre edesshowsthat,forany

x

∈ ω

η

,wehave

1

|d

⋆

(x) ln d

⋆

(x)|

≤

n

X

j=1

1

Q

j

(x)

|d

j

(x) ln d

j

(x)|

.

(10)

Choose a smooth open set

Q

0

⊂ R

3

su h that

Q

0

∪ Q

1

· · · ∪ Q

n

a hieves a overingof

Ω

. Using a partition of unity subordinated to this overing, one may de ompose any

v ∈ H

1

0

(Ω)

in theform

v = v

0

+ · · · + v

n

where

v

j

∈ H

1

0

(Q

j

)

. Thisremark,togetherwith Inequality(10),showsthatitsu estoprovetheexisten eofa onstant

C > 0

su hthat

Z

Q

j

|v(x)|

2

_dx

|d

j

(x) ln d

j

(x)|

2

≤ C

Z

Q

j

|∇v|

2

dx

∀v ∈ H

1

0

(Q

j

) ∀j = 1 . . . n.

(11)

Fromnowon,anduntiltheendoftheproof,wetakea

j

xed. Letusrelabelforamoment

ǫ = ǫ

t

j

,

φ = φ

t

j

andset

Q = ˆ

ˆ

Q

j

= D

ǫ

× I

ǫ

,

Σ := {0} × I

ǫ

and

v = v ◦ φ

ˆ

. Forany

x

ˆ

∈ ˆ

Q

wealso set

d(ˆ

ˆ

x) = inf

y∈Σ

ˆ

|ˆ

x

− ˆ

y|

. Thedieomorphism

φ

and its inverse

φ

−1

are both Lips hitz,sothereare onstants

c

±

> 0

su hthat

c

−

|ˆ

x

− ˆ

y| ≤ |φ(ˆ

x) − φ(ˆ

y)| ≤ c

+

|ˆ

x

− ˆ

y|

. Routineveri ationsshowthatthisimpliesexisten eoftwo onstants

c

′

_{, c}

′′

_{> 0}

su hthat

c

′

d(ˆ

ˆ

x) ≤ d φ(ˆ

x)

≤ c

′′

d(ˆ

ˆ

x)

∀ˆ

x

∈ ˆ

Q.

(12)

Let

r, θ

refertothepolar oordinatesin

R

2

sothatthe ylinder

Q

ˆ

isparametrizedbythe ylindri al oordinates

(r, θ, z) ∈ [0, +ǫ) × [0, 2π] × I

ǫ

. The lassi al Hardy's inequality applied in

Q

ˆ

,see[8℄,showsthat

Z

ˆ

Q

|ˆ

v|

2

_rdrdθdz

|r ln(r)|

2

≤ C

Z

ˆ

Q

|∇ˆ

v| + |ˆ

v|

2

dˆ

x

∀ˆ

v ∈ H

1

( ˆ

Q).

(13)

There only remains to observe that

d(ˆ

ˆ

x) = r

, to use (12), and to apply the hange of variables

x

= φ(ˆ

x)

in the integrals (13). Denoting

Dφ

the dierential of

φ

, sin e

kDφk

_L

∞

( ˆ

Q)

and

kDφ

−1

_k

L

∞

(Q

j

)

arebothbounded,thisnally yieldsinequality(11)with

k∇vk

2

_L

2

_(Q

j

)

+ kvk

2

L

2

_(Q

j

)

insteadof just

k∇vk

2

L

2

_(Ω)

. We on ludeby using Poin are's

in-equalityin

Q

j

.



Wewillalsoneedanother

δ

-dependentweightedinequality. Thisoneinvolvesaweighted

L

2

-normevaluatedonlyovera oronal ylinderofradius

δ

,andisnotprimarilybasedon the lassi alHardy'sinequality. InsteaditisderivedbymeansofKondratiev'sanalysis. Proposition4.2. Denoting

Q

δ

:= {x ∈ Ω | δ < d(x) < 2δ}

wehave

lim sup

δ→0

sup

v∈H

2

_(Ω)∩H

1

0

(Ω)\{0}

n

₁

k∆vk

2

L

2

_(Ω)

Z

Q

δ







v(x)

d

⋆

(x)







2

dx

o

< +∞.

Proof:

Toestablishthisresult,wefollowapathsimilartothatoftheproofofProposition4.1, andusethesamenotations. Takeany

x

∈ ω

η

. A ordingtotherstpartofthepre eding proof,there exists

j

su h that

x

∈ Q

j

and

d

j

(x) = d

⋆

(x)

, hen e

δ < d(x) < 2δ ⇒ δ <

d

j

(x) < 2δ

. As a onsequen e

Q

δ

⊂ ∪

n

j=1

Q

j

δ

where

Q

j

δ

:= {x ∈ Q

j

, δ < d

j

(x) < 2δ}

whi h implies

1

Q

δ

≤

P

n

j=1

1

Q

j

_δ

. Hen e it su es to prove, for ea h

j

, the existen e of onstants

C, δ

0

> 0

independentof

δ

su hthat

Z

Q

j

_δ







v(x)

d

j

(x)







2

dx ≤ C

Z

Ω

|∆v|

2

_dx

_{∀v ∈ H}

2

_{(Ω) ∩ H}

1

0

(Ω),

∀δ ∈ (0, δ

0

).

Fixing

j

, andusinga hangeofvariableslikein thepreviousproof,theinequalityabove boilsdowntoestablishingtheexisten eof onstants

C, δ

0

> 0

independentof

δ

su hthat, forany

v ∈ H

2

_{( ˆ}

_{Q) ∩ H}

1

0

( ˆ

Q)

andany

δ ∈ (0, δ

0

)

wehave

Z

ˆ

Q

δ

|v|

2

r

2

rdrdθdz ≤ C

Z

ˆ

Q

|∆v|

2

+ |∇v|

2

+ |v|

2

dˆ

x.

(14)

where were allthat

Q = D

ˆ

ǫ

× I

ǫ

,thevariables

r, θ, z

refertothe ylindri al oordinates in

Q

ˆ

,and

Q

ˆ

δ

:= {ˆ

x

∈ ˆ

Q | δ < r < 2δ}

. Toestablish(14),set

∆

⊥

v = r

−2

_((r∂

r

)

2

+ ∂

θ

2

)v

. De omposeea h point

x

ˆ

∈ ˆ

Q

as

x

ˆ

= (ˆ

x

⊥

, z)

,sothat

x

ˆ

⊥

= (r cos θ, r sin θ)

and

r = |ˆ

x

⊥

|

. Introdu etheFourierde ompositionof

v

inthe

z

variable,setting

v

p

(ˆ

x

⊥

) :=

1

2ǫ

Z

+ǫ

−ǫ

Then wehave

v

p

∈ H

2

_(D

ǫ

) ∩ H

1

0

(D

ǫ

)

and

−∆

⊥

v

p

∈ L

2

_(D

ǫ

)

for all

p ∈ Z

, if

v ∈ H

2

_{( ˆ}

_{Q) ∩}

H

1

0

( ˆ

Q)

. Applying Kondratiev's analysis in

D

ǫ

\ {0}

, see Chapter 6 of [13℄, we nd the existen eof oe ients

α

p

∈ C

anda onstant

C > 0

independentof

p

su hthat

|α

p

|

2

+

Z

D

ǫ

|v

p

(ˆ

x

⊥

) − α

p

|

2

|ˆ

x

⊥

|

3

dˆ

x

⊥

≤ C

Z

D

ǫ

|∆

⊥

v

p

|

2

dˆ

x

⊥

∀p ∈ Z.

Parseval identity asso iated to De omposition (15), together with the estimate above, indi ates that

P

+∞

p=−∞

|α

p

|

2

< +∞

sothat thereexistsafun tion

α = α(z) ∈ L

2

_(I

ǫ

)

and onstants

C, C

′

_{> 0}

independentof

v

satisfying

kαk

2

L

2

_(I

ǫ

)

+

Z

ˆ

Q

|v(ˆ

x

⊥

, z) − α(z)|

2

|ˆ

x

⊥

|

3

dˆ

x

⊥

dz

≤ Ckf k

2

L

2

_{( ˆ}

_Q)

= Ck∆

⊥

vk

2

L

2

_{( ˆ}

_Q)

≤ C

′

_k∆vk

2

L

2

_{( ˆ}

_Q)

∀v ∈ H

2

_{( ˆ}

_{Q) ∩ H}

1

0

( ˆ

Q).

(16)

The inequality aboveis justied by standardellipti apriori estimates for the Lapla e operator,see[13,Chap.3℄forexample.Finally,letuspi kanarbitrary

v ∈ H

2

_{( ˆ}

_Q)∩H

1

0

( ˆ

Q)

. Plugging(16)into theleft handsideof(14)yields

Z

ˆ

Q

δ

|v|

2

r

2

rdrdθdz =

Z

ˆ

Q

δ

|v(ˆ

x)|

2

|ˆ

x

⊥

|

2

dˆ

x

≤

Z

ˆ

Q

δ

|α(z)|

2

|ˆ

x

⊥

|

2

dˆ

x

+

Z

ˆ

Q

|v(ˆ

x) − α(z)|

2

|ˆ

x

⊥

|

3

dˆ

x

≤ kαk

2

L

2

_(I

ǫ

)

 Z

2δ

δ

dr

r



|

{z

}

=ln(2)

+Ck∆vk

2

_L

2

_{( ˆ}

_Q)

≤ C

′

_k∆vk

2

L

2

_{( ˆ}

_Q)

.

Sin e

v

was hosenarbitrarilyin

H

2

_{( ˆ}

_{Q) ∩ H}

1

0

( ˆ

Q)

,andthe onstant

C

′

_{> 0}

isindependent of

δ

,this on ludestheproof.



5 Norm onvergen e of the resolvent Wewillnowusethepreviousanalysistoshowthat

A

−1

δ

strongly onvergestoward

A

−1

0

in someappropriateoperatornorm. Beforestatingthisresultletusjustpointoutthat,using extension by

0

, we have

H

1

0

(Ω

δ

) ⊂ H

1

0

(Ω)

sothat

H

−1

_{(Ω) ⊂ H}

−1

_(Ω

δ

)

. The expression

A

−1

_δ

(f )

with

f ∈ H

−1

_(Ω)

should beunderstooda ordingto thesein lusions. Proposition5.1.

For

β

⋆

> 0

as in Proposition 3.1, and for ea h

β ∈ R

satisfying

|β| < β

⋆

, there exist onstants

c

β

, δ

0

> 0

independentof

δ

,su hthat

sup

f∈L

2

_(Ω)\{0}

kA

−1

₀

(f ) − A

−1

_δ

(f )k

V

1

β,δ

(Ω)

kf k

L

2

_(Ω)

≤

c

β

| ln δ|

β

∀δ ∈ (0, δ

0

).

Proof:

Firstofall onsidera

C

∞

ut-ofun tion

χ : R → R

su hthat

χ(t) = 0

for

t ≤ 1

and

χ(t) = 1

for

t ≥ 2

,andset

χ

δ

(x) := χ( d(x)/δ )

and

ψ

δ

:= 1 − χ

δ

. Intheremainingofthis proof,weshalldenote

Q

δ

:= supp(∇χ

δ

)

sothat, forany

x

∈ Q

δ

,wehave

δ ≤ d(x) ≤ 2δ

. Weintrodu etheoperator

R

δ

: H

−1

_{(Ω) → H}

1

0

(Ω

δ

)

dened by

R

δ

(f ) := χ

δ

A

−1

0

(f )

∀f ∈ H

−1

_(Ω).

Nowtake any

f ∈ L

2

_(Ω)

, and set

u

δ

:= A

−1

δ

(f ) ∈ H

1

0

(Ω

δ

)

,

u

ˆ

δ

:= R

δ

(f ) ∈ H

1

0

(Ω

δ

)

and

u

δ

− ˆ

u

δ

. There exist onstants

c, c

′

_{> 0}

independentof

δ

su hthat, forany

v ∈ H

1

0

(Ω

δ

)

, wehave

|hA

δ

(u

δ

− ˆ

u

δ

), vi| = |(∇(u

δ

− ˆ

u

δ

), ∇v)

Ω

|

= |(f, ψ

δ

v)

Ω

+ (∇χ

δ

, u

0

∇v − v∇u

0

)

Q

δ

|

≤ kf k

L

2

_(Ω)

kψ

_δ

vk

_L

2

_(Ω)

+ |(∇χ

_δ

, u

₀

∇v − v∇u

₀

)

_Q

δ

|.

(17)

To derive an upper bound for the se ond term in the right hand side above, observe that

δ ≤ d(x) ≤ 2δ

for

x

∈ supp(∇χ

δ

)

. Sin e

|∇d| ≤ 1

, we have

sup

x∈Ω

|∇χ

δ

| ≤

2d(x)

−1

_sup

x∈Ω

|∂

t

χ|

. So Cau hy-S hwarzinequality, together with Proposition 4.1 ap-pliedto

∇u

0

,andProposition 4.2applied to

u

0

,yield theexisten eof a onstant

C > 0

independentof

δ

su hthat

|(∇χ

δ

, u

0

∇v − v∇u

0

)

Q

δ

| ≤ C( k∇vk

L

2

(Q

δ

)

+ kvk

L

2

(Q

δ

)

)k∆u

0

k

L

2

(Ω)

≤ C

′

_{| ln δ|}

−β

_kvk

V

1

−β,δ

(Ω)

kf k

L

2

_(Ω)

.

(18)

Thersttermintherighthandsideof(17) anbeboundedbynotingthat

δ ≤ d

δ

(x) ≤ 2δ

on

supp(ψ

δ

)

. SoapplyingHardy'sinequality(9)yieldsa onstant

C > 0

independentof

δ

su hthat

kψ

δ

vk

L

2

(Ω)

≤ Cδ| ln δ|

1−β

kvk

V

1

−

β,δ

(Ω)

forall

v ∈ H

1

0

(Ω

δ

)

. Pluggingthistogether withEstimate (18)into Inequality(17)providesa onstant

C > 0

independentof

δ

su h that

|hA

δ

(u

δ

− ˆ

u

δ

), vi| ≤ C| ln δ|

−β

kvk

V

1

−

β,δ

(Ω)

kf k

L

2

(Ω)

.

Observethat

A

δ

(u

δ

− ˆ

u

δ

) = (Id − A

δ

R

δ

)f ∈ H

−1

_(Ω)

. Sin e

v ∈ H

1

0

(Ω

δ

)

wasarbitraryin the al ulusabove,whatpre edesshowsthat thereexists a onstant

C > 0

independent of

δ

su hthat

k(Id − A

δ

R

δ

)f k

V

−1

β,δ

(Ω

δ

)

≤ Ckf k

L

2

_(Ω)

| ln δ|

−β

.

(19) Followingthesame al ulus,but hoosing atestfun tion

v ∈ H

1

0

(Ω)

, thesameresultas (19) holdswith

A

δ

, Ω

δ

repla ed by

A

0

, Ω

. To on ludetheproof, there onlyremains to usethestabilityestimatesofProposition3.1thatyield onstants

C > 0

independentof

δ

su hthat,forany

f ∈ L

2

_(Ω)

, wehave

kA

−1

0

(f ) − A

−1

δ

(f )k

V

1

β,δ

(Ω)

≤ kA

−1

0

(f ) − R

δ

(f )k

V

1

β,δ

(Ω)

+ kA

−1

δ

(f ) − R

δ

(f )k

V

1

β,δ

(Ω

δ

)

≤ C( kf − A

0

R

δ

(f )k

V

−1

β,δ

(Ω)

+ kf − A

δ

R

δ

(f )k

V

−1

β,δ

(Ω

δ

)

)

≤ C| ln δ|

−β

_{kf k}

L

2

_(Ω)

.



The proposition above yields onsisten y estimates for the asymptoti sour e problem. Indeed, let

f ∈ L

2

_(Ω)

refer to a xed fun tion not depending on

δ

, let

u

0

∈ H

1

0

(Ω

δ

)

satisfy

−∆u

0

= f

in

Ω

, andlet

u

δ

∈ H

1

0

(Ω

δ

)

satisfy

−∆u

δ

= f

in

Ω

δ

. Then Proposition 5.1 impliesthat

ku

δ

− u

0

k

H

1

_(Ω\U)

= O(| ln δ|

−β

)

for any neighborhood

U

of

Γ

, and for

|β| < β

⋆

.

Sharper results an be obtained in terms of the

L

2

-norm. Observethat there exists a onstant

C > 0

independent of

δ

su h that

kvk

L

2

(Ω)

≤ Ckvk

V

1

β,δ

(Ω)

for any

β ∈ R

. PluggingthisintotheestimateofProposition5.1yieldsthefollowingresult.

Corollary 5.1.

independentof

δ

su hthat

sup

f∈L

2

_(Ω)\{0}

kA

−1

0

(f ) − A

−1

δ

(f )k

L

2

_(Ω)

kf k

L

2

(Ω)

≤

c

ǫ

| ln δ|

β

⋆

−ǫ

∀δ ∈ (0, δ

0

).

Re allthat

L

2

_{(Ω) ⊂ H}

−1

_(Ω

δ

)

,and

H

1

0

(Ω

δ

) ⊂ L

2

(Ω)

,sothat

A

−1

δ

isa ontinuousoperator mapping

L

2

_(Ω)

to

L

2

_(Ω)

. Assu h,itisself-adjointand ompa tand,as anbe he kedby routineveri ations,itseigenvalueswiththoseof

A

−1

δ

onsideredasanoperatormapping

L

2

_(Ω

δ

)

to

L

2

_(Ω

δ

)

. Then, sin e

A

−1

0

: L

2

(Ω) → L

2

(Ω)

is also self-adjoint and ompa t, straightforwardappli ationofTheorem 4.10ofChapter Vof[12℄ yieldsthatthespe tra of

A

−1

δ

and

A

−1

0

are losedtoea hother. Proposition5.2.

For

β

⋆

> 0

asinProposition 3.1,andforany

ǫ ∈ (0, 2β

⋆

)

,thereexist onstants

δ

0

, c

ǫ

> 0

independentof

δ

su hthat

sup

µ∈S(A

0

)

inf

λ∈S(A

δ

)







1

µ

−

1

λ





 +

sup

µ∈S(A

δ

)

inf

λ∈S(A

0

)







1

µ

−

1

λ





 ≤

c

ǫ

| ln δ|

β

⋆

−ǫ

∀δ ∈ (0, δ

0

).

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