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Asymptotics of Arbitrary Order for a Thin Elastic Clamped Plate, II. Analysis of the Boundary Layer Terms Monique Dauge & Isabelle Gruais Abstract. This paper is the last of a series of two, where we study the asymptotics of the displacement in a thin clamped plate as its thickness tends to 0 . In Part I, relying on the structure at in nity of the solutions of certain model problems posed on unbounded domains, we proved that the combination of a polynomial Ansatz (outer expansion) and of a boundary layer Ansatz (inner expansion) yields a complete multiscale asymptotics of the displacement and optimal estimates in energy norm. The \pro les" for the boundary layer terms are solutions of such model problems. In this paper, adapting Saint-Venant's principle to our framework, we prove the results which we used in Part I. Investigating more precisely the structure of the boundary layer terms, we go further in the analysis performed in Part I: the introduction of edge layer terms along the intersections of the clamped face with the top and the bottom of the plate respectively, allows estimates in higher order norms. These edge layer terms are constructed with the help of stable asymptotics, and are the singular parts of the boundary layer terms. As a by-product of all these investigations, we obtain expansions and estimates for the stress tensor in various anisotropic norms, and also estimates in L1 - norm form the displacement eld.

To appear in Asymptotic Analysis

CONTENTS

1 The origin of boundary layer terms. Outline of the paper

4

2 A simple example: the Laplacian

9

1.1 The scaled plate problem . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Multi-scale asymptotics . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Reduced-normal problems . . . . . . . . . . . . . . . . . . . . . . . .

4 5 6

3 Saint-Venant Principle

11

4 Behavior at in nity of solutions in the half-strip

14

5 Boundary layer terms along the edges of the plate

19

6 Estimates for the displacement and the strain and stress tensors

26

References

28

3.1 The main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 The exponential growth . . . . . . . . . . . . . . . . . . . . . . . . . 13

4.1 Fredholm operators in weighted spaces . . . . . . . . . . . . . . . . . 14 4.2 Proof of Theorem 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 4.3 Existence of exponentially decreasing pro les . . . . . . . . . . . . . . 19 5.1 Singular exponents . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 5.2 Stable singular functions . . . . . . . . . . . . . . . . . . . . . . . . . 23 5.3 Expansion of u(") near the edges . . . . . . . . . . . . . . . . . . . . 25

6.1 Error estimates for the displacement eld . . . . . . . . . . . . . . . . 26 6.2 Error estimates for the scaled strain tensor . . . . . . . . . . . . . . . 27 6.3 Error estimates for the scaled stress tensor . . . . . . . . . . . . . . . 27

2

INTRODUCTION

Boundary value problems in thin plates appear as singularly perturbed problems with respect to the thickness parameter " , and the boundary conditions on the part of the boundary transverse to the mean surface give rise in general to boundary layer terms. More precisely, let us introduce thin plates as a family of three dimensional domains of the form " = !  (?"; +") where the two-dimensional domain ! is the mean surface of the plates; an interesting question is the investigation of the asymptotic behavior as " ! 0 of the solution u" of the linear elasticity system on " . In general the boundary conditions on the lateral side ?"0 = @!  (?"; +") cannot be satis ed by any polynomial Ansatz for u" and as usual in singularly perturbed problems, see Il'in [14], there appears a boundary layer in the neighborhood of ?"0 . To our knowledge, the only situation where there is no boundary layer, is the case of periodic boundary conditions on the lateral side ?"0 (when ! is a rectangle, see J. C. Paumier [23]): indeed the periodicity conditions imposed on the right hand sides cancel any boundary layer terms. Boundary layer terms can be roughly described by saying that they behave like ? c e r=" , where r is the distance to ?"0 . For isotropic materials, it is known that the possible values of the constant c are determined by the Papkovich-Fadle eigenfunctions, see R. D. Gregory & F. Y. Wan [13]. Even two-dimensional models for thin plates, as the Reissner-Mindlin model, are singularly perturbed problems: see the papers by D. N. Arnold & R. S. Falk [2, 3] where the boundary layer terms of the Reissner-Mindlin model have been exhibited. In the part I of this paper [11] we constructed an expansion of the solution u" as the sum of an outer (polynomial) part and inner (boundary layer) part, with estimates in H 1 and L2 of the error between truncated expansions and the solution itself. The remaining part of the proof is that relating to the behavior at in nity of the solutions of some model problems on a half-strip, which may be referred as Saint-Venant problems. In this part II, we prove that such solutions can be split into the sum of a rigid displacement and an exponentially decreasing term (this result was stated in [11] and referred as Theorem 3.2 and Corollary 3.4). We also extend the results of [11] by the investigation of estimates in other norms: error on the strain and the stress tensors, or estimates in L1 or H 2 norms for instance, which requires the splitting of the solutions of dierent problems in regular and singular parts in a stable way, cf [9], with respect to the curvilinear abscissa s along the boundary @! , which can be considered as a parameter. The outcome is that the singular part of the expansion of the three-dimensional solution is concentrated in the boundary layer terms. ACKNOWLEDGEMENTS

The authors are glad to thank P. G. Ciarlet and M. Costabel for valuable discussions and remarks. 3

This work is part of the Human Capital and Mobility Program \Shells: Mathematical Modeling and Analysis, Scienti c Computing" of the Commission of the European Communities (Contract no ERBCHRXCT940536). 1 THE ORIGIN OF BOUNDARY LAYER TERMS. OUTLINE OF THE PAPER

1.1 The scaled plate problem In the plate " = !  (?"; ") , where !  R2 is a bounded plane domain with a smooth boundary, we investigate the behavior as " ! 0 of the displacement u" in the case when the plate " is clamped along its lateral face ?"0 = @!  (?"; ") and when the governing equations are those of the linearized elasticity. Denoting by A the rigidity matrix and by e(u") = (eij (u"))ij the linearized strain tensor de ned by eij (v) = 21 (@ivj + @j vi) , the displacement u" of the clamped plate corresponding to volume forces f " is the unique solution of the variational elasticity problem:

u" 2 V ( " ); 8v" 2 V ( " );

Z

"

A e(u") : e(v") =

Z

"

f "
v";

(1.1a) (1.1b)

where the variational space V ( ") is de ned as

V ( ") = fv = (v1; v2; v3) 2 H 1( " )3 j v = 0 on ?"0g: The 9  9 rigidity tensor A , with coecients Aijkl , satis es the usual symmetry relations Aijkl = Ajikl = Aijlk = Aklij and is supposed to be uniformly positive de nite, namely:

8 x" 2 "; 8(tij ) 2 R9 s.t. tij = tji; Aijkl(x") tkltij  c t2ij

(1.2)

where c > 0 is a positive constant. Moreover, we assume that the coecients of A do not depend on the \vertical" coordinate x3 and smoothly depend on the in-plane variables (x1; x2) and, as in [12], [24] that 8 < A 3 = 0 8; ; 2 f1; 2g (1.3) : A333 = 0 8 2 f1; 2g: The relations (1.3) are satis ed for any isotropic material, and allow the uncoupling of the system of elasticity into bending and membrane problems. Problem (1.1) is studied with the help of a scaling in order to set the problem on the reference set = !  (?1; 1) , which is the image of " through a dilatation along the normal direction to the plane containing ! :

x" = (x"1; x"2; x"3) 2 " 7?! x = (x1; x2; x3) = (x"1; x"2; "?1x"3) 2 : 4

The corresponding \scaled" linearized strain tensor denoted by (")(v) is de ned for any function v 2 H 1( )3 by:

 (")(v) = e (v); 3(")(v) = "?1 e3(v); 33(")(v) = "?2 e33(v);

(1.4a)

where we use the convention that the Greek indices  and span the set f1; 2g . Correspondingly, the convenient scaling of the components of u" writes

u(")(x) = u"(x") and u3(")(x) = " u"3(x");

(1.4b)

for which the resulting scaled displacement u(") satis es e(u")(x") = (")(u("))(x) . The corresponding canonical scaling for the body forces writes

f(")(x) := f" (x") and f3(")(x) := "?1 f3" (x"):

(1.4c)

Our analysis holds if the scaled body force f (") satis es an asymptotic property like

f (")(x) ' f 0(x) + " f 1(x) + "2 f 2(x) +


+ "k f k (x) +




(1.5)

Then the scaled displacement u(") solves the new problem:

u(") 2 V ( ); 8v 2 V ( ); where

f (")
v;

(1.6a) (1.6b)

V ( ) = fv 2 H 1( )3 j v = 0 on ?0 = @!  (?1; 1)g:

(1.7)

Z



A (")(u(")) : (")(v) =

Z



1.2 Multi-scale asymptotics In part I of this work, we exhibited an algorithm of construction of a multi-scale asymptotics for u(") . Our result can be compared with many others, relating to various problems, e.g. the homogeneization of periodic elastic structures [22], related problems in thin plates [2, 3], [25], [21]. The outcome of this algorithm is an asymptotics for u(") of the form u(")(x) ' 0(x; "r ) + " 1(x; "r ) + "2 2(x; r" ) +


+ "k k (x; "r ) +


with r the distance to the clamped part of the boundary ( r is also the distance to @! in the in-plane variables (x1; x2) ) and 0(x; t) = u0(x); (1.8a) k k k (x; t) = u (x) ? (r) w (t; s; x3) for k  1 (1.8b) 8 ?1 > < t =k r " ; s is a curvilinear abscissa along @!; with > w (t; s; x3) is uniformly exponentially decreasing as t ! +1; (1.8c) :  is a cut-o function equal to 1 in a neighborhood of r = 0. 5

In thisPexpansion, the polynomial or outer part is Pk  0 "k uk , whereas the remaining sum k  1 "k wk ( "r ; s; x3) is the boundary layer or inner part. The necessity of such a multi-scale Ansatz was proved in Part I [11], where was explained how the Dirichlet traces of the polynomial part of the Ansatz could be compensated by those of the boundary layer part. Roughly speaking, the equations inside and the Neumann boundary conditions involved in (1.6) can be solved by a polynomial Ansatz. Thus the remaining part has to solve a non homogeneous Dirichlet problem, with homogeneous Neumann and interior right hand sides.

1.3 Reduced-normal problems As already hinted, we introduce smooth local coordinates (r; s) in a tubular neighborhood V of @! : r is the distance to @! and the curvilinear abscissa s belongs to S the disjoint union L1S1 [



[
LI S1 , where L1; : : :; LI are the lengths of the connected components of @! . The change of variables (x1; x2; x3) 7! (r; s; x3) maps V onto a domain e = (0; )  S  (?1; 1) with lateral face ?e 0 = f0g  S  (?1; 1) and the problem solved by the terms wk has the form e 3 and w(") = h(") on ?e 0; w(") 2 H 1( ) (1.9a) Z e; 8v 2 V ( ) Ae ~(")(w(")) : ~(")(v) + Ae0(w("); v) = 0; (1.9b)

e where the 9  9 matrix Ae = Ae(r; s) has the same features as A (positivity, smoothness and property (1.3) on the coecients), ~(") is the scaled strain tensor in the variables (r; s; x3) and Ae0 is a rst order integro-dierential form: if i and j denote multi-indices in N3 : X Z 0 e c (r; s) @ w @ v: A (w; v) = e

j j+j j  1 ij

i

i

j

j

Keeping in mind that r is now a fast variable as x"3 , we set t = r "?1 | like x3 = x"3 "?1 . Thus @t = " @r and for purposes of homogeneity we introduce the following change in the displacements (wr ; ws; w3) 7?! ('t; 's; '3) = (" wr ; " ws; w3): (1.10) Now, problem (1.9) is transformed into the problem on e := R+  S  (?1; 1) : e 3 and ' = g on ?e 0 ; ' 2 H 1() (1.11a) Z 4 X e; 8v 2 V () Ae("t; s) e(@t; 0; @3)(') : e(@t; 0; @3)(v) + "k Aek ('; v) = 0; (1.11b) e k=1 where Aek is a second order integro-dierential form: X Z k k e A ('; v) = c ("t; s) @ ' @ v: j j; j j 1 e ij

i

j

6

i

j

With the help of the Taylor expansion of the coecients ck in t = 0 , we obtain for problem (1.11) the formal expansion e 3 and ' = g on ?e 0 ; ' 2 H 1() (1.12a) Z X e; Ae(0; s) e(@t; 0; @3)(') : e(@t; 0; @3)(v) + "k Bk ('; v) = 0; (1.12b) 8v 2 V () e k 1 where for any k  1 , Bk is a second order integro-dierential form: X Z k k B ('; v) = b (t; s) @ ' @ v j j; j j  1 e whose coecients bk are polynomials of t with smooth coecients in s . Thus, the only slow variable is now s and we see that the \principal part" (in the sense of the powers of " ) of this problem does not involve any more the tangential derivation @s : thus the variable s is a mere parameter. The following de nition is now natural (it was referred as De nition 3.1 in [11]). De nition 1.1 The matrix Ae(r; s) being the matrix transformed from A by the change of variables (x1; x2) 7! (r; s) , we denote by B (s) the matrix B (s) = Ae(0; s): The matrix B (s) is positive de nite, depends smoothly on s 2 S and its coecients satisfy the same symmetry properties as A . Let + and 0 denote the semi-in nite strip and its lateral face: + = f(t; x3) j t > 0; ?1 < x3 < 1g and 0 = f(t; x3) j t = 0; ?1 < x3 < 1g: The space of admissible displacements on + is V (+ ) = fv 2 H 1(+)3 j v = 0 on 0g: (1.13) For each s xed in S , the reduced-normal problem is the mixed Dirichlet-Neumann problem in + : ' 2 H 1(+ )3 and ' 0 = g; (1.14a) ij

ij

i

i

j

j

ij

8v 2

V (+ );

Z

+

B (s) e(@t; 0; @3)(') : e(@t; 0; @3)(v) = 0;

(1.14b)

with unknown ' = ('t; 's; '3) and data g = (gt ; gs; g3) . Thus the canonical reduced-normal problem has zero interior (in + ) and Neumann (on R+  f?1; 1g ) data. But, due to the presence of the operators Bk in formula (1.12b), the pro les wk appearing in (1.8b) are associated to functions 'k according to (1.10) which have to solve a recursive system of the form: 8 k k > > < ' (s) 0 = g (Zs) Z k?1 X j k?j k > + B (' ; v) 8 v 2 V ( ) ; B ( s ) e ( @ ; 0 ; @ )( ' ) : e ( @ ; 0 ; @ )( v ) = ? > t 3 t 3 : + + j =1 (1.15) 7

for k  0 , with some (smooth) trace functions gk (s) . Thus we are lead to consider non zero interior right-hand sides and we introduce Sobolev spaces with an exponential weight: De nition 1.2 Let  2 R . For m  0 let Hm(+ ) be the space of functions v such that etv belongs to H m(+ ) . Let V0(+) be the space of distributions w = (wt; ws; w3) such that etw belongs to the dual space of V (+ ) . Let( B satisfying B is a 9  9 positive de nite matrix with constant coecients; B has the symmetry properties Bijkl = Bjikl = Bijlk = Bklij . The general form of problem (1.14) is ' 2 H 1(+ )3 and ' = g ;

8v 2 V (+ );

Z

+

0

0

B e(@t; 0; @3)(') : e(@t; 0; @3)(v) =

Z

+

f v;

(1.16) (1.17a) (1.17b)

for g 2 H 1=2( 0)3 and f 2 V0(+) for an  > 0 . We are going to prove that the solutions of these Dirichlet-Neumann problems are the sum of a term exponentially decreasing at in nity and of a rigid displacement associated to the strain tensor e(@t; 0; @3) . It is straightforward that this space R of rigid displacements is the space of dimension 4 : R = fR = (Rt; Rs ; R3) = (ct; cs; c3) + cn (?x3; 0; t) j ct; cs ; c3; cn 2 Rg: (1.18) Theorem 1.3 Let  > 0 . For all g 2 H 1=2( 0)3 and for all f 2 V0(+ ) there exists a unique rigid displacement R = R(g; f ) 2 R so that the problem ' 2 H 1(+ )3 and ' = g + R ; (1.19a)

8v 2

V (+);

Z

+

0

0

B e(@t; 0; @3)(') : e(@t; 0; @3)(v) =

Z

+

f v;

(1.19b)

has a (unique) solution. Moreover, there exists 0 > 0 , which only depends on the coecients of B , such that if  < 0 , this solution ' belongs to H1 (+ )3 . Remark 1.4  The optimal value of 0 is given in (4.13) below. It is the real part of the \ rst" singularities at t = ?1 of the Neumann problem on the whole strip R  (?1; +1) .  If we de ne (1.20) K1 (+) = \ H1(+)3  < (@tt + @33)w = 0; on  @3w = 0; on x3 = + (2.1) ?1 > : w = g; on t = 0: Taking advantage of the possibility of separation of variables for
, we expand the Dirichlet data g in a basis of eigenvectors of the Neumann problem on the interval (?1; 1) : X + X g = g` cos `x3 + g`? sin(` ? 21 )x3 `0

`1

It is easy to see that the unique temperate solution of the Dirichlet-Neumann problem (2.1) is given by 1 X + X w = g` cos `x3 e?`t + g`? sin(` ? 21 )x3 e?(`? 2 )t: `0

`1

9

Therefore, w is exponentially decaying when t ! +1 if and only if g0+ = 0 , i.e. Z +1

?1

g(x3) dx3 = 0:

Relying on this result, we can solve the general problem corresponding to (1.17) 8 + > < (@tt + @33)w = f; on  @3w = 0; on x3 = + (2.2) ?1 > : w = g; on t = 0; with an exponentially decreasing right hand side f in H?1(+ ) , by  the solution of the problem with f = 0 ,  the odd extension f~ of f to the full strip  = R  (?1; 1) , in order to solve (2.2) with g = 0 ,  the expansion of f~ in the above basis of eigenvectors of the Neumann problem on the interval (?1; 1) which yields coecients f`+ (t) and f`? (t) ,  a Fourier-Laplace transform on these coecients + f^`?(; x3) =

Z +1

?1

+

e?t f`?(t; x3) dt;

for Re  = ?

which yields the elementary equations  2 2 2 +  ? `  w^` ( ) = f^`+ ( ); Re  = ?  2  ? (` ? 21 )22 w^`? ( ) = f^`? ( ); Re  = ?;

 the inverse Fourier-Laplace transform.

The outcome is the following statement corresponding to Theorem 1.3 Theorem 2.1 Let  > 0 . For all g 2 H 1=2( 0) and for all f 2 H?1 (+ ) there exists a unique constant c0 so that the problem 8 + > < (@tt + @33)w = f; on  @3w = 0; on x3 = + (2.3) ?1 > : w = g + c0 ; on t = 0; has a solution in H 1 (+ ) . Moreover, if  < 2 , this solution w belongs to H1 (+) . R Of course c0 = ? 21 ?+11 g(x3) dx3 .

10

3 SAINT-VENANT PRINCIPLE

3.1 The main result This section is devoted to the proof of the Saint-Venant Principle, about the behavior of solutions of the homogeneous elasticity system on any strip (0; L) = (?1; 1)  (0; L) : ' 2 H 1((0; L))3 and ' = 0; (3.1a)

8v 2 V ((0; L));

t=0

Z

(0;L)

B e(@t; 0; @3)(') : e(@t; 0; @3)(v) = 0;

(3.1b)

where V ((0; L)) is the subspace of H 1((0; L))3 of triples v with null traces on t = 0 and t = L and B is a 9  9 positive de nite rigidity matrix with constant coecients. Moreover, we introduce the condition of zero ux against rigid displacements through 0 , namely:

8R 2 R;

(' j R) :=

Z +1

?1

it(')(0; x3) Ri (0; x3) dx3 = 0;

(3.2)

where it(')i=1;2;3 are the components of the normal stress Be(@t; 0; @3)(')n at t = 0 , and R is the set (1.18) of rigid displacements related to the strain tensor e(@t; 0; @3) . Lemma 3.1 If the displacement ' is solution of (3.1), then for any rigid displacement R we have the conservation of ux:

8t 2 [0; L]; (' j R) =

Z +1

?1

it(')(t; x3) Ri(t; x3) dx3:

Proof. As ' satis es (3.1b), Green's formula yields for all t 2 (0; L] Z Z +1 B e(@t; 0; @3)(') : e(@t; 0; @3)(R) = it(')(t;
) Ri(t;
) ? it(')(0;
) Ri (0;
): ?1

(0;t)

Since e(@t; 0; @3)(R) = 0 , the conclusion follows immediately. The Saint-Venant Principle expresses that a solution of the homogeneous elasticity system with zero ux on the semi-in nite strip is exponentially increasing. The following statement and proof are inspired by the corresponding theorem in [22]. Theorem 3.2 (Saint-Venant Principle). Any solution ' of the problem (3.1) with condition (3.2) satis es the estimate: Z

(0;r)

je(@t; 0; @3

j

)(') 2 dt dx

3



e?A(L?r)

Z

(0;L)

je(@t; 0; @3)(')j2 dt dx3 (3.3)

where A > 0 is a positive constant independent of the parameters r and L .

11

Proof. In equation (3.1b) we use a convenient test-function. To this end, we introduce = (t) de ned by : 8 A(L?r) > > < eA(L?t) when 0  t  r (3.4) (t) = > e when r  t  L > : 1 when L  t where A is a positive constant which will be determined later on. The de nition of and the boundary condition (3.1a) yield that v = ( ? 1)' belongs to the test space V ((0; L)) . Using this v in (3.1b), we get: Z

(0;L)

B e(@t; 0; @3)(') : ( ? 1) e(@t; 0; @3)(') = ?

Z

(r;L)

it(') @t 'i

(3.5)

as a consequence of the identity @t = 0 when 0  t  r by construction of . We divide the interval (r; L) into the truncated segments 1r+s = (r + s; r + s + 1) of length 1 . Then, it may be noticed that @t = ?AeA(L?t) = ?A in 1r+s , which yields the estimate: Z Z r+s+1 Z +1 it(') @t 'i  A it(') ('i ? Ri) dx3 dt (3.6) 1r+s

?1

r+s

for any xed element R 2 R because of the assumption that the ux is zero against the rigid displacements and the conservation of ux (Lemma 3.1). It follows: Z Z A ( L ? r ? s ) it(') @t 'i  A e it(') ('i ? Ri) (3.7) 1r+s

1r+s

since we have the inequality eA(L?r?s?1)   eA(L?r?s) in 1r+s : Thus: Z 1 it(') @t 'i  A eA(L?r?s) ke(@t; 0; @3)(')kL2(1r+s ) k' ? RkL2(1r+s) : (3.8) r+s

But as a consequence of Korn's and Poincare's inequalities applied in the rectangles 1r+s whose length and width are independent of the parameters r and s , the rigid displacement R may be chosen so as to satisfy the estimate: k' ? RkL2(1r+s )  C ke(@t; 0; @3)(')kL2(1r+s) (3.9) which yields, with (3.8): Z it(') @t 'i  C A eA(L?r?s) ke(@t; 0; @3)(')k2 1 1r+s

= CA

L2 (r+s )

Z

1rZ+s

 C A eA

eA eA(L?r?s?1) je(@t; 0; @3)(')j2

1r+s

12

je(@t; 0; @3)(')j2:

(3.10)

Summing over the 1r+s , (s = 0; : : : ; L ? r ? 1) we nd : Z Z A je(@t; 0; @3)(')j2 it(') @t 'i  C A e (r;L)

(r;L)

and whence, substituting this inequality into (3.5) : Z

(0;L)

( ? 1) B e(@t; 0; @3)(') : e(@t; 0; @3

)(')  C A eA

But the coercivity of the operator B yields : Z

(0;L)

( ? 1) je(@t; 0; @3

j

)(') 2

 C A eA

Z

(r;L)

which gives: Z

(0;L)

( ? 1) je(@t; 0; @3

j

)(') 2

(r;L)

je(@t; 0; @3)(')j2:

je(@t; 0; @3)(')j2;

Z

? C A eA

Z

(r;L)

(3.11)

(3.12)

je(@t; 0; @3)(')j2  0:

Keeping in mind the expression of in the set (0; r) | cf (3.4), and choosing A such that C A eA  1 , we immediately deduce (3.3).

3.2 The exponential growth Now, we may reformulate the Saint-Venant principle in a way more convenient for our purposes. Corollary 3.3 Let ' be a solution of problem (3.1) on (0; L) for any L , i.e., the displacement ' is solution of the homogeneous Dirichlet-Neumann problem on the semi-in nite strip + = (0; 1) . Moreover, we assume that ' satis es the ux condition (3.2). Then, the following alternative holds for all 0 <  < A=2 : 8 < either e?t e(@t; 0; @3)(') 62 L2(+ ); 8 2 (0; A=2); : or e(@t; 0; @3)(')  0: where A > 0 is the constant appearing in Theorem 3.2. Proof. If e(@t; 0; @3)(') is not identically 0 , there exists r > 0 such that Z

(0;r)

je(@t; 0; @3)(')j2 > 0:

Then, inequality (3.3) shows that there exists a constant c > 0 such that

8L > r;

c eAL



Z

Let 0 <  < A=2 . Since we have e?2L 

8L > r;

c e(A?2)L 

Z

(0;L) e?2t

(0;L)

je(@t; 0; @3)(')j2: on (0; L) , we deduce that

e?2t je(@t; 0; @3)(')j2:

Since c e(A?2)L is unbounded when L ! +1 , e?t e(@t; 0; @3)(') does not belong to L2(+ ) . 13

4 BEHAVIOR AT INFINITY OF SOLUTIONS IN THE HALF-STRIP

Let B be a matrix satisfying (1.16). The aim of this section is to determine the behavior at in nity of the solutions of the two-dimensional problem (1.17) in the half-strip + . We are going to prove Theorem 1.3, i.e. that such solutions are exponentially decreasing towards rigid displacements as the distance to the clamped face becomes \large". The main idea is to perform a splitting of ' similar to the one that occurs when one looks for the singularities of a boundary problem around a corner, cf [15].

4.1 Fredholm operators in weighted spaces For any  2 R and m = + ? 1 , we are going to use the following weighted Sobolev space on  = R  (?1; 1) : Hm() = fv 2 D 0()3 j etv 2 H m ()g; (4.1a) and on + = R+  (?1; 1) : (4.1b) H1(+ ) = fv 2 D 0(+ )3 j etv 2 H 1(+ )g; V (+ ) = fv 2 D 0(+ )3 j etv 2 V (+ )g; (4.1c) (4.1d) V0(+ ) = fv 2 D 0(+ )3 j etv 2 V 0(+ )g;

with V 0(+ ) the dual space of V (+ ) . Then V0(+ ) is the dual space of V? (+ ) . The corresponding elasticity operators are: De nition 4.1 Let B be the operator de ned by

B : V (+ ) 3 ' 7?! B' 2 V0(+ )

(4.2)

where B ' stands for the element of the dual space V0 (+) de ned as:

8v 2 V?

(+ );

hB '; viV0 V? =

Z

+

B e(@t; 0; @3)(') : e(@t; 0; @3)(v);

(4.3)

and similarly we de ne Be : H1 ()3 ! H?1 ()3 . Remark 4.2 It is easily seen that the adjoint of B is B? .

The Fredholm properties of the operator B are closely linked to those of Be : we can prove that Lemma 4.3 For any  2 R , if Be is a Fredholm operator, then B is a Fredholm operator, i.e. the dimension of Ker B and the codimension of Im B are nite.

14

The Fredholm properties of Be depend on the invertibility of its operator-valued symbol B^ in the complex plane through the partial Fourier{Laplace transform with respect to the variable t (which corresponds to the Mellin transform for corner problems): Z +1  7?! '^(; x3) = e?t '(t; x3) dt: (4.4) Any ' 2 Re  = ? .

H1()3

?1

has a Fourier{Laplace transform well de ned in H 1(?1; 1)3 for

De nition 4.4 For every complex number  , we de ne: B^( ) : H 1(?1; 1)3 ?! [H 1(?1; 1)3]0; '^ 7?! B^( )'^

(4.5)

as follows:

hB^( )';^ v^i[H 1]0H 1 =

Z +1

?1

B e(; 0; @3)'^ : e(?; 0; @3)^v dx3:

(4.6)

We know from the general theory, cf M.S. Agranovich & M.I. Vishik [1] that there exists a discrete subset Sp(B) of C , such that: 8 62 Sp(B); B^( ) is an isomorphism. (4.7) Moreover,  ! B^( )?1 is meromorphic with its poles in Sp(B) and in every strip of the form 1 < Re  < 2 :

f 2 C j 1 < Re  < 2g \ Sp(B) is nite.

(4.8)

Similarly to [15] we can show that: Lemma 4.5 For any  2 R such that Re  = ? does not intersect Sp(B) , the operator Be is an isomorphism. As a consequence of Lemmas 4.3 and 4.5 Proposition 4.6 For any  2 R such that Re  = ? does not intersect Sp(B) , B is a Fredholm operator. Here are some additional properties of the above operators.

Proposition 4.7

8  0;

Ker B = f0g:

(4.9)

Proof. It may be noticed that for all   0 : V? (+ )  V (+): (4.10) Therefore, if w 2 Ker B for some   0 , then we may take as test function v = w

in (4.3). From the positivity of the matrix B , it follows that: e(@t; 0; @3)w = 0: 15

Thus w belongs to the space R of rigid displacements. Since moreover w satis es zero Dirichlet boundary condition for t = 0 , it coincides with the null displacement. This yields the result. We also need extra information about the symbol B^ . Proposition 4.8 For all  6= 0 such that Re  = 0 , B^( ) is an isomorphism. Proof. We have, for every w^ 2 Ker B^( ) : Z +1

?1

B e(; 0; @3)w^ : e(?; 0; @3)w^ = 0;

If Re  = 0 we have  = ? , whence: Z +1

?1

B e(; 0; @3)w^ : e(; 0; @3)w^ = 0:

As B is positive de nite with all its coecients real, it follows that e(; 0; @3)w^ = 0 . Therefore, w^ = 0 as soon as  6= 0 and we get the result. If  = 0 , the kernel Ker B^( ) is generated by the constants w^ = (at; as; a3) . Thus B^( )?1 has a pole in  = 0 and the following space P is not reduced to 0 : De nition 4.9 The space P is the space of functions de ned on the whole strip  = R  (?1; 1) by: n o P = W j W (t; x3) = Res et B^( )?1 f^(; x3); f^ holomorphic : (4.11)  =0

From the general theory [1], we know that P is nite dimensional. Its elements P Q have clearly the form q=0 tq wq (x3) . Conversely, we also classically have

Lemma 4.10 n

P = W (t; x3) =

Q X q=0

tq wq(x3) j

8v 2 D ();

Z



B e(@t; 0; @3)(W ) : e(@t; 0; @3)(v) = 0

o (4.12)

and the elements of P are polynomial. This last property comes from the fact that the coecients of B are constant. One should calculate \Jordan chains" for  = 0 to determine P , see [19] for the computation of such a space on R2  (?1; 1) .

4.2 Proof of Theorem 1.3 (i ) First step: Solution up to polynomials. Let 0 be the largest positive

real number such that 8 2 (?0; 0) [ (0; 0);

Sp B \ fRe  = g = ;: 16

(4.13)

As a consequence of property (4.8) of Sp B :

0 > 0:

(4.14)

Lemma 4.11 Let  2 (0; 0) and a right hand side f 2 V0(+ ) be given. Then (i) there exists a displacement ' belonging to \ ' ( s ) = h >

0

0 < Z Z `?1 X j `?j ` > + B (' ; v): 8 v 2 V ( ) ; B ( s ) e ( @ ; 0 ; @ )( ' ) : e ( @ ; 0 ; @ )( v ) = ? > t 3 t 3 : + + j =1 (4.26) Moreover, 'k belongs to C 1 (S; H1 (+ )3) for all  < 0 . Proof. Let  2 (0; 0) be xed. Let 0 > 1 >


> k >


>  be a decreasing sequence. As a consequence of Theorem 1.3 for f = 0 and g = h0 , we obtain '0 in C 1(S; H11 (+ )3) solving (4.26) for ` = 0 . Let us assume that 'j are constructed in C 1 (S; H1j+1 (+ )3) for j = 0; : : : ; ` ? 1 such that (4.26) holds for j = 0; : : P :; `?1 . Then, since the operators Bj are of order 2 with polynomial coecients in t , `j?=11 Bj ('`?j ;
) belongs to C 1(S; V0`+1 (+ )) , and a new application of Theorem 1.3 yields the existence of '` in C 1 (S; H1`+1 (+ )3) such that (4.26) holds for ` . Thus the pro les 'k belong to C 1(S; \ 0 and m +1 let ' be the solution of problem (5.4) with g 2 H ( 0) ; if (5.5a) Re 1+?(s) > m; then ' 2 H m+1 (+ )3; (5.5b) if (5.6a) 8`  1; Re `+?(s) 6= m; then 8 X + X +? +? > < 'sing = +; ? ?  c` S` (s); `; Re ` (s) < m (5.6b) ' = 'reg + 'sing; with > : 'reg 2 H m+1(+ )3; where  +? is a smooth cut-o function equal to +1 in the neighborhood of the corner (t; x3) = (0; + ? 1) of + and the functions S`?(s) are singular solutions of (5.4) whose behavior in terms of    +?(t; x3) := dist (t; x3); (0; + ? 1) is





( +?)` (s) or ( +?)` (s) log  +?; according to the multiplicity of `?+(s) . The complex numbers `+?(s) are the eigenvalues of an analytic family of elasticity operators  7! B +?(s)( ) , i.e. the values of  for which B +?(s)( ) is not invertible. For each  2 C , B +?(s)( ) operates in one variable and is de ned as follows. Let B(s) be the operator governing problem (5.4) and ( +?;  +?) be the polar coordinates centered at (0; + ? 1) . +?The changes of coordinates (t; x3) 7! ( +?;  +?) de nes for each s 2 S an operator B (s) such that   B +?(s)  +?;  +? @ ; @ = 2+? B(s)(@t; @3): Then



B ?+(s)( ) = B ?+(s)  +?; ; @ 21

 :

For an isotropic material with Lame constants  and  , `+ (s) = `? (s) = ` for all s 2 S , where ` are the singular exponents of the two-dimensional Lame operator with mixed Dirichlet-Neumann conditions on an angle with opening =2 . In the following two tables, we give the values of the rst three singular exponents computed by the method of [8] for a few pairs (; ) ; note that the singular exponents depend only on the ratio = . In Table 1, we consider the cases where  > 0 and  > 0 . We note that 1 is always real and less than 1 . Thus, for an isotropic material, solutions u(") do not belong to H 2( ) in general.





1

2

1 1 1 1 1 1 1 1

0.0001 0.001 0.01 0.1 1 10 100 1000

0.594637 0.594874 0.597220 0.619071 0.744750 0.928406 0.990438 0.999005

1.671905 1.672056 1.673714 1.686965 1.718359 1.510215 1.368903 1.354005

3 + ?+ ?+ ?+ ?+ ?

0.808683 0.807980 0.801032 0.741342 0.474498 1.885198 1.989911 1.998999

i i i i i

Table 1 In Table 2, we consider the situation where  < 0 with  > ? , which still ensures the ellipticity in two dimensions. We repeat the last row of Table 1 for the sake of comparison | note that the choice (; ) = (1; 1000) is equivalent to the choice (; ) = (0:001; 1) . Now, Re 1 > 1 , and we have the regularity H 2 .



-

0.001 0. 0.25 0.50 0.75 0.95 0.99 0.999



1

2

1 1 1 1 1 1 1 1

0.999005 1.000000 1.109950 1.061964 1.023139 1.001794 1.000106 1.000001

1.354005 1.352317 0.284274 0.466259 0.691068 1.182371 1.688153 2.419603

+ ?+ ?+ ?+ ?+ ?+ ?

Table 2 22

3 i i i i i i

1.998999 2.000000 2.231620 2.519427 3.094857 3.005457 3.000319 3.000005

+ + + +

0.512300 1.176263 1.687901 2.419601

i i i i

5.2 Stable singular functions As can be seen from these tables, branchings appear in the exponents when the parameters of the rigidity matrix change, for instance+ if the Lame coecients depend on (x1; x2) . So, in general, the singular functions S`?(s) do not depend smoothly on the arc length s . But, relying on [5, 9], we can construct special linear combinations + + ? ? of the S` (s) in order to obtain stable singular functions Sstab; `(s)

C 1 with respect to s and taking values in C 1(+ ):

+ ? (s) s 7?! Sstab ;` +

Simple eigenvalues `?(s) yield directly stable singular functions. + + + + If `?(s0) = `?+1 (s0) is a double eigenvalue at s0 and if `?(s) and `?+1(s) are simple for s 6= s0 in a neighborhood of s0 , then stable singular functions are given in this neighborhood by

S

+ ? ?+ ?+ stab; ` (s) = ` (s) + `+1 (s)

S

S

and S

?+ stab; `+1 (s) =

S`?+(s) ? S`+?+1(s) : s ? s0

Using these stable singular functions, we have (the spaces Hm (+) are those introduced in De nition 1.2): Proposition 5.1 Let g = g(s) and f = f (s) be functions in C 1(S ) with values in H m+1=2 ( 0 )3 and Hm?1 (+ )3 respectively, where 0 <  < 0 . If

8s 2 S; 8`  1;

Re `+?(s) 6= m;

(5.7)

and if for all s , '(s) is solution of

' 2 H 1(+ )3

8v 2 V (+); then



and ' 0 = g (s);

Z

+

B (s) e(@t; 0; @3)(') : e(@t; 0; @3)(v) =

Z

+

(5.8a)

f (s) v;

8 X +? X +? +? > ' ( s ) =  c` (s) Sstab;` (s); sing <  (s) < m + ; ? `; Re  ' = 'reg + 'sing; with > ` : 'reg 2 C 1 (S; H m+1 (+)3);

(5.8b)

(5.9)



with smooth coecients c`? 2 C 1 (S ) . Now we consider problem (1.11) on e := R+  S  (?1; +1) +

e 3 and ' = g on ?e 0; ' 2 H 1() (5.10a) 8Z > > Ae("t; s) e(@t; 0; @3)(') : e(@t; 0; @3)(v) < e  e; Z (5.10b) 8v 2 V () 4 X > k k e > f v: " A ( '; v ) = + : e k=1 23

We note that this problem depends on " through its coecients, but that this dependence is now regular. As is well-known in the edge analysis, the asymptotics near the edges is governed by the reduced-normal part of the operator \frozen" at the edges. Since we start with smooth data, all along the construction of the pro les wk we remain in spaces of functions which are smooth with respect to the tangential variable (the arc length s ) and the following statement is convenient for our purposes: Proposition 5.2 Let g 2 C 1(S; H m+1=2( 0)3) and f 2 C 1 (S; Hm?1 (+)3) , for 0 < m  1 and 0 <  < 0 . If hypothesis (5.7) holds and if '(") is solution + of problem (5.10), then expansion (5.9) still holds with smooth coecients c`?(") 2 C 1(S ) . Moreover, the dependence with respect to the small parameter " is uniform: for all n 2 N 9 Cn(g; f );Cn0 > 0; 8" 2 (0; 1); kc+?` (")kH n(S) + k("t) 'reg(")kH n(S;H m+1 (+))  Cn(g; f ) + Cn0 k'(")kL2(S;H 1(+ )) ; where  is a cut-o function equal to 1 in a neighborhood of t = 0 . The proof combines a classical analysis in the neighborhood of the edges, in the region where t 2 (0; 1) for instance, and uniform a priori estimates in the region where 1 < t < =" . Remark 5.3 The assumption m  1 serves only to avoid the dependence on " of the stable singular+ functions. To handle larger values of m , we have to introduce ?; k (s) of the stable singular parts the \shadows" Ustab ;` + ?; 0 (s) := c +?(s) S ?+ (s) Ustab stab;` ` ;`

which are de ned by induction as solutions of (compare with (1.15)) 8 ?+; k + > > < UZ stab; `(s) 0 = 0 and 8v 2 V ( ); Z k?1 X j +?; k?j + ;k ? > > : + B (s) e(@t; 0; @3)(Ustab; `) : e(@t; 0; @3)(v) = ? + j=1 B (Ustab; ` ; v):

(5.11)

With these new functions, the expansion of the solution ' of problem (5.10) can be written as [mX ?1] X +? X +?; k "k Ustab 'sing(s) =  ;` (s):

+; ? `; Re ` (s) < m k=0 +?; k; d More precisely, there are functions stab ; ` (s) which do not depend on ' , 0 1 [mX ?1] [mX ?1] X +? X @ +? +? + + ; k; d ? A: c` stab;` + "k 'sing =  @sdc`? stab ;` +; ? `; Re ` (s) < m k =1 d=0

S

S

S

24

such that (5.12)

5.3 Expansion of u(") near the edges Let us recall that problem (1.6) is equivalent to problem (5.10) with g = 0 , as far as the behavior in an annulus neighborhood r <  of the clamped part ?0 of the boundary is concerned, cf x1.3. Thus we can deduce from Proposition 5.2: Theorem 5.4 Let u(") be the solution of problem (1.6) and let 0 < m  1 . Then u(") admits the splitting u(") = ureg(") + using("); (5.13a) with 8 X X +? +? r > c` (")(s) +?( r" ; x3) Sstab ; ` ( " ; s; x3); < using(")(r; s; x3) = +; ? `; Re ` (s)  m (5.13b) > : ureg(") 2 C 1(S; H m+1 ( )3): Remark 5.5 We can drop the assumption (5.7) because of the smoothness of the right hand side f (") . As already hinted, in the expansion (5.1) of u(") , the only singular terms are the boundary layer terms wk . Proposition 5.1 applied to wk yield (by induction over k ) that Lemma 5.6 The boundary layer terms wk admit the splitting 8 k X X +?; k ?+ +? wsing(s) = c` (s)  Sstab;` (s); > <  k k k + ; ? `; Re ` (s)  m (5.14) w = wreg + wsing; with > : wk 2 C 1 (S; H m+1 (+)3); reg  for all 0 < m  1 and for all  < 0 . Applying once more the result of Proposition 5.2 to the remainder U N (") de ned in (5.2) and using the technique of pushing forward the asymptotics in order to obtain sharp estimates of the error, we can prove our nal result: Theorem 5.7 The splitting (5.13) of u(") admits the following asymptotic expansion in powers of " : For all N 2 N N X (5.15) c`+?(") = "k c`+?; k + O("N +1) k=1

where O("N +1 ) is in the sense of (5.14), and

kureg(") ?

N X k =0

"k uk + (r)

N X k=1

C 1(S )

+

and the coecients c`?; k are those of

k ( r ; s; x )k N ?m+1=2 "k wreg 3 H m+1 ( )3  C " "

(5.16)

k is de ned in (5.14). where wreg Of course we could extend+ this result to larger values of m by making use of the ?; k; d . shadow singular functions Sstab ;`

25

6 ESTIMATES FOR THE DISPLACEMENT AND THE STRAIN AND STRESS TENSORS

Without further development, we can deduce from all our results reliable estimates for the displacement eld u(") itself in various Sobolev spaces, but also for objects depending linearly from u(") , namely the scaled strain tensor (")(u(")) and the scaled stress tensor (") . More interesting than estimates relative to the remainder U N de ned in (5.2) are estimates relative to the possibility of approximation of u(") by polynomials of " with coecients depending on x only. So we introduce N X (6.1) U N (") = u(") ? "k uk (x): k=0

Another question is the approximation of u(") by hierarchical models, i.e. where the \discrete spaces" are displacements with a polynomial behavior in x3 . Since the boundary layer terms are not polynomial in the variable x3 in general, this question is closely linked to the estimates of U N .

6.1 Error estimates for the displacement eld We deduce from the results of the previous section that for any Sobolev norm k
kN ( ) such that ku(")kN ( ) is nite for all smooth data f , we have | compare with (5.3a) and (5.3b):   kU N (")kN ( )  C "N +1 + "K kwK ( "r )kN ( ) ; (6.2) where wK is the rst non-zero boundary layer term (in general, K = 1 for the in-plane components and K = 2 for the third component). As kw( r" )kH ( ) is O("?+1=2) for any  < 0 , with | cf (5.5a)-(5.5b): (6.3) 0 = sup sup Re 1+?(s) + 1; s2S +; ?

we obtain that, for the in-plane components kUN (")kH ( )2  C ("N +1 + " 23 ?); and for the transverse component kU3N (")kH ( )  C ("N +1 + " 25 ? ): The smaller is  , the better is the estimate. Similarly, for the L1 norm, we obtain kUN (")kL1( )2  C ("N +1 + ") = O("); and kU3N (")kH ( )  C ("N +1 + "2): Thus the estimate of UN (") , resp. U3N (") , does not improve if N > 0 , resp. 26

(6.4a) (6.4b) (6.5a) (6.5b) N > 1.

6.2 Error estimates for the scaled strain tensor The scaled strain tensor is de ned in (1.4a). We are interested by its L2 norm. We have to consider the behavior of the rst terms in the asymptotics. As w31 is zero, we obtain that (6.6) k(")(" w1)kL2( )6 = O("1=2) and k(")("2 w2)kL2( )6 = O("1=2): (6.7) We recall from [11] that u0 and u1 are Kirchho-Love displacements, i.e. cancel e33 and e3 . Thus k(")(" u1)kL2( )6 = O("): (6.8) Finally for k  2 , uk is the sum of a Kirchho-Love displacement and of a term vk which does not cancel e33 in general, thus

k(")("k vk )kL2( )6 = O("k?2 ):

(6.9)

Whence, (")(U 0) and (")(U 1) do not tend to 0 in general as " ! 0 , but k(")(U 2)kL2( )6 = O("1=2): (6.10)

6.3 Error estimates for the scaled stress tensor Following [4], we de ne the scaled stress tensor (") = (kl("))1 k;l 3 by

 = A ij ij (")(u(")); 3 = "?1 A3ij ij (")(u(")); 33 = "?2 A33ij ij (")(u(")): Thus, the equation (1.6b) is equivalent to @k kl(") = fl in and nk kl(") = 0 on ! f+ ? 1g . We have  (") = "?2 A 33 e33(u(")) + A  e  (u(")); (6.11a) ? 2 3(") = 2" A3 3 e 3(u(")); (6.11b) ? 4 ? 2 33(") = " A3333 e33(u(")) + " A 33 e  (u(")): (6.11c) Inserting the asymptotic expansion (1.8) into each term of (6.11), we obtain X k k X (6.12a)  (") = "  + (r) "k k ( "r ; s; x3); k0 k0 X k k X k k r (6.12b) 3(") = " 3 + (r) " 3( " ; s; x3); k0 k  ?1 X k k X k k r 33(") = " 33 + (r) " 33( " ; s; x3); (6.12c) k0 k  ?2 where the ijk are smooth tensors and the kij are exponentially decreasing tensors. The expression of the rst terms ij0 is similar to that given in [4, Ch.3]. We note 27

the strong in uence of boundary layer terms which arise at the degree 0 for  , the degree ?1 for 3 , and the degree ?2 for 33 . Combining (6.12) with the structure of the boundary layer terms in the neighborhood of the edges of the plate, we arrive to the (generically optimal) estimates 0 k 1=2 k (") ?  (6.13a) L2 ( )  C " k3(") ? 0 3kH 1((?1;+1);H ?1(!))  C "1=2 (6.13b) k33(") ? 330 kH 2((?1;+1);H ?2(!))  C "1=2: (6.13c) We can also see that, in a generic way: k3(")kL2( )  C"?1=2 and k33(")kL2( )  C"?3=2:

(6.14)

All these estimates are in accordance with the results quoted in [4, Ch.3]. REFERENCES [1] M. S. Agranovich, M. I. Vishik. Elliptic problems with a parameter and parabolic problems of general type. Russian Math. Surveys 19 (1964) 53{157. [2] D. N. Arnold, R. S. Falk. The boundary layer for the Reissner-Mindlin plate model. SIAM J. Math. Anal. 21 (2) (1990) 281{312. [3] D. N. Arnold, R. S. Falk. Asymptotic analysis of the boundary layer for the Reissner-Mindlin plate model. www preprint, Pennsylvania State University 1995. [4] P. G. Ciarlet. Plates and Junctions in Elastic Multi-Structures: An Asymptotic Analysis. R.M.A. Vol. 14. Masson and Springer-Verlag, Paris and Heidelberg 1990. [5] M. Costabel, M. Dauge. Construction of corner singularities for AgmonDouglis-Nirenberg elliptic systems. Math. Nachr. 162 (1993) 209{237. [6] M. Costabel, M. Dauge. General edge asymptotics of solutions of second order elliptic boundary value problems I. Proc. Royal Soc. Edinburgh 123A (1993) 109{155. [7] M. Costabel, M. Dauge. General edge asymptotics of solutions of second order elliptic boundary value problems II. Proc. Royal Soc. Edinburgh 123A (1993) 157{184. [8] M. Costabel, M. Dauge. Computation of corner singularities in linear elasticity. In M. Costabel, M. Dauge, S. Nicaise, editors, Boundary Value Problems and Integral Equations in Nonsmooth Domains, Lecture Notes in Pure and Applied Mathematics, Vol. 167, pages 59{68. Marcel Dekker, Inc., New York 1994. [9] M. Costabel, M. Dauge. Stable asymptotics for elliptic systems on plane domains with corners. Comm. Partial Dierential Equations no 9 & 10 (1994) 1677{1726.

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