Singular Perturbations I: Spaces and Singular Perturbations on Manifolds Without Boundary (Studies in Mathematics and Its Applications)

SINGULAR PERTURBATIONS I Spaces and Singular Perturbations on Manifolds without Boundary

STUDIES IN MATHEMATICS AND ITS APPLICATIONS

VOLUME 23 Editors: J.L. LIONS, Paris G. PAPANICOLAOU, New York H. FUJITA, Tokyo H.B. KELLER, Pasadena

NORTH-HOLLAND AMSTERDAM NEW YORK OXFORD TOKYO

SINGULAR PERTURBATIONS I Spaces and Singular Perturbations on Manifolds without Boundary

LEONID S. FRANK Institute of Mathematics University of Nijmegen Nijmegen, The Netherlands

1990

AMSTERDAM

NORTH-HOLLAND NEW YORK OXFORD TOKYO

ELSEVIER SCIENCE PUBLISHERS B.V. SARA BURGERHARTSTRAAT 25 P.O. BOX 21 1,1000 AE AMSTERDAM, THE NETHERLANDS Sole distributors fo r the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY, INC. 655, AVENUE OF THE AMERICAS NEW YORK, N.Y. 10010, U.S.A.

L i b r a r y o f Congress C a t a l o g i n g - i n - P u b l i c a t i o n

Data

F r a n k , L . S. ( L e o n i d S . ) . 1934Spaces and s i n g u l a r p e r t u r b a t i o n s on m a n i f o l d s w i t h o u t boundary / L e o n i d S. F r a n k . p. cn. (Singular perturbations ; 1) (Studies i n m a t h e m a t i c s and i t s a p p l i c a t i o n s ; v . 2 3 ) Includes bibliographical references (p. ISBN 0-444-88134-4 (U.S.) 1 . Global a n a l y s i s (Mathematics) 2. M a n i f o l d s (Mathematics) 3. S i n g u l a r p e r t u r b a t i o n s (Mathematics) 4 . Function spaces. I. T i t l e . 11. Series. 111. S e r i e s : F r a n k , L . S . ( L e o n i d S . ) , 1934S i n g u l a r p e r t u r b a t i o n s ; 1. OA372.FB v o l . 1 I PA6 141 5 1 5 ' . 3 5 2 s--dc20 [ 5 1 4 ' .741 90-7631

--

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Dddik 8.

J.-L. LIONS, L. SCHWARTZ, G.E. SHILOV

This Page Intentionally Left Blank

Vii

Contents

Introduction .............................................................

IX

Notation ..................................................................

1

Chapter 1. Manifolds. Functional Analysis. Distributions . . . . . . . . . . . . . . . . . . 5 1.1. Manifolds ......................................................... 5 1.2. Functional Analysis .............................................

10

1.2.1. Metric Spaces and Contractions ..........................

10

1.2.2. Topological Vector Spaces ................................

37

1.2.3. Banach Spaces ...........................................

42

1.2.4. Hilbert Spaces ...........................................

43

1.2.5. Classical Sobolev Spaces W,, and Holder Spaces CkiA. . . .45 1.2.6. Linear Functionals and Dual Spaces ...................... 51 1.2.7. Linear Operators .........................................

54

1.3. Distribution Theory .............................................

85

1.3.1. Test Function Space D ( V ) ................................

85

1.3.2. Distribution Space D'(V ) .................................

86

1.3.3. Some other Distribution Spaces ..........................

90

1.3.4. Parameter Dependent Distributions ......................

95

Notes ..............................................................

107

Chapter 2. Sobolev Spaces of Vectorial Order ...........................

109

2.1. Spaces on Rn ..................................................

109

2.2. Restriction to Hyperplane and Further Imbeddings Results . . . . . 114 2.3. Spaces of Distributions Supported by a Half-Space ............. 120 2.4. Spaces on R z ..................................................

122

2.5. Dual Spaces ...................................................

131

2.6. Sobolev Spaces of Vectorial Order on Manifolds . . . . . . . . . . . . . . . . 133 2.7. Spaces of One Parameter Families of Meshfunctions . . . . . . . . . . . . 138 2.8. Some Spaces of Two Parameter Families of Meshfunctions ...... 194 Notes ..............................................................

211

viii

Contents

Chapter 3 . Singular Perturbations on Smooth Manifolds without Boundary ..........................................................

213 3.1. Singular Perturbations with Constant Symbols ................. 213

3.2. Singular Perturbations with Homogeneous Symbols ............. 224 3.3. Singular Perturbations with Variable Symbols .................. 235 3.4. Continuity of Singular Perturbations in Sobolev Spaces of Vectorial Order ................................................ 3.5. Pseudolocality of Singular Perturbations ....................... 3.6. Asymptotic Expansions of Symbols ............................

239 241 243

3.7. Amplitudes, Adjoints and Products of Singular Perturbations ...250 3.8. The Stationary Phase, Laplace and Saddle Point Methods ...... 258

.................... 295 Diffeomorphisms and Singular Perturbations on Manifolds ...... 307 An Algebra of Difference Operators ............................ 328 Elliptic Singular Perturbations ................................. 366 3.13. Girding’s Inequality ........................................... 437 3.9. 3.10. 3.11. 3.12.

The Fourier Integral Singular Perturbations

3.14. Reduction of Elliptic Singular Perturbations t o Regular Perturbations .................................................. Notes ...............................................................

486 528

............................................................

533

Bibliography

ix

Introduction

Asymptotic analysis, which started as a mathematical tool for the treatment of special problems in mathematical physics affected by the presence of characteristic small or large parameters, has been rapidly developing during the last decennia, acquiring more and more global features and penetrating into different fields of mathematics and applied sciences. Although, originally, asymptotic analysis had a rather heuristic character, it was realized that, in order to guarantee the validity of formal asymptotic expansions, rigorous mathematical theories (especially uniform error estimates) were needed t o ensure further development and the applicability of existing formal techniques. The latter stimulated a vigorous growth of asymptotic analysis as an integral part of pure and applied mathematics. Singular perturbations being one of the central topics in the asymptotic analysis, they play also a special role as an adequate mathematical tool for describing several important physical phenomena, such as propagation of waves in media in the presence of small energy dissipations or dispersions, appearance of boundary or interior layers in fluid and gas dynamics, as well as in the elasticity theory, semi-classical asymptotic approximations in quantum mechanics, phenomena in the semi-conductor devices theory and so on. Elliptic and, more generally, coercive singular perturbations are of special interest for the asymptotic solution of problems, which are characterized by the boundary layer phenomena, as, for instance in the theory of thin buckling plates, elastic rods and beams. A perturbation is said to be singular since its structure and the nature of

X

Introduction

the phenomena which it describes is completely different from the ones which are proper t o the corresponding reduced problem. For instance, considering a gas flow around an obstacle in fluid dynamics in the situation when the di-

mensionless viscosity parameter (the inverse of the Reynolds number) is small, one has a mathematical model (the Navier-Stokes equations) which reflects the physical boundary layer phenomenon in a neighborhood of the obstacle, while setting the viscosity equal t o zero one gets a different mathematical model (the Euler-Lagrange equations), in which this phenomenon is completely lost. Considering a stochastic model which is a superposition of a deterministic process and of a “white noise” of a small level (described as a Wiener process with a small variance), one comes to the Kolmogorov-Chapman parabolic equation with a small diffusion term for the density of the stochastic process in question; it is a singular perturbation of the reduced hyperbolic equation, which describes the deterministic situation. Other examples come from the theory of elastic rods or beams. If an elastic beam at rest is subjected t o a strong pulling out longitudinal force described by a large dimensionless parameter, then using this parameter and setting it equal t o infinity, one can simplify the mathematical model, getting a reduced differential equation, which only partially refects the physical phenomenon. Indeed, for instance, in the case when the beam at rest is simply supported by its end points, the natural boundary conditions would tell that at the end points the displacements and the momenta of the forces applied must be zero. However for the reduced equation (which is of the second order) it is possible t o have only the displacements vanishing, while the momenta of the forces at the end points (not necessarily zero) are determined a posteriori. In fact, a boundary layer phenomenon in a neighborhood of the end points of the beam is present in this situation and should not be neglected. The linear singular perturbation theory and its possible applications is

Introduction

xi

the topic of this volume. Let A' be such a perturbation which is usually a differential (or integro-differential) operator affected by the presence of a small parameter

E

E (0, € 0 ) . One is interested in solving the equation

where f is a given second member., It is (impicitly or explicitly) assumed that the reduced equation (defined in a natural way and usually much simpler than (1)):

can be uniquely solved. Then one is interested in getting a convergent series

(3) for the solution u, of (l),and that is usually not possible, since, as a consequence of a singular nature of the perturbation A', the solutions of (1) do not depend analytically on

of

E

E

[-EO,EO]

E

even in the case when A' is a real analytic function

valued in some operator space.

Giving up the convergence, one asks for an asymptotic convergence of the series on the right hand side of (3), i.e. for each integer N

> 0 one would like

to have in a certain sense:

(4) Usually, formal asymptotic expansion techniques allow t o produce a relatively simple algorithm for computing recursively the coefficients uk, k 2 0, in the asymptotic expansion (3) or, even for more complicated forms of such an expansion, taking into account, for instance, the boundary layer phenomenon.

Introduction

Xii

A very important question, which arises afterwards, is a proof of the asymptotic convergence like (4) (or in a different form, appropriate to the situation considered). The only reasonable way to ensure the asymptotic convergence of approximate solutions to the solution of (1) is to have uniform a prion' estimates for u,, i.e. uniform upper bounds for the norm of the inverse

(d')-' (whose existence is, in fact, a part of the problem) as an operator from an appropriate data space V, into the solution space X,, operator

Such a question is not merely a matter of mathematical rigor, but it is crucial for the entire "raison d'Ctre" of the formal techniques which may, eventually, allow to determine uniquely uk, k

> 0, even in the situation, when the

solution to (1) does not exist or is not uniquely defined without additional restrictions.

For being specific, consider several examples. Example 1. Let q(z) (z E R3) be a real valued infinitely differentiable function and assume that q(z)

q ( o 0 ) for

1x1 2 r , r > 0 being sufficiently large.

Consider the following singular perturbation:

AEu:= u - E2div(q(z)grad u )

(6)

and the corresponding equatior

A'u, = f,

(7)

where f is a given infinitely differentiable function with compact support, i.e. f ( z ) vanishes outside of some ball in

R3,and

u,(z) is supposed t o vanish a t

infinity. The natural reduced operator A' for A' is the identity, so that uo = f, if uE in (7) admits an asymptotic expansion.

Introduction

Xiii

Furthermore, introducing the differential operator:

B ( z ,ax) := div(q(z)grad) = V . (q(z)V),

(8)

one can formally write an asymptotic expansion for u, in the form:

(9)

21,

-

= p k u 2 & ) ,

U2k(Z)

:= (B(.,ax))”(.),

k10

whose right hand side makes sense since f is smooth and has a compact support. Now the crucial question of an asymptotic convergence of the series on the right hand side of (9) to u, arises. Let us make the following basic additional assumption: inf q ( z ) = qo

(10)

> 0.

XER3

Under this assumption (10) (which is an ellipticity condition for the singular perturbation A , ) one can easily show the asymptotic convergence in (9). Indeed, integrating by part after multiplication of (7) by u,, using the Cauchy-Schwarz inequality and the basic assumption ( l o ) , one gets the following a priori estimate

(11)

(IIucII2.(P3)

+ &211vu&yR3J

1/2

I Y(Q)llfllLa(R3),

where

Introducing the norms of vectorial order s = ( ~ 1SZ, , sg) E R3,

where G(E) = Fx,Eu is the Fourier transform of u, one can rewrite (11) in the form:

5

l l ~ l l ( 0 , 0 , 1 ) , ~ Y(~)llfIl(O,O,O),c.

xiv

Introduction

Actually,using (13) and estimating more accurately by the Cauchy-Schwarz inequality, one gets the following sharp a priori estimate:

where y(q)

< 00

is defined by (12).

Differentiating (7) with respect to c and using the same argument, one gets for any integer s2 2 0,

s3

> 0 the following estimate:

where theconstant C ( S ~ , S Z , S ~ may , Eonly O , ~dependon ) s ~ , s ~ , s ~ , E o ,and Y(~) some derivatives of q ( x ) . An estimate like (15) can be established for any s = ( s I , s ~ , sE ~ )W3. Using (15) for each given s E

W3, one finds:

(16) llUe

-

E2kU2kIl(s),c

5 CE

2N

VN

Ilfll(sl,sa+2~,s3-2),,,

> 0,

V E E (O,Eo),

O 0 is a constant.

Then the solution of (6), (7) is given by the formula:

where FX+ and FCJx are the direct and inverse Fourier transform, respectively. Now, if q(z) is not a constant but still satisfies the conditions hereabove and, especially, the condition ( l o ) , one can still define the function: ti!')

(18)

: = ( 4 T ~ ' q ( z ) ) - ~ f ( y ) l ~ - y l - ~exp(-lz-yl/&(q(z))'/2))dy

=

J . 3

= (F;:.(l+

E 2 q ( z ) 1 ~ 1 2 ) - 1 F x - ~ f ):= ( z )(S'f)(z).

It turns out that

where

with a constant C > 0, which does not depend on

E,Q'

and f .

In other words, S' being the operator defined by (18) and introducing

) the functions u whose Fourier transforms are locally the spaces H (S),P (W3of integrable and have the norms (13) finite, VE E ( O , E ~ ) , one can rewrite ( 1 9 ) , (20) as follows:

A'S' = I - EQ',

(21)

I = identity,

where the family of linear mappings Q',

is equicontinuous, i.e. the norm of E

E

(O,EO), VEO

< 00.

&'

is uniformly bounded with respect t o

In troduction

xvi

I Thus, the norm of EQ' (0

(0 < E

< EO)


t),

E R", t E R, = 7Z+,

where Z+ is the set of all non-negative integers and O t , , is the shift operator

on

-+ W, , i.e.

(Ot,,v)(t) = v(t+7).

Note that the corresponding singular perturbation t o be inverted on each step of solving the implicit finite difference problem (36), is again an elliptic singular perturbation having the form:

(37)

A' = 1 - c2A + ( l/2)c4A2, c2 = T ,

I

E R" .

Of course, hereabove one may use the usual finite difference approximation Ah of the Laplacian on the grid R i = hZ":

thus getting (by using the schemes hereabove) unconditionally stable timespace finite difference approximations of the heat equation with one condition at t = 0 (also for the Richardson's scheme), i.e. the approximations hereabove with A replaced by Ah given by (38), are stable (in the sense of nonaccumulation of the errors), whatever the mesh-sizes

T

and h are.

xxii

Introduction

One of the interesting aspects in the theory of singular perturbations is the question of the possibility to reduce such a perturbation to a regular one. This is what can be done for any elliptic singular perturbation. Namely, one can construct explicitly an elliptic singular perturbation SE such that the product

S'A'

will be a regular perturbation of the corresponding reduced operator

A'. Such an operator S' is given by (18) hereabove in the case of the elliptic singular perturbation A' given by ( 6 ) , (7) and satisfying (lo), the reduced operator for A' being the identity. The same kind of reducing operator can be also constructed in the case of elliptic finite difference operators with one

( h > 0) or two

(E

> 0, h > 0) small parameters.

This volume deals with linear singular perturbations (on smooth manifolds without boundary), considered as equicontinuous linear mappings between corresponding families of Sobolev-Slobodetski's type spaces H ( s ) , cof vectorial order s € R3. Chapter 1 provides the necessary (also for the next volume) functional analytic background aiming at the situations, characterized by the presence of a small parameter. Chapter 2 is devoted to the spaces H(,),' and their finite difference versions. Chapter 3 deals essentially with elliptic and hyperbolic singular perturbations and their finite difference counterparts. Here also the classical asymptotic stationary phase, Laplace and saddle point methods are presented and used later on for different purposes, as for instance, the local theory of singularly perturbed Fourier integral operators, diffeomorphisms and symbol transformations in the C*-algebra of the singular perturbations and so on. A special attention is given t o the sharp form of GQrding's inequalities and their applications. The next volume will be devoted to singular perturbations in the elasticity theory and, more generally, to coercive singular perturbations, as well as t o

In trod uction

xxiii

singular perturbations of dissipative and dispersive type. In a work of this kind it is impossible to provide even a first approximation to an adequate list of references, which would, at least remotely, reflect the wealth of publications in the field of the singular perturbation theory and its applications. The only way to produce a first approximation to the solution of the problem of providing a comprehensive bibliography, seemed to me t o restrict the references to publications, which are more or less connected with the topic of this book, thus, taking a chance not to mention some valuable contributions to the asymptotic theory. I would appreciate any suggestion indicating me either inadequate or missing (but still relevant t o the topic) references, so that they could be included in the next volume. I started working on the theory of difference operators in the late fiftiesearly sixties in MOSCOW, being strongly influenced by remarquable achievements in the general theory of partial differential equations and the initiated development towards the theory of pseudodifferential operators. This influence can be easily traced back in my work on the C*-algebra of one parameter families of difference operators. The work on coercive singular perturbations had as its starting point the year of my immigration to Israel (1972) and had been progressing during my staying at the Hebrew University of Jerusalem (1973-1976), my sabbatical leave at the University of Kentucky (1976-1977) and my tenure at the University of Nijmegen (1977-), where Wolfgang Wendt, Guido Sweers, Johannes Heijstek and Henk Norde have been working with me. Writing this book was not an easy task to me, because of a lack of communication and considerable amount of other professional commitments. Still it was enjoyable and challenging. I am sure, many deficiencies (including awkward

linguistics, so characteristic for people whose mother tongue is not English) can be found spread over the text of this and the next coming volume. I should be

xxiv

Introduction

mostly grateful for any suggestion or criticism aiming at the improvement of the book.

I am deeply indepted t o Alain Bensoussan, Bernard Helffer, Denise Huet, Jacques-Louis Lions and Harold Widom who agreed to read the first volume and a part of the second one of this book and helped me by their remarks and criticism t o improve the manuscript. This help has been invaluable to me.

I would like t o express my gratitude t o the Elsevier Science Publishers for encouraging me to carry out the work on this book. Writing it has yet taken much longer than both of us, the Publisher and myself, have ever expected.

Nijmegen, February 1990

Leonid Frank

1

NOTATION

IRn IRE

IRn

5

:

n-dimensional (real) Euclidean space

:

n-dimensional (real) Euclidean space of variables x = (xl,...,x

:

n-dimensional Euclidean space of dual variables 5 = ( 5 l , - - - , t n )

Bnr+ : half-space of x E R: Cn

> 0

n-dimensional complex space of variables

:

c

=

.

(cl,.. ,cn)

x 5 +. . .+x 5 : scalar product between x E IR and 5 E IRn 1 1 n n 5 2 2 i 2 2 3 : Euclidean norms in illRn and illn ( x l + ...+ xn) , 151 = (C1+ ...+ 5,)

<x,O

1x1

with x

)

=

=

5

respectively <x> = (1+/xI2)+for x E mn

0 t h e r e e x i s t s a 0

6 > 0 such t h a t px(f(x),f(xo))
l

for v

C

+

E

f o r each x E M w i t h px(x,x

0

Y , M C

M with px(xv,xo)

)


0

Ea

{UE(0)

(1.2.1)

t E

where f E

C1

=

(E+)

0,

and

$J

E R are given.

Assume that f and 0 satisfy the conditions: (1.2.2)

2

f(t) 2 (1-y ) / 4 ,

$

c

(1-6+y-yo)/2 2

with some y > 0, 6 > 0, where yo = 1 - 4 f ( O ) . We are going to show that problem ( 1 . 2 . 1 )

u

:

R+

-f

has a uniq-ie solution u (t),

R , which can be represented in the form:

and for y

c1

:

IR+

+

W the following estimate is valid with some constant

=

E

= Cl(f,$):

provided that

E~

0

(f,$) is sufficiently small.

It is readily seen, that (1.2.8)

uo(t)

=

uL(t)+f(t) 0

and w(t) is the solution of the initial value problem:

1 , Manifolds, Functional Analysis, Distributions

16

(1.2.9)

a tw(t)+yow(t)-w 2 (t) =

0,

t E IR+,

= $

with y and $ defined by (1.2.6), (1.2.4). 0 Looking for u (t) in the form (1.2.3), one gets for y (t) the following initial value problem:

Notice that as a consequence of (1.2.2), one has: (1.2.12)

A(t)

t y,

v

t

E E+ a

-

Moreover, if $ 5 0 then w(t) 5 0, V t E R + , so that in this case on? has : A(t)-2w(t/E) 2 y, On the other hand, if $

v

t E

zt.

0, then $ < y o - 6 ,

as a consequence of (1.2.2),

and, moreover, one has in this case: 0 < w(t/e) < $. Thus, one finds for 0 < $ < y -6: 0

(1.2.13)

X(t)-2w(t/E) 2 y-2$ = y-2'$+1-yo

2 6,

again as a consequence of (1.2.2). Therefore, one has: (1.2.14)

p(E,t) := A(t)-2w(t/E) t 6,

v

t E

s+.

Now we may rewrite (1.2.10) as an equivalent integral equation in the following fashion:

17

1.2. Functional Analysis

Now consider in C(s+

where r = r(f,$) and

E~

=

)

a ball BrE of radius rE centered at 0 , i.e.

Eo(f,$) will be chosen later on.

One finds for each y E BrE :

where C = C(f,$) is such that

Thus, if r and tz0 are such that

then T defined by (1.2.161 maps BrE into itself. Furthermore, one easily finds for each pair (yl,yz) E BrE

(1.2.22)

X

BrE:

2rE0 < 6 ,

will be an equicontraction in BrE' v E E ( 0 , ~ ~ ) . It is readily seen that the following choice of E~ = s0(f,$) and

then T

r = r(f.9) is compatible with the conditions (1.2.20), (1.2.23)

E~

=

6'/(8C),

With such a choice of

r = 4C(2/2-1)/(2/26). E~

and r, there exists a unique y

(1.2.15), (1.2.16) or, equivalently, of (1.2.10). Thus, one has:

(1.2.24)

(1.2.22):

IIyEllc(~+,5 r(f,$)E,

V

E

E

( 0 , ~ ~ ) .

Now, (1.2.24) and (1.2.10) yield:

i.e. (1.2.7) with c,(f,$) = A(f,$)+c(f,$).

E BrE solution of

I . Manifolds, Functional Analysis, Distributions

18

The argument presented here can be easily extended to the initial value problem for the nxn systems of the form:

under the following assumptions: (i) The equation f(t,u (t)) = 0 has a solutions uo E C1($+). 0

(ii) The Jacobian-matrix f (t,u) has the eigenvalues h.(t,u), 1 5 j 2 n, 3

which in some neighbourhood of the curve (t,u (t)) E Wn+l satisfy the 0 condition: Re X.(t,u) 2 y > 0. 1 (iii) Vector ($-u ( 0 ) ) E Wn lies in a sufficiently small neighbourhood 0 of zero. Then again u (t) exists, is unique and can be represented in the form (1.2.3) with w(t) the solution of the following autonomous system:

a tw(t) +

(1.2.26)

f(O,uo(0)

+ w(t))

= 0,

t E W+ ,

As a consequence of (ii) and Liapunov's theorem, w(t) E 0 is an asymptotically stable stationary solution of system (1.2.26), so that if

0-u (0) lies in an attraction neighbourhood of zero, initial value 0 problem (1.2.26) has a unique solution w(t), which exists for all t Z 0 J, =

and is exponentially decreasing as t

-t

+a-

The same procedure as here above yields an integral equation for y (t) which again can be solved by the equicontraction argument in the family of metric spaces B which are balls of radius r& centred at zero in the space rc C(% ) of vector-functions y(t) with the norm l\ylL(%+) = sup ly(t)/ finite. t2O

I

Example 1.2.12. Consider the following boundary value problem: 2

u

+ f(u,E~ )

(1.2.27)

-E

(1.2.28)

u(E,x') = $(x'), 2

xx

=

0, x E U = (O,l), xt

E au

= {o,i},

where u = a u, u = a u , E E ( O , E ) is a small parameter, $ : a U + W is x x x x x 0 given and f (vl,v2) is a given function of variables v = (v v 1 E W2 with 1' 2

1.2. Functional Analysis real values: f

m2

:

+

w

19

.

We shall assume that f(v ,v satisfies the following conditions: 1 2 lo. f(v) is twice continuously differentiable. 20. f(0) = 0, fv (0,O) = y2 > 0. 1 We shall also assume that $ ( X I ) ,

x' E aU, is sufficiently small, i.e.

($(x')I 5 6 where 6 > 0 will be specified later on.

We are going to show that under the assumptions hereabove, the solution U(E,X) of (1.2.27), (1.2.28) exists, is unique and can be represented in the form:

z

(1.2.29) u(E,X) =

+ V(E,X),

W(X',(X-X'I/E)

x*Eau where w(x',x), x' E aU, are the (well-defined) solutions of the boundary value problems: (1.2.30)

X'

-wxx+f(w,(-l) w(x',o)

=

W )

=O,

X > O ,

$(x'), w(x',x)

+

0

for x

+

+-,

V

au

E

XI

and V(E,X) is at least twice continuously differentiable function of x E such that one has: (1.2.31)

I IvI Ic

I (€a ) kV(E,X)I

c max O5ks2 XE;

:=

(0,2), E )'(

5 CE

with some constant C > 0. Let us start by showing that each boundary value problem (1.2.30) for XI

=

as x

0 and x' = 1 has a unique solution w(x',x) which decreases exponentially +

+-,

provided that

Consider the case

XI

the same way. Denoting a

$ ( X I ) ,

= :=

f v2

rewrit.P ( 1 . 2 . 3 0 ) , fashion :

x' E aU, is sufficiently small.

0, the one of

(o),

w(x)

=

XI

:=

1 can be treated in exactly

w(O,x), $ = $(O), one may

(with x' = 0) esu'valently in the followinq

(1.2.32) w(x) = $ exp(X-x) + m

+

o

2 Law (y)+y w(y) - f (w(y) ,w (y))lG(x,y)dy := y Y

(T(w)1 (x),

where G(x,y) is the Green's function for the operator L(a L(a

)

:=

-a 2

+ aax + y

2

, x

E

X

)

m+,

with Dirichlet boundary condition at x = 0 and where h - is the negative

1. Manifolds, Functional Analysis, Distributions

20

zero of the characteristic equation A2-aA-y2 = 0, the positive one being denoted by A+. It is readily seen that G(x,y) is given by (1.2.33) G(x,y) = ( 2 ~ ) -f

( C2+iaF,+y2)-'exp

(-iyC)(exp(ixs)-exp (1-x) ) dC

R

so that with a = min{lA- , A + }

(1.2.34)

> 0, one has:

lG(x,y)I 5 Cx(

Introduce the notation: 11.2.35)

c

IIwIL;~=

sup exp(ax)Iaxw(x)I, 1

k = 0,1,2

,...

O<jO

and denote by Ck(%+

)

-

the set of all k-times continuously differentiable

functions on R+ for which expression (norm) (1.2.35) is finite. Further let

is a metric space with the distance function: Obviously, Mk r,a Pk;a(w1,w2) :=

IIW1 -w2 4 ; o

For operator T(w) defined by (1.2.32) one finds (1.2.36)

IIT(w1

th;a SItJl

+

2 C!~w~~l;av

where m

(1.2.37)

c

=

c(w) := sup I IG(x,y)lexp(ax-2ay)dy x>o 0

*

(1/2) max Ivl 41wlll;a

1

I a I =2

la:f(v)

-

I.

Furthermore, one easily checks that, in fact, one also has: (1.2-38) IIT(w) with C

=

1 (1.2.37).

I 1;a

5 C1I+I

2 +

CgJIWIIl;a.

max{l,(h-(], C2 = max{lA-l,A+lC(w), where C(w) is defined by

Thus, one finds:

(1.2.39)

IIT(W)l(i;a5 C1l$( + C2(r)r2,

v

1 w E Mria

.

21

1.2. Functional Analysis

Again, an easy computation shows that (1.2.40) IIT(w~)-T(w~)

5 rC3(r)/lW1-W21/1;o, v wj

1

E Mr;a, j

=

1,2,

with some constant C (r) > 0, which is bounded for r E [O,ro] with any 3 ro < m. We shall choose @ and r

=

r($) E (0,1/2] so that the following

inequalities hold:

(1.2.42) Ck =

sup Ck(r), k = 2,3. O 0. 0 Let v(h,x) : + C be any sequence (meshfunction) such that


0 and denote (1.2.155) Q, := {x = (xl,...,xn) E

~ f :I

jxjI 5 a, I

c

j 5 n}.

Let u.(h) be the mean value of u(h,x) over Qa: 7

(1.2.156) u (h) 0

:=

a-n

C u(h,x)hn. xEQa

Then (1.2.154), (1.2.155) yield:

C u(h,y)hn+l. (1.2.157) u (h)-u(h,O) = a-n X 0 XEQ, YETx Yrh It is readily seen (one has to use the induction argument, that

e

(1.2.158) c

< ~ ( y ) , a ~ , ~ > ~ ( h =, y )c c O5t6a xEQt

T(X)

,ax,h u(h,x),

XEQ, YETx

t = kh, 0 < k < a/h. Therefore, (1.2.157), (1.2.158) and HBlder's inequality (for sums) yield:

C a

-n O O , A : 8, + 8,, i s s a i d t o be convergent i n t h e norm topology t o some bounded l i n e a r o p e r a t o r . A

if

1 IA 0 -A v 1 I B,+BB,

I t i s s a i d t o be s t r o n g e l y convergent i f f o r each x

-f

E 8,

0 for v

one has:

+-.

1.2. Functional Analysis

55

I / A O ~ - A v1 ~ I

-Z 0 for v -Z a . The sequence {Av}v>O is weakly convergent to 82 some linear operator A , if for each x E 8 the sequence {Av}v,>O is weakly 1 convergent to A x . 0 The norm convergence implies the strong convergence and the latter implies

the weak convergence. The following result known as the Banach-Steinhaus theorem is important for applications. Theorem 1.2.30.

Let

{ A ~ I ~be> a~

sequence o f continuous Zinear o p e r a t o r s

from t h e Banach space B , i n t o t h e Banach space B,.

:

B,

+

8,

Assume t h a t f o r each

x E 8 , one has:

Then one a l s o has:

i . e . there e x i s t s a constant c IlAVxl I

1

5 C//x/

0

,

such t h a t

V x E B,,

V v > 0.

B2

If A map A-l

:

:

B,

8,

+

+

B2 is a one-to-one linear mapping onto B,,

then the linear

8 , is well defined and is called the inverse of A.

The following result, known as the open mapping theorem, plays an important role in applications. Theorem 1.2.31.

Let

A :

8,

+

a constant r

B , be a continuous Zinear mapping onto B,. 0 such t h a t t h e b a l l Biz) = {y E B , I I

i s contained in t h e image of t h e baZl ~ ( l = ) Ix E B , 1

I

Then t h e r e e x i s t s

/YI I


0 such that one has:

Equation

y is said to be resolvable everywhere if R(A)

Ax =

=

F,

densely resolvable if the closure R(n) in F coincides with F and normally resolvable if R(A) is closed. The correct resolvability of the equation y is equivalent with the existence of the bounded inverse operator

Ax =

A

-1

:

R(A)

-f

D(A).

A linear operator A

:

D(A)

if for each sequence {x 1 k k>l one has: xo E D(A) and yo = D(A) = E, is bounded iff A If A

:

D(A)

+

Axo. :

F, with D(A)

C D(A)

E

+

5 E,

such that xk

-f

is said to be closed

xo and AXk

A linear operator A

:

E

+

+

yo (k

+

-),

F, i.e. with

F is closed.

F is closed, then its kernel N(A) is a closed subspace

+

of E. Equation

Ax =

y with a closed linear operator A is correctly

resolvable iff it is uniquely and normally resolvable. Let A

:

D(A)

+

F

and assume that D(A) is dense in E. Let g E F', i.e.

g is a continuous linear functional on F. Then the linear functional g(Ax)

is defined on D(A). If

g(Ax)

is bounded, i.e. Ig(Ax) I 5

CI

1x1

IE

I

V x E D(A) with some constant C > 0, then g E D(A*) where the linear

operator A*

:

D(A*)

+

E' is defined by the equality (A*g)(x)

V x E D(A). Functional A*g defined by this equality for

v

=

g(Ax),

x E D(A) may be

extended by continuity onto E, since D(A) is dense in E and g is bounded on D(A). Thus, one may assume that A* maps D(A*) space of continuous linear functionals on E). A*

C :

F' into E '

D(A*)

+

(the Banach

E' is called the

adjoint of A; A* is always closed. If F is reflexive, i.e. (F')'=F"=F,

1.2. Functional Analysis

59

then D(A*) is dense in F'. Along with the equation (1.2.172)

AX = y ,

x E D(A) E E ,

y E F

consider the adjoint equation: (1.2.173)

A*g = f,

For ( 1 . 2 . 1 7 3 )

g

E D(A*)

f E E'.

_C F ' ,

one has the same concepts of unique and correct resolvability,

as well as these of the resolvability everywhere and on a dense linear subset in E ' . The equations (1.2.1721,

(1.2.173)

are connected in the following

sense :

Equation ( 1 . 2 . 1 7 2 )

i s densely resolvable i f f equation ( 1 . 2 . 1 7 3 )

is

uniqueZy resolvable; Equation (1.2.172)

i s uniquely resolvable i f ( 1 . 2 . 1 7 3 )

is densely

r e s o lvab l e . The kernel N(A*) of A* is the orthogonal complement to the range R(A) of A.

Thus ( 1 . 2 . 1 7 2 )

i s normally resolvable i f f i t can be solved for each second

member y which i s orthogonal t o N(A*). The linear sets N(A) and R(A*) are orthogonal, as well; however, they are not necessarily orthogonal complements of each other. Equation ( 1 . 2 . 1 7 3 ) is said to be c l o s e l y resolvable if R(A*) is closed in E ' and it is said

normally r e s o l v a b l e , if ( 1 . 2 . 1 7 3 )

can be solved for each second member f,

which is orthogonal to N(A).

If

(1.2.173)

i s normally r e s o l v a b l e , then it i s c l o s e l y resolvable, as

well. The converse of the latter statement is generally speaking false. If A is closed then the closed and normal solvability of ( 1 . 2 . 1 7 3 ) are equivalent. The close solvability of ( 1 . 2 . 1 7 3 ) property of ( 1 . 2 . 1 7 2 ) : M in

there exists a constant C > 0 and a dense set

R (A), such that for each y1 E

(1.2.172),

If ( 1 . 2 . 1 7 2 ) on R(A*).

is equivalent with the following

M

there is a solution x 1 E D(A) of

which satisfies the inequality:

I lxll I E

i s everywhere solvable, then (1.2.173)

Equation (1.2.173) solvable on R(A).

IC I

IyIl

IF.

i s c o r r e c t l y solvable

i s everywehere solvable i f f ( 1 . 2 . 1 7 2 )

i s correctly

1 . Manifolds, Functional Analysis, Disfributions

60

Normally solvable equation (1.2.172) with a closed operator A is called n-normal if N(A) = ker A is finite dimensional: dim ker A


0, is equivalent withtheunique solvabilityevery-

where of the both equations (1.2.172) and (1.2.173). However, in many cases it is possible to establish a priori estimates, which are weaker than (1.2.174). Let E be compactly imbedded in another Banach space Eo and let A

:

Q(A)

+

F, P(A)

5 E,

be closed.

The validity of the following a priori estimate is equivalent with the n-normality of equation (1.2.172):

The d-normality of ( 1 . 2 . 1 7 2 )

is equivalent with the validity of an analogous

a priori estimate for the adjoint operator. Let A be closed, D(A) dense in E and let F' be compactly imbedded in

a Banach space

where

C > 0

G.

Then (1.2.172) is d-normal iff one has:

is some constant.

Equation (1.2.172) (and, respectively, the operator A) is said to be noetherian if (1.2.172) is at the same time n-normal and d-normal. The integer: (1.2.177) K(A)

:=

dim ker A

-

dim coker A

is called the index of equation (1.2.172) and of the operator A. If A and B are both noetherian and D(B) is dense, then BA is again

1.2. Functional Analysis

61

noetherian and, moreover, one has: K(BA)

=

K(A) + K(B)

If D(A) is dense in E and A in (1.2.172)

K

Id A is noetherian, p

(A) :

-K(A).

0.

=

E

=

and the corresponding operator A are called

Noetherian equation (1.2.172) of Fredholm type if

is noetherian, then A* in the

is noetherian, as well, and K(A*)

adjoint equation (1.2.173)

+

F is bounded and has its norm sufficiently

small, then A+Q is noetherian too, and

If A is noetherian and K

E

:

+

F is compact, then again

K(A+K) = K(A).

L e t Ew ' Fw ' w E 0 be two f a m i l i e s of Banach spaces and Fa, w E R, be l i n e a r and equicontinuous. Assume t h a t t h e r e

Corollary 1.2.36.

l e t Aw

:

Ew

+

are two Banach spaces Eo, F ~ ,two f a m i l i e s Jw

s

:

:

xo Yw

+

xw,

+

YO'

-1

Jw

-1

sw

: Xu

+

X

: Yo

+

Yw

and a continuous l i n e a r operator (1.2.178)

1rJw E

"

J

W'

sw,

w E

0,ofequiisomorphisms:

0'

:

E~

+

F~ such t h a t t h e diagramme

Aw

swfs;l A

FO 0

i s c o m t a t i v e , V w E R. i s noetherian i f f A Then

0

i s noetherian and, moreover,

K

( A ~ )=

VwEfl.

Indeed, one has, as a consequence of the commutativity of diagramme (1.2.178):

A O = J A S

w w w'

so that

V w E f l

K

( A ~,)

1 . Manifolds, Funcfional Analysis,

62

since both

Jw ,

Sw

Distributions

are equiisomorphisms of the corresponding families of

Banach spaces.

If diagrame vanishes a s

Iw-oo I

that

is c o m t a t i v e modulo operators whose norm

(1.2.178)

w + oo

a , then t h e same conclusion is s t i l l t r u e , provided

E

i s s u f f i c i e n t l y smaZl.

Indeed, in that case one has: A.

= J

A S + w w w

Q

w’

where

I IQwl

IE +F

0

so that one has for

+

0,

as

w

+

w

0’

0

< 6 with 6 > 0 sufficiently small:

lw-wol

= K ( JA S ) = “ ( A , ) . w w w

K(A )

0

1 (1), E ( S ) E ( 0 . ~ ~ 1be , the family of Hilbert spaces of

Example 1.2.37. Let S1 be the circle of length 1. Let H 1 = (11,12), 11 2 0 integer, E functions u : S1 + C equipped with the inner products:

where

(k) is the Fourier transform on

S

1

,

1

u (k)

:= .f

exp(-2nkix)u (x)dx, k E

0

and ;*(k) is the complex conjugate of

(k).

Let A€

where a E

m

C

:= E

2 2 2 D,a(x)Dx

D + D2 x’ x

=

-id/dx

1 ( S ) , a(x) > 0, V x E S1.

Obviously,

*E

’

1

H(2,2) , E ( ’

-t

is equicontinuous, since u E

H(o,o) , E

E H

(2,2), E

(s’) iff

Z,

1.2. Functional Anulysis

Furthermore, one has: A.

=

D

2

and

is a continuous linear mapping, where H2(S 1 ) Sobolev space of order 2. = coker A.

Obviously, ker A. the index

=

'

1

H(2,0) , E

{c} with c E

C

(S )

is the usual

any constant, so that

0.

K(A )

=

RE : =

E 2Dxa(x) 2

0 Introduce

63

+ 1

Obviously,

is equicontinuous. We are going to show that for

EO

> 0 sufficiently small R

is

invertible and

is equicontinuous. Introduce S E as follows

1

(sEuE)(X)

sE (x,y)uE(y)dy,

=

0 where sE(x,y)

:=

Z (1+4n2E 2k2a(x))-lexp(Znki(x-y)), kEZ

(x,y) E S'xS 1 .

One has: (1.2.179) R s (x,y) = 6(x-y) + EqE(x,y), (x,y) E slxS1 € 6

where 6(x-y) is the Dirac 6-function and (1.2.180) qE(x,y)

1 q(x,E,kE)exp(Znki(x-y)), kEZ (1.2.181) q(x,e,ks) := 4~k€Dx(a(x)/r(x,2nk€))+ ~D~(a(x)/r(x,2i~ks)), 2 :=

r(x,r))

=

2 l+a(x)r) .

I. Manifolds, Functional Analysis, Distributions

64

It is readily seen that the following inequalities hold for q(x,E,kE):

I 6 (1.2.182) (D;~(X,E,~E) where C

P

cP (l+(kEl)-l,V

x E sl, V

E

E

V k E Z, v p E z+,

(O,co),

may depend only on its subscript.

Furthermore, since x

+

q(x,s,kE) is a Cm-function on S1 and (1.2.182)

hold for any integer p 2 0 (uniformly with respect to

E

E ( 0 , ~ ~ )one ) . has

m

for the Fourier transform q(m,s,kE) of q(x,E,ks) as a C -function of x E (1.2.183) Iq(m,e,ks)I

5 CN (l+m2)-N(l+\ksl)-1, V m E Z, V k

E 2, V N

v where the constant C

N

E

E

> 0, (O,E0),

depends only on the integer N > 0.

Now, (1.2.189) yields:

where Id is the identity operator and where 1

(1.2.184) vE(x)

:=

QEuE(x) =

I qE(x,yluE(y)dy. 0

Using (1.2.180), one finds for the Fourier transform jE(m) of VE(X) on

1

S :

(1.2.185) where


0 is a constand and B )=:b(

65

with

(1.2.187) b ' mk = (l+m2 ) (l+k2)-l q(m-k,E,kc).

Now, taking N = 3 in (1.2.183), using (1.2.183), (1.2.186), (1.2.187) and the inequality: (l+m2) (l+k2)-l 5 4(l+(m-k)2 ) , one finds:

where C 1 does not depend on

E

E

(O,E

0

).

Therefore, the linear map

defined by (1.2.184) is equicontinuous, V constand C(E

)

Now,choosing

E~

0

1

, so


0:

0

)

is an equicontruction in

that (I+EQ€)-'

:

H ~ ( 1S)

-f

H ~ ( 1S) ,

E

E

(o,E0),

is equicontinuous. Thus R

-1

:=

SE(Id

+

EQE)-'

(1.2. 188)

is equicontinuous, since the same argument as above using (1.2. 1%)

(the discrete version of Schur's lemma) shows that

is equicontinuous V

E~


0 is sufficiently small.

(O1 s)

is noetherian and its

E ( 0 , ~ ~ Besides, ) . one also has:

Ic(E)}, where C(E) is any complex valued function of

0

(O,E0).

Example 1.2.38. Let a( 0 sufficiently

large (depending on u), introduce the family of finite difference operators

4, on the lattice

I€$,by

First, we show that

A,,

the formula:

may be extended as a linear equicontinuous mapping

from 1 2 ( %) into itself:

3, :

12(\h)

Indeed, one has for v =

h E (O,hol.

12(%),

-t

A,," :

Gh(C) = (a(hc) + hb(h,hc))

+ h(2?r)-'

;h(C)

+

&h (S-q,h,hn) ih(q)dri,

J

lhrl\ 0, then

K

boundary c o n d i t i o n s on u should be imposed i n

a d d i t i o n t o t h e i n t e g r a l equation i t s e l f i n o r d e r t o have t h e unique s o l v a b i l i t y , while f o r

K

< 0 one may introduce

-K

p o t e n t i a l type o p e r a t o r s

with unknown d e n s i t i e s ( i n t h e c a s e considered, unknown r e a l o r complex numbers a s d e n s i t i e s ) f o r t h e same purpose ( s e e [Goh-Kr,

11, [Goh-Feld,

and [Esk, 11 f o r t h e multidimensional c a s e ) . While considering a f i n i t e d i f f e r e n c e approximation of t h e WienerHopf equation h e r e above of t h e form

where P ( 8 ) i s a polynomial of t h e s h i f t o p e r a t o r 8 on %,+, with h 1 c o e f f i c i e n t s C -functions of h E [O,h 1 , P o ( l ) = 1 , one has t o t a k e c a r e 0

11

1.2. Functional Analysis

75

of the index of the family of finite difference (discrete Wiener-Hopf or Toeplitz) operators on the left hand side of (1.2.199), which should coincide with

K.

This may not always be the case, as shows an example of the WienerHopf integral equation with k(x) = exp(-(x[) and its following finite difference approximation: x+(x) (uh(x+h) +

huh(y)exp(-jx+h-yl 1 ) = fh(x),

Z YE%,+

Indeed, it is readily seen that the index equation with k(x) = exp(-lxl) ,sirice 1+;(5)

=

K =

2

0 for the integral 2 -1

( 5 + 3 ) ( 5 +I)

.

For the approximating finite difference equation one finds after the % of kh (x) := exp(-)x-h)):

discrete Fourier transform on ah(0)

:= =

8 + kh(e) -h =

e(B2-2e(Chh+hShh)+1) ( 0 2 -20Chh+l)-l,

where 0 = exp(ih5) is the discrete Fourier transform of the shift operator

,

which is the winding number for ah ('8) when 8 goes around the unit circle r , Thus, the index

Kh

Kh

:=

-1

(217)

[arg ah(e)lr = 1, t/h > 0 ,

while for the integral equation itself

K =

0.

Of course, this kind of difficulty does not appear, if one uses, for instance, the following apprcpriate finite difference approximation:

or any other approximation P (8) of the identity such that P ( 0 ) # 0 for h 0 18 1 = 1 and the winding number of P ( 0 ) along the unit circle 10 I = 1 is 0

zero (see also [Goh-Feld, 11 as far as projection methods for WienerHopf operators are concerned). -1

For k(x) = X+(x)exp(-x), k(S) = (l+iS) the index K = 0. For its -1 h. -h finite difference approximation k (x) such that k (0) = (l+i 0 does not depend on f and

E.

Indeed, one gets (1.2.228) by using (1.2.2241, (1.2.2251, (1.2.227) 2 and the fact that the L (U)-norms of the functions eXp(-X/E), exp(-(l-x)/E) are O ( E 1/2) , as E * +O. Notice that Ao, the inverse of (1.2.223), has its spectrum only at infinity: A ( A ~ )= compact operator A

{ - I , while the spectrum of each AE, the inverse of the , has for each E > 0 only the discrete spectrum:

-1

It will be shown in Chapter 5 that a l l eigenvalues X E of A

are simple,

strictly positive and for each given n > 0 che following asymptotic formula holds for :A:

1.3. Distribution Theory

X E = ( 2 / 4 3 ) ~ - l + v2n2~/J3+ 0 ( s 2 ) ,

85

E +

0.

I Distribution theory

1.3.

In this section some basic facts from classical distribution theory are briefly sketched and several aspects of a possible extension of the distribution theory to a parameter dependent situation are presented, which will turn out to be useful in the following chapters. hlile presenting the

1

classical distribution theory, we follow essentially [Sch, 1 Sh, 1

1

1.3.1.

Test function space D ( U )

Let U

and [Gel

-

where the complete proofs of all the statements can be found.

5 lRn

be a nonempty open set. For each compact K

C

U, the Frechet

space DK was yet described hereabove. The union of the spaces DK when K ranges over all compact subsets in U, is Schwariz's test function space Obviously, D ( U ) is a vector space and $ E D(U) iff @ E

D(U).

m

C

( U ) and

the support of $ (the closure of the set where $(x) # 0 ) is a compact set in U. For @ E D(U) and each integer N L 0 let us introduce the norms: N = 0,1,

...

The restrictions of these norms to any fixed

DK

c D(U) induce the same

topology as do the norms p N hereabove. The same norms (1.3.1) can be used to define a locally convex metrizable topology on D(U). However, D(U) equipped with this topology is not a complete topological vector space, since one can indicate Cauchy sequences in D ( U ) which converge (in the sense of the topology defined by (1.3.1))

m

to C -functions whose support is not compact in U.

Introduce the topology on D ( U ) by saying that a sequence { $ v } v 2 0 C D ( U ) is convergent to zero if for each N = O,l,

norms (1.3.1) uf 4" vanish as v

K

C

+

5 K.

the

and, moreover, there exists a compact

U such that thesupportsof all @v, v = O , l ,

supp $, $v

+ +m

... given

... belong

to K:

This topology being translation invariant, we say that

$ (in D(u)) if $,

= $-$v + 0

(in D ( u ) ) . D(u) is a locally convex

complete topological vector space, where each closed bounded set is compact. Let L be a liiiear mapping of D ( U ) into a locally convex topological vector space X. Then each of the following three properties implies the other two:

1. Manifolds, Functional Analysis, Distributions

86 (i) L

:

U(u) + x

(ii) L

:

D(U)

+

is continuous.

X is bounded.

(iii) The restrictions of L to each DK As

C

D(U) are continuous.

a corollary, one has: every differential operator P(x,ax),

with coefficients a

E C m ( U ) , la1 L m, is a continuous linear mapping of

U ( U ) into itself. Distribution space D'(U)

1.3.2.

A linear functional L

:

D(U)

+

C, which is continuous with respect to the

topology on D(U) (described hereabove) is called a d i s t r i b u t i o n in U. The space of all distributions in U is denoted by D'(U).

The action of

u E D'(U) on a test function $ E D(U) is denoted by a,$>. If L

:

D(U)

+

C is a linear functional, then the following two conditions

are equivalent: (i) L E

D'(u).

(ii) For each compact K C
o

t

D'(U) is said to be convegent

to a distribution L if one has: lim = < L , $ > ,

V $ E

D(u).

V+-

Furthei, { L v j v Z O c D ' ( U ) is a Cauchy sequence if {O 1 support of J , , , j = O,l, ... .

where

U

k. 3

1

Let V be the union of all open sets where a given distribution

L E D'(U) vanishes. The complement W V is said to be the support of L . Let L E D'(U) and denote supp L its support in U. If $ E Cm(U) and J, E 1 in some open subset V c U containing supp L , then $ L = L . If supp

is a compact subset of U, then there exist

constant

C

an integer N 2 0 and a

such that

m

Furthermore, in that case L extends uniquely to a continuous linear functional on p

N

m

C

(U) (equipped with the topology, defined by the seminonns

hereabove). The least integer N 2 0 such that ( 1 . 3 . 2 )

still holds is

called the order of the distribution L . Let L E D(U) and assume that supp L = {xo3, where xo is some point in U; further, assume that the order of L (which is finite in this case) is

1.3. Distribution Theovy

89

N 2 0. Then one has:

where car In1 5 N, are constants,

0, integer m 2 0 and compact set K sets { O ( m , & , K ) ) ,

m E Z+, 6 > 0, K

m

of zero in C (U) is

C

U. The collection of m

C U

defines the topology on C (U). m

The dual space E ' ( U ) of continuous linear functionals on C (U) is the subspace of all distributions in D'(U), which have compact support in U. Obviously, D'6

(x-x,)

,V

c1

E

Zy,

v

xo E Rn , is an element in E' (Pi"

)

.

Important for applications is also the Schwartz space S'(Rn) of tempered distributions which is a subspace of 0 ' ( E n ) m

. The

space

S(Rn ) of

rapidly decreasing C -functions $(x) is introduced as a Frechet space equipped with the topology defined by the norms:

1.3. Distribution Theory where, of course, < x > ~= lilxl

2

91

.

Any differential operator p(a

is a continuous linear mapping from

)

s ( R n ) into itself, as well as the multiplication by any polynomial p(x), p

:

S(Rn )

-f

S(Rn

.

)

Thus, also a differential operator p(x,a

)

whose

coefficients and all their derivatives have polynomial growth of some fixed degree N 2 0 for 1x1

S(W"

is a continuous linear mapping from S(Rn

+ m,

into itself. Obviously, S(IRn)

I>

D(Rn

).

)

Moreover, Z)(Rn) is dense in

).

The space of tempered distributions linear functionals on

s ' (Rn)

is the one of all continuous

s ( R~ ) .

The Fourier transform

is a continuous linear mapping from ~ ( I R " )onto itself, its inverse being:

This is a consequence of the formulae:

9 E s ( R n ) and i(c',CN) can be

The following result is useful in applications. Let assume that supp $I

5

=

{x

(x',xn), x

=

extended as an analytic function of 5 moreover, the following norms of

c$

Ry is denoted by supp Q

5 %!

=

{x

For each u E

so(zy) : =

< 0, V

5 ' E Rn-' , and,

s(lRn )

, whose

support belongs to

similarly, the one of all $ E s(Rn

(x',xn), x

s ' ( W n)

fornIm 5

are finite:

The subspace of all testfunctions @

-

b 0 1 . Then

5 Ol, is denoted by S (Tii") 0 -

),

.

define its Fourier transform by the duality

between S'(Rn) and s(IRn), i.e.

By duality, the Fourier transform is an algebraic and topological isomorphism of S'(Rn )i.e. F S ' ( R n ) linear mapping.

-f

S'(IRn) is a continuous surjective

1. Manifolds, Functional Analysis, Distributions

92

Also by duality, any differential operator p(x,a

)

with coefficients

which(with all their derivatives)grow as a polynomial of some fixed degree N L

o

as 1x1

is a continuous linear mapping from S ' ( R ~into Atself.

+ m,

If the convolution f*g is well defined for f and g in S'(Rn), then one has :

Also, one has for each f Dx =-ia

E S ' ( R n ) and each differential operator p(D

)

,

with constant coefficients: F(p(Dx)f) = p(S)f(5).

For distributions u E S'(Rn) whose Fourier transforms

u(c)

are

locally integrable functions such that 0 such that

(ii) Conversely, if g ( 5 ) is an entire function of 5 E Cn which satisfies the inequality hereabove for some N and C, then there exists f

E st(Rn)

with its support in the ball { 1x1 5 r} such that F ( 6 ) = Fx+Ef, Nbeing its order.

1.3. Distribution Theoy

A

continuous li'near functional on

distribution in

;

which is denoted by

s 3 ( Z n ) is called a

all such functionals form a subspace of

one can also define the partial Fourier transform f(S',x

s'(?ii") by

=

)

s' (Rn)

The restriction of f E an element in S ' ( m n

to R :

is a distribution in

belongs to

s ' (Fy )

s' (xy)

to

.

s(%: )

We shall denote by

s'0 (Zy )

F ,f of x '+S

the formula:

and, using the Hahn-Banach theorem, one can extend each f E

and by

D' (Ry) ,

' being well-defined for each

so(xy) in the usual way,

each f E

tempered

s' (R: ) .

The partial Fourier transform FX 5'

$ E

93

of all $ E

the restrictions to R :

the subspace of distributions in

. The spaces s'o ( g+n 1

and

s(F:

s'

s(Rn) ,

(Rn)r whose support

are dual. Moreover, the

)

is an algebraic and topological partial Fourier transform F X'+E' isomorphism of s ) onto itself.

(x:

Let Uk, k

= 1,2,

be two open sets in IRn and let Ji

:

U1

+

U2

be a

m

C -diffeomorphism. Denote by J $

$-l

: U2

+

the Jacobian of the inverse diffeomorphism

-1

U1. For any f E D'(U ) define the composition foJI E D'(Ul) by 2

the formula: = ,

)/J

V $ E D(Ul).

$-I The mapping D'(U2) 3 f

m

f,$ E u'(U

) is continuous. Since C (U) is 1 dense in D'(U), one can extend the chain rule to the distributions:

ax

(fo$)

k

c

=

l5j5n

+

(ax $ ) ((ax f ) q ) , k k

v

f E

u'(u2).

v f

E

Similarly, one has:

C'$f). Also

has :

if $l

=

: U1

+

($09)(f.$),

U2 and Ji,

v

'$ E C r n ( U 2 ) ,

U' ( U 2 ) .

m

:

U2

+

U3 are two C -diffeomorphisms, then one

1. Manifolds, Functional Analysis, Distributions

94

Now, one can define distributions on a Cm-manifold M. Namely, let be an atlas on M. A distribution f E D'(M) is the collection

(U.,@,), 3 3 JEJ

{fjjjEJ, f . E D ' ( @ j ( U j ) ) , such that

I

If M is an open set in IRn then the new definition of distributions in

D'(M) coincides with the previous one, given hereabove. Of special interest is the case when M is the n-dimensional torus Tn, Tn = { z = ( z

,...,zn), zk =

exp(iCk),

[Ck/ 5

n, 1 5 k 5 nl.

Functions $ on Tn can be identified with functions J, on B i n , which are 2n-periodic in each variable: Denote by

D(Tn)

$ ( t l , ...,En)

=

+(exp(iS,),. . .,exp(iCn)).

the space of all functions @ on Tn such that $ E Cm(Rn).

For each @ E D(Tn) introduce its discrete Fourier transform (Fourier coefficients $(k) of $ ( E ) ) by $(k) =

I @(5)

exp(-i = h-’($(h,O)-(l-q)

0.

-x/h q $(h,x)) =

x,o =

h-y(l-q)

C

q-X/h($(h,O)-$(h,x)) =

x>o =

-hl-Y(l-q) C q-x’h x>o

E a $(h,y). O 0, V x E IE? and q(x) 1x1 b r with r
)exp(i (x/c)E)dC,

R

which solve the following Cauchy problem: = 0, (e2(at-ax)+i)u 2 2

x E IR, t

>

o

(1.3.28) U(E,o;X) = 0 ,

E

a tu(E,o;~)= A M .

First, let x E Ut = {x E I R , 1x1 > t}. We are going to show that -k u(E,t;u) belong to the equivalence class containing the distribution v

which is identically zero in D'(Ut), V k > 0. Indeed, let us rewrite u(~,t;x)in the form:

one has: (1.3.29)

1 at@+/

2 c

~ > 0, , ~v x E Ut.

Thus, using (1.3.29) and integrating by parts, one finds:

Repeating the same argument, one gets the conclusion that u+ - can be represented in the form: u+(~,t;x)=

-

E

k f v (E,~;x), V k 2 0, k

where vi are continuous functions of x E Ut, V t > 0, such that (1.3.30)

sup O<EJEO

sup lvil < Ct,k < XEUt

m,

V t > 0, V k 2 0.

m

Thus, (1.3.30) implies that u = O ( E ) for E + 0 in D'(Ut), i.e. that -ku 1s . in the same equivalence class as zero in D'(Ut), V k 2 0. E Now, let 1x1

t. Using the stationary phase method (see Chapter 3),

107

Notes one can show that the following asymptotic formula holds for U(E,t;X): (1.3.31)

-1 2 2 1/2) sin(€ (t -x )

u(E,t;x) = E1'2(~(t,x)+Ev(E,t;x))

where Ji(t,x) is a smooth function of x E V

t is a continuous function of x E Vt such that (1.3.32)

sup O<E<E~

max / v ( E , ~ ; xI )5 Ct < Ixl 0.

Thus, (1.3.311, (1.3.32) imply that E-YU(E,t;X) and (t,x)sin(E-'(t2-x2) 'I2) belong to the same equivalence class in D'(Vt), t t 0, provided that y < 3/2.

Notes __ For the concise presentation of the smooth manifolds theory have been used [Arnold, 1

1,

[Lang, 1 1, [Loomis, Sternberg, 1

1.

As far as the classical functional analysis is concerned, the brief account on basic functional analytic results needed for the further chapters (also in vol 11) is based on [Riesz-Nagy, 1 [Rudin, 1

1, [Lyusternik - Sobolev ,

1 1,

1.

Example 1.2.11

is a specific case of the general result in [Levinson, 1

1.

Example 1.2.12 illustrates how the linear theory of strongly elliptic and coercive singular perturbations ([Vishik - Lyusternik, 1 1, [Huet, 1 [Lions, 1 1 , [Frank, 15,19,22 1, [Frank - Wendt, 1,5,10

1

1,

and others) can

be extended to the classes of non-linear singular perturbations, whose linearizations are strongly elliptic or coercive singular perturbations (see also vol. I1 of this book). Example 1.2.13 is an illustration of the applicability of the Lindstedt-PoincarG method ([PoincarG, 1.2 Mitropolsky ,

1 1, see also [Nayfeh, 1

1

1

[Bogolyubov-

where formal asymptotic

expansions using this method are given for second order equations) to hamiltonian systems, which are singular perturbations of hamiltonians, describing motions with holonomic constraints. Extension of Banach's theorem (Theorem 1.2.34) to the parameter dependent situation as well as the statements concerning the commutativity of diagrammes in this case (Corollary 1.2.36) seem to be new and are useful in applications. Example 1.2.37 is an illustration of the general result concerning the stability of the index for the elliptic and coercive singular peturbations (see [Frank - Wendt, 1, 5,lO ]).Example 1.2.38 shows how the results concerning the classical

1 . Manifolds, Functional Analysis, Distributions

108

Wiener-Hopf and Toeplitz operators (see LGohberg [Gohberg

-

Feldman, 1

-

Krein, 1

1, [Krein,

1

1,

1) can be extended to one parameter families of

elliptic difference operators (see [Frank, 4,6,9-13

I).

Example 1.2.40

illustrates the complexity of spectral problems for singular perturbations; it is examined with full details in [Frank - Norde, 1 1. We follow [Schwartz,l

1, [Gelfand - Shilov,

1

1, as far as a concise

presentation of the distribution theory is concerned. Parameter dependent test functions and distributions were considered in [Frank, 8

1. The concept

of the singular support for a parameter dependent family of distributions was introduced in [Frank, 14

1, but, of course, was implicitly present in a

previous work by others on this subject.

CHAPTER 2

SOBOLEV SPACES OF VECTORIAL ORDER

3 o f v e c t o r i a l o r d e r s E IR are (5.1 i n t r o d u c e d and t h e i r b a s i c p r o p e r t i e s a r e p o i n t e d o u t . These s p a c e s p l a y I n t h i s c h a p t e r t h e Sobolev s p a c e s H

t h e same r o l e i n t h e s t a b i l i t y t h e o r y of c o e r c i v e s i n g u l a r p e r t u r b a t i o n s , of o r d e r r E IR f o r t h e e l l i p t i c

a s t h e c l a s s i c a l Sobolev s p a c e s H

boundary v a l u e problems w i t h o u t p a r a m e t e r s . 2 . 1 . s p a c e s on

nn

= ( s l , s 2 , s ) E IR3 3 Definition 2.1.1.

Let s

and l e t

E~

be a p o s i t i v e c o n s t a n t

For a f a m i l y o f d i s t r i b u t i o n s

m

(O,Eo1 x

n-1

3

( € , < I )

*

V(E,5',Xn)

E S'(IRX

)

n

whose Fourier t r a n s f o r m s a r e ZocaZZyintegrabZe f u n c t i o n s d e f ~ n ethenorms 3 (FxnlSnv) ( E , 5 ' t S n ) I I L

We say t h a t v E

v

(€,C')

H

E (O,Eo1

The f u m i l y v E H

,E,E'

x

IRn-1.

(s), < '

Obviously, H ( s ) I

(S)

. I

(IR) i f i t s norms ( 2 . 1 . 1 ) a r e f i n i t e ,

(IR)

(IR,

)

'n

if

C,(R)i s a f a m i l y of H i l b e r t s p a c e s w i t h t h e i n n e r

products

where v . = F, 7 n For si,

n

v,. l

j = 2,3 non-negative

i n t e g e r s one c a n a l s o i n t r o d u c e t h e norms

2. Sobolev Spaces of Vectorial Order

110 t h e norms

1 1.11

being e q u i v a l e n t uniformly with

r e s p e c t t o t h e parameters. P r o p o s i t i o n 2.1.2. H

(s). 5 '

i s a family of Banach spaces. s

-s

E .The

maps H (R) i s o m e t r i c a l l y 2s3F (s). 5 ' x +En 2 n o n t o t h e Banach s p a c e B ( ( O , E ~ ] L; (R))of bounded f u n c t i o n s on ( 0 ,0~1 v a l u e d 2 i n L (R). 0

'

operator E

D e f i n i t i o n 2.1.3.

For a family of d i s t r i b u t i o n s 3

(O,E01

E

-t

U(E,X)

E

S'(R;)

d e f i n e t h e norms of order s:

Again,

( O , E ~ ]3

E

+

H

(s), E

i s a Banach s p a c e , s i n c e

( R n ) i s a f a m i l y of H i l b e r t s p a c e s and H

isl < 5 > s 2 < ~ < > s 3x+< F

(S)

(Rn)

i s a n i s o m o r p h i c map of

( I R ~ ) onto B ( ( o , E ~ I ; L ~ ( I R ~ ) ) .

H (S)

Remark 2 . 1 . 4 . For c o n v e n i e n c e w e s h a l l u s e t h e n o t a t i o n [ u ] o f f a m i l i e s of d i s t r i b u t i o n s v a l u e d i n S ' H ( S )

(wn-') ,

X'

[ u ] ( ~ )f o r t h e norms (s), E l t h a t belong t o H

respectively.

Example 2 . 1 . 5 . ( i ) L e t @ E S(Rn-')

and l e t v ( ~ , C ' , x) be t h e f a m i l y

( s ) ,E

W"

2.1. Spaces on

111

-+.

i.e. i f f s2 > ( i i )L e t

(0,11 3 E

(2.1.6)

+ U(E,X)

=

2 ( 4 n ~

W e find, using t h e polar coordinates = (F

;(E,

-2

X+E

Therefore, u

E

E

H

Further, u

H

3

(IR

)

.

i f f s +s

2 3 < + (R3)i f f t h e f u n c t i o n

( s ),E

( S )

-2s (0,11 3 E

+

IS(€) = E

1

m

I

p

2

2 s2

(l+p )

2 2 s3-2 (1+~ p ) dp

0 i s bounded on ( 0 , 1 ] . One h a s f o r

E

+

+0:

i-2s1 -2s1

,

if s

2

In E , i f s2 -3-2(s +s2) 1 , i f s2

I ~ ( E w{ )

< -3/2 =

-3/2 -3/2. 3

Hence, t h e f a m i l y ( 2 . 1 . 6 ) b e l o n g s t o H

(IR )

i f f s belongs t o t h e

(S)

s e t V,

v

3

E R , sl f , / I

f

IIvIl

5 CI

IVI

,E.S'

I ( s ), E , S '

i f s2 = f , i f s2

f.

(R) c a n b e i d e n t i f i e d w i t h t h e

(IR) , it i s enough t o show ( 2 . 2 . 4 ) f o r v v a l u e d

'2+'3 t h e l a t t e r b e i n g dense i n H ( R ) , v r

The Cauchy-Schwartz

holds

, € , E l

(S)

fixed H

u s u a l Sobolev s p a c e H

s(R) ,

E ( 0 . ~ ~x 1R

( s - f e 2 ), E , s ' [n,vl ( s - f e 2 + ( s 2 - f ) e ) , E , 5 '

in

n- 1

v (E,S')

E

IR.

V

E

For such v one h a s :

inequality yields:

where (2.2.6)

de f I ( E , ~ '=) J



-2s

-2s 2 < ~ 5 > 3d5n
0,

IR s i n c e s +s < f . 2 3 F i r s t , c o n s i d e r t h e c a s e : s2 > f . Making t h e change o f v a r i a b l e

5

n

= <S'>t, we g e t :

2. Sobolev Spaces of Vectorial Order

116

with

cs

=

max{/ R

-2s -2(s +s ) *dt,/ dt). IR

Now, let s2 = j . Making the change of variable

en

= E-'2 0 holds:

(0,1],

3'

Further,

v

lLn(E<e'><Ec'>-l)I 5 Ln(l/E),

E (0,ll x IRn-l

(E,C')

and 1 /Ln(E 1.

= j we get:

',

-2s (2.2.8)

I(E,s') 5 C(l+lLn El)

<Eel>

Finally, consider the case: s2 < f . The same change of variable

sn

=

E-'<Ecv>t

yields: 1-2(s +s

2s -1 I(E,S')

(2.2.9)

= E

$ csE

1-2(s +s ) 2s -1 2 3 <Eel>

c

max{/

with =

w

jtl

-2s -S 3(~2 f . Then the r e s t r i c t i o n operators r0

n

( a ),E

: H (s),E (w"-l) a r e uniformly bounded with respect t o E E ( 0 . ~ ~if1

a = (sl,s2-f,s3)f o r s2 > f and a = (s +s - j , o , s 1

2

2

(wn)

+s -j) f o r s2 < f . 3

+

2.2. Restriction to Hyperplane

P r o o f . One

applies (2.2.4)

v(E,S',X

to

s q u a r e s and i n t e g r a t e s o v e r

= F

)

n

x'+S

5 ' E Rn-'

117

,u)

( E , ~ ' , X ~ ) t, a k e s

the

0

Corollary 2.2.4.

The r e s t r i c t i o n operators

( R n ) -F (R) C ( o ) , S , ,T o : H ( 5 ), S ' ( S ) ( m n - l ) w i t h a d e f i n e d i n Theorem 2 . 2 . 3 . , a r e uniformly bounded w i t h (a) r e s p e c t t o 6 ' E mn-l. :

T~

H

+

H

Remark 2 . 2 . 5 . ( R n ) and l e t s2+s3> f . Then t h e f a m i l y

L e t u ( E , x ' , x ~ )€

( O , E ~ ]X R 3 ( € , x n ) + u ( € , x ' , x n ) E with

0

E R3

H

( 0 ), E

is continuous i n x

d e f i n e d i n Theorem 2 . 2 . 3 ,

E

n'

V

E

E

(O,E~].

E iR i s u n i f o r m w i t h

Moreover, t h e c o n t i n u i t y of t h i s map i n x respect t o

(1R;;l)

E (O,E~I

I n f a c t , one f i n d s , u s i n g t h e p a r t i a l F o u r i e r t r a n s f o r m F x , + S , : [ U( E ,x ' ,x +h) -u (E , X ' ,x n ) 1 2

2

5

( 0 ), E

-2s1 2s 2s 3 I S ( ~ , S ' , h ) ~ 2 < ~ S > / ; ( E , S ' , S , ) I

J

82

dS'dS,,

mn where

I s ( ~ , F , ' , h )= 2 s -1 (2~)-~/

2

0'

E (o,Eol,

lRn since s + s > n/2. 2 3 One finds easily when

E

+

+0:

As a consequence of (2.2.12) one gets (2.2.11).

Corollary 2.2.7.

If u E

H(s)

(Rn) w i t h s2+s3 > n/2, then

where t h e c o n s t a n t c may depend onZy on s , n and

E

~

.

Corollary 2.2.8.

Let u E H

(S)

( 0 . ~ ~31 6

+

( R ~ ), s 2 + s 3 > k+n/2, where k > 0 i s i n t e g e r . Then ,E U(E,X) E Ck(Rn) and, moreover, t h e norms l u ( E . . ) ICk(,p)

I

1

2.2. Restriction to Hyperplane

119

s a t i s f y a l t e r n a t i v e l y t h e f i r s t , second o r t h i r d i n e q u a l i t i e s (2.2.11) if s2 > k+n/2, s2 = k+n/2 and s2 < k+n/2. 2 ICk(,p) 5 cllO ( s ),El

t h e same

Remark 2 . 3 . 4 . where s + s 3 = m+y w i t h m > 0 i n t e g e r and IyI < 2 ( 5 ), E ' a consequence of Theorem 2 . 2 . 3 , t h e r e e x i s t s t h e r e s t r i c t i o n

Let u E H+

n u E H

0

( u ) ,E

o f u on t h e h y p e r p l a n e {x

(Rn-')

=

01,

!.

w i t h u E lR3

Then a s

defined

i n t h e f o r m u l a t i o n o f t h e theorem. k- 1 ( R n - l ) , 1 5 k 5 m , is + D u(€,x',xn) E H xn ( s - ( k + ? ) e 2 ),E c o n t i n u o u s (however, n o t u n i f o r m l y w . r . t . E ) , a n d D k - l ~ ( ~ , ~ ) ' , ~0 f o r xn x < 0 , one g e t s t h e c o n c l u s i o n t h a t 71 Dk-lu = 0 , 1 5 k 5 m , V E > 0 . Xn Example 2 . 3 . 5 . S i n c e IR 3 x

The f a m i l y u ( E , x ) s +s

2

3


0 i n t e g e r i n t r o d u c e t h e f o l l o w i n g e x t e n s i o n O m -n o p e r a t o r tN : c O ( R + )+ cN-l ( n n ) , 0

(2.4.3) where H(x

k u = H(Xn)u(E,X',X N

c U(E,X',-PX

)+ff(-Xn)

lSp5N

is t h e Heaviside's f u n c t i o n and C P' s y s t e m of l i n e a r e q u a t i o n s :

(2.4.4)

)

Z (-p) 16p6N

j-1

C

P

= 1,

1 2 p 5 N,

satisfy the

1 5 j 5 N.

Lemma 2 . 4 . 1 .

For s 2 , s3 n o n - n e g a t i v e i n t e g e r s , t h e operator t N w i t h

N 2 s2+s3can

be

extended a s a continuous ( u n i f o r m l y k i t h r e s p e c t t o t h e parameters) mapping from H fs) . € , E l (lR+) ( r a p . H H

( s ) ,E

(S),E(~n+))

into

H ( S ), E L '

(IR) ( r e s p .

(Rn)J.

P r o_ o f . The s t a t e m e n t of Lemma 2.4.1 _ inequalities

i s a n immediate c o n s e q u e n c e of t h e

2.4. Spaces on F?:

where t h e c o n s f a n t C depends o n l y on N. For a f a m i l y of d i s t r i b u t i o n s ( E , S ' ) with s

I'

123

0 +

E

u

S ' ( I R + ) and s

3 E R

2,3 n o n - n e g a t i v e i n t e g e r s , i n t r o d u c e t h e norms:

j =

where t h e infinimum i s t a k e n o v e r a l l e x t e n s i o n s Q of u t o R o r t o IRn, respectively

.

Lemma 2.4.2. 3

Let

5

E R and l e t s .

=

j

3'

2.3 be non-negative i n t e g e r s

. Then

the n o m s

(2.4.1) ( r e s p . (2.4.2)) and (2.4.5) (resp. (2.4.6)) a r e e q u i v a l e n t

uniformly w i t h r e s p e c t t o t h e parameters

E (O,E

0

I

x i ~ ~ - ' .

0

P r o o f . The norm ~

I 1 . I I (s)

d e f i n e d by ( 2 . 1 . 3 ) b e i n g e q u i v a l e n t t o

, € , < I

I I * / I (s), € , E l

uniformly with r e s p e c t t o

where t h e c o n s t a n t C does n o t depend on

( € , E l )

E, 0, one has:

Il

;

v(s)

5 0,

v

s2' s3'

( R + ) is isomorphic €0 the factor-space

n ( s ), E , S ' (JR)/qs),E,5,. The projection II+ n '

2 being orthogonal in L ( R ' n

),

one has

The closed graph theorem implies that there exists a constant C > 0 such

2.4. Spaces on R: that uniformly with respect to ( E , C ' )

I l u l I ((1) S),E,c'

5 CI

We shall use

I 1. I I+

E ( 0 . ~ ~X 1IRn-l holds:

I I(1) I lull (.-) , < '

I U I /;s),E,F'r

127

5

lul

l;s),C,-

as working norms for establishing a priori

( S )

, € , E l

estimates for the s o l u t i o n s of coercive singular perturbations. Definition 2 . 4 . 4 .

A f a m i l y of d i s t r i b u t i o n s t h e space H

( s ) ,E

(BY)

En such t h a t LU E H

( 0 . ~ ~3 1E

+

u(E,.)

E S'(mn) i s s a i d t o belong t o

i f t h e r e e x i s t s an e x t e n s i o n ku E S '

( s ), E

( R ~ ) . The norm on H

( s ) ,E

(HI:)

(nn)o f

u to

i s g i v e n by

where t h e infimum i s taken over a l l t h e e x t e n s i o n s L, or e q u i v a l e n t l y ,

A family of distributions

belong t o

H ( s ) (R+)i

( o , E ~ I3

E + U(E,.)

E s ' ( R ~ )is s a i d t o

f

We shall denote by k o the extension by zero on Rn of functions : defined on R

.

The following result will be needed later on for establishing twosided a priori estimates for elliptic (coercive) pseudodifferential singular perturbations. Theorem 2.4.5. Let s E

n3 be such t h a t

Is2]
0 such t h a t

Proof. Without restriction of generality one can assume that s = (0,s2,s3). Obviously, it suffices to show that

2 . Sobolev Spaces of Vectorial Order

128

(2.4.13)

I b0ul I

v

(S),E, 0

such that

2. Sobolev Spaces of Vectorial Order

130

1 s2 1

Since

< f,

I s 2+s3 1

< f , the last inequality yields:

B

and that proves (2.4.13).

It can be shown, but will not be done here, that c1 in the last inequality can be chosen as 1 6 ( ( f - ~ ~ ~ ) ) - ~ + ( f - l s ~ +(see s ~ ~ )[Fr-Hei]). -' Remark 2.4.6. With s = ( 0 , s 2 ,O),

Is2) < f , Theorem 2.4.5 has as a

consequence the continuity of the extension by zero .Lo in classical SobolevSlobodetsky spaces, i.e. Lo mapping for 1s21
0 ( s ), E (s), E ( R n ) b e i n g isomorphic t o t h e H i l b e r t s p a c e w i t h t h e i n n e r

( s ) .E

product

__ -2s

(u,v)

(2.5.1)

( s ), E

=

( 2 ? ~ ) - * ~1 Rn

;(E,~)V(E,~)E

2s

2 s 2 < ~3d5 p

1

The R i e s z theorem s a y s t h a t any c o n t i n u o u s l i n e a r f u n c t i o n a l @ ( u )on H

(s), E

(2.5.1)

(Rn) and

i s g i v e n by a n e l e m e n t v

I

1

=

ilvli

(s), E

E

H

( s ),E

(IRn)

as an i n n e r product

= (v,v)'

(s), E '

Denote -2s

(2.5.2)

w

= 6

so t h a t w

E

H

2s '2s2 G,

(-s),E

and

2. Sobolev Spaces of Vectorial Order

132 for V u E H Hence,

(Rn), V w E H (2). (-s) , E (s , E ( 2 5 . 2 ) i s a n isomorphism between H*

u E

H

(mn)

( B n ) and H

( s ) ,E

t h e v a l u e of a f u n c t i o n a l w E H

a n d , moreover

(mn) on

,

(-s) E

(-s).E a function

( R n ) i s g i v e n by t h e f o r m u l a ( 2 . 5 . 3 ) . W e s h a l l u s e H

( s ), E

a s an isomorphic r e a l i z a t i o n o f H

*

(mn)

B e s i d e s , t h e form ( 2 . 5 . 3 )

(El").

( s ),E

(-S) ,E

c a n b e a l s o v i e d a s a c o n t i n u o u s e x t e n s i o n of t h e form -2n

J

(u,$) = ( 2 n )

;(~,E)$(e,c)d;, V u E H

mn

( R n ) , V @ E S(IRn).

( s ),€

Theorem 2.5.1.

Let

(i:s) *

isomorphic t o n O +

on u + E

H

(s).E

!L : H

)

* is

(-s), E

(my)

( ~ T ) - ~ " (IE , S ) ! L V ( E , S ) ~ S ,

=

Rn

where

(s). E t h e vaLue of a f u n c t i o n a l v E H

(my) and

(-s),E

w i t h s E m 3 . Then

,€

i s g i v e n by t h e formula:

(u+,v)

(2.5.4)

i:s)

be t h e dual of

,€)

+

i s any continuous Linear e x t e n s i o n

+ H(-s) ,E

(-s) , E

operator; b e s i d e s ( u + , v ) i s we22 d e f i n e d by (2.5.4), i . e .

t h e r i g h t hand

s i d e i n ( 2 . 5 . 4 ) does not depend on t h e e x t e n s i o n Lv of v t o Bn. ~

"-

Proof. F i r s t , n o t i c e t h a t i f u + H;s) , E and E -2n = 0 and, conversely, ( u + , v - ) = (2.rr) (u+,v-)

-

( 0 ) $E

and ( u + , v - ) = 0 f o r V u

+

E H

t h e n v- E H ( - s )

(S),E'

then

,€

if v

-

E

€3

t o t h e d e f i n i t i o n of t h e s u p p o r t of a d i s t r i b u t i o n , one h a s :

.

V $ E C:(Rn)

I

m

i s d e n s e i n H-

S i n c e C0(.(") -2n

(u,@)I S ( 2 n )

I lul I (s), € I 141 I ( - s )

one g e t s t h e c o n c l u s i o n t h a t ( u + , v ) C o n v e r s e l y , l e t v- E H Hence, i n p a r t i c u l a r , supp v-

5

-

.I!,

Let f E H (2.5.5)

(-S)

,E

HT-~),~.

(By), if

(u+,if)( 0 ) , E

=

E

H

(-S) ,E

(2lTY2"1 Rn

v

(IRn),

(u+,$)

=

0,

, € I

v

,€,

and l e t ( u + , v - )

(lRn)

(W")

and

0 , V u + E H;s)

=

(-s),E (v-,$) = (@, v -) = 0 ,

i . e . v- E

(-s) ,E

(-S),E

I n f a c t , according

,€'

=

v- E H-

(-s), E '

0 , V u+ E H+

m

(S)

$ E Co(my)

u+ E H;s)

,E'

, i.e.

,€.

Then t h e form

(E, -n/2

I

=

-n/2

when s2 < -n/2

H ( S ) (Z")

i f f s 2 > -(n/2),

< -(n/2),

s1 5 n-a. < n-a, o r s 1 2 d e n o t e by ITh u i t s r e s t r i c t i o n t o

$.

AS

usual,Hs ( R n ) s t a n d s f o r t h e c l a s s i c a l Sobolev-Slobodetski space of o r d e r s E R.

p r o p o s i t i o n 2.7.4.

If

u E

H s ( ~ n ) w i t h s > 4 2 , then n hu E H ( 0 , t ) (z"), v t

over, t h e following estimates hold:

I

'hUI

I (0, t ), h

5 IIu/lt

(2.7.11)

I lu/ I t

5 [/.a hu

'(0,t)

'

s-t +

Cs,th

Ilul

Is)

< s-n/2

and, more-

2.7. Spaces of One Parameter Families of Meshfundions

fcii:

V h . E ( 0 , h 1, where t h e c o n s t a n t c depends only 0 s,t

on i t s s u b s c r i p t s .

P r o_ o f . I t s u f f i c e s t o c o n s i d e r t h e c a s e , when u E S(IRn) _ i s dense i n

H

(IRn)

,

W e have used t h e f a c t t h a t 1

I
n/2. Then u E C ( Z ) , V k < s-n/2. Y (Y,S)

Proof. _ _ One has: Dau(h,x)

(2n)-" In

=

u (S)dS.

ei<x15'

T

5 .h Hence, for l a / 5 k,the Cauchy-Schwarz inequality and the estimate

151

5

(n/2)/51,V h 2 0, V 5 E Tn

5 rh

l2

IDzu(h,x)



5 C

yield:

-2 (s-k) d5

2 (s-k) I;h(5)12d5 lta/2

I Tn

Tn

5 rh

5 .h

and that proves Proposition 2.7.7. Remark 2.7.8. Proposition 2.7.6 is no longer true when k

=

s-n/2. In fact, the mesh-

function uo(h,x)

=

-1 F ('ln)-1, 5+x,h

obviously belongs to

H ( o , f ) ( 21 ) ,

n

=

but uo(h,O)

1, +

m

when h

+

0.

2

2. Sobolev Spaces of Vectorial Order

144 D e f i n i t i o n 2.7.9.

The f a m i l y o f norms

1 1 . I I (s),h,

h E (0,h0IJiss a i d t o be s t r o n g e r than

~ ~ ~ ~ ~ ( is f t t h ) e,r eh e3x i s t s a c o n s t a n t C, which might depend hO, such

(2.7'16)

' '

that

1 l u l 1 ( s # ), h

' '1

l u / 1 (s),h' V

h E (O,hO], V u E H ( s ) , h ( < )

1 1 . I I (s'), h < 1 1 . 1 1 (s), h 11.1 1 ( ~ ) , h a r e stronger than 1 1 . I I ( s ),h (s'),h-

we w r i t e

I

on s,s' and

Or

.

/ 1 * 1 / ( ~ 8 ) ,i hf t h e norms

as

The f o l l o w i n g r e s u l t c a n b e proved p r e c i s e l y i n t h e same way P r o p o s i t i o n 2.1.8. P r o p o s i t i o n 2.7.10.

One has (2.7.17)

I 1 . 1 I (s'),h


sj,

t h e c o n i c s e t ( s e c t o r ) V* b e i n g c a l l e d , a s p r e v i o u s l y , r e c i p r o c a l t o V C o r o l l a r y 2.7.12. 2

L e t s E IR m d Zet a E i n t

.:v

Then for V

c o n s t a n t c which may depend only on G,T,U,S

T

.

E vs, V 6 > 0 t h e r e e x i s t s a

and ho, such t h a t

2.7. Spaces of One Parameter Families of Meshfundions

I n f a c t , t h e s a m e argument as i n C o r o l l a r y 2.1.12,

145

leads t o (2.7.21).

With 2 . V , j = 1 , 2 , d e f i n e d a s p r e v i o u s l y i n ( 2 . 1 . 1 5 ) ( w i t h s = O ) , and 3 * I S 2 . V b e i n g t h e r e c i p r o c a l h a l f l i n e s , one h a s a l s o t h e f o l l o w i n g

1 s

C o r o l l a r y 2.7.13.

L e t s E n2 and l e t o E i n t a . v * . Then f o r V 1 s

T

E

a 1. vs t h e r e e x i s t s a

c o n s t a n t c such t h a t ( 2 . 7 . 2 1 ) h o l d s . The f o l l o w i n g r e s u l t t u r n s o u t t o b e u s e f u l f o r e s t a b l i s h i n g v a r i o u s k i n d s of a p r i o r i e s t i m a t e s f o r d i f f e r e n c e o p e r a t o r s ( s e e a l s o Theorem 3 . 1 1 . 1 5 ) . Theorem 2.7.14.

Let

@

E S ( Z n ) . Then for V s E R

(2.7.22)

1 l @ u lI

(O,s) , h

where t h e c o n s t a n t

C

'

,

V h E (O,hOl one has:

I 1 IuI

max XERn

s,@,ho

I(0,s), h +C s , $ , h O

'

IUI

(0,s-1) ,h

depends o n l y on i t s s u b s c r i p t s .

Proof. By d e f i n i t i o n , one h a s

where ' d e n o t e s t h e o p e r a t o r a c t i n g on a meshfunction v a c c o r d i n g t o

t h e formula

= [',@I

:

h

1 IBs,$ul

( 0 ) ,h

'

's,@,h0l

lUl

(0,s-1) ,h

'

where t h e c o n s t a n t C may depend o n l y on i t s s u b s c r i p t s . One h a s

:

v(h,E)

def =

Fx+:,h

h Bs,$U

=

with

'

2. Sobolev Spaces of Vectorial Order

146

We estimate the integrand. For doing that, we notice that a routine computation shows that €or any point R = ( Q l , R ) on the unit circle 2 2 2 Q1+R2 = 1 the following inequality holds for V p E I R :

with C

=

max{l,Ip/}.

P

Since both sides of (2.7.24) are homoqeneous in R of order zero, (2.7.24) holds for V Q E n L \ { 0 1 . Applying (2.7.24) with

Q1 = 2,

one obtains

Using the identity

which along with (2.7.25) yields the estimate:

Indeed,

n2

=

= 01

yk b e i n g i n t e g e r w i t h 1 a s t h e i r

g r e a t e s t common d i v i s o r ,

(2.7.32)

{y, ,...,y

} = 1.

F i r s t , w e prove t h e following Lemma 2.7.16.

Let y ( l )

= (yll

,...,y l n )

E zn s a t i s f y t h e c o n d i t i o n (2.7.32). Then it can

be compZeted b y v e c t o r s y ( j ) way t h a t t h e m a t r i x y

= (yjl

= (y.

,..., y I,n )

jk l < j , k 5 n

belong t o S L ( n ; Z ) , i . e . d e t y

=

having

E zn, 2 5 j 5 n , i n such a y ( J ) , 1 5 j 5 n,

as i t s lines,

fl.

Proof. If y ' l )

=

setting y

e (kf

, { e l , ...,e

} b e i n g t h e s t a n d a r d o r t h o n o r m a l b a s i s i n lRn

,

then

1 5 k 5 n , one g e t s t h e i d e n t i t y m a t r i x y = I d E S L ( n ; Z ) .

= ek Now l e t y ' l ) / ek, 1 5 I( 5 n , s a t i s f y (2.7.32). I t i s immediate t h a t

(2.7.32) i s n e c e s s a r y f o r t h e e x i s t e n c e o f y E S L ( n ; Z ) h a v i n g y l i n e . ~ i r s t n, o t e t h a t if y ' l ) f ( l ) = ( f l l ,..., f l n ) E Z n ,

y").

= .f(l)

with s o m e a

as i t s

E SL(n;Z) and Some

t h e n f ( l ) s a t i s f i e s (2.7.32) i f f so i t i s f o r

,...,f I n } = d w i t h d E Z , ( d l > I , t h e 11 = d and v i c e - v e r s a , s i n c e = a f ( l ) y i e l d s { y l l , ...,y

I n d e e d , assuming t h a t { f

equality y ' l )

SL(n;Z) i s a group. Therefore,

if one s u c c e e d s t o complete f ( ' )

by v e c t o r s f

(2)

, . . ., f ( n )

i n such a way t h a t t h e m a t r i x f = ( f . ) h a v i n g f ( 1 ) a s i t s l i n e s w i l l b e l o n g t o lk S L ( n ; Z ) , t h e n t h e m a t r i x y = f a t w i l l s a t i s f y t h e r e q u i r e m e n t o f t h e lemma.

2.7. Spaces of O n e Parameter Families of Meshfunctions Hence, it i s s u f f i c i e n t t o show t h a t f o r a g i v e n y " )

=

149

( y l l , ...,y l n ) E Zn

s a t i s f y i n g ( 2 . 7 . 3 2 ) t h e r e e x i s t s a m a t r i x a E SL(n;Zn) s u c h t h a t = a e . Denote by y # 0 t h e c o o r d i n a t e o f y ( l ) w i t h t h e minimal 1 1, k o a b s o l u t e v a l u e ( i f t h e r e i s more t h a n one c o o r d i n a t e w i t h t h i s p r o p e r t y ,

y(l)

w e t a k e anyone o f t h e m ) . L e t y y"),

Irjo

,

j,

# k o , b e any o t h e r c o o r d i n a t e of

= with integer p . , r . , t h e 1, l o 'j,Y1 r k o + r j o lo lo s a t i s f y i n g t h e i n e q u a l i t y lr , < I Y l , k o ' . Introducing

t h e n one c a n w r i t e y

remainder r .

I

JO

'0

a y ' l ) w i t h a = ( a . ) such t h a t a , , = 1 , Ik I1 1 5 j 5 n , a . = 0 f o r j # k , j # j, o r k # ko and a . = - p . , so Jk 10'kO Jo t h a t d e t a = 1 and a E S L ( n ; Z ) . it i s e a s i l y seen t h a t f ( l )

=

C o n t i n u i n g t h i s p r o c e s s (which i s p r e c i s e l y t h e E u c l i d e a n a l g o r i t h m f o r t h e c o m p u t a t i o n o f t h e g r e a t e s t common d i v i s o r o f i n t e g e r s y l l , ...,y l n ) , one g e t s f i n a l l y a l i n e c o i n c i d i n g w i t h some f e k , 1 5 k 6 n . B e s i d e s , e a c h s t e p o f t h i s p r o c e s s r e p r e s e n t s t h e p a s s a g e from some v e c t o r f E Zn w i t h t h e property (2.7.32)

t o a n o t h e r one f ' E Zn w i t h t h e same p r o p e r t y , f '

w i t h some a E S L ( n ; Z ) . F i n a l l y , it i s q u i t e o b v i o u s t h a t e l

o b t a i n e d from +e i f o n e a p p l i e s t o +e a n a p p r o p r i a t e CY E S L ( n ; Z ) . k k A s a consequence o f Lemma 2 . 7 . 1 6 ,

= af

can be

I

one c a n w i t h o u t r e s t r i c t i o n o f

g e n e r a l i t y , c o n s i d e r t h e t r a c e s o f m e s h f u n c t i o n s on t h e h y p e r p l a n e

II

=

{x

=

01.

AS

usual x

Dznote by (Sh(R;;l a n d , i n t h e s a m e way,by

th

=

) )

(x*,xn), 5 = (6',cn), 5 = ( c ' , ~ , ) . k t h e d i r e c t p r o d u c t o f k c o p i e s o f Sh (IRn-l x',h)'

II l5jSn

'(s.,h) 1

(IRn

x,h

t h e d i r e c t p r o d u c t of s p a c e s

ff(s,),h,

1 6 j 5 k. 1 Theorem 2.7.17.

L e t k > 0 be i n t e g e r . F o r s2> k+f t h e one parameter f a m i l y of mappings (2.7.33)

Hts)

, h (an ) 3 u ( x ' , x n ) x,h

{Dl h u ( x ' , 0 ) j 0 5 j 5 k E x- 1 n-1 0 5 j 5 k '(s1,s2-j-f) , h ( m x ' , h ) -f

is, uniformly w i t h r e s p e c t t o t h e parameter h E [ O , h o ] , a f a m i l y of continuuus l i n e a r mappings. For any g i v e n k s,-f t h e one parameter f a m i l y of mappings

2. Sobolev Spaces of Vectorial Order

150

i s surjective, and there e x i s t s a family o f ( u n i f o m Z y with respect t o the parameter h E (O,hO])bounded linear mappings, l i f t i n g f r o m n- 1 (sl' s2- k - f ) ,h'RxRx',h) to H ( s ().I,",h'.

'

Proof. We first show the continuity of the map (2.7.33) uniformly with respect to h. One has:

Cauchy-Schwarz

n

inequality yields:

Further, obviously, one has for j < s -4: 2 2(1-s2) dS 6 C 1 (2.7.36) I1

2 2 -2 ( 0 and C 2 > C1. 1 Therefore, Crk(h,S') E C ( h h ) , where C ( t ) a r e smooth bounded rk rk 1 f u n c t i o n s f o r a l l t E R+ T h i s c o n c l u s i o n a l l o w s u s t o show t h a t

(2.7.75), perhaps with d i f f e r e n t constants C

.

I n d e e d it s u f f i c e s t o v e r i f y t h e u n i f o r m boundedness i n h E (O,hOl o f the expression

W e have:

D ~ - ' v ( h , E ' , t ) = A m-j-k-h ak(h,5') t,h k

C

Crk(hA)pr t / h ( h X ) z

l5rSm

so t h a t

s i n c e ~ ' ( 0G O<x 0.

Thus t o o b t a i n t h e n e c e s s a r y e s t i m a t e f o r a g i v e n s E (0,+),it s u f f i c e s t o prove t h e i n e q u a l i t y :

where t h e c o n s t a n t C

may depend o n l y on s E ( 0 , i ) .

The l a t t e r w i l l f o l l o w from t h e f o l l o w i n g Lemma 2 . 7 . 3 1 .

Let u E H(O,s),h( 0.

Noticing t h a t kl-s

k+ 1 J yS-'dy k

t ks-l(l+sk-'-s(l-s)

(2k2)-l-l) >

+,

V k 2 1,

we f i n d

NOW,

we show t h a t m

(2.7.93)

J 0

m

p

2 1

(O,t),

may depend o n l y o n s .

where t h e c o n s t a n t C L e t us put x =

2 a (x)dx, V s E

J

(x)dx 5 C

t

,

y

=

e',

B(t)

=

et12a(et)

and R ( t ) = et12p1 ( e t ) .

Then ( 2 . 7 . 9 3 ) r e d u c e s t o t h e f o l l o w i n g e q u i v a l e n t i n e q u a l i t y :

-m

-m

where R ( t ) i s t h e f o l l o w i n g c o n v o l u t i o n

Ks being the following kernel: K s ( t ) function.

=

H(-t)e

t(f-s)

where

H ( t ) i s theHeaviside

2. Sobolev Spaces of Vectorial Order

168 Since m

[ks(c)I

1

=

-it6

J’ K s ( t ) e

-1

(C2+(t-s)21-’ 5 ( 5 s )

dtl =

,

m

w e a t once o b t a i n ( 2 . 7 . 9 3 ) w i t h C

= (4-s)

-1

.

P ( x ) 1s monOtoniCally 1 (2.7.93) y i e l d s :

A s a consequence o f i t s d e f i n i t i o n ( 2 . 7 . 9 1 ) ,

d e c r e a s i n g f u n c t i o n of x E lR+

.

Hence, m

2

2 C

5 4 1 I p l ( n ) 1 2 5 4 1 p ( x ) dx 5 n>Om o 1 2 2 lgnl , 5 4 C J’ a(x) dx = 4Cs C ntO

lGnl

n>O

and t h a t p r o v e s ( 2 . 7 . 9 1 ) . Now, t h e e s t i m a t e s ( 2 . 7 . 9 0 ) ,

( 2 . 7 . 9 1 ) and t h e i d e n t i t y ( 2 . 7 . 8 9 )

p r o v e t h e s t a t e m e n t o f Lemma 2 . 7 . 3 1 and Theorem 2 . 7 . 3 0 . W e s h a l l now c o n s i d e r t h e case

f
o It remains to show that def Q(u)

=

t-(2s+l)h t>O

1 u (t-x)-u (x)I 2h

6 CN (u).

O<xO O<xo t > O

I2

lu(x+2t)-u(x) h,

so t h a t (2.7.114) P ( u ) 5 2

2s+l

X (2t)

-(2s+l)

t>O Now

(2.7.113),

2 2s+l h C l u ( x + Z t ) - u ( x ) ]h 5 2 N(u). x>o

I

(2.7.114) g i v e t h e r e q u i r e d i n e q u a l i t y .

Remark 2 . 7 . 3 7 . The e s t i m a t e ( 2 . 7 . 1 1 2 )

i s t h e o n l y one where t h e c o n d i t i o n s2 #

u s e d , s i n c e we had t o a p p l y Lemmas 2 . 7 . 3 1 , special choice of v

=

2.7.32.

?j

h a s been

Therefore, with t h e

u i n ( 2 . 7 . 1 0 7 1 , t h e a s s e r t i o n of Lemma 2.7.36 i s

t r u e f o r a l l s 2 , 0 < s2 < 1. Theorem 2 . 7 . 3 8 .

Let

w i t h i n t e g e r m 2 1 and E E (O,l).Thenonehasfortheextension

s2 = m-l+e

o p e r a t o r (2.7.55), (2.7.115) 1 m

. .

(2.7.56) u n i f o m Z y w i t h r e s p e c t t o h E (0,h

0

n,+) H ( ~ 1 , ~ 2, h) ( %

1

+ H ( s l,s2), h ( % ) '

and

f o r any s1 E R . Proof. O b v i o u s l y , it s u f f i c e s t o c o n s i d e r t h e c a s e n = 1 , s1 previously, we use t h e notation H ( 0 , s 2 )'

' I -I

( 0 , ~, h~' ) [

-I

Hs,h

and

=

1 1 . I I s,h, [ . I s , ,

so, l e t u E tl

( 0 , ~ , h~ * )

0, so t h a t a s i n s t e a d of

(q*+A S sth ).

p r e v i o u s l y , l e t u . and v , b e d e f i n e d by ( 2 . 7 . 5 7 ) and ( 2 . 7 . 5 8 ) , r e s p e c t i v e l y . 3 3 and Theorem 2 . 7 . 2 6 , one Using t h e d e f i n i t i o n ( 2 . 7 . 7 9 ) o f [ . I ( s ) ,h

g e t s immediately

2. Sobolev Spaces of Vectorial Order

174

where t h e c o n s t a n t C d o e s n o t depend on h and u . I t remains t o o b t a i n t h e estimate

6 C X t t>O

L

- (2e+1)

h ( [ u m - l ( x + t ) - um- 1 ( x ) [

w i t h a c o n s t a n t C which d o e s n o t depend on h and u . W e s h a l l f i r s t prove t h e f o l l o w i n g a u x i l i a r y a s s e r t i o n .

Lemma 2.7.39. =

(2.7.119)

T

=

2

c -

hZ+ and l e t D

2 \,+ . L e t t h e mapping

q llt+q12x+q13hr Y = q21t+q22x+q23h,

be i n v e r t i b l e and t a k e

5

i n t o D;

D~


0

one has:

Di.

xhl ‘ + + c holds:

Proof o f Lemma 2.7.39. D e n o t i n g t h e sum on t h e l e f t hand s i d e o f

( 2 . 7 . 1 2 1 ) by I ( f ) and u s i n g

( 2 . 7 . 1 2 0 ) , one g e t s i m m e d i a t e l y t h a t

I(f)

c

=

t(TrY)

-(2e+1)

If

h ) - f( Y )

.2 2

I

h

6

(T,Y)€D; -

0

(T

1

,y)ED;

c

5 c- ( 2 e + 1 ) 0

2 2 (T+y)-(2e+1) f ( T ) - f ( y ) ] h 6

c

_< C - ( 2 e + 1 )

( T + y ) - ( 2 e + 1 )l f ( T ) - f ( y ) I2h2 =

( T t Y ) EXh,

c

c

y>o

T>Y

+

(T+Y)

(2e+1)

If ( T ) - f ( Y )

I 2h 2 .

2.7. Spaces of One Parameter Families of Meshfundions Making t h e change o f v a r i a b l e s

T

noticing t h a t t+2x b t , V ( t , x )

E

= t + x , y = x i n t h e l a s t d o u b l e sum, and 2

R h , + , we obtain the required inequality

( 2 . 7 . 1 2 1 ) , and t h h t e n d s t h e p r o o f of Lemma 2 . 7 . 3 9 . Now w e c o n t i n u e t h e p r o o f o f Theorem 2 . 7 . 3 8 . Formula ( 2 . 7 . 5 8 ) y i e l d s : (2.7.122) v

(X+t)-Vm-l(~)

m- 1

=

~ ~ + ~ ( x + t ) -(x) u + m- 1

z

um-l(x+t)-um-l(x),

I f x 2 0 then, of course, v

m- 1

(x+t)-vm-,(x)

T h e r e f o r e , f o r proving (2.7.121)

5 c

v

t b 0.

i t s u f f i c e s t o show t h a t

I: t - ( 2 e + i t>O

F i r s t , l e t x 5 -t-(m-l)h

L e t u s p u t z = -x-t-(m-1

h , s o t h a t z b 0. For

such x formula (2.7.122) y i e l d s :

Setting

T

=

pt

y = p z + l c t ] h and n o t i n g t h a t t h i s map t a k e s t h e s e t

{ t > 0 , z 2 01 i n t o i t s e l f , we f i n d

The l a t t e r i m p l i e s

175

2 . Sobolev Spaces of Vectorial Order

176

w i t h some c o n s t a n t C which depends o n l y on m. m Now l e t -t 6 x < 0. L e t u s p u t z = -x, 0 < z 5 t . For s u c h x formula (2.7.122) y i e l d s : v

m- 1

(X+t)-vm-l(x) = u

m- 1

(t-2)-

c c

- ( - 1)

16p';m P

u

I: a'Qm-

m- 1

(pz-(m-l)ph+\alh).

1 ,p

Denoting, as u s u a l , by H(x) t h e H e a v i s i d e f u n c t i o n , H ( x ) : 1 , x 2 0 and

0 , x < 0 , one c a n rewrite ( 2 . 7 . 6 1 )

H(x)

(2.7.125)

(-l)J C C 16p<m

i n t h e following f asi on:

C ~ ( p z - j p h + l a l h )5 1 , V z > ~ E Q .

0, 0 5

j 5 m-1.

I rP

Using i d e n t i t y ( 2 . 7 . 1 2 5 ) w i t h j = m-1, w e o b t a i n t h a t € o r -t 5 x < 0 holds : v

m- 1

(x+t)-v

m- 1

(x)

The t r a n s f o r m a t i o n D

h,arP

=

=

= t-z,

7

y

=

p z - ( m - l ) + l a ( h maps t h e s e t

{ t > 0 , -pt+p(m-l)h-lalh 6 pz < p (m- l ) h- l al hj ,

c D h,a,P

x2

hr+

Moreover, s i n c e 0 5 la\ 2 p ( m - l ) , one h a s \,+. t ( 7 , y ) = ? + y / p + ( m - l ) h - \ a l h / p 2 (T+h)/m, f o r 1 5 p 5 m. Applying Lemma 2.7.39,

we find t h a t

The l a s t i n e q u a l i t y i m p l i e s t h a t t-(2e+i)

(2.7.126)

Z -t<xO

where t h e c o n s t a n t C

m

1

Ivm-l ( x + t ) - v ( x ) 2h2 6 m- 1

depends o n l y on m .

into

2.7. Spaces of One Parameter Families of Meshfundions

171

< x < -t. P u t z = - ( x + t ) , so t h a t 0 < x < (rn-l)h.

F i n a l l y , l e t -t-(m-l)h

I n t h a t c a s e formula (2.7.122) y i e l d s : ( 2 . 7 . 1 2 7 ) ~ ~ - ~ ( x + t ) -( v x) m- 1

=

(-1)

m

z

c

c

16pSm P

+ (-1) m

c

=

z

c

16p6m

(In-')

1 (1-H ( X I

( x + p t )11

aEQm- 1 , p

The e s t i m a t e o f t h e t e r m s

+

( x ) 11

[ H( x ) u m - l ( x + p t ) - u

aEQm- 1 ,p

x=pz-(m-l)ph+lalh

I

x = p z - (m-1 ) p h + a1 h

H ( X ) U ~ - (~x + p t ) - u m - , ( x ) , w i t h x = p Z - ( m - l ) p h + l a l h

i s o b t a i n e d j u s t a s above i n t h e c a s e x + t + ( m - l ) h 5 0 . I t r e m a i n s t o e s t i m a t e t h e l a s t sum o n t h e r i g h t hand s i d e of

( 2 . 7 . 1 2 7 ) . Denote

H- ( x )

= l-H(x),

and n o t i c e t h a t f o r m u l a ( 2 . 7 . 6 2 ) (2.7.128)

(-1)J

1

H - ( p z - j p h + l a j h ) E 0 , V z > 0 , 0 2 j 6 m-1.

1

C

c a n b e r e w r i t t e n i n t h e form

~ E .Q 1 .P

l

The same a r g u m e n t , u s i n g

(2.7.132)

and (2.7.79).

I

g e n e r a l i z e t o t h e space

t.

(Sl'S2)

Theorem 2 . 7 . 4 3 .

Let s2 (*

>

f . Then t h e farniZy o f mappings h

. 7 . 136)

ti ( s1,

+

\,

s2), h

i s u n i f o m Z y continuous w i t h r e s p e c t t o h E ( O , h O l . Moreover, t h e map Rh is s u r j e c t i v e and t h e r e e x i s t s a continuous

2 . Sobolev Spaces of Vectorial Order

180

(unifomnly w i t h r e s p e c t t o h) f a m i l y o f l i f t i n g operators Lh '

Proof. __ The f i r s t p a r t o f Theorem 2.7.43 c o n c e r n i n g

Rh, immediately f o l l o w s from

( e x t e n s i o n theorem) and Theorem 2.7.17 ( t r a c e t h e o r e m ) . I t

Theorem 2 . 7 . 3 8

r e m a i n s t o prove t h e e x i s t e n c e o f a l i f t i n g o p e r a t o r a s s t a t e d i n t h e second p a r t of Theorem 2.7.43. Without r e s t r i c t i o n of g e n e r a l i t y , one c a n assume t h a t s1

1 1.1

0 , so t h a t w e s h a l l w r i t e s i n s t e a d of s 2 and

=

Hs, Hsej-+,

i n s t e a d of v e c t o r i a l s u b i n d e x n o t a t i o n . F u r t h e r , w e u s e t h e

notation

1 1 . I 1' 0 ,h

f o r t h e norms o f m e s h f u n c t i o n s on

a . ( x ' ) EHs-j-t,h 1

(IR~-')

,o

s2-f. Denote

T

0

u = u(x',O).

'Theorem 2.7.45.

Let

s = ( s l , f ) , w i t h s1 E R .

Then t h e follow

0.

It is easily seen that E (x,y) is the discrete fundamental solution for h -l), which vanishes the difference operator l+Dx,hD:,h, Dx,h = (ih)

-'

(@X,h

at infinity. Indeed, an easy computation shows that (2.7.174) Fxt5 ,hEh

=

0 holds

Further, assuming that s2+s3 < 3/2 and that either s1 < 1 , s 2 5 $ , or that f < s2 < 3/2, sl+s2 < 3 / 2 , one gets using the last inequality and carrying

out a straightforward computation the following inequality:

where y = 1-s -max{O,s - 4 ) > 0 . 1 2 Hence, the meshfunctions u and v above belong to the same equivalence 1 class in any H ( s ) ( Z ) with any s as described above. However, they are not in the same equivalence class when, for instance, s the difference derivative of w

is of order X,E

=

E - ~ in

(0,2,0).Indeed 1

H (0),E,h(Rh), since

2. Sobolev Spaces of Vectorial Order

200

aXwx,€ =

(E

e-X/E-he-hX)H(x).

Notice that u(~,h,x)is an asymptotic solution of the boundary value -AX

problem: (Eax+l)y = e

,

x E lR+

,

y(0)

=

4 , extended by zero on R - ,

while

v(E,h,x) is the solution of the difference boundary value problem: (l-e-')-'(l-e-'O-')v

=

e-Ax , x € R h +\{O},

v(E,h,O)

=

4,

extended by zero on R the difference problem being an approximation h,- ' of the differential one with the accuracy O(h). The fact that u and v belong to the same equivalence class in

H

(s)

(Zl), with s, as described above

(i.e. either s < 1, s2 4 t , s2+s3 < 3/2,or s +s < 3/2, t < s 2 3/2, 1 1 2 s 2 + s 3 < 3/2) (and also the fact that I ly-ul I 0, as E + 0, h + G ) ( s ) ,E,h means that v, as a solution to the approximating difference scheme -f

converges to the solution y of the differential problem with small para1 ( 2 ) with s as described above. meter E in any ff (S)

It is easily seen that the meshfunction w(&,h,x), solution to the difference boundary value problem (€8 +l)w = e , x E %,+ , extended x,h does not belong to the same equivalence class as u and v by zero on IR hr-l' abc-ve in tf ( Z ) , V s , although this difference boundary problem too is an -AX

(s)

approximation to the differential one with the same accuracy O(h). However, this scheme differs essentially from both the differential problem for Y(E,X) and difference problem for v(E,~,x),since the corresponding has the solutions of the form $(l-p) x/h , homogeneous equation on 37 h.+ which do not vanish when p > 2 , & + 0, h + 0 , x 2 xo > 0 , i.e. they are not of boundary layer type. Definition 2 . 8 . 9 .

1 1 . 1 1 ( s ) ,E.h a r e s u i d to be s t r o n g e r than t h e n o r m 1 1 . 1 1 ( s ' ) ,E,h i f t h e r e e x i s t s a constant c = c which may depend onZy on i t s

The noms

' ,cO ,ha'

s ,s

s u b s c r i p t s and such t h a t

It is quite obvious that a necessary and sufficient condition for (2.8.15) to hold is the inequality s -s'

(2.8.1G)

(E+h)

' s!.

Proof. first 151

Taking alternatively in ( 2 . 8 . 1 6 ) c = h =

l€,l-',

=

1, E

0, h

h

immediately the conclusion that the condition s (2.8.16)

-f

c * 0 and finally c = 1 , h =

>

0, afterward

-+

one gets

+ 0,

s' is necessary for

(and, therefore, for ( 2 . 8 . 1 5 ) ) to hold.

On the other hand, it suffices to show (2.8.16) with s' = 0, s

>= 0 .

One gets in this case: -S

(E+h)5>s3 =

(E+h)

(

(E+h) f ,

n

f.

A slight generalization of the previous lemma stated here below, is proved

by the same argument, Lemma 2.8.14.

Let u E H

1

(wh)w i t h

s > f , s2+s3 > 2 k < minIs2+f,s2+s3+f} t h e mapping

(2.8.21)

( s ) ,c,h

1 3 u ti ( s ) ,c,h (Rh)

+

t . Then f o r each

n 0 {DPx,h -'} I 0 , V 5 E T5,h, one can estimate the integral on the right hand side of the last inequality, in

the following fashion: f1 T

2Sl -2s -2s (E+h) 2 1 and V 6 < 0, the right hand

u E ff(o),E,h(\)

side of (2.8.40)

X

-t

vanishes identically in this case, since it is a limit

-0 of the corresponding integral over T1 3 X with the integrand -1 -h (e6-h) u (h,X).

when 6

+

one gets an analoguous conclusion for u E and 'I i.e. II+ Gh 5 0. u : 0 for x E W 1 h,+ h' h Hence, one has the orthogonal projections: h

'iT O ) ,E,h

(2.8.45)

-h

where H-

( s ),E,h

HTs), E , h '

h =

4

T O ) ,E,hr

H-(0),E,h'

1 i.e* '(0) ,E,h(%)

h T O ) ,E,h

= o

stands for the Mellin transform of the meshfunctions in

respectively.

One checks easily the following more general formula:

t Introduce the functions 1, V h > 0.

E R , V h E R+ , the function Ct

vanishing when 181 > 1 (including 8 = analytic and non-vanishing for 18 Now introduce for each s

=

I

m)

( 8 ) is analytic and non+ rh t and the function 5- ( 8 ) is ,h

< 1.

(sl,s2,s3) E R3 the norms of order s , as

follows:

V u E H

1

(s),~,h(wh,+)'

1 where, as previously, p = h/E and 1 u stands for the extension of u on IRh 0 1 by putting u Z 0 for x E IR . h.-

'

2.8. Spaces of Two Parameter Families o j Meshfunctions

Indeed, 5 -s2hS-:p((lu)" h

" h ) -(leu)

H-h

E

209

so that (2.8.45) yields (2.8.49).

( 0 ) ,E,h'

Lemma 2.8.23.

The norms (2.8.48) and (2.8.27) are equivaZent in H

.

uniformZy with respect t o ( ~ , h E ) Il

1 ( s ), ~ , h ( ~ h , + ')

EO'ho Proof. -

-

2 1 Since Il+ is an orthogonal projection in L (T ) and, obviously 16 I 0, in L (T ) onto the functions in L (T ) which can be extended

1 analytically on the exterior of T1 in C , including the infinity: of course, II- is an orthogonal projector in L2 (T1 ) , V h > 0, onto the functions in h 1 L2(T1), which can be extended analytically on the interior of T 1 Let U = (x+,x-) C R be a finite interval and let Uh = U n % . We

.

assume that x+ E Uh.

-

Consider first the case s2 L 0, covering of

s3 L

by two open intervals U

0 integer. Let

and

U-

the corresponding partition of unity, i.e. X? E X,(X)+X_(X)

: 1,

v

x E

5,

U+ U U-

be a

and denote by {X+(x) ,X- (x)1 C

m

o (U* )

and

X+(X+) = 0. - - = 1, X+(X-) - +

2 . Sobolev Spaces of Vedorial Order

210

For a meshfunction u(E,h,-)

:

Uh

+

, denote

C, ( ~ , h )E JIE

0rh0

U+(E,h,X)

=

(X+u)(E,h,x++x), u-(E,~,x)= (X-

and define the norm of u of order s E R

[

where

1 - 1 1' ( s ) ,E,h are

3

,

U)

(E,h,x--x)

as follows

the norms (2.8.49).

Definition 2.8.24.

The space

H (s).E,h(Uh ) is d e f i n e d t o c o n s i s t o f a l l meshfunctions

n o m s (2.8.50)

.

finite f o r each ( ~ , h )E Il

The space

H

u with

€Ofh0 (u) c o n s i s t s o f a l l meshfunctions u E

(S)

H

(u

(s),E,h h

)

such

that

One defines the quotient-space H where

Ho( S ) (U)

' ''

(S)

is again the subspace of

functions u whose norms

For functions Q(e,h,-)

:

( S ),5

,h,Uh

aUh

C

+

(u)

=

ff

H ( S ) (U),

( S )

(U)/Hys) (U) as previously,

consisting of all mesh-

vanish When (E,h) -t ( 0 , O ) .

the norms of order 1 E IR are

introduced as previously

H1 , ~ ,(aU h h ) the family of spaces of functions Q with norms finite, V (&,h) E JIE

We denote by (2.8.52)

0gho

Further, introducing the norm

H1(aU) the space of Q with norm previously H (au) stands for the quotient 1 denote by

(2.8.53)

space

the subspace of functions Q whose norms (2.8.52)

finite, while, as

0 H1(au)/Hl(au)

with

0 H,(au)

vanish, when (E,h) + ( 0 , O ) .

Remark 2.8.25. The norms, defined by (2.8.50)

depend on the partition of unity {X+,X-}.

However, it can be shown that the norms, corresponding to different choice of partition of unity are all equivalent uniformly with respect to

, so that they define the same topology in the corresponding

(E.h) E Il spaces.

'0tho

Notes

21 1

Notes Weighted spaces with a small parameter of vectorial order (0,k ,k ) with 1 2 integer k l 2 0, k 2 0, were already implicitly present in the work by 2 Vishik and Lyusternik 1 1 1 , where they appear through quadratic forms related to singularly perturbed strongly elliptic operators with homogeneous Dirichlet boundary conditions (see also Huet [l]) . This kind of spaces is also useful Lions [l]).

in singularly perturbed control problems, (see

Spaces of order (0,sl,s2)with s1 E IR,

s2

E R , have been

introduced in Demidov [l] and used for the investigation of the first boundary value problem for some classes of elliptic pseudodifferential singular perturbations

.

HBlder type spaces with a small parameter have been used in Fife [11 for establishing one-sided a priori estimates for elliptic singular perturbations with constant coefficients. 3 Sobolev type weighted spaces of vectorial order s = ( s 1 ’ s 2 ’ s 3 ) E R have been introduced in Frank [221 in order to establish two-sided a priori estimates (uniform with respect to the small parameter) for the solutions of coercive (elliptic up to the boundary) singular perturbations. Results concerning equivalence of norms in spaces of vectorial order (Lemma 2.4.2) have been obtained in Frank and Wendt [lo], as well as theones concerning the extension by zero (Theorem 2.6.6.)

(see also Frank and Heijstek [l]).

Sobolev type meshfunction spaces of any vectorial order have been introduced in Frank 171 in order to establish two-sided a priori estimates (uniform with respect to the mesh-size) for elliptic and coercive finite difference operators (Frank [8,9,11,13], see a l s o Thomee and Westergren [l], where meshfunction spaces have been used in order to get interior regularity results for a subclass of elliptic finite difference operators). Sobolev type spaces of meshfunctions, which depend on two small parameters, have been introduced in Frank [16,21,23] and ussd in order to establish two-sided a priori estimates for solutions of elliptic finite difference equations, which approximate elliptic and coercive singular perturbations (Frank [15,19,22]).

This Page Intentionally Left Blank

CHAPTER 3 SINGULAR PERTURBATIONS ON SMOOTH MANIFOLDS WITHOUT BOUNDARY

Several classes of pseudodifferential and difference singular perturbations are introduced here and the elliptic theory of these parameter dependent operators developed. A special attention is given to parameter dependent hyperbolic pseudodifferential and difference singular perturbations and corresponding classes of Fourier integral operators. 3.1.

Singular perturbations with constant symbols

With any differential operator Q ( D ) , D

=

(D1,

...,D

),

Dk = -ia/axk, with

constant coefficients one can associate the polynomial Q( v"),

v(l) = (0,1,4) c a n n o t b e compared, s i n c e

n o r v(l)

> v").

A l l t h i s j u s t i f i e s t h e following

D e f i n i t i o n 3.1.1.

A function Q

:

( 0 . ~ ~x 1x n + c i s s a i d t o belong t o t h e class Pv w i t h

v = (v1,v2,v3) E R ~ i ,f ( i )Q ( E , ~ )i

s a polynomial i n 5 E

R~

w i t h c o e f f i c i e n t s belonging t o

m

c

((O,EO1).

( i i )There

e x i s t s a decovposition Q

=

such t h a t

Q+,R

Q,

can be extended a s

a homogeneous f u n c t i o n of ( E - I , ~ )E R + x mn of degree v1+v2, (3.1.5)

-1 Q o ( tE , t C )

= t

satisfying the inequality

v 1+v 2

Q,,(E,S),

t/ t

E R+, t/

(E,c)

E

R + x Rn,

3 . 1 . Singular Perturbations with Constant Symbols

and t h e remainder

R

215

s a t i s f i e s the inequality -V

(3.1.7)

151 v 2 < E c > v 3 ~ 1 5 ~ - 1 + E ) v,

/ R ( E , S L 5 CE

w i t h some p o s i t i v e c o n s t a n t

E

E

v 5

(O,Eo1,

E En,

15)

2 1,

C.

The f u n c t i o n Q ~ ( E , ~ ): IR

x

IR~.-+

c i s c a l l e d t h e p r i n c i p a l symbol o f

Q ( E , ~ ) .

- (v +u2)

Putting t =

E

in

go(€,1c'/-l5',

=

1

/L(E,S)I 5 CE

'<EF,>'

( i i ) t h e r e e x i s t s a decomposition L a homogeneous f u n c t i o n of (E-',S) E

(3.1.10)

Lo(t

-1

3

,V

E ( 0 . ~ ~X 1l R n \ S ,

= L~+R, IR+X

such t h a t L 0 can be extended a s of degree v1+v2,

(R~,{o})

v +v ~ , t 5 )= t

2Lo(~,S), V t

E IR+,

V

(E,E)

€ IR+

x

(IR~\{O})

s a t i s f y i n g t h e ineyuazity -v (3.1.11)

iLO(E,S)l

and t h e remainder

R

5 CE

v

IR(E,S)I

x

(lRn\{O})

s a t i s f i e s t h e estimate: -v

(3.1.12)

3

'151 2 < ~ 5 > v , V (E,S) E R+

5 CE

v

'151

2 < ~ 5 > v 3 ( 1 5 1 - 1 + t ) ,V E E ( 0 . ~ ~ 1V, 5 E Rn\S,

151 21 w i t h some p o s i t i v e c o n s t a n t c . The f u n c t i o n L O ( ~ , S ) : E+ x I R ~+ c i s c a l l e d t h e principaZ symbol of L.

3 . Singular Perturbations on Smooth Manifolds without Boundary

216 Lemma 3 . 1 . 3 . (1)

If L . E l u ( j )j,

( i i )If

1 L~ E

1

=

1,2, then L1L2 E L" w i t h v

IL1(E,S)I 6 C I E

w i t h a p o s i t i v e c o n s t a n t cl, (iii)L

(1)

( i v ) 1" ( 1 )

uL

n

" (2) 5 L\,

1"

V

P r o o f . With -

u(1)+v(2).

s a t i s f i e s the inequality: -V

(3.1.13)

=

(1) 1

u(l)



<ES>

" (1) 3

,v

E (O,Eo1

(E,5)

t h e n L ~ ( E , ~ ) - 'E 1 w i t h u

with the least u

>

.(I),

=

x R

n y

-u ( 1 ) .

j = 1,2.

c 1 w i t h t h e l e a s t v =< u ( j ) , j = 1,2. (2) - I, b e t h e s u b s e t where L . i s n o t d e f i n e d , m . 1 1 and l e t L . = L + R . be t h e decomposition accor1 jrO I ( i i )of D e f i n i t i o n 3 . 1 . 2 . I t i s immediate t h a t t h e symbol

j = 1 , 2 l e t S . c Rn

3

the constants in

(3.1.9)

ding t o t h e p a r t

L = L1L2 s a t i s f i e s ( 3 . 1 . 1 0 ) w i t h v = v ( 1 ) t v ( 2 ) on Rn\S

For showing

(3.1.9)

with S

= S

1

w e e s t i m a t e L ( E , S ) - L ( E , ~ )a s f o l l o w s :

"

s2-

Further, the inequality

214l4,

5

<S>a-a

so t h a t ( 3 . 1 . 5 ) h o l d s f o r L

m

=

= L L

1 2

v (5,n)

with u

v a E

E n n x lRn,

R

= u ( ~ ) + v ( ~ and )

rnaxtm2 'm1 + ~ v2( ~ ) l + l v ~ ~ ) l } . O b v i o u s l y , t h e f u n c t i o n Lo

=

LloLz0 s a t i s f i e s t h e c o n d i t i o n s ( 3 . 1 . 1 0 ) ,

(3.1.11). The r e m a i n d e r R (3.1.14)

IH(E,E)

One h a s

I

6

=

L-Lo

can b e e s t i m a t e d a s f o l l o w s :

lR1(E,5)R2(E,5) /+IR1(E,5)L2,0(E,c)

I+IR2(E,c)L

3.1. Singular Perturbations with Constant Symbols -v

/R1(E,5)R2(E,t)~5

cE

217

v llcj 2<Et>v3(~5~-i+i)8

v

(E,c)

E

(o,Eol

nn\S,

151 2

1

w i t h v , = v ! 1 ) + v ! 2 ) s i n c e ( / ~ l - l + E ) -5~ C ~ S I - ~ + fEo r 151 2 1 , E 5 E 0' 1 3 3 F u r t h e r , it i s q u i t e o b v i o u s t h a t t h e t w o l a s t t e r m s i n t h e r i g h t hand -V

s i d e of

15 1 u 2 < ~ c > u( 31 5

( 3 . 1 . 1 4 ) a r e bounded by C E

a s Well.

T h i s p r o v e s t h e f i r s t p a r t of Lemma 3 . 1 . 3 .

If L 1 s a t i s f i e s (3 1 . 1 3 ) t h e n o b v i o u s l y ( 3 . 1 . 8 ) h o l d s f o r -1 L ( E , ~ )= L 1 ( c , < ) with v =

-u(').

The d i f f e r e n c e L ( E , S ) - L ( E , I ? ) c a n b e

estimated a s follows:

where C 1 and C a r e t h e same p o s i t i v e c o n s t a n t s a s i n ( 3 . 1 . 1 3 ) respectively.

v w i t h some p o s i t i v e c o n s t a n t c

2

.

E

E

R+,

v

5 E nn\iO})

and (3.1.12),

3. Singular Perturbations on Smooth Manifolds without Boundary

218

Thus, t h e f u n c t i o n L (E,S) = L l 0 ( t , t ) - ' 0 (1)

with v =

s a t i s f i e s (3.1.10).

(3.1.11)

.

-w

The remainder R = L-L

0

can b e e s t i m a t e d a s f o l l o w s :

I t i s l e f t t o t h e r e a d e r t o check

( i i i ) and

I

(iv).

D e f i n i t i o n 3.1.4.

If

L 6

then t h e family of operators

L

(3.1.15)

L(E,D)u(x)

E

-f

-1

V u

= FEex L ( E , ~ )Fx+[u,

d e f i n e d by t h e formula

L(E,D)

E

S(Rn),

i s s a i d t o belong t o O P L ~ and i s c a l l e d a s i n g u l a r p e r t u r b a t i o n . Obviously,

L(c,D)

:

S(iRn)

+

S' (Rn).

D e f i n i t i o n 3.1.5.

A singular perturbation L ( E , D )

:

i f for

v

v e c t o r i a l order v E IR' L(€,D)

: H

( s ) ,E ( n n )

S ( I R ~ -f ) S'

(nn) is s a i d t o have t h e

s € 1~~ i t can be exLended as

( m n ) uniformly w i t h r e s p e c t t o + H(s-v) ,E

E

E ( 0 , ~ ~ l .

Proposition 3.1.6.

If

L ( E , ~ )

E Lv,

then

: s(#)

L(E,D)

-f

s 8 ( m n ) has t h e v e c t o r i a L order w.

P r o_ of. _

L

If L ( E , ~ )E

(3.1.8)

then

yields L(E,D)

E

V s E Rn u n i f o r m l y w i t h r e s p e c t t o E Along w i t h L ( E , D )

E OPLw

: S(R)

+

S'(R)

D e f i n i t i o n 3.1.7. 0

(3.1.17) L

0

(E

v1



-v

0

(Rn)

1.

-f

,

0

( € , e l )

-f

L(E,E',D,),

i s c a l l e d a reduced symbol of

-V

Lv

e = (l,-l,l),

( 5 ) being a reduced symbol of L ( E , ~ ) t, h e operator OpLO OPL = L (E,D).

L(E,E) E

-V

2 < ~ < > 3 L ( ~ , 5 ) - < 5 > 2 L o ( 5 ) ) E L-,,

t h e reduced operator o f

H(s-v),E ( I R n )

d e f i n e d by t h e f o r m u l a :

The f u n c t i o n L ( 5 ) E L ( o , v 2 , 0 )

if

(O,E

( s ) ,E

w e s h a l l a l s o c o n s i d e r l a t e r t h e f a m i l y of

one d i m e n s i o n a l s i n g u l a r p e r t u r b a t i o n s L(E, denotes t h e d u a l i t y betweenIl(u,2) , E

, 2 lU/

(Rn)

,

l((v-e2),2) v

(Bn)and H

E

,€,

E (O,EO1'

(-u/2) ,E

(En).

Proof. Without r e s t r i c t i o n of g e n e r a l i t y one c a n assume t h a t 8 L (6,F)

0

=

0, y

b e i n g t h e same a s i n t h e p r o o f of P r o p o s i t i o n 3 . 1 . 1 1 ,

u s i n g t h e same argument a s i n t h e p r o o f of t h i s p r o p o s i t i o n :

=

1.

one g e t s

3.1. Singular Perturbations with Constant Symbols

223

P r o p o s i t i o n 3.1.18.

Let

L ( E , D ) E OPE be s t r o n g l y e l l i p t i c o f order v w i t h p r i n c i p a l symbol L ~ ( E , ~ )Assume . t h a t L admits t h e reduced symbol L0 ( 5 ) E L and l e t

(0 'V2 , O )

0 L~

( 5 ) be t h e corresponding reduced p r i n c i p a l symbol. L e t

(3.1.25)

0 -1 0 P ( F , ~ =) L ( E , S ) - L ~ ( E , ~ ) L ~ ( LS ) ( 5 ) .

Then t h e r e e x i s t s a c o n s t a n t

c such t h a t

Proof. ___ O b v i o u s l y , t h e symbol P E identically: P

0 (3.1.12), P ( E . 5 )

consequence of

(E,S)

Z

Lu and i t s p r i n c i p a l and r e d u c e d symbols v a n i s h

0 , P o ( S ) E 0 . T h e r e f o r e , a s a consequence of

E L(u-el)

u L(u-e2),

(3.1.17), P ( E , S )

s i n c e P0 ( E , S )

E L(u-e),

. . T h i s l e a d s to t h e c o n c l u s i o n t h a t P ( E , ~ E) ).

e

=

L

Z

0. F u r t h e r , a s a

(1,-1,1),

(we)

"

0

s i n c e P ( 5 ) :0 .

('("-el)

"

L(u-e2)) =

I t i s immediate t h a t

w i t h some c o n s t a n t C > 0 and t h e l a t t e r h a s ( 3 . 1 . 2 6 ) a s i t s immediate

I

consequence. C o r o l l a r y 3.1.19.

Let

L(E,D)

E

OPEv be s t r o n g l y e l l i p t i c o f order u and l e t L O ( ~ , c )be i t s 0

0

principaZ symbo2, L (D) i t s reduced symbol and

L ~ ( D )i

symbol. Assume t h a t t h e r e i s a c o n s t a n t yo

such t h a t

Then t h e r e e x i s t s such a c o n s t a n t

c that

> 0

t s principal reduced

3. Singular Perturbations on Smooth Manifolds without Boundary

224

(3.1.28)

2 1 I U I ](v,2),E~

0 Re<eieL(E,D)u,u> 2 (YoY -CE)

V u E H

(v/2)

,E

(2) ,v

E

E

(O,Eo1.

I n f a c t , (3.1.27) is an immediate consequence o f (3.1.25)-(3.1.27). Remark 3. 1.20.

0

I f L ( E , D ) 6 OPE

a d m i t s t h e r e d u c e d symbol L ( D ) E OPE

(0,v2I O )

'

then

i n t r o d u c i n g t h e symbols

(3.1.29)

-1

0

R(E, '1

mn

E

a (x,E ,5)

( 5 )d5

c a n b e d i f f e r e n t i a t e d under t h e i n t e g r a l s i g n and t h e c o r r e s p o n d i n g i n t e g r a l s are a b s o l u t e l y and u n i f o r m l y c o n v e r g e n t w i t h respect t o x on any compact K c U , Further, l e t v

E

E

E

(O,E~]. m

The i n t e g r a t i o n by p a r t y i e l d s :

Co(U).

Using t h e l a s t i n e q u a l i t i e s , one shows t h a t t h e f u n c t i o n a l s

are w e l l d e f i n e d , i f u

E

V

E'(U),

E

E

( O , E ~ ] and, moreover, t h i s f a m i l y of

f u n c t i o n a l s i s equibounded i n t h e s e n s e o f D e f i n i t i o n 1.2.33.

In f a c t ,

w r i t i n g formally (3.3.12)

<E v

la ( x , E , D ) u , v > = ( 2 ~ ) I- ~ I

v(x)E"1a(x,E,S);(S)ei<x'5>

dsdx =

IRn lRn =

I ~ ( c (I ) mn mn

(2.rrl-n

and u s i n g ( 3 . 3 . 1 0 ) , defined f o r V u

E

Ev

1 i<x,S> a(x,E,, a2(x,€,S) = c ( x ) ,

= (O,O,O),

t h e c o n d i t i o n (3.3.14)

being

3.4. Continuity of Singular Perturbations

239

obviously s a t i s f i e d .

~ ~ / < / ~ + 1 < 1b ~e l o+n g~s ~ / c 1 ~

On t h e o t h e r hand, t h e symbol a ( x , ~ , E ; )= ~n ( R n ) , with v = ( 0 , 2 , 3 ) b u t does n o t belong t o S ( R ) t o Sv 1# O admit a n a t r u a l g r a d u a t i o n .

and d o e s n o t

3 . 4 . C o n t i n u i t y of s i n g u l a r p e r t u r b a t i o n s i n Sobolev s p a c e s of v e c t o r i a l o r d_ er _ The a i m of t h i s s e c t i o n i s t o p r o v e t h e f o l l o w i n g Theorem 3.4.1.

L e t a ( x , E , c ) E sV

1,O

the

( m n ) and Let

a ( x , s , c ) E s(IR:). Then f o r

v

s E R

3

foZlowingfamiLy of Linear mappings i s equicontinuous ( s e e D e f i n i t i o n 1 . 2 . 3 3 )

(3.4.1)

a ( x , E , D ) : H ( s ),E (7Rn)

-f

H(S-V)

,E

(Rn)

,

(o,Eol.

E

Proof. F i r s t assume t h a t s = v = 0 , s i n c e t h e argument i n t h i s c a s e i s p a r t i c u l a r l y simple.

The

assumption

a ( x , E , S ) E So 1no

(mn) n

S ( m n ) yields the

inequalities

/ a ( r l , ~ , S I) 5 CN-N,

(3.4.2)

v

N

since

naa(n,~,E)

D"a(x,E,S). x-trl x

= F

Therefore,

II

(3.4.3)

I

~ ( ~ , E , D ) (~0 I) , E

Since a(x.E.5)

= F

-1 a(rl.E.5) n+x

5

c ~ < ~ >I U-I ~ I ( oI)

and j e x p ( i x . n )

/

;E

G

+

5 1 one g e t s u s i n g ( 3 . 4 . 3 ) :

i s equivalent t o the inequality

Now c o n s i d e r any s and v . Then ( 3 . 4 . 1 )

where

V N.

L2(IRn),

5

KG i s a n i n t e g r a l o p e r a t o r w i t h t h e k e r n e l

and t h e c o n s t a n t C d o e s n o t depend on

E

and u.

Again t h e a s s u m p t i o n s on a ( x , E , S ) imply t h a t

:

3 . Singular Perturbations on Smooth Manifolds without Bounda y

240

The f o l l o w i n g i n e q u a l i t y h o l d s :

<E>T-T

(3.4.7)

4 21TI<E-T)>lTl

(3.4.7) h o l d s with (3.4.7)

with

Thus, (3.4.8)

N

T

and t h a t i m p l i e s t h a t

< 0 , one s w i t c h e s

5 and n and a p p l i e s

> 0.

-T

(3.4.7)

(3.4.6), /K(~,E,TI)

where C

Z 0. I f

T

2 4 2 '

'

I n d e e d , e v i d e n t l y one h a s :

E W.

t/ T

I

lead t o the inequality:

5 CN<S-n>-N,

depends o n l y on N ; h e r e N = N

-1

1

s

2

-v

2

I-1

s

3-v 3

I.

T h e r e f o r e , f o r V N one h a s :

where f ( 5 ) = C

N

-N and f *

N

g s t a n d s f o r t h e c o n v o l u t i o n of f and g .

Applying t h e i n e q u a l i t y :

and t a k i n g N > n , one g e t s t h e c o n c l u s i o n t h a t t h e f a m i l y o f i n t e g r a l 2 n o p e r a t o r s w i t h k e r n e l s K ( C , E , ~ ) are c o n t i n u o u s i n L ( W ) and t h e i r norms

a r e u n i f o r m l y bounded w i t h r e s p e c t t o

E

E (O,E

0

1.

I

C o r o l l a r y 3.4.2.

Then t h e foZZowing f a m i l y o f l i n e a r mappings i s equicontinuous for each 3 s = (Sl,S2,S3) E R : a(X,E,D)

In f a c t , a ' ( x , E , D )

: II(s)

( S )P E

u n i f o r m l y bounded i n am(E,D)

,E(IRn)

: H

+

(Wn)

13

(s-u) , €

H(S-V) ,E

(2) , E (IR

E

(O,Eo1.

) and have t h e i r norms

E ( 0 ,3 ~a c c o r d i n g t o Theorem 3.3.1,

E

O

: H

n

(IR )

( s ) r E ( R n ) + H(S-V) ,E a s f o l l o w s from P r o p o s i t i o n 3.1.6.

while

uniformly with r e s p e c t t o

D e f i n i t i o n 3.4.3.

A singtilar pel-tilrbation

a(X,E,D)

v = ( v ~ P ~ ~ i ,f "f o~r )each

a(x,E,D)

: H

( S ) ,E

given

i s s a i d t o have v e c t o r i a l order E~

m

the family

( I R R n ) + H(S-V)

is equicontinuous and, moreover, f o r each

,E

(Wn),

(E

E

(o,Eol)

< v the mppings

E

E (O,E~I,

3.5. Pseudolocality of Singular Perturbations (Rn)

a(x,E,D) : H ( s )

H

+

(s-v)

o r have unbounded norms when

E

(E

24 1

E (O,E?I) e i t h e r are n o t continuous

,E

0.

+

C o r o l l a r y 3.4.4.

A f a m i l y o f p e r t u r b a t i o n s a ( x , E , D ) has order v E R

and a ( x , ~ , c )$?

S'

1 ro

( n n ), V

3

.

zf a ( x , E , c ) E s"

1,o

p < v.

(Rn)

Example 3.4.5. with A t h e Laplace o p e r a t o r , belongs

( i )A f a m i l y of p e r t u r b a t i o n s E'A'-A

( R n ) with V

t o S'

>

LI

and i t s o r d e r i s

(0,2,2)

1r o ( i i )A f a m i l y of p e r t u r b a t i o n s

a

1

E

= (0,2,2).

(x)D D with a ( x ) E C z ( R n ) kj k 1 kj

lsk,j v ( l ) t...> v ( J ) t . . , 1 u (1)1 c --, e r i s t symbols a . E s" 7 1.0 1 [ v ( J ) ] c -m f o r j + m, which a r e homogeneous of degree / v ( J ) I i n (E- , S ) ' g o

3.6. Asymptotic Expansions of SymOols for

E

x n , 151 t

E IR+, 5 E

I , i.e. ( j1

-1

a.(x,t 3

(3.6.18)

~ , t 5 )= t

241

+”( j )

v1

2

a . (x,E,t), 3

v

t t 1,

v

(E,a(x,E,S)f(S-E

-1

q)dg =

mn =

(2n)-n J

e

i < x , E>

a (x,E

,S+E

-1

rl)

2 ( 5 )d5.

lRn

We a r e g o i n g t o u s e T a y l o r ' s f o r m u l a (3.6.23)

-1 a ( x , ~ , c + n) ~

where a ( a )

=

aaa

5

=

'a1 a

i'

1

c

=

a(")

-1 (X,E,E

n)cn

+ qN(x,E,5,q)

(0)

l 4 < N

D a and qN i s t h e r e m a i n d e r . S i n c e a E

5

Kv

(U) one

has def (3.6.24)

rN

=

(N)

(a-

z a,) OSjf ( x ) )

E

+

o(E1-'v'),

U, f E B C C ; ( U ) ,

=

E +

0,

where a 0 ( x , E , C ) i s d e f i n e d

by (3.3.2), (3.3.3). One h a s t h e same a s y m p t o t i c formula (3.6.31) a l s o f o r ( u n i f o r m l y w i t h r e s p e c t t o q i n any g i v e n compact K2

C

V r- E IRn\{O]

IRn\{O}), b u t , o f

c o u r s e , a o ( x , E , C ) s h o u l d b e r e p l a c e d i n t h i s c a s e by i t s e x t e n s i o n t o

U

x

R+ X ( R n \ { O j ) as a homogeneous f u n c t i o n of (c-l,C) of d e g r e e

1\11

v +v

2'

1

3.7. Amplitudes, A d j o i n t s and P r o d u c t s of S i n g u l a r P e r t u r b a t i o n s W e s t a r t w i t h i n t r o d u c i n g c l a s s e s of s i n g u l a r p e r t u r b a t i o n s , which

w i l l l o o k a s more g e n e r a l , b u t i n f a c t , c o i n c i d e w i t h t h e c l a s s e s d e f i n e d p r e v i o u s l y i f some n a t u r a l c o n d i t i o n s a r e s a t i s f i e d . D e f i n i t i o n 3.7.1.

The f u n c t i o n a ( x , y , E , < ) E cm(u x u order v

=

u

e x i s t s a constant c

v

( O , E ~ ]x

nn)i s c a l l e d amplitude of

(vl,v2,v3) E m3 and i s said t o belong t o t h e c l a s s s v

i f f o r each compact K c

for

x

( x , y ) E K,

v

u and

x

> 0

(E,c)

The s i n g u l a r p e r t u r b a t i o n

(U

x

U)

there

such t h a t

E (o,cO1 A~

1 ,O

each t r i p l e of i n d i c e s a,B,y,

x

nn.

w i t h an amplitude a E S y , o ( U

x

U) i s d e f i n e d

a s t h e double i n t e g r a l

f o r V u E C:(U), in



F

is given by the f o m l a ( x ,E , E , )

,

sv+~-(O,l,O) (U). 1.0

3

P r o_ of. _ We s h a l l f i r s t assume t h a t b ( x , E , F ) h a s compact s u p p o r t i n x b ( x , ~ , S )? 0 , V x E W K ,

E

E ( O , E ~ ] , 5 E l R n , where K

C

E

U, 1.e.

U i s some compact

set.

I n t h a t c a s e F u b i n i ' s theorem y i e l d s (3.7.19)

(a(x,E,D)

0

b ( x , E , D ) u ) ( x ) = (271

where (3.7.20)

c ( x , E , ~ )= (2n)-nJnei<X'TI'

a ( x , ~ ,

where t h e m a t r i x S ( x )

=

I / S k j ( x ) I15k,jsn

263

i s d e f i n e d by t h e f o r m u l a e :

S ( x ) b e i n g , o f c o u r s e , symmetric, S ( O 1 = d i a g ( p l , ..., u n )

00’

function in

I d e n t i f y i n g t h e symmetric nxn m a t r i c e s w i t h R c o n s i d e r S ( x ) a s a Cm-map from

0,

m

matrix-

and C

n ( n + 1 ) / 2 one can

i n t o R n ( n + 1 ) / 2 . W e s e e k a smooth ( C m )

upper t r i a n g u l a r m a t r i x - f u n c t i o n Q ( X )

:

0O + f l R n ( n + 1 ) / 2 s u c h t h a t

where Q ( x ) * i s t h e c o n j u g a t e of Q ( x - ) . Denote F(Q,X)

=

Q*S(O)Q-S(X)

where Q i s any u p p e r t r i a n g u l a r m a t r i x and x E m

00 ’

One c a n c o n s i d e r F ( Q , x ) a s a C -map from R

n(n+1)/2

X

0,

into the

s p a c e o f a l l symmetric m a t r i c e s . One h a s F(Id,O)

=

and, given t h a t a l l

0

u,, I

1 5 j 6 n a r e d i f f e r e n t from z e r o , one f i n d s

easily that

QF ( Q r x ) / Q = I d , x = O = where J i s t h e m a t r i x which h a s t h e e n t r i e s : kj a k j = 1 , alm = 0 , V ( l , m )

# (k,j).

O b v i o u s l y , t h e m a t r i c e s J k j + J j kform a b a s i s i n t h e s p a c e o f a l l symmetric nxn m a t r i c e s , so t h a t

3 . Singular Perturbations on Smooth Manifolds without Boundary

264

t h e I m p l i c i t F u n c t i o n Theorem c a n b e a p p l i e d , which g u a r a n t e e s t h e e x i s tence of a (well-defined) C

m

upper t r i a n g u l a r m a t r i x Q ( x ) i n a

oo

s ~ i f i c i e n t l ys m a l l neighbourhood

of t h e o r i g i n , t h a t s a t i s f i e s t h e

conditions :

Obviously t h e map y

h(x)

=

def -

Q(x)x

i s a Cm diffeomorphism of in

0,

i n Rn o n t o some neighbourhood of t h e o r i g i n

which r e d u c e s g ( x ) t o t h e form

pln

Y’

(g

h)(y) = g ( O ) + $

0

2 C U.Y. I<j K

=

+QK(p),

~ . ( x , a ) f ( x ) ~~ ~, (=x ~, a~being ), some l i n e a r d i f f e r e n t i a l 3 2j a t most, and whsre

m’

c - c o e f f i c i e n t s of order

o being some i n t e g e r s and

c K ( g ) > 0 being some constants,which may depend

on K and 9. The asymptotic f o m m * ~ a

can be d i f f e r e n 6 i a t e d wi+h m s p e c t t o

p

i n f i n i t e l y many t i m e s .

3.8. The Stationary Phase Proof. -

265

I t s u f f i c e s t o c o n s i d e r t h e c a s e when supp f b e l o n g s t o some s m a l l

0

neighbourhood

o f t h e p o i n t x o , s i n c e t h e p a r t i t i o n o f u n i t y and

XQ

P r o p o s i t i o n 3.8.4

can be used i n o r d e r t o reduce t h e g e n e r a l s i t u a t i o n

t o t h i s case. Assuming t h a t

0

i s so s m a l l t h a t Morse's Lemma c a n b e

XO

a p p l i e d , w e can rewrite I ( p ) a s follows:

Using r e p e a t e d l y t h e one d i m e n s i o n a l s t a t i o n a r y phase method n t i m e s , one g e t s (3.8.27) f o r I ( p ) and t h e c o r r e s p o n d i n g f o r m u l a e f o r i t s derivatives

.

I

W e are g o i n g t o g e t e x p l i c i t

formulae f o r t h e c o e f f i c i e n t s a . ( f , g ) i n 1

(3.8.27). Denote by Q

(3) the differential operator 4 (xo)

2 -1 . 2 i s t h e i n v e r s e m a t r i x f o r t h e m a t r i x D g ( x ) of t h e where D g ( x o ) 0 second d e r i v a t i v e s of g ( x ) a t x o , and i n t r o d u c e

2

u 9 ( x0 1

where

i s t h e s i g n a t u r e of D g ( x

).

0

Theorem 3.8.7.

Under t h e assumptions o f Theorem 3.8.6. t h e foZZowing asymptotic formula holds for

where

Q

g i v e n by (3.8.1):

I(p)

9 (xo)

(a),

h (x,xo), y 9

a r e given r e s p e c t i v e z y by (3.8.28), (3.8.29),

(3.8.30), where (3.8.32)

\

and where f o r

=

p

n/2 + k

-

[2k/3]

>= 1 holds:

w i t h some i n t e g e r

M k ' O

3. Singular Perturbations on Smooth Manifolds without Boundary

266 Proof. -

Denote iph (x,xo) v(p,x0,x) = f(x)e g

and

For a symmetric regular matrix A one has the following formula

where

- (2T)-n/21det Al-f e(in/4) sign A 'n,A

-

sign A being the signature of A. Using Parceval's identity and the last formula, one can rewrite I ( p ) as follows: 1

Expanding the exponential under the integral sign into Taylor's series, one gets:

and that is precisely (3.8.31). Since h ( x , x ~ )has at x = x0 a zero of order 3 at least, the degree 4

of the polynomial

is [2j/?] at most. Writing I ( p ) in the form

using the fact that the coefficients of an asymptotic expansion are uniquely d e f i n e d and that the degree of q . ( p ) is at most [2j/3 , one gets the 7 p-' with p 5 n/Z+k-[Z(k-1)/31

conclusion that all the terms containing have to be included in the sum

c

-1

L-

Sj'P),

O<j g ( x ) , V x E v\{xo}.

Then t h e asymptotic expansions ( 3 . 8 . 2 6 1 ,

and t h e corresponding

(3.8.31)

e s t i m a t e s f o r t h e remainders are s t i l l v a l i d when such t h a t I m p

5 0,

/p

2 1,

p

is a complez parameter,

uniformly w i t h r e s p e c t t o a r g

p.

If (3.8.45)

g(xo) < g(x)

v

x

E v\{xol,

then t h e s t a t e m e n t s ab ve a r e t r u e f o r I m p L

o,

Ip

1

2 1.

Proof. W e s h a l l assume t h a t ( 3 . 8 . 4 4 ) h o l d s , t h e case when one h a s ( 3 . 8 . 4 5 ) __

c a n b e t r e a t e d i n t h e c o m p l e t e l y analoguous way. i t s u f f i c e s t o p r o v e t h i s theorem f o r n

Evidently,

=

1 , s i n c e Morse’s

Lemma r e d u c e s t h e m u l t i d i m e n s i o n a l case t o n = 1. I n one d i m e n s i o n a l

s i t u a t i o n i t i s enough t o show t h a t ( 3 . 8 . 1 6 ) h o l d s f o r complex p s u c h t h a t I m p S 0 . For d o i n g t h a t one i n t e g r a t e s i n ( 3 . 8 . 1 4 ) a l o n g t h e r a y

r

Y

= {z

I

z E

c,

i0 z=y+oe , a 2 0, 8

where y = a r g p . Then f o r z E

r

Y

one f i n d s

and t h a t y i e l d s ( 3 . 8 . 1 6 ) .

I

Remark 3.8.13. If

( 3 . 8 . 4 4 ) h o l d s , t h e n one h a s f o r p +

m

=

n/4-y/2},

3.8. The Stationary Phase

27 1

uniformly with r e s p e c t t o a r g p . The p r o c e d u r e above f o r computing a s y m p t o t i c s o f t h e form ( 3 . 8 . 4 6 ) i s c a l l e d t h e L a p l a c e method. I f f , g a r e a n a l y t i c and U i s a m a n i f o l d i n Cn of r e a l dimension n ,

t h e n t h e p r o c e d u r e above h a s t h e name: t h e mountain p a s s method ( o r t h e s a d d l e p o i n t method). W e s h a l l s t a t e t h e main r e s u l t i n t h i s case which w i l l b e u s e d l a t e r on for d e s c r i b i n g a s y m p t o t i c b e h a v i o u r of s i n g u l a r l y p e r t u r b e d Poisson type o p e r a t o r s .

E C n , b e holomorphic a t t h e p o i n t z o , which i s supposed t o

Let g ( z ) , z

b e a r e g u l a r c r i t i c a l p o i n t f o r g ( z ) . L e t U be a s u f f i c i e n t l y s m a l l neighbourhood o f z o , U

=

{z

1

Iz-zo/




-ul,

LG

u n

{Re(ig(z)-ig(z ) )

-GI,

=

0

=

where u > 0 i s s u f f i c i e n t l y s m a l l . The r e l a t i v e homology g r o u p H ( Gu , L u )

i s isomorphic t o Z . Denote

by y t h e g e n e r a t i n g c y c l e o f t h i s t r o u p . I f g ( z ) =

C 22 t h e n one l Z j 6 n I'

can d e f i n e Y a s f o l l o w s

(3.8.47)

y

=

=...=

{yl

yn

=

01 n u

Theorem 3.8.14.

L e t f ( z ) , g ( z ) be holomorphic a t z o E y, z o being a reguZar s t a t i o n a r y p o i n t f o r g ( z ) and l e t J f ( z ) eiPg(')dz, Y y being t h e generating c y c l e d e f i n e d above. ~ ( p )=

Then one has f o r (3.8.48)

I(p)

p

-

-f

-:

p -n/2

eipg(zo)

I: a . p - 1 . jto

'

The main term i n t h e asymptotic expansion ( 3 . 8 . 4 8 ) has t h e form: (3.8.49)

I(p)

-

f ( z o ) ( 2 n / p ) " l 2 ( d e t DZq(zO))-' 2

rJhere ( d e t D2g(z z

o

))

eipg(zo),

' I 2i s d e f i n e d by t h e o r i e n t a t i o n o f

y.

3 . Singular Perturbations on Smooth Manifolds without Bounda y

212 Proof. -

I n a s u f f i c i e n t l y s m a l l neighbourhood of zo one c a n u s e a holomor-

p h i c d i f f e o m o r p h i s m z = h ( w ) (holomorph v e r s i o n o f Morse's Lemma) which r e d u c e s g ( z ) t o a sum of squares: (g

0

h ) (w) = g ( z o ) +

b e s i d e s , one h a s 2

h'(0)

I: 1sj 1. Then 5

+

213

g ( x , t ; C ) d o e s n o t have s t a t i o n a r y

p o i n t s on R and t h e p a r t i a l i n t e g r a t i o n i n t3.8.50)

yields:

u n i f o r m l y w i t h r e s p e c t t o ( x , t ) i n any compact s e t K b e l o n g i n g t o t h e halfplane x / t

> 1.

let a

Now,

=

x / t < 1. Then

5

+

g ( x , t ; c ) h a s two s t a t i o n a r y p o i n t s :

which are r e g u l a r , s i n c e a c: 1.

,f

L e t f ( 5 ) E C:(lR)

:

R

+

R b e such t h a t f ( 5 ) : 1 f o r

5 E [ 1 , i . e .

1x1 > I t / , on: has: E ( x , t ) = O ( h ) , h h u n i f o r m l y w i t h r e s p e c t t o ( x , t ) i n any compact b e l o n g i n g t o t h e s e t IIxj

'

-f

0

It\}.

Assume now t h a t 1x1 < Itl. L e t f ( < ) E C m ( R ) be s u c h t h a t 0

supp f c ( - 2 , 2 ) ,

E

f E 1 for 5

[-3/2,3/2].

Then one h a s :

One c a n c o n s i d e r t h e i n t e g r a l

2 m a s a n e x t e n s i o n o f E ( x , t ) t o R up t o a n e r r o r O(h ) , f o r h + 0 . h Applying t h e s t a t i o n a r y p h a s e method ( f o r m u l a ( 3 . 8 . 3 1 ) ) t o t h e i n t e g r a l ( 3 . 8 . 5 6 ) one g e t s t h e f u l l a s u m p t o t i c e x p a n s i o n f o r Eh ( x , t ) when h + 0 . Again w e w r i t e h e r e o n l y t h e f i r s t t e r m i n t h i s e x p a n s i o n : (3.8.57)

Eh ( x , t )

where la1 < 1 , a

=

-1 2 -d 2*(nh) ' / t / - f ( l - a ) cos(h-'t(aarccosa-Jl-a

x/t;

t h u s , s i n g s u p p Eh =

Now assume t h a t (1-a ) -ha

with

0

{\XI

2

)+n/4),

5 Itl}.

E (2/5,2/3).

Then ( 3 . 8 . 5 3 ) c a n b e r e w r i t t e n a s f o l l o w s :

where $ ( S )

= 5-(1/6)5

3

-sin

5

5

= O(5 )

A f t e r t h e change o f v a r i a b l e

Since p

=

( 1 - a ) 3/2h-1

each i n t e g r a l

+ m

(h

+

5

=

when

5

(1-a)

t n,

+ 0.

( 3 . 8 . 5 8 ) becomes

0 ) i s a l a r g e p a r a m e t e r , one c a n a p p l y t o

3. Singular Perturbations on Smooth Manifolds without Boundary

216

t h e S t a t i o n a r y Phase Method. Hence, w i t h 1-a = Ch',

2/5 < u < 2/3, one f i n d s :

and moreover,

T h e r e f o r e , t h e main t e r m i n t h e a s y m p t o t i c b e h a v i o u r of E ( x , t ) i n t h e h c a s e c o n s i d e r e d i s g i v e n by t h e f o r m u l a :

-

The a p p l i c a t i o n o f t h e S t a t i o n a r y Phase Method t o t h e i n t e g r a l o n 3/2 h-l + the large ( 3 . 8 . 5 9 ) w i t h P = (1-a)

t h e r i g h t hand s i d e o f

parameter l e a d s t o t h e following asymptotic formula for E ( x , t ) : h

where c . > 0 , j = 1 , 2 are any g i v e n c o n s t a n t s 7 y = m i n I 1 - 3 ~ / 2 , 5u/2-11 > 0.

< c 2 , 2/5 < a < 2/3 and

C

Example 3.8 . 1 7 . Consider t h e i n t e g r a l

0 i s an i n t e g e r . where h E ( 0 , h ] i s a s m a l l p a r a m e t e r and p 0 The f u n c t i o n ( 3 . 8 . 6 1 ) i s t h e s o l u t i o n of t h e Cauchy problem:

lim

t++o

G

(x,t) = 6(x).

Using t h e change of v a r i a b l e 5

where

1 -

prh +

(x/t-1)

2p- 1

,€ one f i n d s :

3.8. The Stationary Phase

277

2p (3.8.63)

a

= t(x/t-l)

2p- 1

For p = 1 u s i n g Cauchy‘s theorem one f i n d s e a s i l y : -

( 2 n t h ) - f e - ( x - t ) 2/ ( 2 t h )

-

Gl,h(~,t)

W e a r e g o i n g t o compute t h e f i r s t t e r m i n t h e a s y m p t o t i c e x p a n s i o n of G

Pth

( x , t ) when h

0 , u s i n g t h e S a d d l e P o i n t Method.

-f

The polynomial

g (5) P

=

i ~ - ( 2 p ) - ~ 5 ’ P , gp(5) :

c

1

+

c

1

has t w o regular stationary points n i_ ni -__ 2 (2p-1) Z(Zp-1) (3.8.64) = e , C 1 = -e

c0

w i t h t h e same imaginary p a r t , I m 5 .

I

g

P

(Lo)

= g

P

(ill

=

=

sin(;

~i 2(2p-1)

)

> 0 , where

i ( 1 - i ) e 2 ( 2 p - 1 ) , R e g (5.)< 0 , 2P P I

and f o r any o t h e r s t a t i o n a r y p o i n t 5

one h a s :

I t i s e a s i l y seen t h a t

max R e g ( 5 ) = R e g ( < , ) , P I

j = 0,l.

<EY

I n d e e d , t h e s t a t i o n a r y p o i n t s of t h e f u n c t i o n R e g ( 5 ) : y + R a r e t h e 2p- 1 P p o i n t s , where R e 5 = 0, i . e . the points C j , j = 0 , l i n (3.8.64) o n l y , so t h a t one h a s

The Cauchy theorem y i e l d s :

Applying t h e S a d d l e P o i n t Method t o t h e i n t e g r a l on t h e r i g h t hand

3 . Singular Perturbations on Smooth Manifolds without Boundary

278

s i d e o f ( 3 . 8 . 6 5 ) a t each s t a t i o n a r y p o i n t < , , j = 0 , 1 , one g e t s t h e f u l l 3 asymptotic expansion f o r G ( x , t ) when h + O . Wewritedown e x p l i c i t l y the P,h f i r s t t e r m of t h i s e x p a n s i o n : I

-1 2p-1 -f (3.8.66)G ( x , t ) = 2(2nh)-+(xt-l-l) a R e { J ; ; - - e a g P ( S O ) ] ( l + O ( h a - l ) )= 9 ( C 0 ) P,h P _ _P_ 1 2p- 1 -f -c h - l a = (2nth) 2 ( x / t - l ) (2p-1) w p ( c o s ( b h-'a+a ) ( l + O ( h a - l ) ) ~

P

P

Note t h a t one h a s : s i n g supp G

P*h

= ix = t}.

Example 3 . 8 . 1 8 . C o n s i d e r t h e Cauchy problem

d K (x,t) +h-l(%(x,t)-%(x-h,t)) dt

= 0, x

E \,

t

E

W+

h

(3.8.67) 1i m

t++o where

%(x,t)

= Ah(X)

"i, and 6h ( x ) a r e d e f i n e d i n Example 3.8.16. Using t h e F o u r i e r t r a n s f o r m i n x E "i, , one f i n d s e a s i l y :

(3.8.68)

% ( x , t ) = (2nh)-'

J

e

h

-1

(ix 0, t > 0, h

(1/3,1/2).

-f

0.

Rewriting t h e

i n t h e form =

(21rh)-'(a-l)


0. As a consequence o f

=

289 (p,q)-parabolicity,

the

main c o n t r i b u t i o n i n t o t h e a s y m p t o t i c b e h a v i o u r o f t h e i n t e g r a l (3.8.108)

i s g i v e n by a neighbourhood o f z e r o R e a ( q ) 5 -C6,

in1

5 6 s i n c e f o r 1q1 2 6 one h a s :

w i t h some p o s i t i v e c o n s t a n t C 6 , depending on 6 . Hence,

E ( x , t ) = (2rrh)

-1

h

J

h - l ( i x q + t a ( q ) ) d r lt O ( e - t h - l C

6).

lrlIS6

One h a s i n t h e case c o n s i d e r e d : ixq+ta(q)

t a rl Pb ( q ) , P

=

E C"([-6,61), b ( 0 )

where b ( n )

R e a

P

0 , j = 1 , 2 , which depend o n l y 1 i n (3.8.101). P

on w and t h e c o e f f i c i e n t a 2'.

+inh b e i n g p - h y p e r b o l i c , assume a d d i t i o n a l l y t h a t t # 0 (mod Z I T ) . Then t h e f o l l o w i n g a s y m p t o t i c f o r m u l a e

The a p p r o x i m a t i o n D

# 0, V n

a'(n)-iw hold:

a ) f o r p even

V ( x , t ) s u c h t h a t h l x + w t /- o / ( P - l ) + h - l A,

I'

B., 1

j

=

Ix+wt/

1 , 2 , are some non-vanishing

which depend o n l y on w and a

P

(p+l)'(p-l)

+ 0 when h

constants, I m A . 1 i n (3.8.101).

=

-f

0; h e r e

0 , j = 1,2,

b ) f o r p odd, one h a s t h e same f o r m u l a ( 3 . 8 . 1 1 4 ) a s i n t h e p - p a r a b o l i c case f o r sgn(x+wt) t h e p-parabolic

=

-sgn I m ap, a n d , t h e same f o r m u l a ( 3 . 8 . 1 1 5 ) a s i n

c a s e w i t h even p , when s n g ( x + w t ) = sgn I m a

P

.

The f o l l o w i n g argument g i v e s an i d e a how t o p r o v e t h e f o r m u l a e ( 3 . 8 . 1 1 4 ) , (3.8.115).

3 . Singular Perturbations on Smooth Manifolds without Bounda y

292

Introducing the new variable n = /(x/t)+o(l’(p-l)E,

one rewrites the

integral (3.8.108) in the form where the phase function in the exponent is as follows:

where $ ( n ) = a(q)-iwn-a np

P

=

for rl

O(np+l),

-f

0.

One can consider O(x,t,E) as a small perturbation of the phase function

when a =

I (x/t)+wIP-l

is small, the remainder being of order

+1 p+l $ ( a S ) = O(ap 5 ) , when a

One has the natural conditions on large parameter when h

+

+

0.

1 (X/t)+ W /

:

h-ll (X/t)+W lp’(p-l) has to be a

0 in order to apply saddle point or stationay

phase method to the corresponding integral with the phase function iS+a SP, while h-ll(x/t)+wl(p+l)/(p-l) has to be a small parameter when

P

h

+

0, for being able to use the Taylor expansion for the function

exp(ih-’t$(aE)) as in Example 3.8.16. Thus, the computation of the main term in the asymptotic expansion of E (x,t) for (x,t) such that -1

h 1 (x/t)+wl p/(p-l) + m, h-ll (x/t)+wl t p + l ) / ( p - l ) + O the case of the polynomial phase function iS+a 5’.

P

(h+O) is reduced to The use of the saddle

point (as in Example 3.8.17) or stationary phase method (in the corresponding p-hyperbolic case), leads to the formulae (3.8.114), (3.8.115) above. Example 3.8.23. 1 Let 5 E C , 151 < 1 and let:

where 5

*

is the complex conjugate of 5.

Introduce s;(c;e)Je=l = r,

o


) .

(Co,5) E R

n+l

-

, x

=

(xo,x) E Rn+',

one can rewrite (3.9.9) as follows:

Let n = 2m+2, where rn 2 0 is integer. One has in that case (see [ G - Sh, I ] ) : (3.9. where C12m+2

6 /:(I

is the area of the unit sphere in R

t) is the &-function on the sphere

(3.9.12)

(6(l;l-t),$)

=

t

t

=

{y 1

and the distribution

x E iR2m+3, 1y1

= t}, i.e.

.

J $(:)do_, S

S

2m+3

V Ji E C m ( ~ R 2 m + 3 ) X

Using (3.9.10), (3.9.11), one finds:

Furthermore, one has

where, as usual, s

=

max {s,O}.

Hence, (3.9.131, (3.9.14) yields: -1

(3.9.15) E 2 m + 2 ( ~ , t ;=~) and for m = 0 one finds:

Therefore the FISP (3.9.6), (3.9.7) with n equivalent representation:

=

2m+2 admits the following

3 . Singular Perturbations on Smooth Manifolds without Boundary

298 and f o r n

2 m + l one c a n u s e t h e c l a s s i c a l d e s c e n t method f o r g e t t i n g a

=

s i m i l a r r e p r e s e n t a t i o n f o r ( E : (~ t )~u )~( x ) ( s e e , [Cour.,

1

I.

One f i n d s t h e f o l l o w i n g f o r m u l a f o r t h e c o r r e s p o n d i n g k e r n e l d i s t r i b u t i o n :

where J ( p ) i s t h e B e s s e l f u n c t i o n of o r d e r z e r o and H ( T ) , T C R , i s 0 H e a v i s i d e ' s f u n c t i o n (see, f o r i n s t a n c e , [Grad.-Ryz., 11 f o r t h e B e s s e l f u n c t i o n s . N

Indeed, l e t a g a i n 5 =

(5,,5) E R

2m+2

-

, x

= (xO,x) C R

Using t h e d e s c e n t method, one f i n d s f o r

t h e following formula

Again a p p l y i n g ( 3 . 9 . 1 0 ) , one g e t s :

since

1

1

e-ipe(1-82)-4d8

=

71

Jo(p),

V p 6 IR

-1 (see [ G r a d . - R y z . ,

1

1).

I n p a r t i c u l a r , one h a s :

L e t B b e symmetric p o s i t i v e d e f i n i t e nxn m a t r i x . The s o l u t i o n of t h e s i n g u l a r l y p e r t u r b e d Cauchy problem: (E

2 2

at-E

lim

two

2

= t 1. A s a consequence o f t h e l a s t f o r m u l a f o r E 2 m + 2 , B ( ~one , t ;g~ e t )s ,t h e c o n c l u s i o n t h a t i t s s i n g u l a r s u p p o r t , s i n g supp E

2m+2,B whose boundary i s

c o i n c i d e s with t h e set

*

v

Dv*

=

( t , x ) E R n + ' , , so t h a t t h e c o r r e s -

ponding S i n g u l a r F o u r i e r I n t e g r a l O p e r a t o r i s j u s t a s i n g u l a r p e r t u r -

sv ( U ) i n t r o d u c e d i n D e f i n i t i o n 3 . 3 . 1 . 1,o One f i n d s e a s i l y i n t h i s case

b a t i o n of t h e c l a s s

A s a consequence, one g e t s a g a i n t h e p s e u d o l o c a l i t y p r o p e r t y o f

singular perturbations: s i n g supp(E"la(x,E,D)u)

5

s i n g supp u .

Example 3 . 9 . 9 . Let n

=

(3.9.24)

n2

=

n + l , U1 = U2

$(x,y,t,X,E)

=

=

{(x,t) E R

n+ 1

},

and l e t

< x - y , C > + ( t - T ) ( A 2+)t ,

where B i s symmetric p o s i t i v e d e f i n i t e nxn m a t r i x . One h a s :

Q (X,C) r$

=

{(X,t,y,T) E U

X

2 U, X-y+(t-T) ( A +)-*BC = 0).

Hence i n t r o d u c i n g rl = B t 5 , one f i n d s e a s i l y :

3. Singular Perturbations on Smooth Manifolds without Boundary

302

The phase f u n c t i o n ( 3 . 9 . 2 4 ) a p p e a r s w h i l e s o l v i n g t h e Cauchy problem f o r t h e s i n g u l a r l y p e r t u r b e d wave o p e r a t o r : 2 2 2 a t - € CBa X '

ax>+]

(see Example 3 . 9 . 4 ) , so t h a t t h e s i n g u l a r i t i e s o f t h e c o r r e s p o n d i n g

fundamental s o l u t i o n

a r e c o n t a i n e d i n Q $ . A c t u a l l y , f o r n = 2mc3, t h e

s i n g u l a r s u p p o r t of t h e fundamental s o l u t i o n i s e x a c t l y t h e whole Q$, a s Example 3.9.4

shows.

Now w e s h a l l i n t r o d u c e t h e c l a s s of

hyperbolic singular perturbations

and i n v e s t i g a t e t h e s i n g u l a r i t i e s of t h e k e r n e l d i s t r i b u t i o n s a s s o c i a t e d w i t h t h e s i n g u l a r F o u r i e r i n t e g r a l o p e r a t o r s which s o l v e t h e Cauchy problem f o r h y p e r b o l i c s i n g u l a r p e r t u r b a t i o n s . D e f i n i t i o n 3.9.10.

With a ( x , t , E , c , < ) E S v ( ~ y + l,) R:+'=

R

~

x

XR

t,+'

O p a i s s a i d t o be a

s t r i c t l y hyperboZic s i n g u l a r p e r t u r b a t i o n i f v 2 2 0 , v3 2 0 a r e i n t e g e r , a ( x , t , c , S , < ) i s polynomial i n 5

of degree v 2 + v 3 and t h e principa2 symbol

a ( x , t , ~ , S , < )s a t i s f i e s t h e c o n d i t i o n : 0 The zeros < . ( x , t , ~ , S ) ,1 5 j 6 v2+v3,0f t h e equation 1

(3.9.25)

a0(x,t,E,S, 5 . )

I

=

0

n+ 1

are r e a l and d i s t i n c t , v ( x , t ) E R +

,v

(E,c)

E R + x ( B ~ l\ o } ) .

H y p e r b o l i c s i n g u l a r p e r t u r b a t i o n s whose symbols a r e polynomial i n ( E , t ) , a r e o f s p e c i a l i n t e r e s t . W e s h a l l c o n s i d e r t h e Cauchy problem f o r t h e c o r r e s p o n d i n g h y p e r b o l i c d i f f e r e n t i a l s i n g u l a r p e r t u r b a t i o n s whose symbols have c o n s t a n t c o e f f i c i e n t s and c o i n c i d e w i t h t h e i r p r i n c i p a l symbols. L e t ao(t,S, 0 , v2 2 0. C o n s i d e r t h e Cauchy problem:

n+l

aO(E,Dx,Dt)u = 0 ,

( x , t ) E R+

(3.9.26)

where 6 . = 0 , 1 5 j < m, bm = 1 i s t h e Kronecker symbol and v E C m ( R n ) 1m 0 Let O,j

'j

A s a consequence of Theorem 3 . 9 . 7 t h e s i n g u l a r s u p p o r t of t h e k e r n e l

d i s t r i b u t i o n a s s o c i a t e d w i t h t h e s o l u t i o n u ( E , x , ~ )o f

(3.9.26),

is

For c l a s s i c a l h y p e r b o l i c o p e r a t o r s ( v 3 = O ) , t h e s e t K*

c o n t a i n e d i n K*.

i s n o t h i n g e l s e b u t t h e d u a l c o n e of r a y s f o r t h e c h a r a c t e r i s t i c cone of

(see f o r i n s t a n c e [Cour I]).

t h e o p e r a t o r a,(E)

I n t h i s c a s e dim K* = 2n,

i

w h i l e i n t h e g e n e r a l case of h y p e r b o l i c s i n g u l a r p e r t u r b a t i o n s onemay h a v e : dim K

*

=

2 n + l , a s it was t h e case i n Example 3 . 9 . 4 .

Using ( 3 . 9 . 3 2 ) one g e t s a f t e r t h e change o f v a r i a b l e s E C

+

5 the

f o l l o w i n g formula f o r t h e d i s t r i b u t i o n a l k e r n e l o f t h e o p e r a t o r , t h a t s o l v e s t h e Cauchy problem ( 3 . 9 . 2 6 ) (3.9.341

:

-1 im-1(2n)-nE1-n/n c a . (l,c)eiE ' j ' t ~ x ~ y ~ l ~ ' ) d ~ , R 1 c j s v +v 2 3 a r e g i v e n by ( 3 . 9 . 3 3 ) .

E(E,x-y,t)

=

where a . and 4 . 7 7 Let E . ( ( x - y ) / t ) be a s t a t i o n a r y p o i n t f o r 1 (3.9.35)

V5cj(l,S.) 1

=

(x-y)/t,

0.

1' l - e .

1 5 j 5 m.

Definition 3.9.11.

The p o i n t ( x , y , t ) E

K*

i s s a i d t o be non-focaZ i f t h e corresponding

s t a t i o n a r y p o i n t s 5 . ( ( x - y ) / t ) d e f i n e d by ( 3 . 9 . 3 5 ) are r e g u l a r , i . e . t h e 5

3.9. The Fourier Integral Singular Perturbations

matrices

D

2

305

5 ( 1 , c . ) are non-singular.

5 ,

I

Applying t h e s t a t i o n a r y p h a s e method t o t h e i n t e g r a l ( 3 . 9 . 3 4 ) a t e a c h n o n - f o c a l p o i n t ( x , y , t ) a n d assuming a 6 l i t i o n a l l y t h a t t h e c o r r e s ponding s t a t i o n a r y p o i n t s S . ( ( x - y ) / t ) a r e w e l l - d e f i n e d by ( 3 . 9 . 3 5 ) , one

I

g e t s t h e following assymptotic formula f o r E ( E , x - Y , ~ ) a t each such p o i n t (x,y,t): (3.9.36)

E(~,x-y,t)

N

v

+I,

-1 (2Tr)

i

-n/2

t

-1/2

E

1-n/2

~

q . ( ( x - y ) / t )e 1 6 j 6 v +v 7 2 3

itE-lAj

((x-y)/t)

A . ( 2 ), z E R n , i s t h e Legendre t r a n s f o r m of t h e f u n c t i o n 5 , ( 1 , c ) , 5 E 3 1 i . e . A.(z) and < . ( l , E ) a r e r e l a t e d by t h e Lagrange t r a n s f o r m a t i o n 1 3

where

(3.9.37)

A.(z) = < Z , S > + < . ( l , S ) , 3

1

v65 3 (1,5)

mn,

= -2,

and where

2

X . ( 5 . ) b e i n g t h e s i g n a t u r e o f t h e m a t r i x D E E < , ( 1, 5 , ) . I

1 ( 3 . 9 . 3 6 ) , E ( ~ , x - y , t )a t e a c h n o n - f o c a l p o i n t i s

3

A s a consequence o f

r a p i d l y o s c i l l a t i n g when For e a c h ( x , y , t ) multi-indice

c1

K

*

E +

0.

one f i n d s , a p p l y i n g P r o p o s i t i o n 3 . 8 . 4 ,

t h a t f o r each

t h e following asymptotic formula h o l d s :

2 2 For t h e wave o p e r a t o r a ( E . S . ~ ) = 5 -151 one h a s c1 = 151, c 2 = -1c1, 0 $ l ( t , x , y , l , S ) = < x - y , C > + t / < I , $ 2 ( t , x , y , 1 , 5 ) = < x - y , C > - t / E and t h e

s t a t i o n a r y p o i n t s a r e n o t w e l l - d e f i n e d by ( 3 . 9 . 3 5 ) so t h a t ( 3 . 9 . 3 6 ) c a n not hold. Notice t h a t i n t h i s case, i n f a c t , $

E

s(orlto)(UxU).

For t h e s i n g u l a r l y p e r t u r b e d wave o p e r a t o r w i t h t h e symbol ao(E,S,S)

c2

= E 2 5 2 -E 2 [ 5 1 2 + 1 ( s e e Example 3 . 9 . 4 )

= - E - ~ < E ~ >and


,

t h e c o r r e s p o n d i n g s t a t i o n a r y p o i n t s a r e w e l l - d e f i n e d by

( 3 . 9 . 3 5 ) p r o v i d e d t h a t Ix-yl

and

one h a s :

< t . One f i n d s e a s i l y i n t h i s c a s e

3 . Singular Perturbations on Smooth Manifolds without Boundary

306

2 2 -2 I D s S 5 j ( l , c3. ) I = ( l - l x - y ] t

Hence, one has K* (x,y,t)

E

K

*

=

{ ( x , y , t ) , Ix-yI

l+n/2

< t} i n t h i s c a s e , e a c h p o i n t

i s n o n - f o c a l and t h e f o r m u l a e ( 3 . 9 . 3 6 ) - ( 3 . 9 . 3 8 ) a r e v a l i d

f o r t h e d i s t r i b u t i o n a l k e r n e l E ( ~ , x - y , t )of t h e corresponding o p e r a t o r s o l v i n g t h e Cauchy problem ( 3 . 9 . 2 6 ) f o r t h e S i n g u l a r l y P e r t u r b e d Wave 2 2 2 O p e r a t o r E at-€ A + l ( s e e ( 3 . 8 . 1 0 0 ) where t h e f i r s t t e r m i n t h e a s y m p t o t i c e x p a n s i o n f o r E ( E , x , t ) i s e x p l i c i t e l y w r i t t e n down). N o t i c e t h a t i n t h e c a s e of t h e s i n g u l a r l y p e r t u r b e d wave o p e r a t o r

a t- & 2 A x +1 -

2 2

t h e c o r r e s p o n d i n g p h a s e f u n c t i o n s a r e $ ( t , x , y , ~ , < )= 1 < x - y , S > + t ~' < E S > , G 2 ( t , x , y , & , S ) = < x - y , S > - - t ~ - ~ < & and c > $ j € S ( l ' o r l ) ( U x U) E

but

$jtr

S (o.l,o)

(UX

U).

Hence, it i s n a t u r a l t o i n t r o d u c e t h e f o l l o w i n g D e f i n i t i o n 3.9.12.

A Fourier I n t e g r a l S i n g u l a r Perturbation A'

$

i s s a i d t o be a Proper Fourier

I n t e g r a l S i n g u l a r Perturbation (PFISP) i f i n ( 3 . 9 . 2 ) t h e corresponding phase -1

function $ ( x , Y , E , t ) s a t i s f i e s t h e conditions: 1'

$

b u t $(X,Y,E 2"

i s real-valued, $ ( x , Y , E -1 , c ) E S ( l ' o ' l ) ( u l -1

, S ) ?! S(o,l,o)

(x,y,h,t!

variabzes ( A , 5 ) f o r

(U1

x

x

u2),

U2).

i s p o s i t i v e l y homogeneous o f degree 1 w i t h r e s p e c t t o

1 A I 2+ 15 I

2

+ and, moreover, $ s a t i s f i e s

vx,x,S$ f 0 , vy,h, 0 , R ' > 0 , such t h a t 6 ' 5 IVg(y)l 5 R ' ,

V y C supp f . L e t 6 < 6 ' .

x 1 ( 5 ) : 0 f o r 151 > 6 , x , ( S )

!

1 for

and l e t x , ( S )

151 < 6 / 2 . Denote

E C;(lRn)

,

3.10. Diffeomorphisms and Singular Perturbations

Q 1 ( x , ~ , p )=

J 1 mn u

309

qx,eip'dydS.

L e t K C U b e any g i v e n compact. W e s h a l l show t h a t f o r any m u l t i i n d i c e s

a,B and any i n t e g e r N 2 0 h o l d s :

Denote

Given t h e c h o i c e o f t h e c u t - o f f

x l ( S ) , one o b v i o u s l y h a s :

function

IVg(y)-SI 2 c o > 0 , V ( x , y , S ) E K

X

X

supp

xl.

Therefore, the

c o e f f i c i e n t s o f t h e o p e r a t o r L ( y , C , a ) a r e smooth and bounded on Y U X supp x . F u r t h e r m o r e , s i n c e a ( x , E , S ) E sv (U) one o b v i o u s l y h a s f o r 1 1,o any a : -V

(3.10.9)

lD;q(x,y,E,P,S)\

5 C

a,K

E

l



V2

<EPS>

v3

,

V ( x , y , ~ , p , S )E K x

X

(O,E

0

1

x

l R + x lRn,

depends o n l y on a and K .

where t h e c o n s t a n t C U,K

Using t h e f o r m u l a

L ( Y , s , a y ) ei P J l

= ipeiP',

and d e n o t i n g by Lt(y,S,a t

L (y,c,a

Y

)

=

Y

t h e f o r m a l a d j o i n t o p e r a t o r of L ,

- w i t h I? E lRn and f o r any i n t e g e r N > 0 one h a s :

where t h e remainder i s g i v e n by t h e formula

a given vector

3. Singular Perturbations on Smooth Manifolds without Boundary

312

m

w i t h some 8 E ( 0 , l ) and E l E C O ( U ) . ( U ), one g e t s f o r R N f t h e f o l l o w i n g and a E Sv 1 ,o

E C;(U)

S i n c e Daf e st i m a t e :

-V

(3.10.13)

IRN(x,E,p,rl,Dx)f(x)I 5 C N ( f l ) E

w i t h some c o n s t a n t C ( f l ) > 0 N S e t t i n g n = Vg(x), f l (x)

=

V

-N

V <EP>

P'

3

f ( x ) ei p h 9 ( x ' y ) and a p p l y i n g t h e f o r m u l a

(3.10.12) t o t h e i d e n t i t y e-ipg ( x ) a( x , E , D X ) ( ei W f )

~

ip iphg(x,y)) -ipg (x) f(y)e a ( y , E , D 1 (e Y=x Y

I

= e

one g e t s t h e f o r m u l a ( 3 . 1 0 . 1 ) . The e s t i m a t e ( 3 . 1 0 . 2 ) f o r t h e r e m a i n d e r R N ( x , E , p ) f o l l o w s from

I

grows a t most l i k e ( 3 . 1 0 . 1 3 ) and t h e f a c t t h a t t h e c o n s t a n t C ( f e i p h l N Y=x P "/21 f o r p + m, g i v e n t h a t t h e f u n c t i o n h ( x , y ) h a s a z e r o o f o r d e r 2 when y-x

+

9

I

0.

C o r o l l a r y 3.10.2. Taking N = 1 i n ( 3 . 1 0 . 1 ) , ( 3 . 1 0 . 2 ) , one g e t s t h e f o l l o w i n g aSymptOtiC formula : ( 3 . 1 0 . 1 4 ) e - i p g ( x )a ( x , E , D

) (f

( x ) ei p g ( x ) )

=

f ( x ) a(x,E,pvxg(x))

-

2 2 - i { + ( p / 2 ) T r (DE5a( x ,E , 5 ) Dxxg ( x ) ) }

5

-v

+

O(E

1p"2-2<Ep>v

C o r o l l a r y 3.10.3. -1 . Taking p = E i n (3.10.1),

3

).

(3.10.21,

one f i n d s

iE-lg(X) ( 3 . 1 0 . 1 5 ) e-iE - l g ( x ) a ( x , E , D x ) ( f ( x ) e ) =

N-[N/2]-V

+

O(E

-V2

1

)

u n i f o r m l y w i t h r e s p e c t t o x i n any compact s e t K C C U and w i t h r e s p e c t t o

3.10. Diffeornorphisnzs and Singular Perturbations

313

m

f in any bounded set in

Co(U).

= v, In particular, if a E Kv(U) and ar = C a . ord a . = v(1), v(') j z o I' 1 is its graduate symbol, then the following asymptotic formula holds:

where, as usual a(a)= aaa, the asymptotic relation (3.10.16) being uniform

5

with respect to x I n any compact set K c c

U,

with respect to f in any

m

given bounded set in C ( U ) . 0 We are going to apply Theorem 3.10.1 in order to show how the symbols of singular perturbations are transformed under the coordinates diffeomorphisms. This will enable us to introduce singular perturbations on smooth manifolds without boundary. Theorem 3.10.1 will also be applied for giving a different proof of the formulae for the symbols of product of properly supported singular perturbations.

We shall need the following useful auxiliary result. Theorem 3.10.4. (j)

(u), j b 0 w i t h v ( 1 ) 1,o L e t ( O , E ~ I 3 E + a(x,~,c)E cm(ux

Let a E

1

E Sv

=

(j) (vl,v2 ,v3), ViJ) c

::

R~ ) be continuous w i t h r e s p e c t t o

E ( 0 . ~ ~and 1 assume t h a t t h e r e e x i s t c o n s t a n t s

c

a. 5

-a

andp

for

=

j

+

m.

u ( a , B ) such

that

f o r any x E U,

E

E (o,c01,

5 E R ~ .

Further, assume t h a t t h e r e e x i s t u ( N )

J-

-m

for

N +

and such t h a t on every

compact s u b s e t K cc u one has:

uniformly w i t h r e s p e c t t o x on any compact s u b s e t K

a U and

6

E ( 0 . ~ ~ 1

3. Singular Perturbations on Smooth Manifolds without Boundary

314 Proof. -

(01

According t o Theorem 3 . 6 . 2 ,

t h e r e e x i s t s a symbol b ( x , ~ , S )C Sv

1# O

(U) such

that

(N) (b(x,E,S) -

Z O - N < ~ S >', V x E

K C C U, V E

E (O,E~],

V N Z O . W e have t o check t h a t (3.10.20) h o l d s f o r t h e d e r i v a t i v e s B w D D r ( x , E , S ) . F o r d o i n g t h a t , o n e may u s e t h e Kolmoqorov i n e q u a l i t y

x s

z

SUP

/ a l = l xEK 1

2 IDa43(x)( 5 C s u p I43(x)( Z 1Da@(x)I, xEK2 1452

where K , a r e compact s e t s , K CC i n t K 2' 7 1 O b v i o u s l y , i n o r d e r t o show t h e v a l i d i t y of t h i s l a s t i n e q u a l i t y i t s u f f i c e s t o c o n s i d e r t h e one d i m e n s i o n a l case. Using t h e T a y l o r formula i n t h e form:

43 ' ( x )

= 'clp

26

6

( x + 6 )-43 (x-6) 1 + -(a" ( y + 6 )- $ ' I 4

and c h o o s i n g 6 = ( 2 s u p I @ ( x )I/sup

I

(y-6) )

,

I

I$"(x) ) ' ,

one g e t s t h e Kolmogorov

i n e q u a l i t y above. Indeed, a p p l y i n g t h i s l a s t i n e q u a l i t y t o t h e f u n c t i o n s @c(X,E,T))

with K

r(X,&,c+rl)

1 = K X { O } , K 2 = K 6 X Irl g e t s , using (3.10.17), (3.10.20)

/ 1111

5 61, K6 = { x

I

d i s t ( x , K ) 6 6}, one

3.10. Diffeornorphisrns and Singular Perturbations Now w e g i v e a d i f f e r e n t p r o o f of Theorem 3.7.6,

315

which i s b a s e d on t h e

s t a t i o n a r y phase method. Theorem 3.10.5.

L e t a ( x , ~ , ~: )c;(u)

+

c;(u),

a(x,~,<E )

m

b(x,E,D) :

c;(u)

+

C o ( u ) , b ( x , E , c ) E S'

1 .o

Sv

(u), and

1 PO ( U ) be p r o p e r l y supported

s i n g u l a r p e r t u r b a t i o n s . Then t h e i r product c ( x , E , D )

=

a ( x , ~ , D 0) b ( x , ~ , D ) ,

i s a properZy supported singuZar p e r t u r b a t i o n , whose symbol E Sv+' ( u ) has t h e f o l l o w i n g asymptotic expansion: 1,o

c(x,E,S)

t h e asymptotic r e l a t i o n (3.10.21) being understood in t h e f o l l o w i n g sense

u n i f o m l y w i t h r e s p e c t t o x i n any compact s u b s e t K C C to

E

E

U and w i t h r e s p e c t

(O,Eo1.

Proof. Let u

m

E C o ( U ) . S i n c e a ( x , ~ , D ) :C;(U)

Applying (3.10.1) w i t h p

=

151,

+

C E ( U ) , one h a s

g ( x ) = <x,w> w

=



compute t h e a s y m p t o t i c e x p a n s i o n o f a ( x , 6 , D ) ( b ( x , E , < ) e -f

m,

one g e t s t h e formulae

c ( x , E , ~ )of

)

when

(3.10.21), (3.10.20) f o r t h e symbol

the singular perturbation c = a

o

b . Now Theorem 3.10.4 i m p l i e s

E S"+'(U). I 1P O For some classes of s i n g u l a r p e r t u r b a t i o n s t h e c o m p o s i t i o n f o r m u l a C(X,E,S)

N

(3.10.21) i s v a l i d w i t h a remainder which i s of o r d e r O ( E < E S > - ~ )when 6 ' 0 ,

5'm.

D e f i n i t i o n 3.10.6.

A singular perturbation A o p J~

1 ,o

(u) i f

v

E op

= (vl,0,v3) and

( u ) i s s a i d t o b e in t h e c l a s s 1,o i t s symbol i s a cm-function of t h e form

Sv

a ( x , s , E ( ) , which s a t i s f i e s t h e i n e q u a l i t i e s

3 . Singular Perturbations on Smooth Manifolds without Boundary

316

and f o r V ( x , E , ~ ) E K

x

( 0 . ~ ~x 1iRn

where K is any compact s e t in

U.

Theorem 3.10.7.

L e t AE

=

Op a E Jv ( u ) , BE = Op b E Sy,o(u) be properly supported. Then 1r 0 C = AE o BE is a p r o p e r l y supported singuZar p e r t u r b a t i o n

t h e i r composition

i n OP Sv+' ( U l whose symbol c(x,E,~)has t h e f o l l o w i n g asymptotic expansion: 1,O

where t h e asymptotic r e l a t i o n (3.10.24) i s i n t e r p r e t e d in t h e f o l l o w i n g sense: f o r each N

= O,l,

... t h e

remainder

moreover,

Proof. ~

m

m

Cn

Let u E C (K). Since A

: CO(U)

0

-f

C

(U)

, one has

so that (3.10.1), (3.10.2) yield

+ (271)-n

where r

N

i rN(x,E,5)ei<X"> mn

u ( 5 )dS ,

satisfies the inequality IrN(X,E,S)I 5

N-(V +Ul) 1

cN , K ~

Y

t x' is the transpose of

the Jacobian matrix

x i(x).

Proof. It suffices to consider properly supported singular perturbations. Let m

u E Co(V) and denote y

= x(x), v(x) = ( u

X) (x). One finds

where we have denoted (3.10.30) b (x,E,a(x,E,Dx)(ei<x(x)rS>).

X

Notice that Vx<x(x),S> = tx'(x)5 # 0, V (x,S) E U Hence, formula (3.10.1) with p = 151, g(x)

=

x

(lRn\{O)).

<x(x),w>, w

=

5 / 1 5 ] yields

(3.10.28), (3.10.29), while Theorem 3.10.4 implies that m

a (x,E,5) E sy,o(V). Therefore, any C -diffeomorphism

X

bijection

x,

:

x

:

U

+

V induces a

Sy,o(U) ++ sy,o(V) according to the formula (3.10.27), the

corresponding symbols being transformed according to the formulae (3.10.281, (3.10.29).

I

3. Singular Perturbations on Smooth Manifolds without Bounda y

318

C o r o l l a r y 3.10.10.

E S"(U)

L e t AE = O p a , a ( x , E , < )

of a ( x , € , D x ) . L e t

and l e t a o ( x , E , E ) b e t h e p r i n c i p a l symbol m

x

: U

V b e a C -diffeomorphism and l e t a X , O ( x , ~ , Eb)e

-f

t h e p r i n c i p a l symbol o f t h e t r a n s f o r m e d s i n g u l a r p e r t u r b a t i o n d e f i n e d by ( 3 . 1 0 . 2 7 ) . Then ( 3 . 1 0 . 2 8 ) ,

( 3 . 1 0 . 2 9 ) imply:

C o r o l l a r y 3.10.11. Let

m

x

: Rn

Rn b e a C -diffeomorphism which r e d u c e s t o a l i n e a r map

-t

o u t s i d e of some b a l l i n Rn Then f o r V s

E

R3

.

t h e diffeomorphism

x

i n d u c e s a b i j e c t i o n of each s p a c e m

H (S)

( R n ) o n t o i t s e l f . I n d e e d , u n d e r t h e C -diffeomorphism t h e i d e n t i t y

o p e r a t o r i s t r a n s f o r m e d i n t o a n o p e r a t o r J = op j , whose symbol j ( x , < ) E So ( E n ) and i s c o n s t a n t f o r 1x1 s u f f i c i e n t l y l a r g e . T h e r e f o r e , 1,o w i t h v ( x ) = (u o x ) ( x ) Theorem 3 . 4 . 1 i m p l i e s t h a t v E H ( I R n ) whenever

.

u E Hs(Rn)

Hence, t h i s p r o v e s Lemma 2 . 6 . 1 ,

w h i l e Lemma 2.6.2

is a direct

consequence o f Theorem 3 . 4 . 1 . Remark 3.10.12. m

x

A C -diffeomorphism

i s o m o r p h i c mappings: (3.10.32)

x

*

: U

x*$

V w i t h U and V open sets i n R n i n d u c e s * -1 m = $ 0 x , x*$ = ( x ) $, V $ E C (U), V IJJ E C m ( u ) , -f

m

m

: Co(V)

+

Co(U),

x*

: C m ( U ) + Co0(V)

m

a n d , by d u a l i t y between C ( V ) and D'(V) t h e isomorphism 0 (3.10.33)

x,

:

D'(U)

*

D'(V).

F u r t h e r m o r e , it g i v e s r i s e t o a map t o t h e formula ( 3 . 1 0 . 2 7 ) ,

i.e.

( 3 . 1 0 . 3 4 ) X,a

x -1 ,

= (a

o

X)

D

x,

:

op

V a(x,E,Dx)

S y , o ( u ) * Op S"

1,o

E

(V) according

Op . q y , , ( U ) ,

a s w e l l a s a symbol homomorphism (which i s a n isomorphism u p t o t h e smoothing symbols o f o r d e r (3.10.35)

-m):

(X,a) ( y , ~ , n )= a ( y , ~ , n ) , X

where a ( y , ~ , q )i s w e l l d e f i n e d a s y m p t o t i c a l l y by ( 3 . 1 0 . 2 8 )

X

smoothing symbol of o r d e r

-m.

up t o a

3.10. Diffeomorphisms and Singular Perturbations

319

F u r t h e r m o r e , t h e p r i n c i p a l symbols a r e t r a n s f o r m e d a c c o r d i n g t o t h e formula ( 3 . 1 0 . 3 1 ) , which c a n b e r e w r i t t e n a s f o l l o w s :

(x,ao) ( y , ~ , r ~ = )a o ( x , E , S ) l x = x - ' ( y ) . 5 = tX ' ( X ) l l .

(3.10.36)

m

W e c a n now d e f i n e s i n g u l a r p e r t u r b a t i o n s on a C

paracompact m a n i f o l d M.

D e f i n i t i o n 3.10.13. m

A f a m i l y of operators A €

i s s a i d t o be a s i n g u l a r p e r t u r -

: c ~ ( M ) -f c-(M)

b a t i o n i n t h e c l a s s o p Sv

1 to

and a cm diffeomorphism x

i f f o r any coordinate neighbourhood x c

(MI,

:

x

+

u c n n , t h e f a m i l y o f operators

M

A€

U

d e f i n e d by t h e commutative diagram

i s a singular p e r t u r b a t i o n i n op Sv

1to

I n o t h e r words A' -1 .

A E = (A'

o

x)

x

0

: C;(M)

+ Cm(M)

: X + U,

i s i n t h e c l a s s Op

1 s a s i n g u l a r p e r t u r b a t i o n i n Op

U consequence of Theorem 3 . 1 0 . 9 , :A

x

(u).

i s i n Op

sv

sv 1,o

sv (M), 1,o

if

(U). A s a

( U ) f o r any C -diffeomorphism

1 PO so t h a t D e f i n i t i o n 3.10.13 i s c o r r e c t , i . e . t h e c o n c e p t of

s i n g u l a r p e r t u r b a t i o n on m a n i f o l d d o e s n o t depend of s p e c i f i c c h o i c e of neighbourhoods and d i f f e o m o r p h i s m s . Furthermore, given t h e t r a n s f o r m a t i o n r u l e (3.10.36) f o r t h e p r i n c i p a l

*

symbols, t h e y a r e w e l l d e f i n e d f u n c t i o n s on t h e c o t a n g e n t b u n d l e T ( M ) . I n t h e same way one d e f i n e s s i n g u l a r p e r t u r b a t i o n s classes O p S " ( M ) and Op K v ( M ) ,

u s i n g t h e d i a g r a m above and t h e i n v a r i a n c e of the classes

S"(U), K V ( U ) (see D e f i n i t i o n s 3.3.2 and 3 . 6 . 4 ) w i t h r e s p e c t t o t h e

Cm-

diffeomorphisms. However,

one c a n g i v e an i n d e p e n d e n t i n t r i n s i c d e f i n i t i o n o f s i n g u l a r

p e r t u r b a t i o n s on a smooth m a n i f o l d . D e f i n i t i o n 3.10.14.

Let

M

be a

Cm

paracompact manifold. A f a m i l y of operators m

E

-f

: C;(M)

-f

c

(M)

i s s a i d t o be a s i n g u l a r p e r t u r b a t i o n on

i f t h e f o l l o u i n g c o n d i t i o n s are s a t i s f i e d : 1

'.

There e x i s t s a sequence k

,

7

J.

--, f o r

j

+

m,

such t h a t

M

3. Singular Perturbations on Smooth Manifolds without Bounda y

320

2O. E

There e x i s t s a sequence s . 3

+

-m

for j

+ m,

such t h a t f o r each given

E ( 0 . ~ ~one 1 has:

( 3 . 1 0 . 3 9 ) e- ip g (x)AE ( f ( x )e i p g ( x ) )

f o r v f E c;(M), v g E

-

S .

’

C b E ( f , g ) p I, p + m , j>o c ~ ( M ) , dg f 0 on supp f , g being r e a l

3O. There e x i s t s a sequence m . C 3

for

-m

j +

a,

valued.

such t h a t

E c ~ ( M ) , V g E c - ( M ) , dg # 0 on supp f , g being r e a l vaZued. Furthermore, t h e order v = ( v l , v 2 , v 3 ) of t h e singuZar p e r t u r b a t i o n is t h e l e a s t v E m3 such t h a t

for V

f

> v (’I dEf

(3.10.41) v

(k.,m.-k.,s.-m.+k.), V j 1 3 3 3 3 1

=

0,1,

...

Furthermore, t h e f u n c t i o n ( 3 . 1 0 . 4 2 ) uA ( f , g ) = e

-ig(x)A (feig(x)

E

) #

E

which is asymptoticalZy represented by (3.10.39) when I V g ( x ) [ each g i v e n

+

m

for

i s s a i d t o b e t h e symbol o f t h e operator A ~ . E R n and i n t r o d u c i n g t h e f u n c t i o n

E,

Taking g ( x ) = < x , < > , 5

one c a n r e w r i t e t h e o p e r a t o r A (3.10.44)

m

: CO(M) + C

m

(M)

i n t h e following fashion:

4

( ~ ~ u ) ( =x )( 2 7 ~ ) -1~ e IRn

where i t i s u n d e r s t o o d t h a t supp u b e l o n g s t o some p a t h V

x

: V

-f

C

U i s a diffeomorphism o f V c M o n t o some open s e t U

Furthermore,

homogeneous f u n c t i o n s o f

f u n c t i o n s on

M

X

Taking g ( x ) a


o

E

-1 , < ) f u n c t i o n s o f d e g r e e m . which where a . ( x , E , S ) are homogeneous i n ( E 3 3' a r e smooth f o r x E M , E Rn\{O}, f E R + .

I f v ( O ) = ( k , m -k ,s -m +k ) s a t i s f i e s t h e c o n d i t i o n 0 0 0 0 0 0 (3.10.47)

v(O) t "(1)

=

(k.,m.-k. s.-m.+k.), 3 3 3 ' 3 3 3

j = 1,2

,...,

t h e n t h e zero c o e f f i c i e n t on t h e r i g h t hand s i d e o f ( 3 . 1 0 . 4 6 ) i s s a i d t o be t h e p r i n c i p a l symbol o f A (3.10.48)

If

and i s denoted by o

AE.O'

U , ~ , ~ ( ~ , E , C )= a O ( f , E , 5 ) .

(3.10.47)

i s well defined,

i s s a t i s f i e d i . e . t h e p r i n c i p a l symbol o f

t h e n A~ E Op Sv

(0) (M)

.

I f , moreover, one has (3.10.50)

v ( O ) t v ( l ) t...> v ( 1 ) t

.. . ,

One h a s t o u s e Theorem 3 . 1 0 . 1 i n o r d e r t o show t h a t e a c h s i n g u l a r i n t h e sense of D e f i n i t i o n 3.10.13 i s a s i n g u l a r

perturbation A

p e r t u r b a t i o n i n t h e sense of D e f i n i t i o n 3.10.14 a l s o . W e s h a l l n o t e l a b o r a t e on t h i s p o i n t and l e a v e t h e d e t a i l s t o t h e r e a d e r . Example 3 . 1 0 . 1 5 . L e t R 1 denote t h e u n i t c i r c l e ,

nl

=

{eie E C

1

1

101 5 n } .

Define t h e

operator

n+

:

c;(nl)

by t h e formula (3.10.51)

( I I + ~ )( 8 )

1 lim 211

= -

6++0

and l e t

n-

= Id

71

Jr1-e

i (e-y+iS)

-1

1

u(y)dy

-Tr

-.'II

Consider t h e family o f o p e r a t o r s E

-f

A

:

Cm(nl)

+

Cm(Ql),

3. Singular Perturbations on Smooth Manifolds without Bounda y

322 (3.10.52)

A

=

Id

+

+

-

E D ~ (- ~Il ) ,

where I d i s t h e i d e n t i t y o p e r a t o r and D

e

=

-ia/ae.

The F o u r i e r series e x p a n s i o n

e s t a b l i s h e s a n isomorphism of C m ( R ) o n t o t h e s p a c e s of r a p i d l y d e c a y i n g 1 s e q u e n c e s , {u 1 -m 0 s u c h t h a t lun\< CNRs2 N of t h e s p a c e D(Tm), m b 1 i n t e g e r ) . U s i n g t h e F o u r i e r s e r i e s e x p a n s i o n o p e r a t o r F o n e c a n r e w r i t e (3.10.52) as f o l l o w s : (3.10.54)

AEu = F - l ( l + E / n l ) F u .

We s h a l l see t h a t AE i s a s i n g u l a r p e r t u r b a t i o n i n t h e c l a s s Op S ( o r o ' l ) ( R 1 ) and even i n O p K

( O r O r l ) (R1).

L e t U C R1 b e any p a t h

U

22

( i t s u f f i c e s t o t a k e f o r i n s t a n c e (-n/2,n/2)

E

m

m

x E

x

(R 1 , : 1 on some p a t h 0 1 U , one c a n r e w r i t e t h e o p e r a t o r AE a s f o l l o w s ( w e d e n o t e 8 and y by

and i t s r o t a t e s ) and l e t u

Co

(U). With

C

1 x and y , r e s p e c t i v e l y ) :

Introducing b(x,y)

=

-1 i(x-yl (i(x-y)) [l-e IX(Y)

I

one h a s : (3.10.55)

A

u

=

u(x)

+

+ ( ~ I T ) - ~ E Dl i m

I

i b ( x , y ) [ ( x - y + i S ) - l - ( x - y - i G ) -1 ] u ( y ) d y .

6+0 R Since

m

(x-y+i6)-1

= -i/ ei(x-y+iG)c d c ,

0 one c a n r e w r i t e ( 3 . 1 0 . 5 5 ) a s f o l l o w s :

(x-y-id)

-1

=

O

i J e -m

i(x-y-iS)c dE

3.10. Difleomorphisms arid Singular Perturbations

323

where

N o t i c e t h a t t h e p r i n c i p a l symbol o f t h e s i n g u l a r p e r t u r b a t i o n A

is

a o ( x , E , C ) = 1+E151. Using ( 3 . 1 0 . 5 4 ) , one c a n d e f i n e t h e i n v e r s e o p e r a t o r A - l :

O f course, A

R1

: A-lu(f?) =

-1

i s a f a m i l y of c o n v o l u t i o n o p e r a t o r s on

(rE * u ) ( f ? ) ,

where t h e c o n v o l u t i o n k e r n e l r (8) i s g i v e n

by t h e f o r m u l a :

rE(e)

=

z

(~n)-l

(l+Elnl)-leinf?.

nE Z Introducing t h e d i s t r i b u t i o n

m

one checks t h a t r

=

(e)-E-'r(f?/E)

EqE(f?), where q ( 0 ) E C

(R 1) u n i f o r m l y

w i t h r e s p e c t t o E. One c h e c k s i n t h e s a m e way as p r e v i o u s l y f o r A p e r t u r b a t i o n i n Op

s ( o r o ' - l )(R1)

Notice t h a t A i l

E'

-1 . t h a t AE i s a singular

whose p r i n c i p a l symbol i s ( l + ~ I f l ) - ' .

allows t o solve t h e following s i n g u l a r l y perturbed

boundary v a l u e problem Av(x) = 0 , (3.10.58)

x E B1 = { X E R

a ( 1 - E--)V(X) aNe

lQl

2

, 1x1

< l},

= u(f?)8

where Nf? i s t h e inward normal a t e

if?

E

Ql,

and E > 0 .

I n d e e d , u s i n g t h e F o u r i e r series e x p a n s i o n , one f i n d s t h a t

(3.10.59)

v(x)

=

?

(211)-'

--71

where x E B ' ,

x

=

1-1x1 2

I I

1-2 x cos ( e-y)

+

I

2 ( . % i 1 u )( y ) d y ,

if? (xie

.

N o t i c e t h a t t h e problem ( 3 . 1 0 . 5 8 ) where t h e inward normal i s r e p l a c e d by t h e outward o n e , i s n o t w e l l posed f o r a l l E > 0 , s i n c e i n t h a t case 1 t h e o p e r a t o r A = F- ( l - E l n l ) F i s n o t always i n v e r t i b l e when E + 0 . I t i s -E

3. Singular Perturbations on Smooth Manifolds without Boundary

324

s t i l l a s i n g u l a r p e r t u r b a t i o n i n Op l-ElC1

i s no l o n g e r i n v e r t i b l e ,

s ( O ' O t l ) (n,)

N o w we check t h a t t h e s i n g u l a r p e r t u r b a t i o n

i n t h e c l a s s Op K ( o n o f 1 ) ( S 2

E

A

(f)

=

E S"(n,)

i . e . t h e r e i s no symbol r

such

1 w i t h ro t h e p r i n c i p a l symbol o f r .

t h a t ( l - ~ l c l )0r( x , E , ~ )

one h a s f o r e a c h f

whose p r i n c i p a l symbol

1

)

is, i n f a c t ,

(3.10.52)

i n t h e s e n s e o f D e f i n i t i o n 3.10.14.

Indeed,

Cm(al)

f(8)+Efl(e),f,(e)

so t h a t ( 3 . 1 0 . 3 8 ) h o l d s w i t h ko Now, w e check ( 3 . 1 0 . 3 9 ) .

=

= D

e (n+-n-)f(e) E cm(nl),

0 , k l = -1, k . I

First, we notice t h a t

=

--, j

> 1.

(3.10.52) f o r u

E

m

C

(R1)

can be r e w r i t t e n as f o l l o w s

f o r each g Moreover,

E

,

Cm(lR)

g' ( 8 )

# 0 on supp f

( 8 ) , g being real valued.

(3.10.61) can b e d i f f e r e n t i a t e d with r e s p e c t t o

I n d e e d , assuming t h a t g ' ( 8 ) > 0 on supp f

e

and p .

( t h e case g ' ( 8 ) < 0 c a n b e

t r e a t e d i n a c o m p l e t e l y analoguous w a y ) , i n t r o d u c e t h e new v a r i a b l e t = g ( y ) - g ( 8 ) and l e t y

=

x e ( t ) .Then

where

al(e,t) al(B,-)

E

C:(R)

=

,V 0

F u r t h e r , with $ ( t )

E

t(e-y)-'a(e,y)

E

Q1,

m

CO!R)

(g*(y))-lf(y), y =

and a l ( B , O )

,

=

xe(t),

f(8).

$(t) F 1 f o r It1 S a , $ ( t )

0 f o r It1 2 2 a ,

a b e i n g s u f f i c i e n t l y s m a l l , one c a n r e w r i t e I ( 8 , p ) as f o l l o w s

3.10. Diffeomorphisms and Singular Perturbations

325

The function

-03

Therefore, one has I1(e,p) = O ( p part (or Proposition 3.8.1).

)

for

p +

m,

as shows the integration by

Furthermore, the function

is analytic for complex t # 0 , It1 < a. Using the Cauchy theorem, one can rewrite I ( 8 , p )

2

in the following

fashion Il(e,p) = e ipg(8)f(8) lim [

6++0

J t-'$(t)eiPtdt + lt/>6

dt + nil,

+ 6+6 t-leiPt where

C i =

{t E

C

1

I

It1 = 6, Im t > 0).

Therefore, with

ri

the contour consisting of intervals [-2a,-6],

[6,2a] and the half circle C i , one has (3.10.63) I 1 ( B , p )

=

eipg(e)f(8) [ J + t-'$(t)eiptdt +nil.

r6 The integration by part yields: (3.10.64) /+ t-l (t)eit dt

-m

= O(p

),

forp

+

m,

r6 so that one gets combining (3.10.62)-(3.10.64) the asymptotic formula

(3.10.61). Differentiation with respect to p or 8 leads to the same kind of integrals. Combining (3.10.611, (3.10.62), one find -m

(3.10.65) ewipgA (feipg) = (l+Epjgl(e)j)f(e)+EDef(e)+O(p

),

so that (3.10.39) holds, as well, with s = 1, s1 = 0, s . 0 3 -1 in . Setting p = E (3.10.611, one finds

=

p + m

-m,

j > 1.

3. Singular Perturbations on Smooth Manifolds without Boundary

326 when

E +

0 , so t h a t (3.10.40)

h o l d s w i t h mo

=

0, ml = -1, m .

3

= -a,

3 > 1.

One c o u l d have u s e d t h e f a c t t h a t t h e c o n v o l u t i o n o p e r a t o r ( n i )

-1

v . p . x-'*

can be considered a s t h e s i n g u l a r p e r t u r b a t i o n ( p s e u d o d i f f e r e n t i a l o p e r s t o r of o r d e r z e r o ) w i t h symbol sgn

c,

and have a p p l i e d Theorem 3.10.1

However,wehavepreferedtogivehere

i n o r d e r t o g e t t h e expansion (3.10.65).

a n i n d e p e n d e n t argument u s i n g s o m e d i f f e r e n t The o r d e r of A "(1) =

(-m,--,-m),

technique.

i s " ( O ) = ( 0 , 0 , 1 ) , f u r t h e r m o r e , one h a s " ( l )

7"

>

1. Hence, A E

E op

P ( O ) (fill,

=

(-l,O,O),

i t s p r i n c i p a l symbol

being the function

and i t s symbol b e i n g

u

(f,c)

E/clf(B)+(f(e)+EDef(e)).

=

AE

The r e a s o n f o r which t h e e x p a n s i o n s ( 3 . 1 0 . 3 8 ) - ( 3 . 1 0 . 4 0 ) perturbation A

d e f i n e d by (3.10.52)

f o r the singular

( o r e q u i v a l e n t l y , by ( 3 . 1 0 . 5 4 ) )

c o n t a i n o n l y two t e r m s i s t h a t it d i f f e r s o n l y by a smoothing o p e r a t o r

i s a l i n e a r f u n c t i o n of

1)

( 1 + I~D

from t h e s i n g u l a r p e r t u r b a t i o n

E Op

s (o'ofl)(pi) ,

whose symbol

and a p i e c e - l i n e a r f u n c t i o n of 5 , so t h a t

6

0 f o r l a [ > 1 and 5 # 0. 1 m 1 = {z(s) E C , s E [ 0 , 1 ] } b e any c l o s e d Jordan C c u r v e i n C . One

Da(l+EISl)

5

Let

r

can c o n s i d e r t h e corresponding o p e r a t o r A

: Cm(r) + Cm(r)

g i v e n by t h e

formula (3.10.66)

where D

(A u ) ( z )

=

-1 -1 (ri) I(z-5) u ( < ) d < , z E

u (z)+ E a(z)D v.p.

r

is the tangential derivative a t z E

r.

The same c o n c l u s i o n s are v a l i d f o r t h e o p e r a t o r (3.10.661,

K(ofo'l)

a s i n g u l a r p e r t u r b a t i o n i n Op

(r)

r

A

being

w i t h t h e p r i n c i p a l symbEl

/el.

a o ( z , ~ c )= 1 + E a ( z )

I-' 5 Cis>-1 , V ( z , n ) E r x R , t h e n a Op s ( o f o ' -(lr)) w i t h t h e p r i n c i p a l symbol

F u r t h e r m o r e , i f jao(z,c) singular perturbation R

E

r ( Z , E ~ )= ( l + ~ a ( zlt1)41 ) and s u c h t h a t i t s symbol r ( z , ~ < s) a t i s f i e s t h e 0 c o n d i t i o n s o f Theorem 3.10.7 h a s t h e f o l l o w i n g p r o p e r t y : (3.10.67)

A

0

E

R

E

=

I d + EQA'),

R

0

€

A

6

=

Id

+ EQ ( 2 ) E

'

3.10. Diffeornovphisrns and Singular Perturbations

E

where Q ( 1 )

Indeed,

r

if

s ( o ' o r o () r )

and I d i s t h e i d e n t i t y . 1,o ( 3 . 1 0 . 6 7 ) i s a consequence of C o r o l l a r y 3.10.8.

Op

= R , a(z) f a

(3.10.67)

i f RE

321

=

# 0 , l+alnl # 0 , V

Op ( 1 + E l C l ) - ' .

convolution o p e r a t ors R

E

I n t h e l a t t e r c a s e R:

= E-lr(x/E)

*

Notice t h a t

R , t h e n Q ( 1 ) : 0 , j = 1.2 i n

u , V u E C;(R)

i s a f a m i l y of

,

where t h e

d i s t r i b u t i o n a l k e r n e l r ( x ) h a s t h e form (3.10.68)

r(x)

=

(2n)

-1

J (l+lg])

-1 i x g e dg.

R One c a n u s e ( 3 . 1 0 . 6 8 ) f o r c o n s t r a c t i n g an o p e r a t o r R (3.10.67)

on a smooth c l o s e d c u r v e

r

of a p i e c e of z E

r

r

with p r o p e r t y

above. I n d e e d , t a k i n g t h e l e n g t h t

between a g i v e n f i x e d p o i n t zo

a s a g l o b a l c o o r d i n a t e , so t h a t z

E r

and a g e n e r i c p o i n t

z ( t ) , one r e w r i t e s A

=

i n the

following fashion ( A v ) ( t ) = v ( t ) + E a ( z ( t ) ) D tv . p .

J (t-s)-lK(t,s)v(s)ds,

(Ti)-'

R

where v ( t )

(u

=

m

o

K(t,s)

z) ( t ) E C O ( R ) , and =

(t-s)( z ( t ) - Z ( S ) ) - l Z ' ( S ) .

I n t r o d u c inq

one d e f i n e s t h e o p e r a t o r R

(REu

o

2)

a s follows:

(t)= J R

The p r i n c i p a l symbol of A p r i n c i p a l symbol of R

K E ( t , s ) (U

o

Z)

(s)ds.

b e i n g t h e f u n c t i o n a.

b e i n g ro

=

(1+Ea( 2 )

=

( l + ~ a ( z151 )

and t h e

15 I ) - l , C o r o l l a r y 3.10.8 i m p l i e s

the property (3.10.67). I t i s l e f t t o t h e r e a d e r t o check t h a t t h e s i n g u l a r p e r t u r b a t i o n

i s t i g h t l y connected with t h e o p e r a t o r B

(3.10.66)

U

C

R2

t h e i n t e r i o r of

problem: Aw z

E

r,

=

where N

mappings.

)

0 i n U, w

d e f i n e d a s f o l l o w s . With

r

and w i t h w t h e s o l u t i o n Ef t h e D i r i c h l e t

=

u on

r,

l e t (B u)

i s t h e inward normal a t z E

(2)

r.

=

(l-Ea(z)a/aNZ)u(z), ( H i n t : u s e conformal

3. Singular Perturbations on Smooth Manifolds without Boundary

328

3 . 1 1 . An a l g e b r a o f D i f f e r e n c e O p e r a t o r s I n t h i s s e c t i o n s e v e r a l c l a s s e s o f one p a r a m e t e r f a m i l i e s o f d i f f e r e n c e o p e r a t o r s are c o n s i d e r e d .

I n some r e s p e c t t h e y are s i m i l a r t o t h e c o r r e s -

ponding c l a s s e s o f s i n g u l a r p e r t u r b a t i o n s introduced i n Sections 3.1

-

( p s e u d o d i f f e r e n t i a l operators1

3 . 3 , 3 . 9 and a r e i n t e n d e d f o r n u m e r i c a l

t r e a t m e n t o f boundary and i n i t i a l v a l u e problems f o r t h e s e o p e r a t o r s . Throughout h i s s e c t i o n h

E (O,ho]

w i l l b e a s m a l l p a r a m e t e r and U

h a one p a r a m e t e r f a m i l y of u n i f o r m meshes ( g r i d s ) w i t h meshsize h i n an open s e t

u c - mn.

W e s t a r t w i t h t h e c a s e when U = R n

and Uh

=




0 one has

< v. The j u n c t i o n ao(x,q) : I R ~x (T~\IO:)

f o r some Symbol

Fv (u), i f t h e r e e x i s t s

such t h a t uniformly on each compact

(TY \ { O j ) )

Cm(U x

1-1

Of

+

c is c a l l e d t h e p r i n c i p a l

F V ( 2 ) , h r l - ( v 1 + v 2 )a0 (x,hE) being

t h e symbol a E

homogeneous i n

(h-l,F) of degree v l + v 2 . Proposition 3 . 1 1 . 3 .

If a(x,h,hS) C F V ( u ) ther, t h e p r i n c i p a l symbol a (x,n) 0

:

U

x

(Tn \{01) 1,n

+

c is a cm-function which on every compact

and for each m u l t i - i n d i c e s a,B s a t i s f i e s t h e i n e q u a l i t i e s : (3.11.6)

ID:D:ao(x,n)

1

5 Ca , O , K I w ( q )

I

v2-lal

, v (X,ri) E

K

(T>{Oj),

K c

u

3. Singular Perturbations on Smooth Manifolds without Boundary

3 30

where t h e c o n s t a n t s c

depend onZy on t h e i r s u b s c r i p t s . v +v -In1

a,B,K

Proof. Multiplying ( 3 . 1 1 . 3 ) by h -

and letting h

-f

0, ( 3 . 1 1 . 4 )

I

yields ( 3 . 1 1 . 6 ) .

m

The class of functions a (x,n) E C ( U

0

.

(TY,n\{O})) satisfying (3.11.6)

X

is denoted by HV2 (U)

Definition 3.11.4.

A symbol a E F V ( u ) i s s a i d t o beZong t o t h e cZass G' ( u ) , i f t h e r e exist a (j)= (j) (j)) =2 , v = v (0)> v ( l ) > . . _ ,v ( j ) + v ( J )4. -m, sequence v ( v l ,v2 1 2 .(I)

for j

and f u n c t i o n s a .(x,n) E H

+ m,

(u) such t h a t f o r any given

7

0 holds:

integer N

>

(3.11.7)

(a(x,h,h5)-$6(1 ) , Ba(hS) E FY,oAIRn)

The c o r r e s p o n d i n g d i f f e r e n c e o p e r a t o r i s t h e f a m i l y o f s h i f t o p e r a t o r s 0

h hZn, Oau(x) = u ( x + h a ) . Of c o u r s e , t o t h e complex c o n j u g a t e symbol h Ba(hS) * = e x p ( - i < h a , S > ) c o r r e s p o n d s t h e i n v e r s e s h i f t o p e r a t o r h ' O-au(x) = u ( x - h a ) . h : E Op Fo (U), V a E Z n . Obviously, 0

on

=

a. 2 . L e t < . ( h , h S ) = ( i h ) - l ( e J ( h < ) - l ) and l e t S * ( h , h S ) b e i t s complex 7 c o n j u g a t e . Obviously < , , c * E F ( O ' ' ) ( R n ) and t h e c o r r e s p o n d i n g f a m i l i e s 3 7 1;o of d i f f e r e n c e o p e r a t o r s D ,., h , D j , h a:-e ( m u l t i p l i e d by - i ) f o r w a r d and back-

ward f i n i t e d i f f e r e n c e d e r i v a t l v e s , D

3th

u = ( i h ) - l ( u ( x + h e . ) - u ( x ) ) ,D* u = ( i h ) J J rh

The symbols 5

a,h

=

-1 (ih) (e"(hS)-l),

-1

(u(x)-u(x-he.)). 7

v ( l ) t

1

_ _.. Then

-(v;j)+v(J)) k t ay(x,h,n) (j) v2

H

f o r each

i h - <x

1 h jt0

(u), ( v ~ J ) + v ~ J )4)

rl

a.(x,n) 7 -m

E Tn\{O) and each f

tQ>f ( x ) )

P r o o f . The d e f i n i t i o n ( 3 . 1 1 . 2 0 ) y i e l d s : __

The T a v l o r f o r m u l a v i e l d s :

=

-

-, E c i (U) hoZds

for j

+

3 . Singular Perturbations on Smooth Manifolds without Bounda y

336

where R

M

i s t h e remainder,

(3.11.25) R (x,h,n,hS) = M

C

hM ( a ) a (x,h,rl+yhS)Sa, y E [ 0 , 1 1 .

IaI=M '!

Furthermore, s i n c e a ( x , h , h S ) E G v ,

(3.11.7) y i e l d s :

v a

V x E K C C U , V h E (O,ho],

E

Zy,

v

N 2 0

Now w e e s t i m a t e t h e r e m a i n d e r . On t h e s e t

one h a s f o r

1011

= M >

v

x E

2'

K C C U, h

f (O,hol:

Further, ( 3 . 1 1 . 2 8 ) / h M a ( a )( x , h , n + y h S )

-V

/

5 CM,K h

V

'


ih-l<xtl^l>f( x ) )

'i,( e

u n i f o r m l y on any compact s e t i n

u

x

T~

O,,,a,,

=

~

\{oI.

1,ri One p r o c e e d s as f o l l o w s . By S c h w a r t z ' s k e r n e l t h e o r e m , one c a n w r i t e

%

:

E ~ ( u -+) crn(u)a s f o i i o w s

(3.11.32) K u ( x ) h

=

,

V u E E'(U), m

where t h e k e r n e l f u n c t i o n k

h

( x , y ) E C (U

U) uniformly with r e s p e c t t o

X

h E [O,hol. Let

x

m

E Co(U),

x

: 1 on some open subset U

kh(XrE)

1

s u p p f . Then i n t r o d u c i n g

2

Fy+s k h ( x r Y ) X ( Y )

=

one can w r i t e , a s p r e v i o u s l y f o r a ( x , h , h < ) : -ih-l<x,n> K

ih-l<x,ri> f ( x ) )

=

h(e

S i n c e k h ( x , s ) i s r a p i d l y d e c r e a s i n g (uniformlywithrespecttohandx)w h e n 5 - t m , o n e g e t s conclusion ( 3 . 1 1 . 3 1 ) , u s i n g t h e sameargumentas i n t h e e s t i m a t e o f t h e remainder R M

.

Now a n a n a l o g u e o f t h e a s y m p t o t i c f o r m u l a ( 3 . 1 0 . 1 ) w i l l b e s t a t e d and proved f o r d i f f e r e n c e o p e r a t o r s . Theorem 3.11.13.

Let f E

E

Cm(U), g

0

Cm(E),

LeR a ( x , h , h D ) E Op F"

1.0

integer

N 2 0

: C:(U)

holds: -1

( 3 . 1 1 . 3 3 ) e-ih

and d g ( x ) # 0 , V x E s u p p f . + C m ( U ) . Then for each

g being real-valued

(U), a(x,h,hD)

9(X)a(x,h,hD

(f(x) e

ih-lg (x)

)

=

where (3.11.34) $ (x,Y) 9

= g(X)-g(y)-

and where t h e remainder s a t i s f i e s on any compact K c u ( w i t h a c o n s t a n t

3. Singular Perturbations on Smooth Manifolds without Boundary

338

c

depending only on i t s s u b s c r i p t s ) t h e f o l l o w i n g e s t i m a t e

NrK

(3.11.35)

( R (x,h) N

1

5 C

N,K

,

hN-"/21-vl-"2

P r o o f . A f t e r t h e change o f v a r i a b l e s ~

5

-f

V (x,h) E K

x

(O,hol.

h c , one g e t s t h e f o r m u l a

-1 ( 3 . 1 1 . 3 6 ) e-ih

g ( X ) a ( x , h , h D x )( f ( x ) e i h - l g ( x ) ) =

where

and t h e i n t e g r a t i o n on t h e r i g h t hand s i d e o f

5.

r e s p e c t t o y and a f t e r w a r d w i t h r e s p e c t t o Then f o r

151 2 R ,

(3.11.35) i s f i r s t with Let

IV g(y)[ 5

C,

Y

V y E

u.

R sufficiently large, the operator

i s w e l l d e f i n e d a n d , moreover, one h a s :

S p l i t t i n g t h e i n t e g r a l on t h e r i g h t hand s i d e of where t h e i n t e g r a t i o n w i t h r e s p e c t t o {

151

< R} and

> R},

(3.11.36)

i n t o two p a r t s

5 i s r e s p e c t i v e l y over t h e sets

u s i n g t h e o p e r a t o r ( 3 . 1 1 . 3 8 ) and t h e f o r m u l a

( 3 . 1 1 . 3 9 ) , one g e t s , a f t e r [ n / 2 ] + l p a r t i a l i n t e g r a t i o n s i n t h e second p a r t , a r e p r e s e n t a t i o n o f t h e l e f t hand s i d e o f

( 3 . 8 . 3 6 ) by a b s o l u t e l y and

uniformly convergent i n t e g r a l s . Using t h e same argument a s i n t h e p r o o f o f Theorem 3 . 1 0 . 1 ,

one g e t s t h e

c o n c l u s i o n t h a t t h e s t a t i o n a r y p h a s e method i s a p p l i c a b l e t o t h e i n t e g r a l ( 3 . 1 1 . 3 6 ) , t h e o n l y s t a t i o n a r y p o i n t of t h e p h a s e

on t h e r i g h t hand s i d e of function

+

( d e f i n e d by ( 3 . 1 1 . 3 7 ) ) b e i n g t h e p o i n t M(x) = ( x , V x g ( x ) ) ; b e s i d e s ,

t h e same argument a s p r e v i o u s l y i n t h e p r o o f of Theorem 3 . 1 0 . 1 M(x) i s r e g u l a r . Now u s i n g ( 3 . 1 0 . 2 3 ) w i t h q proof of

(3.10.11,

C o r o l l a r y 3.11.14. graded

t h e formula (3.11.33). If

4,

=

=

V x g ( x ) , one g e t s , a s i n t h e

I

a ( x , h , h D ) , with a(x,h,hS)

symbol a r i s w e l l - d e f i n e d .

shows t h a t

E G"(U),

then i t s

Indeed, t h e c o e f f i c i e n t s of t h e

asymptotic expansion (3.11.24) a r e uniquely determined.

3.11. An Algebra of Difference Operators

339

The n e x t r e s u l t i s concerned w i t h t h e c o n t i n u i t y p r o p e r t i e s of d i f f e r e n c e o p e r a t o r s as mappings between s p a c e s ff Theorem 3.11.15.

v

-v

L e t a E Fv

1t o

(Rn),

v

(s),h("h) * = ( v , , v 2 ) E R2 , and

h l < < > 2 a ( - , h , h S ) E S ( R z ) uniformly w i t h r e s p e c t t o ( h . 5 ) E (O,hol x Rn

Then t h e f a m i l y (3.11.40)

is

5 '

%,

,,(.lf:)

a(x,h,hD) : H(s+V) 2 equicontinuous, V s E R :=

-f

H

.

( s ), h

h E (O,hol,

(, V y E R ,

one f i n d s f o r any i n t e g e r N 2 0 : (3.11.45) where C

lK(h,c,q)l 5 C

N rho

Ntho

( x , h , h < ) ) ,

/a(?O

where t h e asymptotic r e 2 a t i o n (3.11.47) i s i n t e r p r e t e d in t h e fol2owing sense: f o r each i n t e g e r N > 0 one has: (3.11.48)

with hN

=

rN(x,h,h5)

def = (c(x,h,hS)-

Z / 4 < N

1

=aa

5

a(x,h,hS)D" b ( x , h , h E ) ) E X

V+U-(~,N).

We g i v e h e r e a n h e u r i s t i c argument, which c a n b e e a s i l y made r i g o u r o u s by a p p l y i n g t h e s t a t i o n a r y p h a s e method w i t h p = h - l I 5 I

as a l a r g e parameter

I n d e e d , one h a s f o r e a c h u E C m ( U ) and h s u f f i c i e n t l y s m a l l

0

3.1 1. A n Algebra of Difference Operators

34 1

The complete proof of (3.11.47), that is the justification of i3.11.48), can be done in the same way as the proof of Theorem 3.10.5. It is left to the reader (see also [Fr, 9 , l o ] ) , where the proof is given without use of the stationary phase method). A different commut'ition formula for the symbol c(x,h,h,

where the parameter r satisfies the (hyperbolicity) condition:

-t (3.11.64) 0 5 r 5 n

.

3.11. A n Algebra of Difference Operators A s a consequence o f

v

ri

349

( 3 . 1 1 . 6 4 ) , one h a s p 2 5 4 , so t h a t I f 3 ( p ) f

1

=

1,

E T ~ v, r E [ o , n t I . rl

Consider t h e f o l l o w i n g d i f f e r e n c e F o u r i e r o p e r a t o r s :

where (3.11.66)

++- ( x , y , t ; h , h S )

=

-1

<x-y,C>

I n f3 ( p (hg)1 .

f t (irh)

h

h

-

-

I t i s e a s i l y s e e n t h a t t h e f u n c t i o n s v + ( x , t ) = ( E + ( t ) u )( x ) s o l v e t h e

f o l l o w i n g Cauchy problems

where, a s p r e v i o u s l y , IR+

=

+

TZ ,

Z

+

=

{k E Z

t , T

I

k > 01.

One c h e c k s e a s i l y u s i n g t h e s t a t i o n a r y p h a s e method, t h a t a g a i n i n t h i s

case t h e s i n g u l a r s u p p o r t of t h e c o r r e s p o n d i n g k e r n e l s of t h e o p e r a t o r s

E:(t), -

i s contained i n

(3.11.68)

7

=

( a c t u a l l y , c o i n c i d e s with) t h e cone:

2n+l { ( x , y , t ) E iR+

1

Ix-yl

5 t}.

N o t i c e t h a t when r + 0 one g e t s a g a i n t h e d i f f e r e n c e F o u r i e r o p e r a t o r s h E + ( t ) from Example 3.11.27.

Eh ( t ) t h e d i f f e r e n c e F o u r i e r o p e r a t o r s d e f i n e d by ( 3 . 1 1 . 6 3 ) , f ,A (3.11.66) with

Denote by (3.11.65), (3.11.69)

p2 = p

2 A

(ri)

=

2

r

where A i s a symmetric p o s i t i v e d e f i n i t e nxn m a t r i x and r s a t i s f i e s t h e (hyperbolicity) condition: (3.11.70)

o

5 r 5

(nj (A1

I)-+.

The c o r r e s p o n d i n g f u n c t i o n s vh

f ,A

of t h e d i f f e r e n c e equation r e p l a c e d by . Our c o n j e c t u r e i s t h a t t h e s i n g u l a r s u p p o r t of t h e k e r n e l s

E+ A ( h ; x , y , t ) of - I

the operators

Eh+,A ( t ) c o i n c i d e s w i t h t h e cone ( 3 . 1 1 . 6 2 ) .

is

3 . Singular Perturbations on Smooth Manifolds without Boundary

350

If n = 1 , r = 1 , one f i n d s an e x p l i c i t formula f o r t h e k e r n e l s

E+(h;x, y, t) of the d i fferen ce Fourier opeators

-

(3.11.65), (3.11.66).

h

E -+ ( t ) d e f i n e d by ( 3 . 1 1 . 6 3 1 ,

Indeed, i n t h i s c a s e O + ( p ( q ) ) = c o s q f i l s i n q l , -

SO

t h a t a n e a s y c o m p u t a t i o n shows t h a t (3.11.71) E + ( h , x , y , t )

=

(ZTih)-'

;(e ,.

-ih

-1

(x-y-t)n+eih

-1

(x-y+t)q

)

dn

and t h e same formula ( u p t o t h e s i g n ) h o l d s f o r E - ( h , x , y , t ) . Of c o u r s e , i n t h i s case t h e c o r r e s p o n d i n g d i f f e r e n c e F o u r i e r o p e r a t o r c a n b e r e w r i t t e n a s a d i s c r e t e c o n v o l u t i o n o p e r a t o r i n t h e form:

w i t h E + ( h , x , y , t ) g i v e n by ( 3 . 1 1 . 7 1 ) . AS a consequence o f

h

-t

( 3 . 1 1 . 7 1 ) , E + ( h , x , y , t ) converges i n D ' ( I R )

when

0 t o the distribution

t h e l a t t e r b e i n g t h e s o l u t i o n of t h e Cauchy problem

I

= Op(lSI) i s t h e s i n g u l a r p e r t u r b a t i o n ( i n f a c t , t h e where o f c o u r s e ID p s e u d o - d i f f e r e n t i a l operator) w i t h t h e symbol 151 E S ( 0 , l r O ) ( R ) .

W e a r e g o i n g t o r e s t r i c t t h e c l a s s of p h a s e f u n c t i o n s and s h a l l

c o n s i d e r , from now on t h e p h a s e f u n c t i o n s # ( x , y , h , h c ) which s a t i s f y t h e following C o n d i t i o n 3.11.29.

There e x i s t s a c o n s t a n t 6 ( h , n ) E IR+

x Tn

> 0

w i t h I < ( h , q )I

such t h a t for any ( x , y ) E u1 2 6

x

u2 and any

holds:

-1 (3.11.72) $ ( x , y , h , q ) = h $ ( x , y , l , n ) .

For such phase f u n c t i o n s t h e c o n d i t i o n (3.11.57) w i l l be s t a t e d a s foZlows:

3.11. An Algebra of Difference Operators

35 1

On t h e o t h e r hand, we s h a l l e x t e n d t h e c l a s s o f t h e o p e r a t o r s c o n s i d e r e d i n t h e f o l l o w i n g way. We s h a l l d e n o t e by

Kh t h e c l a s s o f o p e r a t o r s o f t h e form

where t h e f a m i l y o f k e r n e l f u n c t i o n s h s e t in Cm(U1

x

U2)

E

for h

: Cm(U

4

0

2

K ( h , x , y ) b e l o n g s t o a bounded

[O,hol w i t h a g i v e n h

We s h a l l u s e t h e n o t a t i o n l i n e a r mappings Ah

+

F"

m l,o

+ C

0' f o r t h e f a m i l i e s of continuous

(4)

(U1),

v

h C ( 0 , h 1, which c a n b e 0 4 satisfying

r e p r e s e n t e d i n t h e form ( 3 . 1 1 . 5 8 ) w i t h t h e p h a s e f u n c t i o n C o n d i t i o n 3.11.29,

i.e.

a ( x , y , h , h S ) E Fy,,CU,

x

(3.11.72),

(3.11.731, and with an amplitude

U2) (mod K h ) .

h One can a s s o c i a t e w i t h a d i f f e r e n c e F o u r i e r o p e r a t o r A + E -V

its distribution kernel h

FY,O(d)

'A4 ( h , x , y ) , where

V ( x , y , h ) E U1

x

U2

(O,hol.

X

W e are g o i n g t o l o c a l i z e t h e s i n g u l a r i t i e s o f A ( h , x , y ) .

d

F i r s t , introduce t h e set

where t h e u p p e r b a r , a s u s u a l d e n o t e s t h e c l o s u r e o f t h e c o r r e s p o n d i n g s e t . I n t h e same way, a s p r e v i o u s l y f o r Theorem 3 . 9 . 7 ,

one p r o v e s t h e

following Theorem 3.11.30.

The s i n g u l a r support o f t h e f a m i l y h

+

A (h,x,y)

0

of t h e d i s t r i b u t i o n a l

k e r n e l s d e f i n e d by ( 3 . 1 1 . 7 5 ) i s contained i n t h e s e t (3.11.76), (3.11.78)

(3.11.77): s i n g supp A ( h , x , y )

d

5

Q4.

Q

4

d e f i n e d by

3. Singular Perturbations on Smooth Manifolds without Boundary

352

Now t h e c l a s s of H y p e r b o l i c D i f f e r e n c e O p e r a t o r s w i l l b e i n t r o d u c e d and t h e c o r r e s p o n d i n g F o u r i e r D i f f e r e n c e O p e r a t o r s w i l l b e c o n s i d e r e d f o r Hyperbolic D i f f e r e n c e Operators with c o n s t a n t c o e f f i c i e n t s . One d i s t i n g u i s h e d ( t i m e - l i k e ) v a r i a b l e i s g o i n g t o p l a y a s p e c i a l n+l - R n X R and c o n s i d e r g r i d s x,t x + = (hZn) x ( T Z ) w i t h two mesh-sizes h and T = r h , where r > 0 i s a

r o l e . Hence, w e s h a l l d e n o t e W IRE::

g i v e n c o n s t a n t . The d u a l v a r i a b l e s w i l l b e d e n o t e d by ( S , E ; ) 0

A s usual, 0

E

Rn x R

5

50

and 0-1 a r e , r e s p e c t i v e l y , t h e f o r w a r d and backward s h i f t

o p e r a t o r s on t h e g r i d R

while 8

= TZ,

t , T

symbols. F u r t h e r , D

=

(iT)-'

t t T

-1

( T S ~ ) ,B0 ( T S0 )

*o ( e T - l ) , Dt,-,

-1

= (iT)

stand f o r t h e i r are

(I-@-')

( m u l t i p l i e d by -i) forward and backward d i f f e r e n c e d e r i v a t i v e s and

A(T,TS

*

-

(T,?< ) d e n o t e t h e i r r e s p e c t i v e symbols. A s u s u a l , 0 ORn x n+l = with R = { t E R t > 0 ) and x t,+ t ,+ n+ 1 n+l w i t h R+ t h e c l o s u r e o f R ),

*;+I=

I

%,-,,+

~

.

D e f i n i t i o n 3.11.31. n+ 1

A symbol a ( x , t , h , h E , ~ < ~E ) Fv(P(+

)

,

is s a i d t o be s t r i c t l y

v = (vl,v2)

hyperbolic i f t h e following conditions are s a t i s f i e d :

i s an i n t e g e r ;

1". v2 > 0 2'.

a can be r e p r e s e n t e d i n t h e form:

(3.11.79) a ( x , t , h , h S , T S O ) = h

'1

*

-k

( ~ E ; ~ ) p ( x , t , h , X , C) ,,
0, V (x,h, 0 , V x E IR+

3

,...,w

V w ' = (wl

),

l+ir w =exp(ir 5 k k k k

,

# 1.

Consider several specific examples. With u

=

exp(i 0,

co > 0 being the density and the

compressibility characteristic of the medium, respectively; here, as previously a = a/at, ax = a/ax. 2 t Let %,+ be the greed in =

z:

(h,T),

T =

(x,t) E R

X

z+1

with the meshsize

rh, r > 0 being a given constant.

Consider the following finite difference approximation of the system 2 above on the greed B,,+ :

3.11. An Algebra of Difference Operators

363

where, as previously, 0 is the shift operator in the greed h I$, = {x E IR I x =kh, k E Z } , (0 v) (x) = v(x+h), and iDt,,, iDx,h are h forward finite difference derivatives in t and x, respectively, on the 2

greed ph,+ * Multiplying the second equation by ( p c

-1

0 0

,

adding it to and after-

wards subtracting it from the first one, one gets the following finite 2 + -1 difference equations on If$,+for the Riemann'sinvariants u- = u+(poco) p of the differential system above:

where we have denoted: L2(rc0,Oh,Dt,, ,Dx,h) = if ( 4 ) (l?rco)+(lfrco)@h)Dt,TfcoDx,h}, the latter being hyperbolic finite difference approximations of the operators at+coax. If the initial conditions for u and p are highly oscillatory functions 0)

of the form f (x)exp(ih-l$o(x)) with $o E C ( I R ) , fo(x) = p 0(x) for

o

m

f (x) = u (x) in CO(R 1 , then the initial conditions for the Riemann's 0 O + (x)). invariants u- will be of the same form u:(x)exp(ih-'$ 0 Seeking the asymptotic expansion of u'(xlt)

one gets for $'(x,t)

as h

+

0 in the form

the following Hamilton-Jacobi equations:

+

with the initial conditions $-(x,O) = $ (x). 0

The transport equations for u'(x,t) k as above.

are derived in the same way

Finally, for the multidimensional wave equation U

x0x0

-

16k,j(n

akjUx x k j

with positive definite matrix A approximation

=

= o

I [ akj 1 I

and for its finite difference

3. Singular Perturbations on Smooth Manifolds without Boundary

364

1 akjDxk,hDx,,h)uh=O,h0 =rh,h1=...=h =h, h D* x ,h (DxO' 0 0 0 lO

symbol, where a . s a t i s f i e s on each compact K c c u t h e i n e q u a l i t y 3 (j) .(I) la4(x,p,n) J

and where v

= v")

I

cK l w ( r i )

1'"



3

,

>v

g i v e n p E R+ , P = h/E ho Zds :

3.12. E l l i p t i c Singular P e r t u r b a t i o n s In t h i s s e c t i o n a p r i o r i e s t i m a t e s a r e e s t a b l i s h e d and parametrix c o n s t r u c t i o n s a r e c a r r i e d o u t f o r E l l i p t i c p s e u d o d i f f e r e n t i a l and d i f f e r e n c e s i n g u l a r perturbations. We s t a r t with a g l o b a l v e r s i o n of t h e d e f i n i t i o n of symbols S v ( U ) introduced i n Section 3 . 3 . D e f i n i t i o n 3.12.1. a ( x , E , < ) i s s a i d t o be i n t h e symbol class L'(R") following conditions are s a t i s f i e d

A function

i f the

( i )a E s'(R") ( i i )t h e r e

x

-f

e x i s t s a symbo2 a,(E,c) E s'(R") such t h a t t h e @ n e t i o n : = a ( x , ~ , c ) - a - ( ~ , cbeZongs ) t o s(R:), i . e . t h e folZowing

a'(x,c,c)

i n e q u a l i t i e s hold:

3.12. Elliptic Singular Perturbations

367

where C > 0 are some constantswhichmay depend on t h e i r s u b s c r i p t s . With a (x,E,S) the corresponding symbol in the representation (3.3.2) (with 0 a . and r satisfying (3.3.3), (3.3.4)),we shall denote (with some abuse of notation) also by a (x.E.~)the homogeneous extension of a . (as a function 0 of ( E -, C~) ) to (x,E,S) E Rn X IR+ x Rn , which is also called the principal symbol of a. Definition 3.12.2.

A symbol a E L'(IR~)

i s s a i d t o be e l l i p t i c of order v E

m3

i f i t s principal

symbol a0(x,E, v 3 , V

(E, 0

+ 0,

is some constant. The

same argument as previously in the case M = R n , shows that the ellipticity condition is equivalent with the a priori estimate (3.12.19) where the norms are in corresponding spaces H

( s ), E

(M).

One can also show (but it will not be done here), that if a(x,&,D) E Op S"(M)

is elliptic and M is compact, then the kernel of m

a(x,E,D)

:

H

V 6 E (O,Eol;

( s ) , )E'(

+ H ( s - v ) ,E

(M) belongs to a bounded set in C (M),

furthermore, the range of a(x,~,D) is the orthogonalcomplement

of thekernel ofthe adjoint a(x,~,D)*. Therefore, theindex dim coker a of an elliptic singular perturbation a(x,&,D) H

(s-v)

fact, 0

( M ) does not depend on s

X(E) :

H

=dim ker a ( s ),E

(M)

*

E 1R3. It will be shown later, that, in

,E

X(E)

5

X ( 0 ) if a(x,s,D) has the reduced pseudodifferential operator 0

) the index of a (x,D). a (x,D), where ~ ( 0 is

3.12. Elliptic Singular Perturbations

38 1

Example 3.12.24. The singular perturbation (3.10.52) on the unit circle

a1

is elliptic of

order v = (0,0,1), since its principal symbol a (x,E,S) = l+EIE;I 0

-

<ES>.

As a consequence of (3.10.54) kernel and cokernel of this singular

perturbation are both trivial and its index

X(E)

=

0 , t/

E

2 0.

The next example is concerned with Stokes' singular perturbation (3.2.43). Example 3.12.25. Consider the singular perturbation S (E,D)whose symbol is the following

e

nxn matrix s

E

(E,S):

S ~ ( E , S )= where w = 5 / 1 5 / , w

E

2ie+E21 = 0 is an eigenvector of s (E,S) e 2 2iE 2 = E (e + 15 ) , while w is an eigenvector with zero

I

with the eigenvalue u eigenvalue. Thus,

is not elliptic. However, considered on the

se(E,S)

orthogonal complement of

w

in Cn it becomes elliptic of order (-2,0,-2).

We are going to introduce a more general ellipticity concept for systems of singular perturbations in order to include in the elliptic theory Stokes' singular perturbation (3.2.44). Introduce again the notation

and introduce

Definition 3.12.26.

A matrix symbol a(x,E,S) = 1 lakj(x,E,S) 1 1 with a E LvkJ(mn) is said to kj be elliptic in the sense of Douglis-Nirenberg, if there exist vectors vk E IR 3 , LI; E n 3 , 1 5 k 5 p, such that the symbol ;(X,E,~), N

(3.12.40) a(x,E,S) = q,,(E,~)a(x,E,S)q,(E,~)

is elliptic of some order v in the sense of (3.12.2) where the norm in Hom(CP;CP) .

I. 1

stands f o r

3. Singular Perturbations on Smooth Manifolds without Boundary

382

A matrix singular perturbation a(x,e,~)is said to be elliptic in the sense of Douglis-Nirenberg, if its symbol a(x,~,S)is elliptic i n the same sense. The following result is proved by repeating word for word the proof of Theorem 3.12.12.

Theorem 3.12.27.

Let a(x,s,D) be e2Ziptic in the sense of Douglis-Nirenberg and let 1-1k' 1 5 k 5 p and v be the corresponding vectors in Definition 3.12.26. Then uniformly with respect to (3.12.41) 119

-P

where s' E R

3

(E,D)u/

1 (s)

E

,E

E (O,E I 0

-j

one has:

I

Iqu,(E,D)a(x,E,D)ul ( s - v )

is any order s.t. s' < s-uk, 1

,E

+

I lul I

(s'), €

2 k 5 p.

Example 3.12.28. Consider Stokes' singular perturbation the symbol a(A,S):

where, as previously,

ST

is the row-vector transpose of 5 f lRn

.

With any u = ( u t,O)T such that <S,u'> = 0, one finds that 2 T a(A,S)u = ( A + l < l ) u , so that on the orthogonal complement of Span{(S,O) ,en+l] 2 n+ 1 a(A,S) reduces to ( A + / < ] )I in C (n-l) . Furthermore, an easy (n-l) computation shows that the other two eigenvalues of a(A,S) are, respectively,

Hence, if u - = (S,p-) is an eigenvector associated with the eigenvalue restricted to the orthogonal complement of u- in CnTi has 2 (X+lSl 2), while X/El2 when its norm of order (X+lS ) , since x+

X-,

then a ( A , S )

I

(A+l s'+s'

1 2 '

>

C

$

is some constant, which

0, given that s ' < s ,

3 . Singular Perturbations on Smooth Manifolds without Bouliday

402

Now, letting

0, h

E -+

-f

0, p = h/E being a given positive constant,

(3.12.106) yields:

the constant

C

here being the same, as on the right-hand side of (3.12.99).

Thus, (3.12.107) yields (3.12.72) with q(x)

S

C-l > 0, i.e. a is

globally elliptic. Examples 3.12.57. 1”. Let q(x)

E

m

C

(U), U _C R1 and let q(x) > 0, V x E U.

The symbol b(x,h/E,hE),

is elliptic of order

I ,

(0,0,1) and it is globally elliptic of the same

order if q(x) 2 qo

0, V x E

=

(in the case U = R1 , in addition, it is 1

assumed that q(x) = q_+q’(x) with q‘(x) E S(R

))

.

The corresponding difference singular perturbation b(x,h/E,hD) is an approximation by finite differences of the elliptic singular perturbation (l+iEq(x)Dx) E op s”(u), w = (o,o,I). Denote by b*(x,h/E,hD) the formal adjoint of b(x,h/e,hD).

The difference

singular perturbation (3.12.109) a(x,E,h,hD) = b*(x,h/E,hD)

0

b(x,h/E,hD)

is elliptic of order u = ( 0 , 0 , 2 ) , a E Op F W ( u ) , its principal symbol being ao(x,P,n)

=

l2

(b(x,p,q)

with b(x,p,n) given by (3.12.108); furthermore, the

difference singular perturbation (3.12.109) is a three point approximation by finite differences of the formally self-adjoint differential singular perturbation a(x,E,D) = (1-iED q(x)) (l+ieD q(x)) E Op S ( o ’ 0 ’ 2 ) ( U ) .

Besides,

if U is a finite interval, say U = (O,l), then the equations a(x,s,D)u = 0, 1 x E U and a(x,E,h,hD)u = 0, x E Uh = &$,fl U, have the same asymptotic x ’ E a U , in the following sense: solutions uo = exp(-jx-x’//(q(x’)~)), supla(x,~,D)u x o XEU with some constant C

I

S CE,

sup[a(x,E,h,D)u,-l6 C(E+h) XEU

0 which depends only on q(x).

Notice that a(x,y,~,h,hS)= b(y,h/E,hS)* b(x,h/E,hE) with b(y,p,n)* the complex conjugate of b ( y , p , n )

is the amplitude of the difference

sinqular perturbation (3.12.109) (see Remark 3.11.20).

3.12. Elliptic Singular Perturbations

403

The difference singular perturbation b+(h/E,hD) with the symbol -1 = l+p (exp(in)-l), b+ E F(of081)(~) , is a non-elliptic

b+(p,n)

approximation by finite differences of the singular perturbation l+i-ED. Indeed, b+(p,n) being also the principal symbol of b+(h/E,hD), one has: b+(2,1~)= 0 . Notice that b-(h/E,hD) = l+p-'(l-CI-') operator to the left) whose symbol is b- ( p , r l )

=

(with 0-1 the shift l+p-' (l-exp(-iq)) is an

elliptic approximation of l+iED. Consider a family of difference singular perturbations: t [0,1] 3 t + a (~,h,hD)defined by their symbols

2'.

(3.12.110) at (~,h,h5)= -~l 0.

The latter requirement leads to the following recurrence formulae for the coefficients a;'(x,o,q)

in (3.12.113):

3 . Singular Perturbations on Smooth Manifolds without Boundary

406

-1 where a-l = 0.

Now, choosing a cut-off function $(t) and a sequence 6 . 7

J.

0 as in the

proof of Theorem 3.6.1, we can write the following formula for the symbol of a parametrix for a(x,~,h,hD): (3.12.116) a-'(x,c,h,hS)

C

=

-1 k -1 + ( 6 k ~ )E ak (x,h/~,hS),

k>O with a ; '

defined by recurrence formulae (3.12.115).

It is left to the reader to check that for any integer N > 0 one has:

-1 (3.12.117) a(x,~,h,hD)o a (x,E,h,hD) - Id = E ~ R ~ ' ~ N '

(i.e., if A(x) = Am+A'(x) with A'(x) E S(lRn)) and is globally elliptic (of -1

order v = (0,0,2)),then a

(x,~,h,hD) constructed above is, in fact, a m

quasi-inverse operator for a with accuracy O ( E

),

that is (3.12.17) holds

for any integer N > 0 with R E P h which is uniformly bounded with respect to N E ( 0 .~ ~ 1 h ,E (O,hol fromH(s),E,h(-~ with V N > 0):

3.12. Elliptic Singular Perturbations

a(x,E,h,hS)

-1 -1 a . (x,~,h/~,hS) = l+r(x,~,h,hS) - a o (x,~,p,hS)

*

so that modulo an operator in Op F(-"'o'-'")

( U ) one has:

1 r0

(3.12.118) Op a

o

407

Op a ; '

=

Id + Op r

-1 Op . a

o

.

Since ~~l~ := op r op a;1 E op F(-lr0r-l) ( U ) , the operator (1+KEfh)-l 1 ,o exists and can be represented as a convergent Neumann series, so that E Op

(I+K'lh)-'

Fo

1 to

Let l+k-'(x,e,h,hS)

(U).

be the symbol of (I+KEfh)-l.

Since (3.12.119) Op a

-1

o

Op a .

1 Op(l+k- ) = Id

o

and there is in this case a one to one correspondence between the symbols and the operators in (3.12.119) (modulo a symbol of order ( - - , O , - - )

),

one

gets the conclusion that the singular difference perturbation with the -1 (1+k (x,e,h,hS))is a parametrix and at the same symbol a-'(x,E,h/E,hS)

-

0

time a quasi-inverse with accuracy

N

O(E ),

V N

> 0, for the elliptic

singular perturbation a(x,E,h,hD) above. Finally, we outline the extension of the elliptic theory Of singular perturbations in spaces ff

( s ) .E

to elliptic differential singularperturbations

in analoguous spaces with L -structure, 1 < p


.-lll~ll

, so

and denote by I s.

with volume of each cube

1

over every cube of

L1(R 1

this mesh becomes

t

I

that the mean value of ju(x)

c

S.

Further, divide each cube ofthe mesh into 2

. . those of them, over which

,Il2,.

Obviously, one has:

n

equal Cuke

1.1

the mean value of

is

3.12. Elliptic Singular Perturbations

415

1 lu(x) /dx < 2nsui(Ilk).

(3.12.147) sv(Ilk) 5

Ilk Indeed, if Ilk is obtained as a result of the subdivision of a cube I' from the mesh above, then, according to the construction, one has: SU(Ilk)

5

J Ilk

lU(x) Idx 2 1 lu(x) [ d x <su(1') I'

=

Znsu(Ilk).

Define

wherexI V x

(XI is the characteristic function of Ilk, i.e. xlk(x) E 1, lk

E Ilk and

xI

(x) 0 elsewhere. lk Next, making again the same subdivision of the cubes from the mesh

above, which are not among I ll,I12,..., one singles out those new cubes over which the mean value of lu(x) I is L s and extends the

..

I Z 1 ,I22,.

definition by (3.12.148) of w ~ ~ ( xto) these cubes. Continuation of this process leads to theconstruction of disjoint cubes I . and corresponding 3k

functions w . ~ ( x ) ;one rearranges them, for convenience, as a sequepce, 3

denoted again by Ik, wk(x), and defines v(x) by setting v(x) V x f! Q

=

=

u(x),

U I . It is clear, that (3.12.146) holds and (i) is valid, as k

k

well. For showing (ii), first notice that

1 (Iv(x)I+lwk(x) 1 )dx

5 31 /u(x)ldx.

I

k

k '

Indeed, the last inequality is an immediate consequence of (3.12.148) and the definition of v(x). Since the cubes Ik, k = 1,2,..., are disjoint, supp wk

J Rn\Q

Iv(x) [dx =

In R ' 9

one gets immediately (ii):

lu(x) ldx,

5 Ik

and

3 . Singular Perturbations on Smooth Manifolds without Bounda y

416

Further, (iii) follows from (3.12.147) if x E Q. On the other hand, if x $! Q, then there are arbitrary small cubes containing x, over which the mean value of lu(x) 1 is
1 and (ii) from Lemma 3.12.71:

3.12. Elliptic Singular Perturbations

417

This last inequality yields:

1

(3.12.150) u{x E Rn\Q*

I

1 w(x) t (4)s) 5 6Cs- /uII

Hence, it follows from (3.12.149), (3.12.150) that w(x) < ( f ) sexcept on a set of measure at most

1

The assumption k E

P

w th some p E (1,m) and (ii), (iii) from Lemma

3.12.71 yield:

Therefore, one gets,using the last inequality:

N

Since the measure of the set, where w(x) 2 s/2 is bounded by constant (3.12.151). one findsrusing the last inequality, that UIX E Rn

I

jc;(x)

I

> s

and that is precisely (3.12.145) with a = s . with C3 = (2C2)p+6C+u(I~)/~(10),

I Next, using Theorem 3.12.70

an argument of Marcinkiewicz (see [ Z , l ] )

and Corollary 3.12.68, the following statement will be proved: Theorem 3.12.72.

f o r a l l p E ( l , m ) , o r e l s e f o r no such p. P for some p E (1,m). It will be shown that k E fir,

L e t k E K. Then e i t h e r k E Proof. Let k E __

v

k

P

r E (1,pl. For u E L (Rn) define v

E Lr(Rn) , ws E L (Rn) such that u

I

u(x) = ws(x) if /u(x) > s and u(x) = vs(x) if lu(x)

I

u +v s s' 6 s, where s > 0 is a =

given fixed number. Obviously, w

has compact support, ws E L1(Rn) ,

I lwsl

5

I1uI ILr(Rn)

3. Singular Perturbations on Smooth Manifolds without Bounda y

418

and, moreover, one has:

E Lp(Rn), j lvs 1 ]

By a similar argument one finds:v

5

ll~ll

and moreover,

For a given Lebesque-measurable function f ( x ) denote by m(t) the measure of the set where I f ( x )

I

>

t. Then for each f E L (Rn) with q E 1 one has: 9

m

m

(3.12.152)

1 If ( x ) Iqdx

=

Rn

-1 tqdm(t) 0

=

q

tq-'m(t)dt. 0

Further, one has for each f E L (Rn) : 9

so that

(3.12.153) m(t) = u{x E Rn

I

I

/f(x) > t} S I ' t

If1

1'

L q The last inequality will be used for estimating Introduce

!Jaw = (3.12.154) uls(t)

VIX =

E Rn 1

V{x E

JZ(X) 1 > Rn 1 ] G s ( X ) 1

N

N

where, as previously, u = k*u, w

t), >

t],

N

=

k*ws, v

=

k*vc. I

Applying Theorem 3.12.70 to k E

Since by assumption k E M

bSlI

P'

p, ws E L 1 ( R n ) , one finds:

one has with some constant C

0:

, so that using (3.12.153) one finds for L (mn) P ! ~ ~ ~ defined (t) by (3.12.154), the following inequality: 5 CI

L (R")

P

3.12. Elliptic Singiilar Perturbations

419

N

Now applying (3.12.152) with f = u , one finds: m

Further, since Iw (x)[ 5 t, Iys(x) 1 5 t implies lu(x) 1 5 2t. one has: N

N

p0(2t) 9 ~ ~ ~ ( t ) + p ~ so ~ (that t ) ,the following inequality holds: m

(3.12.156) 1

Iy(x)lrdx

Eln

=

2rr 1 tr-lp0(2t)dt 9 0 m

5 r2r(7 tr-'uls(t)dt+ 1 tr-1u2s(t)dt). 0 0

These two integrals on the right-hand side of the last inequality will be estimated by setting s = t in the definition of w

and v

Using again (3.12.152) and (3.12.155). one finds:

By definition of w (x t m

1 0 m

p t x E IRn

.

3. Singular Perturbations on Smooth Manifolds without Boundary

420

Combining (3.12.156)-(3.12.158), one gets finally, that

with some constant C > 0 , and that proves k E Now, Corollary 3.12.68 yields: k E

fir, V

Mr,

V r E (l,p].

r E (1,m).

I

Now, one needs only two more lemmas in order to be able to prove Theorem 3.12.62. Lemma 3.12.73.

There e x i s t s a f u n c t i o n

+ E Cm(Rn) , supp $ c 15 E 0

Rn 1 0) 6 IS1 < 21,

such t h a t

c

(3.12.159)

+(2-k0 E 1 ,

v

5 E Rn\{Ol.

kEZ

Proof of Lemma 3.12.73. Let 0 2 0 be a function in Cm(R+) , supp 0 0 O(r) # 0, V r E [1//2,/2]. Defining

+ ( S ) = @ ( / 5 /( )

z

r

C

Q(2-klcl))-1,

v

I

f < r < 2), and let

5 E Rn\t01,

+(O)

=

0,

kEZ

it is easily seen, that + ( 5 ) satisfies (3.12.159).

a

Lemma 3.12.74.

L e t T be c d i s t r i b u t i o n i n S'(xn) and l e t p E (1,m). Then t h e following t w o c o n d i t i o n s a r e equivaZent:

(3.12.160)

and (3.12.161)

3.12. Elliptic Singular Perturbations where

c is a constant and, a s u s u a l ,

p-'+p'-'

=

1.

proof of Lemma 3.12.74. HGlder's inequality yields:

Proof of Theorem 3.12.62, The proof consists of three steps. First it will be shown that an approximation of f(6) with compact support is in M2. Next, it will be proved that this approximation belongs to the class K defined above by (3.12.142). Finally, one shows that it is allowed to take the limit of these approximations. Step 1. Let $ ( 5 ) be the function constructed in Lemma 3.12.73 and let fk(F)

=

f(5)$(2-kg). Using Leibnltz forxula, one can estkate the

derivatives of f ( 5 ) by those of f(5) : k

Hence, (3.12.1211, (3.12.162) yield:

0 5 j 5 2, are some constants which depend only on n. where C n,j'

Introducing gk(x) = (Fgixfk)(x), Parceval's identity along with (3.12.163) yields:

42 1

3. Singular Perturbations on Smooth Manifolds without Boundary

422

(3.12.164)

1 (1+22k 1x1 2) X Igk(x) I 2dx

5

Rn

wiiere we have denoted X = [n/2]+1 and where C n,j' which depend only on n .

j

=

3,4, are some constants,

Using Cauchy-Schwarz' inequality and (3.12.164), one finds: (3.12.165)

lgk(x) Idx 5

mn

where, as previously, Since fk(5)

=

X = [n/21+1 and C depends only on n. n,5 (F g ( 5 1 , one can estimate f ( 5 ) in the following x+S k k

fashion:

almost everywhere on IRn. Introduce

Since at each point 5 E Rn at most two functions f ( 5 ) do not vanish, k one gets, using 3.12.1661,the following estimate almost everywhere on Rn : IFN(
2t

for Iyi 5 t.

First, we esti-nate such ar. integral for each gk(x) in the definition

3.12. Elliptic Singular Perturbations

423

of GN(x). Dropping the additive term 1 between the paranthesis on the left-hand side of (3.12.164), and using again Cauchy-Schwarz' inequality, one finds:

with the same X = [n/2]+1 and some constant

which depends only on n. n.7 Further, using (3.12.167), one gets for IyI < t the estimate: C

k Since n/2-X = n/2-[n/2]-1 5 -f, (3.12.168) can be used when 2 t t 1. k If 2 t < 1 , then one uses Cauchy-Schwarz' inequality, Parceval's identity and (3.12.162), respectively, in order to estimate the integrals on the left-hand side of (3.12.168). One has, using Cauchy-Schwarz' inequality with X = [n/2]+1:

Further, Parceval's identity and (3.12.162) yield:

Hence, using (3.12.169) and the last inequality, one gets:

3. Singular Perturbations on Smooth Manifolds without Bounda y

424

N

where t h e c o n s t a n t C

depends o n l y on n.

Combining ( 3 . 1 2 . 1 6 8 ) ,

( 3 . 1 2 . 1 7 0 ) , one f i n d s f o r G

I:

=

Ikl6N

I

/ G (x-y)-GN(x) IdxSCnB

)x/>2t

= {y

E Rn

1

min{t2k,(t2k)n'2-X1

C

= C(l)B.

-m 0 , p E ( 1 , m ) .

Then t h e r e e x i s t s a c o n s t a n t

> 0 such t h a t holds:

Proof. Without restriction of generality, one can assume that -

s1 = 0.

Further, as a consequence of (3.12.177) and Corollary 3.12.63,

fl(c)

-S

=



is a Fourier-multiplier in L (IR")

using again the Leibnitz f2(5) = s2(1+151s2)-1

P

, t/

s2 2 0 . Furthermore,

formula, one gets immediately that and f3( 0:

On the other hand, one finds:

3 . Singular Perturbations on Smooth Manifolds without Boundary

428

and

Lemma 3.12.81.

Let

c

s = (sl,s2,s3)E R

> 0

3

,

s 3 > 0 , p E (I,-).

-S

Proof. (E

to

One checks easily that <ES>

15 I <EC>-') E

Then t h e r e e x i s t s a c o n s t a n t

such t h a t h o l d s :

E (O,E

3

, <E'~ (1+( E 15 I ) s 3 )

-'

and

s3 are Fourier-multipliers in L ( R n ) uniformly with respect 0

1

Lemma 3.12.80.

P

and uses the same argument as previously in the proof of

I

For s = (sl,s2,s3)E R

X

R+

X

R + , p E ( 1 , ~ )and u E S ( R n ) introduce

the family of norms

Corollary. L e t s E R x R+ x R + , p E (I,-). Then t h e f a m i l i e s of norms

-

and I I I I I I ( s ) there e x i s t s a constant c >

a r e e q u i v a l e n t uniformly w i t h r e s p e c t t o

o

E

I I .I 1 ( S ) , P , E E (O,E

0

I i.e.

such t h a t

Indeed, using the definition of

1 1 . I 1 ( s ) .P,E

and the previous two

3.72. Elliptic Singular Perturbations

429

lemmas, one finds:

Therefore, in order to prove (3.12.184). one has only to show, that with some constant C > 0 one has:

is defined by (3.12.183). For that purpose, it suffices to show that f ( 5 ) = 15

I s3 (I+ 15 I

s +s

2

3 -1 )

with V s2 2 0, s 3 2 0 , is a Fourier-multiplier in L (Rn) . One checks easily, using the Leibniz

formula, that the condition (3.12.123) for

fs(c) is satisfied, so that Corollary 3.12.63 applies. This gives:

1 Corollary 3.12.83.

L e t s E R+

z+, p E (I,-). Then t h e r e e x i s t s a c o n s t a n t C

x Z+ x

> 0

such

t h a t one has: -S

(3.12.186) C-lI lul s I I E

+

I

5

E

( s ) ,P,E s +s

3D 2 3 x , UI

I

1

(1

/u/

I

s2

I

+ 1 (/IDx,UI L (IR*') l < = j s n 2 L

P

P

I

) )

I

5 CI lul (.-)

(w")

+

,P,EI

L (Rn)

P

v

E

E

(o,Eo],

v

U

E H

( s ) rP,E

(Rn) .

Indeed, one uses Corollary 3.12.82 and Corollary 3.12.63 for showing (3.12.186). For doing that, one has to show that for each function f

.(F)

or1

=



0 the

satisfies (3.12.123). This can be easily seen by

3 . Singular Perturbations on Smooth Manifolds without Boundary

430

induction argument. Therefore, one has for any positive integer a:

On the other hand, with any positive even integer a holds:

and for a positive odd integer one has:

using (3.12.187)-(3.12.189), one gets immediately (3.12.186).

I

Remark 3.12.84. For s = (sl,s2,s3)E R families of norms 1 1 1 . 1 are equivalent with to

E

x

lR+

x

R+

,

s

,

1 I I ( s ) ,E,P without

1 1 . I 1 ( s ) ,P.E

E Z+, j = 2,3, one can define

using the Fourier-transform, which

introduced above, uniformly with respect

I

E (O,Eol-

-1 Let u E (0,l) and let 2n(n+a) < p
0 and some u E (O,l), then one

has the following equivalency (see [Ste, 1,2])

Now, let u

=

(0,u2,u3) with u , E 3

(O,l), j

=

2,3. Then holds:

-

\-.

I

and it is quite obvious, how the last equivalency extends to the case when s = (sl,s2,s3),s1 E I R , s . = m.+u. j = 2,3 with integer m . 2 0 , m 2+m3 > 0

1

and

U.

E (O,l), j

=

3

2,3.

I' I

3

3

Now we are in a position to outline the proof of the two-sided a priori estimates in spaces H

(s),P,E

(Rn) for elliptic singular perturbations

We need the following technical Lemma 3.12.85.

L e t a E L'(R")

c and

be e l l i p t i c of order v . Then t h e r e e x i s t p o s i t i v e c o n s t a n t s

R such t h a t

(3.12.192) la(x,E,E)I 2 CE

-ul

151 "2'3,

v

x E

Rn, V

provided t h a t E~ i s s u f f i c i e n t l y s m a l l . Moreover w i t h xR(S) E Cm(Rn) , xR t 0, f o r 1 5 ) 2 ZR, t h e symbo2 xR(S)a(x,E,5)-'

couple of m u l t i - i n d i c e s

a,B

holds:

xR

E

o

v

E

E

(o,EO1.

5 E R n , 151

f o r 151 5 R,

beZongs t o L-~(R")

2 R,

xR

, i.e.

:1

for each

3. Singular Perturbations on Smooth Manifolds without Boundary

432 Proof. Since -

v n (see D e f i n i t i o n 3 . 1 2 . 1 ) , one h a s a E S ( R )

a E L'(R")

.

F u r t h e r m o r e , a ( x , E , S ) b e i n g e l l i p t i c , i t s p r i n c i p a l symbol a 0 ( x , E , S ) s a t i s f i e s ( 3 . 1 2 . 2 ) , so t h a t one g e t s u s i n g ( 3 . 1 2 . 2 ) and ( 3 . 3 . 4 ) w i t h a = B = O :

la(x,E,c)

1

-v 2 (C/~)E

2 lao(x,c,S)

1

1-1

(a-ao) ( x , E , < )

1

2

-vl

151

CE

2<E

v3

( 1 - c l ( E + < ~ > - l ) )2

v

151 2<EF,>v3, V x E

V

IRn,

E

E

(o,Eo],

v 5 E

En,

151

b R,

( E +RP1) S f. 1 0 N o w , w e check (3.12.193) u s i n g i n d u c t i o n w i t h r e s p e c t t o a and 5. With-

provided t h a t C

o u t r e s t r i c t i o n o f g e n e r a l i t y , one c a n assume t h a t v1 = 0 . F o r a = B = 0 (3.12.193)

i s an immediate consequence o f

(3.12.192). W e use i n d u c t i o n i n

o r d e r t o show, t h a t f o r 151 > 2R h o l d s :

I n d e e d , assuming t h a t ( 3 . 1 2 . 1 9 4 ) ,

(3.12.195)

hold for l a l + ( B (

=

m, one

finds that

where b a + e k , B

=

aD

5b a , g - ( l + l a ~ + ~ B l ) b a , B D S ksaa t i s f i e s

i n s t e a d o f a , as it c a n e a s i l y b e s e e n . Bcek a -1 The Same argument a p p l i e s t o D D a

x

s

,

(3.12.195)

w i t h a+ek

t h e d e t a i l s being l e f t t o t h e

reader. Next, w e p r o v e t h e main r e s u l t , s t a t e d i n t h e f o l l o w i n g Theorem 3.12.86.

Let a E

L'(R~)

and p E

(l,m),

E

E

( 0 , ~ ]

0

(3.12.196)

be e l l i p t i c o f order v. Then f o r each s E I R ~ ,s ' E alvs t h e foZZowing equivalency holds uniformly w i t h r e s p e c t t o

w i t h c o n s t a n t s , which may depend on s , s ' , p and

1 / u / 1 (s),P,E

provided t h a t

E~

- j lop

I

( a ) u l ( s - v ) ,P,E

i s s u f f i c i e n t l y small.

+

E

~

:

I l u l I ( s ' ) ,P,E'

3.12. Elliptic Singular Perturbations

433

Proof. First, we prove (3.12.196) for a = a(E,S) which does not depend on ___ x. Since s' E a l V s , i.e. s; = sl, one can assume without restriction of generality that s ' = s1 = u1 = 0. With a E a(E,F,),s = (0,s2,s3), 1 v = (0,u2,u3) and x ( 5 ) as in Lemma 3.12.85, one can write R

Now, using Leibnitz' formula and (3.12.193), it is easily seen that f

(E,

5)

=

xR ( 5 ) a( E, 5 ) -' I < 1 v2u3satisfies the

condition (3.12.123)

uniformly with respect to E E ( O , E 01 , i.e. with a constant Bo on the righthand side of (3.12.123), which does not depend on E E ( O , E 01 . Hence, Corollary 3.12.63 appliy to fl(E,S). The same, obviously, holds for s -s s3-sj m , since f2(E,S) E C 0 (Rn) . Therefore, f2(E,S) = (l-xR)<S> '<ES> Corollary 3.12.63 yields:

where the constant C depends only on s,s',p and

E 0' On the other hand, Proposition 3.12.78 yields:

I lul I

( s t )

,P,E 5 CI

lul I ( s ) ,P,E-

Furthermore, one uses again the splitting

xR as in Lemma 3.12.85, and the previous argument,in order to show that as a consequence of Corollary 3.12.63, the following inequality holds:

with

I lo?(a) with a constant

C,

UI

I (s-") ,P,E 5

I

cl lul (s),P,E

which does not depend on

E

and u.

Now, if a=a(x,E,S), then one has to use a partition of unity argument, that is to prove analogues of Lemma 3.12.9

and Lemma 3.12.10 in

3 . Singular Perturbations on Smooth Manifolds without Boundary

434 spaces H

( s ) ,PrE

(Rn) , which can be shown again by means of Corollary

I

3.12.63, but will not be done here. Example 3.12.87.

Consider the singular perturbation which appears in the dislocation theory and whose symbol a(E,S) is

5, (3.12.197) a(E.5) = (l+exp(-~/Sl))sgn

E

> 0, 5 E R

(see also Example 3.2.6). This singular perturbation is elliptic of order zero but a f L0 (IR1 ) 1) , since it is not smooth at 5 = 0. However, with x6 E Cm (R x6 E 0 for

151

5 6,

x6

E 1 for I S / 2 26 the singular perturbation O p ( x (S)a(E,u2<ES>"3 Id), where the constant y is the one, appearing in (3.13.1). Denoting Rr = Re Ac-(y/2)€ -"1u2<ED>u3 Id B*B E

E'

one can write for

v

u E

m

c0 (K):

I IUI (Y/2) I l u !

Re(AEu,u) = (Y/2) 2

(V/2),E

+(B u,B u)+(REu,u) k E

E

3

V s E l R ,

I

where c1 may depend on s and K. Remark 3.13.6.

Definition 3.13.1 carries over in an obvious way to singular perturbations ona smooth manifold M without boundary, their principal symbols being well defined for V

E

E R+ as a mapping from the cotangent bundle T*(M)

into Hom(CP;CP) in the case of matrix-valued symbols. Hence for a strongly 3 elliptic singular perturbation a(x,E,D) : Cm(M) + Cm(M) of order u E R 3 one has for each s E R :

where the constant co depends only on y in (3.13.1) and the constant C

1

does not depend on u E Cm(M). Example 3.13.7. Consider again the singular perturbation A

from Example 3.10.15 defined

by (3.10.54) or, equivalently by (3.10.58) on the unit circle R

1

C

C.

Its principal symbol a0 ( x , E , S ) = 1 + E / < 1 satisfies (3.13.1) with u = (0,0,1) is strongly elliptic of order (O,O,l). Defining

and yo = 1, so that A

the singular perturbation B

=

F-l(l+Elnl)fF with F the Fourier series E

one has in this case (A u,u)

=

(B u,B u) 2 c E

E

0

1

2

and AE = BE. Besides, 2 with some /U/ 1

expansion operator, it is easily seen that B* = B

E

(0,Ort)

,E

3.13. Gdrding's Inequality

443

constant co > 0 which may depend upon the partition of unity in the definition of the norms 1 if3 E c I I e l 5 TI. fil = Ie

1. I [

on the circle

(O,O,!)

,E

Singular perturbation (3.10.66) is strongly elliptic of the same order v = (0,0,1) BE

iff Re a(z) t . a

having ( I + € Re a(z)

/

r,

0, V z E

the corresponding operator

as its principal symbol.

It will be shown later that for the singular perturbation A

defined

by (3.10.66) the following inequality holds:

where the constant C does not depend on u and

E.

Now several forms of sharp Gzrding's type inequalities will be considered and we start with the one for one parameter families of difference operators introduced in 3.11. We follow essentially [Friedr, 1 1

1

and [Vai, 1

for proving this form of sharp Gzrdinq's inequality.

The symbols considered are valued in Hom(CP;CP), i.e. are pxp matrixsymbols.

As usual, we denote by a,(x,n)

the principal symbol of a symbol

a E F"(U), ao(x,n) = lim h"l:(x,h,n). h+O Theorem 3.13.8. O-ii

Let p E F ( n

)

Assume t h a t t h e primcipal symboZ pO(x,ri) E n e g a t i v e hermitian m a t r i x :

1,I?

)

and is a non-

(3.13.22) Po(X,I1) 2 0 , v (x,r?)E lRnxTy,n.

Moreover, assume t h a t t h e r e e x i s t s a c o n s t a n t

B

c such t h a t 1

I

(3.13.23)~*~lD~p(x,h,ri)-D~p,(x,n) B 6 Clh, V (x,h,n) E Rn x(O,ho]xTy,ri,

IB/

v 8, Then f o r ph

=

op(p) one has:

2

(3.13.24) Re(Phu,u)t -C2h/l u l

lo,

V u

m

n

E CO(R

),

V h

E (O,hOl

where c 2 i s some c o n s t a n t , which may depend on ho, and where

I 1. I 1

5 n+l

2

are t h e i n n e r prod-uct and t h e norm in L ( n n)

.

(

,

)

and

3 . Singular Perturbations on Smooth Manifolds without Boundavy

444

Proof. As

a consequence of assumption (3.13.23),it suffices to prove (3.13.24)

for the difference operator Ph,D

-

Op po, whose acting on a function

u E Cm(IRn) is given by the formula: 0

Indeed, if p

pm(h,hE) then, as a consequence of (3.13.23) and Parceval's

identity, one has:

I I Ph-Ph,ol lL2,L2

5 C h. Thus, without restriction of 1

generality, one can assume, that pm(h,h5) x E IRn , belongs to S(IR:)

5

0, so that p, as a function of

and so it is for po, as well. We use again the

same argument, as in the proof ot Theorem 3.11.15. Denote r(x,h,hE) One has for

=

V(5)

F

x+S

=

=

%".

u the following representation:

J

=

p(x,h,hS)-po(x,hC), Rh = Op r, v

G(E-n,h,hn)G(Il)dIl,

IRn where ;(C,h,hrl)

=

Fx+Er.

For any function 41 E S(Rn

)

and for each multi-index a one has the

following inequality, as a consequence of Parceval's identity:

where R

is the area of the unit spheere in IRn. n Applying the last inequality with 161 5 n+l to r(x,h,n) (as a function

of x E lRn

)

and using (3.13.23), one gets the conclusion that there exists

a constant C such that

Therefore, one has

I"( denotes the inner product in Rn.

Obviously, one has:

Further, choosing J, such that

/

2

$ ( z ) z dz = 0,

k

R

1 5 k 5 n

(for instance, choosing $ ( z ) to be even), one finds, that

6

R (u) and, as a consequence of (3.13.34), the optimal h choice of 6 is: 6 = h f . f 2 : L2(lRn) -t L ( R n ) in the same Now, estimating the norm of ((F’:-Ph,o)u)

(x)

=


0 is a given constant and where the difference operator Qh,v - O P v' has the symbol: (3.13.46) qv

=

C

Ikl 0, i.e. the coefficients ak on the right han? side of (3.13.46) are to be chosen to satisfy the condition: (3.13.47) exp(irqA)-

1 ak exp(ikrl) (k(5v

=

o[n

2 -v

)

for

rl

+

0.

3 . Singular Perturbations on Smooth Manifolds without Boundary

452

Taylor‘s expansion for exponents in (3.13.47) yields the following system of equations for the matrix coefficients ak:

It is readily checked that ak are given by the formula: (-l)v-k (3.13.49) a = k (V-k) ! (V+k)!

n (rA-pI), IPlZV PZk where I is the identity matrix in Hom(CP;CP) Indeed, since ak are polynomials in A, it suffices to show the identities

It is sufficient to show (3.13.40) for any 2V+l distinct values of h ,

since the left hand side of (3.13.50) is a polynomial of degree at most 2v. Taking h

=

m/r, m = @,fl, ...,+v

(3.13.51) qv = q (rA;rl) =

‘

Ik( 0 holds:

First, we notice that

2 (3.13.78) I/ n + l < ~ ~ ~ lB- u ~ ~ X ( P ( X , ~ , ~ ) - p5 Oc E( ,x , 1E8~1 )5) n + l ,

v

(X,E,S)

E ~ n x ( O , E o ]E n .

.

470

3. Singular Perturbations on Smooth Manifolds without Bounda y

Then f o r pE

= op(p)

Re(P u,u) E

and f o r each

(S),E

s

3

E IR holds: 2 (s1,s2,s3+(v-1)/2)

2 -csl/u//

where t h e c o n s t a n t c does n o t depend on

E

.El

v

E

E (O,EO1'

and u.

Corollary 3.13.24. (0.2m) m E R be such t h a t i t s p r i n c i p a l symbol Let b(x,h.W) E F l , o ho(x,n) i s a non-negative h e m i t i a n m a t r i x : h (x,n) 2 0, 0

{O}). Further, assume t h a t t h e order o f b(x,h,hc)i s 5 2m-1. Then t h e r e e x i s t s a c o n s t a n t k such t h a t f o r t h e

V ( x , n ) E IRn x(T:

h-2%o(x,h-l E Op S(orlr-l)

( R X)

r

and considering the initial value problem:

(3.13.149)

i(D +a(x,t,€,Dx))uE(x,t) = f(x,t) t u (X,O) = u (x),

0

one shows in the same way, as previously, that the same a priori estimate, as ( 3 . 1 3 . 1 4 8 ) , with C

holds for u (x,t) uniformly with respect to

which does not depend on u,uo,f and

t Since for each given

E

E

E (O,E

0

1,

i.e.

6.

0

> 0 one has: a(x,t,E,Dx) E Op S ( R x ) , one

shows using the Picard method and the contruction mappinq arqument that the solution u (x,t) to ( 3 . 1 3 . 1 4 9 ) unique. Now, using ( 3 . 1 3 . 1 4 8 )

exists for t E [-T,T] and is

with s = (O,r,O) with appropriate r E R , and

E 0 that for each 2 uo E H ( R x ) , each f E L ([-T,T];H (R ) ) there exists a unique solution

the compactness argument, one shows by letting

u E

o

O'

-f

x

([-T,T];H (IR ))of initial value problem ( 3 . 1 3 . 1 4 5 ) . a x We are going to consider the following finite difference approximation

C

of ( 3 . 1 3 . 1 4 5 )

which can be successfully used for its numerical treatment.

Let, as previously, let I

= imT1

sT,T

5

= hZ C IR be a grid with meshsize h >

0 and

be the grid on [-T,T] with meshsize T .

Consider the two parameter family of solutions u (x,t), (x,t)E Rh XIT h,T of the following discrete initial value problem

and where the difference operators B hrT following symbols:

qnrT have,respectively, the

3.13. Girding's Inequality

483

b(x,t,h,~,h 0 a g i v e n c o n s t a n t , t h e f a m i l y of def d i f f e r e n c e o p e r a t o r s Bh = B H ( 0 ), h ( % ) has an h , r h ' Bh ( 0 ) ,h(iRh) inverse B-l(iR ) whose norm i s u n i f o r m l y bounded, h . H ( 0 ), h ( % ) -t H ( 0 ) , h h w i t h r e s p e c t t o h E ( 0 , h 1, p r o v i d e d t h a t ho i s s u f f i c i e n t l y s m a l l .

'

'

0

I n d e e d , f o r e a c h pxp m a t r i x $ ( x ) f1 f o r t h e commutator [$,Oh I d ] = $ ( x ) O f l

E

C1

-Oil$

(z) , $

.

=

I / @ j k ( x1) 1 ,

one h a s

(x) t h e following estimate:

where $ ' ( x ) i s t h e m a t r i x , whose e n t r i e s are ( d / d x ) $ . ( x ) a n d , a s u s u a l , lk s t a n d s f o r t h e norm i n Hom(CP;CP).

1 1

Hence, one f i n d s , u s i n g ( 3 . 1 3 . 1 5 2 ) :

(1) h '

= Id+TR

and t h e s a m e argument y i e l d s : i Im B

h

=

( f ) ( B -B*) h h

= =

r ( A ( x , t ) 8 +O A ( x , t ) - A ( x , t ) o - ' 4 h h h (2) TR h '

where t h e d i f f e r e n c e o p e r a t o r s R i J )

:

H(o) ,h(%

norm u n i f o r m l y bounded w i t h r e s p e c t t o h Thus, f o r h o ,

To

E

(O,h

0

H(?,

)

-f

1

by C / A ' ( x , t )

S u f f i c i e n t l y s m a l l Bh = B h , r h

]) i n v e r s e Br e s p e c t t o ( h , T ) E ( O , h o l ~ ( O , ~ obounded h'

,h(lF$,

OhL 1 A ( x , t ) ) =

have t h e i r

has a (uniformly with

'

H ( 0 ),h("h)

H ( 0 ), h ( % -1 . N o t i c e , t h a t t h e p r i n c i p a l symbol b i l ( x , t , r , h < ) of Bh i s g i v e n by t h e formula:

,

3 . Singular Perturbations on Smooth Manifolds without Boundary

484

-1 (3.13.154) bo (x,t,r,h)dp}. 0

Furthermore, since I(w;-x) = I(w;x), one can rewrite the last formula in the form: I(w;x) =

(+)

(2n)-3~2(w)~i( (<x,w>+io)-l-(<x,w>-io)-l

-

m

-Q2(w) /(p2+02(w)

exp(ip<x,w>)dp},

-m

for the distributions -1 (xfiO) and the residuum calculus for computing the integral on the right

so that the Plemelj-Sokhotski formulae

hand side of the last formula, one finds: 2 3 2 -1 ) I20 (w)A(<x,w>)-Q (w)exp(-Q(w)I<x,w>\)}, I(w;x) = (161~

3. Singular Perturbations on Smooth Manifolds without Boundary

488

where 6(y), y E R is the Dirac's 6-function. Further, noticing that 2 -(d/dy)2 exp(-@(o)/yl) = 4 (w) exp(-@(w)/yl) + 2@(w)6(y), and using the fact, that

=

Zwk2

1 , one gets finally the following

=

formula for I (w;x): I(w;x) where A

=

=

-(l6n2)-'A

O(w)

exp(-@(w) I<x.w>/),

Z(a/ax.)L is the Laplace operator.

I

Thus, one finds for S(x) when n=3: S(x) = -(16n2 -1 A For a = Id, one has: @(a)

exp(- f /<x,w>l)dw

'

n3 1 and the last formula yields:

5

the latter coinciding with A02(1,x) defined by (3.2.23) when n=3. The same kind of computation carries over for any odd dimension n, since it allows to evaluate I(w;x) by replacing the integral over R

= {p>O) in its expression by the one over R = {-m 0 (which Since c - ~ < E ~ >5 -~~( € 5 4) c <ES> -2 with 0

0

depends only on the smallest and greatest eigenvalues of the matrix

I 1 akj 1 1 ) , one gets (3.14.6)

SE :

H

the conclusion that (Rn)

(0,0,-2), E

uniformly with respect to

E

E

( 0 . ~ ~ 1V,

e0
'

3 It

Using the representation I

bl(X,E,~)--b'(~,E,n)=

I

(a/ae)bi(X,~,q+e(T-n))de,

0

the inequalities (3.12.1) with v

=

(-l,l,-1) and inequality (3.4.7), one

gets:

V ( x , E , T , ~ ) ,V a E N n ,V k L 0,

with m = 3 + / v 1 , and with some constants Ca,k > 0, which may depend only 3 on their subscripts. Hence, one has for j inequalities: E,O) (3.14.25) I~'(y,s,~)-~'(y

I

1-v

2 C < T - ~ > ~ E'<ET>'~-~, t/ y,E , T

k

,n ,k.

Similarly, one has:

Using (3.14.25) and the estimate

' :1

(y,E,n)1 5 CkE

-lJ

lJ lJ ' 2 < ~ q >3-k,

V y , ~ , ,k, n

(which follows from the fact that a'(.,~,n) E S(lR:)

and from (3.12.1)),

one can estimate the integral on the left hand side of (3.14.23) in the following fashion:

3 . Singular Perturbations on Smooth Manifolds without Boundary

494

where

One chooses k > 0 to be sufficiently large in order to guarantee convergence of the integral on the right hand side of (3.14.27), and that ends the proof of (3.14.23). The same argument can be used to show (3.14.24). Hence, we have proved (3.14.191, (3.14.211. Using (3.14.26) instead of (3.14.25), one gets (3.14.23), (3.14.24) with K replaced by the kernel: K,(~,E,I?) = (j,(E,5)-j,(€,~))at(5-n,E,q).

This yields (3.14.20) with Q, satisfying (3.14.21).

1

Remark 3.14.5. A slight modification of the proof shows that Lemma 3.14.4 is still true if

u2 is not necessarily zero

(j(x,E, 0 , i.e. iff Re a ( 0 ) > 0 ,

fi1.

one can use ( 3 . 1 4 . 4 )

Assuming AE to be elliptic of order (0,0,1),

in

order to reduce ( 3 . 1 4 . 4 2 ) to a regular perturbation equation on R 1 , the a reduced symbol here being, evidently, :

=

a (8).

However, it is more convenient in this case to use the construction of a reducing operator indicated in Remark 3.14.3.

Actually, one can take

as a reducing operator here any singular perturbation whose principal symbol is: so(e,Ec)

= a ( e )( a ( e ) + E / c I )

-1

,

SE

E Op

S(oro‘-l)

(a1)

3 . Singular Perturbations on Smooth Manifolds without Boundary

502

and whose reduced symbol

s o F 1.

One of the possible definitions is the following one:

c

(sEv)( 8 ) = ( 2 n I - l

a ( e ) (a(e)+EikI)-' G(kjexp(ik8)

=

kEZ = v(B)-E(2n)

-1

X Ik/(cx(e)+E/kl

G(k)exp(ike)

kEZ where G(k) are the Fourier coefficients of v. Anyhow, since the reduced operator Ao (which is the multiplication by

a ( @ )with Re a ( e ) > 0) is invertible, one gets the conclusion (see Corollary 3 . 1 4 . 1 2 ) v

=

(0,0,1),

that A' given by ( 3 . 1 4 . 4 3 1 ,

AE

:H

(S),E

is an isomorphism uniformly with respect to

(0 ) + H E

1 (S-V),E E (O,EO1,

(a1) I

provided that E~ is sufficiently small. m

If g E Cm(R1), then the solution u ( B ) of ( 3 . 1 4 . 4 2 ) with respect ot

E)

is in C (fl,)

(uniformly

and, moreover, it can be represented by the following

asymptotically convergent series:

so that, for instance,

One can consider for a harmonic function u in the unit disk B

1

another

boundary condition on R1 of the form: (3.14.44)

where II,

B(8)

(-a/ak,+a(e)-EB(e)a =

2

/ae 2 ) u E ( e ) =

g(e),

E

E ( 0 . ~ ~ 1

( E 1 ( 8 ) , I 1 2 ( 8 ) ) is a given smooth vector field on R1,

, g(B)

and a ( B ) ,

are given smooth functions.

Using the same argument, one can easily show that under the conditions:

a ( B ) > 0 , B ( B ) > 0, II ( 8 ) case+ !L2(8) sine c 0 , V 8 E R 1 , there exists a 1 uniquely defined harmonic function u satisfying boundary condition ( 3 . 1 4 . 4 4 ) , V

E

E (O,E

0

1, provided that

E~

is sufficiently small. In the latter case

the corresponding singular perturbation on the boundary turns out to be elliptic of order ( O , l , l ) EB(e)c

2

-(.tl(e)

cos e

+

and to have the following principal symbol:

i2(e)

sine) 151.

Furthermore, the reduced boundary condition

3.14. Reduction of Elliptic Singular Perturbations

503

for a harmonic function u in B1, defines an isomorphism from H (B1) onto s2 (R if a ( 9 ) > 0 (see [Fr.-W.,1,31), which means that reduced equation Hs2-3/2 1 (3.14.45) is uniquely solvable in Hs2-+(R1) for each g E Hs2-3/2 (R1). Hence, under the assumptions above on R e ,

(R

H(sl,s2-3/2,s3) 1 small.

V

),

B ( e ) , a ( B ) , the perturbed equation

(R1) (s1,s2-t,s3+1)

(3.14.44) is solvable in H

for each

E ( 0 . ~ ~ 1provided , that

E

E~

is sufficiently

It is obvious that a convenient choice of the spaces H

( s ) ,E

( B 1 ) for

the harmonic functions u , satisfying the boundary condition in (3.14.39) (respectively,boundary condition (3.14.44)) is the one, where s = (s1,s2,s3) satisfies the conditions s2 > f, s2+s3 > 3/2 (respectively,s2 > 3/2, s2+s3 > 3/2) (see also Theorem 2.2.21, since this choice of s guarantees

existence of traces of u and its first transversal derivatives on R 1 = aB1 uniformly with respect to E). Consider for a harmonic function uE in B1 the following boundary condition on a. = aB 1'. (3.14.46) (-Ea/aNe+i)(exp(iKe)n++n-)uE(9)= g(e), where TI+ and II- = Id-II'

are the same, as above, and

Since the trace operator (-Ea/aN +l)u(r,B)

e

singular perturbation AE E Op (AEv)( 9 )

=

(2rrI-l

S(o'otl)

K

E Z.

I r=l defines an elliptic

(R,),

(I+Elkj);(k)exp(ikB)

C

kEZ (see Example 3.10.15),

equation (3.14.46) can be rewritten in the equivalent

form:

where gE(e) = AElg(9)

def =

C

1

(l+Ejkj)- g(k)exp(ik9).

kEZ The operator on the left hand side of (3.14.47) (well known in the literature as Noether's example of an operator with index), has a nontrivial kernel of dimension dimension

K

iff

K

> 0.

-K

iff

K

< 0 and a non-trivial cokernel of

In both cases its index (difference between the

dimension of its kernel and the one of its cokernel) is

-K.

The reduction of

3. Singular Perturbations on Smooth Manifolds without Boundary

504

(3.14.46) to (3.14.47) shows that the singularly perturbed operator on

the left-hand side of (3.14.46) has the same index, VE. The same is true, when one considers the boundary condition for the harmonic function uE in B1 of the form:

(3.14.48) (-Ea(e)a/aNe+B(e))(exp(iKe)n++K-)u ( 8 ) = 9(e),

provided that a ( e ) ,

B(e)

are smooth functions satisfying the ellipticity

condition:

where C is some positive constant. Indeed, if (3.14.49) is satisfied, then the singular perturbation in Op S ( o r o r l(Q,) ) associated with the trace operator defined by (3.14.48) on harmonic functions uE, is elliptic of order (O,O,l), its principal symbol being

with H ( C ) the Heaviside function. Since the singular perturbation A

(-Ea(e)a/aNe+B(e))uE(r,e) of order (0,0,1)

jr=l

associated with the trace operator

on harmonic functions u in B ~ is , elliptic

and has an invertible reduced operator (which is just the

multiplication by B ( 8 ) # 0, V 0 E R1), the inverse A-’ exists and is an elliptic singular perturbation in Op S ( o c o r - l(al), ) V that

E~

E

E

( 0 . ~ ~ 1provided ,

is sufficiently small (see Corollary 3.14.12).

Thus, the index of the singular perturbation defined by (3.14.48) is the same as the one for the reduced operator, the latter being again that of Noether on the left hand side of (3.14.47). It is readily seen that (3.14.17) can be reformulated in the following equivalent way: find a function $ + ( z ) analytic in B1 = { z E C, I z I < 1 1 and a function $ - ( z ) analytic in CB1 = { z E C, / z / > 1 ) such that exp(i~e)~+(exp(iB))+$-(exp(ie))= g ( € J ) ,

which is the Riemann-Hilbert problem for the circle Q l [:4uskh.,

1

C

C (seefor instance

I,.

Example 3.14.12. We come back to the singular perturbation from Example 3.13.26:

3.14. Reduction of Elliptic Singular Perturbations (3.14.50) a(x,s,D

)

505

-~A~-, x E R n .

=

With +(x) defined by (3.13.124) and b(x,E,DX) defined by (3.13.126), one reduces the equation (3.14.51) a(x,E,D )uE(x) = fE(x) to the following one: (3.14.52) b(x,E,D )vE(x) = gE(X),

X

E Rn,

where

-1 -1 (3.14.53) v (x) = u (x)exp(E +(x)), gE(x) = Ef€(x)exp(~ Q(x)). Assuming that g v

E H E

(s1,s2,s3+2) ,E

E

E H

( s ) ,E

( R n)

,

one gets the conclusion that

(R"), since b(x,E,S) E !-(o'of2)(Rn) is elliptic of

order ( 0 . 0 . 2 ) . 0

Furthermore, the reduced operator b (x,Dx) is just the multiplication =f by q ( x ) de jVX+(x) 2 qo > 0 , V x E R n , q(x) = l$m\2+qo(x) with

l2

q' E S(Rn). Thus, using (3.14.14), one defines a reducing operator

E s(x,E,D~)

L (o'of-2)(Rn) and gets the conclusion that (3.14.52) has a well--

( 1 ~ ~for ) any g E H (En). (s1,s2,s3+2), E E ( S ) ,E One can also use as a reduclng operator the singular perturbation

defined solution v E H E

s (x,ED~) with the symbol: 0

. (3.14.54) so(x,~S)= q(x) ( 1 ~ 521+lO C ( q ( x , ~ , D ) being )~ convergent in the Banach space k>O (M)) of all continuous linear mappings in H (M) L(H(s) ,E(M);H (s), E ( s ) .E uniformly with respect to E E (O,E 1.

the series

0

Hence, for each f E H

(s-V)

formula for the solution u E H (3.14.63) u =

,E

(M), one has the following asymptotic

( s ) ,E

(M) of the equation a(x,s,D)u = f:

k O N (q(x,C,D)) (a (x,D))-lS(X,E,D)f+ E g,

Z

O5k 0.

E ( O , E ~ ] and symbols

a(x,~,c)admits an asymptotic expansion: a(x,E,5)

-

z

E

k k a (x,S)

k>O

then (3.14.63) can be simplified. We give here an example to illustrate this situation. Example 3.14.14. On the unit circle R 1 with 8 E [ - i ~ , n ) as a coordinate consider the following singular perturbation a(e,E,D,),

m

where q E C (R1) and satisfies the ellipticity Re(q(0))L > 0, tl 0 E R1. We choose the branch q(0)

=

509

((q(0))2)1’2 such that Re q(0) > O , V 8 E R l .

3 . Singular Perturbations on Smooth Manifolds without Bounda y

510

The reduced operator ao being identity, the singular perturbation defined by (3.14.64) is a family of isomorphic linear mappings uniformly with respect to that E~ given

6

E

( O , E ~ ] of H

(R

(S),E

1

onto itself, V s E R 3 , provided

)

is sufficiently small (if q(8) > 0, V 8 E SZ

E~

> 0 uniformly with respect to

E

1

then it holds for any

E (O,E~]).

One can use a partition of unity and the definitlon of singular perturbations on a smooth manifold without boundary for constructing a reducing operator for a(e,~,D) given by (3.14.64) (see Remark 3.14.9).

e

However, given that 8 E [ - r , n I isa coordinateonR1, one can give a different definition of a left reducing operator for a(B,E,DA) without using a partition of unity. Indeed, define the kernel s(@,E,e-e')

=

as follows:

S(e,E,e-e')

(2n)-1 c (1+E2(q(e))2k2)-1exp(ik('t3-8')). kEZ

Using the Poisson formula (1.2.180), one can represent S ( O , E , f 3 - 8 ' )

in

the form: (3.14.65)

S(e,E,e-e')

=

(2€q(e))-l I: exp(-1e-e'-2nk(/(Eq(e))), kEZ

given that with Re q ( 8 ) > 0 one has:

Define the singular perturbation

SE =

s ( ~ , ED

e)

as an integral

operator with the kernel S: def (3.14.66) (S(8,E,~e)g) (8) = J S(0,E,e-ei)g(e')de, v g E

Crn(Ql).

12 1

Since the principal symbol s

0

0

E s ( o , o , - 2 ) (R 1 ) , one has:

( 8 , 6 , < ) of

s(@,c,Da

is: s,=

"

2 2 2 -1 ( l + ~ (q(8)) ) ,

s(S,E,D ) E Op S(ogo*-2) (nl). Furthermore,

e

SE = s(B,E,De)has identity as its reduced operator. Therefore, as a

consequence of Theorem 3.14.6 and Remark 3.14.9, SE = s(B,E,De ) defined by (3.14.651, (3.14.66), is a reducing operator for a(e,6,De) given by (3.14.64) (in fact, it is a quasiinverse operator for a(B,E,Dg)),i.e. Id+ q (~B ) ,E,D S ( ~ , E , D ~ ) ~ ~ ( ~=~ , E,D where

e) ,

3.14. Reduction of Elliptic Singular Perturbations

51 1

is a family of continuous linear mappings uniformly with respect to E

E

(O,E~], V s

E

R 3 , provided that

E~

is sufficiently small.

Besides, (a(e,E,D,)) -1 =

X (Id-s(e,E,De)oa(B,E,De))k .,s(8,E,De), ktO

the series on the right-hand side of the last formula being convergent ( Q ) ) of all continuous linear 1 mappings in H ( Q ) uniformly with respect to E E (O,E 1 , V s E R3 (s),E 1 0 Furthermore, if fE E C m ( Q ) (uniformly with respect to E E (O,E 1 ) 1 0

in the Banach algebra L ( H

( s ) , E ( Q l ) ' H ( s ) ,E

.

then the solution u (8) of the equation ~(@,E, D (8) = fE(e), e~)U

E

E

(O,E

01, m

is well defined and belongs to a bounded set in C ( i l l ) , V provided that

E~

E

E

(0.~~1,

is sufficiently small (see Remark 3 . 1 2 . 2 3 ) .

Moreover, if f (8) admits an asymptotic expansion,

then so it is also for uE(8), (3.14.67)

(8)

-

c

E

k

uk(e), uk E

m

c (a1),

kZO

where the coefficients uk can be defined by recursion. For instance with f (8)

5

f (8), one finds easily: 0

Besides, the asymptotic convergence in ( 3 . 1 4 . 6 7 ) , space H

( 3 ) ,E

(R

1

(3.14.68)

in each

is an immediate consequence of the a priori estimate

with a constant C which might depend only on s and q, the latter resulting from Theorem 3.14.6 and Remark 3.14.9. A

special attention deserves the case when f(8) = fs(8)+g(8) where

3 . Singular Perturbations on Smooth Manifolds without Boundary

512 m

g E C (Q1) and f (8) is the singular part of f, having a finite number of points in R1 as its singular support. For instance, consider the case, when f (8) = sgn 8, 8 E Ql. Since sgn 6 E H

(0),E = sgn 8:

au

(Q ) ,

1

one has for the solution u (6) of the equation

Using the reducing operator s ( € l , ~ , D ) defined by (3.14.65), (3.14.66), 6 one can represent u (6) in the form:

u where on

E

(el

= s ( e , E , ~ )f (e)+EVE(e), 6 s

I

/v C E 1 1 (0,0,2) ,E E (O,eo1.

C

with some constant

C >

0, which does not depend

Using (3.14.65), (3.14.66), one finds:

, def (3.14.69) ~ ( ~ , e= l

s ( e , E , ~ )f

6

= (2Eq(e))-l c

s

sgn

(6) =

el

exp(-/e-e'-2nk//(Eq(e)))de',

e E nl.

kEZ Ql

Computing integrals on the right-hand side of (3.14.69), one finds explicitly the first term u ( E , ~ ) in the asymptotic expansion u (6) the following formula:

the point

e

= IT

being identified with 8 =

-IT.

Hence, also for u(E,e) the singular support consists of the same two points as a consequence of the pseudolocality property of singular perturbations (see Theorem 3.5.4). Now the construction of a right reducing operator for a given elliptic singular perturbation will be given and some applications will be considered.

3.14. Reduction of Elliptic Singular Perturbations

513

We are going to show that for a given elliptic singular perturbation a(x,E,D 1 E Op Sv(M) the left factorizing and reducing operators r(x,E,D and s(x,E,D

)

defined by their symbols ( 3 . 1 4 . 1 4 ) ,

)

are also right factorizing

and reducing operators, respectively. The essential part here is the following analogue of Lemma 3 . 1 4 . 4 : Lemma 3 . 1 4 . 1 5 .

Let j(x,~,S)E Lv(Rn) w i t h w 0

j (x,~) E I,

v (x,~) E

L e t a(x,c,S) E L ! ' ( R ~ ) .

R~

x

=

( 0 , 0 , v 3 ) and assume t h a t t h e reduced symbol

mn. Then one has:

i . e . f o r each s E m3 t h e folZowing i n e q u a l i t y h o l d s :

w i t h a c o n s t a n t c > 0 , which does n o t depend on

E

and u.

Proof. First a transparent heuristic argument will be presented, which explains why ( 3 . 1 4 . 7 0 1 ,

(3.14.71)

should be true.

Using Theorem 3 . 7 . 6 , one finds:

5

as a consequence of Definition 3 . 1 2 . 3 . Thus,

1, one has:

3. Singular Perturbations on Smooth Manifolds without Bounda y

514

given t h a t t h e o r d e r of t h e symbol on t h e l e f t hand s i d e of t h e l a s t

< v+p, v

inclusion i s v+p-(o,/crj , o ) + ( o , I , I )

IcrI > 0.

QF,

Besides, t h e amplitude of t h e o p e r a t o r which i s t h e k e r n e l of t h e o p e r a t o r

QE,

(see Definition 3.7.1),

coincides with t h e k e r n e l of t h e

operator

t h e o r d e r of t h i s amplitude being a t most v+p. Now a rigorous proof of t h e lemma w i l l be given, which i s very s i m i l a r t o t h e one of Lemma 3.14.4.

Without r e s t r i c t i o n of g e n e r a l i t y , one can

assume t h a t v1 = p l = 0. L e t a'(X,E,S)

def = a(x,E, iR

onehas: -v

s -p

2<En2

-v

-s 3

-s 2<ET>

v

' IE

3/K(ll,E,T)

iRn,

v

E

Idr? 6 CE,

E

(o,Eolp

3.14. Reduction of Elliptic Singular Perturbations

-v

s -p (3.14.76) In 0. Now the mean value theorem and the inequalities in Definition 3.12.1

of classes Lp(Rn ) yield:

where we have used Peetre's inequality: <S>p-p

5

2'pl, one can estimate the integral on the left hand side of (3.14.75) in the following fashion:

where ml = 1+1u2-11+/u31+js21+1s31+1u 2 +v2 1 and m2 =

l u 2+v 2 I + / s 2 ] + / s 3 / .

We choose k to be sufficiently large in order to guarantee the convergence of the integral on the right hand side of (3.14.81), and that

.

yields (3.14.75)

The same argument can be used in order to prove (3.14.76). Hence, we have proved (3.14.72). In order to show (3.14.73), one will have to estimate the kernel

-

def K _ ( ~ , E , T ) = j'(T-n,&,n) (arn(~,T)-arn(&,n)). Using (3.14.80) instead of (3.14.79), one gets (3.14.75), (3.14.76) with

K replaced by Km. This yields (3.14.73). We have shown that the singular perturbations Q ? , j = 2,3, in (3.14.72),(3.14.73) have atmostorder I that QE = Q:+Q~ E Op LU+' (iRn1, as well.

v+p,

so

Now, using exactly the same argument, as in the proof of Theorem 3.14.6, one shows its analogue with r(x,&,D), s(x,E,D) as right factorizing and reducing operators:

the same kind of inequalities being also true for elliptic singular perturbations in S'(M)

with M a smooth manifold without boundary.

3.14. Reduction of Elliptic Singular Perturbations

517

Corollary 3.14.16.

L e t M be a smooth manifold w i t h o u t boundary and l e t a E o p s'(M) of order v

=

f a c t , a0

H

be e l l i p t i c

(v1,v2,v3)and a s s m e t h a t t h e reduced operator ( 0 '"2 , O ) (M! maps isomorphicaZZy M (M) o n t o L~(M), so t h a t , i n ao E op s V-

:

-v

( M ) + HS

s2

2

(M)

is an isomo$icm

f o r any giuen

s2 E

w.

2

Consider t h e equation a(x,~,D)u(x) = f (x) w i t h f E H

(s-v), E

(M)

giuen.

Then U(X) can be represented i n t h e form: 0

(3.14.84) u(x) = S(x,E,D) (a (x,D))-'f(x)

1 IvI 1 ( s ) , E

where

depend on

E;X

+ EV(X),

w i t h some c o n s t a n t c > 0 , which does n o t

and f.

Indeed, using (3.14.82) and the corresponding version of (3.14.30),

(3.14.31) on a smooth manifold (see Remark 3.14.9), one gets immediately (3.14.84). Example 3.14.17. Let Tn be the n-dimensional torus, Tn = Rn/Zn, with the global coordinates m n x . E [0,1], 1 5 j 6 n. Let q(x) > 0, q E C ( T ) be given, and consider the 7 equation on T ~ :

(3.14.85)

(E

2 2 A -A+q(x))u'(x)

=

f(x),

where f E L2(Tn) is given and A is the Laplace operator on Tn. Using (3.14.84), one can represent the solution u(x) of (3.14.85) in the form:

where s(x,E,D) is a right reducing operator for the singular perturbation on the left hand side of (3.14.15) and

v

=

(0,2,2)

I / v s I I (v),E

I

, with

5 CI/fl

and a constant C > 0 which does not depend on

L (T

E,U

and f.

Moreover, as a right reducing operator, one can choose the one, which is globally defined as follows:

where S(E,x) =

Z kEZn

2 2

(1+4rr E [kI2)-l exp(2rri).

3. Singular Perturbations on Smooth Manifolds without Bounda y

518

Using again the Poisson formula ( 1 . 2 . 1 W ) , S(E,X) =

A.

Z

-2

one finds:

(E,x-k),

kEZn where A.

-2

-1

(€,XI = FS,x(1+c252)-1

is given by ( 3 . 2 . 2 3 )

F o r n = 3 , one finds using ( 3 . 2 . 2 4 ) :

and for the solution u of ( 3 . 1 4 . 8 5 )

one gets the asymptotic formula:

0

where (lu ) (x) is the periodic extension of uo with a constant 5ClIfll 2 L (T )

6,U.f.

c

:

Tn

-f

C onto Rn and

> 0, which does not depend on

Example 3 . 1 4 . 1 8 . Let U aU

C

R2 be a bounded convex domain, whose boundary aU is a Cm-curve,

{x=y(s), 0 5

=

s 5

L}, the parameter s being the arc length on aU

between a fixedpint xo = y(0) and the current point y(s). Let u E ( x ) be a harmonic function in U, which satisfies the following boundary condition on

au

(3.14.87)-Eau~(x)/aNs+a(s)UE(X) = g'(s),

where N

x=y(s), 0 5 s 5 L,

is the unit inward normal to aU at x = x(s), a ( s ) > 0 ,

a ( 0 ) = a(L) and the periodic extension of a

:

v

S

E [o,L],

[O,L] -+ R+ Onto R iS

m

supposed to be in

C (R).

We shall also assume that gE E H ( o ,s2-f ,s3-l) s2+s3> 3/Z

sought in H

caul with

s2 > f ,

(3), E

will be

(U), so that the trace on the left hand side of ( 3 . 1 4 . 8 7 )

is well-defined, V

E

>

0 (see Theorem 2 . 2 . 3 ) .

Denote by P the following integral operator: (Pa)(x) where (3.14.88)

,€

and the harmonic functions uE(x) satisfying ( 3 . 1 4 . 8 7 )

P(s,x-Y

-2

3.14. Reduction of Elliptic Singular Perturbations

519

Introducing

with

T

=

dy/ds the tangential unit vector to aU at y(s), one can rewrite

P(s,x-y(s)) in the form: P(s,x-y(s)) = F-lI~x

(3.14.89)

I I)

exp ( -P 5

1

the right hand side of (3.14.89)

#

being well-defined, since p > 0 , V x E

i,

x#y(s), as a consequence of the convexity of U. Thus, one finds for x

p(s',s)

> 0,

for

s'

+

y(s'),

# s.

Notice that 2

x'(s',s) =s'-s+(s'-s) a(s',s), p ( s ' , s ) =tk(s',s)( s ' - s ) ~k(s',s) , > 0, s'

where k(s,s) = k ( s ) b 0 is the curvature of m

k(s',s) being C

au

#

s,

at y(s), a(s',s) and

in ( s ' , s ) .

Further, since y(0) = y(L), one can consider the operator Q,

with the distributional kernel Q(s,s'-s) defined by (3.14.901, operator acting on functions $ defined on the circle R R , R

=

as an L/2r

parametrized by s E [O,LI. Let {6k}15ksm be a covering of RR by intervals Ak of length 6 sufficiently small and let {

x

~ be} a subordinate ~ ~ ~ partition ~ ~ of unity.

Then one has:

where (Q$

k

) (s')

is the operator of the form:

3 . Singular Perturbations on Smooth Manifolds without Boundary

520

f f qk(s ' , s , S ')exp(i ( s ' - s ) 5 ' ) $k ( s ')dE'ds

(Qa,) ( s ' ) = (2r)-'

R3R

with the amplitude q given by the formula: k (-.14.91)qk(s',s,€,') =

as, ,NS> 1 5

1 exp (i( s ' - s 1 2 a ( s ' ,s ) 5

I-

(f )k( s

I ,

s ) ( s '- s )

It is readily seen that with any cut-off function $ ( S ' ) ,

jc'l

2

15

$( f ,

s2+s3 > f . If g E Cm(aU) then to E

(O,E~], V

E~




E

> 0 and the mesh-size h > 0 ) is useful for the numerical treatment of

singularly pertrubed hyperbolic operators. There is a hope to be able to elaborate on this topic in forthcoming volumes of this book. The ellipticity concept for singular perturbations was introduced in [Fife, 1

1

and, independently in the form as it is formulated in 3.12

(Definition 3.12.2) in [Frank,15,19,221. C h a r a c t e r i s t i c t w o - s i d e d a priori estimates for elliptic and coercive singular perturbations (uniform with respect to the small parameter) were stated in [Frank, 15,19,22] and proved in [Frank, 22

1.

The strong ellipticity concept for finite difference

3. Singular Perturbations on Smooth Manifolds without Boundary

530

1.

operators is already present in [Lax - Nirenberg, 1

The concept of

ellipticity for one parameter families of difference operators was introduced in [Frank, 4

1

and independently (for a subclass of finite difference schemes,

which approximate elliptic differential operators) in [Thomee - Westergren, 1

1.

Two-sided a priori estimates for elliptic finite difference operators

were stated in [Frank, 41 and proved in [Frank, 13

1.

For difference

singular perturbations analoguous concepts were introduced in [Frank, 16, 17,21

1

and corresponding two-sided estimates established in [Frank, 23

1.

A priori estimates (uniform with respect to the small parameter) in L P norms for elliptic singular perturbations were established in [Sweers, 1.21. Theorem 3.12.62 is due to Hdrmander and we follow esstentially the scheme

1

in [Hdrmander, 1

for proving it. In order to establish uniform a priori

estimates in L -norms for elliptic singular perturbations, the classical

P

result in [Michlin,l,2 1 (Corollary 3.12.63) suffices. The guide lines for proving results similar to Theorem 3.12.62 are to be traced back, for instance, in [Zygmund, 1

1.

As far as L -estimates for classical pseudo-

P

differential operators and more general classes of operators are concerned, the reader is refered to [Beals, 2

1

1 ,

[Triebel, 1

1

1,

[Fefferman - Stein, 1

1,

[Strichartz,

and others.

For the strongly elliptic differential systems introduced in [Vishik,

11 M.I. Vishik proved that the symmetric quadratic forms associated with the real part of such a system of order 2m (in a bounded domain R

C

lRn

with sufficiently small diameter) defines an equivalent norm in the Sobolev space Hm(fi). A proper extension of this result to arbitrary bounded domains was given in [Gsrding, 1.2

1

and came into use in the mathematical

literature under the name of Ggrding's inequality. This kind of inequality

1.

for singular integral operators was established in [Calderon - Zygmund, 1 Several important results toward the sharp form of G&ding's are to be found in [Seely, 1

1.

inequality

The sharp form of Girding's inequality for

the classical pseudodifferential operators, as introduced in [Untergerger Bokobza, 1 1, [Kohn-Nirenberg, 1

1

-

and for one parameter families of

difference operators (appearing in the stable approximations of well posed evolution problems), was first established in [Lax - Nirenberg, 1

1.

A

simplified proof of the sharp form of Ggrding's inequality was given in [ Friedrichs, 2

1

and [Vaillancourt, 1

3.

For classes F"(Zn

)

(see Definition

3.12.32) of one parameter families of difference operators the sharp form

of Gsrding's inequality was proved in [Frank, 9,lO

1.

For singular

Notes

53 1

perturbations whose reduced symbol is identity it was stated in [Frank, 241 (see also [Helffer

-

SjGstrand, 1

1

for the case of semi-classical pseudo-

differential operators). As far as Gsrding's inequality is concerned in the case of pseudodifferential (and more general) operators without small or large parameters, the reader is also refered to [Beals - Fefferman, 1 [Fefferman - Phong, 1

1

1,

and others. Example 3.13.14 is taken from [Frank, 3

and the last part of Example 3.13.15 from [Frank, 24

1

1. For proving the

Lax-Nirenberg theorem (Theorem 3.13.19), we follow essentially [Friedrichs, 2

1,

[Vaillancourt, 1

relevant

1

(with some minor modifications).Example 3.13.28 is

for the probability theory (see [Friedlin - Wentzel, 1

1

and for

quantum mechanics (see [Helffer - Robert, 1 I). A s far as the difference methods for the conservation law systems are concerned, the reader is refered to [Lax, 2,3 1, [Godunov, 1 1, [Harten - Hyman - Lax, 1 1, [Osher,

1

1,

[Van Leer, 1

1

and others.

The idea of a constructive reduction of coercive singular perturbations to regular perturbations was put forward in [Frank, 19

out in [Frank - Wendt, 1-5,6,10

1

and fully worked

1 and [Wendt, 1,2 1. The Wiener-Hopf

factorization was used in [Eskin, 1

1

in order to reduce an elliptic pseudo-

differential singular perturbation with homogeneous Dirichlet boundary conditions to a regular perturbation. Example 3.14.11 is taken from [Frank - Wendt, 1

1.

devices (see [Mock, 1

Example 3.14.13 comes from the theory of semiconductor

1,

[Markowich, 1 1, [Smith, 1

1,

[Sze, 1 1). Example

3.14.17 is of interest for the linear elasticity theory and Example 3.14.18 is relevant for the diffraction theory. The idea of reducing elliptic finite difference singular perturbations to regular perturbations, put forward in Examples 3.14.19 and 3.14.20, can be consistently worked out for the general elliptic finite difference operators in the same way and spirit as it has been done for the elliptic singular perturbations in Op L v ( M ) .

This Page Intentionally Left Blank

533

Bibliography

Adams, R.A.: [l] Sobolev spaces. Academic Press, 1975. Agmon, S.: [l] Lectures on elliptic boundary value problems. Van Nostrand, Princeton-London-Toronto 1965. Agmon, S . , A. Douglis and L. Nirenberg: [l]Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12,623-727 (1959); 11. Comm. Pure Appl. Math. 17, 35-92 (1964). Agranovich, M.S. and M.I. Vishik: [l]Elliptic problems with a parameter and parabolic problems of general type. Usp. Mat. Nauk, 19 (3), 53-161 (1964) (Russian). Airy, G.B.: [l]On the intensity of light in a neighborhood of a caustic. Trans. Cambr. Phil. SOC.6, 379-402 (1838). Andronov, A.A., A.A. Witt and S.E. Hailin: [l]Theory of oscillations.“Nauka”, Moscow 1981 (Russian). Arnold, V.I.: [l] Mathematical methods in classical mechanics.

“Nauka”,

Moscow, 1974 (Russian). English transl.: Springer-Verlag, Berlin-Heidelberg-New York 1978. Aronszajn, N. and K.T. Smits: [l]Theory of Bessel potentials I. Ann. Inst. Fourier (Grenoble) 11, 385-475 (196 1). Aubin, J.P.: [l] Approximation of elliptic boundary value problems. Wiley, New York, 1972. Bachvalov, N.S.: [l]On the optimization of methods for the solution of ordinary diffeential equations with highly oscillating solutions. J . of Comp. Math. and Math. Phys., 11:5, 1318-1322 (1971) (Russian). Beak, R.: [l]A general calculus of pseudo-differential operators. Duke Math. J. 42, 1-42 (1975).

534

Bibliography

- [2] LP boundedness of Fourier integral operators. Memoirs Amer. Math.

SOC.,Providence, R.I. 1982. Beals, R. and C. Fefferman: [l]On local solvability of linear partial differential equations. Ann. of Math. 97, 482-498 (1973). Beals, R., C. Fefferman and R. Grossman: [l] Strictly pseudoconvex domains in 43”. Bull. Amer. Math. SOC.8, (2), 125-322 (1983). Bellman, R.: [l]Perturbation techniques in mathematics, physics and engineering. Holt, New York 1964. Bensoussan, A., J.L. Lions and G. Papanicolaou: [l] Asymptotic methods in periodic structures. North-Holland, Amsterdam-New York-Oxford 1978. Bensoussan, A., J.L. Lions and R. Temam: [l]Sur les methodes de decomposition, de d6centralisation et de coordination.

Cahiers IRIA, 11, 5-190

(1972). Berezin, I.S. and N.P. Zhidkov: [l]Computing methods. Pergamon Press, 1965. Berger, M.S. and L.E. F’raenkel: [l]On the asymptotic solution of a nonlinear Dirichlet problem. J . of Math. and Mech., 19, 553-585 (1970). Bers, L.: [l] Mathematical aspects of subsonic and transonic gas dynamics. Surveys in applied mathematics. 111. J. Wiley, New York, 1958. Besjes, J .G.: [l] Singular perturbation problems for linear elliptic differential operators of arbitrary order, I , Degeneration to elliptic operators. J . Math. Anal. Appl. 49, 34-46 (1975); 11, Degeneration to first order operators. J . Math. Anal. Appl., 49, 324-346 (1975). Birkhoff, D.: [l]Quantum mechanics and asymptotic series. Bull. Amer. Math. SOC.39, 681-700 (1933). Birkhoff, G.D.: [l]On the asymptotic character of the solutions of certain linear differential equations containing a parameter. Trans. Amer. Math. SOC. 9, 219-231 (1908). Bogolubov, N.N. and Yu.A. Mitropolskii.1.: [l] Asymptotic methods in the theory of nonlinear oscillations. “Nauka” , Moscow 1974, (Russian), English transl.: Gordon and Breach, New York 1961. Borel, E.: [l] Sur quelques points de la thkorie des fonctions. Ann. Sci. Ecole Norm. Sup. 12 (3), 9-55 (1895).

535

Bibliography

Boutet de Monvel, L.: [l]Boundary problems for pseudo-differential operators. Acta Math. 126, 11-51 (1971). Brailovsky, M. and L.S. Frank: [l] Stationavy flow by solar wind around a multipole. Report No. 77, 1-35, Institute of Space Research of USSR Academy of Sciences, 1971. Brauner, C. and B. Nicolaenko: [l] Singular perturbations and free boundary problems. Computing methods in Applied Sciences and Engineering, North-Holland, Amsterdam (1980). BrCzis, H.: [l]Analyse fonctionelle. Masson. Paris, 1983. Burgers, T.M.: [l] The nonlinear diffusion equation: asymptotic solution and statistical problem. North-Dordrecht-Holland, Amsterdam] 1974. Calderbn, A.P. and A. Zygmund: [l]On the existence of certain singular integrals. Acta Math. 88, 85-139 (1952). - [2] Singular integral operators and differential equations. Amer.

J . Math.

79, 901-921 (1957). Carleman, T.: [l] PropriCtCs asymptotiques des fonctions fondamentales des membranes vibrantes. C.R. Cong. des Math. Scand. Stockholm 1934, 34-44, Lund (1935). Chazarain, J. and A. Piriou: [l]Introduction

la thCorie des Cquations aux

dkrivCes partielles linkaires. Gauthier-Villars, 1981. Ciarlet, P.G. and H. le Dret: [l]Justification of the boundary conditions of a clamped plate by an asymptotic analysis. Asympt. Anal., 2: 2, 257-278, (1989). Ciarlet, P.G. and P. Rabier: [l] Les equations de von KBrmBn. Springer-Verlag, Berlin-Heidelberg-New York, 1980. Cole, J.D.: [l] Perturbation methods in applied mathematics. Blaisdell Publ., New York 1968. Comstock, C.: [l] Singular perturbations of elliptic equations. SIAM J . Appl. Math., 20, 491-502 (1971). Courant, R.: [l]Partial differential equations. Interscience, New York, 1962.

-

[2] Variational methods for the solution of problems of equilibrium and vibrations. Bull. Amer. Math. SOC.49, 1-23 (1943).

536

Bibliography

Courant, R., K.O. Friedrichs and H. Lewy: [I] Uber die partiellen Differentialgleichungen der Mathematischen Physik. Math. Ann., 100, 32-74 (1928). Daletski, Yu.L.: [l] An asymptotic method for the solution of some differential equations with oscillatory coefficients. Dokl. Acad. Nauk USSR, 143, no. 5, 1026-1029 (1962) (Russian). Daletski, Yu.L. and M.G. Krein: [l]Stability of solutions of differential equations in Banach spaces. “Nauka”, Moscow 1970 (Russian). Davis, R.B.: [l]Asymptotic solutions of the first boundary value problem for a fourth-order elliptic partial differential equation. J. Rat. Mech. and Anal., 5, 605-620 (1956). De Bruijn, N.G.: [l]Asymptotic methods in analysis. North-Holland, Amsterdam, 1958. De Giorgi, E. and T. Franzoni: [l] Su un tip0 di convergenza variazionale. Rend. Sem. Mat. Brescia, 3, 63-101 (1979). Demidov, A.: [l] Elliptic pseudo-differential boundary value problems with a small parameter in the coefficients of the leading operator. Math. USSR

Sb., 20, 439-463 (1973). Dieudonnk, J.: [l]Elkments d’Analyse, Gauthier-Villars, 1978. - [2] Foundation of modern analysis. Academic Press, New York, 1964.

Dixmier, J.: [l]Les C*-algBbres et leurs reprksentations. Gauthier-Villars, Paris 1964. Dobrokhotov, S.Yu. and V.P. Maslov: [l] Finite gap almost periodic solutions in asymptotic expansions. Anal. and Numer. approaches to Asympt. Problems (0. Axelsson, L.S. Frank, A. van der Sluis, eds.), 1-15, NorthHolland, Amsterdam-New York-Oxford (1981). Duistermaat, J.J.: [l] Fourier integral operators, Lecture Notes, Courant Inst. of Math. Sci., New York, 1973. Duistermaat, J.J. and L. Hormander: [l]Fourier integral operators. 11. Acata Math. 128, 183-269 (1972). Dunford, N. and J.T. Schwarz; [l]Linear operators 1-11. Interscience Publishers, New York-Lon don, 1958-1963. Eckhaus, W.: [l]Boundary layers in linear elliptic singular perturbation prob-

Bibliography

537

lems. SIAM Rev. 14, 225-270 (1972). Eckhaus, W. and E.M. De Jager: [l]Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type. Arch. Rat. Mech. and Anal., 23, 26-86 (1966). Egorov, Ju.V.: [l]The canonical transformations of pseudedifferential operators. Uspekhi Mat. Nauk 24 : 5, 235-236 (1969). Erdklyi, A.: [l]Asymptotic representation of Fourier integrals and the method of stationary phase. SIAM J . 3 : 1, 17-27 (1955). - [2] General asymptotic expansions of Laplace integrals. Arch. Rat. Mech.

and Anal., 7 : 1, 1-20 (1961). Eskin, G.I.: [l] Boundary value problems for elliptic pseudo-differential equations. Amer. Marth. SOC.Transl. of Math. Monographs 52, Providence,

R.I., 1981. Evgrafov, M.A. and M.V. Fedorjuk: [l] Asymptotics for the solutions of the equation

w N ( z ) - p ( z ,X)w(z) = 0 for X Nauk, 21 : 1, 3-50 (1966).

+ 00

in the complex domain. Uspekhi Mat.

Fedorjuk, M.V.: [l]The saddle point method, “Nauka”, Moscow, 1977. - [a] The stationary phase method for multidimensional integrals. Journ.

Vichisl. Mat. and Mat. Fisica 4 : 4, 671-681 (1964) (Russian).

- [3] The stationary phase method and pseudo-differential operators. Uspekhi Mat. Nauk 26 : 1, 67-122 (1971) (Russian). Fefferman, C. and D.H. Phong: [l]The uncertainty principle and sharp Gdrding inequalities. Comm. Pure Appl. Math. 34, 285-331 (1981). Fefferman, C. and F.M. Stein: [l]HP spaces of several variables. Acta Math., 129, 137-193 (1972). Fife, P.: [l]Singularly perturbed elliptic boundary value problems I: Poisson kernels and potential theory. Annali di Mat. Pura Appl. Ser. 4, 90, 99-148 (1971). - [2] Dynamics of internal layers and diffusive interfaces. SIAM, Philadelphia

1988. Flatto, L. and N. Levinson: [l]Periodic solutions of singularly perturbed sys-

Bibliography

538

tems. J . Rat. Mech. and Analysis, 4, 943-950 (1955). Fleming, W.H.: [l] Stochastic control for small noise intensities. SIAM J . Control, 9, 472-517 (1971). Fok, V.A.: [l] Problems of diffraction and propagation of electremagnetic waves. “Sovietsk. Radio”, 1970. Frank, L.S.: [l]On Cauchy’s problem for some equations with variable coefficients which are not of Kovalevski’s type. Vestnik Moscov. Univ., 5, 13-23 (1963) (Russian, English summary).

- [2] Difference methods for the solution of the ill-posed Cauchy problem systems of the first order. Vestnik Moscov. Univ., 21,3-11 (1966) (Russian, English summary).

- [3] The difference schemes of high order of accuracy for systems of first order. Vestnik Moscov. Univ., 21, 26-33 (1966) (Russian, English summary). - [4] Difference operators in convolutions. Dokl. Akad. Nauk SSSR, 181, no.

2, 286-289 (1968); English transl. Soviet Math. Dokl., 9, 831-840 (1968).

- [5] Hyperbolic difference operators. Dokl. Akad. Nauk SSSR, 189, no. 3, 486-488 (1968); English transl.: Soviet Math. Dokl., 10, no. 6, 1449-1451 (1969). - [6] Coercive boundary value problems for difference operators. DokLAkad.

Nauk SSSR, 192, 42-46 (1969); English transl.: Soviet Math. Dokl., 11, 591-594 (1970). - [7] The asymptotic behavior of rapidly oscillating solutions of hyperbolic

difference equations. Uspekhi Mat. Nauk, 25, 252-254 (1970) (Russian).

[8] Spaces of network functions. Mat. Sb., 86(128) : 2(10), 187-233 (1971); English transl.: Math. USSR Sbornik, 15, 183-226 (1971). - [9] An algebra of difference operators. Institute of Space Research of USSR, Acadmy of Sciences, Moscow, 1971. -

- [lo] Algkbre des opCrateurs aux diffCrence finies. Israel J. Math., 13, 24-55

(1972). - [ll]Fonctions de mailles et thdorie elliptique des opkrateurs aux diffkrences

finies. Seminaire Goulaouic-Schwartz (1972-1973): equations aux dCrivkes partielles et analyse fonctionnelle: exposks Nos. 5-6, Paris, Centre de

5 39

Bibliography

mathhmatique, Ecole Polytechnique, 1-15 (1973). - [la] Sur la fonction spectrale d’un ophrateur aux diffhrences finies elliptique.

C.R. Acad. Sci. Paris, 276, 1521-1523 (1976). - [13] OpCrateurs elliptiques aux diffkrences finies.

J . Math. Anal. Appl.,

45, 260-273 (1974). - [14] Comportement asymptotique et support singulier de la fonction de

Green des opkrateurs diffhrence - diffkrentiels stables. C.R. Acad. Sci. Paris, 278, 1051-1054 (1974). - [15] Problbmes aux limites coercifs avec un petit parambtre. C.R. Acad.

Sci. Paris, 282, 1109-1111 (1976). - [16] Perturbation singulibres et difftkences finies. C.R. Acad. Sci. Paris,

283, 859-862 (1976). - [17] Perturbazioni singolari alle differenze finite.

Ren. Sem. Univ. di

Torino, 35, 129-142 (1976/77). - [l8] General boundary value problems for ordinary differential equations

with a small parameter. Annali Mat. Pura Appl., 114, 26-67 (1977). - [19] Perturbzioni Singolari Ellittiche. Ren. Sem. Mat. Milanepavia, 47,

135-163 (1977).

- [20] Factorisation for difference operators. J . Math. Anal. Appl., 62, 170-185 (1978). -

[all Coercive singular perturbations:

stability and convergence. Numeri-

cal analysis of singular perturbation problems, 181-198, Academic Press, London (1979). - [22] Coercive singular perturbations. I, A priori estimates. Annali Mat.

Pura Appl., 119, 41-113 (1979). - [23] Elliptic difference singular perturbations. Annali Math. Pura Appl.,

122, 315-363 (1979). - [24] InCgalitC de Gkding et perturbations singuli6res elliptiques aux diffh-

rences finies. C.R. Acad. Sci. Paris, 296, 93-96 (1983).

- [25] Perturbation singulibres coercives IV: probkme de valeurs propres. C.R. Adac. Sci. Paris, 301, 69-72 (1985).

540

Bibliography

- [26] Remarque sur le phCnombne de bifurcation pour les perturbations sin-

gulibres coercives quasi-lin6aires. C.R. Acad. Sci. Paris, 302, 17-20 (1986). - [27] On a singular perturbation in the kinetic theory of enzymes and the

corresponding nonlinear time optimal control. Analysis and optimization of systems (A. Bensoussan and J.-L. Lions, eds.), 653-667, Springer-Verlag,

Berlin-Heidelberg-New York-Tokyo (1986). - [28] Perturbations singulikres coercives: m6thode de factorisation et appli-

cations. Equations aux d6rivCes partielles, expos6 110.18, Ecole Polytechnique, Palaiseau, 1986-87. - [29] Coercive singular perturbations: eigenvalue problems and bifurcation

phenomena. Annali Mat. Pura Appl., (IV), CXLVIII, 367-395 (1987). - [30] Sharp error estimates in the reduction procedure for the coercive sin-

gular perturbations. Asympt. Analysis, 3 : 1, 37-42 (1990). - [31] Toeplitz operators and the theory of one parameter families of differ-

ence operators. Rep.9008, Univ. Nijmegen, February 1990. Frank, L.S. and L.A. Chudov: [l] Difference methods for solving the incorrect Cauchy problem. Numerical methods in Gas Dynamics, Moscow, nr. 4, 3-27 (1965) (Russian). Frank, L.S. and E. van Groesen: [l] Singular perturbations of an elliptic operator with discontinuous nonlinearity. Analytical and numerical approaches to asymptotic problems in analysis, North-Holland, Amsterdam, 289-303 (1981). Frank, L.S. and J.J. Heijstek; [l]On the continuation by zero in Sobolev spaces with a small parameter. Asympt. Analysis, 1 : 1, 51-60 (1988). - [2] On the reduction of coercive singular perturbations to regular per-

turbations. Operator Theory: Advances and Applications, 41, 279-297, Birkhauser-Verlag, Base1 (1989). Frank, L.S. and H. Norde: [l]On a singular perturbation in the linear soliton theory. Rep. Univ. Nijmegen, 1990 (to appear in Asympt. Anal.) Frank, L.S. and W.D. Wendt: [l] Isomorphic elliptic singular perturbations. Rep. no. 8003, Univ. Nijmegen, 1980. - [2] Coercive singular perturbations: asymptotics and reduction to regularly

Bibliography

541

perturbed boundary value problems. Analytical and numerical approaches to asymptotic problems in analysis, 305-318, North-Holland, Amsterdam

(1981). -

[3] Coercive singular perturbations: reduction to regular perturbations and applications. Comm. Partial Diff. Equations, 7, 469-535 (1982).

-

[4] Coercive singular perturbations: reduction and convergence, Theory and applications of singular perturbations. Lecture notes in mathemat-

ics,942, 54-64, Berlin-Heidelberg-New York (1982). -

[5] Coercive singular perturbations 11: reduction and convergence. J. Math. Anal. Appl., 88, 464504 (1982).

-

[6] Sur une perturbation singulikre en thCorie des enzymes. C.R. Acad. Sci. Paris, 294, 741-744 (1982).

- [7] Solutions asymptotiques pour une classe de perturbations singulikres elliptiques semilinkaires. C.R. Acad. Sci. Paris, 295, 451-454 (1982). -

[8] Sur une perturbartion singulikre parabolique en thCorie cinCtique des enzymes. C.R. Acad. Sci. Paris, 295, 731-734 (1982).

[9] On an elliptic operator with discontinuous nonlinearity. J. Diff. Equations, 54, 1-18 (1984). - [lo] Coercive singular perturbations 111: Wiener-Hopf operators. J. d’Analyse Math., 43, 88-135 (1983/84). - [ll] On a singular perturbation in the kinetic theory of enzymes. Delft -

Technische Hogeschool, Delft Progress, 53-66 (1985).

[12] Elliptic and parabolic singular perturbations in the kinetic theory of enzymes. Nonlinear partial differential equations and their applications: Collkge de France seminar, Vol. VII, 84141, (H. BrCzis and J.L. Lions (eds.)), Pitman, Boston 1985. - [13] Elliptic and parabolic singular perturbations in the kinetic theory of enzymes. Annali Mat. Pura Appl., 144, 261-301 (1986).

-

Freidlin, M.I. and A.T. Wentzel; [l] Random perturbations of dynamical systems. Springer-Verlag, Berlin-Heidelberg-New York, 1984. Friedman, A.: [l]Singular perturbations for the Cauchy problem and for boundary value problems. J. Diff. Equations, 5, 226261 (1969).

542

Bibliography

Friedrichs,K.O.: [l] Asymptotic phenomena in mathematical physics. Bull. Amer. Math. SOC.,61 : 6, 79-94 (1955).

- [2] Pseudo-differential operators: an introduction. Courant Inst. of Math. Sci., Lecture Notes, New York 1970. Friedrichs, K.O. and J . Stoker: [l] The nonlineair boundary value problem of the buckled plate. Amer. J. Math. 63, 839-888 (1941). Galpern, S.A.: [l] Cauchy problem for general systems of linear partial differential equations. Dokl. Akad. Nauk SSSR (N.S.), 119, 640-643 (1958). Gkding, L. [l] Linear hyperbolic partial differential equations with constant coefficients. Acta Math. 85, 1-62 (1951).

- [2] Dirichlet’s problem for linear elliptic partial differential equations.Math. Scand. 1, 55-72 (1953). - [3] On the asymptotic distribution of the eigenvalues and eigenfunctions of

elliptic differential operators. Math. Scand. 1, 237-255 (1953).

- [4] Transformation de Fourier des distributions homogknes. Bull. SOC. Math. France, 89, 381-428 (1961). Gelfand, I.M. and G.E. Shilov: [l]Generalized functions, vol. 1-111, “Fizmatgis”, 1958. English transl.: Academic Press, New York, 1964. Godunov, S.K.: [l]A difference method for numerical calculation of discontinuous solutions of the equation of hydrodynamics. Mat. Sb. (N.S.) 47 (89), 271-306 (1959) (Russian). Gohberg, I. and I.A. Feldman: [l] Convolution equations and projection methods for their solution. “Nauka”, Moscow 1971 (Russian). Gohberg, I. and M.G. Krein: [l]Systems of integral equations on a half-line with kernels depending on the difference of arguments. Uspekhi Mat. Nauk, 13 : 2, 3-72 (1958). - [2] Defect numbers, root numbers and indecies of linear operators. Amer.

Math. SOC.transl. Ser. 2, 13, 185-264 (1960). Goldenveizer, A.A.: [l] Thin elastic shells. Gos. Izdat. Tehn.-Ter.

Lit.,

MOSCOW, 1953 (Russian). Grabmuller, H.: [l]Singular perturbation techiniques applied to integro-differential equations. Pitman, San Francisco, 1978.

Bib Ziography

543

Gradshtein, I.S.: [l] Linear equations with variable coefficients and small parameters in the highest derivatives. Mat. Sb. (N.S.) 27 (69), 47-68 (1950) (Russian). Gradshtein, I S . and I.M. Ryzhik; [l] Table of integrals, series and products. Academic Press, 1980. Greenlee, W.M.: [l]Singular perturbations of eigenvalues. Arch. Rat. Mech. and Anal., 34, 143-164 (1969). Groen, P.P.N.: [l]Singular perturbations of spectra. Asymptotic Analysis (F. Verhuls, ed.), Springer-Verlag, Berlin-Heidelberg-New York (1979). Grubb, G.: [l] Functional calculus of pseudo-differential boundary problems. Progress in Mathematics, 65, Birkhauser, Boston 1986. Hadamard, J.: [l]Le problkme de Cauchy et les Cquations aux dCrivCes partielles 1inCaires hyperboliques. Paris 1932. Handelman, G.H., J.B. Keller and R.E. O’Malley Jr.: [l]Loss of boundary conditions in the asymptotic solution of linear ordinary differential equations. I. Eigenvalue problems. Comm. Pure Appl. Math. XXI, 243-261 (1968). Harris Jr., W.A.: [l]Singular perturbations of eigenvalue problems. Arch. Rat. Mech. and Anal., 7, 224-241 (1961). Harten, A. van: [l] Nonlinear singular perturbation problems: proof of correctness of a formal approximation based on a contruction principle in a Banach space. J . Math. Anal. Appl. 65, no. 1, 126-168 (1978). Harten, A., J.M. Hyman and P.D. Lax: 111 On finite difference approximations and entropy conditions for shocks. Comm. Pure Appl. Math. 29, 297-322, (1976). Heijstek, J.J.: [l]On a singular perturbation in the theory of elastic rods. Ph.D. Thesis, Univ. of Nijmegen, 1989. Helffer, B. and D. Robert: [l]Calcul fonctionel par la transformCe de Mellin et applications. J. Funct. Anal. 53, (1983). - [2] Riesz means of bound states and semiclassical limit connected with a

Lieb-Thirring’s conjecture. Asympt. Anal. 3 : 1, (1990). Helffer, B. and J. Sjostrand: [l]Puits multiples en limite semi-classique. Ann. Inst. H. PoincarC (section physique), 42, no. 2, 127-212 (1985).

5 44

Bibliography

Herglotz, G.: [l]Uber die Integration h e a r e r partieller Differentialgleichungen mit konstanten Koeffizienten 1-111. Berichte Sachs. Akad. d. Wiss. 78, 93-126, 287-318 (1926); 80, 69-114 (1928). Hirth, J.-P. and J . Lothe: [l]Theory of dislocations. Mc Graw-Hill, New York 1967. Hoppensteadt, F.: [l] On quasi-linear parabolic equations with a small parameter. Comm. Pure Appl. Math., XXIV, 17-38 (1971). IJormander, L.: [I] Estimates for translation invariant operators in LP spaces. Acta Math. 104, 93-140 (1960).

- [2] Pseudedifferential operators. Comm. Pure Appl. Math. 18, 501-517 (1965).

- [3] The spectral function of an elliptic operator. Acta Math. 121, 193-218 (1968).

- [4] The analysis of linear partial differential operators. Vol. I-IV, SpringerVerlag, Berlin-Beidelberg-New York, 1985. Huet, D.: [l] Singular perturbations of elliptic problems. Annali di Mat. Pura Appl. XCV, 77-114 (1973). - [2] Dkcomposition spectrale et opkrateurs lint5aires. Press Universitaire de

France, 1976. - [3] StabilitC et convergence dans Ies problkmes de perturbation singulikre.

Asympt. Anal. 2 : 1, 5-20 (1989). - [4] Remarque sur un thCor5me d’hgmon et applications b quelques problk-

mes de perturbation singulikre. Boll. Unione Mat. Ital., 3, XXI, 219-227 (1966). Il’yn, A.M.: [l]Difference scheme for the differential equation whose highest derivative is affected by the presence of a small parameter. Math. Notices, 6 : 2, 237-248 (196s) (Rcssian).

- [2] Matching of asymptotic expansions for solutions of boundary value problems. ”Nauka”, Moscow, 1989 (Russian). John, F.: [l] Plane waves and spherical means applied t o partial differential equations. New York 1955. - [2] A priori estimates, geometric effects and asymptotic behavior. Bull.

Bibliography

545

Amer. Math. SOC.,81 : 6, 1013-1023 (1975). Kamin (Kamenomostskaya), S.L.: [l]The first boundary problem for equations of elliptic type with a small parameter in the highest derivatives. Izv. Akad. Nauk SSSR, Ser. Mat., 19, 345-360 (1955) (Russian).

- [2] Perturbation elliptique d’un opCrateur du premier ordre avec un point singulier. C.R. Acad. Sci. Paris, 285, 677-680 (1977). Kaplun, S.: [l]Fluid mechanics and singular perturbations. Academic Press, New York 1967. KBrmAn, von T.: [l] Festigkeitsprobleme im Maschinenbau. “Encyklopadie der Mathematischen Wissenschaften”, vol. IV-4, C, 14-15, Leipzig (1907). Kato, T.: [l]Perturbation theory for linear operators. Springer-Verlag, BerlinHeidelberg-New York 1976. Keller, J.: [l]Corrected Bohr-Sommerfeld quantum conditions for nonseparable systems. Ann. Physics, 4, 100-188 (1958). Keller, J.B. and R.E. O’Malley Jr.: [l] Loss of boundary conditions in the asymptotic solution of linear ordinary differential equations. 11. Boundary value problems. Comm. Pure Appl. Math., XXI, 263-270 (1968). ICifer, J u . I.: [l] On the spectral stability of invariant tori under small random perturbations of dynamical systems. Dokl. Akad. Nauk SSSR, 235, 512515 (1977); Sov. Math. Dokl. 18, 952-956 (1977). Kohn, J.J. and L. Nirenberg: [l]On the algebra of pseudo-differential operators. Comm. Pure Appl. Math. 18, 269-305 (1965). Kolmogorov, A.N.: [l] Zufdlige Bewegungen. Ann. of Math., 35, 116-117 (1934). Korteweg, D.J. and G. de Vries: [l]On the change of form of long waves acting in a rectangular channel and on a new type of long stationary waves. Philos. Mag., 39, 422-443 (1895). Krein, M.G.: [l]Integral equations on a half-line with kernels, depending on the difference of arguments. Uspekhi Mat. Nauk, 13 : 5, 3-120 (1958). Krein, S.G.: [l] Linear differential equations in Banach spaces.

“Nauka”,

Moscow 1967 (Russian). Kucherenko, V.V.: [l] Asymptotic solutions of equations with complex char-

5 46

Bibliography

acteristics. Mat. Sb. 95, 164-213 (1974); Math. USSR Sbornik 24 : 2,

159-207 (1974). Kuman-go, H.: [l]Algebras of pseudodifferential operators. J. Fac. Sci. Tokyo

17, 31-50 (1970). Ladyzhenskaya, O.A.: [l] On equations with a small parameter in the highest order derivatives in linear partial differential equations. Vestnik Leningrad Univ. Ser. Mat. Mech. Astr., 12 : 7, 104-120 (1957) (Russian, English summary). Landau, L.D. and E.M. Lifschitz: [l] Elasticity theory. “Nauka”, Moscow 1965 (Russian); ThCorie de 1’ClasticitC. “Mir” , Moscou 1967.

- [2] Electrodynamics in continuous media. Gosizdat, Moscow, 1957 (Russian). Lang, S.: [I] Differential manifolds. Addison Wesley Publ., 1972. Lax, P.D.: [l] Asymptotic solutions of oscillatory initial value problems. Duke Math. J. 24, 627-646 (1957).

[2] Hyperbolic systems of conservation laws. 11. Comm. Pure Appl. Math., X,537-566 (1957). - [3] Shock waves and entropy. Contributions t o nonlineair functional analysis, Academic press, New York, (1971). -

Lax, P.D. and C.D. Lawrence; [l]The small dispersion limit of the Kortweg-de Vries equation. I. Comm. Pure Appl. Math., XXXVI, 253-290 (1983). Lax, P.D. and L. Nirenberg: [l]On stability of difference schemes: a sharp form of Giirding’s inequality. Comm. Pure Appl. Math., 19, 473-492 (1966). Lax, P.D. and R.D. Richtmyer: [l]Survey of the stability of linear finite difference equations. Comm. Pure Appl. Math. 9, 267-293 (1956). Lax, P.D. and B. Wendroff: [l] Systems of conservation laws. Comm. Pure Appl. Math. 13, 363-375 (1960). Leray, J.: [l] Analyse lagrangienne et mhcanique quantique. Univ. L. Pasteur, Strasbourg 1978; English transl.: MIT Press, Cambridge, Mass., 1982. Levin, J.J. and N. Levinson: [l]Singular perturbations of nonlineair systems of differential equations and an associated boundary layer equation. J. Rat. Mech. and Analysis, 3, 247-270 (1954).

Bibliography

5 47

Levinson, N.: [l]The first boundary value problem for EAU+ A(z,y)uz + B ( z ,y)u, C(z, y)u = D ( z ,y) for small E . Ann. of Math. (2) 51, 428-445

+

(1950). Lions, J.-L.: [l]Perturbations singulikres dans les probkmes aux limites et en control optimal. Springer-Verlag, Berlin-Heidelberg-New York, 1973.

- [2] ContrblabilitC exacte et homogCnCisation. Asympt. Anal., 1 : 1, 3-11 (1988). Lions, J.-L. and E. Magenes: [l]Problkmes aux limites non homogcnes et applications. 1-111. Dunod, Paris 1968-1970; English transl.: Springer, BerlinHeidelberg-New York, 1972-1973. Loomis, L.H. and S. Sternberg: [l]Advanced calculus. Addison-Wesley, %ading-London-Amsterdam-Don Mills, 1980. Lopatinski, Ya.B.: [l] On a method of reducintg boundary problems for a system of differential equations of elliptic type to regular integral equations. Ukrain. Math. Z. 5, 123-151 (1953). Amer. Math. SOC.Transl. (2) 89, 149-183 (1970). Ludwig, D.: [l]Exact and asymptotic solutions of the Cauchy problem. Comm. Pure Appl. Math. 13, 473-508 (1960). Lyusternik, L.A.: [l] On difference approximations of the Laplace operator. Uspekhi Mat. Nauk 9, No. 2 (60), 3-66 (1954) (Russian). Lyusternik, L.A. and V.I. Sobolev: [l]Elements of functional analysis. “Nauka”, MOSCOW, 1965 (Russian). Markovich, P.A.: [l]The stationary semiconductor devices equations. SpringerVerlag, Wien-New York, 1985. Maslov, V.P.: [l]Theory of perturbations and asymptotic methods. MOSCOV.,

Gos. Univ., Moscow 1965 (Russian); Thkorie des perturbations et m & h e des asymptotiques. Dunod, Paris 1968; English transl.: Springer-Verlag, 181-182, 1972. - [2] Operational methods.

“Nauka”,Moscow 1973 (Russian); MCthodes

opkrationnelles. “Mir” , Moscou 1987.

- [3] Complex WKB method for nonlinear equations “Nauka” , Moscow 1977 (Russian).

548

Bibliography

Maslov, V.P. and M.V. Fedorjuk: [l] Semi-classical approximations for equations of quantum mechanics. “Nauka”, Moscow 1976 (Russian; English transl.: D. Reidel Publ., vol. 7, Dordrecht 1981). Matkowsky, B.J. and E.L. Reiss; [l] On the asymptotic theory of dissipative wave motion. Arch. Rat. Mech. and Anal., 43(3), 194212 (1971). Mauss, J.: [l] Approximation asymptotique uniforme de la solution d’un probkme de perturbation singulihe de type elliptique. J. de MCcanique, 8,

373-391 (1969). Melin, A. and J. Sjostrand: [l] Fourier integral operators with complex phase functions, Springer Lecture Notes no. 459, 120-223 (1974). - [2] Fourier integral operators with complex phase and application to an

interior boundary problem. Comm. Partial Diff. Eqns., 1, no. 4 , 313-400

(1976). Michelson, D.: [l] Stability theory of difference approximations for multidimensional initial boundary value problems. Math. Comp. 40, 1-45

(1983). Mikhlin, S.G.: [l] On the multipliers of Fourier integrals. Dokl. Akad. Nauk

SSSR 109, 701-703 (1956) (Russian). -

[2] Multidimensional singular integrals and integral equations. Pergamon Press, 1965.

Mimura, M., M. Tabata and Y. Hosono: [l] Multiple solutions of twepoint boundary value problems of Neumann type with a small parameter. SIAM J. Math. Anal., 11, 613-631 (1980). Miranker, W.L.: [l] The computational theory of stiff differential equations. Rome 1975. Mises, von R.: [l] Grenzschichte in der Theorie der gewonlichen Differentialgleichungen. Acta Univ. Szeged., Sect. Sci. Math., 12, 29-34 (1950). Mityagin, B.S.: [l] Differential equations with a small parameter in the Banach space. Izv. Azerbaijan Acad. of Sciences, 1, 23-28 (1961) (Russian). Mock, M.S.: [l] Analysis of mathematical models of semiconductor devices.

Boole Press, Dublin 1983. Morawetz, C. and D. Ludwig: [l]An inequality for reduced wave operator and

Bib liogaphy

549

the justification of geometrical optics. Comm. Pure Appl. Math. 21 : 2,

187-203 (1968). Morgenstern, D.: [l] Singulare storungstheorie partieller Differentialgleichungen. J. Rat. Mech. and Anal., 5, 203-216 (1956). Moser, J .: [l]Singular perturbation of eigenvalue problems for linear differential equations of even order. Comm. Pure Appl. Math., vol. VIII, 251-278

(1955). Muskhelishvili, N.I.: [l] Singular integral equations. Third ed. “Nauka” 1968 (Russian); English transl. of the first ed.: P. Noordhoff N.V., GroningenHolland, 1953.

+

f(z,y,y’,A) = 0 0. Proc. Phys. Math. SOC.Japan, 21, 529-534 (1939). Naimark, M.A.: [l] Linear differential operators. “Nauka”, Moscow 1969 (Russian).

Nagumo, M.: [l] Uber das Verhalten des Integrals von Ay” fur A

Nayfeh, A.H.: [l] Introduction t o perturbation techniques. Wiley Interscience Publ., New York-Toronto 1981. Nirenberg, L.: [l] Remarks on strongly elliptic partial differential equations. Comm. Pure Appl. Math. 8, 648-675 (1955). Nishiura, Y. and H. Fujii: [l] Stability of singularly perturbed solutions to systems of reaction-diffusion equations. SIAM J . Math. Anl. 18, 1726

1770 (1987). Nother, F.: [l] Eine Klasse singularer Integralgleichungen. Math. Ann., 82,

42-63 (1920). Oleinik, O.A.: [l] On equations of elliptic type with small parameter. Mat. Sb. (N.S.), 31 (73), 104117 (1952) (Russian).

- [2] Mathematical problems of boundary layer theory. Uspekhi Mat. Nauk, 23, 3-65 (1968); English transl.: Univ. of Minnesota Lecture Notes, 1969. Oleinik, O.A. and A.I. Zhizhina: [l] On the boundary value problem for the equation EY” = f(z, y, y’) for small

E.

Math. Sb., 31 (73), no. 3, 707-717

(1952) (Russian). Olver, F.W.J.: [l] Asymptotics and special functions. Academic Press, New York-London 1974.

Bibliography

550

Osher, S.: [l] Numerical solution of singular perturbation problems and hyperbolic systems of convervation laws. Anal. and Numer. Approaches to Asymptotic Problems, North-Holland, Amsterdam-New York, 179-204 (1981). Ott, H.: [l]Die Sattelpunktsmethode in der Umgebung eines Pols. Ann. Phys. 43, 393-394 (1943). Palais, R.S.: [l] Natural operations on differential forms. Trans. Amer. Math. SOC.,92, 125-141 (1959). Pazy, A.: [l] Asymptotic expansions of solutions of ordinary differential equations in Hilbert space. Arch. Rat. Mech. and Anal. 24, 193-218 (1967). Peetre, J.: [l] Another approach to elliptic boundary problems. C o r n . Pure Appl. Math. 14, (1961).

- [2] Mixed problems for higher order elliptic equations in two variables. I. Ann. Scuola Norm. Sup. Pisa, 15, 337-353 (1961). Petrovsky, I.G.: [l] Uber das Cauchysche Problem fur ein system h e a r e r partieller Differentialgleichungen im Gebiete der nichtanalytischen Funktie nen. Vestnik Univ. Moscow SCr. Int. 1, No. 7, 1-74 (1938).

- [2] Sur I’analyticitC des solutions des syst5mes d’kquations diffCrentielles. Mat. Sb. 5 (47), 3-70 (1939). Poincark, H.: [l] Les mCthodes nouvelles de la mCcanique celeste. GauthierVillars , 1892- 1893. - [2] LeGon de mCcanique cCleste. Gauthier-Villars, Paris, 1905-1910.

Raviart, P.A.: [l]Sur la rksolution numirique de 1’Qquationu~+uu,--E((u,(u,),

= 0. J. Diff. Equations, 8 (l),56-94 (1970). Rayleigh (Lord Rayleigh): [l]On waves propagated along the plane surface of an elastic solid. Proc. London Math. SOC.17, 4-11 (1885). - [2] Theory of sound.

I. London, 1937.

Rellich, F.: [l]Perturbation theory of eigenvalue problems. Gordon and Breach, New York 1969. Rempel, S. and B.-W. Schulze: [l]Index theory of elliptic boundary problems, Akademie-Verlag, Berlin 1982. Richardson, L.: [l]Weather prediction by numerical process. Cambridge 1922.

551

BiblioTaphy

Richtmyer, R.D. and K.W. Morton: [l] Difference methods for initial value problems. Interscience, New York 1967. Riesz, F. and B.Sz.-Nagy: [l]LeCons d’analyse fonctionelle. Acad. des sciences de Hongrie, Budapest, 1953. Roseau, M.: [l] Asymptotic wave theory. North-Holland, Amsterdam-New York-Oxford, 1976. Rothe, E.: [l]Uber asymptotische Entwiklungen bei Randwertaufgaben elliptischer partieller differentialgleichungen. Math. Ann., 108, 578-594 (1933). Rudin, W.: [I] Functional analysis. Tata Mc Graw-Hill, New Delhi, 1973. Saito, Y.: [l]On the asymptotic behavior of the solutions of the Schrodinger equation (-A+Q(y)-k2)V

= F . Osaka J . Math., 14, 11-35 (1977).

Sanchez-Palencia, E.: [l]Non-homogeneous media and vibration theory. Springer-Verlag, Berlin-Heidelberg-New York, 1980. Schlichting, H.: [l]Boundary layer theory. Mc Graw Hill, 1968. Schuss, Z.: [l] Asymptotic expansions for parabolic systems. J . Math. Anal. and Appl., 44, 136-159 (1973). Schur, J .: [l] Bemerkungen zur Theorie der beschrakten Bilinearformen mit unendlich vielen Verinderlichen. J . fur reine und angewandte Mathematik, Bd. 140, 1-28 (1911). Schwartz, L.: [l]Thdorie des distributions, 1-2. Hermann, Paris 1950-1951). - [2] Thdorie des noyaux. Proc. Int. Congr. Math.,

I, 220-230, Cambrdige

(1950). - [3] Transformation de Laplace des distributtions. Comm. SCm. Math. Univ. Lund, Tome suppl., 196-206 (1952). Seeley, R.T.: [l] Extension of Cw-functions defined on a half-space. Proc. Amer. Math. SOC.15, 625-626 (1964).

- [2] Refinement of the functional calculus of Calderon and Zygmund. Koninkl. Nederl. Akad. v. Wet. Proceedings, Serf. A, 68, 521-532 (1965).

- [3] Complex powers of an elliptic operator. Proc. Symp. Pure Math., 10, 288-307 (1968). Serrin, J.: [l]Recent developments in the mathematical aspects of boundary layer theory. Int. J. Engng. Sci. 9, 233-240 (1971).

Bibliography

552

Shamir, E.: [l] Mixed boundary value problems for elliptic equations in the plane, the &-theory. Ann. Scuola Norm. Super. Pisa, 17, 117-139 (1963).

- [2] Elliptic systems of singular integral operators I. Trans. Amer. Math. SOC.,127, 107-124 (1967). Shapiro, Z.Ya.: [l]On general boundary problems for equations of elliptic type. Izv. Akad. Nauk SSSR, 17, 539-562 (1953) (Russian). Shilov, G.E.: [l]Mathematical analysis. I. Fizmatgiz, Moscow, 1961 (Russian). -

[2] Mathematical analysis. 11. “Nauka”, MOSCOW,1965 (Russian).

- [3] Generalized functions and partial differential equations. Gordon and Breach, Science Publ. New York, 1968 (Russian original, 1965). Sirovich, L.: [l] Techniques of asymptotic analysis. Springer-Verlag, BerlinHeidelberg-New York, 1971. Smith, R.A.: [l] Semiconductors, Cambridge University Press 1978. Smoller, J.A.: [l]Recent developments in the mathematical aspects of boundary layer theory. Int. J . Engng. Sci. 9, 233-240 (1971). Sobolev, S.L.: [l] MQthodenouvelle A rQsoudrele probkme de Cauchy pour les dquations linkaires hyperboliques normales. Mat. Sb. 1 (43), 39-72 (1936). -

[2] Some applications of functional analysis in mathematical physics.Leningrad, 1950 (Russian); English transl.: Amer. Math. SOC.Transl., Providence, R.I. 1963.

- [3] Cauchy’s problem for a special case of systems not belonging to the Kowalevsky type. Dokl. Akad. Nauk SSSR, 82, 205-208 (1952) (Russian). Sobolevskii, P.E.: [l] On second order equations with a small parameter multiplying the highest order derivative. Uspekhi Mat. Nauk, 19 : 6, 217-219

(1964) (Russian). Sommerfeld, A.: [l] Optics. Lectures on theoretical physics. Academic Press, New York, 1969. Srubschchik, L.S. and V.1 Yudovich: [l] The asymptotic integration of the system of equations for the large deflection of symmetrically loaded shells of revolution. J. of Appl. Math. and Mech. (P.M.M.) 26, 913-922 (1962)

(Russian). Stein, E.M.: [l] Note on singular integrals. Proceedings of Amer. Math. SOC.

Bibliography

553

8, 250-254 (1957). - [2] Singular integrals and differentiability properties of functions. Prince-

ton Univ. Press 1970. Strang. G.: [l]On strong hyperbolicity. J. Math. Kyoto Univ., 6, 397-417 (1967). Strichartz, R.S.: [l] Multipliers on fractional Sobolev spaces J . Math. Mech. 16, 1031-1060 (1967). Sweers, G . : [l] Estimates for elliptic singular perturbations in Lp-spaces. Doctoraalscriptie, Univ. of Nijmegen, 1984.

-

[2] Estimates for elliptic singular perturbations in Lp-spaces. Asympt. Analysis, 2 : 2, 101-138 (1989).

Sze, S.M.: [l]Physics of semiconductor devices. Wiley, New York, 1981. Taylor, M.E.: [l] Pseudodifferential operators. Princeton University Press, 1981. Temam, R.: [l] Sur la stabiliti et la convergence de la mithode des pas fractionnaires. Ann. Mat. Pura Appl., 4, LXXIV, 191-180 (1968). Thomie, V. and B. Westergren: [l] Elliptic difference equations and interior regularity. Numer. Math., 11 : 3, 196-210 (1968). Tikhonov, A.N.: [l] On the dependence of the solutions of differential equations upon a small parameter. Mat. Sb., 22 (64), no. 2, 193-204 (1948) (Russian). Trkves, F.: [l] Introduction to pseudodifferential and Fourier integral operators. I, 11. Plenum Press, New York-London, 1980. Triebel, H.: [l] Interpolation theory, function spaces, differential operators. North-Holland, Amsterdam, 1978. Unterberger, A. and J. Bokobza: [l] Opkrateurs de Caldkron-Zygmund prkcisks. C.R. Acad. Sci. Paris, 259, 1612-1614 (1964). Vaillancourt, R.: [l] A simple proof of Lax-Nirenberg theorems. Comm. Pure Appl. Math., 23, 151-163 (1970). Vainberg, B.R.: [l]Asymptotic methods for equations of mathematical physics. Moscow State Univ., 1982 (Russian). Van Dijke, M.: [l] Perturbation method in fluid mechapics. Academic Press,

554

Bibliography

London-New York, 1964. Van Leer, B.: [l] Upwind differencing for hyperbolic systems of conservation laws. Numerical methods for engineering, I, 137-149, Dunod, Paris 1980. Viahik, M.I.: [l]On strongly elliptic systems of differential equations. Mat. Sb. 29, 615-676 (1951) (Russian). Vishik, M.I. and G.I. Eskin: [l] Normally solvable problems for elliptic systems of equations in convolutions. Mat. Sb. 78, 326-356 (1967) (Russian).

Vishik, M.I. and L.A. Lyusternik: [l] Regular degeneration and boundary layers for linear differential equations with a small parmeter. Uspekhi Mat. Nauk 12, no. 5 (77), 3-122 (1957); English transl.: Amer. Math. SOC. Transl., (2), 20, 239-364 (1962). - [2] Asymptotic behavior of solutions to differential equations, whose co-

efficients or these ones of the boundary conditions are large or vary fast. Uspekhi Mat. Nauk, 15, 3-80 (1960). Volk, I.M.: [l] On periodic solutions of autonomous systems. Prikl. Mat. Mekh., 12, no. 1, 29-38 (1948) (Russian). Volosov, V.M.: [l]The averaging for systems of ordinary differential equations. Uspekhi Mat. Nauk, 17, no. 6, 3-126 (1962) (Russian). Wasow, W.: [l] Asymptotic expansions for ordinary differential equations. Wiley, Interscience Publ., New York-London-Sidney, 1965. Weinstein, A.: [l]The order and symbol of a distribution. Trans. Amer. Math. SOC.,241, 1-54 (1978). - [2] On Maslov’s quantization condition.

Fourier integral operators and

partial differential equations. Springer Lecture Notes in Math. 459, 341372 (1974). Wendt, W.D.: [l] Coercive singularly perturbed Wiener-Hopf operators and applications. Ph.D. Thesis, Univ. of Nijmegen, 1983. - [2] The reduced operators of singularly perturbed Wiener-Hopf matrices.

Asympt. Anal., 1 : 1, 61-93 (1988). - [3] Stability and convergence results for a class of one-dimensional homog-

enization problems. Preprint no. 701, Univ. Bonn, 1984. Wilcox, C.H.: [l] Perturbation theory and its applications in quantum mechan-

Bibliography

555

ics. Wiley, New York, 1966. Yamaguti, M. and T. Nogi: [l]An algebra of pseudodifference schemes and its application. Publ. Res. Inst. Math. Sci. Ser A.3, 151-166 (1967). Zygmund, A.: [l]Trigonometric series. I, 11. The University Press, Cambridge, 1959.

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