The Mosco convergence of Dirichlet forms approximating the Laplace operators with the delta potential on thin domains
aa r X i v : . [ m a t h . A P ] J un THE MOSCO CONVERGENCE OF DIRICHLET FORMSAPPROXIMATING THE LAPLACE OPERATORS WITH THE DELTAPOTENTIAL ON THIN DOMAINS
HIROTOSHI KURODA
Abstract.
We consider the convergent problems of Dirichlet forms associated with the Laplaceoperators on thin domains. This problem appears in the field of quantum waveguides. We studythat a sequence of Dirichlet forms approximating the Laplace operators with the delta potentialon thin domains Mosco converges to the form associated with the Laplace operator with thedelta potential on the graph in the sense of Gromov-Hausdorff topology. From this results wecan make use of many results established by Kuwae and Shioya about the convergence of thesemigroups and resolvents generated by the infinitesimal generators associated with the Dirichletforms. Introduction
In this paper we consider the Mosco convergence of energy functionals associated with Laplaceoperators in thin domains with respect to the Gromov-Hausdorff topology. Roughly speaking,we define thin domains Ω ε , which sometimes called fat graphs, a simple graph G = ( V, E )and take some suitable weighted measure dµ ε on Ω ε . Here we call the simple graph if thegraph has only one junction point (see Figure 1). Moreover we define the energy functional ϕ ε associated with the Neumann Laplacian on the thin domain Ω ε . These notations are defined insection 3.1 and 3.2. Our main topic is to prove that the energy functional sequence { ϕ ε } Moscoconverges to the functional ϕ associated with Kirchhoff type Laplacian on the graph in the senseof Gromov-Hausdorff topology. These definitions of functionals ϕ ε and ϕ are given in section3.3.Many mathematicians studied quantum waveguides under various boundary conditions. First,Hale and Raugel [6] considered the squeezing problems in thin domains. Recently, Bouchitte etal. [3] showed the Mosco convergence of the energy functional sequence associated with DirichletLaplacian on thin tubular domains in the case which the graph is only one finite space curve.Kosugi [8] considered the convergence of solutions to elliptic equations with Neumann boundaryconditions on thin domains. Kuwae and Shioya [10] proved that the Mosco convergence ofthe quadratic forms implies the convergence of the semigroups and the resolvents of self-adjointoperators associated with these forms. So our results are more convenient to apply to the problemin the quantum physics. We refer the papers [4, 12] for more details about Γ-convergence, Moscoconvergence and their applications. Also many researchers considered the Dirichlet boundaryproblems in thin domains[2, 5, 11]. In this Dirichlet problems more complicated structuresappear in the gluing conditions at the vertices on the limiting graph.We explain the table of contents in this paper. In section 2 we recall ideas about the Gromov-Hausdorff topology established by Kuwae and Shioya [10]. In section 3 we prove that a sequenceof thin domains converges to a graph with respect to the Gromov-Hausdorff topology and defineenergy functionals on the space of all square integrable functions on thin domains. In section4 we prove that our energy functional sequence satisfies asymptotic compactness condition. To This research was supported by JST, CREST: A Mathematical Challenge to a New Phase of Material Science,Based on Discrete Geometric Analysis.2010
Mathematics Subject Classification.
Primary 31C25, Secondary 35J10, 81Q10.
Key words and phrases. thin domain; Dirichlet form; delta potential; Mosco convergence; Gromov-Hausdorfftopology. obtain the continuity of the limit function on the graph is most difficult part in this story.In section 5 we consider our main theorem that Dirichlet forms, which associated with theNeumann Laplace operators added the potential function approximating the delta function, onthin domains Mosco converges to Dirichlet form, which associated with the delta type Laplaceoperator. In this paper we treat a simple graph to avoid the complicated notations which maydisturb to understand essential ideas. For that reason, we remark that our method works aboutmore general network shaped graph in last section 6.2.
Preliminaries
We recall some definitions and propositions about Gromov-Hausdorff topology introduced byKuwae and Shioya [10]. We refer to that paper for more details.First, denote by M c the set of isomorphism classes of triples ( X, p, m ), where X is a locallycompact separable metric space such that any bounded subset of X is relatively compact, p ∈ X and m is a positive Radon measure on X . Let A and B be any directed sets. Definition 2.1 (cf. [10, Remark 2.2]) . ( measured Gromov-Hausdorff convergence ) We say thata net { ( X α , p α , m α ) } α ∈A of spaces in M c converges to a space ( X, p, m ) ∈ M c in the sense ofpointed, measured, and compact Gromov-Hausdorff convergence if and only if there exist netsof positive numbers { r α } α ∈A , { r ′ α } α ∈A and { ε α } α ∈A such that r α , r ′ α ր ∞ , ε α ց , and m α -measurable maps f α : B ( p α , r α ) → B ( p, r ′ α ) called ε α -approximations for α ∈ A , such that | d ( f α ( x ) , f α ( y )) − d α ( x, y ) | < ε α for any x, y ∈ B ( p α , r α ) , α ∈ A ,B ( p, r ′ α ) ⊂ B ( f α ( B ( p α , r α )) , ε α ) for α ∈ A , lim α Z B ( p α ,r α ) u ◦ f α dm α = Z X u dm for any u ∈ C ( X ) , where d α , d denote the distance functions on X α , X , B ( A, r ) the open metric r -ball of a set A in a metric space, and C ( X ) the set of real valued continuous functions on X with compactsupport in X . Definition 2.2 (cf. [10, Definition 2.3]) . ( L -strong convergence ) Let { ( X α , p α , m α ) } α ∈A be anet of spaces in M c and ( X, p, m ) ∈ M c be a space. A net { u α } α ∈A with u α ∈ L ( X α , m α ) is said to strongly L -converges to a function u ∈ L ( X, m ) if { ( X α , p α , m α ) } α ∈A converges to ( X, p, m ) with respect to the pointed, measured, and compact Gromov-Hausdorff topology and ifthere exists a net { ˜ u β } β ∈B of functions in C (supp m ) tending to u in L ( X, m ) such that lim β lim sup α k Φ α ˜ u β − u α k L ( X α ,m α ) = 0 , where f α : B ( p α , r α ) → B ( p, r ′ α ) are ε α -approximations, and for v ∈ C (supp m ) , Φ α v := ( v ◦ f α on B ( p α , r α ) , X α \ B ( p α , r α ) . We define the Hilbert spaces H α := L ( X α , m α ) and H := L ( X, m ) and assume that { ( X α , p α , m α ) } α ∈A converges to ( X, p, m ) in the sense of Definition 2.1. We remark that wecan define the following concepts for general Hilbert spaces, but in this paper we need only L spaces. More general definitions are treated in [10]. Definition 2.3 (cf. [10, Definition 2.5]) . ( L -weak convergence ) We say that a net { u α } α ∈A with u α ∈ H α weakly converges to a function u ∈ H if lim α h u α , v α i H α = h u, v i H for any net { v α } α ∈A with v α ∈ H α tending strongly to v ∈ H . HE MOSCO CONVERGENCE OF DIRICHLET FORMS ON THIN DOMAINS 3
Next, we recall properties about quadratic forms on Hilbert spaces.
Definition 2.4 (cf. [10, Definition 2.8]) . (Γ -convergence ) We say that a net { F α : H α → R := R ∪ {±∞}} α ∈A of functions Γ -converges to a function F : H → R if and only if (F1) and (F2) hold: (F1) If a net { u α } α ∈A with u α ∈ H α strongly converges to u ∈ H , then F ( u ) ≤ lim inf α F α ( u α ) . (F2) For any u ∈ H there exists a net { u α } α ∈A with u α ∈ H α which strongly converges to u and F ( u ) = lim α F α ( u α ) . Definition 2.5 (cf. [10, Definition 2.11]) . ( Mosco convergence ) We say that a net {E α } α ∈A of closed quadratic forms with E α on H α Mosco converges to a closed quadratic form E on H ifand only if both (F2) and the following (F1 ′ ) hold: (F1 ′ ) If a net { u α } α ∈A with u α ∈ H α weakly converges to u ∈ H , then E ( u ) ≤ lim inf α E α ( u α ) . Definition 2.6 (cf. [10, Definition 2.12]) . ( Asymptotic compactness ) The net {E α } α ∈A is saidto be asymptotically compact if for any net { u α } α ∈A such that u α ∈ H α and lim sup α ( E α ( u α ) + k u α k H α ) < + ∞ , there exists a strongly convergent subnet of { u α } α ∈A . Lemma 2.7 (cf. [10, Lemma 2.15]) . Assume that {E α } α ∈A is asymptotically compact. Then, {E α } α ∈A Γ -converges to E if and only if {E α } α ∈A Mosco converges to E . Definition 2.8 (cf. [10, Definition 2.13]) . ( Compactly convergence ) We say that E α → E compactly if {E α } α ∈A Mosco converges to E and if {E α } α ∈A is asymptotically compact. Lastly, we recall the relations between the convergence of densely defined closed quadraticforms E α and a behavior infinitesimal generators A α associated with E α . For a densely definedclosed quadratic form E , A , { T t } t ≥ and { R ζ } ζ ∈ ρ ( A ) denote the infinitesimal generator associatedwith E , the instead of the strongly continuous contraction semigroup, and the strongly continuousresolvent, that is R ζ = ( A − ζ ) − , where ρ ( A ) denotes the resolvent set of A . Theorem 2.9 (cf. [10, Theorem 2.4]) . The following are all equivalent: (1) E α → E with respect to the Mosco topology (resp. E α → E compactly). (2) R αζ → R ζ strongly (resp. compactly) for some ζ < . (3) T αt → T t strongly (resp. compactly) for some t > . Gromov-Hausdorff convergence of thin domains
Notations about graphs and thin domains.
We write a point x ∈ R n like x = t ( x , x ′ ) ∈ R × R n − = R n and denote by O the origin.Let G = ( V, E ) be a graph where V is the set of all vertices and E is the set of all edges asfollows: V = { O, v j | j = 1 , . . . , N } , E = { e j = −−→ Ov j | j = 1 , . . . , N } .l j ∈ (0 , + ∞ ) denotes the length of e j for j = 1 , . . . , N . We identify each edge e j by an interval(0 , l j ) = { s ∈ R | < s < l j } . Throughout this paper s is the coordinate of the graph. Moreover dµ denotes the 1 dimensional Lebesgue measure on the graph G . HIROTOSHI KURODA
Next, we define a thin domain Ω ε . For j = 1 , . . . , N , we take an orthogonal matrix R j ∈ O ( n )satisfying(3.1) det R j = 1 , R j a = l − j −−→ Ov j , where a = t (1 , , · · · , ∈ R n is the unit vector. Firstly we define tubular domains D j,ε as D j,ε := { x = R j y ∈ R n | εl ≤ y < l j , | y ′ | < ε } , for ε ∈ I := (0 , ε ] and j = 1 , . . . , N . Here we take positive constants l and ε such that D i,ε ∩ D j,ε = ∅ for i = j and ε ∈ I . Now, we shall denote by B ε the normal section of thetubular domain D j,ε , that is(3.2) B r := { y ′ ∈ R n − | | y ′ | < r } for r > ∂D j,ε as follows:Γ j,ε := { x = R j y ∈ R n | y = l j , | y ′ | ≤ ε } , Γ ′ j,ε := { x = R j y ∈ R n | y = εl, | y ′ | ≤ ε } . We use the following notations D j , Γ j , . . . instead of D j,ε , Γ j,ε , . . . for simplicity. e (s=0)(s=l ) v e e v v O (s=l )(s=l ) Figure 1. graph G OD ε v y y’ l j ε l R j ε Figure 2. tubular domain D j,ε Thirdly we fix an open set J in R n satisfying the following conditions: O ∈ J, J ∩ D j = ∅ , ∂J ∩ ∂D j = Γ ′ j for j = 1 , . . . , N and we define a domain Ω in R n and a part of its boundary Σ asΩ := J ∪ N [ j =1 D j , Σ := ∂ Ω \ N [ j =1 Γ j . Now we suppose that the surface Σ is C .Lastly we define a family of thin domains for ε ∈ I as(3.3) J ε := { x = ( ε/ε ) z ∈ R n | z ∈ J } , Ω ε := J ε ∪ N [ j =1 D j,ε , Σ ε := ∂ Ω ε \ N [ j =1 Γ j,ε . We remark that the boundary Σ ε is also C and \ ε ∈ I Ω ε = G .By the definition of domains we obtain that the volume of each domain is(3.4) | J ε | = ( ε/ε ) n | J | = O ( ε n ) , | D j,ε | = ( l j − εl ) ωε n − = O ( ε n − ) , where ω denotes the volume of the ( n −
1) dimensional unit ball B defined by (3.2). HE MOSCO CONVERGENCE OF DIRICHLET FORMS ON THIN DOMAINS 5 D ε J ε D ε D ε O Figure 3. thin domain Ω ε O Γ ε Γ ε Γ ε Σ ε Σ ε Σ ε Figure 4. boundary ∂ Ω ε Proof of the convergence of thin domains.
We define a Radon measure dµ ε on Ω ε for ε ∈ I as(3.5) dµ ε := 1 ωε n − dx, where dx denotes the Lebesgue measure on R n .We define projections { f ε : Ω ε → G } ε ∈ I as follows. First, we fix a continuous function f : J → G ∩ J such that Range( f | Γ ′ j ) = G ∩ Γ ′ j , where J is the closure of J . Next, we define a continuous function f ε : Ω ε → G for ε ∈ I as f ε ( x ) := εε f (cid:16) ε ε x (cid:17) if x ∈ J ε = ( ε/ε ) J,π ( R − j x ) if x ∈ D j,ε where π : R n → R is the orthogonal projection with respect to the first component, that is π ( y ) = y . Proposition 3.1. (Gromov-Hausdorff convergence of thin domains)
The sequence ofpointed measured spaces { (Ω ε , O, dµ ε ) } ε ∈ I defined by (3.3) and (3.5) converges to the pointedmeasured space ( G, O, dµ ) as ε → +0 in the sense of pointed, measured and compact Gromov-Hausdorff topology. proof. For any test function ψ ∈ C ( G ), we consider the following limit(3.6) lim ε → +0 Z Ω ε ψ ◦ f ε dµ ε = lim ε → +0 Z J ε ψ ◦ f ε dµ ε + N X j =1 ωε n − Z D j,ε ( ψ ◦ f ε )( x ) dx . By the definition of dµ ε , see (3.5), at the first term of (3.6) we have (cid:12)(cid:12)(cid:12)(cid:12)Z J ε ψ ◦ f ε dµ ε (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ψ k L ∞ ( G ) | J ε | ωε n − , HIROTOSHI KURODA then(3.7) lim ε → +0 Z J ε ψ ◦ f ε dµ ε = 0holds because of (3.4).At the second term of (3.6), ψ j = ψ | e j denote the restriction to each edge e j . By thetransformation of variables x = R j y we obtain that1 ωε n − Z D j,ε ( ψ ◦ f ε )( x ) dx = 1 ωε n − Z ( εl,l j ) × B ε ψ j ( y ) dy = Z l j εl ψ j ( y ) dy . Hence(3.8) lim ε → +0 ωε n − Z D j,ε ( ψ ◦ f ε )( x ) dx = Z l j ψ j ( s ) ds holds for j = 1 , . . . , N .Since (3.6), (3.7) and (3.8), we get the following limit condition:lim ε → +0 Z Ω ε ψ ◦ f ε dµ ε = 0 + N X j =1 Z l j ψ j ( s ) ds = Z G ψ dµ for any ψ ∈ C ( G ). Therefore the proof is completed. (cid:3) Definition of energy functionals on thin domains and graphs.
Let V ∈ C ( R n ) be a nonnegative valued function with supp V ⊂ ε − J . We define a constant C V and a functional sequence { V ε } ε ∈ I on R n as C V := 1 ω Z ε − J V ( x ) dx and V ε ( x ) := 1 ε V (cid:16) xε (cid:17) for x ∈ R n . Here the function V ε has the compact support in J ε .Next, we define functionals ϕ Kε , ϕ Vε , ϕ ε : L (Ω ε , dµ ε ) → [0 , + ∞ ] as ϕ Kε ( u ε ) := Z Ω ε |∇ u ε | dµ ε if u ε ∈ H (Ω ε , dµ ε ) , + ∞ otherwise ,ϕ Vε ( u ε ) := Z Ω ε V ε | u ε | dµ ε , (3.9) ϕ ε ( u ε ) := ϕ Kε ( u ε ) + ϕ Vε ( u ε )for ε ∈ I . Also we define functionals ϕ K , ϕ V , ϕ : L ( G ) → [0 , + ∞ ] as ϕ K ( ψ ) := N X j =1 Z l j | ψ ′ j ( s ) | ds if ψ ∈ H ( G ) , + ∞ otherwise ,ϕ V ( ψ ) := ( C V | ψ ( O ) | if ψ ∈ H ( G ) , + ∞ otherwise , HE MOSCO CONVERGENCE OF DIRICHLET FORMS ON THIN DOMAINS 7 (3.10) ϕ ( ψ ) := ϕ K ( ψ ) + ϕ V ( ψ )where ψ j is the restriction to each edge e j . Here we define functional spaces on G as follows: L ( G ) := { ψ : G → C | ψ j ∈ L ( e j ) f or j = 1 , . . . , N } ,H ( G ) := { ψ ∈ C ( G ) | ψ j ∈ H ( e j ) f or j = 1 , . . . , N } , where C ( G ) is the set of all continuous functions on G .In this paper we prove that ϕ ε Mosco converges to ϕ as ε → +0. This statement is our maintheorem. We discuss more details in section 5.4. The compactness condition
To prove the Γ-convergence of our energy functionals, we have to study the behavior as ε → { u ε } ε ∈ I with u ε ∈ L (Ω ε , dµ ε ) satisfying the following condition:there exists a constant M > ε ∈ I n ϕ Kε ( u ε ) + k u ε k L (Ω ε ,dµ ε ) o ≤ M. To do so, we divide the thin domains and consider the problem on each junction domain J ε andtubular domain D j,ε separately.4.1. In the tubular domains.
First, we consider the behavior of { u ε } ε ∈ I in the tubular domain D j,ε . It is convenient tostudy the problem in some domain independent to ε . So we define a fixed tube Q j in R n as Q j := (0 , l j ) × B and define a functional sequence { w εj } ε ∈ I ⊂ L ( Q j ) as follows:(4.2) w εj ( y ) := u ε ( R j α ε ( y )) for y ∈ Q j , where R j is the orthogonal matrix satisfying (3.1) and α ε : Q j → ( εl, l j ) × B ε is a transformationof variables defined by(4.3) α ε ( y ) := t (cid:18) l j − εll j y + εl, εy ′ (cid:19) for y = t ( y , y ′ ) ∈ Q j . It is easy to calculate the Jacobian of the transformation of variables α ε :det( ∇ α ε ( y )) = ( l j − εl ) l − j ε n − Lemma 4.1.
Suppose that a functional sequence { u ε } ε ∈ I with u ε ∈ L (Ω ε , dµ ε ) satisfies thecondition (4.1). Let { w εj } ε ∈ I be the functional sequence defined by (4.2). Then the followingproperties hold. (1) For j = 1 , . . . , N , it follows that lim ε → +0 Z Q j |∇ y ′ w εj ( y ) | dy = 0 , where ∇ y ′ means the derivative with respect to y ′ , that is |∇ y ′ w εj ( y ) | = n X i =2 (cid:12)(cid:12)(cid:12)(cid:12) ∂w εj ∂y i ( y ) (cid:12)(cid:12)(cid:12)(cid:12) . (2) There exists a subsequence { w ε m j } ∞ m =1 of { w εj } ε ∈ I such that { w ε m j } ∞ m =1 converges to somefunction ψ ∞ j = ψ ∞ j ( s ) ∈ H (0 , l j ) in L ( Q j ) for j = 1 , . . . , N , that is lim m →∞ Z Q j | w ε m j ( y ) − ψ ∞ j ( y ) | dy = 0 . HIROTOSHI KURODA proof.
By the transformation of variables y = α − ε ( R − j x ) defined by (4.3), we obtain Z Q j |∇ y ′ w εj ( y ) | dy = l j ( l j − εl ) ε n − Z D j,ε |∇ x ′ u ε ( x ) | dx = l j ωε l j − εl Z D j,ε |∇ x ′ u ε | dµ ε ≤ l j ωM ε l j − εl since the condition (4.1). So the claim 4.1 (1) holds.Also by the similar calculations we have Z Q j (cid:12)(cid:12)(cid:12)(cid:12) ∂w εj ∂y ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dy = l j − εll j ε n − Z D j,ε (cid:12)(cid:12)(cid:12)(cid:12) ∂u ε ∂x ( x ) (cid:12)(cid:12)(cid:12)(cid:12) dx = ( l j − εl ) ωl j Z D j,ε (cid:12)(cid:12)(cid:12)(cid:12) ∂u ε ∂x (cid:12)(cid:12)(cid:12)(cid:12) dµ ε ≤ ωM, (4.4)and Z Q j | w εj ( y ) | dy = l j ( l j − εl ) ε n − Z D j,ε | u ε ( x ) | dx = l j ωl j − εl Z D j,ε | u ε | dµ ε ≤ l j ωMl j − ε l . From these calculations, { w εj } ε ∈ I is bounded in H ( Q j ). Therefore there exist a subsequence { ε m } ∞ m =1 ⊂ I, ε m ց w ∞ j ∈ H ( Q j ) such that w ε m j → w ∞ j strongly in L ( Q j ) , ∇ w ε m j ⇀ ∇ w ∞ j weakly in L ( Q j ) , as m → ∞ for j = 1 , . . . , N . Because of Lemma 4.1(1), it follows that ∇ y ′ w ε m j → L ( Q j ) ( m → ∞ ) , then ∇ y ′ w ∞ j = 0, which in terms implies that we can take some function ψ ∞ j = ψ ∞ j ( s ) ∈ H (0 , l j )satisfying(4.5) w ∞ j ( y ) = ψ ∞ j ( y ) for y ∈ Q j . This function ψ ∞ j satisfies the claim in Lemma 4.1(2). (cid:3) Next, we prove that the function ψ ∞ j is the limit of u ε m in the tubular domain D j,ε m as m → ∞ . Lemma 4.2.
For each j = 1 , . . . , N , the functional sequence { u ε m } ∞ m =1 converges to the function ψ ∞ j defined by (4.5) in the tubular domain D j,ε m as m → ∞ , that is lim m →∞ Z D j,εm | u ε m − ψ ∞ j ◦ f ε m | dµ ε m = 0 . HE MOSCO CONVERGENCE OF DIRICHLET FORMS ON THIN DOMAINS 9 proof.
By using the transformation of variables x = R j α ε ( y ), it follows that Z D j,εm | u ε m − ψ ∞ j ◦ f ε m | dµ ε m = a j,m ω Z Q j | w ε m j ( y ) − ψ ∞ j ( a j,m y + ε m l ) | dy ≤ a j,m ω Z Q j | w ε m j ( y ) − ψ ∞ j ( y ) | dy + 2 a j,m Z l j | ψ ∞ j ( s ) − ψ ∞ j ( a j,m s + ε m l ) | ds (4.6)where a constant a j,m is defined by(4.7) a j,m := ( l j − ε m l ) /l j and converges to 1 as m → ∞ .Since Lemma 4.1 (2) and ψ ∞ j ∈ H (0 , l j ) ⊂ C ([0 , l j ]), the right hand side of (4.6) convergesto 0 as m → ∞ . Therefore we get the conclusion. (cid:3) Hereafter we take the subsequence { u ε m } ∞ m =1 , which satisfies the claim in Lemma 4.1 (2), of { u ε } ε ∈ I . So we obtain the limit functions { ψ ∞ j } Nj =1 on edges. Next we consider the connectingconditions at the origin O about these limit functions.4.2. In the junction domain.
Next, we study the behavior of { u ε } ε ∈ I , which satisfies the condition (4.1), in the junctiondomain J ε . Especially we consider the continuity of the limit function at the junction point O .To do so, we fix a constant a ∈ ( l, min l j /ε ) and define D aj,ε := { x = R j y ∈ R n | εl ≤ y < εa, | y ′ | < ε } , J aε := J ε ∪ N [ j =1 D aj,ε for ε ∈ I and j = 1 , . . . , N . Next, we define a fixed domain J a as J a = ε − J aε and a functionalsequence { v ε } ε ∈ I ⊂ L ( J a ) as follows:(4.8) v ε ( z ) := u ε ( εz ) for z ∈ J a = ε − J aε . Since the junction domain J ε squeezes to the origin O , we expect that the functional sequence { v ε } ε ∈ I converges to some constant as ε → +0. Lemma 4.3.
Suppose that a functional sequence { u ε } ε ∈ I with u ε ∈ L (Ω ε , dµ ε ) satisfies thecondition (4.1). Let { v ε } ε ∈ I be the functional sequence defined by (4.8). Then the followingproperties hold. (1) It follows that lim ε → +0 Z J a |∇ v ε ( z ) | dz = 0 . (2) There exists the sequence { ξ ε } ε ∈ I such that lim ε → +0 Z J a | v ε ( z ) − ξ ε | dz = 0 , where ξ ε is defined by (4.9) ξ ε := 1 | J a | Z J a v ε ( z ) dz. proof. By using the transformation of variables z = ε − x , we have Z J a |∇ v ε ( z ) | dz = 1 ε n − Z J aε |∇ u ε ( x ) | dx = ωε Z J aε |∇ u ε | dµ ε ≤ M ωε.
So Lemma 4.3 (1) holds.The complex number ξ ε denotes the mean value of v ε in J a for ε ∈ I . Then we obtain theclaim in Lemma 4.3 (2) by using the Poincar´e inequality and Lemma 4.3 (1). (cid:3) Continuity of the limit function.
In this part we mention that the limit function { ψ ∞ j } Nj =1 belongs to the effective domain of ϕ defined by (3.10). Lemma 4.4.
Suppose that a functional sequence { u ε } ε ∈ I with u ε ∈ L (Ω ε , dµ ε ) satisfies thecondition (4.1). Then there exists a subsequence { u ε k } ∞ k =1 of { u ε } ε ∈ I and a function ψ ∞ ∈ H ( G ) satisfying the following condition (4.10) lim k →∞ Z Ω εk | u ε k − ψ ∞ ◦ f ε k | dµ ε k = 0 . proof. First, we take a sequence { ε m } ∞ m =1 ⊂ I ( ε m ց
0) such that all claims in Lemma 4.1 andLemma 4.3 are satisfied.To show that ψ ∞ j (+0) is independent to j , we consider the following limit:lim m →∞ ωε m Z ε m aε m l | ψ ∞ j ( s ) − ξ ε m | ds, where { ξ ε } is the sequence defined by (4.9). Here, we have ωε m Z ε m aε m l | ψ ∞ j ( s ) − ξ ε m | ds = 1 ε nm Z ε m aε m l Z B εm | ψ ∞ j ( y ) − ξ ε m | dy ≤ ε nm Z D aj,εm ( | ψ ∞ j ( π R − j x ) − u ε m ( x ) | + | u ε m ( x ) − ξ ε m | ) dx. (4.11)We know already that the second term of the right hand side of (4.11) converges to 0 as m → ∞ because of Lemma 4.3. In fact(4.12) 1 ε nm Z D aj,εm | u ε m ( x ) − ξ ε m | dx ≤ Z J a | v ε m ( z ) − ξ ε m | dz −→ m → ∞ ) . At the first term of the right hand side of (4.11), we use the transformation of variables x = R j α ε m ( y ). Then1 ε nm Z D aj,εm | ψ ∞ j ( π R − j x ) − u ε m ( x ) | dx = a j,m ε m Z a − j,m ( a − l ) ε m Z B | ψ ∞ j ( a j,m y + ε m l ) − w ε m j ( y ) | dy ′ dy (4.13) HE MOSCO CONVERGENCE OF DIRICHLET FORMS ON THIN DOMAINS 11 where a j,m is the constant defined by (4.7). By taking a new coordinate s = a j,m y ( a − l ) ε m , the righthand side of (4.13) equals to(4.14) ( a − l ) Z Z B | ψ ∞ j ( g m ( s )) − w ε m j ( h m ( s ) , y ′ ) | dy ′ ds where g m ( s ) := ε m { ( a − l ) s + l } , h m ( s ) := a − j,m ε m ( a − l ) s. Firstly, it follows that(4.15) lim m →∞ Z | ψ ∞ j ( g m ( s )) − ψ ∞ j (0) | ds = 0since ψ ∞ j ∈ C ([0 , l j ]) and g m ( s ) → m → ∞ .Secondly, { w ε m j } ∞ m =1 ⊂ H ( Q j ) is bounded, then { w ε m j | ∂Q j } ∞ m =1 ⊂ H ( ∂Q j ) is relativelycompact in L ( ∂Q j ). So w ε m j | ∂Q j converges to w ∞ j | ∂Q j in L ( ∂Q j ). Therefore we get(4.16) lim m →∞ Z B | ψ ∞ j (0) − w ε m j (0 , y ′ ) | dy ′ = 0 . Thirdly, we have Z Z B | w ε m j (0 , y ′ ) − w ε m j ( h m ( s ) , y ′ ) | dy ′ ds ≤ Z Z B h m ( s ) Z h m ( s )0 (cid:12)(cid:12)(cid:12)(cid:12) ∂w ε m j ∂y ( τ, y ′ ) (cid:12)(cid:12)(cid:12)(cid:12) dτ dy ′ ds ≤ a j,m ε m ( a − l ) Z s Z B Z l j (cid:12)(cid:12)(cid:12)(cid:12) ∂w ε m j ∂y ( τ, y ′ ) (cid:12)(cid:12)(cid:12)(cid:12) dτ dy ′ ds ≤ a j,m ε m ( a − l ) Z Q j (cid:12)(cid:12)(cid:12)(cid:12) ∂w ε m j ∂y ( y ) (cid:12)(cid:12)(cid:12)(cid:12) dy ≤ a j,m ε m ( a − l ) ωM since (4.4). Therefore we obtain(4.17) lim m →∞ Z Z B | w ε m j (0 , y ′ ) − w ε m j ( h m ( s ) , y ′ ) | dy ′ ds = 0 . From (4.14), (4.15), (4.16) and (4.17), it follows that(4.18) lim m →∞ ε nm Z D aj,εm | ψ ∞ j ( π R − j x ) − u ε m ( x ) | dx = 0 . So the following limit(4.19) lim m →∞ ε m Z ε m aε m l | ψ ∞ j ( s ) − ξ ε m | ds = 0holds by (4.11), (4.12) and (4.18) for j = 1 , . . . , N .Moreover it follows that(4.20) 1 ε m Z ε m aε m l | ψ ∞ i ( s ) − ψ ∞ j ( s ) | ds ≤ ε m Z ε m aε m l ( | ψ ∞ i ( s ) − ξ ε m | + | ξ ε m − ψ ∞ j ( s ) | ) ds and lim m →∞ ε m Z ε m aε m l | ψ ∞ i ( s ) − ψ ∞ j ( s ) | ds = lim m →∞ Z al | ψ ∞ i ( ε m s ′ ) − ψ ∞ j ( ε m s ′ ) | ds ′ = ( a − l ) | ψ ∞ i (+0) − ψ ∞ j (+0) | . (4.21) Hence we obtain | ψ ∞ i (+0) − ψ ∞ j (+0) | = 0for i, j = 1 , . . . , N since (4.19), (4.20) and (4.21). Then it follows that(4.22) ψ ∞ (+0) = · · · = ψ ∞ N (+0) = lim k →∞ ξ ε mk =: v ∞ by taking some suitable subsequence { ξ ε mk } ∞ k =1 of { ξ ε m } ∞ m =1 . Therefore we can define a contin-uous function ψ ∞ ∈ H ( G ) as ψ ∞ := ( ψ ∞ j ( s ) on e j = { s | < s < l j } for j = 1 , . . . , N,v ∞ at O. Lastly, we prove (4.10). Here we replace ε m k to ε k for simplicity. Because of Lemma 4.2, wehave to prove only this limit condition:(4.23) lim k →∞ Z J εk | u ε k − ψ ∞ ◦ f ε k | dµ ε k = 0 . Now, we have Z J εk | u ε k − ψ ∞ ◦ f ε k | dµ ε k ≤ Z J εk ( | u ε k − ξ ε k | + | ξ ε k − ψ ∞ ◦ f ε k | ) dµ ε k ≤ ε k ω Z J a | v ε k ( z ) − ξ ε k | dz + 2 k ξ ε k − ψ ∞ k L ∞ ( G ) | J ε k | ωε n − k . Hence (4.23) holds since (3.4), (4.22) and Lemma 4.3 (2). (cid:3) Convergences of energy functionals
We discuss our main theorem 5.3 in this section.
Theorem 5.1.
The functional sequence { ϕ ε } ε ∈ I defined by (3.9) Γ -converges to the function ϕ defined by (3.10) as ε → +0 in the sense of Definition 2.4. proof. We have to prove that the two conditions (F1) and (F2) in Definition 2.4 are satisfied.First, we check (F1). To do so, let { u ε } ε ∈ I with u ε ∈ L (Ω ε , dµ ε ) be any functional sequencewhich converges strongly to some function ψ ∈ L ( G ) as ε → +0 in the sense of Definition 2.2.Now we have to show that(5.1) ϕ ( ψ ) ≤ lim inf ε → +0 ϕ ε ( u ε ) . If lim inf ε → +0 ϕ ε ( u ε ) = + ∞ , then (5.1) is trivial. So we can assume thatlim inf ε → +0 ϕ ε ( u ε ) < + ∞ . Hence we are able to take some suitable subsequence { u ε m } ∞ m =1 for ε m ց C > ϕ ε m ( u ε m ) ≤ C ( m ∈ N ) , lim inf ε → +0 ϕ ε ( u ε ) = lim inf m →∞ ϕ ε m ( u ε m ) . In this case, { u ε m } ∞ m =1 satisfies the condition (4.1), that issup m ∈ N n ϕ Kε m ( u ε m ) + k u ε m k L (Ω εm ,dµ εm ) o < + ∞ . By Lemma 4.4, some subsequence of { u ε m } ∞ m =1 converges strongly to a function ψ ∞ ∈ H ( G )in the sense of Definition 2.2. Therefore it follows that ψ = ψ ∞ ∈ H ( G ) , lim m →∞ k u ε m − ψ ◦ f ε m k L (Ω εm ,dµ εm ) = 0 . HE MOSCO CONVERGENCE OF DIRICHLET FORMS ON THIN DOMAINS 13
We fix a positive constant δ . Then, for 0 < ε < δ/l we can divide domains Ω ε as follows:Ω ε = J δε ⊔ N G j =1 D δj,ε , D δj,ε := { x = R j y ∈ R n | δ < y < l j , | y ′ | < ε } . Here we prepare a useful lemma about functionals on tubular domains. This lemma is provedby only direct calculations. So we omit it here and prove in the end of this proposition’s proof.
Lemma 5.2.
Let Q ε := (0 , L ) × B ε be a tubular domain and E ε be a following bilinear form on L ( Q ε , dµ ε ) : E ε ( g ε ) := Z Q ε |∇ g ε | dµ ε if g ε ∈ H ( Q ε , dµ ε ) , + ∞ otherwise for < ε < . Suppose that a functional sequence { g ε } <ε< with g ε ∈ L ( Q ε , dµ ε ) con-verges strongly to some function g ∈ L (0 , L ) as ε → +0 in the sense of Definition 2.2 and { E ε ( g ε ) } <ε< is bounded. Then g ∈ H (0 , L ) and the following inequality Z L | g ′ ( s ) | ds ≤ lim inf ε → +0 E ε ( g ε ) holds. Because D δj,ε are tubular domains for fixed δ , we can apply this results. So we obtain Z l j δ | ψ ′ j ( s ) | ds ≤ lim inf m →∞ Z D δj,εm |∇ u ε m | dµ ε m . So the following inequality N X j =1 Z l j δ | ψ ′ j ( s ) | ds ≤ lim inf m →∞ N X j =1 Z D δj,εm |∇ u ε m | dµ ε m ≤ lim inf m →∞ ϕ Kε m ( u ε m )holds for δ >
0. By tending δ → +0, we obtain that(5.2) ϕ K ( ψ ) = N X j =1 Z l j | ψ ′ j ( s ) | ds ≤ lim inf m →∞ ϕ Kε m ( u ε m ) = lim inf ε → +0 ϕ Kε ( u ε ) . On the other hand, it follows that by the transformation of variables x = ε m zϕ Vε m ( u ε m ) = Z J εm ε m V (cid:18) xε m (cid:19) | u ε m ( x ) | dxωε n − m = Z ε − J V ( z ) | v ε m ( z ) | dzω . From the same arguments in the proof of Lemma 4.4, we obtain that(5.3) lim m →∞ ϕ Vε m ( u ε m ) = Z ε − J V ( z ) | ψ ( O ) | dzω = ϕ V ( ψ ) . So (5.2) and (5.3) imply (5.1), that is the condition (F1).Next, we check (F2). For any ψ ∈ D ( ϕ ) = H ( G ), we have to make a functional sequence { u ε } ε ∈ I with u ε ∈ L (Ω ε , dµ ε ) such that(5.4) lim ε → +0 k u ε − ψ ◦ f ε k L (Ω ε ,dµ ε ) = 0 , lim ε → +0 ϕ ε ( u ε ) = ϕ ( ψ ) . Since ψ j ∈ H (0 , l j ), we can define the functions ψ εj ∈ H ( εl, l j ) as ψ εj ( s ) := ψ j (cid:0) γ ε ( s ) (cid:1) , γ ε ( s ) := l j l j − εl ( s − εl ) ( εl < s < l j )for j = 1 , . . . , N . Moreover we can define the function u ε ∈ C (Ω ε ) as u ε ( x ) := ( ψ εj ( π R − j x ) ( x ∈ D j,ε ) ,ψ ( O ) ( x ∈ J ε ) . Hereafter we show that { u ε } ε ∈ I satisfies (5.4) below.To prove the first condition of (5.4), we estimate the following norm k u ε − ψ ◦ f ε k L (Ω ε ,dµ ε ) = Z J ε | u ε − ψ ◦ f ε | dµ ε + N X j =1 Z D j,ε | u ε − ψ ◦ f ε | dµ ε = 1 ωε n − Z J ε | ψ ( O ) − ( ψ ◦ f ε )( x ) | dx + N X j =1 Z l j εl | ψ εj ( s ) − ψ j ( s ) | ds ≤ k ψ ( O ) − ψ k L ∞ ( G ) | J ε | ωε n − + N X j =1 Z l j εl | ψ j ( γ ε ( s )) − ψ j ( s ) | ds. Hence this limit condition lim ε → +0 k u ε − ψ ◦ f ε k L (Ω ε ,dµ ε ) = 0holds since (3.4) and γ ε ( s ) → s as ε → +0.To prove the second condition of (5.4), we use the transformation of variable t = γ ε ( s ). Sowe have Z Ω ε |∇ u ε ( x ) | dµ ε = N X j =1 Z l j εl (cid:12)(cid:12)(cid:12)(cid:12) dψ εj ds ( s ) (cid:12)(cid:12)(cid:12)(cid:12) ds = N X j =1 Z l j εl | ψ ′ j ( γ ε ( s )) γ ′ ε ( s ) | ds = N X j =1 l j l j − εl Z l j | ψ ′ j ( t ) | dt. Therefore u ε ∈ H (Ω ε ) = D ( ϕ Kε ) and(5.5) lim ε → +0 ϕ Kε ( u ε ) = N X j =1 Z l j | ψ ′ j ( t ) | dt = ϕ K ( ψ ) . Also we obtain that ϕ Vε ( u ε ) = Z J ε ε V (cid:16) xε (cid:17) | ψ ( O ) | dxωε n − = | ψ ( O ) | ω Z ε − J V ( z ) dz = ϕ V ( ψ )(5.6)for ε ∈ I . Since (5.5) and (5.6), the sequence { u ε } ε ∈ I satisfies (5.4). HE MOSCO CONVERGENCE OF DIRICHLET FORMS ON THIN DOMAINS 15
By above argument, the proof is completed. Lastly, we prove Lemma 5.2.(Proof of Lemma 5.2)We define a functional sequence { h ε } ⊂ L ( Q , dµ ) as h ε ( y ) := g ε ( y , εy ′ ). Then by thedirect calculations we have Z Q | h ε − g ◦ π | dµ = Z Q ε | g ε − g ◦ π | dµ ε −→ ε → +0)and E ( h ε ) = Z Q |∇ h ε | dµ = Z Q ε (cid:12)(cid:12)(cid:12)(cid:12) ∂g ε ∂x (cid:12)(cid:12)(cid:12)(cid:12) + ε |∇ x ′ g ε | ! dµ ε ≤ Z Q ε |∇ g ε | dµ ε = E ε ( g ε ) . Hence { h ε } is bounded in H ( Q , dµ ). Therefore we obtain that g ∈ H ( Q , dµ ) and Z Q | g ′ | dµ ≤ lim inf ε → +0 Z Q |∇ h ε | dµ . Since above inequalities it follows that Z L | g ′ ( s ) | ds = Z Q | g ′ | dµ ≤ lim inf ε → +0 E ( h ε ) ≤ lim inf ε → +0 E ε ( g ε ) . This is the conclusion of lemma which we omit to prove before. (cid:3)
Theorem 5.3.
The functional sequence { ϕ ε } ε ∈ I defined by (3.9) Mosco converges to the function ϕ defined by (3.10) as ε → +0 in the sense of Definition 2.5. proof. This statement follows from Proposition 5.1 (Γ-convergence), Lemma 4.4 (asymptoticcompactness) and Lemma 2.7. (cid:3)
Therefore we can apply Theorem 2.9 established by Kuwae and Shioya [10].6.
Remarks about more general network case
We consider about simple thin domains, which have a single junction point, on the previoussection. So we remark more general case.It is not difficult to see that our previous arguments work (by refining the transformation ofvariables α ε in the tubular domains which both side are junction domains) if a connected graph G = ( V, E ) satisfies the following conditions:(1) The set of all edges E is finite.(2) All edges have a finite length.Because under these assumptions, thin domains Ω ε is bounded, we can apply the compactlyembedding theorem in each part. Figure 5. network shaped graph G Acknowledgment
The author would like to express my gratitude to S. Albeverio and P. Exner for insightfulcomments and suggestions, and thanks to C. Cacciapuoti for illuminating discussions.
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