The limiting equation for Neumann Laplacians on shrinking domains ∗

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The limiting equation for Neumann Laplacians on shrinking domains ^∗

Yoshimi Saito

Abstract

Let {Ω}0<≤1 be an indexed family of connected open sets in R², that shrinks to a tree Γ asapproaches zero. LetHΩ be the Neumann Laplacian and f be the restriction of an L²(Ω₁) function to Ω. For z ∈ C\[0,∞), set u = (HΩ−z)⁻¹f. Under the assumption that all the edges of Γ are line segments, and some additional conditions on Ω, we show that the limit function u0 = lim_→0u satisfies a second-order ordinary differential equation on Γ with Kirchhoff boundary conditions on each vertex of Γ.

1 Introduction

Let Ω be a connected open set in R². Consider a family of the Neumann LaplaciansHΩ, 0< ≤1, on the sub-domain Ω such that{Ω} shrinks to Γ in the sense that

Ω = Ω₁⊃Ω₂⊃Ω₁ (1> 2> 1>0), (1.1) lim_→0Ω= Γ,

where the bar over a set means the closure of the set. We continue here the study started in [11] regarding the following question: In what sense does the operator HΩconverges to an operator on Γ as→0? That is, we try to find conditions under which, given a thin domain and an operator on the domain, an operator on an imbedded tree or network gives a good approximation of the operator on the domain. This investigation is part of the general question on replacing the study of a thin domain by the study of an imbedded tree or network, which has been proposed in many branches of science such as physics and chemistry. For references on this problem, see for example Ruedenberg-Scherr [9], Exner-Seba [5], Kuchment [7], Schatzman [12], Rubinstein-Schatzman [10], and Kuchment- Zeng [8]. In [12], a family of “fattened” domains Ω of a C² manifold M are considered. It is shown that thek-th eigenvalueλk() of the Neumann Laplacian

∗Mathematics Subject Classifications: 35J05, 35Q99.

Key words and phrases: Neumann Laplacian, tree, shrinking domains.

c2000 Southwest Texas State University and University of North Texas.

Submitted March 9, 2000. Published April 26, 2000.

Partially supported by Deutche Forschungs Gemeinschaft grant SFB 359.

1

HΩ on Ω converges asymptotically to the k-th eigenvalue λk of the Laplace- Beltrami operator of M. In [10] the above results are extended to the case where the manifoldM is replaced by a graphG; see for example [8], and for a simplified proof [10].

In [11], we discussed the convergence of the resolvent (HΩ −z)⁻¹ as → 0. After introducing the Hilbert spaces L2(Γ) and H¹(Γ) on the tree Γ and defining the selfadjoint “Neumann Laplacian” operatorHΓinL²(Γ) as in [4], we presented a set of general conditions under which the the resolvent (HΩ−z)⁻¹f

converges as as → 0, where f is the restriction of a function f ∈ L²(Ω) to Ω([11], Theorem 4.5). Also in [11], we studied the case where Ω is a bounded convex set and the tree Γ is a straight line segment (ridge).

Let Ω be a bounded convex set inR² and suppose that Ω and Ω are given by

Ω ={x= (x1, x2) :−`−(x1)< x2< `+(x1), a < x1< b},

Ω={x= (x1, x2) :−`−(x1)< x2< `+(x1), a < x1< b}, (1.2) Γ ={x= (x1,0) :a≤x1≤b},

where −∞ < a < b <∞, 0 < ≤1 and `±(t) are positive C¹ functions on [a, b]. Then we have, forf ∈H¹(Ω) andz∈C\[0,∞),

→0limγ

(HΩ−z)⁻¹f

= (HΓ−z)⁻¹(γf), (1.3)

in a weighted Hilbert spaceL²_a0(Γ), whereHΓ is the “Neumann Laplacian” on Γ defined in [4] (see§2), γis the trace operator on Γ, and

L²_a0(Γ) =L²(Γ;a0(σ)dσ), (1.4) a0(σ) =`−(σ) +`+(σ)

([11], Theorem 5.5).

In this work, we consider the case when Γ is a tree such that all the edges are line segments (Assumption 4.2, (i)). Suppose that the family{Ω} is given by

Ω={(σ, s) :−`<sub>−</sub>(σ)< s < `+(σ), σ∈Γ}, (1.5) whereσis the arc length along the edges of Γ andsis the arc length along the curveCσ=τ⁻¹(σ),τbeing a map from Ω into Ω∩Γ which is Lipschitz contin- uous almost everywhere in Ω (see§2). Then we assume that, forσbelonging to the edge ej of Γ, the curveCσ is perpendicular to the edge near ej except its vertices (Assumption 4.2, (ii)). Set

u(x) =u(σ, s) = (HΩ−z)⁻¹f, (1.6) wheref ∈H¹(Ω)∩C¹(Ω). Then there exists a subsequence{uk}^∞_k=1 such that {uk(σ,0)}, the restriction ofukon the tree Γ, converges tou0weakly inL²_a0(Γ) ask→ ∞, and u0 satisfies the equation

−a⁻¹₀ (σ) d

dσ(a0(σ)u⁰)−zu=f(σ,0) (1.7)

on each edge with the Kirchhoff boundary condition at each vertex (Theorems 4.5 and 4.8), wherea0 is given by (1.4) andu⁰ means the derivative ofu0 with respect to the arc lengthσ along the edge.

In§2, after introducing the tree Γ imbedded in the open connected set Ω, we discuss the change of variablesx= (x1, x2) →(σ, s). In §3 some estimates of u(σ,0) are given. These estimates will be used to guarantee the weak conver- gence of{uk}^∞_k=1. §4 is devoted to showing the above convergence of{uk}^∞_k=1 to a solutionu0 of the equation (1.6) (Theorems 4.5 and 4.8). The main tools are Lemmas 4.1 and 4.4 whose proof will be given in§6. We shall discuss the continuity of the limiting functionu0at each vertex in§5.

2 Preliminaries

In this section we are going to introduce a domain Ω inR², a tree Γ contained in Ω and a family{Ω}0<≤1 of sub-domains of Ω.

Let Ω be a domain (i.e., a connected open set) inR². Let Γ⊂Ω be a tree, that is, a connected graph without loops or cycles, where Γ is the closure of Γ.

Its edgesej, j ∈J, are non-degenerate open curve such that the closureej is a smooth curve, whereJ is an index set. The endpointsej\ej are the vertices.

Here we should note that we allow these edges to be smooth curves, not just line segments. We shall assume that Γ has, at most, a countably infinite number of edges, and hence the index setJ is a subset of the natural numbersN. We also assume that each vertex of Γ is offinite degree, that is, only a finite number of edges emanate from each vertex, and that only one edge emanates from a vertex c if c belongs to the boundary ∂Ω of Ω. For every x, y ∈ Γ there is a unique path in Γ joiningxand y. Thus, by introducing the distance betweenxandy by the length of a unique path connectingxand y, Γ becomes a metric space.

Also, if Γ is endowed with the natural one-dimensional Lebesgue measure, it is aσ-finite measure space. The tree Γ is rooted at an arbitrary fixed pointa∈Γ.

We definet_ax(or equivalentlyx_at) to mean thatxlies on the path from ato t.

Throughout this work we assume the following: (I)Assumptions on Ω and Γ:

(1-i) Ω be a domain (i.e., a connected open set) inR²and Γ⊂Ω be a connected tree which has at most countable number of edgesej,j ∈ J, where Ω is the closure of Ω. Each edgeej is an open curve with finite length such that the closure ej is a C² curve. The endpoints ej\ej are called the vertices. When any two edges are connected, they are connected only at their vertices. Also they are not tangential at the vertex from which the two edges emanate.

(1-ii) We haveE(Γ)⊂Ω, where E(Γ) is the set of all edges of Γ.

(1-iii) Forv ∈V(Γ)∩∂Ω, only one edge emanates from v, where V(Γ) is the set of all vertices of Γ, and∂Ω is the boundary of Ω.

(II)Assumptions onτ. There exists a mapτ from Ω into Ω∩Γ which satisfies the following:

(2-i) The subsetτ⁻¹(V(Γ)) is a (2-dimensional) null set. For eachej ∈E(Γ), set Ω_j = τ⁻¹(ej). Then Ω_j is an open set and τ is locally Lipschitz continuous on Ω_j, that is, for each x ∈ Ω_j there exists a neighborhood V(x)⊂Ω_j ofxand a positive constantγ(x) such that for ally∈V(x)

dΓ(τ(x), τ(y))≤γ(x)|x−y|, (2.1) wheredΓ denotes the metric on Γ and| · | the Euclidean metric (for defi- niteness) onR².

(2-ii) LetC(t) =τ⁻¹(t) fort∈E(Γ). ThenC(t) is a rectifiable curve. Further, C(t)∩Γ = {t} and C(t)\{t} has two components, C±(t) say. Also we assume that C(t) is not tangential to Γ at t. Let C+(t) and C−(t) be parameterized by arc lengthswhich is measured fromtwith 0≤s≤`+(t) onC+(t) and−`₋(t)≤s≤0 onC−(t). Letτ(x) = (τ1(x), τ2(x))∈Γ for x∈Ω. Then, fort∈E(Γ), there exists a null sete(t)⊂C(t) with respect to ds, the measure induced by the arc length parameterson C(t), such thatτ1 andτ2 are differentiable atx∈C(t)\e(t).

(2-iii) Let |∇τ(x)| = [|∇τ1(x)|² +|∇τ2(x)|²]^1/2. For t ∈ E(Γ) fixed, define

|∇τ(s)| on C(t) by |∇τ(s)| = |∇τ(x)| with x ∈ C(t) and d_C(t)(t, x) = s, where dC(t)(t, x) is the distance between t and x along C(t). Then

|∇τ(s)|,|∇τ(s)|⁻¹∈L¹(C(t), ds).

(2-iv) For any vertexv∈Ω the functions`± are bounded below from 0 around v, i.e., for a vertexv∈Ω, there exists a neighborhoodU(v)⊂Γ ofvsuch that

inf_t∈U(v)\{v}`−(t)>0, (2.2) inf<sub>t∈U(v)\{v}</sub>`+(t)>0.

Some examples of the triples (Ω,Γ, τ) are given in [2, 3, 4, 11] including horn- shaped domains, room and passages domains and fractal domains.

For j ∈ J let the edgeej have the verticesaj and bj such that bj a aj, where the tree Γ is rooted ata. Then we parameterizeej byσj(t) =dist(aj, t), wheredist(aj, t) is the arc length fromaj to t∈ej alongej. From now on we may drop the subscript j in σj if there is no danger of misunderstanding. If x= (x1, x2)∈Ω_j=τ⁻¹(ej)⊂Ω is such thatτ(x) =t(σ) anddist(t(σ), x) =s, where dist(t(σ), x) is the distance between x and t(σ) along the curveCt(σ), then a co-ordinate system on Ω_j is defined by

x=x(σ, s), τ(x) =t(σ), s∈(−`−(σ), `+(σ)), (2.3) where`±(σ) =`±(t(σ)). A family{Ω}_0<≤1of sub-domains of Ω is defined as follows:

Definition Forj∈J and 0< ≤1 let

Ω⁽⁾_j ={x=x(σ, s)/ σ∈ej, −`−(σ)< s < `+(σ)} (2.4) and

Ω=

∪j∈JΩ⁽⁾_{j ◦}

, (2.5)

whereA^◦ is the interior ofA. By definition we have Ω = Ω₁.

Definition For each 0< ≤1 letHΩ be the Neumann Laplacian on Ω. It is known ((2.4) in [4]) that

∂(x1, x2)

∂(σ, s) = 1

|∇τ(σ, s)|. (2.6)

LetI∈ej. Then we have, forf ∈L¹(τ⁻¹(I)), Z

τ⁻¹(I)f(x)dx= Z

I dσ

Z _`+(σ)

−`−(σ)f(σ, s)|∇τ(σ, s)|⁻¹ds. (2.7) Note that we have again simplified the notation by writing (σ, s) forx(σ, s). Of particular importance is the case whenf =F◦τ in (2.7) with F∈L¹(I):

Z

τ⁻¹(I)F◦τ(x)dx = Z

I

F(σ)dσ

Z _`+(σ)

−`−(σ)|∇τ(σ, s)|⁻¹ds (2.8)

=:

Z

I

F(σ)α(σ)dσ, where

α(σ) :=

Z _`+(σ)

−`−(σ)

1

|∇τ(σ, s)|ds. (2.9)

IfI=ej in (2.7) and (2.8), thenτ⁻¹(I) should be replaced by Ω⁽⁾_j .

3 Evaluation of u = (H

− z)

f on Γ

We shall start with an additional assumption on the tree Γ and the family {Ω}_0<≤1. Then we shall show some evaluation for the restriction of

u=u(f, z)) = (HΩ−z)⁻¹f (3.1) on Γ, wherez∈C\[0,∞) andf is the restriction off ∈L²(Ω) on Ω.

Assumption 3.1. (i) For eachj ∈J `±(σ) are positiveC<sup>1</sup>function onejand are continuously extended onej. Also`±(σ) satisfy

sup_ej(`−(σ) +`+(σ))≡Lj<∞, (3.2) sup_ej(|`<sup>0</sup><sub>−</sub>(σ)|+|`⁰₊(σ)|)≡Rj<∞,

where`<sup>0</sup><sub>±</sub>(σ) = <sub>dσ</sub><sup>d</sup> `±(σ).

(ii) Forj ∈ J there exists j ∈(0,1] such that |∇τ(σ, s)| is continuous on Ω⁽_j^j),

0< mj ≡inf

x(σ,s)∈Ω⁽_j^j)|∇τ(σ, s)| ≤sup

x(σ,s)∈Ω⁽_j^j)|∇τ(σ, s)| ≡Mj<∞,(3.3) supx(σ,s)∈Ω⁽_j^j)∂x(σ,s)

∂s ≡Kj <∞.

Now we introduce a positive function on eachejwhich will play an important role.

Definition 3.2. For eachj∈J, set

a^(j)₀ (σ) = `−(σ) +`+(σ)

|∇τ(σ,0)| (σ∈ej). (3.4) Note that a^(j)₀ is a bounded positive function onej. From now on we may drop the superscriptjina^(j)₀ if there is no risk of misunderstanding, i.e. a0(σ) = a^(j)₀ (σ). Also note that

a0(σ) = lim

→0⁻¹α(σ) (σ∈ej), (3.5) whereα is given by (2.9).

For a subset Ω⁰ of Ω,kfk_Ω⁰ denotes theL² norm off on Ω⁰. Also we set kψk²_ej,ao =R

ej|ψ(σ)|²a0(σ)dσ, (3.6) kψk²_ej =R

ej|ψ(σ)|²dσ, Lemma 3.3. We have

ku(·,0)k²_ej,a0≤2 Mj

mj

L²_jK_j²k∇uk²_Ω()

j +⁻¹kuk²_Ω()

j , (3.7)

and

kuk²_Ω()

j ≤2 Mj

mj

L²jKj²k∇uk²_Ω()

j +ku(·,0)k²_ej,a0 (3.8) foru∈H¹(Ω⁽⁾_j )∩C¹(Ω⁽⁾_j ), wherek kA,A⊂R², is the norm of L²(A).

Proof. (I) LetI be a closed subset of ej. SinceI⊂ej ⊂Ω⁽⁾_j , u(σ,0) and u⁰(σ,0) are bounded on I. Then we have

ku(·,0)k²_I,a0 = Z

I

⁻¹|∇τ(σ,0)|⁻¹

Z _`+(σ)

−`−(σ)|u(σ,0)|²ds dσ

≤ 2⁻¹m⁻¹_j Z

I

Z _`+(σ)

−`−(σ)|u(σ,0)−u(σ, s)|²ds dσ +2⁻¹m⁻¹_j

Z

I

Z _`+(σ)

−`−(σ)|u(σ, s)|²ds dσ (3.9)

≡ 2⁻¹m⁻¹_j (I1+I2). Using the second inequality of (3.3), wee see that

∂u

∂s≤Kj|∇u|, (3.10)

which is combined with (2.7) to give I1 ≤

Z

I

Z _`+(σ)

−`−(σ)

Z _s

0

∂u

∂s(σ, η)dη²ds dσ

≤ Z

I

Z _`+(σ)

−`−(σ)

Z _`+(σ)

−`−(σ)

∂u

∂s(σ, η)dη₂ ds dσ

≤ Lj Z

I

Z _`+(σ)

−`−(σ)

∂u

∂s(σ, s)ds₂

dσ (3.11)

≤ (Lj)² Z

I

Z _`+(σ)

−`−(σ)

∂u

∂s(σ, s)²ds dσ

≤ (Lj)²Kj²Mj

Z

I

Z _`+(σ)

−`−(σ)|∇u(σ, s)|² ds dσ

|∇τ(σ, s)|

≤ (Lj)²K_j²Mjk∇uk²_Ω() j ,

where we have used the fact thatτ⁻¹(I)⊂Ω⁽⁾_j . As forI2 we have I2≤Mj

Z

I

Z _`+(σ)

−`−(σ)|u(σ, s)|² ds dσ

|∇τ(σ, s)| ≤Mjkuk²_Ω()

j . (3.12) Thus we have from (3.11) and (3.12)

ku(·,0)k²_I,a0 ≤2 Mj

mj

L²_jK_j²k∇uk²_Ω()

j +⁻¹kuk²_Ω()

j . (3.13)

SinceI⊂ej is arbitrary, (3.7) follows from (3.13).

(II) As in (I), we have kuk²_Ω()

j ≤ 2

Z

ej

Z _`+(σ)

−`−(σ)|u(σ, s)−u(σ,0)|²|∇τ(σ, s)|⁻¹ds dσ +2

Z

ej

Z _`+(σ)

−`−(σ)|u(σ,0)|²|∇τ(σ, s)|⁻¹ds dσ (3.14)

≡ 2(J1+J2).

Then we can proceed as in evaluationIj to obtain J1 ≤ Mj

mj

L²_jK_j²²k∇uk²_Ω()

j , (3.15)

J2 ≤ Mj

mj

ku(·,0)k²_ej,a0, (3.16)

which completes the proof. ♦

Proposition 3.4. Suppose that Assumptions2.1and3.1hold. Letf ∈H¹(Ω).

Set

u=u(f, z) = (HΩ−z)⁻¹f, (3.17) wherez∈C\[0,∞)andf is the restriction off onΩ⁽⁾_j . Suppose that

lim sup

→0

X

k∈J

Mk

mk

L²_kK_k²k∇fk²_Ωk+kf(·,0)k²_ek,a0 <∞, (3.18)

wheref(σ,0) on each edge ej is given by the trace off on ej, Then, for suffi- ciently small ∈(0,1]andj∈J,

ku(·,0)k²_ej,a0 ≤ 4 Mj

mj

|z|⁻¹h

L²_jK_j²kfk²_Ω (3.19) +|z|⁻¹X

k∈J

Mk

mk

L²_kK_k²k∇fk²_Ω

k+kf(·,0)k²_ek,a0 i .

Remark 3.5 It has been known (see,e.g., Gilbarg-Trudinger [6], Theorem 8.10) that the conditionf ∈H¹(Ω) implies u ∈ H³(Ω)_loc, since u ∈H¹(Ω).

Then, by the Sobolev imbedding theorem (see, e.g. Adams [1], Theorem 5.4), we haveu∈C¹(Ω).

Proof of Proposition 3.4. (I) It is easy to see that kuk²_Ω()

j ≤ kuk²_Ω≤ |z|⁻²kfk²_Ω, (3.20) k∇uk²_Ω()

j ≤ k∇uk²_Ω ≤ |z|kuk²_Ω+kfkΩkukΩ≤2|z|⁻¹kfk²_Ω.

(II) It follows from (3.20) and (3.7) withureplaced byu that ku(·,0)k²_ej,a0≤2 Mj

mj

2L²_jK_j²|z|⁻¹kfk²_Ω()

j +⁻¹|z|⁻²kfk²_Ω()

j . (3.21) The inequality (3.19) is obtained from (3.21) and (3.8) withureplaced byf. ♦

4 The limiting equation

Letu= (HΩ−z)⁻¹fbe as in (3.17). Sinceku(·,0)kej,a0is uniformly bounded for∈(0,1] by Proposition 3.4 iff ∈H¹(Ω) satisfies (3.18),uconverges weakly along some subsequence{m}^∞_m=1with the limiting functionu0. In this section we shall prove thatu0is a solution of a second-order ordinary differential equa- tion on Γ with the Kirchhoff boundary condition on each vertex (Theorems 4.5 and 4.8). The equation is independent from choice of the subsequence{m}^∞_m=1. First we shall state two lemmas (Lemmas 4.1 and 4.4) which will play crucial roles in this section. These lemmas will be shown in §6. In order to prove Lemma 4.4, we need another important assumption (Assumption 4.2). Let Γ₀ be a measurable subset of the tree Γ and letabe a positive measurable function defined on Γ₀∩∪_j∈Jej. Then the Hilbert spaceL²(Γ₀, a) is a weightedL²space with inner product

(F, G)_Γ0,a=X

j∈J

Z

Γ0∩ej

F(σ)G(σ)a(σ)dσ (4.1)

and normkFkΓ0 = [(F, F)_Γ0,a]^1/2. We denoteL²(Γ₀,1) byL²(Γ₀).

Lemma 4.1. Suppose that Assumptions 2.1and 3.1 are satisfied. Let j ∈J and let {u}0<≤1 be a family of functions such that

u∈H¹(Ω⁽⁾_j )\

C¹(Ω⁽⁾_j ) (0< ≤1). (4.2) Let F ∈L²(ej, a0), i.e., kFkej,a0 <∞. Setv(x) =F(τ(x)). Then there exists Cj =Cj(Mj, mj, Lj, Kj), a positive constant depending only onMj, mj, Lj, Kj, such that

1 Z

Ω⁽⁾_j u(x)v(x)dx− Z

ej

u(σ,0)F(σ)a0(σ)dσ (4.3)

≤ Cj

√k∇uk_Ω()

j kFkej,a0+ku(·, s)kej,a0

Z

ej

|F(σ)|²ψ(σ)²a0(σ)dσ_1/2 , where

ψ(σ) = 1 a0(σ)

Z _`+(σ)

−`−(σ)(|∇τ(σ, s)|⁻¹− |∇τ(σ,0)|⁻¹)ds. (4.4) Here we need another assumption.

Assumption 4.2. (i) All the edgesejof the tree Γ are finite (non-degenerate) line segments. (ii) Letj ∈ J. Let E be a closed subset of the (open) edge ej. Then there exists a positive numberj(E)∈(0,1], depending only onjandE, such that, for anyt∈E, the portion of the curveCt∩Ω⁽_j^j(E)) is a line segment which isperpendicularto ej.

Remark 4.3. (i) Roughly speaking, (ii) of the above assumption claims that the curveCtis perpendicular toej nearej except the vertices. (ii) Assumption 4.2, (ii) also implies

|∇τ(σ, s)|= 1 ((σ, s)∈τ⁻¹(E)∩Ω⁽_j^j(E))),

|∇τ(σ,0)|= 1 (σ∈ej), (4.5) a0(σ) =`−(σ) +`+(σ) (σ∈ej).

Lemma 4.4. Suppose thatAssumptions 2,1, 3.1 and4.2 hold. Let j∈J and let{u}0<≤1 be a family of functions such that

u∈H¹(Ω⁽⁾_j )\

C¹(Ω⁽⁾_j ) (0< ≤1). (4.6) Let F ∈C²(ej)with F⁰ ∈C₀¹(ej), where F⁰ is the derivative of F with respect toσ. Let 0=j(supp F⁰). Then, by setting v(x) =F(τ(x)), the inequality

1 Z

Ω⁽⁾_j ∇u· ∇v dx− Z

ej

∂u

∂σ(σ,0)F⁰(σ)a0(σ)dσ

≤ √

C(Lj, Rj)(kF⁰kej,a0+kF⁰⁰kej,a0)k∇uk_Ω()

j (4.7)

holds for ∈(0, 0), whereC(Lj, Rj)is a positive constant depending only on Lj andRj.

Theorem 4.5. Suppose that Assumptions2,1,3.1and4.2hold. Letf ∈H¹(Ω) which satisfies (3.17). Let f be the restriction of f on Ω⁽⁾_j for ∈(0,1]. Let u= (HΩ−z)⁻¹fbe as in(3.16). Letj∈J. Let{m}^∞_m=1⊂(0,1]be a decreas- ing sequence such that{m}^∞_m=1converges to0and the sequence{um(·,0)}^∞_m=1 converges weakly inL²(ej, a0). Then the limit function u0 satisfies

Z

ej

u0(σ)

−(a0(σ)F⁰(σ))⁰−zF(σ)a0(σ)−f(σ,0)F(σ)a0(σ) dσ= 0 (4.8)

for anyF ∈C₀²(ej), i.e., u0 is a weak solution of the equation

−1

a0(a0u⁰)⁰−zu=f(·,0) (4.9) onej.

Remark 4.6. (i) The sequence{m}^∞_m=1 which satisfies the conditions in the above theorem does exist sinceku(·,0)kej,a0 is uniformly bounded for∈(0,1]

by Proposition 3.4. (ii) Thus, the limit functionu0is, not only a weak solution of (4.9), but also a strong solution withu0∈C²(ej).

Proof of Theorem 4.5. (I) Let v(x) = F(τ(x)). We extend F on Γ by settingF = 0 outsideej. Then we havev ∈H¹(Ω) andv= 0 outside Ω⁽⁾_j . Let 0 be as in Lemma 4.1. Note that ψ(σ) = 0 onej for∈(0, 0], whereψ(σ) is given by (4.4). Therefore, replacingu by zu in Lemma 4.1 and using the second inequality in (3.20), we obtain

1

Z

Ω⁽⁾_j zu(x)v(x)dx− Z

ej

zu(σ,0)F(σ)a0(σ)dσ

≤ Cj|z|√

k∇uk_Ω()

j kFkej,a0 (4.10)

≤ 2Cj

√kfkΩkFkej,a0

for∈(0, 0], which implies that 1

Z

Ω⁽⁾_j zu(x)v(x)dx= Z

ej

zu(σ,0)F(σ)a0(σ)dσ+O(√

) (→0), (4.11) whenF∈C₀²(ej) is fixed. Next we setu=f in Lemma 4.1 to obtain

1 Z

Ω⁽⁾_j f(x)v(x)dx= Z

ej

f(σ,0)F(σ)a0(σ)dσ+O(√

) (→0), (4.12) (II) Similarly we have from Lemma 4.4

1

Z

Ω⁽⁾_j ∇u(x)· ∇v(x)dx= Z

ej

∂u

∂σ (σ,0)F⁰(σ)a0(σ)dσ+O(√

) (→0), (4.13) (III) It follows from (4.11), (4.12) and (4.13) that

1 Z

Ω⁽⁾_j

∇u· ∇v−zuv−fv dx

= Z

ej

∂u

∂σ(σ,0)F⁰(σ)a0(σ)−zu(σ,0)F(σ)a0(σ) (4.14)

−f(σ,0)F(σ)a0(σ) dσ+O(√ ).

Noting that the domain of integration in the left-hand side of (4.13) can be extended to Ωand thatv∈H¹(Ω), by the definition of the Neumann Laplacian HΩ

0 = 1

Z

Ω

∇u· ∇v−zuv−fv dx

= Z

ej

∂u

∂σ(σ,0)F⁰(σ)a0(σ)−zu(σ,0)F(σ)a0(σ) (4.15)

−f(σ,0)F(σ)a0(σ) dσ+O(√ ).

Thus, by using partial integration, we have 0 =

Z

ej

u(σ)

−(a0(σ)F⁰(σ))⁰−zF(σ)a0(σ)−f(σ,0)F(σ)a0(σ) dσ+O(√ ). (4.16) Set=min (4.15) and letm→ ∞. Then we have (4.7), where we should note that, sincea0is positive and bounded below from zero on any closed subset of, um(·,0) converges tou0 weakly inL²(I) as well as inL²(I, a0) with any closed subsetI ofej. This completes the proof. ♦

Corollary 4.7. Let u0 be as inTheorem 4.5. Then following limits exist

σ→alimj+0a0(σ)u⁰₀(σ), lim

σ→bj−0a0(σ)u⁰₀(σ). Proof. The proof is obvious. Withσ, σ0∈ej we have

a0(σ)u⁰₀(σ) =− Z _σ

σ0

zu(η,0)−f(η,0)

dη+a0(σ0)u⁰₀(σ0) (4.17)

♦Noting that Γ consists of at most countably infinite edges, we may assume that there exists a sequence{m}^∞_m=1 such that there existsu0∈L²(Γ, a0)_locsuch that

m→0 (m→ ∞), (4.18)

um(·,0)→u0 inL²(ej) (j∈J),

Theorem 4.8. Suppose that Assumptions 2,1, 3.1 and 4.2 hold. Let um = (HΩ−z)⁻¹fm be as in(4.18),wheref is as inTheorem 4.5. Letc be a vertex ofΓ and set

J(c) ={j∈J :aj =c or bj=c}, (4.19) whereaj andbj are the endpoints of ej withbjaaj. Then it follows that

X

j∈J(c)

η(j)a0(c)u⁰₀(c) = 0, (4.20)

i.e., The Kirchhoff boundary condition is satisfied at each vertex ofΓ, where η(j) =

1 ifc=bj,

−1 ifc=aj, (4.21)

and

a0(c)u⁰₀(c) =

lim_{σ→bj,σ∈ej}a0(σ)u⁰₀(σ) ifη(j) = 1,

lim_{σ→aj,σ∈ej}a0(σ)u⁰₀(σ) ifη(j) =−1. (4.22)

Proof. (I) LetF be a function defined on Γ such that suppF ⊂(∪_j∈J(c)ej)∪ {c},

F = 1 in a neighborhood ofc, (4.23) F isC² on eachej withj∈J(c).

LetFj be the restriction of F onj. Then Fj ∈C²(Γ_j) andF_j⁰ ∈C₀¹(ej). Set vj(x) =Fj(τ(x)). Then, using Lemmas 4.1 and 4.4, and proceeding as in the proof of Theorem 4.5, we have

1

Z

Ω⁽⁾_j

∇u· ∇v_j−zuvj−fvj dx

= Z

ej

∂u

∂σ(σ,0)Fj(σ)a0(σ)−zu(σ,0)Fj(σ)a0(σ) (4.24)

−f(σ,0)Fj(σ)a0(σ) dσ+O(√ ).

By summing up both sides of (4.24) with respect toj ∈J(c), which is a finite set since Γ is of finite degree, we obtain, as in (4.14),

0 = X

j∈J(c)

Z

ej

∂u

∂σ(σ,0)F⁰(σ)a0(σ)−zu(σ,0)F(σ)a0(σ) (4.25)

= −f(σ,0)F(σ)a0(σ) dσ+O(√ ),

where, and in the sequel, we shall use F(σ) in place of Fj. Here, by partial integration,

Z

ej

∂u

∂σ (σ,0)F⁰(σ)a0(σ)dσ=− Z

ej

u(σ,0)(a0(σ)F⁰(σ))⁰dσ, (4.26) where we should note thatF⁰ has a compact support inej. Combine (4.25) and (4.26) and let→0 alongm to give

0 = X

j∈J(c)

Z

ej

−u0(σ,0)(F⁰(σ)a0(σ))⁰−zu0(σ,0)F(σ)a0(σ) (4.27)

−f(σ,0)F(σ)a0(σ) dσ+O(√ ).

(II) Suppose that c = aj. Then, repeating partial integration, and noting thatF = 1 nearc, we obtain

− Z

ej

u0(σ,0)(F⁰(σ)a0(σ))⁰dσ

= Z

ej

u⁰₀(σ,0)F⁰(σ)a0(σ)dσ

= lim

σ→aj

Z _bj

σ u⁰₀(σ)F⁰(σ)a0(σ)dσ (4.28)

= − Z

ej

(a0(σ)u⁰₀(σ,0))⁰F⁰(σ)dσ−a0(c)u⁰₀(c)

= −

Z

ej

(a0(σ)u⁰₀(σ,0))⁰F⁰(σ)dσ+η(j)a0(c)u⁰₀(c) Similarly, we have, forc=bj,

− Z

ej

u0(σ,0)(F⁰(σ)a0(σ))⁰dσ (4.29)

= −

Z

ej

(a0(σ)u⁰₀(σ,0))⁰F⁰(σ)dσ+η(j)a0(c)u⁰₀(c), where we should note thatu0∈C²(ej) ((ii) of Remark 4.6).

(III) It follows from (4.26), (4.27) and (2.28) that

0 = X

j∈J(c)

Z

ej

−(a0(σ)u⁰₀(σ,0))⁰−zu0(σ,0)a0(σ,0) (4.30)

−f(σ,0)a0(σ,0) F(σ)dσ+ X

j∈J(c)

η(j)a0(c)u⁰₀(c)

Sinceu0is now a strong solution of the equation (4.9), the first term of the left- hand side of (4.30) is zero, and hence the Kirchhoff boundary condition (4.20)

follows from (4.30). ♦

5 Continuity of the limit function

Letu0 be a limit function on Γ given by (4.17). Since u0 is a solution of the differential equation on eachej, j ∈J, u0 is smooth on eachej (Remark 4.6, (ii)). In this section, we shall show, under some additional conditions, that{u} converges to u0 in stronger senses, and thatu0 is continuous at the vertices of Γ.

The proof of the following proposition will be given in§6.

Proposition 5.1. Suppose that Assumptions 2,1, 3.1and 4.2hold. Let u = (HΩ −z)⁻¹f, where z ∈ C\[0,∞), f ∈ H¹(Ω), and f is the restriction of f on Ω. Let f satisfy (3.18) Then there exists a positive constants Cj = Cj(Kj, K_j⁰, Lj, Mj, mj, z), depending only onKj, Lj, Mj, mj andz, such that

ku⁰(·,0)k²_ej,a0 (5.1)

≤ Cj

kuk²_2,Ω+X

k∈J

Mk

mk

L²_kK_k²k∇fk²_Ω(k)

+kf(·,0)k²_ek,a0 ,

wherek · k2,Ω is the norm of the second-order Sobolev space H²(Ω)on Ω.

Theorem 5.2. Suppose thatAssumptions 2,1, 3.1and4.2hold. Suppose that lim sup

→0

√kuk_2,Ω<∞. (5.2)

Let f ∈ H¹(Ω) which satisfies(3.18), and let u = (HΩ−z)⁻¹f, where z ∈ C\[0,∞], andfis the restriction off onΩ⁽⁾_j . Let{m}^∞_m=1such thatm→0 as m → ∞ and um(·,0) converges weakly in each L²(ej, a0). Then the limit functionu0 is continuous at any vertexc ofΓ such that c∈Ω.

Example 5.3. Let

Ω ={x= (x1, x2) :−`−(x1)< x2< `+(x1), a < x1< b},

Ω={x= (x1, x2) :−`<sub>−</sub>(x1)< x2< `+(x1), a < x1< b}, (5.3) Γ ={x= (x1,0) :a≤x1≤b},

where −∞ < a < b <∞, 0 < ≤1 and `±(t) are positive C¹ functions on [a, b]. and the mapτ is given by

τ(x1, x2) = (x1,0) ((x1, x2)∈Ω). (5.4) Suppose that Ω is a bounded convex set. Note that each Ωis a convex set, too.

Then it has been known ([11]) that

kuk2,Ω≤C(z)kfkΩ≤C(z)kfkΩ, (5.5) whereC(z) is a positive constant depending only onz. Thus the condition (5.2) is satisfied in this case.

Proof of Theorem 5.2. Let

J(c) ={j∈J :aj =corbj=c}, (5.6) For eachj ∈J(c) letcj∈ej and letej0be all points onejbetweencandcj ∈ej

(includingcandcj). Set Γ(c)⁰ =∪_j∈J(c)ej0. Sincea0is bounded below from 0, it follows from (3.19) and (5.2) that there exists0∈(0,1] such that the sequences {ku(·,0)kej}_0<<0 and{ku⁰(·,0)kej}_0<<0 are uniformly bounded, where the normk kej is given in (3.6). Therefore there exist a sequence {m}, m → 0 (m→ ∞), andc⁰_j∈ej0, (j∈J(c)), such that lim_m→∞um(c⁰_j,0) exists for each j∈J(c). Then we see from

um(σ,0) =R_σ

c⁰_ju⁰(η,0)dη+um(c⁰_j,0) (σ∈Γ⁰_j), (5.7) um(σ,0)−um(σ⁰,0) =R_σ

σ⁰u⁰(η,0)dη (σ, σ⁰ ∈Γ⁰_j),

(3.19) and (5.1) that{um} is uniformly bounded and equicontinuous on Γ(c)⁰. Therefore there exists a subsequence of{m}, which will be denoted again by {m}, such that {um} converges to u0 uniformly on Γ(c)⁰, and hence u0 is continuous on Γ(c)⁰. This completes the proof. ♦

Example 5.4 (Rooms and passages domain). Let {hk}, {δ2k}, k= 1,2,· · ·, be infinite sequences of positive numbers such that

X∞ k=1

hk =b≤ ∞, 0<const. ≤hk+1

hk

≤1, 0< δ2k ≤h2k+1, (5.8)

and letHk := P_k

j=1hj, k = 1,2,· · ·. Then Ω⊂ R² is defined as the union of the roomsRk and passagesPk+1 given by

Rk = (Hk−hk, Hk)×(−^h₂^k,^h₂^k), (5.9) Pk+1= [Hk, Hk+hk+1]×(−^δk+1₂ ,^δk+1₂ ),

fork= 1,3,5, . . .. In§6.1 of [2], this was analyzed as an example of a generalized ridged domain with generalized ridge Γ = [0, b] (b <∞) or Γ = [0,∞) (b=∞).

In order to make Γ a tree, each point on Γ which connects a room and the adjacent passage can be called a vertex (V0, V1, V2, . . .in Fig. 1)

Figure 1: A domain of Rooms and passages

A mappingτ is defined as follows: (i) in a passageP: τ(x1, x2) =x1; (ii) in the first half of the roomRsucceeding the passageP:

τ(x1, x2) = max(x1,|x₂| − δ

2), 0≤x1≤ h

2, (5.10)

wherePis of widthδand 0≤x1≤hinRafter translation. Henceτis Lipschitz and|∇τ|= 1 almost everywhere in Ω. It is easy to see thata0(x1) in this case is a bounded continuous function on Γ, and hence the Kirchhoff boundary condition will be imposed only atx1= 0, b(b <∞) orx1= 0 (b=∞). Sincea0is positive on Γ, the limit functionu0is continuous on Γ, and the differential equation for u0 can be explicitly written usinghk andδ2k.

Finally we are going to show, under some additional conditions. thatu(·,0) converges as→0 without taking a subsequence. We are now in a position to introduce another weightedL²spaces on the tree Γ.

Definition 5.5. Suppose that a tree Γ satisfies (I) of Assumption 2.1. Let a(σ) and b(σ) be positive functions defined on∪_j∈Jej such thata(σ) andb(σ) are bounded from 0 near each vertexv∈Ω. Then letH¹(Γ, a, b) be a subspace ofL²(Γ, a) such thatF ∈H¹(Γ, a, b) satisfies the following conditions.

, ,

, ,

@

@

@

@

@

@

@

@

@

, ,

, ,

,

?

6

?

6

Figure 2: Ctfor a domain of rooms and passages (a)F is continuous on Γ∩Ω.

(b)F is absolutely continuous on eachej∩Ω.

(c)F satisfies

kFk²_Γ,a,b,1=X

j→J

Z

ej

|F⁰(σ)|²b(σ)dσ+kFk²_Γ,a<∞, (5.11)

whereF⁰ denotes the derivative ofF with respect toσ.

Note that it is assumed that Γ∩∂Ω consists only of the vertices of the tree Γ. The next lemma guarantees thatH¹(Γ, a, b) is a Hilbert space with inner product

(F, G)_Γ,a,b,1=X

j∈J

Z

ej

F⁰(σ)G⁰(σ)b(σ)dσ+ (F, G)_Γ,a. (5.12) and norm

kFkΓ,a,b,1= [(F, F)_Γ,a,b,1]^1/2 (5.13) Lemma 5.6. Suppose thatH¹(Γ, a, b)be as inDefinition 5.5.ThenH¹(Γ, a, b) is a Hilbert space with its norm and inner product given by(5.12)and(5.13).

The proof of this lemma will be given in§6. SetH¹(Γ, a0) =H¹(Γ, a0, a0).

Then, under Assumption 3.1, it follows from Lemma 5.6 that H¹(Γ, a0) is a Hilbert space.

LetHΓ,0be the selfadjoint operator inL²(Γ, a0) associated with the sesquilin- ear form

`0[F, G] = Z

ΓF⁰(σ)G⁰(σ)a0(σ)dσ (F, G∈H¹(Γ, a0)) (5.14)

Theorem 5.7. Suppose thatAssumptions 2,1, 3.1, 4.2 hold. Suppose that lim sup

→0

√kuk_2,Ω<∞. (5.15)

Suppose that the tree Γ has a finite number of edges. Let f ∈H¹(Ω), and let u= (HΩ−z)⁻¹f, wherez∈C\[0,∞], and f is the restriction off onΩ⁽⁾_j . Then we haveu(·,0)∈H¹(Γ, a0)and

u(·,0)→(HΓ,0−z)⁻¹f(·,0) (→0) (5.16) weakly inH¹(Γ, a0).

Proof. (I) Let N be the number of the edges of Γ. Then it follows from Propositions 3.4 and 5.1 that

XN j=1

ku(·,0)k²_ej,a0+ku⁰(·,0)k²_ej,a0

≤ 4|z|⁻¹h X^N

j=1

MjK_j²L²_j mj

kfk²_Ω (5.17)

+|z|⁻¹X^N

j=1

Mj

mj

X^N

k=1

Mk

mk

L²_kK_k²k∇fk²_Ω(k)

+kf(·,0)k²_ek,a0 i +

XN j=1

Cj

hkuk²_2,Ω+ XN k=1

Mk

mk

L²kKk²k∇fk²_Ω(k)

+kf(·,0)k²_ek,a0i ,

which, together with the fact thatu∈C¹(Ω), implies thatu(·,0)∈H¹(Γ, a0) for each 0 < ≤ 1. Also, by replacingk∇fk_Ω(k)

by k∇fkΩ in (5.15), we see thatku(·,0)k_Γ,a0,1 is uniformly bounded for∈(0,1].

(II) Let{n}^∞_n=1⊂(0,1] be a sequence such thatn→0 asn→ ∞andun

converges weakly inH¹(Γ, a0). Let u0∈H¹(Γ, a0) be the limit function. Since un is a bounded sequence in L²(Γ, a0) and H¹(Γ, a0) is dense in L²(Γ, a0), u0 is also the weak limit of un in each L²(ej, a0) (j ∈ J). Therefore from Remark 4.6, (ii) and Theorem 4.7 we see thatu0 is a solution of the equation

−a0(σ)⁻¹(a0(σ)u⁰)⁰ −zu = f(σ,0) with the Kirchhoff boundary condition at each vertex of Γ. LetF ∈H¹(Γ, a0). Then, by using partial integration and the fact thatu0 satisfies the above equation with Kirchhoff boundary condition, we have

`0[u0, F] = XN n=1

Z

ej

u⁰₀(σ)F⁰(σ)a0(σ)dσ

= −

Z

Γ(a0(σ)u0(σ)⁰)⁰F(σ)dσ (5.18)

= (zu0−f(·,0), F)_Γ,a0,

where`0[u0, F] is the sesquilinear form used to define the operatorHΓ,0((5.15)).

Then, by the definition ofHΓ,0,u0belongs to the domain ofHΓ,0 and

u0= (HΓ,0−z)⁻¹f(·,0). (5.19) Since the limiting functionu0 is now independent of the subsequenceun, we can conclude that the sequence{u(·,0)}itself converges to (HΓ,0−z)⁻¹f(·,0) weakly inH¹(Γ, a0). This completes the proof. ♦

Remark 5.8. Let (Ω,Γ, τ) be as in Example 5.3. Then not only Theorem 5.7 can be applied to this case, but also it has been shown in [11],§5 that the sequence{u(·,0)} converges to (HΓ,0−z)⁻¹f(·,0) strongly in L²(Γ, a0).

6 Proofs

Proof of Lemma 4.1. (I) By (2.6) and (2.7) we have Z

Ω⁽⁾_j u(x)v(x)dx

= Z

ej

F(σ)

Z _`+(σ)

−`−(σ)u(σ, s)|∇τ(σ, s)|⁻¹ds dσ

= Z

ej

F(σ) Z ^`+(σ)

−`−(σ)u(σ,0)|∇τ(σ,0)|⁻¹ds (6.1) +

Z _`+(σ)

−`−(σ) u(σ, s)|∇τ(σ, s)|⁻¹−u(σ,0)|∇τ(σ,0)|⁻¹ ds dσ

≡ Z

ej

u(σ,0)F(σ)a0(σ)dσ+G1+G2, where

G1=R

ejF(σ)R_`+(σ)

−`−(σ)(u(σ, s)−u(σ,0))|∇τ(σ, s)|⁻¹)ds dσ, (6.2) G2=R

eju(σ,0)F(σ)R_`+(σ)

−`−(σ)(|∇τ(σ, s)|⁻¹− |∇τ(σ,0)|⁻¹)ds dσ.

(6.3) (II) Proceeding as in (3.11), we obtain

1

|G1| ≤ 1

Z

ej

|F(σ)|

Z _`+(σ)

−`−(σ)

Z _`+(σ)

−`−(σ)

∂u(σ, η)

∂s dη

|∇τ(σ, s)|⁻¹ds dσ

≤ Mj

mj

Z

ej

|F(σ)|a0(σ)

Z _`+(σ)

−`−(σ)

∂u(σ, s)

∂s ds dσ (6.4)

≤ Mj

mj

Z

ej

|F(σ)|a0(σ) Z ^`+(σ)

−`−(σ)

∂u(σ, s)

∂s ²|∇τ(σ, s)|⁻¹ds_1/2

×