A spectral problem for the Laplacian in joined thin films

We consider a 3d multi-structure composed of two joined perpendicular thin films: a vertical one with small thickness hna\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^a_n$$\end{document} and a horizontal one with small thickness hnb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^b_n$$\end{document}. We study the asymptotic behavior, as hna\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^a_n$$\end{document} and hnb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^b_n$$\end{document} tend to zero, of an eigenvalue problem for the Laplacian defined on this multi-structure. We shall prove that the limit problem depends on the value q=limnhnbhna.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=\displaystyle {\lim _n\dfrac{h^b_n}{h^a_n}.}$$\end{document} Precisely, we pinpoint three different limit regimes according to q belonging to ]0,+∞[\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$]0,+\infty [$$\end{document}, q equal to +∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$+\infty $$\end{document}, or q equal to 0. We identify the limit problems and we also obtain H1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^1$$\end{document}-strong convergence results.


Introduction
Let n , n in N, be a 3d multi-structure composed of two joined perpendicular thin films (see Fig. 1): a vertical one a n with small thickness h a n and a horizontal one b n with small thickness h b n (from now on, the exponent 'a' stands for above, while 'b' for below). In n consider the following eigenvalue problem with mixed boundary conditions ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ − U n = λU n in n , U n = 0 on n , ∂U n ∂ν = 0 on ∂ n \ n , (1.1) where n denotes the part of the boundary of n having small thickness (see dotted area in Fig. 1) and ν denotes the exterior unit normal to n (see Sect. 2 for the rigorous definition of n and n , and for the weak formulation of problem (1.1)). For any n in N, problem (1.1) has a discrete positive spectrum {λ n,k } k∈N with corresponding eigenfunctions {U n,k } k∈N forming an orthonormal basis in L 2 ( n ) (see Sect. 2), equipped with the inner product This means that the following normalization U n,k 2 L 2 ( n ) = h a n , ∀k ∈ N, is considered, but it does not restrict the generality of our results. Problem (1.1) arises, for instance, from the Fourier analysis in the study of the heat problem or the propagation of sound waves (cf. [23], see also [14] in connection with elastic waves).
For reasons of simplicity and economy, especially from a numerical point of view, one tries to remodel the 3d problem with a problem defined on a multi-structure composed of 2d components. In this paper, it will be obtained by an asymptotic process based on the so-called "dimensional reduction", i.e., by the study of the asymptotic behavior of problem (1.1) as h a n and h b n tend to zero. Precisely, we pinpoint three different limit regimes according to q belonging to ]0, +∞[, q equal to +∞, or q equal to 0.
• When q belongs to ]0, +∞[, i.e., when the thicknesses of the two thin films vanish with the same rate, we obtain the following limit eigenvalue problem, where ω a is the cross-section of the vertical film, ω b + and ω b − are the two parts into which ω b , the cross-section of the horizontal film, is divided by the intersection with ∂ω a (see Fig.  2 Problem (1.3) is a 2d − 2d − 2d eigenvalue problem with coupled conditions on γ (see the last two lines of (1.3)).
The weak formulation of (1.3) is given by (3.4) (see also (3.1), (3.2), and (3.3)). This problem has a discrete positive spectrum {λ k } k∈N with the corresponding eigenfunctions {(u a k , u b k+ , u b k− )} k∈N forming a basis in L 2 (ω a )×L 2 where δ h,k denotes the Kronecker delta. In Theorem 3.1 we prove the convergence of the eigenvalues of problem (1.1), as n → +∞, to the eigenvalues of problem (1.3) with conservation of the multiplicity. We prove also a strong H 1 -convergence result for the corresponding eigenfunctions (see (3.5), (3.6), (3.7), and Corollary 3.2).
• When q = +∞, i.e., when the thickness of the vertical thin film vanishes faster than the thickness of the horizontal thin film, the limit spectrum is the union of the spectra of the following two uncoupled 2d eigenvalue problems with homogeneous Dirichlet boundary 129 Page 4 of 31 A. Gaudiello et al.
Precisely, one has to collect together the eigenvalues of these two problems and order the obtained set in an increasing sequence {λ k } k∈N with the convention of repeated eigenvalues. The corresponding eigenfunctions form an orthonormal basis in L 2 (ω a ) × L 2 (ω b ). In Theorem 3.3 we prove the convergence of the eigenvalues of problem (1.1), as n → +∞, to the sequence {λ k } k∈N with conservation of the multiplicity. Moreover, by means of renormalization in b n , we prove a strong H 1 -convergence result for the corresponding eigenfunctions (see (3.10), (3.11), (3.12), and Corollary 3.5).
• When q = 0, i.e., when the thickness of the horizontal thin film vanishes faster than the thickness of the vertical thin film, we choose the sequence {U n,k } k∈N of eigenfunctions associated to the discrete positive spectrum {λ n,k } k∈N of problem (1.1) such that it forms an orthonormal basis in L 2 ( n ) equipped with the inner product i.e., the following normalization is considered. In this case, the limit spectrum is the union of the spectra of the following three uncoupled 2d eigenvalue problems, the first one with mixed boundary condition, while the other two with homogeneous Dirichlet boundary condition As above, one has to collect together the eigenvalues of these three problems and order the obtained set in an increasing sequence {λ k } k∈N with the convention of repeated eigenvalues. The corresponding eigenfunctions form an orthonormal basis in In Theorem 3.6 we prove the convergence of the eigenvalues of problem (1.1), as n → +∞, to the sequence {λ k } k∈N with conservation of the multiplicity. Also in this case we prove a strong H 1 -convergence result for the corresponding eigenfunctions, but by means of a renormalization in a n (see (3.16), (3.17), (3.18), and Corollary 3.8). Notice that, when q belongs to ]0, +∞[, choosing (1.2) or (1.4) as normalization leads to the same limit result. Instead, to obtain a meaningful result, normalization (1.2) must be used when q is +∞ and normalization (1.4) when q is 0.
In Sect. 2, following [3], problem (1.1) is rescaled on a fixed domain. Section 4 is devoted to obtaining a priori estimates of the eigenvalues λ n,k of problem (1.1): below by a positive constant independent of n and k, and above by an explicit constant independent of n but dependent on k (see also Remark 4.2). The upper bound of λ n,k implies H 1 -a priori estimates of the eigenfunctions. In Sect. 3, the main results are stated. Section 5 contains some results that are crucial for proving the main results, i.e., Theorems 3.1, 3.3, and 3.6. Precisely, in Proposition 5.1 we give a trace convergence result, written in a very general way, which will allow us to identify junction and boundary conditions in the limit problems. In Proposition 5.2, we prove a density result for approximating the elements of the space of setting of the limit problem, when q belongs to ]0, +∞[, with regular functions. Although this result was used in other works, to our knowledge, there are no previous proofs of it. Our proof is rather technical and it works also for domains which are not "symmetric". Proposition 5.3 is devoted to building a recovery sequence which will be used in the proof of all three main results. Sections 6, 7, and 8 are devoted to proving the main results in the case where q belongs to ]0, +∞[, q equal to +∞, or q equal to 0, respectively. The three proofs follow the same pattern. In them, we highlight the novelties and refer to [9] and [22] for the classical parts.
In this paper we consider the Laplace operator in order to investigate the effect of the junction condition on the limit problem. It is of course possible to replace the Laplacian by an elliptic operator with a symmetric and positive definite thermal conductivity matrix. Taking into account our analysis and arguing as in [10] easily lead to the limit problem. Moreover, we just considered two perpendicular thin films. Of course, the whole analysis works with the appropriate modifications if the two films form an angle other than π 2 . We leave the study of these cases to an interested reader.
The asymptotic behavior of a spectral problem for an homogeneous isotropic elastic body consisting of two folded and perpendicular plates with the same thickness h but with the requirement of large elastic coefficients, of order O(h −2 ), was studied in [12] (see also [13]). This assumption technically avoids a rescaling of the eigenvalues and gives very different asymptotic behaviors from our problem. Also, we refer to [4], [11], [16], and [17] for different eigenvalue problems in plate theory.
The modelling of spectral problems for the Laplace operator in joined 1d −1d and 1d −2d multi-structures were obtained in [7], [9], [10], [15], and [20]. The modelling of the spectrum for the linear water-wave system in a joined 1d − 2d multi-structure was obtained in [1].

Position of the problem and rescalings
Let l + 1 , l − 1 , l 2 , and l 3 be four positive real numbers such that Set (see Fig. 2) Let h a n n∈N , h b n n∈N be two sequences in ]0, 1[ such that For every n in N set (see Fig. 1) For every n in N, consider the space L 2 ( n ) equipped with the inner product and the space equipped with the inner product The classical spectral theory (for instance, see [21]) ensures the existence of an increasing diverging sequence of positive numbers {λ n,k } k∈N forming the set of all the eigenvalues of Problem (1.1), i.e., (2.5) Moreover, there exists a L 2 ( n )-Hilbert orthonormal basis {U n,k } k∈N such that, for every k in N, U n,k belongs to V n and it is an eigenvector of (2.5) with eigenvalue λ n,k ; hence, n,k U n,k k∈N is a V n -Hilbert orthonormal basis.

Set now
From now on, As it is usual (see [3]), problem (2.5) will be reformulated on the fixed domain a ∪ b ∪ ] − 1 2 , 1 2 [×]0, l 2 [ through the following maps To this aim, for every n in N, let H n be the space and let V n be the space defined by equipped with the inner product a n : (2.8) Moreover, for every n and k in N, set (2.9) Then, for every n in N, {λ n,k } k∈N is an increasing diverging sequence of positive numbers forming the set of all the eigenvalues of the following problem ⎧ ⎨ ⎩ u n ∈ V n , a n (u n , v) = λ(u n , v) n , ∀v ∈ V n , (2.10) {u n,k } k∈N is a H n -Hilbert orthonormal basis such that, for every k in N, u n,k belongs to V n and it is an eigenvector of (2.10) with eigenvalue λ n,k . Moreover, λ − 1 2 n,k u n,k k∈N is a V n -Hilbert orthonormal basis. In particular, one has ⎧ ⎨ ⎩ u n,k ∈ V n , a n (u n,k , v) = λ n,k (u n,k , v) n , ∀v ∈ V n , ∀n, k ∈ N, Furthermore, for every k in N, λ n,k is characterized by the following min-max Principle where F k is the set of the subspaces E k of V n with dimension k (for instance, see [21]).
Problem (2.10) is obtained from (2.5) by means of rescaling of variables, once multiplied by 1 h a n .

The main results
This section is devoted to stating the main results of this paper.
The limit problem will depend on q defined by (2.1) which acts as a weight on ω b in the scalar product. Precisely, three different limit regimes will appear according to q belonging to ]0, +∞[, q equal to +∞, or q equal to 0.

The case q in ]0, +∞[
be equipped with the inner product Both are Hilbert spaces. Moreover, the norm induced on V by the inner product α q (·, ·) is equivalent to the usual Then, all classic results hold true for the eigenvalue problem (see [21]) 4) and the following result will be proved.
equipped with the inner product (·, ·) n defined by (2.6) and V n be the space defined by (2.7) equipped with the inner product a n (·, ·) defined by (2.8 n,k u n,k k∈N is a V n -Hilbert orthonormal basis and, for every k ∈ N, u n,k = (u a n,k , u b n,k ) is an eigenvector of Problem (2.10) with eigenvalue λ n,k . Assume Then, there exists an increasing diverging sequence of positive numbers {λ k } k∈N , depending on q, such that lim n λ n,k = λ k , ∀k ∈ N, and {λ k } k∈N is the set of all the eigenvalues of Problem (3.4). Moreover, there exist an increasing sequence of positive integer numbers {n i } i∈N and a (L 2 (ω a ) × L 2 (ω b ), [·, ·] q )-Hilbert orthonormal basis {u k } k∈N (depending possibly on the selected subsequence {n i } i∈N and q) such that, for every k in N, u k belongs to V and it is an eigenvector of Problem (3.4) with eigenvalue λ k , and as n diverges. Furthermore, {λ − 1 2 k u k } k∈N is a (V , α q )-Hilbert orthonormal basis. As far as the original problem (1.1) is concerned, one has the following result which is an immediate corollary of Theorem 3.1, by change of variable.  n,k U n,k k∈N is a V n -Hilbert orthonormal basis and, for every k ∈ N, U n,k is an eigenvector of Problem (2.5) with eigenvalue λ n,k .
Then, there exists an increasing diverging sequence of positive numbers {λ k } k∈N , depending on q, such that lim n λ n,k = λ k , ∀k ∈ N, and {λ k } k∈N is the set of all the eigenvalues of Problem (3.4). Moreover, there exist an increasing sequence of positive integer numbers

belongs to V and it is an eigenvector of Problem (3.4) with eigenvalue λ k , and
where, from now on, − a n i means 1 | a n i | a

The case q = +∞
Let [·, ·] 1 be the inner product on L 2 (ω a ) × L 2 (ω b ) defined by (3.1) with q = 1. Moreover, still denote by α 1 the inner product on (3.8) Then, both are Hilbert spaces and all classic results hold true for the eigenvalue problem (see [21]) and the following result will be proved when q is equal to +∞.

Theorem 3.3
For every n in N, let H n be the space L 2 ( a ) × L 2 ( b ) equipped with the inner product (·, ·) n defined by (2.6) and V n be the space defined by (2.7) equipped with the inner product a n (·, ·) defined by (2.8). n,k u n,k k∈N is a V n -Hilbert orthonormal basis and, for every k ∈ N, u n,k = (u a n,k , u b n,k ) is an eigenvector of Problem (2.10) with eigenvalue λ n,k .
and it is an eigenvector of Problem (3.9) with eigenvalue λ k , and ⎛ ⎝ u a n i ,k , Remark 3.4 Notice that (3.10) and (3.12) imply that As far as the original problem (1.1) is concerned, one has the following result which is an immediate corollary of Theorem 3.3, by change of variable. Assume that (2.1) holds true with q = +∞. Let L 2 (ω a ) × L 2 (ω b ) be equipped with the inner product [·, ·] 1 defined by (3.1) and be equipped with the inner product α 1 (·, ·) defined by (3.8).
and it is an eigenvector of Problem (3.9) with eigenvalue λ k , and
and still denote by α 1 the inner product on H 1 γ a (ω a ) × W 0 defined by (3.3) with q = 1, i.e., (3.14) Then, both are Hilbert spaces and all classic results hold true for the following eigenvalue problem (see [21]) and the following result will be proved when q = 0.  n,k u n,k k∈N is a V n -Hilbert orthonormal basis and, for every k ∈ N, u n,k = (u a n,k , u b n,k ) is an eigenvector of Problem (2.10) with eigenvalue λ n,k .

16)
as i diverges, 1 h a n h a n h b n ∂ x 1 u a n,k → 0 strongly in L 2 ( a ), ∀k ∈ N, 1 h a n ∂ x 1 u a n,k → 0 strongly in L 2 ( a ), ∀k ∈ N.
As far as the original problem (1.1) is concerned, one has the following result which is an immediate corollary of Theorem 3.6, by change of variable. Assume that (2.1) holds true with q = 0. Let L 2 (ω a ) × L 2 (ω b ) be the space equipped with the inner product [·, ·] 1 defined by (3.1), W 0 be defined by (3.13), and H 1 γ a (ω a ) × W 0 be the space equipped with the inner product α 1 (·, ·) defined by (3.14). Then

A priori estimates on the eigenvalues
This section is devoted to proving lower and upper bounds for the eigenvalues of Problem (2.10) Proposition 4.1 For every n in N, let H n be the space L 2 ( a ) × L 2 ( b ) equipped with the inner product (·, ·) n defined by (2.6), V n be the space defined by (2.7) equipped with the inner product a n (·, ·) defined by (2.8), and {λ n,k } k∈N be the increasing diverging sequence of all the eigenvalues of Problem (2.10). Then,

2)
where l 2 is the positive real number involved in the definition of ω a and ω b (see Sect. 2).
Proof As far as the proof of (4.1) is concerned, at first note that the boundary conditions on u a n,k and u b n,k provide that u a n,k L 2 ( a ) ≤ l 2 ∂ x 2 u a n,k L 2 ( a ) , Combining now (2.13), (4.3), and (2.12) gives λ n,k = a n (u n,k , u n,k ) ≥ a |∂ x 2 u a n, i.e., (4.1) holds true. As far as the proof of (4.2) is concerned, let {λ j } j∈N be the increasing diverging sequence of all the eigenvalues of the following problem For every j in N, set Fix k in N and set Then, for every n in N, Z k is a subspace of V n with dimension k. Consequently, the min-max Principle (2.14) provides that i.e. (4.2) holds true with c k = λ k .

Remark 4.2
It is possible to give an estimate of the constant c k in Proposition 4.1. Indeed, it is well known that the set of all the eigenvalues of problem (4.4) is given by Then, Recall that A im sin iπ l 2 x 2 sin mπ l 3 x 3 , with A im in R, is an eigenfunction of (4.4) with eigenvalue Remark 4.3 Proposition 4.1 is independent of the asymptotic behavior of h a n n∈N and h b n n∈N .
Choosing k = h in (2.13) and taking into account Proposition 4.1 provide the following result.

Corollary 4.4
For every n in N, let H n be the space L 2 ( a ) × L 2 ( b ) equipped with the inner product (·, ·) n defined by (2.6) and V n be the space defined by (2.7) equipped with the inner product a n (·, ·) defined by (2.8).

Some preliminary results
This section contains some results that are crucial for proving Theorems 3.1, 3.3, and 3.6. Precisely, Proposition 5.1 will give a trace convergence result, written in a very general way, which will allow us to identify junction and boundary conditions in the limit problems. Proposition 5.2 will give a density result for approximating the elements of V defined in (3.2) by regular functions. Although this result was used in other works, to our knowledge, there are no previous proofs of it. Our proof is rather technical and it works also for domains which are not "symmetric". Proposition 5.3 is devoted to building a recovery sequence which will be used in the proof of all three main results.

5.4)
Notice that assumption (5.2) ensures that the function w given by (5.3) is independent of x 3 , i.e., Then, it makes sense to write w(0, x 2 ) in (5.4).
Proof At first, one proves the existence of x 3 in ] − 1, 0[ and of an increasing sequence of positive integer numbers {i j } j∈N such that as j diverges. Indeed, set Then, Fatou's Lemma combined with assumption (5.3) provides that Consequently, there exist two constants c in ]0, +∞[ and x 3 in ]−1, 0[, and an increasing sequence of positive integer numbers {i j } j∈N such that which provides (5.6), thanks to (5.3) and (5.5). Now, for proving (5.4), fix ϕ in C ∞ 0 (]0, l 2 [) and split the first integral in (5.4), written with index i j , as (5.7) One will pass to the limit, as j diverges, in each term of this decomposition.
As far as the first integral on the right-hand side of (5.7) is concerned, assumption (5.2) implies that As far as the second integral on the right-hand side of (5.7) is concerned, assumption (5.1) and (5.6) imply as j → +∞.

(5.9)
As far as the last integral on the right-hand side of (5.7) is concerned, (5.6) implies Eventually, passing to the limit in (5.7), as j diverges, and taking into account (5.8), (5.9), and (5.10) give (5.4) for the subsequence {n i j } j∈N . Notice that (5.4) holds true for the whole subsequence {n i } i∈N too, since the limit ϕw does not depend on {n i j } j∈N .
The following proposition is devoted to approximating the elements of the space V defined by (3.2) by more regular functions belonging to the space V reg defined by The proof of (5.12) will be split into two steps.
Step 1. The first step is devoted to proving (5.12) when Split v b in the even part and in the odd part with respect to x 1 , i.e., where v e (x 1 , As far as the approximation of v o is concerned, since it belongs to Then, setting for every and As far as the approximation of v a and v e is concerned, set which implies, thanks to definition (5.18), that and Set now, for every n in N, v a n : ( Then, the sequence v a n n∈N is included in C ∞ 0 (]0, l 2 [×[0, l 3 [) and, thanks to (5.21), (5.20), and (5.17), it converges strongly in H 1 (ω a ) to the function given by Moreover, setting for every n in N, and by virtue of (5. 19) v e n → v e strongly in H 1 ω b . (5.25) Now, setting for every n in N, Eventually, the sequence v a n , v b n n∈N , so built, is in V reg and satisfies (5.12).
Step 2. The second step is devoted to proving (5.12) when For instance, assume By virtue of the previous step, there exists a sequence v a Then, the sequence v a n , v b n n∈N belongs to V reg and satisfies (5.12). The proof of (5.12) is similar if l − 1 < l + 1 .
This section concludes with the building of a recovery sequence for functions in V reg with functions in V n defined by (2.7).

Proposition 5.3
Let V reg be defined by (5.11).
as n → +∞, (5.26) Proof For every n ∈ N set Obviously, {g n } n∈N is included in H 1 a ( a ) and the last line of (5.26) is satisfied. Moreover, by the definition of V reg , it is easy to see that as n diverges, which imply the convergences in (5.26).
Eventually, introduce the space (5.27) which will be used in the following sections.

Proof of Theorem 3.1
The proof will be split into several steps.
Step 1. The first step is devoted to proving the existence of an increasing sequence of positive integer numbers {n i } i∈N , an increasing sequence of positive numbers {λ k } k∈N , a sequence u k = (u a k , u b k ) k∈N in V , where V is the space defined by (5.27), and a sequence (ξ a k , ξ b k ) k∈N in L 2 ( a )×L 2 ( b ) (depending possibly on the selected subsequence {n i } i∈N ) such that, for every k in N, 1 h a n i ∂ x 1 u a n i ,k , as i diverges, and Estimates in (4.1) and in (4.5), assumption (2.1) with q in ]0, +∞[, and a diagonal argument ensure that (6.1), (6.2), and (6.3) hold true for a suitable increasing sequence of positive integer numbers {n i } i∈N and for suitable sequences . Eventually, (6.4) follows by passing to the limit in as i diverges, thanks to assumption (2.1) with q in ]0, +∞[ and the strong L 2 -convergence in (6.2).
For asserting that u k = (u a k , u b k ) belongs to V , it remains to prove the following result.
The transmission condition in (2.7) gives As far as the first integral in (6.6) is concerned, the weak H 1 -convergence in (6.2) and the fact that u a k is independent of x 1 imply (6.7) As far as the last integral in (6.6) is concerned, note that estimate in (4.5) provides that 1 h a Then, combining (6.8) with the weak H 1 -convergence in (6.2) and using Proposition 5.1 yield (6.9) Eventually, the junction condition in (6.5) follows from (6.6), (6.7), and (6.9).
Step 3. This step is devoted to proving that ∀k ∈ N, (6.10) Fix k in N. To prove (6.10), by following the classic idea of −convergence, a recovery sequence will be constructed for regular function as test function in (2.11) written with index n i yields a 1 h a n i ∂ x 1 u a n i ,k 1 h a n i ∂ x 1 g n i + ∂ x 2 u a n i ,k ∂ x 2 g n i + ∂ x 3 u a n i ,k ∂ x 3 g n i dx (6.13) Passing to the limit, as i diverges, in (6.13) and using (2.1) with q in ]0, +∞[, (6.1), (6.2), (6.3), and (5.26) provide that which implies (6.10), thanks to the density of V reg in V proved in Proposition 5.2. Relations in (6.11) follow from (6.10), (6.4), and from the fact that λ k are all positive. As far as (6.12) is concerned, either (6.12) holds true, or {λ k } k∈N is a finite set. In the second case, by virtue of (6.4), Problem (6.10) would admit an eigenvalue of infinite multiplicity. But this is not possible, due to the Fredholm's alternative Theorem.
, +∞[ and it is an increasing and diverging sequence of eigenvalues of Problem (6.10), {u k } k∈N is an orthonormal sequence in (L 2 (ω a )× , for every k ∈ N u k is an eigenvector for Problem (6.10), with eigenvalue λ k , and convergences (3.5), (3.6), and (3.7) hold true.
Moreover, arguing as in [9] (see step 2 in the proof of Theorem 2.5) or as in [22] (see Theorem 9.2), one can prove that there does not exist (u, λ) ∈ V × R satisfying the following problem [u, u] q = 1.
As in [9] (see step 3 in the proof of Theorem 2.5), this implies that the sequence {λ k } k∈N forms the whole set of the eigenvalues of Problem (3.4), that {λ − 1 2 k u k } k∈N is a (V , α q )-Hilbert orthonormal basis, and that {u k } k∈N is a (L 2 (ω a )×L 2 (ω b ), [·, ·] q )-Hilbert orthonormal basis.
In conclusion, since the sequence {λ k } k∈N can be characterized by the min-max Principle, for every k ∈ N convergence (6.1) holds true for the whole sequence {λ n,k } n∈N .

Proof of Theorem 3.3
The proof will be split into several steps.
Step 1. The first step is devoted to proving the existence of an increasing sequence of positive integer numbers {n i } i∈N , an increasing sequence of positive numbers {λ k } k∈N , a sequence u k = (u a k , u b k ) k∈N in V , where V is the space defined by (5.27), and a sequence (ξ a k , ξ b k ) k∈N in L 2 ( a )×L 2 ( b ) (depending possibly on the selected subsequence {n i } i∈N ) such that, for every k in N, . Eventually, (7.4) follows by passing to the limit in as i diverges, thanks to the strong L 2 -convergence in (7.2).
For asserting that u a k belongs to H 1 0 (ω a ), it remains to prove the following result.
As far as the first integral in (7.6) is concerned, the weak H 1 -convergence in (7.2) and the fact that u a k is independent of x 1 imply As far as the last integral in (7.6) is concerned, note that estimate in (4.5) provides that 1 h a n b |∂ x 3 u b k,n (x)| 2 dx ≤ c k h b n → 0, as n → +∞, (7.8) moreover, weak H 1 -convergences in (7.2) and assumption (2.1) with q = +∞ provide u b n i ,k → 0 strongly in H 1 ( b ), (7.9) as i diverges. Then, combining (7.8) with (7.9) and using Proposition 5.1 yield Eventually, boundary condition (7.5) follows from (7.6), (7.7), and (7.10).
Step 3. This step is devoted to proving that To obtain (7.11), it is enough to prove that (7.13) and to add (7.12) and (7.13). Equation (7.12) follows immediately by passing to the limit, as i diverges, in (2.11) written with index n i and with a test function v = (v a , 0), v a in H 1 0 (ω a ), and using (7.1) and (7.2). As far as the proof of (7.13) is concerned, h a n i ∂ x 1 u a n i ,k 1 h a n i ∂ x 1 g n i + ∂ x 2 u a n i ,k ∂ x 2 g n i + ∂ x 3 u a n i ,k ∂ x 3 g n i dx (7.14) Passing to the limit, as i diverges, in (7.14) and using (2.1) with q = +∞, (5.26), (7.1), (7.2), and (7.3) provide (7.13) with v b in C ∞ 0 (ω b ). Then, (7.13) holds true for any v b in H 1 0 (ω b ), by a density argument. Moreover in a classical way (for instance, see [9] or [22]) one can prove that the sequence {λ k } k∈N forms the whole set of the eigenvalues of Problem (3.9), that {λ − 1 2 k u k } k∈N is a (H 1 0 (ω a ) × H 1 0 (ω b ), α 1 )-Hilbert orthonormal basis, and that {u k } k∈N is a (L 2 (ω a ) × L 2 (ω b ), [·, ·] 1 )-Hilbert orthonormal basis.
In conclusion, since the sequence {λ k } k∈N can be characterized by the min-max Principle, for every k ∈ N convergence (7.1) holds true for the whole sequence {λ n,k } n∈N .

Proof of Theorem 3.6
The proof will be split into several steps.
Step 1. The first step is devoted to proving the existence of an increasing sequence of positive integer numbers {n i } i∈N , an increasing sequence of positive numbers {λ k } k∈N , a sequence u k = (u a k , u b k ) k∈N in V , where V is the space defined by (5.27), and a sequence (ξ a k , ξ b k ) k∈N in L 2 ( a )×L 2 ( b ) (depending possibly on the selected subsequence {n i } i∈N ) such that, for every k in N, as i diverges, and using the strong L 2 -convergence in (8.2).
For asserting that u b k belongs to W 0 , it remains to prove the following result. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.