Remark 3.2.10. One can prove similar results for approximations by surfaces Mcn :=
(x, y, z)∈R:x∈M, y2+z2 =αn(x)2 , n∈N.
equipped with metric gn induced by the euclidian metric of R3, written in local coordinates (t, ϕn) as
gn:=dt2+αn(t)2dϕn2
.
The measures mbn will have βn(t)/2π as weight w.r.t the Lebesgue measure.
Defining now the Dirichlet form Ebn(un) :=
Z
Mcn
|Oun|2dmbn, un∈H1(Mcn,mbn),
under the assumptions that {αn}n∈N ⊂ C1(M), {βn}n∈N ⊂ C(M) with αn > 0, βn ≥k >0 on M, n∈N such thatαn&0,αn0 →0,α2nβn→1 uniformly on M we have
(i)Ebn → E in the sense of Γ-convergence;
(ii) The sequence {Ebn}n∈N is asymptotically compact.
(iii) Rbnζ →Rζ compactly for some ζ <0;
(iv)Tbtn →Tt compactly for some t >0.
3.3. THE CASE OF THEN-SPIDER
whereui :=u|ei ∈L2(Ii, ρi(x)dx).
We consider also the open sets Mn⊃
x∈R3 :d(x, M)< n1 with the euclid-ian metric and such that there exists a sequencern&0 with
Mn\B3(O, rn) =
x∈R3 :d(x, M)< 1 n
\B3(O, rn).
Additionally we shall assume in the sequel that r3n goes to 0 faster than 1/n2:
n2rn3 →0 as n→ ∞. (3.3.1)
This assumption ensures the fact that the vertex-neighborhood part ofMn enclosed in the ball B3(O, rn) shrinks to the vertex O faster than the edge-neighborhood part shrinks to the union of edges. Notice that the edge-neighborhood part of Mn consists of the union of N cylinders around interior segments of the edges. We decompose Mn = Mntube ∪(Mn \Mntube) where Mntube = SN
i=1Mn,itube is the union of thoseN cylindrical tubes of radius 1/n contained in Mn.
Let us equip the metric space Mn with a measure mn which is absolutely continuous with respect to the 3-dimensional Lebesgue measure λ3:
mn := n2
πθn·λ3|Mn.
We choose the corresponding density θn : Mn → (0,∞) to be smooth and such that (Mn,itube, mn|Mtube
n,i ) is isomorphic with (Ii×B2(0,1/n),nπ2ρi(x)dxdydz) as metric measure spaces. Suppose also that 0< a≤θn ≤b uniformly onMn.
Figure 3.2
Proposition 3.3.1. (Mn, mn)→(M, m)in the sense of measured Gromov-Hausdorff convergence.
Proof. We denote by Mn,i the full tube around the edge ei, i = 1, ..., N and by fn,i : Mn,i → Ii, i = 1, ..., N the projection that we used in subsection 3.2.1. We definefn :Mn →M the continuous projection of the whole setMn on the graphM such that fn=fn,i on Mn,itube,i= 1, ..., N.
Foru∈C0(M) we have
n→∞lim Z
Mn
u◦fn dmn= lim
n→∞
Z
Mntube
u◦fn dmn+ lim
n→∞
Z
Mn\Mntube
u◦fn dmn =
=
N
X
1=1 n→∞lim
Z
Mn,i
u◦fn,i dmn+ lim
n→∞
Z
Mn\Mntube
u◦fn dmn
−
N
X
i=1 n→∞lim
Z
Mn,i\Mn,itube
u◦fn,i dmn, but
Z
Mn\Mntube
u◦fndmn ≤ kukL∞mn(Mn\Mntube)≤ kukL∞mn(B(O, rn))
= kukL∞
n2 π
4πr3n
3 sup
B3(O,rn)
θn→0 as n → ∞ and similarly
Z
Mn,i\Mn,itube
u◦fn,i dmn→0 as n → ∞.
Therefore, by the measured Gromov-Hausdorff convergence that we have got for tubes within 3.2.1, we have
n→∞lim Z
Mn
u◦fn dmn =
N
X
i=1
Z
ei
u dm= Z
M
u dm, u∈C0(M).
We shall work in the sequel with stronger assumption on the approximating sets Mn, namely that our εn-approximations fn : Mn → M may be chosen to be Lipschitz. See that at least for Mn =
x∈R3 :d(x, M)< n1 the maps fn can be chosen Lipschitz.
Let us define the Dirichlet forms E(u) :=
N
X
i=1
Z
Ii
|u0(x)|2ρi(x) dx, for u∈C(M) with u|Ii ∈H1(Ii), ∀i (3.3.2)
3.3. THE CASE OF THEN-SPIDER
En(un) :=
Z
Mn
|∇un|2 dmn, forun ∈H1(Mn), n∈N. (3.3.3) Then we have the following convergence result
Theorem 3.3.2. (i) En→ E in the sense of Γ-convergence.
(ii) The sequence {En}n∈N is asymptotically compact.
Proof. (i) Let us consider the sequence {un}n∈N with un ∈ L2(Mn, mn) and u ∈ L2(M, m) such that un→u strongly. We have to prove that
E(u)≤lim inf
n→∞ En(un). (3.3.4)
We may suppose lim infn→∞En(un) < ∞; replacing {un}n∈N by a subsequence if necessary we may assumeun∈ D(En),n∈N. We decomposeMn =Mnδ∪(M\Mnδ), M = Mδ ∪(M \Mδ) where Mnδ := Mn \B(0, δ), M := M \B(0, δ) for δ > 0 arbitrarily fixed. Denote
Eδn(vn) : = Z
Mnδ
|∇vn|2 dmn, vn ∈H1(Mnδ) Eδ(v) : =
Z
Mδ
|∇v|2 dm, v ∈H1(Mδ)
For δ > 0 fixed and n large enough Mnδ is the disjoint union of N cylinders for which we have proved already the Γ-convergence Eδn → Eδ. Since un → u implies un|Mnδ →u|Mδ we conclude that
Eδ(u|Mδ)≤lim inf
n→∞ Eδn un|Mδ
n
≤lim inf
n→∞ En(un), δ >0. (3.3.5) Obviously u|Ii ∈ H1(Ii, dx) for each i = 1, . . . , N because u ∈ H1(Mδ) for each δ >0. If we knew that u lies in C(M) then
E(u) = Eδ(u|Mδ) +
N
X
i=1
Z
Ii\Mδ
|u0(x)|2dx
and the last term tends to 0 for δ → 0, which together with (3.3.5) yields (3.3.4).
Therefore it remains to prove u∈C(M).
Let us consider a set Mcn ⊂ Mnwith Mcn ∩Mn,itube = ∅ for i = 3, ..., N, with Mn,1tube, Mn,2tube ⊂ Mcn and such that there exist the maps Ψn : Jn → Mcn with Jn cylindrical tube of radius 1/naround the segmentJ, Ψnbijection with Ψn ∈C1(Jn), Ψ−1n ∈C1(Mcn) and Ψn bi-Lipschitz. We identify the segmentJ withe1∪e2∪{O}by the continuous bi-Lipschitz map Ψ :J →e1∪e2∪ {O}. We consider the projections fbn :Mcn →e1∪e2∪ {O},fbn:= Ψ◦ϕn◦Ψ−1n whereϕn:Jn→J the projections that we used for cylindrical tubes in 3.2.1. Because Mn,1tube, Mn,2tube ⊂ Mcn are cylindrical
Figure 3.3
tubes of radius n1, one hasfbn =fn on Mn,1tube∪Mn,2tube.Let m0n be the measure on Jn given bym0n(A) :=mn(Ψn(A)) for any Borel setA⊂Jn. Denote bym0 the measure obtained in a similar way on J: m0 =m◦Ψ.
We shall prove that un|Md
n → u|e1∪e2∪{O} strongly for the measured Gromov-Hausdorff convergence (Mcn, mn|
Mdn) → e1∪e2∪ {O}, m|e1∪e2∪{O}
with εn -appro-ximations fbn. Since un → u strongly for the measured Gromov-Hausdorff con-vergence (Mn, mn) → (M, m) with εn-approximations fn, there exists a sequence {vk}k∈N⊂C0(M) with vk→u inL2(M, m) such that
k→∞lim lim sup
n→∞
Z
Mn
|vk◦fn−un|2 dmn= 0 (3.3.6) and therefore
k→∞lim lim sup
n→∞
Z
Mdn∩Mntube
vk◦fbn−un
2
dmn
= lim
k→∞lim sup
n→∞
Z
Mdn∩Mntube
|vk◦fn−un|2 dmn= 0.
We have Z
dMn\Mntube
vk◦fbn − un|2dmn
≤ 2 Z
Mdn\Mntube
vk◦fbn
2
dmn+ 2 Z
Mdn\Mntube
|un|2 dmn
≤ 2kvkkL∞mn(Mcn\Mntube) + 2 Z
Mdn\Mntube
|un|2 dmn,
3.3. THE CASE OF THEN-SPIDER
and from (3.3.6) we deduce lim sup
n→∞
Z
Mdn\Mntube
|un|2 dmn ≤ 2 lim
k→∞lim sup
n→∞
Z
dMn\Mntube
|vk◦fn−un|2 dmn + 2 lim
k→∞lim sup
n→∞
Z
dMn\Mntube
|vk◦fn|2 dmn
≤ 2 lim
k→∞lim sup
n→∞
kvkkL∞mn(Mcn\Mntube) = 0 and consequently
k→∞lim lim sup
n→∞
Z
Mdn∩Mntube
vk◦fbn−un
2
dmn= 0.
Since {vk|e1∪e2∪{O}}k ⊂ C0(e1 ∪e2 ∪ {O}) we conclude that un|dM
n → u|e1∪e2∪{O}
strongly for the measured Gromov-Hausdorff convergence (Mcn, mn|
dMn)→ e1∪e2∪ {O}, m|e1∪e2∪{O}
withεn-approximationsfbn, which is equivalent to the strong convergenceun◦Ψn→ u◦Ψ for the measured Gromov-Hausdorff convergence (Jn, m0n)→(J, m0) with εn -approximations ϕn. Becauseun ∈ H1(Mn) we get un◦Ψn ∈H1(Jn) and from the Γ-convergence that we proved for cylindrical tubes within the subsection 3.2.1 we deduce u◦Ψ ∈ H1(J) ⊂ C(J) and thus u ∈ C(e1∪e2∪ {O}). In a similar way we proveu∈C(ei∪ej∪ {O}) for i,j = 1, ..., N, i6=j, thereforeu∈C(M), which ends the proof of (3.3.4).
In order to prove the second property (3.1.9) from the definition of the Γ-convergence we consideru∈L2(M, m) and we defineun:=u◦fn,n∈N.Obviously un→u.
Since our εn-approximations fn : Mn → M may be chosen Lipschitz, they satisfy the following two properties:
1. u∈C(M) withu|Ii ∈H1(Ii) ∀i ⇔ u◦fn∈H1(Mn)∀n (the implication
”⇐” was proved above)
2. |∇(u◦fn)|2 ≤k(u0)2◦fn ∀n with k >0 constant.
Because one inequality from (3.1.9) was proved above, it remains to show E(u)≥lim sup
n→∞
En(un) (3.3.7)
and we may suppose that E(u)< ∞ ⇔u ∈ C(M) with u|Ii ∈H1(Ii), ∀i. Accor-ding to our assumption we haveu◦fn∈H1(Mn)∀n. We know thatEMntube
n (u◦fn) = EM∩Mtube
n (u) from the cylindrical case and then lim sup
n→∞
(En(un)− E(u)) = lim sup
n→∞
EMnn\Mtube
n (un)−
N
X
i=1
Z
ei\Mntube
|u0|2dm
!
= lim sup
n→∞
Z
Mn\Mntube
|∇(u◦fn)|2dmn−
N
X
i=1
Z
ei\Mntube
|u0|2dm
!
≤klim sup
n→∞
Z
Mn\Mntube
|u0|2◦fndmn−
N
X
i=1
Z
ei\Mntube
|u0|2dm
!
= lim sup
n→∞
k Z
Mn
|u0|2◦fndmn− Z
Mntube
|u0|2◦fndmn N
X
i=1
Z
ei\Mntube
|u0|2dm
!
= lim sup
n→∞
k Z
Mn
|u0|2◦fndmn− Z
M∩Mntube
|u0|2dm−
N
X
i=1
Z
ei\Mntube
|u0|2dm
!
= 0.
(ii) Let us suppose that {un}n∈N is a sequence with un ∈ L2(Mn, mn) such that supn∈N(En(un) +kunk2L
2(Mn,mn))<∞. We have to find a subsequence{unk}k∈N
strongly convergent. We denote by Mfn,1 =fn−1(e1) which is an open subset of Mn
and we consider the maps Ψn :Jn →Mfn,1with Jnthe open cylindrical tube of radius 1/n around the segment J, Ψn bijections with Ψn ∈ C1(Jn), Ψ−1n ∈ C1(Mfn,1), Ψn
bi-Lipschitz maps. The segment e1 is identified with J by the map Ψ :J →e1. We have proved already the asymptotic compactness for cylindrical tubes. According to Lemma 3.3.3 stated below we have
sup
n∈N
EJn(un◦Ψn) +kun◦ΨnkL
2(Jn,mn)
<∞
and therefore there exists a subsequence{unk◦Ψnk}k∈Nstrongly convergent to av1 ∈ H1(J) or equivalently n
unk|
Mfn,1
o
k∈N
is strongly convergent to v1◦Ψ−1 ∈ H1(e1).
We replace now the initial sequence {un}n∈N by {unk}k∈N for the simplicity of the notation and we repeat the procedure for the edges e2 and e3, ...,eN−1 and eN, and we obtain a subsequence {unk}k∈N such that n
unk|
Mfn,i
o
k∈N
is strongly convergent to ui :=vi◦Ψ−1 ∈ H1(ei), i = 1, ..., N. We define u = ui on ei, i = 1, ..., N. For each i= 1, ..., N there exists a sequence {vij}j∈N⊂C0(ei) L2-convergent to ui such that
j→∞lim lim sup
k→∞
Z
Mfn,i
vji ◦fnk −unk|Mf
n,i
2
dmnk = 0
Then the functionsvj :=vij onei,vj(O) = 0 belong toC0(M), the sequence{vj}j∈N is L2-convergent to u and
j→∞lim lim sup
k→∞
Z
Mn
|vj◦fnk−unk|2 dmnk = 0, which proves the strong convergence of {unk}k∈N tou.
3.3. THE CASE OF THEN-SPIDER
Lemma 3.3.3. Let Qn, n ∈N be open subsets of R3 such that there exist the maps Ψn : Jn → Qn with Jn the open cylindrical tube of radius n1 around the segment J (Jn = J ×B2(0,1/n)), Ψn bijections with Ψn ∈ C1(Jn), Ψ−1n ∈ C1(Qn) and Ψn are bi-Lipschitz maps. Suppose that Jn and J are equipped with metrics and measures mn and m like in Proposition 3.2.6, Q is either a segment or the union of two segments of R3 and denote Ψ : J → Q the isometry between J and Q that preserves the singularity. We consider the maps gn :Qn →Q, gn := Ψ◦ϕn◦Ψ−1n where ϕn : Jn → J are the projections that we used in subsection 3.2.1. If we denote m0n := mn ◦ Ψn and m0 := m ◦ Ψ then (Qn, mn) → (Q, m) in the sense of measured Gromov-Haussdorf convergence with εn-approximations gn. Moreover, L2(Qn, m0n) 3un →u∈ L2(Q, m0) strongly if and only if L2(Jn, mn)3un◦Ψn→ u◦Ψ∈L2(J, m) strongly, un∈H1(Qn) if and only if un◦Ψn ∈H1(Jn) and there existsC > 0 such that
1
Ckun◦ΨnkL
2(Jn,mn)≤ kunkL
2(Qn,mn”) ≤Ckun◦ΨnkL
2(Jn,mn) (3.3.8) 1
CEJn(un◦Ψn)≤ EQn(un)≤CEJn(un◦Ψn) (3.3.9) Proof. We showed in Proposition 3.2.6 that (Jn, mn) → (J, m) in the sense of the measured Gromov-Hausdorff convergence, and our hypothesis about the maps Ψn, n ∈ N and Ψ ensures us that (Jn, mn) and (Qn, m0n), n ∈ N, (J, m) and (Q, m0) respectively have the same isomorphism classes. The fact that un ∈ H1(Qn) if and only if un ◦Ψn ∈ H1(Jn) is well-known (see for instance Proposition IX.6 in [Bre92]). The proof of inequalities (3.3.8) and (3.3.9) consists in applying the formula of changing the variables for the integrals that appear and using the fact that the partial derivatives of Ψn and Ψ−1n are bounded, since Ψn are bi-Lipschitz maps.
Corollary 3.3.4. The sequence {En}n∈N compactly converges to E.
There exists a unique self-adjoint and non-negative operatorLassociated with E, whose domain consists of thoseu∈C(M) withu|ei ∈H2(Ii), ∀i.
On an edge ei the operator L is given by Lu=−1
ρi
(ρiu0i)0. (3.3.10) Moreover,L satisfies the Kirchhoff boundary condition in the vertex O:
N
X
i=1
ρi(O)u0i(O) = 0, (3.3.11)
where the derivative is taken on each edge in the direction away from the vertex O.
Indeed, from the condition
E(u, u) =hLu, ui for any uin the domain of L we derive
N
X
i=1
Z li
0
(u0i)2ρidx=−
N
X
i=1
Z li
0
(ρiu0i)0uidx =−
N
X
i=1
(ρiu0iui)|l0i+
N
X
i=1
Z li
0
(u0i)2ρidx for any u∈C(M) with u|ei ∈ H2(Ii), ∀i. Since u is continuous in O we obtain the Kirchhoff boundary condition (3.3.11) in the vertex O, plus a Neumann boundary condition in the loose vertices u0(li) = 0 for each i.
From Theorem 3.1.10 and Corollary 3.3.4 the following result is straightfor-ward.
Corollary 3.3.5. For the corresponding strongly continuous contraction semigroups and the strongly continuous resolvents associated with E and En we have:
(i) Rnζ →Rζ compactly for some ζ <0;
(ii) Ttn→Tt compactly for some t >0.
Our approximating domains Mn don’t necessarily have a smooth boundary, but they are at least Lipschitz, in the sense that locally, ∂Mn can be written as the (euclidian) graph of a Lipschitz function withMnlying on one side of the graph. On such domains a Rellich compact embedding theorem still holds (see [Ros98]). Since the Rellich compact embedding theorem gives the compactness of the resolvent for bounded domains, from Theorem 3.1.10 we obtain also the convergence of spectra of the associated generators Ln:
Corollary 3.3.6. The kth eigenvalue of Ln converges to the kth eigenvalue of L as n → ∞ for any k.
Remark 3.3.7. The convergence of spectra of the Neumann Laplacian on graph-like compact manifolds has been treated quite extensively in the paper [EP05]. They analyze there graph-like manifolds that around the edges behave like cylindrical neighborhoods with weights and three different cases of vertex-neighborhoods, that produce different operators in the limit. The limit operator on the graph depends on whether the vertex neighborhood decays (in volume) faster, slower, or at the same rate with the edge-neighborhood.
Our study considered open subsets ofR3, a more divers class of edge-neighbor-hoods and a decay of the volume of the vertex-neighborhood that should be faster than the one of the edge-neighborhoods. Besides, the Kuwae-Shioya approach
3.3. THE CASE OF THEN-SPIDER
gives the convergence of the whole structure heat kernel-Dirichlet form-resolvent-semigroup on our open domain towards the one on the graph. The other two cases that [EP05] solved give in the limit some operators that are not defined onL2(M, m), but on a more general Hilbert space that containsL2(M, m) as a subspace. There-fore, we cannot expect that the Kuwae-Shioya theory in its actual form could handle those two cases too.
In fact, the convergence of spectra has been investigated intensively in the last years, also for boundary conditions other than Neumann. Mixed boundary condi-tions for the approximating sequence of manifolds have been considered in [Po05], [Gr07]. In [Po05] for instance the main result states the convergence of the spectra of a family of approximating open sets from R2 with small vertex neighborhoods and with a mixed boundary condition towards the spectrum of the Laplacian on the graph with Dirichlet boundary condition, which is actually a graph operator without coupling between edges. The paper [Po06] studies the approximations with non-compact manifolds and in the Neumann case gives, besides the convergence of spectra, the norm convergence of resolvents.
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Acknowledgements
The following is my heartfelt appreciation to those people, both past and present, who gave me spirit and encouragement to start and complete this thesis:
I would like to thank in the first place to my advisor Prof. Dr. Karl-Theodor Sturm for giving me the chance to perform my PhD in Bonn, for his scientific support, for constructive criticism, for his patience and advice.
I am grateful and indebted to Prof. Dr. Lucian Beznea, for all the help and for his guidance starting with the university years and lasting throughout the time I spent at my home institute in Romania.
My thanks go also to my former colleagues in the research team here in Bonn:
Kathrin Bacher, Tom Christiansen, Atle Hahn, Martin Hesse, Ryad Husseini, Nico-las Juillet, Gustav Paulik, Robert Philipowski, Max (’Kostja’) von Renesse, Ann-Kathrin R¨ower, Hendrik Weber for interesting discussions and for a nice working environment.
Special thanks go to my family in Romania, to my son, to my husband and to my mother for their moral support and understanding through those difficult times when I was away, for their unconditional love.
Finally, I wish to thank all my friends whose names are not mentioned here, they all know who they are.