Eigenfrequencies of a tube bundle immersed in a fluid

Appl Math Optim 18:1-38 (1988) Applied Mathematics and Optimization © 1988 Spdngex-Verlag New York Inc.

Eigenfrequencies of a Tube Bundle Immersed in a Fluid

Freddy Aguirre and Carlos Conca

Universit6 Pierre et Marie Curie, Laboratoire d'Analyse Numrrique, Tour 55-65, 5~me 6tage, 4 place Jussieu, 75252 Paris Cedex 05, France, and Departamento de Matemzlticas, Universidad de Chile, Casilla 170/3, Santiago, Chile

Abstract. In this paper we study a simplified version of a mathematical model that describes the eigenfrequencies and eigenmotions of a coupled system consisting of a set of tubes (or a tube bundle) immersed in an incompressible perfect fluid. The fluid is assumed to be contained in a rectangular cavity, and the tubes are assumed to be identical, and periodically distributed in the cavity. The mathematical model that governs this physical problem is an elliptic differential eigenvalue problem consisting of the Laplace equation with a nonlocal boundary condition on the holes, and a homogeneous Neumann boundary condition on the boundary of the cavity. In the simplified model that we study in this paper, the Neumann condition is replaced by a periodic boundary condition. Our goal in studying this simple version is to derive some basic properties of the problem that could serve as a guide to envisage similar properties for the original model. In practical situations, this kind of problem needs to be solved for tube bundles containing a very large number of tubes. Then the numerical analysis of these problems is in practice very expensive. Several approaches to overcome this difficulty have been proposed in the last years using homogenization techniques. Alterna- tively, we propose in this paper an approach that consists in obtaining an explicit decomposition of the problem into a finite family of subproblems, which can be easily solved numerically. Our study is based on a generalized notion of periodic function, and on a decomposition theorem for periodic functions that we introduce in the paper. Our results rely on the theory of almost periodic functions, and they provide a simple numerical method for obtaining approximations of all the eigenvalues of the problem for any number of tubes in the cavity. We also discuss a numerical example.

0. Introduction

This paper is motivated by the physical problem of determining the eigenfrequencies and eigenmotions of a coupled system consisting of a set of tubes (or a tube

2 F. Aguirre and C. Conca

Figure 0.1

bundle) immersed in a perfect fluid (Figure 0.1). The fluid is assumed to be incompressible, and contained in a rectangular cavity. On the other hand, the tubes are assumed to be identical, periodically distributed in the cavity, and assembled in an elastic way.

For small oscillations of the fluid around a state of rest, the mathematical model describing the eigenfrequencies of the tube-bundle-fluid system consists in the Laplace equation with a nonlocal boundary condition on the boundaries of the tubes, and a homogeneous Neumann boundary condition on the external boundary of the cavity. The model is an elliptic differential eigenvalue problem, in which the eigenfunctions represent the potential velocity of the fluid, and the eigenvalues represent the frequencies of vibration of the system. The nonlocal boundary conditions on the tubes model the fluid-tube-bundle interactions, and the Laplace equation governs the motion of the fluid, which for small oscillations can be assumed to be irrotational. Finally, the homogeneous Neumann condition means that the normal velocity of the fluid is zero on the external walls of the cavity. This is the standard boundary condition verified by a perfect fluid on a rigid wall.

Our aim in this paper is to study this eigenvalue problem, but with periodic boundary conditions instead of the homogeneous Neumann condition on the lateral sides of the cavity. The mathematical problem that we study is a simplified version of the first model, in which the above physical interpretation is somewhat lost, but in which the general structure of the problem is preserved. Our goal in studying this simpler version is to establish some basic properties of the eigenvalues and eigenfunctions of the problem that, in our opinion, may serve as a guide to envisage new methods for tackling the Neumann case.

In practical applications one requires, to solve this kind of eigenvalue problem, tube bundles containing a large number of tubes (several hundreds or thousands of tubes). The numerical analysis of the problem and the effective computation of the eigenfrequencies (and eigenmotions) are therefore very

Eigenfrequencies of a Tube Bundle Immersed in a Fluid 3

expensive or in some cases even impossible, since the domain in which the problem is formulated is too complicated to be discretized. This is the central difficulty of this class of problem, and it must be taken into account in any theoretical or numerical study.

Several approaches to overcome this difficulty have been proposed for the Neumann case. Most of them are based on homogenization techniques. Planchard et al. [8], Planchard [4]-[6], and Planchard and Ibnou Zahir [7] have investigated these problems in this direction. Alternatively, we consider here an approach that consists in obtaining an explicit decomposit ion of the problem into a finite family of subproblems. Each one of these problems provides two of the eigenvalues and eigenfunctions of the original problem. Our analysis is based on a generalized notion of periodic function, and a general decomposit ion theorem for periodic functions that we establish in this paper. These results are inspired in the theory of almost periodic functions developed by Bohr [1]. They provide a simple numerical method for obtaining approximations of all the eigenvalues and eigenfunctions of the problem, and they show how the eigenvalues depend on the number of tubes of the bundle.

To conclude this introduction, let us briefly discuss the contents of the paper. In Sections 1 and 2 we first give a precise formulation of the problem that we shall study, and we establish the main theorems related to our study. Next, Sections 3-5 are devoted to the proofs of these theorems, and to discussing a numerical example. At the end of Section 2 we give a detailed outline of Sections 3-5.

1. Formulation of the Problem

1.1. Preliminary Notations. Complex-Valued Functions

Throughout this paper we shall systematically deal with functions u, v, z, . . . . that are defined in a domain of R 2 and that take values in the complex field C. We will refer to these as complex-valued functions. The real part of a complex-valued function u will be denoted by Re(u), and its imaginary part by Im(u) . We shall frequently use the standard decomposition of u in its real and imaginary parts, i.e.,

u = Re (u )+ i Im(u) ,

where i denotes the imaginary unit of C. The absolute value of u will be denoted by lu[.

Moreover, if ~ is any linear (differential) operator acting on real-valued functions, then we shall write ~ u for

,~(Re(u)) + i~ ( Im(u) ) .

Finally, to avoid confusion in the notation, the real-valued functions will be denoted by greek letters: ~0, v, . . . , etc.


1.2. Formulation and Physical Motivation of the Problem

Let a- = (zl, 7-2) be a given vector in R 2 with strictly positive components 7-1 and 7-2. We associate to 7- the elementary cell Y, defined by

Y = ]0, 7 - , [x]0 , 7-2[.

Let T be a connected open subset of Y, strictly contained in Y (i.e., ~ c y ) , with a smooth boundary y. We set (see Figure 1.1)

y * = y - ~ .

I f A is any subset of Y, and i = (11,12) is any integer vector of Z 2, we shall denote by A(I) the translated image of A by the vector (1~7-~, 127-2), i.e.,

A(i) = (117"1,127-2) -Jl- A.

According to this definition we introduce (3, defined by

G = R 2 - i ~ 2 T ( I ) .

Let us observe that G represents the region of R 2 consisting of the whole space R 2 in which we made an infinite number of perforations or holes. All of them have the same shape T and they are periodically distributed in R 2, with period 7-1 along the yl-axis and 7-2 in Y2 (see Figure 1.2).

Besides that, let m, n be two given positive integers, and let 12 be the subregion of G defined by

m--1 n--1

l l = G c ~ = ~ - U U T(I), Ii=O /2=0

where ~ is the rectangle

= ]0, m7-1[x]O, n7-2[

Y2

y , l$

Figure 1.1. The unit cell Y*.

v

Yl


Y2

Figure 1.2. The unbounded region G.

(see Figure 1.3). Thus l'l consists of the rectangle ~ in which we make (m x n) periodic perforations. The number (m x n) of perforations in fl will be frequently referred to throughout. We shall denote this number by L, i.e.,

L = m x n .

Under these geometrical conditions, the aim of this paper is the study of the following periodic eigenvalue problem: find those real numbers /z's for which there exists a nonidentically constant real-valued function ~ = ~(y) , defined in

Yl

n'g

. . . . . . . . . . O

. . . . . . . . . O

© O . . . . . . . . . O m'g

Figure 1.3. The region £/.

Q

Y2


G, such that

A~=O in G, (1.1a)

a._~= f~ au i~v • ~ov ds on each 3,(1), Vie ][2, ( i . lb) (i)

is fLperiodic, (1.1c)

where, in (1.1b), a/Ov denotes the external normal derivative, and the vector v, with components vl, v2, represents the outward unit normal to 3,(1). By (1.1c) we understand that

q~(yl+ml"r l ,y2+n2"r2)=cp(yl ,y2) , V ( y l , y 2 ) E G , V(ml,n2)ET/2.

Remark 1.1. Since the integral of v on 3,(1) is equal to zero for all i, it is clear that trivial solutions of (1.1) can simply be constructed by setting ~ = constant, and/x any (real) number. Certainly, these solutions are of no interest. They have been eliminated in the formulation of the problem imposing the condition that the solutions are not identically constant. We note however that any nontrivial solution of this problem can only be defined up to an additive constant.

Our main motivation for the study of this eigenvalue problem is to obtain preliminary theoretical results in the hope that they could later be applied to the (numerical) study of a modified version of this problem, in which the periodicity condition (1.1c) is replaced by an homogeneous Neumann boundary condition on the lateral sides o f fL ~ The Neumann version of this problem plays an important role in engineering. As is shown by Planchard [4]-[6] or by Planchard et al. [8] this problem is a mathematical model describing, in the two-dimensional case, the frequencies of vibration and the eigenmotions of a set of L identical tubes immersed in an inviscid incompressible fluid.

In this physical problem, the fluid and the tubes are assumed to be contained in a (rectangular) three-dimensional cavity of transversal section ~. The fluid occupies, in ~ , the region f~, and the sections of the tubes are represented by the holes of f~, i.e., by the union of the T(l)'s, for I e 7/2 such that 0-< 11 -< m - 1, 0 - / 2 < - n - 1 (see Figure 1.3). As well as that, the tubes are assumed to be rigid but elastically mounted in such a way that they can move transversely (to fix ideas, one can imagine, for example, that they are interconnected by springs; see Figure 1.4). In this mechanical system, one considers small oscillations of the fluid around a state of rest. Since the tubes have been assumed to be elastically supported, the whole bundle of tubes will also oscillate together with the fluid, and the problem is to determine the resonance eigenfrequencies of the fluid-tube- bundle system.

We shall not go into the details here to explain how equations (1.1a) and (1.1b) are derived from this physical problem. For a complete study of this model refer to Planchard [5], [6]. Nevertheless, let us mention that in these equations

Of course, in this case the problem is formulated in l~, and not in G.


etc. . .

Figure 1.4. The spring system.

the function ~ represents the potential velocity of the fluid, and the eigenfrequencies f ' s of the system are represented in/x, by means of the following formula (see p. 66 of [4]):

py2 Ix K - M f 2' (1.2)

where p, K, M are given real constants; p is the specific density of the fluid, K represents the stiffness constant of the spring system interconnecting the tubes, and M is the mass, per unit of length, of each tube.

Before concluding with the physical aspects of this kind of problem, let us remark that the investigation of fluid elastic vibrations of tube arrays arises in the study of several different engineering machines. Examples of such tube arrays are the heat exchangers, the condensers, and the fuel element assemblies in nuclear reactor cores. A common feature which is always present in all practical situations is the fact that the tube array is very large (several hundreds or thousands of tubes). The numerical computation of the eigenfrequencies is very expensive and sometimes impossible, because the domain II is too complicated to be discretized by finite elements or finite differences. In this context, we shall see that our theoretical results will allow us to propose a numerical method that overcomes this difficulty, at least in the present case.

1.3. The Complex-Valued Formulation o f the Problem

To begin the study of problem (1.1), let us remark that this problem has been formulated imposing the eigenvalues/x 's to be real numbers and the associated eigenfunctions to be real-valued functions. This is the natural f ramework for this problem if one is looking for solutions with a physical meaning. When one deals, however, with eigenvalue problems it is often convenient to formulate the problem in the complex field. For reasons that will soon become apparent, we will do this in our case.


Let us reformulate (1.1) as follows: find those complex numbers A's for which there exists a nonidentically constant complex-valued function u, defined in G, such that

Au =0 in G, (1.3a)

O U = A v . f u v d s on each T(l), V l e Z 2, (1.3b) av Jv(I)

u is f~-periodic. (1.3c)

From the spectral theorem and the properties of the eigenfunctions that we shall later establish for this problem, it will be proved that both formulations are equivalent. Therefore, the study of (1.1) reduces to that of (1.3), and conversely. For technical reasons, in what follows we will concentrate our attention on the complex-valued formulation of the problem, and not on the real one.

1.4. Variational Formulation of the Problem

In order to establish the variational (or weak) formulation of the problem, let us introduce the following Sobolev space of periodic (complex-valued) functions:

H I ( O ) = { v : G -~ C I v e H ~oc( G ) and v is fl-periodic}

equipped with the norm

,,,o=(i, .li v IVv(y)l 2 dy) .

In H i ( l ) ) the subspace of constant functions will be identified with C, and we shall denote by ~ ( f ~ ) the quotient space H~ ( fD /C . If v is any function in H i ( l ) ) , its equivalence class in ~ ( 1 ~ ) will be denoted by 1).

Multiplying (1.3a) by any function v in a smooth equivalence class t~ of ~ l ( f l ) , and integrating by parts in 11, it is elementary to check using (1.3b, c) (and density arguments) that the variational formulation of (1.3) is:

Find A e C, and (1.4a)

ti e ~ ( f / ) , ti # 0, such that (1.4b)

fo "-' ) V f ~ . V f f d y = h Z Z f tvds • fJvds , V b e ~ ( f l ) . 11=0 12 =0 (1) (I)

(1.4c)

It is useful to remark that the left-hand side and the right-hand side of (1.4c) define continuous sesquilinear forms in H~,(Iq) which vanish if u or v are constants, and which thereby induce continuous sesquilinear forms in ~ 1 ( ~ ) . Moreover, using standard arguments it is easy to prove that the left-hand side in (1.4c) defines a coercive sesquilinear form in ~ ( 1 1 ) . In fact, it induces a norm in ~ ( 1 2 ) , which is equivalent to the quotient-norm induced by H~(12) on ~ l ( f~) .


The (equivalence classes of) eigenfunctions of (1.4) that a priori are defined up to a multiplication factor will sometimes be normalized imposing the following condition:

fn[ Vul: d y = l . (1.5)

1.5. The Spectral Theorem

Our starting point for the study of (1.4) (or (1.3)) is the following theorem that gives detailed theoretical information about the solutions (or eigenvalues and eigenfunctions) of (1.4):

Theorem 1.1 (The Spectral Theorem). There exist 2L real and strictly positive numbers A1, . . . , AEL (not necessarily distinct), and 2L equivalence classes of functions ! i l , . . . , fi2L in Y(~ (fl) with the following properties:

For each j = 1 , . . . , 2L, the pair (Aj, tij) is a solution of (1.4). (1.6a)

The set {tij}j=l.2L is orthonormal, i.e.,

fl_LVlij" V~kdy=t~jk, Vj, k= I , . . . , 2 L ,

where 8jk denotes the Kronecker' s 8-symbol. (1.6b)

The pairs {(Aj, fij)}j=l,2L are the unique solutions of (1.4) in the following sense: if (A, ti) in C x Y(~(ll) is any solution of (1.4), then there exists at least one j, 1 <-j<-2L, such that A = Aj, and fi can be written as a linear combination of all the fij's for which Aj is equal to A. (1.6c)

Using the terminology of eigenvalues and eigenfunctions, this theorem asserts that the eigenvalues of (1.4) form a finite set of real positive numbers, that the total number of eigenvalues (including their multiplicities) is 2L, and that the eigenfunctions can be chosen in such a way that they form an orthonormal set in ~ ( 1 ) ) . This particular property of this kind of eigenvalue problem was outlined for the first time by Planchard [5] while studying the Neumann case. The proof of Theorem 1.1, and of the analogue of this theorem in the Neumann case, consists in proving that problem (1.4) can be stated as an eigenvalue problem in C 2£, i.e., a finite-dimensional eigenvalue problem. In Section 3 we give the proof of this result for the periodic case. For the Neumann case we refer to Planchard [5] or Ibnou Zahir [3].

Before proceeding further, let us state the following corollary to Theorem 1.1 that enables us to verify that the real and complex formulations of our eigenvalue problem are equivalent:

Corollary 1.2. Let A be any of the eigenvalues of (1.4). Assume the multiplicity of A to be s, l <-s<-2L, and let {fh . . . . , 6s}c Y(~((I) be a basis of the eigenspace associated with A. Then A is an eigenvalue of (1.1), and all the nonzero real and


imaginary parts of fq , . . . , f~s are (real-valued) eigenfunctions of (1.1) that span a (real) eigenspace of dimension s associated with the eigenvalue A.

From Theorem 1.1 and its corollary we can now conclude that (1.1) and (1.3) (or (1.4)) are equivalent. Actually, both problems have exactly the same eigenvalues, the eigenvalues in both formulations have the same multiplicity, and, for each eigenvalue A, a real-basis of the corresponding eigenspace can be obtained from a complex-basis, and conversely, as follows: in the first case, it suffices to extract from the real and imaginary parts of the complex-valued eigenfunctions the greatest linearly independent subset of functions, and, in the second c~/se, it suffices to consider the real-valued eigenfunctions as pure real complex-valued functions. Certainly, at least in the first case, it is clear that if the original complex-basis is orthogonal, then this property may not remain true for the real-basis obtained by this procedure. However, it can be orthogonalized by the Gram-Schmidt procedure.

2. The Orthogonal Decomposition of the Problem

As was mentioned while presenting the physical interpretation of our eigenvalue problem, in all practical situations the number L of perforations in II is very large, and an effective computation of the eigenvalues is therefore very expensive and sometimes impossible. Our aim in this section consists in establishing some theoretical results that provide an explicit characterization of the eigenvalues and eigenvectors of (1.4), and that simplify its numerical analysis for large values of L. These results are based on a finite orthogonal decomposition of the space H I ( l l ) into L subspaces. The main property of this decomposition is the fact that each of the subspaces in this decomposition contains one or, at most, two eigenspaces of the problem, and their corresponding eigenvalues and eigenfunctions can be easily computed numerically. In fact, as we shall see, in order to obtain the eigenspaces of (1.4) contained in each of these subspaces, it suffices to solve a "small" eigenvalue problem of the same type as (1.4), but posed in the unit cell Y* instead of II. In other words, by means of this decomposition of H~(l l ) , the eigenvalue problem (1.4) is decomposed into L (small) eigenvalue problems in Y*. We shall refer to these as the subproblems associated to (1.4).

By applying this decomposition method to the numerical solution of (1.4) the amount of computational effort required is reduced, and it allows us to treat the problem for large values of L. A second interesting feature of these results is the fact that they assist in elucidating how the eigenvalues of (1.4) depend on rn and n (or on L). Roughly speaking, these results show that the eigenvalues of (1.4) essentially only depend on the unit cell Y*.

2.1. Generalized Periodic Functions

Let D be the unit circle of the complex plane C, i.e.,

D={o~ ~Cll~ol= 1}.


We begin this section by giving the following definition that generalizes the classical concept of a periodic function:

Definition 2.1. Let to = (wl, w2) be a given vector in C 2 with components wl, w2 on the unit circle D. A complex-valued function v in H~oc(G) is said to be an (to, Y*)-periodic function or an (to, ~r)-periodic function if the following conditions hold for (almost) all y in G:

v(yl + rl, Y2) = tol v(yl, Y2), (2.1a)

V(yl, y2+ T2) = to2v(y,, Y2). (2.1b)

Let us remark that if to is equal to (1, 1), then this definition coincides with the standard definition of a Y*- (or ~r-) periodic function. Furthermore, if there exist two integer numbers p~, P2 in Z+ such that w~ is a pl-root of the unity in C, and to2 is a p2-root of the unity, i.e., if

(tOl) p' -- (w2) p2 = 1,

then using (2.1a, b) it is easy to see that every (to, Y*)-periodic function is (p~r~,pz~-2)-periodic. On the other hand, if there is no Pl, P2 with the above properties, then the (to, Y*)-periodic functions are not periodic functions for any period, and all that can be said in this case is that they are almost periodic functions in the sense of Bohr [1].

Using the definition of (to, Y*)-periodic functions, let us now introduce the spaces in which the analysis of our eigenvalue problem will be carried out.

Definition 2.2. Let to be given in D x D. We denote by Hi ( to , Y*) the set of all complex-valued functions v in H~oc(G) that are (to, Y*)-periodic functions,

H#(to, Y*) with an inner product and a norm as follows: and we equip 1

(U, V) l ,y* = f y . u(y)O(y)dy + fly. Vu(y)" V~5(y)dy, (2.2a)

II u I1, y . = (u, u ) , . y . = I u(Y)12 dy+ fy. I V . ( y ) 12 dy. (2.2b)

Let us note that H~((1, 1), Y*)= H I ( Y * ) , and that, for all to, H i ( t o , Y*) is a (complex) vector space. It is not reduced to the zero-function because it contains at least the following family of (linearly independent) functions:

= {Vk}k~z: (2.3a)

vk(y) = (Wl)Y'/"(w2) y:/': exp{ i(2~'klyl/~'l + 2~rk2Y2/ Z2)}, (2.3b)

as well as the functions with compact support in Y*. The following theorem summarizes the main basic properties of these spaces:

Theorem 2.1. Let to be given in D x D. Then the following statements hold: 1 H~(to, Y*) is a Hilbert space for the inner product (2.2a). (2.4a)


I f to # (1, 1), then the only identically constant function contained in H~,(to, Y*) is the zero-function. The constant functions are all contained in H1((1, 1), Y*). (2.4b)

I f to # (1, 1), then the map

v~H~( to , r*)-,Ivl~,y.= f ]Vv(y)12 dy y* is a norm in Hi ( to , Y*), and it is equivalent to (2.2b). I f to = (1, 1), then this map is a norm, equivalent to (2.2b), in the quotient space H~((1, 1), Y*)/C. (2.4c)

2.2. The Orthogonal Decomposition of the Space H i 02)

Using the definitions and preliminary results of Section 2.1 we can now proceed toward our primary goals, the orthogonal decomposition of H~(I~) and its application to the solution of the eigenvalue problem (1.4).

Let tOo,. = 1, tO~,.,.. . , tO,,_~,,, be the m complex mth roots of the unity, and let too, = 1, tO~ . . . . . ,tO,_~,, be the n nth roots of the unity. For each j = 0, 1 , . . . , m - 1 and for each k = 0 , 1 , . . . , n - 1 we define tojk by

tojk = (tOi,-, tOk,), , (2.5)

i.e., the tojk's are the vectors of D x D whose first components are one of the ruth roots of 1, and its second components one of the nth roots of 1. Next, let

' H#(to~k, Y*). us consider, together with the to~k S, their corresponding spaces Since the first and second components of tojk verify

( tOjm ) m = ( tOk, ) n = 1, Vj, k,

then it is clear from (2.1a, b) that the following inclusion

H~(to~k, Y*) c H~(Y~)

holds for all j and k. Moreover, we have:

Theorem 2.2 (The Orthogonal Decomposition Theorem of H~(f l ) ) . The spaces 1 H# ( to j k , Y*), f o r j = 0 , . . . , m - 1, k = 0 , . . . , n - 1, and tojk defined by (2.5), form

a family of subspaces of H i (lq) with the following properties: m--I n--1

Hi(11) = • 2 H~(tojk, Y*), (2.6) j=O k = O

H~(tojk, Y*) c~ Hl(toj,k ,, Y*) = {0}, V(j, k) ~ (j ' , k'). (2.7) 1 1 I f (j, k) # (j', k'), then, for all u ~ H #( to~k, Y*), v ~ H #(tOj,k,, Y*), the following

( orthogonality ) conditions hold:

auO dy = O, (2.8a)

f a V u . V~dy = 0, (2.8b)

m--l l(f.y ) ( f l y ) ~ uv ds • Or ds =0. (2.8c) /1 = 0 /2 = 0 \ (I) (1)


This theorem states that the space H i ( l ) ) can be decomposed into a direct sum of the subspaces 1 H#(tojk , Y*), this family of subspaces being orthogonal with respect to the inner products of L2(I~) and HI(I~), and verifying (2.8c). Every function v in H i ( l ) ) can therefore be uniquely written in the form

m--I n--1

v= Y, ~ V~k, (2.9) j = 0 k=0

1 where Vjk ~ H#(tojk, Y*). This implies that any O-periodic function in H1(12) can be written as a sum of functions that only depend on their values on the unit cell Y*.

2.3. Decomposition of the Eigenvalue Problem (1.4) into (m × n) Elementary Eigenvalue Problems

The spaces 1 H#(tojk, Y*), f o r j = 0 . . . . . m - 1, k = 0 , . . . , n - 1, play a key role in the numerical solution of (1.4). As will be seen throughout this section, these spaces allow the eigenvalue problem (1.4) to be decomposed into (m × n) subproblems. Each one of them is formulated in a different space 1 H #(tojk, Y*), and e a c h one provides two of the 2L eigenvalues and eigenfunctions of (1.4).

The variational formulations of these problems have slightly different forms i f j and k are not both equal to zero or i f j = k = 0. In this first case, the variational formulation of these problems is:

Find t~jk ~ C , and

U~k C Hi(O~jk, Y*), Ujk ~ O, such that

f Y* V Ujk " V Vjk dy = AJk ( f ~ " ujk v ds ) " ( f v Vjk V dS ) ,

(2.10a)

(2.10b)

H #(to~k, Y*), ~ ~)jk ~ 1

(2.10c) and, i f j = k = 0, the corresponding subproblem is:

Find Ao0e C, and (2.11a)

tioo6 H i ( ( 1 , 1), Y*)/C, tioo¢ 0, such that (2.11b)

fg. Vfioo" V~oody= Aoo(f ftoov ds) " (fv f~oov ds),

Vt~oo~ H i ( ( 1 , 1), Y*)/C. (2.11c)

If j = k = 0 (in which case to = tooo = (1, 1)), the corresponding subproblem (i.e., (2.11)) is exactly of the same type as (1.4), but posed on the unit cell Y* instead of fL Applying Theorem 1.1 to (2.11), it follows that this problem admits two linearly independent (equivalent classes of) eigenfunctions and two (real positive) eigenvalues. Let us denote the eigenvalues by Alo, A2o, and their corresponding eigenfunctions by rico, ti2o, respectively. We assume the eigenfunctions are orthonormalized in such a way that they verify

f Vti~o. Vff~o dy = 6s,, Vs, t = 1, 2. (2.12)

14 F. Agui r re and C. C o n c a

From the differential interpretation of (2.11) (which is (1.3) with Y* in place of f~), and since every Y*-periodic function is also l)-periodic, it is straightforward to see that every solution of (2.11) is also a solution of (1.3) or (1.4). Thus A~o, A2o are two of the 2L eigenvalues of (1.4), and U~oo, ti~o are (the) eigenfunctions associated with these eigenvalues. Hence (2.11) provides two solutions of (1.4).

The next results show that the remaining 2(L-1) solutions of (1.4) can be obtained by a similar procedure. They summarize the main properties of the other subproblems associated with (1.4) (i.e., the problems (2.10), for j and k not both zero).

Proposition 2.3. For all j = 0 , . . . , m - 1, k = 0 , . . . , n - 1, such that j and k are not both zero, there exists two real and strictly positive numbers A)k, Aj2k, and two

2 1 functions U~k, U~k in H#(tOjk , Y*) with the following properties:

The pairs (Ajlk, 1 2 2 Ujk), (Ask, Ujk) are solutions of (2.10). (2.13a)

f VUjk" V~k dy = 8st, Vs, t = 1, 2. (2.13b)

1 I f (Ajk, Ujk) in C x H#(tOjk, Y*) is any solution of (2.10), then A = AJk or A = Aj2k, and Ujk is a linear combination of Ujk,1 Ujk.2 (2.13C)

Let b ° be the set consisting of the two eigenvalues 1 2 Aoo, Aoo of (2.11), and each one of the two eigenvalues A)k, Aj2k of (2.10), for j # 0 or k # 0 , i.e.,

{;r io , A o, , , 2 = A10, . . . , A m _ l , n _ l , A m _ l , n _ l } . (2.14a)

On the other hand, let M be the subset of ~ ( 1 ) ) that includes the equivalence classes of eigenfunctions fio~o, ti2o of (2.11) (orthonormalized by (2.12)), and of

• 1 fi~k that the eigenfunctions Ujk, Ujk of (2.10) the (two) equivalence classes Ujk , 1 2 induce in ~ ( 1 ) ) . That is,

~/~ = { / ~ 1 0 , 4 2 0 , .1 .1 .2 U I 0 , . . . , U m - - l , n - 1 , U m - l , n - - 1 } . (2.14b)

The following theorem characterizes the eigenvalues and eigenfunctions of (1.4) in terms of the solutions of the subproblems (2.10) and (2.11):

Theorem 2.4. Then the following conclusions hold true:

ff~ constitutes the entire spectrum of the eigenvalue problem (1.4).

is an orthonormal basis in ~ (1) ) of the eigenfunctions of (1.4). . s Foreachj = 0 , . . . , m - 1, k = 0 , . . . , n - 1, ands = 1, 2, Ujk is an eigen-

function of (1.4) associated with the eigenvalue Aj~k.

Let 5 ~ and tilt be the sets defined by (2.14a) and (2.14b), respectively.

(2.15a)

(2.15b)

(2.15c)

This is the central theorem of this paper. It shows that the eigenvalue problem (1.4) admits 2L eigenvalues, that to these eigenvalues there corresponds an orthonormal basis of eigenfunctions, and that the effective computation of them can be carried out by solving a finite sequence of problems that only admit two


eigenvalues. This is, using the terminology of matrices, a sort of "block diagonali- zation" (of two eigenvalues) of (1.4). In our opinion, the main advantage of using these results in the solution of (1.4) is the fact that the subproblems do not require too much computational effort to be (numerically) solved. In their formulations they only involve spaces of functions that can be completely known from their values on the unit cell Y*. Therefore, any numerical method used in solving them only needs the unit cell Y* to be discretized. To close this section, let us mention that in Section 5 we discuss a numerical example related to these results.

2.4. Complementary Properties of the Eigenvalues of (1.4)

In this section we study a continuous parametrization of the subproblems (2.10) that will enable us to explicitly state in detail the dependence of the eigenvalues of (1.4) on m and n. This parametrization consists in associating to each to in D x D - { ( 1 , 1)} an eigenvalue problem of the same type as (2.10), and then to study how the solutions of these problems depend on to. For reasons that will become clear later, the point to = (1, 1) is expressly excluded in what follows.

For each to in D x D - { ( 1 , 1)}, let us consider the following eigenvalue problem:

Find /3 =/3(to) ~ C, and (2.16a)

z = z(to) ~ H i ( t o , Y*), z ~ 0, such that (2.16b)

Iy . V Z ' V ~ d y = / 3 ( I z v d s ) ' ( I ~'J, ds), V v E H I ( t o , Y*). (2.16c)

Actually, (2.16) defines a continuous family of problems that depends on the parameter to. Let us remark that if to = tojk, took defined by (2.5) and j ~ 0 or k ~ 0, then the problems of this family are exactly (2.10). Therefore, by means of (2.16) the subproblems (2.10) can be regarded as particular elements of a continuous family of problems which is independent of m and n; and which only depends on the unit cell Y*.

The following proposition generalizes Proposition 2.3 to all to in D x D - {(1, 1)}. It shows that for each to, (2.16) has properties very similar to (2.10), i.e., that it also admits two eigenvalues with their corresponding eigenfunctions:

Proposition 2.5. Let to be given in D x D - {(1, 1)}. Then there exist two real and strictly positive numbers fll = ill(to), f12 = fl2(to), and two functions zl = zl(to), z2 = zz(to) in Hi(to, Y*) with the following properties:

The pairs (ill, zO, (/32, z2) are solutions of (2.16). (2.17a)

f Vzs. V~, dy = 6st, Vs, t = 1, 2. (2.17b) y*

I f (/3, z) in C x Hi(to, Y*) is any solution of (2.16), then/3 =/31 or/3 =/32, and z is a linear combination of zl, z2. (2.17c)


Our next step consists in studying the dependence of the eigenvalues of (2.16) on to. To this end, for each to, let us number the (two) eigenvalues and eigenfunctions of (2.16) in such a way that the following inequality holds:

0 </3,(to) ---/32(to), (2.18)

and so that the eigenvalue/3s(to) corresponds to zs(to), s = 1, 2. Numbering the solutions of (2.16) as below for all to, the eigenvalues /3~(to), /32(to) define on D x D -{(1, 1)} two positive real-valued functions. The following theorem shows an important property of these functions, which states that the eigenvalues of (2.16) depend continuously on the parameter to:

Theorem 2.6. For each t o ~ D x D - { ( 1 , 1)}, let ~31(to), /32(to) in R+ be the two eigenvalues of (2.16). Assume that they are numbered in such a way that (2.18) holds for all to. Then/31( ' ) , /32( ' ) define two functions from D x D - { ( 1 , 1)} into R+ with the following properties:

/31(" ), /32(" ) are continuous in D x D -{(1, 1)}. (2.19a)

There exists two real constants s¢1 = ~ I ( Y * ) , ~2 = ~2(Y*) , that are independent on to, such that

0 < ~, <<- ~3,(to) </3~(to) <- 6

for all to in D × D - { ( 1 , 1)}. (2.19b)

This theorem, together with Theorem 2.4, clarifies the dependence of the eigenvalues of our original problem (1.4) on m and n. They show the existence of two continuous functions/3~(. ),/32(" ) from D × D - { ( 1 , 1)} into R+ that are independent of m and n. For m, n given, these functions evaluated at the points to=tojk, j = 0 , . . . , m - l , k = 0 , . . . , n - l , j ~ O or k # 0 , provide 2 ( L - l ) eigenvalues of (1.4). The remaining two eigenvalues can be obtained by solving (2.11).

Remark 2.1. It may seem natural to extend/3~(. ), /32(.) to all D x D defining /31(1, 1),/32(1, 1) by

/3~(1, 1) = A~o, (2.20a)

/32(1, 1) = h2o, (2.20b)

where, in (2.20), h~o is the smallest eigenvalue of (2.11), and h2oo is the greatest. Nevertheless, we cannot prove that these extensions of/31( ') and/32( ' ) define continuous functions in D x D. Further, the numerical results that we discuss in Section 5 suggest that at least/3~(.) is not continuous in to= (1, 1) (see Figure 5.2). In this paper we do not study the limit behavior of/31,/32 as to goes to (1.1). This could be an interesting subject for further studies in this problem.


From Theorems 2.4 and 2.6, it is straightforward to prove the following corollary which states that the eigenvalues of (1.4) are bounded, independently of m and n:

Corollary 2.7. Let ~rnin, ~rnax in g~+ be defined by

~min = min{A10, ~:1}, (2.21a)

~:m,x = max{A2o, ~:2}, (2.21b)

where A ~o, A 20 are the two eigenvalues of (2.11 ). Then for each given pair of integers m, n, the entire spectrum b ~ of the eigenvalue problem (1.4) lies in the interval [~min, ~:m,x]. Neither ~min nor ~:max are necessarily eigenvalues of (1.4).

With this result we close our study of the eigenvalue problem (1.4). Before proceeding with the proofs of the results stated in this section, let us emphasize that this study is only valid for the case of periodic boundary conditions on the external boundary of 11. But we hope that the solution of problem (1.4) may serve as a guide to the solution of problem (1.4a, b) with a homogeneous Neumann boundary condition on the lateral sides of 11, which is a problem in several engineering applications.

2.5. Outline of Sections 3-5

The next three sections of the paper are devoted to the proofs of the results formulated in Sections 1 and 2, and discuss a numerical example. We shall adhere to the following outline:

Section 3. The Spectral Theorem. 3.1. Proof of Theorem 1.1. 3.2. Proof of Corollary 1.2.

Section 4. The Orthogonal Decomposition Theorems. 4.1. Proof of Theorem 2.1. 4.2. Proof of Theorem 2.2. 4.3. Proof of Propositions 2.3 and 2.5. 4.4. Proof of Theorem 2.4. 4.5. Proof of Theorem 2.6.

Section 5. A Numerical Example.

3. The Spectral Theorem

In this section we shall prove Theorem 1.1 and its corollary (Corollary 1.2) that were formulated in Section 1.5.

3.1. Existence and Uniqueness of 2L solutions of (1.4). Proof of Theorem 1.1

The proof of this theorem consists in showing that problem (1.4) can be reduced to finding the characteristic values (i.e., inverses of eigenvalues) of a strictly positive self-adjoint operator from C 2L onto C 2L.


The p r o o f requires the use of vectors in C 2L, and a c o m m e n t on the nota t ion to be used must be made. Let r be a vector in C 2L. Following a number ing similar to that used for the per fora t ions of l~, we shall group the componen t s of r by subvectors rE, 1= (l~, 12), Ii = 0 , . . . , m - l , 1 2 = 0 , . . . , n - 1 . Each subvector will have two componen t s rl, r 2, which are complex numbers , and in r we shall assume that the rl's are a r ranged as follows:

r = (r(o,o), ro,o ), • • •, r(m-l,O), r(o, l), • • . , r(m-l,n-1))"

A vector r o f C 2L will also somet imes be written, when specifying its components, (rl).

Let Q : C 2L --~ C 2L be the opera to r defined by

VI, (3.1)

where ti is the solut ion of the following var ia t ional problem:

ti c ~1(1~), (3.2a)

I n m-I n-1 fit Vft. V~dy = ~ E rl" f)-~ ds, V t ~ ~ ( f ~ ) . (3.2b) 11=0 12~0 (1)

Since the lef t -hand side o f (3.2b) defines a coercive sesquil inear fo rm in ~ l ( l q ) , and the r ight -hand side defines, for each r e C 2L, a l inear cont inuous form in ~(~(f~), it fol lows f rom the L a x - M i l g r a m l e m m a that (3.2) has a unique solut ion ti in ~ l ( f / ) . Therefore Q is well defined, and it is clearly l inear (and continuous) . The fol lowing lemma, that we shall p rove later in this section, summarizes the main proper t ies of Q:

Lemma 3.1. The operator Q is strictly positive and self-adjoint.

Since Q is also finite d imensional , its characteris t ic values fo rm a (finite) set o f 2L real and strictly posit ive numbers A1,. • . , A2L (not necessari ly distinct), and there exists an or thogonal basis {rj = (rjl)}j-l,2L of C 2L, fo rmed by eigenvectors o f Q, such that

h~Qrj = rj, I<-j<-2L. (3.3)

We assume that the eigenvectors are (o r tho)normal ized in such a way that the following condi t ion holds:

m--I n--I rj- rk = ~. ~ rjl" rkl = Aj6jk = Akt~jk, V j, k = 1, 2 , . . . , 2L. (3.4)

i1=0 •2=0

To each e igenvector rj we associate ~ij in ~ ( f / ) , defined as the (unique) solution o f (3.2) for r = rj.


Let us verify that the pairs {(Aj, f4t)b=~,2L satisfy (1.6a, b, c). First, since tij is the solution of (3.2) for r = ri, it satisfies

Vfij. V~dy = Y. • r~,. ~v ds, V~) ~ ~ ( a ) . (3.5a) 11=0 12=0 (I)

But r t = AjQrj, then (3.5a) is equivalent to

L m--lnl(ffy ) ( f l y ) Viii. V~dy = A i E E f~jv as • ~v as , vf2 c ~ ( ~ ) , 11=0 12=0 (I) (I)

(3.5b)

i.e., the pair (At, ti~) is a solution of (1.4) for all j, I<-j<-2L, since tij #0 . This proves (1.6a).

Let us now prove that the set {~/t}j=L2L is orthonormal in ~ l ( f l ) (i.e., (1.6b)). To prove this, let us take 6 = rig in (3.5a). We obtain

V#~lj" V~ kdy = ~ Y. rjl . Uklll dy, (3.6a) I1 =0 12~0 (I)

and using (3.3) and (3.2), it follows that the boundary integral in the right-hand side of (3.6a) is equal to Aklrkl, and (3.6a) becomes

f ~ m--1 n--1 Vfit. Vffkdy=Ak I ~ ~ rtL.rkl, (3.6b)

11~0 •2=0

which implies that

f Vzij. Vffkdy=Stk

because the r / s verify (3.4). Property (1.6b) is therefore proved. To conclude the proof of the theorem, let us verify that (1.6c) also holds.

Let (A, fi) ~ C × ~ ( f l ) be any solution of (1.4). We define r in C 2L by

r , = ( I ~(') fivds), VI.

Since (A, ti) satisfy (1.4), it can be easily checked that (A, r) verify

AQr = r,

and that fi is a solution of (3.2). Thus k is a characteristic value of Q, and r is an eigenvector of Q associated with A. Then there exists at least one j, 1 -<j-< 2L, such that A = At, and r can be written as a linear combination of the r / s that span the eigenspace associated with A. Let r j , , . . . , rt~ be these eigenvectors. Then there exists complex numbers a ~ , . . . , a~ such that r can be written as follows:

r= ~ atrj,. t= l

We define ~b in )~(12) by

~b = ~ atilt,. t= l


Multiplying (3.5a) by a,, and adding the results from t = 1 to t = s, it follows that ~b verifies

Ill m--1 n--1 I~ / Vw. V f d y = ~ Z r,. fvds,

11 =0 /2 =0 (I) w ~ ~ , ( n ) .

Thus ~b is a solution of (3.2). Since ti is also a solution of (3.2) (for the same r), and this problem admits a unique solution, we conclude that ti = ~b. This proves (1.6c), and it completes the proof of Theorem 1.1. []

Proof of Lemma 3.1. First, we prove that Q is self-adjoint. Let r ~ , r 2 E C 2L be any two given vectors, and let u~, ti: ~ Yg~(fl) be the solutions of (3.2) associated with rl, rz, respectively. Taking ~3= fi2 in (3.2b) for r = r l , it follows that

But

I~ m--I n--1 f), V t i l ' V f f 2 d y : E E rll" ~2 lyds"

11=0 /2=0 (I) (3.7a)

and

r . Or=O iff t i=0 . (3.8b)

Since the solution of (3.2) is zero if and only if r is equal to zero, (3.8b) is equivalent to

r . Q r = 0 if r = 0 , (3.8c)

which proves that Q is strictly positive, since (3.8a) holds. This completes the proof of Lemma 3.1. []

( Qr2)~= ( f~o) tiEr ds),

then (3.7a) can be rewritten as

f Vti~ • dy = r l " Q~2. (3.7b) Vff2

Interchanging the roles of rl , r2 in (3.7b), we conclude that

rl" Qr2= Qrl" r2,

which proves that Q is self-adjoint. Let us now prove that Q is strictly positive. Let r be any given vector in C :L,

and let ~/be the solution of (3.2). Taking r 1 = r 2 = r in (3.7b), it follows that

r . Q r = f IVul ~ d y _ O (3.8a) Ja


3.2. Proof of Corollary 1.2

Let k be an eigenvalue of (1.3) or (1.4) o f mult ipl ici ty s, and let { ~ 1 , . . . , z),}c N ~ ( I I ) be a basis o f the e igenspace associa ted with k. Since k is a real number , and (A, t~j) is a solut ion of (1.3) or (1.4) for all j = 1 , . . . , s, then the pairs {(k, Re(t~j))}j=l.s and {(A, Im(~j))}j=~,s are all solutions of (1.1a, b, c). Moreover , since 6j # 0, then, for each j = 1 , . . . , s, ei ther the real or the imaginary par t o f t~j is not zero. Thus k is an eigenvalue of (1.1).

Let us next prove that the real and imaginary parts o f the t~j's span a f ini te-dimensional space of d imension s. To prove this, we denote by Er and Ei the spaces spanned by the real and imaginary parts o f the t~j's, respectively. Let I r : R s'-'~ Er, I i : R s ~ Ei be two operators , defined by

Ir~t = i aj Re0)j), j = l

Ii . = ~ ~j Im(6j), j=l

where ot = ( a l , • • . , as) in R s. Let us assume for the m o m e n t that

ker(Ir) c~ ker(Ii) = {0}. (3.9)

Then using classical results f rom linear algebra, it fol lows that

d im L(R~) + dim I~(R') = s,

which proves the required result. To conclude, we prove that (3.9) is true. Indeed , i f / r ( ~ ) = Ii(ot) = 0, then

aj Re (~ j )= i t~j Im(1)j)=0. j = l j = l

Hence we have

j = l

which implies that a l = a2 . . . . . as = 0 (i.e., a = 0), since the t~j's are l inearly independent . This comple tes the p roo f of Corol la ry 1.2. []

4. The Orthogonal Decomposition Theorems

In this sect ion we shall p rove all the results fo rmula ted in Section 2.

4.1. The Spaces of (to, Y*)-Periodic Functions. Proof of Theorem 2.1

We shall divide the p r o o f of this theorem into three parts:

(a) Proof of (2.4a). Let to = (o)1, to2) in D x D be given. We begin by noting H#( to , Y*), defined by (2.2a), can also be defined as that the inner p roduc t o f 1


follows:

(u, v)l,y*= fy.(,) u(y)v(y) dy fy (,)Vu(y) " Vv(y) •

Vu, v ~ H i ( to , Y*), (4.1)

for any l~ 7/2, because the right-hand side of this expression is independent of I. Indeed, since u and v verify (2.1a), changing y~ by x~ - (Yl-/~r~), and y: by x2---(Y2-/2"r2) in these integrals, we obtain

f (u(y)~(y)+Vu(y). VtT(y)) dy Y*(I)

- - I - - I 2 f = (w,w,) '(wzw2) (u(x)O(x) +Vu(x)" Ve(x)) dx. J Y*(O, O)

But (tolo3~) - - ( ¢ o 2 o ) 2 ) = 1 , then the right-hand side of (4.1) is independent of l. Let us now prove that H i ( t o , Y*), equipped with the inner product (4.1), is

a Hilbert space. It is easy to check that H i ( t o , Y*) is pre-Hilbert. To prove completeness, let {vj} be a Cauchy sequence in H i ( t o , Y*). Then {vj} is a Cauchy sequence in H~(Gc), for any G~, a compact subset of G, and there exists therefore a function v in H~oc(G) such that

vj~v strongly in H~(Gc), as j , o o . 1 The next step consists in proving that v belongs to H#(to, Y*). To this end, let

Gc be any compact subset of G. Since {vj} converges in H~(Gc) to v, then {vj} converges in L2(Gc), and we can extract from {vj} a subsequence {vj,} such that

vj,(y)~v(y) for almost all y in G~, as j'->oo.

Then

v(yl + zl, Y2) = lim vj,(yl + zl, Y2) a.e. in G~. j ' ~ o o

Since vj, belongs to H i ( t o , Y*), it follows that

v(yl + ~'l, Y2) = Wl lim vj,(yl, y:) a.e. in G~, j'--~oo

i . e . ,

v(yl + r l , Y2) = tOlV(yl, Y2)

Similarly,

v(yl , Y2 + T2) = w2v(yl , Y2)

a.e. in Go.

a.e. in Go. 1 Therefore v belongs to H#(to, Y*), since Gc is an arbitrary compact subset of

G. Statement (2.4a) is therefore proved. []

(b) Proof of (2.4b). Let to in D x D be given. Assume that an identically constant function C belongs to H i ( t o , Y*). From (2.1) we have C = tolC = to2C. Then C = 0, if to # (1, 1). On the other hand, it is obvious that H i ( ( 1 , 1), Y*) contains all the constants. This completes the proof of (2.4b). []


(C) Proof of (2.4c). Let to ~ (1, 1) be given in D x D. From (2.4b), it is straightforward to prove that the map defined in (2.4c) is a norm in H i ( t o , Y*).

Let us next prove that there exists two real constants a l , 42 such that

VveH~(~, Y*). (4.2)

The second inequality is obviously verified with az = 1. Thus it suffices to prove the first inequality.

Assume that a, verifying the first inequality in (4.2) does not exist. Then H#(o~, Y*) such that there exists a sequence {vj} in x

l II vj Ill,Y.> I Vjll,Y.. J

We define uj = vj/II vj II,,Y.. Then the sequence {uj} verifies

II uj Ill,Y* = 1, (4.3a)

lu l l ,Y*< l / j , (4.3b)

and we can therefore extract from the sequence {uj} a subsequence, still denoted by {uj}, such that

uj~u weakly in H l ( y*) , as j ~ o o . (4.4a)

1 Since uj e H i ( t o , Y*), then u also belongs to H#(to, Y*). Moreover, from (4.3b) it follows that

V u j ~ 0 strongly in L2( y*) 2, as j ~ o o . (4.4b)

Combining (4.4a) with (4.4b), we deduce that Vu = 0, i.e., u is a constant. Using (2.4b), and the fact that u belongs to H i ( t o , Y*), it follows that u is zero.

On the other hand, since the canonical embedding of H i (Y * ) in L2(Y*) is compact, (4.4a) implies that we can extract from {uj} a subsequence, still denoted by {uj}, such that

uj ~ u strongly in L2(Y*), as j ~ oo. (4.4c)

Therefore, combining (4.4b) with (4.4c), it follows that

uj~u strongly in H l ( Y*), as j ~ .

This contradicts (4.3a), and the fact that u =0. Statement (2.4c) is therefore proved for the case t o e (1, 1). The proof of (2.4c) for the space H~((1, 1), Y*) is similar to this, and hence we shall omit it. This completes the proof of Theorem 2.1. []

4.2. The Orthogonal Decomposition of H~(l-l). Proof of Theorem 2.2

We shall also divide the proof of this theorem into three parts:

(a) Proof of (2.6). Since, for each j = 0 , . . . , m - l , k = 0 , . . . , n - l , the spaces 1 H#(~jk, Y*) are subsets of H I ( O ) , the right-hand side of (2.6) is included in


H ~ ( I I ) . Therefore , in order to prove (2.6) it suffices to prove that every O-per iodic funct ion v in H~,(I)) can be d e c o m p o s e d into a sum of funct ions Vjk'S in

1 H#(t%k, Y*), i.e., we have to prove that for each j and k there exists Vjk in H l ( ~ k , Y*) such that

m--I n--I

v = ~ ~ vjk. (4.5) j = 0 k=0

To prove (4.5), let us define ~)jk = t)jk(Y), for e a c h j = 0 , . . . , m - 1, k = 0 , . . . , n - 1, by

1 ~1 n--1 1)jk(Yl, Y2) : - - E (tOjm)-ll(tOkn)-121)(Yl -}-/17"1, y2W/272). (4.6)

mn 1,=o t2=o

First, let us verify that Vjk belongs to H~(tOjk, Y*). Indeed, a.e. in G, we have

Vjk(Yl+ ~'1, Y2) 1 ,,-1 n-l ~ ' - - E E (O)jm)--ll(c~Okn) '2v(yx+IITI+rl,y2+I2r2) mn t,=o 12--0

rl--I

=tOjm ~ ~ (tOjm)-l~(tOkn)-121)(ylq-l~7"l,Y2A-1272). (4.7) my/ /~=l 12=o

Since v is O-per iod ic , and %,. is an mth root o f the unity, the terms of this sum for 1'1 = m verify

(t%,) -m = (c%,) ° = 1, (4.8a)

v(y~+mr~,y2+Izr2)=v(y l ,y2+l:2) , V / 2 = O , . . . , n - 1 . (4.8b)

Combin ing (4.7) with (4.8), we deduce that

Vjk(yl + rl , Y2) -- O')jm m-1 n-1 - - - - ~ E (tojm)-t:(tOk,)-12v(yl+ll'rl,y2+12r2) • mn q=o 12=o

That is,

Vjk(y, + Zl, Y2) = tOjmVjk(Yl, Y2)"

Similarly,

Vjk(Yl, y2 + ~'2) = Wk,Vjk(ya, Y2). 1 Therefore , Vjk C H #(t%k, Y*).

Next, let us prove that (4.5) holds true. Indeed , for a lmost all y in G, we have

m--1 n--I 1 m--1 n--I m--1 n--1

Y, ~ V jk (Y l ,Y2)=- - ~ ~ ~ ~ (Wj,,)-t'(tOk,)-t2v(yl+ll~'l,y2+12"C2) j = 0 k = 0 / ' nn j = o k = 0 I t=0 •2=0

1 " - ~ " - ' ( ~ ' )("k~_-' -12) - E E (~j~) 1, (,ok.) my/ 11=o 12=o \ j = o o

x v(y, + ICra, y2+ 1:'2). (4.9)


Let us assume for a moment that the following identities hold:

,,-1 {O if 11=0, Y. (coj,,)-q = (4.10a) ~=o if 11 ~ 0,

" - ' {0 if 12=0, E (t0k,) - '2= (4.10b) j=o if /2#0.

Combining (4.9) with (4.10), it follows that m--1 n-- t

Y~ E Vjk(yt, Y2) = v(y,,y2), j = O k=O

which proves the required decomposit ion of v. To conclude, let us verify that (4.10) is true. First, using that %" is equal to

exp{i2~rj/m}, it follows that m--1 m--1

5~ ( tot , , ) -"= Y. exp{-i27rjlt/m} j = 0 j = 0

m - I

= • (tot,,,) -j. (4.11) j=o

Let S be the last sum in (4.11), i.e., nl--1

S = Z (t%,~) -j. (4.12) j = O

Using the fact that tot,,, is an ruth root of the unity, it is straightforward to prove that S satisfies the following equation:

--1 S=t%mS. (4.13)

Therefore, it follows that

{~n if to , , , ,= l ,

S = , v if Wilt? 1 ~1~ 1. (4.14)

But tot,,, = 1 if[ it =0 , then (4.10a) follows immediately from (4.11), (4.12), and (4.14). The proof of (4.10b) is similar. Then (2.6) is proved []

(b) Proof of (2.7). Let (j, k), ( j ' , k') be given. Assume that (j, k) ~ ( j ' , k'), and let v be a function in H~(tOjk, Y*) n H~(tOj,k,, Y*). Since v is (t~jk, Y*)-periodic and (%'k', Y*)-periodic, it follows from (2.1) and (2.5) that v verifies

v(yt + zl, Y2) = %,,,v(yl, Y2) = W~,r,,V(y,, Y2)

and

v(y~, y2+ ~'2) = Wk,,V(y~, Y2) = Wk',,V(yt, Y2)

for almost all y in G. Therefore, by difference we have

(%,, - w~,,,,)v(yt, Y2) = 0, (4.15a)

(tOkn --tOk'n)13(Ya, Y2) = 0 a.e. in G. (4.15b)

Since (j, k) ~ (j', k'), it follows from (4.15) that v = 0 a.e. in G. This proves (2.7). []


(c) Proof of (2.8). We begin by proving (2.8a). Let (j, k), (j ' , k') be given. Assume that (j, k) # (j ' , k'), and u c H~(~Ojk, Y*), v ~ H~(tOj,k,, Y*) be two given functions. Since u is (O~jk, Y*)-periodic, v is (tOj,k,, Y*)-periodic, then using (2.1) the integral in the left-hand side of (2.8a) can be decomposed as follows:

In m--l n-l l u~ dy = Z Y, ug dy Ii=O 12=0 Y*(I)

m--1 n - 1 f f = Y, X (tO~m)-I'(tOk,)-I:(~j',,)-"(d~k',) -h uedy, /t =0 /2 =0 Y*(0,0)

i.e.,

fn u ~ d y = ( f y , u~dY)(lm~i (toj,~j,,,)-I')(l~=l ° (tOk, d~k,n)-12). (4.16)

Since (toj,,o3j,m) is an mth root of unity, and (tOk,d~k,,) is an nth root of unity, then, using the same arguments used to prove (4.14) from (4.12), we have that

,.-1 ~ m if toj,,dJj,,. = 1, (tojmdJj,m) -1'= (4.17a)

t,=o [ 0 otherwise,

~-' { o i f - 1, (tOk,tOk,n) 2 = (4.17b) - --I O)knO)k'n =

1~=o otherwise.

But

and

~oj~o3j,,,, = 1 iff j = j '

m t tOknfDk, n = 1 iff k - k ,

then

m--I n--1 Y~ (tOj"O3j,m)-" = 0 or Y, (~OkntOk,,) -12 = 0,

I1=0 •2=0 (4.18)

4.3. The Subproblems Associated with (1.4). Proof of Propositions 2.3 and 2.5

We begin by noting that Proposition 2.3 is a particular case of Proposition 2.5, when tO=tOOk,j=0, . . . , m- - l , k = 0 , . . . , n - l , and j # 0 or k # 0 . Therefore, we shall only prove Proposition 2.5. The proof is very similar to that of Theorem 1.1 (see Section 3.1), and hence we shall not go into complete details of the proof.

because (j, k) # (j ' , k'). Combining (4.16) with (4.18), it follows that

a uv dy =O,

which proves (2.8a). The proofs of (2.8b) and (2.8c) follow the same pattern as the proof of (2.8a). For this reason, we shall omit them here. This completes the proof of Theorem 2.2. []


Let to in D x D - {(1, 1)} be given, and let U: C 2"-~ C 2 be the operator defined by

= Ir zv ds, ( 4 . 1 9 ) Ur

where r = (r~, r2) , and z = z(to) is the solution of the following problem:

z ~ H i ( t o , Y*), (4.20a)

f V z . V ~ d y = r . ~ v d s , Vv~Hi( to , Y*). (4.20b) y *

As in the proof of Theorem 1.1, it can be easily checked that U defines a linear strictly positive and self-adjoint operator from C 2 onto C 2. Thus U admits two real and strictly positive characteristic values/31,/32, and there exists an orthonormal basis {r~, rE} o f C 2, formed by eigenvectors of U, such that

s = 1, 2, (4.21a)

Vs, t = 1, 2. (4.21b)

/3sUrs = rs,

and

r~ • ~, = 8s , ,

To each eigenvector rs we associate zs in H i ( t o , Y*), defined as the unique solution of (4.20) for r = r~, s = 1, 2. Following the same arguments used in Section 3.1 to prove Theorem 1.1, it can be verified that the pairs (/31, z~), (/32, z2) satisfy (2.17a, b, c). Proposition 2.5 is therefore proved. []

4.4. The Orthogonal Decomposition Theorem of (1.4). Proof of Theorem 2.4

To prove this theorem we use the orthogonal decomposition theorem of H i ( f l) (see Theorem 2.2) and Proposition 2.3. First, let us note that ~ is an orthonormal set in ~ # ( ) . Effectively, let Ujk, ft~k be two elements in d/. If (j, k) ~ (j ' , k'), then

W;k'Vg,k, dy=0,

s 1 t 1 because Ujk ~ H #(tojk, Y*) and Uj,k, C H #(to~,k,, Y*), and hence (2.8b) is verified. On the other hand, if (£ k) = ( j ' , k'), then

In Vti]k • Vffjk dy = 62,

because in this case U]k, U~k verify (2.13b) or (2.12). JR is therefore an orthonormal set in Hi( f~) .

Let us now prove that each pair (Aj~k, tij~k), with A~k in 6e, ti]k in ~t, is a solution of (1.4). I f j = k = 0, this was proved in Section 2.3, before establishing Proposition 2.3. Therefore, let us assume that j and k are not both zero.


First, since u j ~ H~(COjk, Y*), and H~(OJik , y * ) c H~(f~), it follows that ti]k~ Y(~(I~). Besides that, let v he any test function in H~(I~). According to Theorem 2.2, v can be uniquely written in the form

m--I n--I v= ~, E Vj'k' (4.22)

j'=O k'=O

with 1 Vj,k,~ H¢(COyk,, Y*). Using this decompsoition of v, it follows that

VU;k" V~dy = ~, Y~ Vuj~. V~j, k, dy, (4.23) j '=0 k'=0

which, using (2.8b), implies

fc VUjk" VVdy= fnVujsk" VVjkdy. (4.24a)

Since Uj~k and V~k are (tOjk, Y*)-periodic functions, the integral in the right-hand side of (4.24a) can be decomposed in the form

f l ) m- I n-1 f VU]k" V~,k = E E Vu~" V~,k dy

It =0 12=0 Y*(I)

m--1 n--1 f y y - - 1 - - 1 = (O)jgtl(.,Ojrtl) l(O)kn(..Okn) 2 VUjk" V~jk dy, It =0 12=o Y*(O,O)

which implies that (see (4.17))

fa VU]k" V~jkdy=mn fy . Vl, ijSk " V~jkdy. (4.24b)

Combining (4.24a) with (4.24b), we obtain

f V u j k ' V ~ d y = m n f y . VUjk'V~jkdy. (4.25)

On the other hand, let A be defined by

/(I, / A= 2 • U]k V dS • bv ds . (4.26) 11=0 12=0 (1) (I)

Using again the decomposition (4.22) of v, it follows that A can be rewritten as follows:

A= Y. Z Z Z u2~vds • ~,k, Vds , 11=0 12=0 j'=O k'=O (I) (I)

which, using (2.8c), implies

A = 2 ~ U]kV ds • ~jkll ds . (4.27) 11=0 12=0 (I) (I)


Since U]k and Vjk are (tojk, Y*)-periodic functions, each term of the sum in (4.27) can be rewritten as follows:

(I~m uJ ~v ds) " (I~m ~J ku ds )

=(tOJmO)jm)-I'(~t)kn~)kn)-Iz(I~,(O,o) UjSkv dS ) " ( f,~O,O) ~Jk v ds)

Substituting this expression into (4.27), it follows that

A = m n ( fv Uj~kv dS ) . ( f , ~jkV dS) (4.28)

and using the fact that (Aj~k, Uj~k) is a solution of (2.10), (4.28) implies that

mnf A = - 7 VU;k " VVjkdy. (4.29) Ajk y*

Combining (4.26) and (4.29) with (4.25), we deduce that

V U]k " V ~ dy = Aj~k ~ Uj~kV ds • ~v ds , (4.30a) /1 =O 12:O \ J ) ' ( 1 ) ( |)

which implies that

Io )(I, ) Vft;k" V fdy = Aj~k Z ~ fij~kV ds • fv ds , (4.30b) 11=0 12=o XJy(I ) (I)

because (4.30a) holds for all U]k in tij~k and for all v in b (note that the sesquilinear forms occurring in (4.30a) are both invariant under addition of constants). Therefore, (4.30b) proves that Aj~k is an eigenvalue of (1.4), and that tij~k is an eigenfunction of (1.4) associated with Aj~k. This proves (2.15c). It also proves (2.15a, b), since it has already been proved that ~ is an orthonormal set in ~ ( f l ) . Theorem 2.4 is therefore proved. []

4.5. The Continuous Parametrization of the Subproblems. Proof of Theorem 2.6

First we prove that/31(" ),/32(" ) are continuous. Let tOo be given in D x D - {(1, 1)}, and let {tOj} be a sequence of vectors in D x D - { ( 1 , 1)} such that

tOj+tOo in C x C , as j-->oo. (4.31)

Our aim consists in proving that/31(tOj) converges to/31(tOo) in •, as j--> oo, and that/32(tOj) converges to/32(tOo). The proof of these results is based on a priori estimates of the eigenvalues /31(tO),/32(tO) of (2.16), and on an approximation lemma for (tO, Y*)-periodic functions.


We begin by establishing the following lemma that provides a priori estimates for the eigenvalues:

Lemma 4.1. There exists two real constants st1 = ~1(Y*), st2 = st2(Y*), that are independent of 110, such that

0 < ~1 ~/31(110) ~/32(t['O) ~ ~2 (4.32)

foral l 110 in D x D - { ( 1 , 1)}.

It follows from (4.32) that {/31(toj)}, {/32(toj)} are bounded, and we can therefore extract from {/31(to;)}, {/32(toj)} subsequences, still denoted by {/31(to;)}, {/32(toj)}, such that

/3,(toj) --> /3* inR, a s j-~ce, (4.33a)

/32(toj)~/32* inR, as j~oo , (4.33b)

0 < - /3* -</3*. (4.33c)

On the other hand, let z~ i = zl(toj), ZEj = z2(toj) be eigenfunctions associated with fll(to~), fl2(toj), respectively. Without loss of generality, we can assume that the eigenfunctions Zlj, ZEj are normalized in such a way that they satisfy (see (4.1))

II z,~ II1.Y*.~ = II z2j II1.Y*.~ = 1 (4.34)

for any l c Z 2. Using (2.1), and the fact that H ~ ( Y *) is reflexive, it follows that we can extract from {z~j}, {z2j} subsequences, still denoted by {zl~}, {z2j}, respectively, such that

z l ~ z * weaklyin HI(Y*(I)), as j~oo , (4.35a)

z2j~z*2 weakly in HI(Y*(I)), as j~oo , (4.35b)

for all I c Z 2. To the sequences {z~}, {z2j} we associate the sequences {rlj}, {r2j} in C 2, defined by

rsj = j~ zsjv ds, s = 1, 2. (4.36)

Since the Zlj'S, z2j's are bounded in Hi(Y*), it follows that the r~j's, r2~'s are bounded in C 2, and we can therefore extract convergent subsequences in C 2. Moreover, since the trace map is continuous, we have

rl~--> r*l = j~ z*v ds in C 2, as j ~ oo, (4.37a) #

r2j~r*2=j~z*2vds in C 2, as j ~ . (4.37b)

The next step of the proof consists in showing that (fl*, z*) and (/3", z*) are solutions of (2.16) for 110 = too. To this end, let us use the first part of the following lemma that we shall prove later in this section:

Lemma 4.2. Let too be given in D x D, and let {tot} be a sequence in D x D such that

~j-~too in C x C , as j~oo.


Then for all v in Hi(tOo, Y*) there exists a sequence {v~} properties:

vis H~(toj, Y*) for all j,

Iloj-vlI, as j~oo, ~IET/2.

with the following

(4.38a)

(4.38b)

Remark 4.1. We can note that this lemma also applies for the case too = (1, 1).

Let v be given in H~,(too, Y*) (too# (1, 1)), and let {vj} be the sequence given by Lemma 4.2. Since (/31(toj), z u) is a solution of (2.16) for to = tot, it follows that

fy Vz,j dy=/3,(to)(f z,,v ds) . ~iv ds). (4.39)

Using (4.33a), (4.35a), and (4.38b), we can pass to the limit in (4.39), as j ~ ~ , and we obtain

fy.Vz* . Ve ay = /3*( f,, z*, v ds) . ( ev ds). (4.40)

Besides that, applying the same arguments used in part (a) of the proof of Theorem 2.2 (see Section 4.1), it can be easily verified from (4.35a) that z* belongs to Hi ( too , Y*). Moreover, taking v = z u in (2.16c), and using (4.34a) and (4.36), we deduce that

2 [ II z,~ II1,,- -- 1 =/3,(toj)lrul 2+ Izul=dy, d y ,

and passing to the limit, as j ~ o o (we use (4.33a), (4.35a), (4.37a), and the fact that the canonical embedding of H~(Y*) into L2(Y*) is compact), we obtain

1=/3"1[I Z*lVdS 2+Sr. lz*12dy , (4.41)

which implies that z* ~ 0, since/3*-> 0. Therefore, (4.40) implies that (/3", z*) is a solution of (2.16) for to = too, since v is arbitrary in H i (too, Y*). In a completely analogous manner, one can prove that (fiE*, Z*) is also a solution of (2.16) for to = too- Finally, since z u and z2j are orthogonal in Hi (Y*) , for all j, it follows that ru, r2j are orthogonal in C 2, and passing to the limit we obtain that r* and r* are also orthogonal in C 2. This implies that z* and z* are orthogonal in Hi (Y*) . Since/3~(too),/32(too) are the unique eigenvalues of (2.16) for to = too, and they verify /31(too)_</32(too), it follows from (4.33c) that /3* =/31(too) and /3* =/32(too). Moreover, in (4.33a, b) all the sequences converge. This proves that /31( ') , /32( ') are continuous in too. The proof of (2.18a) is completed, since too is arbitrary in D x D - { ( 1 , 1)}.

Remark 4.2. Let/31( ' ) , f12(') be extended to to = (1, 1) by (2.20) (see Remark 2.1). Then the beginning of the proof can be applied to these functions until (4.41) is obtained. It cannot however be concluded from (4.41) that z* is not a constant, and then it is not certain that (/3", £*) is a (nontrivial) solution of (2.11).


To conclude the proof of Theorem 2.6, let us now prove Lemmas 4.1 and 4.2.

Proof of Lemma 4.1. To prove this lemma we use the fact that the eigenvalues of (2.16) can be characterized with the aid of the Rayleigh quotient. Let to in D x D-{ (1 , 1)} be given. The Rayleigh quotient is defined by

Vv e Hi ( to , Y*),

where

r~ = .I vfa) vv ds

v ~ 0 , R(v) = ~12 1 [Yvl 2 dy, (4.42a) Y*(I)

(4.42b)

and 1 is any vector in Z 2. Since the (to, Y*)-periodic functions verify (2.1), it is easy to verify that the definition of the Rayleigh quotient is independent of i in Z 2.

The minimax principle (see Courant and Hilbert [2]) states that

/31(to) = Min R(v), (4.43a) v~Hl(to, Y *)

/32(to) = Min Max R(v), (4.43b) E2~J;(Hl (~t~,y*)) vEE2

where

~:(Hi( ta , Y*)) = {E21 E2 is a subspace of Hi ( to , Y*), dim E2 = 2}.

First, we prove the existence of an upper bound for the eigenvalues. For each to in D × D-{ (1 , 1)}, let Vo(to) be the space defined by

Vo(to) = { v e H#(to, Y*)[ v = 0 on 0 Y(I), Vlc Z2}.

We remark that Vo(to) is a closed subspace of 1 H#(to, Y*). Besides that, let us intoroduce the space Wo, defined by

Wo = {v'e Hl(Y*)lv'=OonOY}, and let us consider in Wo the following eigenvalue problem:

Find a e C, and (4.44a)

u ~ Wo, u ~ 0, such that (4.44b)

fy, VU'V~dy=a(f uvds) "(I ~vds), Vve Wo. (4.44c)

This problem is of the same type as (2.16). Following similar arguments used in the proofs of Propositions 2.3 and 2.5, or in Theorem 1.1, it is an easy matter to prove that the eigenvalues of (4.44) form a finite set of two real and strictly positive numbers al - a2.


By virtue of (4.43b), and the fact that Vo(tO) is a vector subspace of H ~ ( ~ , Y*), it follows that

/32(to) --- Min Max R(v). (4.45) E2~ ~(V0(to)) V~E 2

Let us assume for a moment that the right-hand side of this inequality is independent of to, and that it is equal to

Min M a x R ( v ) = a 2 , V t ~ D x D - { ( 1 , 1)}, (4.46) E2C ~:(V0(to)) v ~ E 2

where a2 is the greatest eigenvalue of (4.44). Therefore, (~.45) proves the upper inequality in (4.32) with ¢2 = t~2.

To prove (4.46) we again use the minimax principle. We have

a2 = Min Max R(v). (4.47) E~(Wo) wE~

Let E2~ ~(Vo(tO)) be given. We define ¢ __ E2 - {v'l v' = V ly* for some v ~ E2}.

From the definitions of the spaces Vo(tO), Wo, it follows that

E~ ~ ~:(Wo). (4.48a)

Moreover, since the Rayleigh quotient only depends on the values of the functions on Y*, we have

R ( v ' ) = R ( v l y . ) = R ( v ) , Y v ~ E 2 . (4.48b)

From (4.48a) and (4.48b), we deduce that

Min Max R(v) >- a2. (4.49) E2e ~(Vo(tO)) v ~ E 2

On the other hand, let E~ he given in ~(W0). We set

E2 = {vl v(y, , Y2) . . . . l,,. l:,,q,, w 1 ~'2 ~ ~1 - !1 r l , Y2 -/272) for some v' c E ~}.

Using the definitions of Vo(t~) and Wo, it can be easily proved that

E2 s ~(Vo(~)) . (4.50a)

Moreover, since ]Wll t, = I,o=1'= = 1, we have that

R(v) - 1 2 fy.(,) IVy'(x) 12 dy = R(v'), (4.50b)

f~, v '(x)v dy (I)

where x = (Yl - llrl, Y2- 12z2). Therefore, using (4.50a) and (4.50b), it follows that

Min Max R(v) -< ot2, (4.51) E2e ~:(Vo(oJ)) rE/5 2

which proves, together with (4.49), the required result. The proof of the lower bound in (4.32) is simpler. Effectively, using the

minimax principle and similar arguments as in the first part, it follows that

/31(to)--- t~l, Vto~ D x D - { ( 1 , 1)}, (4.52)

34 F. Aguir re and C. Conca

where ~ is the smallest eigenvalue of the following problem:

Find ~ ~ C, and (4.53a)

u c H~(Y*), u nonidentically constant, such that (4.53b)

fv.Vu. Wdy=, (f u,,ds) .(f e,,as), VvcHI(y* ) . (4.53c)

Since 51> 0, (4.52) proves the lower bound in (4.32). This completes the proof of Lemma 4.1. []

H#(tOo, Y*) be given. Assume that too# (1, 1). Proof of Lemma 4.2. Let v in For each j, we define

e j = (Wlj/¢.O10 , W2j /W20) , (4.54)

where (w~j, w2j), (W~o, W2o) are the components of ¢oj, ¢Oo, respectively. Define vj by

vj = %v, (4.55a)

where q~j is the following (smooth) function:

~Pj(Y,, Y2) = (elj)Yl/"(e2j) yj~. (4.55b)

Let us prove that the sequence {vj} so defined verifies (4.38a, b). To this end, we first note that ~j belong to cg°°(G)c~ H~(ej , Y*), and that vj therefore belongs

1 1 to H~o¢(G). Besides that, since ~j~ H#(I~j, Y*), and v~ H#(tOo, Y*), it follows that

vj(Yl-k- ,rl, Y2) = %(Yl -.b ,.rl , y2)v(yl-.b ,1.1, Y2)

= EljC'OIo~Oj(Yl, y2)v(yl, Y2),

which implies (using (4.54) and (4.55a))

vj(y~ + r,, Y2) = w,jvj(y,, Y2). (4.56a)

Analogously,

I ) j (Yl , Y2 ''b "/'2) = O)2jl)j(Yl , Y2 ) . (4.56b) 1 Therefore, vj belongs to H#(coj, Y*), since (4.56) holds a.e. in G. This proves

that the sequence {vj} verifies (4.38a). To prove (4.38b) we first note that

I[ (1- ~j) IIL~(y.)-, 0, as j + ~ ,

because ej ~ (1, 1), as j ~ ~ . Moreover, we have

V~j=~oj(llog e,j, l l o g e2j)

and hence

I1V~j IIL~(~,.)2-~ 0, as j ~ .

(4.57a)

(4.57b)


Using (4.57a) and (4.57b) we can prove that the Ha(Y*( l ) ) -norm of ( v - v j) converges to zero, as j ~ ~ , for all ! in Z 2. To simplify the notation we take 1 = (0, 0) but the same holds for any 1. We have

I Iv -v j l l 2 ,,Y* = II v (1 - ~j)

which implies that

II v - vj 11,2,i . - < II (1 - ~j)II ~=¢ y.)II v II,~,y. + II v 11~2¢ y.)II v ~ j 11~=¢ ~.)2.

Then using (4.57), it follows that

11 v - vj I1,~,~.-, o, as j--, oo,

which proves (4.38b). This completes the proof of the lemma. []

5. A Numerical Example

The test problem that we have studied from a numerical point of view consists of the unit cell Y* shown in Figure 5.1.

To simplify, throughout this section we shall assume that the parameter to = (w,, w2) in D × D is written in the following equivalent form:

O ) 1 = exp{i0~}, (5.la)

to2 = exp{ i02}, (5.1b)

where 01, 02 are real numbers in [0, 27r[. This means that in what follows the set D x D will be identified with [0, 2~-[ 2 by means of the homeomorphism

0 = (01, 02) c [0, 2~r[ 2-~ to = (w,, w2) 6 D x D,

defined by (5.1). For the numerical solution of the subproblems (2.16), and (2.11), we have

used Lagrange finite elements of degree 1 on triangles. The corresponding linear

Y2

2/3

1/3

y°

I I

0 113 213

Figure 5.1


,-A

VI

v

o

t ~

E Z

~d

ao

m


10.89

8.60

6.97

Figure 5.2.

27r

10.89 ~ ~ ~ : : i

6.97

(b)

Graphical representation of fl~(. ) under two different points of view.

systems associated with the discretization of these problems have been solved by using a conjugate-gradient method where, at each step of the algorithm, the gradient is projected over the space of (to, Y*)-periodic functions. To solve the linear systems it is not convenient to use a direct method, because the (to, Y*)- periodicity condition destroys the band structure of the matrices. An interesting advantage of using a conjugate-gradient method to solve the systems is the fact that it can be implemented in such a way that the matrices do not have to be stored.

In order to obtain accurate approximations of the solutions of the subproblems, a fine mesh of Y* is required, since we are using polynomial interpolation functions of degree 1. We have therefore used a uniform triangulation of Y* consisting of 256 triangles. This means that each lateral side of Y* contains 13 mesh-points, and each lateral side of the hole contains five mesh-points.

The subproblems (2.16) were solved for 1024 different values ofoJ or 0 (which implies that the eigenvalue problem (1.4) was solved for a domain 11 containing 1024 holes; m = n = 32). The different values of 0 were taken in a uniform mesh of [0, 217"[ 2, having a step size equal to 7r/16. Using the symmetries of Y*, it is not necessary to solve effectively all the problems. Actually, since Y* is symmetric with respect to Yl =1, y2=½, and Yl =Y2, it can be easily verified that it suffices to solve (2.16) for those values of 0 such that 0 - < 01- < 02- < ~.

A survey of the numerical results that we have obtained for the eigenvalues is given in Table 5.1 and Figures 5.2 and 5.3. Table 5.1 contains the values of

10.89

8.60

10.89

Figure 5.3. Graphical representation of ~2(' ) under two different points of view.


/3~(.), /32(" ) at the mesh-points of [0, ,//.]2. They represent a quarter of all the eigenvalues of (1.4) for m =32 and n = 32. The remaining eigenvalues can be obtained by symmetrizing Table 5.1 with respect to 01 = ~ and 02 = zr. On the other hand, Figures 5.2(a) or (b) (resp. 5.3(a) or (b)) give graphical representations of the function/31( ' ) (resp. /32(" )) under two different points of view. In these figures the black points denote the eigenvalues 1 2 A o0, A oo (see (2.11 )) corresponding to the cases 01 = 02 = 0 or 2~'. Figure 5.2 clearly suggests that/3~ is not continuous at the point (1, 1). To close, let us mention that, for each to, the conjugate-gradient method requires about 25 iteration to solve the subproblem (2.16).

Acknowledgments

The authors would like to express their gratitude to F. Mignot and J. P. Puel for raising the questions that motivated our paper, and for many written communicat ions during the course of this research. The thanks extend also to F. Murat for reading the manuscript and for helping us to get over some rough spots.

References

1. Bohr H (1947) Almost Periodic Functions. Chelsea, New York 2. Courant R, Hilbert D (1962) Methods of Mathematical Sciences. Interscience, New York 3. Ibnou Zahir M (1984) Probl~mes de valeurs propres avec des conditions aux limites non locales.

Th~se de 36me cycle, Universit6 Pierre et Marie Curie, Paris VI 4. Planchard J (1982) D6veloppements asymptotiques de fr6quences propres d 'un faisceau de tubes

61astiques plong6s dans un fluide incompressible. EDF Bull Direction Etudes Rech S6r C 2:65-76 5. Planchard J (1982) Eigenfrequencies of a tube bundle placed in a confined fluid. Comput Methods

Appl Mech Engrg 30:75-93 6. Planchard J (1984) Comportement vibratoire des assemblages combustibles d 'un r6acteur: un

module simplifi6. EDF Bull Direction l~tudes Rech Ser C 1:5-15 7. Planchard J, Ibnou Zahir M (1983) Natural frequencies of tube bundle in an incompressible

fluid. Comput Methods Appl Mech Engrg 41 :47-68 8. Planchard J, Remy F N, Sonneville P (1982) A simplified method for determining acoustic and

tube eigenfrequencies in heat exchangers. Pressure Vessel Technol 104:175-179

Accepted 5 March 1987

Eigenfrequencies of a tube bundle immersed in a fluid

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