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
--
CIP ISBN: 0 444 88 134 4 OELSEVIER SCIENCE PUBLISHERS B.V. 1990 All rights reserved. N o part of this publication may be reproduced, stored in a retrieval system or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V.1 Physical Sciences and Engineering Division, P.O. Box 103, 1000AC Amsterdam, TheNetherlands.
Special regulations fo r readers in the U.S.A. - This publication has been registered wdth the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the U.S.A. All other copyright questions, including photocopying outside of the U.S.A., should be referred to the publisher.
No responsibility is assumed by the publisher fo r any injury andlor damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. PRINTED IN THE NETHERLANDS
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
where the constant C
> 0 may only depend
on N , s,€ 0 , y ( q ) and some deriva-
tives of q ( x ) . Thus, (16) implies the asymptotic convergence in (9) and one may differentiate the asymptotic relation (9) with respect to
3:
any number of times
without loosing the asymptotic convergence, provided that f E C r ( R 3 ) , i.e. f is smooth and has compact support. Note that (9) is not satisfactory in the sense, that it provides the asymptotic approximations to ue, which all have their support coinciding with the support o f f . This is not the case for the solution u, of (6), (7) under the assumptoin
(lo), the support of u,
being the entire R3.
A more appropriate asymptotic formula for the solution u, of (6), ( 7 ) under the assumption (10) is provided by the following argument. First, assume that
xv
In trod u c tion q
q(z)
> 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)
<
E
<
EO)
is strictly less than 1, i.e.
is an equicontraction, provided t h d
Hence, (21) yields for
EO
> 0 sufficiently
EO
EQ'
> 0 is sufficiently small.
small:
the series on the right hand side of (23) being convergent with respect to the operator norm in L(H(8-u),,(R3))uniformly with respect t o
E
E(0,~).
One can show that in fact one has also the following asymptotic relation:
SEA' = I - E Q ~ ,
E
E (O,EO),
where the family of linear mappings:
is again equicontinuous. Of course, for the zero approximation uLo) defined by (18) one has:
and moreover, both u, and u!)'
are supported by the entire W3.
Introducing
where the constant C > 0 does not depend on
E,
u, and f .
The basic difference between (16) and (26) is the fact, that the norm off on the right-hand side of (26) is the same, i.e. u i N - l ) defined by (25) converges
Introduction
xvii
at order O ( E ~ t o)u, in H(s1,,(R3),Vf E H ( s - v ) , ,(R3), thus, also for non-smooth second members f , while in (16) the larger is N
> 0 the smoother
the second
member must be in order t o have the asymptotic convergence of order O ( E ~ ) . Now, let us turn to the situation when q(x) does not satisfy condition
( l o ) , for instance, let us consider the case of q(x) z -1. Note that the formal asymptotic expansion (9) in this case takes the form: U,
-
C(-1)'E2'Akf, k10
where A is the Laplace operator, A = C a;,
aj
= 8/8xj.
However, equation (7), which ir. this case takes the form:
does not have a unique vanishing at infinity solution for f E CF(R3). One can consider two different classes of solutions satisfying at infinity the so-called Sommerfeld radiation conditions:
(29)
u,'(x) = O ( T - ' ) ,
(iaa,. f l)u,f(x)
= O ( T - ' ) , as 1x1 = T + CO,
where 8,. = d / d r is the derivative with respect t o
T
= 1x1.
Each solution u : is given by the formulae:
Thus, the right hand side of (27) does not converge asymptotically, since it does not 'know' t o which solution u,f or u; it should converge. In fact, different methods are needed, in order t o get convergent asymptotic approximations for the solutions u,f of (6), (7) when q(x)
5
qo
< 0,V x
E R3,
of course, u , f ( z ) being defined as solutions of ( 6 ) , (7) satisfying the respective Sommerfeld radiation conditions (29).
xviii
Introduction
Example 2. One of the efficient methods for solving approximately differential (and pseudodifferential) equations is the use of their finite difference approximations, which are, of course, perturbations of the approximated operators, the small parameter being the mesh-size of the uniform grid where the discretized difference equations are considered. Let us have a look at the boundary value problem:
c
-a,"u(c)
u = (0,l) = p ~ ( ~ ~ ) ,E au = {o,i}, = f (c),
cE
where 8, = d / d x , f is a given smooth function of
3:
E
r and p(x'), x' E d U ,
are given real or complex numbers. Since the solution u ( z ) of (31) is a smooth function of c E
8,too,
one
is templed t o use a higher order approximation of (31) on the grid v h =
{ c = k h , k=O, 1 , . . . , N }with integer N = h-' > 0. For instance, the following finite difference approximation of -8: has the
accuracy O ( h 4 ) on smooth functions:
where
a,,*
and a i , h are the forward and backward finite difference derivatives,
respectively, i.e.
Indeed, a straightforward computation shows that for any smooth function
u(x),c E
r,one has:
One is tempted to use (32) for solving numerically (31). However, one can not use (32) for all points on the grid the points
Xk
u h
but only for
= k h with 1 < k < N . Of course at the points 2 0 = 0 and
IN
=1
Introduction
XiX
one can use the boundary condition in (31). Still one extra boundary condition, say at the points x1 = h and XN-1 = 1-h, is missing and one has t o find this boundary condition in an appropriate way, since otherwise a discrete version of a boundary layer behavior in a neighborhood of the boundary dU = ( 0 , l ) will appear, i.e. solutions of the homogeneous equation
of the form C+q-"/h+C-q-(l-x)/h, with q = (7+@)
will emerge with non-
negligible coefficients C* . Thus, the approximation (32) along with appropriate boundary conditions is a singular perturbation of (31), h being the corresponding small parameter. Some perturbations by finite differences might destroy the basic structure and make disappear the fundamental properties of the operators which are being approximated. For instance, the basic property of the differentiation operator
8, = d/dx (x E R) is the fact that all the solutions of the homogeneous
equation &u(x) = 0 are smooth (in fact, constants). This property of 8, is preserved on the grid Rh = hZ = { x = t h , k E Z} by both the forward and the backward
finite difference approximations of 8,.
centered finite difference derivative
&,h
= (1/2)(&,h
+ a:,,)
& ,h
However, the does not enjoy
such a property anymore, since, besides the constants, also the non-smooth meshfunction (-l)"lh is a solution of the homogeneous equation &,hu(x)
=
0 on the grid Rh. The centered finite difference derivative is a non-elliptic approximation of
a,.
For the Hilbert transform
,
( H u ) ( z ):= (7ri)-1 v.p. the approximation
J. f(y)(z
- y)-ldy
Introduction
xx
is non-ellaptic (Hh is no longer invertible in (XhU)(2)
:= (Ti)-%
c
12(Rh)),
f(y)(z-y)--l(l-
while the approximation:
COS(7rh-1(z-y)),
2
E Rh
Y c R A \{s)
preserves on the grid Rh all the fundamental properties of the Hilbert transform. Example 3. Finite difference approximations of the heat equation provide a wealth of interesting situations from the point of view of singular perturbations. Let us consider the Cauchy problem:
(34) where cp(z), f ( z , t )are given smooth functions with compact support. Using the backward finite difference discretization as an approximation of
8, = a / d t on the grid W? = { t = k r , k>O}, one gets for the corresponding approximation ur(z, t ) , t E
W$
of u ( z , t ) the following singularly perturbed
recursion equations (implicit finite difference scheme):
For finding ur(x, t ) on each step of the recursion one has to invert the singular perturbation (1- &’A), r = E’
, which is precisely of the elliptic type discussed
hereabove. One might be tempted to use the centered finite difference proximating
at,,
for ap-
at since the accuracy in this case is O ( r 2 ) . This is the so-called
Richardson’s scheme. However,
at,, being a non-elliptic approximation of at,
one should not expect to have a finite difference scheme, reflecting the basic properties of the heat operator. First, there is a problem of imposing (or not) an extra initial condition, since the Richardson’s approximation is a three-step finite difference scheme. An easy Fourier analysis shows that actually one has to consider the corresponding discretized problem in this case on
W$ x RZ not as
xxi
Introduction
an initial, but as a boundary value problem with just one boundary condition at t = 0 and another one either a t t = co (the solution vanishes as t
< 03.
or at t = T
--+
+co)
In both cases the smooth part of the solution will be 'pol-
luted' by the non-smooth part generated by the solutions of the homogeneous equation having the form: v,(i,t)
= ( - l ) ( T - * ) /(T( ~ + T ~ A-~~)A' )/ -~( ~ - * ) l ~=$ ( z ) = (-i)(*-*)/~ (~;-,((1+~21t14)1/2
+T1t12)-(T-*)/r~z+E~)
(I,t),
> 0, the oscillatory non-smooth factor ( - l ) ( * - * ) / ,being always
for any finite T present.
A good scheme having the accuracy O ( T ~is )the following one: (36)
{
t
7
+
("
(T/2)A2 - A)@t,,)u,(z,t)
(
U(t,O)
= cp(I),
2
-+
= (l +
-A))f(i>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
<*
=
...,<:
(<;,
:
*
<
l
=
1 1
(lC1l
=
...+ 5 ' 5 n n
< I < * +
2
:
+...+l
the hermitian product in Cn :
Euclidean norm in Cn
U
:
open subset of Euclidean space Rn
uc
:
complement of a subset U in illn
U
<
the complex conjugate of
X
CC
V
means that the closure n
:
of
u is a compact subset
of an open set
" c RX z z+
:
set of all integers
:
set of non-negative integers
Zn =
ZX..
:
2:
.xZ
:
set of all the n-tuples m
set of n-tuples a
Znf+: set of n-tuples m
8
=
.
(a1,. .,a
(my,.-.,m
)
(ml,.. . ,m
7
1
=
...a n !
al!
:
length of a E Zn
)
with m . E 2 , 1 5 j 5 n I
with a , E Z + 3
)
with m , E Z, 1 5 j < n. mn E 1
5 a : means a . 5 B ~ I,< j i n for a , E~ zn
la/ = a +...+a
a!
=
=
z+
Notation
2
...xa
xa = x"1
sa
c1a~ ...E nan
=
h E (0,h0 1,
<
hZn
=
:
E
mn,a
for x E
for E, E I R ~
cL
5'
E : Z
E : Z
E (O,E 0 1 small positive parameters
one parameter family of grids {x =(xl,...,xn), x 1 5 j S n } with m . E Z, 1 5 j < n , h
7
Tn Erh
mn
:
dual of
a"
:
<,+
,...,
aEl. *-at;
'j
= Dal.. .Dan
X.
El...~cI a
For a u(c()
ern
1
with D
-ia/ax ., 1 5 j 5 n 1
-ia/aSj, l < j ~n
=
function u ( x ) :
aau
=
=
=
(sometimes denoted by D) with D
xn
Da = D"1
LI
3
, i.e. one parameter family of n-dimensional toruses
family of grids in IF?
x1
a
hm.,
(also denoted 1 , i.e. of all X,+ m . E Z, l S j < n , mn x = (x, x 1 , x . = hm '+ n 1 j' I a 15 j S n =.':2 . .a (sometimes denoted by a) with a X . = a/ax,, I 1 xn 1 al a with a = = a/aSj, l < j < n
5 :D
=
{ 5 = ( 5 , ,..., S n ) E I R n , ( h S 3 . ( S n ,1 5 j ~ n ~ , h E ( 0 , h o ~ , T ~ = T ~ , 1 5-
x,h,+
a"
%
j
E (O,hol
v
X u =
D u = (D
U,
x1
grad
u =
...,DX
U)
..,au/axn)
(au/ax,, . =
(-iau/axl,...,-iau/axn)
n m
Taylor series for u E C
:
Leibnitz'formula:
Q(x,D)
:
linear partial differential operator polynomial in D with coefficients in
m
C
(U), i.e.
3
Notation
If some q (x) with la1 = m does not vanish identically, then m is said to be the order of Q(x,D).
When q (x)
qa,
va,
la1 5 m, do not depend on x E U,
then we write Q(D). t Q(x,D)
:
transpose of Q(x,D), defined as follows
Q(x,D)*
:
adjoint of Q(x,D) defined by Q (x,D)*U (x)
C
=
Da (q: (x)u (x)1
la14m where qi(x) x E 2
A = (a/ax,)
supp u
:
is the complex conjugate of q, (x),
=
((11
5 m,
u.
+...+ (a/axn)’
:
the Laplace operator in variables x E Rn
the support of the function u(x), i.e., the closure of the set of points at which u(x) does not vanish
k -
C (U)
:
space of k times continuously differentiable complex-valued functions in the closure of U, where either k E Z or k =
Ckny(;)
:
subspace of C k ( t ) of functions whose derivatives of order k satisfy the
HBlder condition with 0 < y < 1, i.e.
~ x - ~ ~ - ~ I D ~ ( x ) - D<~ m, u ( V~a), Ila1 SUP Y X,YEU,X#Y m
=
k
m
C (U) 0
:
subspace of C ( U ) of functions with compact support in U; the m
elements of S(nn
+m
:
)
C
0
(U) are also called test-functions in U
the Schwartz space of complex-valued
m
C
functions u(x), x E En,
satisfying the conditions:
L (U)
P
:
Lebesgue space of (equivalence classes) of functions u such that lu(x)
m
L (U)
:
Ip
is integrable over U, 1 5 p 5
m,
i.e. the following norm is
Lebesgue space of (equivalence classes) of essentially bounded functions u in U, i.e. the following norm is finite:
Notation
4
where ess sup is the supremum of (u(x)1 over X E W E with E the set of Lebesgue measure zero where lu(x) I may be +-.
"
= f u(x)v*(x)dx (u,v) L2(W
:
the scalar or hermitian product in L2(U), also
denoted (U,V)~,where v*(x) is the complex-conjugate of v(x), x € U. 0
x,h
(Ox
=
,h' '. . "xn,h
)
:
the shift operator 3n the family of grids
4 , (Ox
hu) (x) = u(x+he ) . k k' -1 2 = h ((Ox h-l),...,(Ox - 1 ) ) = h (Ox,h-l) : forward finite x.h 1' nth difference derivative with respecc to x E IF(. (lattices)
-1
a*
-1 h-l((l-0-l ), ..., (1-0 h)) = h-'(l-O-' ) : backward finite x,h 3,h xn difference aerivative with respect to x E
=
x,h
-i2 x,hD* = ia* x.h x,h' D
x ,h
=
<(h,S) = (51(h,S1),..-,5n(h,Sn))ICk(h,Sk) = (ih)-'(exp(ihC,)-l)
:
Fourier
transform of D x,h' 5*(h,5)
(5;(h,S1) ,...,5~(h,Sn)),5;(h,Sk)
=
=
(ih)-'(l-exp(-ihE k) )
:
Fourier
transform of D* x,h'
~ ( 5 =) w*(s)
F x-ts -1 F
1 ( 51)
(0
,...,wn(Sn)),
(W~(C~),...,W~(C~)),
=
~ ~ ( 5 =, )-i(exp(iSk1-1). w;(ck)
=
-i(l-exp(-iEk)).
:
Fourier transform, Fx-tSu = ; ( 5 ) , ;(E)
:
F-l inverse Fourier transform, S+xu
=
=
f u(x)exp(-i<x,<>)dx. Rn
u(x),
5-tX
u(x) = (2n)-" f
;(<)exp(i<x,S>)dS.
an F x-tS,h
:
discrete Fourier transform on the lattice
Gh(<)
=
hn Z
X(
, Fx-t5,h~=
-h u (5),
u(x)exp(-i<x,C>).
X€<
-1 5 -1 a,h;h F : inverse discrete Fourier transform, F E-tx,h
u(x)
-h
=
u(x),
(21~)-~fu (5)exp(i<x,5>)d5. Tn 5 ,h The Poisson formula : relates the integral and discrete Fourier transforms =
as follows: iih(<) =
C
kEZn
;(5+21rh-'k).
5
CHAPTER 1
MANIFOLDS, FUNCTIONAL ANALYSIS, DISTRIBUTIONS
In this chapter basic definitions and results concerning smooth manifolds, classical functional analysis and distributions are presented. Since mostly this material belongs to the standard background, the proofs are given only
of the statements involving one or several small parameters, which are extensions of classical results to the situation characterized by the presence of these parameters and where the uniformity (equicontinuity) is needed for further applications to singular perturbations and their finite difference approximations.
1.1.
Manifolds
Let M be a set. Definition 1.1.1.
An a t l a s o f
CZUSs
Cp (p 2 1) on M is a cOZk?CtiOn
Of
p a i r s ( u , , + , ) ( j ranging in some indexing s e t ) s a t i s f y i n g t h e foZZowing 3
3
conditions: (i) Each
u . i s a s u b s e t of M and t h e { u . } cover 3
3
M.
(ii) Each 4 . is a b i j e c t i o n of U . o n t o an open s u b s e t +.(u.)of an 1
EucZidean space Bn and f o r any j (iii) The map
3 ,k
the set
+ 1 ( U1.n U,) ,
3
1
is open in
mn .
is a c P - b i j e c t i o n of t h e open s e t +,(uj n u,) i n nnonto t h e open s e t , ) i s c a l l e d a c h a r t of t h e a t l a s . (U fl u,) cnn. Each p a i r ( U . k j 3 1 I f a p o i n t x E M l i e s i n U . t h e n ( u , , + , )is s a i d t o be a c h a r t a t x.
+
,+
3
3'
3
It will be assumed that each t.wo points x,y E M have non-intersecting charts at x and y , respectively. Definition 1.1.2. Given an open s u b s e t
u
c M and a b i j e c t i o n
+
:
u
-P
u'
w i t h U' an open s u b s e t in m n , (u,+) is s a i d t o be compatible w i t h t h e
1. Manifolds, Functional Analysis, Distributions
6
a t l a s {(u.,$.)} if each map Oj$-' i s a c P - b i j e c t i o n o f t h e corresponding 3 3 d' ( a s d e f i n e d i n ( i i i ) ) .
open s u b s e t s in
Two a t l a s s e s a r e s a i d t o be compatible i f each c h r t of one a t l a s i s compatible w i t h t h e o t h e r a t l a s . An equivalence c l a s s o f a t l a s s e s o f c l a s s
s t r u c t u r e o f CP-manifold on
M,
on
Cp
M
is s a i d t o d e f i n e a
t h e dimension n of Euclidean space Rn i n
D e f i n i t i o n 1.1.1. being t h e dimension o f t h e cP-manifold A chart $ : U
x E
+
M.
. ..,on (x),
i s given by n coordinate f u n c t i o n s $ ( x ) , 1
Wn
u.
Examples 1.1.3. m
i s a C -manifold, whose a t l a s c o n t a i n s j u s t one
lo. Euclidean space Wn chart.
2O. Sphere
n-1
S
C
n IR
,n
Sn- 1
> 2,
= {x E
nn!
1x1 = 11 i s again a
m
c -
manifold, whose a t l a s c o n t a i n s a t l e a s t two c h a r t s ( U ,,$ , ) I , j = 1,2 t h e -1 1 1 maps $ . , j = 1 , 2 , being t h e s t e r e o g r a p h i c p r o j e c t i o n s onto two copies of n-1 B
.
m
Further c i r c l e S1 i s a C -manifold,whose a t l a s contains one c h a r t given by t h e angle coordinate 8 =
0-l
:
n1
-f
S
1
.
nn i s s a i d t o be a k-dimensional manifold n , i f in each c h a r t u a t x, t h e r e imbedded in Euclidean space Wn , k
D e f i n i t i o n 1.1.4. A s e t M c
e x i s t c o n t i n u o u s l y d i f f e r e n t i a b l e f u n c t i o n s f.: U 1
-f
W ,
1 5 j 5 n-k such
t h a t t h e i n t e r s e c t i o n u Cl M is d e f i n e d by t h e equations f . = 0 , 1 5 j 5 n-k 1 and, moreover, t h e v e c t o r s IVfi)lbj5n-k are l i n e a r l y independent.
If
f ,E 3
cP, p
2 1, t h e n M is a k-dimensional
cP-manifoZd imbedded in mn
.
In each c h a r t U 3 x one may introduce l o c a l coordinates by solving t h e equations f . ( xl,...,x
= 0 , 1 2 j 5 n-k
)
1
a s f u n c t i o n s of o t h e r
k
and by expressing n-k coordinates x . 1 c o o r d i n a t e s , t h e i m p l i c i t f u n c t i o n theorem being
a p p l i c a b l e h e r e , s i n c e v e c t o r s { V f . ( x ) ) l b j j n - k a r e l i n e a r l y independent, 1 V X E U .
Example 1.1.5. The r o t a t i o n group S O ( 3 ) of Euclidean space W3
m
i s a C -manifold 3 9 homeomorphic with ( r e a l ) p r o j e c t i v e space P ) imbedded i n t o W
Let M
mx
C
Wn
be a jc-dimensional CP-manifold
imbedded i n Wn
(which i s
.
. Tangent
a t each p o i n t x E M i s defined a s t h e orthogonal complement of n-k Euclidean subspace R spanned by v e c t o r s {Of, (x))lsj,
1.1. Manifolds
Vectors in TM
7
are also said to be tangent of M at point x.
Tangent space TM
can be also defined in a different way,
A map x = $(t), t E (t t ) 5 W , tl < 0 < t2, $(t) E M is said to be a 1' 2 differentiable curve on 14, if $(t) is a differentiable function of t.
are said to be equivalent, if $ . ( O ) = x I
Two curves $.(t), j = 1,2, 3
and lim t-'(Q1(t)-Q2(t)) = 0. t 4 Definition 1.1.6.
The equivalence c l a s s o f d i f f e r e n t i a b l e curves $(t), $ ( O )
=
x, i s s a i d t o
be a tangent v e c t o r t o M a t p o i n t x E M. The s e t of a l l tangent v e c t o r s a t x E M equipped w i t h t h e n a t u r a l s t r u c t u r e of a v e c t o r space is s a i d t o be
tangent space t o M a t p o i n t x E M. m
Let U
C
M be a chart on
C
-manifold M and let $(t)={$l(t),...,$n(t,)
be
acurveonM, $ ( O ) =x.ThenthenumbersF,.= $ . ( O ) ,
l<jZn with $.(t) = (d/dt)$.(t) 3 3 3 1 are said to be coordinates of the tangent vector 5 E TM at point x E M.
Union U TM of tangent vector spaces at each point x E M has a xEM x natural structure of a differentiable manifold of dimension 2n, where n is the dimension of M. Definition 1.1.7.
u xEMTMx i s c a l l e d tangent bundle o f cP-rnanifold M and i s denoted by
TM.
TM is again a CP-manifold whose dimention is twice the one of M.
A generic point in TM is the pair (x,S) where x E M and 5 E TM tangent vector of M at
X.
Local coordinates on TM may be introduced in the
following fashion. Let x l , ...,x Ux) and
El, ...,E
iS a
be local coordinates on M (in some chart
the components of a tangent vector 5 of M at x. Then 2n
...,xn,S1,...,Sn)
real numbers (xl,
are local coordinates of a point
(x,E) E TM. The map
is said to be the natural projection. -1 Inverse image p (x) of x E M is tangent space TM
X'
called also the
fiber at x of tangent bundle TM. If M is imbedded in Euclidean space, then the Euclidean metric naturally generates arc length and angle measurements, volume measure etc. on M. All these quantities can be expressed in terms of length of tangent
1 . Manifolds, Functional Analysis, Distributions
8
vectors, i.e. in terms of a positive definite quadratic form (in 5 ) given on each fiber TM
:
TMx 3 5
(1.1.2)
-f
<<,6> E IR+.
For instance the arc length l ( y ) of a curve y:
[tl,t21 3 t
+
x(t) E M on
M, is given by the formula: l(y)
(1.1.3)
t f J2 dt.
=
Definition 1.1.8.
A cP-manifold M with a given positive definite quadratic form <<,c> on each tangent space TM is said to be a Riemannian manifold, quadratic form < 5 , 5 > being the Riemannian metric on M. Remark 1.1.9.
...,x
Let U be a chart of an atlas on M with local coordinates xl,
in U.
Then Riemannian metric is given by the formula: 2 ds =
(1.1.4)
’’
Z a . (x)dx.dx lSj,ksn k’
...,dx
where (dxl,
I lajk(x) I I
)
a . (x) = a (x) kj
are the coordinates of a tangent vector x, matrix
is posqtive definite and of class CP-’.
Let f
:
M
+
N be a map from differentiable manifold M into differentiable
manifold N (bothof class $)
. Then
f is said to be differentiable (of class Cp) if
it is given in local coordinates on M and N by differentiable (of class
C)‘
functions. Definition 1.1.10.
Let f
M
+
N be differentiable (of class
and let Ji
:
iR
:
-f
cP) map. Let (x,c) E
M be a curve on M such that $(O) = x, $ ( O )
=
TM be given 6. Define
The map
is called the derivative of differentiable map f
: M
+
N
at point x E M.
It is readily seen that f ( 5 ) does not depend on the choice of a *x.
curve $(t) on M with $(O) = x,
Ji(0) =
5.
1.1. Manifolds
Let x =
...,yn) be local coordinates on M and N, ...,Sn) and ( q 1 , ...,q ) be the components of
(Xl,...,~n),
respectively and let ( E l ,
9
y = (yl,
respectively. tangent vector 5 E TMx and tangent vector TN f(x) ' Then one has:
,
Taking the union of all maps f,
x E M, one gets a single map from
X
tangent bundle T M into tangent bundle TN: f,
:
TM
f,(S)
TN,
-f
=
f, (6) for 5 E TMx. X
The map f, g
:
N
-+
:
TM
TN is again differentiable. Furthermore, if f : M
-f
K, h = g
f
o
:
M
+
K, then it holds: h,
= g,
o
-f
N.
f,.
Now a useful concept will be introduced. Let M be a $-manifold (p t 1). Definition 1.1.11. A p a r t i t i o n of u n i t y (of c l a s s cp) on a cp-manifold M c o n s i s t s of an open
covering { u . } of M and a system of functions $ .
1
3
:
M + R satisfying the
conditions: (i) For aLZ x E M one has:
I$.
1
(x) t 0 .
(ii) The support of $ , i s contained i n U.. 3
1
{u.} of
(iii) The covering
1
M i s finite.
(iv) For each point x E M one has:
z i If
$.(XI = 1.
1
{vk } i s another covering of
t o { u , } i f each v
M, then
i s contained i n some
it i s said t o be subordinated
u..
1 k 1 Each compact differentiable manifold admits a differentiable partition
of unity. One can consider CP-manifolds with boundary which are defined in the same way as previously with the only difference that the bijections $I. in 3
Definition 1.1.1
in this case are either between a chart U . and an open set
in Rn or between U .
...,x
If (xl,
1
)
n
3
M and some set in Rn in Euclidean space Wn
.
are local coordinates on a CP-manifold M in a neighbour-
hood of one of its boundary points, then the boundary aM of M in this neighbourhood may be given as the collection of the points (x~,...,x~-~,~).
10
1. Manifolds, Functional Analysis, Distributions
The boundary a M is a CP-manifold, whose dimension is n-1. A typical example of a CP-manifold with boundary is an open domain U
in Euclidean space Rn whose boundary aU is a CP-hypersurface of dimension n-1. 1.2. Functional Analysis In this section a (part of the) classical functional analytic background and its extension to the situation with a parameter dependance are presented, which will be useful later on in the following chapters. 1.2.1. Metric Spaces and Contractions Definition 1.2.1.
An a b s t r a c t s e t
x i s s a i d t o be a (complete) m e t r i c space, or simply, a p : x x x m+ i s w e l l - d e f i n e d , which X
m e t r i c space, i f a d i s t a n c e f u n c t i o n
-f
s a t i s f i e s the following conditions: (i) I d e n t i t y p r o p e r t y : px(x,y)
0 i f f x = y.
=
(ii) Symmetry: pX(x,y) = px(y,x). (iii) T r i a n g l e i n e q u a l i t y
p (x,y)+p (y,z) 2 p (x,z). X X X c x such t h a t p (x ,x ) + 0 (iv) Completeness: For each sequence {x v v21 X V l J f o r v , -t~ +m, t h e r e e x i s t s an element x E x such t h a t p (x,x ) -t 0
x
for v
v
+ +m.
Only complete metric space will be considered. If a set X ' equipped with a distance function p is not complete, then one
may also take the completion X of X' with respect to the convergence concept (topology) defined by
p
on X'.
Definition 1.2.2.
L e t x and Y be two m e t r i c spaces and l e t t h e map x 3 x + f ( x ) = y E Y be d e f i n e d on some s e t M c x and valued i n some s e t i n Y . The map f(x) i s s a i d t o be continuous a t some p o i n t x E M, i f f o r each E > 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)) <
In other words, f : M sequence {x }
v v>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)
)
<
6.
X is continuous at xo E M iff for each -f
0 as v
+ m,
one has: p Y (f(xv),f(xo))-to
+ m.
Definition 1.2.3.
Two m e t r i c spaces
one-to-one
map f
x and :
x
+
Y Y
are s a i d t o be homeomorphic i f t h e r e e x i s t s a
o f x onto
Y
such t h a t f
:
x
+
Y and f-l
: Y
+
x
1.2. Functional Analysis
are c o n t i m o u s a t each p o i n z o f
X,
11
respectively of
Y.
Definition 1.2.4.
Let G. 5 I R ~be an open s e t . We s h a l l c o n s i d e r continuous f u n c t i o n s x(w), x
:
G.
+
x valued i n a m e t r i x space x w i t h t h e d i s t a n c e j i m c t i o n
p
Assume
X'
t h a t x is a d d i t i o n a l l y equipped w i t h a ,family of d i s t a n c e f u n c t i o n s
p
w'
which have t h e f o l l o w i n g p e r y e r t i e s : (i) For each w E G. t h e topology d e f i n e d by p
i s e q u i v a l e n t w i t h t h e one
induced by p x , i . e . t h e r e e x i s t s a c o n s t a n t y ( w ) such t h a t (y(w))
-1
PX(X,Y) 5 PU(X,Y) 5 y ( w ) p X (x,y),
(ii) The f u n c t i o n 51 3 w
+
p,(x(w),y(w))
v
(x,y) E
x
x
x
i s continuous f o r each couple o f
continuous f u n c t i o n s x ( w ) ,y(w), valued i n X. The m e t r i c space x equipped w i t h such a d i s t a n c e f u n c t i o n
p
i s said
t o be c o n t i n u o u s l y parametrized by t h e parameter s e t G. and i t is denoted by xw, w E G.. Example 1.2.5. kl o . Let X = C (U), U = (O,l), k 2 0 integer, be the set of all k-times continuously differentiable functions and let p (u,v) be the following X
distance function:
c max laJ(u(x)-v(x))I . O<j
=
with all the derivatives up to the order k. Let G. = (0,l)be the parameter set and let functions of
E
E
E (0,l). For continuous
valued in Ck(E) introduce the following distance functions
Pk,E (u,v):
Obviously, P ~ , ~ ( u , vsatisfies ) the conditions (i), (ii) in Definition 1.2.4.
The set of all continuous functions ( 0 , l ) 3
with distance function p ( 0 , k ) ,E
imbedded in Co to
E
u E Ck(6) equipped
k,E (0,k), E (U) has a structure of a vector space and it is -
Obviously, C
E +
hereabove, will be denoted by C
-
(U)
.
(e), the imbedding being continuous uniformly with respect
E (0,l).
2'. Let IRl C lR1 be a family of grids and let Ct(lR1 ) be the set of x,h x ,h all continuous functions of h E (0,l) defined on R 1 = hZ and valued in x,h
1. Manifolds, Functional Analysis, Distributions
12
there exists a constant C > 0 which does not depend on u E C
(0,O)
(ii) ,E
E
E (0,l) and
( u , O ) , the latter being
(AEu,O) 6 Cp
and such that p
O,E
OJE
valid with C = 1, since
I
(2~J-lexp(-(x-yj/~)dy < 1,
V
E (0,l).
E
u
-
Furthermore, it is readily seen that AE
: C
(0,O)
(U)
+
,E
c ( 0 , 2 ) ,E (U)
is an
ecpihomeomorphism (equiisomorphism in this linear case), i.e. that there exists a constant C > 0 which does not depend on U E C
-
(0,O)
E
E (0,l) and
(U) and such that ,E
-1
PO,€
(u,O) 6 p
2t E
V
(u,O),
(AEu,O) 5 Cpo
E
E (0,1),
,E
V u E C
2
. Let xh =
( 0 , O ) ,E
Ck(Rk,h) be as in Example 1.2.5.
the backward finite difference derivative, With a* x,h -1 1 (a:,hu) (x) = h (u(x)-u(x-h)), x E Rx,h , V
introduce the symmetric finite difference operator $,h,
Xx,h
=
(112)(ax,h+a:,h)
It is readily seen that V
ax,h
:
Ck+l 1 h (mx,h
+-
k 1 'h("x,h
' k
2 0, h E ( 0 , l )
is a family of equicontinuous mappings. h The family of mappings A , h is a family of equiisomorphisms, V k 2 0. Indeed, one has:
where
and it is easy to check that
h E (0,1)
(U)
I
1.2. Functional Analysis the metric space of the sequences u : Z
CL(mi,h
-f
13
C with the distance function p
X'
being equipped with the following family of distance functions
)
'k,h'
-1
(u(x+h)-u(x)) is the forward finite difference 1 nx,h Obviously, C;(ni,h) is a family of metric spaces with the natural
where (aXlhu)(x) = h
derivative on the grid
structure of a vector space, parametrized by a parameter set (0,l)3 h. Furthermore the space Ck (E) of all k-times continuously differentiable
?ii
is continuously imbedded in Ck(R1 ) by taking the h Fc_.h of each u E C ( R) , this imbedding being restriction TI u(x) onto x E lR1 h xth continuous uniformly with respect to h E ( 0 , l ) . functions on
Definition 1.2.6.
Let x
be two families of metric spaces parametrized by a c
,yw
with families of distance functions A family of continuous maps,
p,
mn, equipped
and uw, respectiueZy.
is said to be equicontinuous at a point x0 ( w ) , if f o r any E > 0 there exists a 6 > o such that p (A~(X),A~(X ) ) < E as soon as p (x,x ) < 6, uniformly 0 w o with respect to w E n, i.e. 6 = ~ ( E , x ) can be chosen independently of 0 w E a. x , Y ~ ,w E a, are said to be equihomeomorphic, if there exists an equicontinuous family of one-to-one maps A" : xu + yw, such that : yw + xw is equicontinuous, as well.
CAY-'
Example 1.2.7.
lo. With X
= E
C
(0,0),E
(u)defined
in Example 1.2.5
,
consider the following
family of maps A€: 1 (A'u)
(x) :=
I
-1 u(y) ( 2 ~ ) exp(-Ix-y//E)dy.
0
men C
(0,O)
: .E
c
( 0 , O ) ,E
6) c ( 0 , O ) +
(u)is a vector
-
,E
(U) is equicontinuous. Indeed, since
space and A E is linear, it suffices to show that
1 , Manifolds, Functional Analysis, Distributions
14
(Ah)-'
:
Ch(Rx,h) k 1
+
Ch k+2 (Rx,h), 1 h E (0.1)
is an equiisomorphism. Furthermore, one has: 2 (1/2)exp(-lxl)-G (x)=O(h ) , h
h
+
0,
uniformly with respect to x on each compact set in R
1 X
.
Definition 1.2.8.
Let
x be a metric space.
A map
T : X + X
such t h a t pX(T(x) ,T(y)) 2 BpX(x,y),
V (x,y) E X x X
w i t h some 8 E [o,I), i s said t o be a contraction. Let Xw be a family of metric spaces parametrized b y of maps: T
:
Xu
+
Xu,
w E
5
Rn
. A family
R
i s c a l l e d an equicontraetion i f t h e r e e x i s t s a nwnber e E not depend on w E n and such t h a t
[0,1)
which does
The following result is due to S. Banach and is known as the fixed point theorem. Theorem 1.2.9.
Let T
:
X
x be a contruction. Then there e x i s t s a unique element
-+
xo E X,
such t h a t xo = T(xo).
The following statement is a useful extension of the Banach contruction mapping theorem, which can be proved by exactly the same argument as Theorem 1.2.9. Theorem 1.2.10. I f TW
:
Xw
-f
(i.e. R 3 w
Xu is an equicontraction and T depends continuously on w E R,
+
p,(T,(x(w),y(w))
is continuous f o r each couple of p o i n t s
1.2. Functional Analysis (x(w),y(o)) E
x u ) , then the fixed point x
x
0
15 (w)
of T
o
:
x
w
+
x
w
depends
continuously on w E R as well. Example 1 . 2 . 1 1 . As an application of Banach's fixed point theorem (Theorem 1 . 2 . 9 1 ,
consider
the following initial value problem with a parameter E E ( 0 , l ) : 2
u (t)+u (t)-UE(t)-f(t) = 0, t > 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
Now, choose
I$[
so small that it holds:
With r and @ chosen to satisfy (1.2.41) with Ck, k = 2,3 defined by (1.2.42), operator T defined by (1.2.32) is a contraction in metric space 1 so that (1.2.32) has a well-defined solution w(x) which decreases exponentially as x -f +-, along with its derivative, as a consequence of the 1 construction of M r;a' Furthermore, it is easily seen that, in fact, w(x) is twice continuously differentiable (in order to check it, one just has to differentiate twice the equation (1.2.32)) and, as a consequence of (1.2.30) and the fact that
.
f ( 0 ) = 0, one finds that w E C2 (F ) a + Thus, for I $(XI) I sufficiently small the boundary value problems (1.2.30) have well-defined solutions w(x',x), (x',x) E aU O(exp(-ax)), as x
x
z+,which are
+-.
+
Besides, as a consequence of (1.2.41), one has for w(x',x) the following estimate: (1.2.43)
Il k ;o
I~w(x',*)
< 2C1C21$lr
k = 0.1,
where the constants C1 and C2 are defined hereabove, so that one also has: (1.2.44)
Iwx(x',O)
I
< 2C1C21$1.
As a consequence of (1.2.30) and the fact that f(0) = 0, estimate
I . Manifolds, Functional Analysis, Distributions
22
(1.2.43) also holds for k = 2. It can be shown, but will not be done here, that, in fact, the boundary value problems (1.2.30) are uniquely solvable in the class Of exponentially decreasing functions for x
-f
+-,
v2 E R in El2, intersect the stable (for t
-f
iff the lines ($(x'),v2), +-)
invariant manifold (Curve)
X- of the first order autonomous system (1.2.45)
1
(t) = v (t), 2
2
(t) = f(v(t)),
t > 0,
the solutions v(t) belonging to X- being asymptotically stable for t Obviously v(t)
-f
+a.
0 is a saddle point type stationary solution of
(1.2.45), so that (1.2.45) has also one unstable invariant manifold X+ in a neighbourhood of the origin. Now, looking for U(E,X) in form (1.2.29) with w(x',x)
as above, one gets for (1.2.46)
:
au
x
z+defined
the following boundary value problem:
V(E,X)
L(E,x,~~) = vF(E,v,Ev~), x E
u , v(E,x')
=
0, x' E
au,
where we have introduced the notation: (1.2.47)
+ a(E,x)Eax
L(E,x,~,) := -E2a:
+
b(sA
with a(E,x) := f V
(1.2.48)
(w(E,x),EW
z
W(E,X) =
(€,XI)
2
t
b(E,x) := f (w(E,x)rEwX(ErX)) I v1
w(x',lx-x'I/E)
X'Eau and with (1.2.49)
F(EV,EV
)
=
f(w(E,x)+v,~w (E,X)+EV )-f(W(E,X),EWX(E,x))-
-EV
f
x v
2
(W(E,X),EWx(E,X))-Jf,J (W(E,X),EW (€,XI). 1
Obviously, one has (uniformly with respect to 2 2 F(Ev,Ev~) = O(V + ( E v ~ )1, i.e. for v E C
(0,1); E
El:
2 as v +(Ev~)
-f
0,
(a),with < r
one has:
s
Cr2
1.2. Functional Analysis
23
wirh some constant C = C(r) which is bounded for r E (O,ro] with V r0 < Furthermore, as a consequence of (1.2.48) and the exponential decreasing of W(E,X) for
E +
0 uniformly with respect to x E U ' c c U, one finds:
where we have denoted: (1.2.53)
=
$(XI)
lim w
(€,XI)
= w (x',O),
X'
E aU,
E"0
and the arrows mean the uniform converqence on (1.2.44),
Thus,
(1.2.52), (1.2.53) yield:
2 b(E,x) 2 yo > 0,
(1.2.54)
i'.
v x
E
u,
v
E
E (O,Eol,
(v) is continuous and fv (0) = y2 > 0. V1 2-1 Furthermore, partial integration yields for V v E C (U), v(x') = 0,
provided that E~ > 0, given that f
x' E
au,
-
v : U
(1.2.55)
+
R and L ( E , x , ~ ~ defined ) by (1.2.47):
2 2 Iv(x)L(E,x,~)v(x)dx = I[(Ev~)+ ( ~ ( E , x ) - ( E / ~ ) ~ ~ ( E , x ) )Idx, )v U
U
where a(E,x) and b(E,x) are defined by (1.2.48). The chain rule yields: (1.2.56)
Ea
(€,XI
f (W(E,X),EW (E,X))EW (E,x)+ v v 1 2 +f (W(E,X),EW (€,XI) E 2w (E,X). v 2v 2 xx
=
Therefore, using (1.2.43) with k = 1,2 and (1.2.56), one finds:
where the constant C > 0 does not depend on Choosing (1.2.58)
I $ (x') 1
6
E (O,Eo] and $(XI).
so small that
2 b(~,x)-(~/2)a(E,x) 2 (1/2)y0 > 0,
one gets:
Therefore, the solution of the boundary value problem: (1.2.60) is unique.
L(E,x,~~)v(x)= g(x),
x E
u,
v ( x ~ )= 0 ,
E
au
I. Manifolds, Functional Analysis, Distributions
24
-
We are going to show that for each g(x) continuous function onE there exists a solution V(E,X) with
SUP llvlic (6) 5 C < O<E<E (0,2),E
provided that
E~
is
0
sufficiently small. Let us rewrite (1.2.60) equivalentlv in the following fashion: -E
2 2 2 vxx+asvx+y v = g(x)+(a-a(c,x))sv + ( y -b(E,x))v,
x E U,
(1.2.61)
E au,
v(E,x') = 0 ,
(01, y2 = fv ( 0 ) are the same as hereabove. 2 1 Notice that the following estimates hold with some constant C > 0:
where a = f
V
as a consequence of the properties of w(x',x) constructed hereabove. Let GE(x,y) be the Green's function for the boundary value problem: (1.2.63)
L ~ ( E ~ ~ ):=V
2 -E2v +aEv +y v = h(x), xx x
x E
u , v(x')
= 0,
It is readily seen that GE(x,y) is given by the formula (1.2.64)
GE(x,y) =
E
-1 (G((X-y)E-I)-AE(y) exp(h-xE-l)-
-~~(y)exp(-h+(l-x)€-l)), (x,y) E
where (1.2.65)
G(x) := (2n)-'I
(C2+iaC+y2)-l exp(ixc)dc, R
AE(y),
B (y) satisfy the linear system AE
(1.2.66)
(y)+BE (y)exp
i,
= G
(-YE-')
(y)exp(X-E-') +BE (y) = G( (I-y)E-')
and h -, are the zeros of the characteristic equation (1.2.67)
2
A -aA-y2
= 0,
A- <
0 < A+,
introduced also hereabove. The residuum calculus gives immediately:
ux
U,
x' E
au.
1.2. Functional Analysis (1.2.68)
25
iG(x)
with a = min{ 1 A I , A + }
> 0, so that (1.2.64)-(1.2.68) yield:
where C > 0 is some constant, which may depend only on a and y. Moreover, it is easily seen that also the following estimates are valid (perhaps with a different constant C > 0 , which still depends only on a and y )
:
for (x,y) E ij x
U.
Thus, as a consequence of (1.2.62), (1.2.69), (1.2.70), one has the
following estimates with k = 0,l:
(1.2.71)
k SUE I ja-a(E,y) 1 j (€ax) G~(X,Y)/dy s CE, xEU U k {sup 1 IY -b(E,y) I I (€ax) GE(x,y) Idy 5 CE XEU u
and the same estimates (1.2.71) for ( € 3
) GE(x,y), as it is easy to check. Y Now, using the Green's function G (x,y) we may rewrite boundary value
problem (1.2.61) equivalently in the following fashion: (1.2.72)
V(E,X) = ( T
where we have denoted: (1.2.73)
(TE(V))
(x) ::
One has for each pair vl,v2 in the metric space C 1
distance p (v1'v2) =
I Iv
-v
II c(o,l),E
(0,1)
,E
(5)with
the
(if,:
where, of course, (1.2.71! have been used. Furthermore, for g E Co(<)
and each v E C
(0,1), E
(c),one also finds:
1. Manifolds, Functional Analysis, Distributions
26 Thus, if (1.2.76)
E~
is chosen in such a way that
C E < ~ 1,
where C is the same constant, as on the right hand side of ( 1 . 2 . 7 4 ) ,
then
the family of (linear) mappings
-
(1.2.77)
TE : C
(0,1)
rE
(U) -t c
(0,1)
,E
(U)
will be an equicontraction. Thus, ( 1 . 2 . 7 2 ) has a well-defined solution v E C Moreover, as a consequence of ( 1 . 2 . 7 5 ) ,
(G, v E
E
(O,E0).
the following estimate with some constant C > 0, which does not
of ( 1 . 2 . 7 2 ) depend on
(0,1) , E
one a l s o has for the solution V(E,X)
E
E ( 0 , ~ ~v)and ’ g:
provided that CEO < 1, where the constant C is the same as on the right hand side of ( 1 . 2 . 7 5 ) . one gets immediately the conclusion that for each 0(t)(= C ( U ) 1 , the well-define solution v of ( 1 . 2 , 6 1 ) (or,
Using ( 1 . 2 . 6 1 ) , 9
c(o,o)
,E
equivalently, of (1.2.60)
in fact, belongs to
C
(0.2) , E
(6)(the metric
space
of twice continuously differentiable functions with the distance function defined by ( 1 . 2 . 3 1 ) ) depend on
E
and, moreover, with some constant C > 0 which does not
E ( O , E ~ ) , v and g the following estimate is valid:
provided that
E~
is sufficiently small (i.e. satisfies the inequalities
specified hereabove). Denote by
( G ( 9 ) ) (x) =
V(E,X) the solution of ( 1 . 2 . 6 0 ) ,
may be regarded as a family of maps (linear operators) form C
(0.2) ,E
(u).In fact, if one takes instead of C0 -
(U)
of continuous functions g ( E , x ) (depending on
E
C
so that E 0- . ( U ) into
the matric space C
-t-
( 0 . 0 ) ,€
G
(5
E ( 0 , ~ ~with ) ) the same
distance function, as on C o 6 ) , then the (linear) maps
are a family of homeomorphisms (isomorphisms) which are Lipschitz equicontinuous, as a consequence of ( 1 . 2 . 7 9 ) .
i.e. one has for each pair g1 ’ 9 2
1.2. Functional Analysis
27
where the constant C > 0 is the same as on the right-hand side of (1.2.79)' 1 and g 0 > 0 is sufficiently small.
i.e. does not depend on that
E~
E
E (O,E
E
-
C (o,o),E(U),
k = 1.2, provided
Now, we come back to boundary value problem (1.2.46) with L ( E , x , ~ ) and F(E,V,EV ) defined by (1.2.471, (1.2.48) and (1.2.49), respectively. Using the operators G
constructed hereabove and having the property
(1.2.80), one may rewrite (1.2.46) equivalently in the following fashion:
where we have denoted:
Introduce the following family of metric spaces X rE (1.2.84) XrE
=
{v E C
5 rsl,
(0,2),E
where r > 0 will be chosen later on. One has:
as a consequence of (1.2.51) and (1.2.84). E XrE, k
Furthermore, one finds for each pair v
where C1 > 0 does not depend on
E
E (O,E
0
)
and v
=
1,2:
E XrE, k = 1,2.
Of course, again we have used (1.2.51). Now, choosing r > 0 so small that the following inequality is satisfied:
1. Manifolds, Functional Analysis, Distributions
28
(1.2.87) Crso < 1 , where C > 0 is the maximum of the constants C1, C 2 on the right-hand side of (1.2.85), (1.2.86), respectively, one gets the conclusion that the family of maps:
(1.2.88) Q, : MrE
+
MrE,
E
E
(o,Eo)
is a family of equicontractions (provided that
E
0
> 0 is chosen, as
specified hereabove), so that there exists a well-defined solution v E Mrs of (1.2.82). As a consequence of the construction, one has:
with any constant r
0 satisfying (1.2.87).
Using (1.2.461, (1.2.511, one also finds that, in fact, v E
C(o,z),E
(U)
and, moreover,
with some constant C > 0, which does not depend on
6 .
Finally, (1.2.891, (1.2.90) imply (12.31). One should emphasize that using exactly the same argument for the completely non-linear boundary value problem:
(1.2.91)
F(E
2UXX,EUX,U) = 0 , x E
u,
U(E,X') = $ ( x i ) ,
XI
E
au
one can prove the existence and uniqueness of the solution U(E,X) to
(1.2.91) for
6
E
( 0 , ~ and ~ ) j $ ( x ' ) I <,
6, x' E au with
E~
> 0, 6 > 0
sufficiently small, if the following conditions are satisfied: (i) F(v), v
=
(vl,v2,v3)is sufficiently smooth (at least twice continuously
differentiable in some neighbourhood of the origin in lR (ii) F(0) = 0, F
V
L(S)
(0)
# 0 and the polynomial
3
).
L(5),
1
:= -ac2+ibS+c, 5 E
R
does not have real zeros and has precisely one zero in the upper and the lower complex halfplanes, where it is depoted: (a,b,c) = (VvF)( 0 )
1.2. Functional Analysis
29
Moreover, under assumptions l o , 2' again U(E,X) may be represented in the form (1.2.29) with
V(E,X)
satisfying (1.2.31) and w(x',x) the solutions
of the following boundary-value problems: X'
F ( w ~ ~ , ( Wx,W) ~ ~ ) = 0,
X > 0,
W(X',O)
the latter being well-defined, provided that Indeed, since F V1
=
$(XI),
I $ (x')[
is sufficiently small.
(0) # 0 so it is also in some neighbourhood of zero 2 ( E uxx,~ux,u),V E E ( O , E ~ ) ,
in B 3 , to which belong the values of the vectors
provided that E~ is sufficiently small. Consequently, one may solve the 2 2 thus, reducing boundary equation F ( E u , E U , u ) = 0 with respect to E u xx x xx' value problem (1.2.91) to the form (1.2.27), (1.2.28). condition 2" being satisfied as a consequence of condition (ii). The essential requirement (ii) hereabove is the condition for the linearized (at zero) equation:
+ bEvX + cv = 0 aELv xx to be a proper elliptic singular perturbation (see vol. I1 of this book), which amounts in the situation considered to condition (ii).
I
Example 1.2.13. A
different nature of dependence upon a small parameter, namely a highly
oscillatory behaviour, is exhibited by the motion of a mass point in the presence of a strong force field in the vicinity of a smooth surface y in R3
,
the force field being directed towards y. Heuristically, it is obvious
that the mass point stays in a narrow neighbourhood of y and, as the force field becomes infinitely large, the motion occurs on y itself giving rise to a Lagrangian dynamical system on the smooth manifold y. One
says in this
case that the mechanical system is subjected to a holonomic constraint. Consider the motion of a mass point of mass 1 in B3
,
whose potential
energy in cylidrical coordinates (r,z,$) has the form: (1.2.92)
2 -1 2 UE(r) = ( 2 )~ (r-1) ,
the Lagrangian L
E
E (O,E~),
of this mechanical system being:
Here, as it is customary in the mechanics of mass points, the dot stands for the derivative with respect to the time variable t.
1 , Manifolds, Functional Analysis, Distributions
30
When
E
cylinder y
-f
=
0 the motion with initial data in a small vicinity of the {(l,z,$)
1
0 5 $ < 27,
-m
< z <
takes place on the cylinder
itself and i s characterized by the reduced Lagrangian Lo given by
which is a free motion (no potential energy) on y and, obviously, one has: $(t) = $0+C2t, z(t) = z +vt, where $ 0
0
=
$(O), C2
= i(O),
zo
=
~ ( 0 1 ,v
=
l(0)
are the initial conditions for $(t) and z(t). We shall analyze the perturbed motion described by the perturbed Lagrangian L Since
z
given by (1.2.93). and $ are cyclic coordinates ( L E does not depend on z and $ ) ,
the axial and angular momenta are preserved, the latter yielding the following first integrals for the mechanical system considered: (1.2.95) p (t) = z(t) E v,
2 . pm(t) = r (t)$(t) E p ,
so that for the radial component r(t) one finds the following Lagrangian
function
where we have denoted:
The Newton's equation for r = r(c,t) has the form: (1.2.98) 'Q(E,t) + (d/dr)VE(r) = 0 , where, as usual, the double dot stands for the second derivative with respect to t. From now on we shall assume that the motion starts from the cylinder y and has no initial radial velocity, i.e. that r(E,t) is subjected to the
following initial conditions: (1.2.99)
r(E,O) = 1,
r(E,O)
=
0.
Introducing the new unknown function (1.2.100) p(E,t)
=
E-2(r(c,t)-1),
p
(E,t),
1.2. Functional Anaiysis (1.2.98),
y i e l d f o r p ( ~ , t ) t h e following i n i t i a l value problem:
(1.2.99)
- n2 (1+E2 p ) - 3
p + p
E
31
=
0,
(1.2.101) P(E,O)
where S?
=
= 0
0, P(E,O)
=
$(O) i s t h e (given) i n i t i a l angular v e l o c i t y .
The c r i t i c a l p o i n t s (d/dr)VE( r ) , i . e .
S ( E2 ) of equation
a r e t h e zeros of
(1.2.101)
t h e s o l u t i o n s of t h e equation:
and only t h e p o s i t i v e one i s of i n t e r e s t . Obviously, i t has t h e following asymptotic behaviour: (1.2.103)
S ( E ~ )= G.
2
+ O(E
2
),
E
-f
0.
Furthermore, one has: (1.2.104)
2
2
( d /dr ) V E ( r ) > 0 ,
so t h a t a l l t h e motions i n t h e neighbourhood of t h e s t a t i o n a r y s o l u t i o n r(E ,t )
2 E S ( E ) of
a r e p e r i o d i c f u n c t i o n s of t with some period
(1.2.101)
21TE/W(E).
We a r e going t o apply t h e Poincar6-Lindstedt method i n o r d e r t o analyze t h e asymptotic behaviour of t h e s o l u t i o n of Since p ( ~ , t ) i s p e r i o d i c with period
(1.2.101).
~ T ~ E / w ( Ew ) e
s h a l l seek p i n t h e
form:
which y i e l d s t h e following i n i t i a l value problem f o r (1.2.106)
2 2 2 w (E)';+u-~ (1+~ u ) - ~= 0,
t h e frequency
W(E)
u(E,O)
Let
{
U(E,T)
W(E)
where
=
=
0,
being unknown, a s w e l l ; here t h e double d o t s t a n d s f o r
t h e second d e r i v a t i v e d2/d.r2.
(1.2.107)
= ;(E,O)
u(E,T):
2
= V ( T ) + E W(T)
1
+
E
2
v
+
E
4
*
U(€),
E4q!E,T!
32
1. Manifolds, Functional Analysis, Distributions
{u
= 0
V(T)
(1.2.108)
2
T),
(1-COS
W(T)
=
4 -3R (l-COS
T)
2 (3/2)Q .
=
we are going t o show t h a t with some c o n s t a n t C > 0 one has:
I lqll
SUP
2 C ([O,a])
O<E<E
:= sup c max l akT q ( E , . r ) / 5 C , O < E < E ~ OCkC2 OStLa
(1.2.109) sup O<E<E
I!.l(E)I
5 c,
0 provided t h a t a = a ( € ) i s such t h a t
a(€) = 0(E-4), as E -+ 0 . 2 Writing w ( c ) = 1+2E2u+E4K(E), s u b s t i t u t i o n of (1.2.107) i n t o (1.2.101)
(1.2.110)
y i e l d s f o r q(E,T) t h e following i n i t i a l value problem:
..
(1.2.111)
2 4 2 2 (1t2UE + E K ( E ) ) q + q = (K(&)t3n (2UtE K ( E ) ) ) V 2 2 2 4 6 -a (K(E)+3a (2U+E K ( E ) ) ) - F (2E V,E WIE q ) , = 0
q(E,o) = i ( E , O ) where
We choose
(1.2.113)
K(E)
K (E)
t o be:
2 2 -1
= - 6 ~ R ~ ( 1 + 3 c ~ R=~ -9R4(1+3~ )-~ R )
,
where JA(E.0)
(1.2.115)
2t3R 4 ( 2 v t2~K ( E ) )
:= K ( E ) R
2 4 L 2 ( E ) := 1t2uE + E K ( E )
a r e well-defined by (1.2.1081,
(1.2.113).
Notice t h a t f o r F(v,w,q) defined by (1.2.112) one has:
(1.2.116) F(v,w,q) = O ( V 2 + / W \ + / q l )a, s IVl+lW\+\ql
-f
0
1.2. Functional Analysis
33
and, moreover, one also has for F and F = (a/aq)F with some constant C > o : q 2 (1.2.117) IF (v,w,q)[ S C, IF(v,w,q)[ 5 C(v +Iwl+lql), 9
V(v,w,q) with IvI+[wl+lql 5 1/2. Let us rewrite (1.2.114)
equivalently in the following fashion:
where we have denoted: T
(TE(q))(T) :=
(1.2.119)
I'q0(E,T)
I
2
6
4
Sin((T-S)/W(E))F(E V(S),E W(S),E q(E,s))ds,
0 := A ( E , n ) (l-COS(T/W(E)).
Now introduce a family of metric spaces as follows: (1.2.120) xE := Cq E c"([o,aI) [ llq-qolI s 11 c0( o a]) wnere, as previously, Co([O,a]) is the set [v;?ctor space) of all continuous functions on [O,al and Ilgll = max Iq(c,T) I . c'([o,~I) O
Using (1.2.117),itisreadilyseenthatonehaswithsomeconstantC
1
> 0:
2
4
6
),E*w(s),E q(E,s) Ids 5 C , ~ E, V provided that
5
where A
O
E
E
E
( 0 , ~ ~ ) ~
> 0 is so small that
0
2 2 2 4 4 (20, +6s Q +E ( 1 + 2 A O ) ) 5 ( 1 / 2 ) , 0 0 0
A(E,n).
max
=
6
OSESE
0
Furthermore, again using ( 1 . 2 . 1 1 6 ) , constant C
2
(1.2.117),
one finds with another
> 0:
under the same condition ( 1 . 2 . 1 2 2 ) . Let a = a(€) be as in ( 1 . 2 . 1 1 0 ) , lim
E
4
so that one also has:
a(€) = 0.
€4
Thus, choosing
E
0
> 0 so small (if needed) that
1. Manifolds, Functional Analysis, Distributions
34 sup
E
4
-1
a(€) < C
O<E<E
with
c
=
0 max{C1,C2}, one gets the conclusion that T
E
:
X
-f
E
X
E
E'
E
(O,EO)
is a family of equicontractions, so that ( 1 . 2 . 1 1 8 )
E X
has a well-defined solution q(E,T), q
for
T
and also ( 1 . 2 . 1 1 4 )
E tO,a(E)1.
Thus, one also has:
with A.
defined as in ( 1 . 2 . 1 2 2 ) .
Using ( 1 . 2 . 1 1 4 ) ,
(1.2.113)
one gets immediately the first of the
(1.2.124),
the second one being an immediate consequence of
inequalities ( 1 . 2 . 1 0 9 ) ,
and the fact that
is well-defined by
V(E)
K ( E ) .
In fact, q(E,T) is infinitely differentiable function of with respect to E +
E),
i.e. all the derivatives of q(E,T) in
T
(uniformly
T
are bounded as
0 for T E [o,~(E)].
Finally, let us write the asymptotic formula for r(E,t) that we have established:
(1.2.125)
w
2
(E)
2 2 = 1 + 3 ~0 - 9 ~ ~ 0 ~ ,
provided that a(€) =
O(E
-3
)
and
0 is chosen small, as above.
>
E~
One also has the following asymptotic formula: r(s,t)
= I+E
2 2
4
0 COS(W(E)~/E)+O(E) ,
V t
E [~,a(~)l
(1.2.126) (,(E)
= 1 + ( 3 / 2 ) ~ ~ f i ~V,
provdided that a(E) = o ( E - ' )
and
E
E (0,~~)'
E~ >
0 i s sufficiently small as indicated
hereabove. One may differentiate the asymptotic relations ( 1 . 2 , 1 2 5 ) , with respect to t multiplying it after each differentiation by
(1.2.126) E.
Notice that the perturbed motion apporches the reduced one (the free motion on the cylinder y ) for t =
-3 O(E
)
performing high frequency
oscillations with periods of order O ( E ) and amplitudes of order E +
0.
2 O(E ),
as
1.2. Functiorial Analysis
35
This argument may be extended to mechanical systems effected by the presence of a strong force field directed towards some hypersurface
r
in
Rn of codimension one. Namely, with (r,$) the appropriate local coordinates
( 4 on the hypersurface and r the distance to the hypersurface), one may extend the argument hereabove to the Lagrangians of the form:
where a($) > 0 and L(r,$,;) L(r,$,;) = O(Ir1
3
is supposed to satisfy the condition: )
as r
+
0.
Furthermore, as usual, L($,$) is supposed to be a positive definite quadratic form with respect to the local velocity $ with coefficients smooth functions of the local coordinate $ and, moreover, the potential energy of the mechanical system (with the holonomic constraint forces) on the hypersurface
r
in Rn described by L($,+) is supposed to be a non-
negative smooth function of the local coordinates $ on
r.
Thus, these
conditions being fulfilled, the solution $(t) of Newton's equations, which correspond to L($,$), exists globally for all t E R . Introducing p
= E
-2
r, one finds the following Newton's equations for
the motion of the mechanical system:
where the subindices $ and
6 stand
for the gradients of the corresponding
scalar functions with respect to $ and
6, respectively. 2
Again in the layer of thickness O(E
)
around the hypersurface the
dynamics of the mechanical system (with the initial condition on the hypersurface) is mainly a superposition of a slow motion on the hyperssurface, described by the local coordinates $ ( ~ , t )with the main term $(t) solution of the system
and of a fast oscillatory movement in the normal direction to the hypersurface described by the coordinate r(E,t) = E 2 p(E,t) and subjected to the
1. Manifolds, Functional Analysis, Distributions
36
2
initial conditions r(E,O) = ;(E,O)
,i
,+) ,
= O(E ),
whose main term has the form:
,i
B(g ), C(g ) and W ( E ) are defined by the initial 0 0 0 . 0 0 conditions g ( 0 ) = 4 and g ( 0 ) = go; besides, one also has: 0
where A ( g
0
W(E) =
2 a(gO) ( ~ + O ( E) ) ,
E -t
0,
where a($) > 0 is the same as on the left-hand side of (1.2.127). To estimate the remainders in the asymptotic formulae for g(E,t) and 2 r(E,t) = E p(c,t) one uses again the argument based on Banach's
theorem concerning the equicontractions.
x w i t h the distance function p i s said c x such t h a t f o r each t o be separable i f there e x i s t s a sequence {x 1 k k2O 6 > o and f o r each x E x one can find an element x of t h e sequence
Definition 1.2.14. A metric space
kg
Ixk}k>O such t h a t P(X.Xk0) < 6 .
Any sybset of
x having t h e property hereabove i s said t o be dense i n
X.
Examples 1.2.15.
lo. The set
C
0
(b) of functions which are continuous on
= [0,11 with the
distance p ,
is separable. Indeed, the countable set of all polynomials with rational coefficients may be chosen as the sequence {x k k>O in Definition 1.2.14. The set 1 of all sequences of the form x = (ao,al, 2 an,.
.
...,
P
.)
with a k
real numbers equipped with the distance p ,
is separable.
...,rn,O,...)
Indeed, the sequences of the form (ro,rl,
with n 2 0 any
integer rk, 0 5 k I n, any rational numbers, form a countable subset of 1
P
and, thus, may be chosen as the sequence { x } in Definition 1.2.14. k k2O 3 O . The set L (U), u = (0,l) with p E [ O p ) , of all functions x(t) such that /x(t)1'
P
is integrable over
U,
is separable, the distance p
P
on L ( U )
P
1.2. Functional Analysis
37
being defined as follows:
Indeed, the countable subset of all polynomials with rational coefficients is dense in L (U).
P
4'.
of all almost everywhere on U = ( 0 , l ) bounded functions
The set L,(U)
with the distance p,,
is not separable. Here vrai max stands for the maximum of lu(t)-v(t) I over V \ E with E the set of Lebeargue's measure zero where u(t)-v(t) may be infinite. A
subset K in a metric space M is called compact, if each sequence
{ x ~ } c~K>, contains ~ a subsequence {x nk~k>O,whichconveqesto some elementxEM. Compactness criterium. A s e t K
5
M i s compact i f f f o r each 6 > 0 t h e r e
e x i s t s a f i n i t e + n e t i n M, i . e . a f i n t i e s e t o f elements {xk}lsksn C M such t h a t f o r each x E M t h e r e i s an element xk o f t h e &net whose d i s t a n c e t o x i s l e s s than 6 : p(x,xk) 1.2.2.
< 6.
Topological vector spaces
Definition 1.2.16. A set
x i s s a i d t o be a v e c t o r space o v e r t h e f i e l d o f r e a l o r complex
nwnbers, i f f o r i t s e l e m e n t s , c a l l e d a l s o v e c t o r s , two a l g e b r a i c o p e r a t i o n s , a d d i t i o n and s c a l a r m u l t i p l i c a t i o n , a r e d e f i n e d , w i t h t h e f o l l o w i n g usual algebraic properties: (i) To every p a i r (x,y) E x a way, t h a t x+y = y+x
and
x
x corresponds a v e c t o r
(x+y) E
x,
i n such
x+(y+z) = (x+y)+z;
b e s i d e s x c o n t a i n s a unique v e c t o r 0 ( t h e zero v e c t o r or o r i g i n of X) such t h a t x+O
=
x, V x E
-x such t h a t x+(-x)
x
and t o each x corresponds a w e l l - d e f i n e d v e c t o r
= 0.
(ii) To every p a i r (a,x) w i t h x E X and a r e a l or complex number
corresponds a v e c t o r ax E Ix = x,
X,
i n such a m y t h a t
a(f3x) = (aB)x,
1. Manifolds, Functional Analysis, Distributions
38
and such t h a t the two d i s t r i b u t i v e laws hold:
The symbol 0 i s a l s o used f o r t h e zero element of t h e s c a l a r f i e l d . A v e c t o r space over R
( r e a l numbers) i s s a i d t o be a r e a l v e c t o r space,
and t h e one over C ( t h e complex numbers) i s c a l l e d a complex v e c t o r space. D e f i n i t i o n 1.2.17.
A set
x
i s said t o be a topoZogical vector space, i f t h e following four
conditions are f u Zfi Z led: ( i )A
colZection
Y
of subsets of x i s singled out w i t h the following
properties:
0 and X beZong t o the system
lo. The empty s e t
2O. The union of any number 3'.
of subsets i n
Y
Y;
i s again a subset i n Y ;
The i n t e r s e c t i o n of any f i n i t e number of subsets i n
set i n
Y
i s again a sub-
Y.
The elements of t h e c o l l e c t i o n
Y
are c a l l e d open s e t s and t h e
co2lection Y i t s e l f i s said t o be a topoZogy on X. Euch open s e t containing a point x E
of
x i s said t o be a neighbourhood
X.
A point x E M c x i s called an i n t e r i o r point of the s e t M, i f M contains a neighbourhood u(x) of X.
Obviously, a set M i s open i f f every x (ii)x
i s a Hausdorff space and
of p o i n t s x
1
and x
2
Y
E
M i s an i n t e r i o r p o i n t of M.
i s a Hausdorff topclogy, i . e . each p a i r
in x have d i s j o i n t neighbourhoods u(xl) and u(x,)
:
u(xl) n u(x2) = 0. A point x E x i s said t o be a Z i m i t point of a s e t M c X, i f each
neighbourhood of x contains a t Zeast one point of M other than A set M c
x i s said t o be closed,
Every p o i n t x E
X
X.
i f i t s complement fiM i s open.
is a cZosed s e t . The closure k of a s e t M c
-
i n t e r s e c t i o n of a l l closed s e t s t h a t contain M; M
=
X
i s the
M U M' where M' i s the
c o l l e c t i o n of a l l l i m i t p o i n t s of M. 0 The i n t e r i o r M o f a s e t M i s the union of a l l open s e t s t h a t are
subsets o f M. A sequence Ix 1 i n a Kausdorff space (write: l i m x
= x)
i f every neighbourhood
nf i n i t e l y many of the p o i n t s xn.
x E x of x contains a l l but
x converges t o a p o i n t U(X)
1.2. Functional Analysis
39
It is readily seen that x = lim x is well-defined. n(iii) x is a r e a l or complex v e c t o r space. (iv) The v e c t o r space o p e r a t i o n s a r e continuous w i t h r e s p e c t t o t h e
topology
i . e . by d e f i n i t i o n t h e mapping
Y,
x
x
x 3
(x1,x2)
-f
is continuous (if x 1. E X, j
(Xl+X2) E
x
1,2 and if V is a neighbourhood of x +x 1 2' then there should exist neighbourhoods V.(x.) such that V1(x1)+V2(x2) c V) I 1 =
and sirni Zarly t h e mapping x
ip
x
3 (a,x) + ax E
i s continuous, where x E X, a E
0
0
x
i s t h e f i e l d of t h e r e a l or complex numbers (if
and V is a neighbourhood of ax E X, then for some r > 0 and
some neighbourhood W(x) 3 x one should have 6W c V whenever 15-a A set A
C
I
<
r)
.
X is said to be symmetric if x E A implies that (-x) E A,
as well. If U c X is a neighbourhood of zero in X, then (-U) n U is a symmetric neighbourhood of zero in X. A collection Y' c Y is said to be a base for the topology Y if every
member of Y (that is, every open set) is a union of members of YO.
A collection Z of neighbourhoods of a point x E X is a local base at x if every neighbourhood of x contains a member of Z. If E c X and if
2
is the collection of all intersections E
V with
V E Y, then Z is again a topology on E inherited from Y.
If a topology Y is induced by a metric p , then
p
and Y are said to be
-
compatible with each other.
Two topologies Y and Y on X are said to be equivalent, if for each neighbourhood U E Y there exists a neighbourhood and, conversely, for each
c
?
E
?
such that
c U
there exists V c Y such that V c V.
Example 1.2.18. Let X be the set of all infinitely differentiable functions in an interval U
=
(a,b) C R , which have a compact support in U, i.e. for each function
x(t) in X there exists an interval U' cc U such that x(t) vanishes outside U'. Obviously, X is a vector space. Let the neighbourhoods of zero be
defined as follows: for each 6 > 0 and each n 2 0 integer a neighbourhood ) I < 6, for U(n,6) is the set of all functions x(t) E X such that / x ( ~(t)
1. Manifolds, Functional Analysis, Distributions
40
k = O,l,.. .,n. With the neighbourhoods of each element xo(t) E X defined as the sets of functions x(t) such that x(t)-xo(t) form a neighbourhood of zero,
x
becomes a topological vector space, which is usually denoted either
m
by D(U) or by c;(U). Example 1.2.19.
Let Xh, h E (O,l), be the family of vector spaces of sequences u(h,x), x E F$l,= hZ with real or complex values, which depend continuously on + -, i.e. for any integer k one has: <x> lu(h,x) I + 0 as
h E (0,l) and are rapidly decreasing as 1x1
k 2 0 and for each sequence u 1x1 + -, where < x > ~= l+lxl 2
.
:
R1 h
+ C
Introduce the neighbourhoods of zero in Xh as follows: for each 6 > 0 and every integer k > 0, n 2 0 , a neighbourhood of zero U (6,h,n) is the set h of all sequences u(h,x) such that
..,
p ~~ < x > ~ u(h,x) / a ~I < 6 , for m = 0,1,. n (1.2.128) / ~ l ~ :=, s ~~ ; x,h xERh and for h E (0,1), With the topologies defined in such a way, h
+
Xh becomes a family
of topological vector spaces. Considering each u(h,x) E Xh as the Fourier coefficients of the periodic function C(h,S), ;(h,c)
=
h X u(h,x) exp (-ix<), xERh
one gets another family of vector spaces Xi of all periodic functions ;(h,5) with period 2n/h, which are infinitely differentiable in 5 . For each 6 > 0 and every integer k 2 0 , n 2 0 define a neighbourhood U (6,k,n) of h zero in Xi as the set of all periodic functions <(h,S) such that
where we have denoted: -1 2 (1.2.130) sh(c) := (ih) (exp(ihS)-l), * = l+lSh(C)
I .
It is easy to show that the (dual) topologies defined by (1.2.128) and (1.2.129), (1.2.130), respectively, are compatible. I
1.2. Functional Analysis
41
Tile following types of topological vector spaces are useful in applications: 1
.X x
2 ' . 3
.
is locally convex if there is a local base whose members are convex: is metrizable if its topology Y is compatible with some metric
and the corresponding metric respect to in
p.
X is said to be a Frechet space if X is locally convex, metrizalbe
p
is complete (i.e. each Cauchy sequence with
p
has a limit in X) and invariant with respect to the translation
x: p(x+z,y+z) = p(x,y), V x,y,z
in
X.
The following example of a Frdchet space plays an important role in the distribution theory. Let U
m
C
IRn be an open set and denote by C (U) the vector space of
all complex valued infinitely differentiable functions on U. For each m n the Schwartz space of all u E C (IR ) , whose C IRn denote by D K support lies in K.
compact K
m
Introduce a topology on C (U) which makes it into a Frechet space such that
DK
each bounded
is a closed subspace of Cm(U) whenever K
To do this, let us choose compact sets KN, N = 1,2, in the interior of KN+l and U = N = l,Z,
C
U and, moreover,
m
set in C (U) equipped with this toopology is compact.
..., by
U
...,
such that KN lies m
KN. Define seminorms PN on C (U),
Nd 1
setting:
P (u) = max max /D:u(x) N lal
I. m
They define a metrizable locally convex topology on C (U), a compatible translation invariant metric p(u,v)
:=
c Nt 1
p
being the following one:
2-NP (u-V)(l+P (u-v))-l. N N
Furthermore, for each compact K
C
U,
DK
m
is a closed subspace of C ( U ) , and
a local base is given by the sets:
0
N
m
:=
{u E Cm(U)
1
P ( u ) < N-'}, N
N = 1,2,
...
Each I) C C (U) is again a Frechet space with the same local compactness K m property as C ( U ) itself.
1. Manifolds, Functional Analysis, Distributions
42
1.2.3.
Banach spaces
Definition 12. 20 .
A vector space B is said t o be a Banach space if t o every x E B a non-
negative nwnber I ( x (1 i s associated, caZZed the norm of
X,
i n such a way
that ( i ) IIx+yli 5
I
( i i ) lax1 (iii) 11x1
1
1
=
>
I I x t l + l l y l I, l a / 11x1 1 , v
o if
x
+ 0,
V (x,Y) E B x 8, x E
B , a r e a l (or complex) n u d e r ,
( i v ) every Cauchy sequence I x } c n n>O
B has a l i m i t x E B, i . e . B is
complete in the metric defined .Ey the norm on 8 . Example 1.2.2 1.
lo. For each given i n t e g e r k t 0 t h e space C k ( E ) of a l l k-times continuously d i f f e r e n t i a b l e f u n c t i o n s u on some closed i n t e r v a l
5=
[a,b],
equipped with t h e norm: (1.2.131)
IuI
=
Ck(6)
z ma5 OSj
Ia:u(x)
1,
i s a Banach space, t h e topology defined by t h e norm (1.2.131)
equivalent with t h e uniform convergence on
being
with t h e d e r i v a t i v e up t o t h e
o r d e r k. 2 ' . The spaces C (0,k) ,E t h e family of norms:
(u)introduced
i n Example 1.2.5 and equipped with
(1.2
i s a family of Banach spaces. A l s o t h e spaces C k ( R 1
h x,k with t h e family of norms: (1.2.133)
:=
) introduced i n Example 1.2.5 and provided
Z s u p 1 Ia:,hu(h,x)I, O<j
i s a family of Banach spaces.
h E (O,l),
I
The following r e s u l t i s known a s Kolmogorov's theorem: A t o p o l o g i c a l v e c t o r space X i s normable ( i . e . a norm compatible with topology can be introduced on X) i f f X possesses a bounded convex neighbourhood of zero. Two Banach spaces B1 and B2 are s a i d t o be isomorphic i f t h e r e e x i s t s a one-to-one
continuous map f : B1 -+ B 2 , which p r e s e r v e s t h e a l g e b r a i c
o p e r a t i o n s on B 1 and B2, i . e .
f (ax+%y)= af ( x ) + %(fy ) , V (x,y) E B1
and V a , % r e a l o r complex numbers.
X
B2
1.2. Functional Analysis Let B be a Banach space and let Bo Two elements xk E B, k = 1,2, class X, if (x -x 1
2
)
E
Bo.
43
B be a closed subspace of B.
C
are said belonging to the same equivalence
The linear structure and the norm on the set of
the equivalence classes are induced by the linear structure and the norm on B and the Banach space B/BO, thus obtained is called the quotient space of B modulo B
0'
1.2.4.
Hilbert spaces
Definition 1.2.22.
A vector space H i s c a l l e d a (complex) Hilbert space (or unitary space) i f t o each ordered p a i r of vectors x and y i n H is associated a complex number (x,y), called the inner product or scalar product of x and y, such t h a t
t h e following properties are v a l i d : (i) (x,y) = (y,x)* (The upper star denotes complex conjugation.)
, .
(ii) (x+y,z) = (x,z)+(y z ) (iii) (Ax,y)
=
A(x,y),
(iv) (x,x) 2 0 ,
V x E
H
V (x,y,A) E
H,
(x,x)
(v) Every Cauchy sequence {x 1
=
x
H
x
0 iff x
t
C. = 0.
ti converges t o some element x E H , i.e.
H is comp2ete with respect t o t h e norm:
a complex vector space w i t h an inner product over t h e f i e l d
If H is
o f the r e a l x w n b e r s , then H is c a l l e d a r e a l Hilbert space. The following basic properties of the Hilbert spaces can be easily established: (1.2.134) (1.2.1353 (1.2.136)
1 (x,y)1 S (1x1 ( y (I H (Cauchy-Schwarz inequality), 1 /x+y(I H 6 ( ( x (IH+( / y (I H (The triangle inequality), I ly[I H 5 I [Ax+y(I H for every A E C iff (x,y) = 0.
Every nonempty closed convex set V c H contains a unique element x of minimal norm. An
element x is said to be orthogonal to a set E c H if (x,y) = 0, V y E E. If M is a closed subspace of H and M I is the set of all elements
orthogonal to M, then (1.2.137)
H
=
M B MI,
h is
a closed subspace of H and, moreover, one has:
1. Manifolds, Functional Analysis, Distributions
44
i.e. each element x E ff can be uniquely represented in the form: x = x M If M
C
+M' f xM
EM,
xiEd.
ff is a closed subspace, then (Ml)'
=
M.
If {xnInzO is a sequence of pairwise orthogonal vectors in
ff, then
each of the following three statements implies the other two: (i) E x nzO
converges, in the norm topology of
ff.
(iii) X (xn,y) converges, for every y E H. ntO A collection of vectors {e 1 C ff is said to be orthonormal system 1, k = j n if (e e.) = dkj = {o, k z j . k' I
Let L
5
be the closed subspace of ff spanned by an orthonormal
system {e 1 and let x E L. The numbers n n2l
,...
(1.2.138) f. = (x,e.), j = 1,2 7
3
are called the Fourier-coefficients of x E L with respect to the orthonormal basis {e 1 in L. n n2l One has Parceval's identity for every x E L:
Moreover, for every x E ff one has the Bessel's inequality:
If ff does not contain a non-zero element which is orthogonal to an
ff, then {en>n2l is an orthonormal basis in ff and in that case Parceval's identity (1.2.139) holds for every x E ff with
orthogonal system {en}n2O
C
f. given by (1.2.136).
I
A
Hilbert space
ff which has an orthonormal basis {enlnZ1 is called
separable. Every separable complex (real) Hilbert space ff is isometric and isomorphic to the complex (real) Hilbert space l2 of the sequences u
=
{u 1 n ntO
equipped with the inner product (u,v) =
z nZO
u v* n n'
where again the upper star stands for the complex conjugation.
1.2. Functional Analysis
45
In particular, the space L2(U), U = (0,l) of all square integrable complex valued functions u(x) on U with the inner product := (1.2.140) (u,v) L*(U)
J u(x)v(x)*dx
and l2 defined hereabove are isomorphic and isometric (Riesz-Fischer). 1.2.5. Classical Sobolev spaces W and HBlder spaces Ck" Plr Let U
Rn be a bounded open set. For p E
C
[l,m),
any r-times continuously differentiable function u
:
U
integer r 2 0, and + C
define the
Sobolev norm:
where
Taking the closure of the set of all r-times continuously differentiable functions in
with respect to the norm defined by (1.2.141), (1.2.142),
one gets the Sobolev space W (U). P.r The elements of W (U) which are not r-times continuously differentiable Pfr functions on are said to have generalized derivatives up to the order
u,
r t 0 which are in L ( U ) .
P
One can give a different definition of a generalized derivative a"u E L (U) of a function u E L (U), la1 5 r. Namely, let $ E Cr(?J) and X P P let J, have a compact support in U (i.e. $ vanishes identically in some neighbourhood of the boundary aU of U). A function v E L (U) is said to be the generalized derivative aau of
P
u
E
L
P
( u ) if for any function
J, E Cr(u) with compact support in
u one has:
(U) as the vector space of the Thus, one can define the Sobolev space W PPt functions u : U + C which have all generalized derivatives up to the order
r >= 0 in L (U). P Notice, that two functions which are elements of W (U) and coincide a.e. PCr have also all their generalized derivatives up to the order r, which (U). Thus, coincide a.e. and, therefore, are indistinguishable in W Ptr (U) are (equivalence) classes of functions, prr
actually the elements of W
1 . Manifolds, Functional Analysis, Distributions
46
which coincide a.e. and have all their generalized derivatives up to the order r 2 0 in L (U).
P
It can be shown but will not be done here, that both definitions of the Sobolev spaces W (U) are equivalent. Of course, W (U) is a Banach P,r Prr space. Define the Sobolev space W ( l R n ) as the closure of the vector space p,r of all r-times continuously differentiable functions with compact support in X i R n , with respect to the norm
Sobolev spaces are very useful in applications. Aslo very useful are k,X Hdlder spaces C (U). Again let U u
:
U
+
C Xin be a bounded open set. A continuous function 0,X C is said to belong the Hdlder space C (U) with A E ( 0 , l ) if the
following norm is finite: (1.2.144) IIull
o,x c
(U)
. .=
SUP-
-
Ix-Yl-Alu(x)-u(Y)
I.
(x,y)Euxu,x#y
k,X Hdlder space C ( u ) with integer k 20 and X E (0,l) consists of all ktimes continuously differentiable functions u(x) such that
or equivalently,
where, of course,
I Iul I
=
COG)
max lu(x) 1 . XEU
The following classical result is due to S.L. Sobolev and Morrey. Theorem 1.2.23.
L e t u c =“be bounded and l e t
au b e a smooth rnanijold.
u has a l l g e n e r a l i z e d d e r i v a t i v e s of order l e s s
If u E
w
PrZ
(U) t h e n
than r and, moreover, i f
rp > n , 0 2 k < r-(n/p), t h e n u E ck(,) and t h e r e e x i s t s a constand which does not depend on u E w (u) and such t h a t
C > 0
P#r
More p r e c i s e l y , t h e r e e x i s t s a f u n c t i o n v E u a.e.
and such t h a t ( 1 . 2 . 1 4 7 )
which c o i n c i d e w i t h ck,A (u)
holds w i t h a c o n s t a n t
C >
0 , which a l s o
47
1.2. Functional Analysis coes n o t depend on u and v.
Very useful for the numerical ixeatment of partial differential equations are finite difference versions of the classical Sobolev spaces k,X W (U) and Hdlder spaces C (U). P,r Again, let U C lRn be a bounded open set with a smooth boundary aU, p E [ 1 ,m) and r LO an integer. Denote
n
Uh = U where n\
=
<,
h E (O,ho)
n . hZ LS a one parameter family of grids in lRn, and introduce
the discrete version of L -norms for sequences (meshfunctions) u(h,x)
:
X$
P
+
C on Uh in the following fashion:
lu(h,x) Iphn)'",
h E (O,ho)
P
h Furthermore, a family of discrete norms similar to (1 2.141), are defined in the same way:
Remark 1.2.24. It can be shown but will not be done here that the following norms are equivalent with (1.2.149) uniformly with respect to h E (0,h 1 , V ho > 0. 0 Let v(h,x) : + C be any sequence (meshfunction) such that
<
u(h,x) : u(h,x), V x E U Introduce for u(h,x)
:
h
X$
<
is finite. and (1.2.149) for v with U = h C and with Uh as defined hereabove the -f
following norm:
where the infimum is taken over all meshfunctions v with
1 \ v /1 P,r;uh
<
m
and such that v(h,x) E u(h,x), V x E Uh. Then, there exists a constant C > 0, which does not depend on h and u(h,x) and such that
<
(U ) , h E (0.5 ) , consist of all meshfunctions The spaces W p,r h 0 u(h,x) : + C, which depend continuously on h E (0,h ) in the sense of 0 the topology defined by the norms (1.2.149) and, moreover, such that
1. Manifolds, Functional Analysis, Distributions
48
Obviously, W ( U ) is a family of Banach spaces and the vector p,r h space E (U ) for which semi-norm (U,) of all meshfunctions u E W P,r p,r h (1.2.150) vanishes, is a closed subspace of W (Uh), i.e. again a family P*r of Banach spaces. Two meshfunctions u and v in W (U,) are said being equivalent or Per belonging to the same equivalence class, if /u-vI = 0 where 1 - 1 p,r;u p,r;u is the norm defined by ( 1 . 2 . 1 5 0 ) . In other words, u and v are said to
be equivalent, if they belong to the same equivalence class in the family (U ) modulo of quotient spaces Wp,r(Uh)/Ep,r(Uh), h E (O,ho), of W p,r h
2 2 2 w (x;y) := afi-lh-nexp(lx-yl /(]x-yl -h ) ) ,
h
w (x,y) E 0 ,
for Ix-yl < h
for (x-yl L h,
h
where
1 a =
I
2
exp(p2(p - 1 )
-1
)dp
0 is chosen in such a way that
1
w
V y
(x;y)dx E 1,
E
lRn,
V h > 0.
Rn For every function u E W (V) define its averaged u (x) with the PPr h averaging kernel w (x;y) hereabove as follows: h
where 1 u(x) is the extension of u 0
outside the open set V
C
Rn
:
V
-f
C
by settint it identically zero
.
Let U C C V. It can be shown but will not be done here that for each u E W
p,r
(V) one has for its averages uh as defined hereabove:
I 1Uh-J
Ip,r;u - t O
ash+O.
Obviously, u (x) is infinitely differentiable function with respect h , V h > 0.
to x E lRn
With each u E W (V) associate the meshfunction u(h,x) : n\ -t C, Per which is the trace of the averaged u (x) of u ( x ) on the family of grids h Using HBlder's inequality one can show that u(h,x) belongs to W (Uh) P.r
$.
1.2. Functional Analysis
49
and, moreover, with some constant C > 0, which does not belong on
u E W (V) and h E (O,l), it holds: P,r lblp,r;u
I juI \p,r;Uh5
CI / u /
1 p.r;V'
1.1
is the norm defined by (1.2.148)-(1.2.150). p,r;U Thus, the traces of the averaged of all u E W (V) is a linear - p,r subset in w (U 1, provided that U C C V, i.e. U C V. p,r h The finite difference analogy of Hdlder spaces C k n X are defined in
where
(u)
<
an obvious way by introducing the families of norms for the meshfunctions u(h,x)
:
-f
C as follows:
-
-
Obviously, traces of functions u E CkpX(v) on Uh = U I UCC
V, form a linear subset in the spaces
C
k,X -
<
with
(Uh), h E (O,l), of the
meshfunctions u(h,x), whose norms (1.2.151) are uniformly bounded with respect to h E ( 0 , l ) (i.e. are bounded by a constant which does not depend on h E (0,l)). The following result is a finite difference version of the Sobolev'sMorrey result stated in Theorem 1.2.23. Theorem 1.2.25.
Let u c R~ be an open bounded s e t w i t h a smooth boundary k i s the l a s t i n t e g e r such t h a t 0 5 k < r-(n/p), X
e x i s t s a constant c
> 0
=
au.
I f rp > n,
r-k-n(p), then there
which does not depend on h E (O,ho)and u E W
P,r
and such t h a t
Proof. __ First consider the case when Uh = RE and r
A
=
=
1, so that p > n, k = 0 ,
1-(n/p). We shall show that
where the constant c > 0 depends only on p and n; here, of course,
(u,)
1. Manifolds, Functionai Analysis, Distributions
50
a x , h ~=
(ax,huf
Obviously, for each x E Rn\{O} h (1.2.154) u(h,x)-u(h,O) = h
C
one has:
YET where
r
Y.
,>u(h,y),
c Rf: is the set of points whose distance to the line tx,
0 5 t 5 1, in Rn is minimal, and ~ ( y )is the tangent vector to the broken
line passing through the points of Tx whose each piece is parallel to one of the coordinate axes x .
I'
< j 5n.
Let a > 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
where p ' is such that l/p + l/p' = 1. BY translation and by triangle inequality one also has:
for each pair of points (x,y) E Rf:
X
RE
such that a
=
2 Ix-yl , so that
1.2. Functional Analysis
51
(1.2.160) y i e l d s (1.2.153) with c = 22-n/P(n+l-n/p). Using t h e induction argument, one g e t s (1.2.152) f o r Uh
=
and
RE
using Remark 1.2.24, one f i n a l l y e s t a b l i s h e s (1.2.152) f o r each U
h'
Remark 1.2.26. In t h e proof of Theorem 1.2.25 we have used an a d a p t a t i o n t o t h e d i s c r e t e s i t u a t i o n of t h e method of prving t h e Morrey's r e s u l t f o r c l a s s i c a l Sobolev space, which is t o be found i n [ B r e z . l ] 1.2.6.
Linear f u n c t i o n a l s and dual spaces
8 be a r e a l ( o r complex) Banach space.
Let from
B
A l i n e a r mapping f ( x )
i n t o t h e f i e l d of r e a l ( o r complex) number i s c a l l e d a l i n e a r
functional. A l i n e a r f u n c t i o n a l f ( x ) defined on t h e whole Banach space
t o be continuous a t a p o i n t xo E {xnlnZl c 8 such t h a t
1 Ix
-xoI
1
B i f f(xn)
+
0 for n
+
+
f ( x ) f o r any sequence
E B i f it
0
m.
A l i n e a r f u n c t i o n a l f ( x ) on a Banach space
point x
8 is said
8 i s continuous a t each
i s continuous a t j u s t one p o i n t x
0
E 8.
A l i n e a r f u n c t i o n a l f ( x ) i s s a i d t o be bounded on
8 i f there exists
number c 2 0 such t h a t (1.2.161)
If(x)I 5 c / I x I / , V x
E 8.
The l e a s t number c such t h a t (1.2.161) s t i l l holds i s c a l l e d t h e norm of t h e f u n c t i o n a l f ( x ) and i s denoted by
\
If
I I.
Thus
A l i n e a r f u n c t i o n a l f ( x ) on a Banach space
8
s continuous i f f f ( x )
i s bounded.
D e f i n i t i o n 1.2.27.
A f a m i l y of l i n e a r f u n c t i o n a l s f ( x ) on a f a m i l y o f Banach spaces 8
w'
depending upon a parameter w i n some parameter s e t fi 5 Rn i s s a i d t o b e equicontinuous ( o r equiboundedl i f t h e r e e x i s t s a c o n s t a n t c 2 0 , which does n o t depend on w E 0 and such t h a t
a
1. Manifolds, Functional Analysis, Distributions
52
The least constant c such t h a t (1.2.163)
s t i l l holds i s c a l l e d t h e
of l i n e a r f u n c t i o n a l s f norm of t h e family {f ( x ) loEn by sup on
wcn B .
1 I f w /[
where
1 I f w (1
( x ) and i s denoted
i s t h e norm of f ( x ) a s a l i n e a r f u n c t i o n a l
Example 1.2.28. U = ( 0 , l ) be t h e family of Banach spaces introduced i n
Let C (o,o) Example 1.2.21.
Let y E U be given. Consider t h e following gamily of l i n e a r functionals f
E
( v ) on C E
( O , O , ) ,E
6 ):
-'I vE( y )exp
(1.2.164) f E ( v E ) := ( 2 ~ )
( - Ix-y
1 /E) dy.
U
Then t h e family (1.2.164) i s equiuontinuous on C
( 0 , O ) ,E
(5). Indeed,
one
has :
One of t h e fundamental r e s u l t s i s t h e following theorem of HahnBanach. Theorem 1.2.29.
Let B be a Banach space equiped w i t h the norm 1 I 1 I and l e t M c B be a subspace of 8. Let f ( x ) be a l i n e a r functional on M such t h a t (1.2.165)
/ f ( x )1 5
Ci
1x1
IB,
v
x
E
M.
Then there e x i s t s an extension of f ( x ) onto the whoZe Banach space 8 such that /f(x)1 5
where c 2
0
C ( 1x1
18,
V x E
8
i s t h e same constant as on t h e r i g h t hand-side of (1.2.165).
Obviously, The Hahn-Banach theorem i s s t i l l t r u e f o r t h e f a m i l i e s of equicontinuous f u n c t i o n a l s . L e t 8 be a Banach space. The s e t
f u n c t i o n a l s on
8' of a l l continuous l i n e a r
equipped with t h e norm (1.2.1621,
c a l l e d t h e dual space of
i s given a Banach space,
8.
Existence of a convex neighbourhood of zero i n a Banach space
B
is
t h e necessary and s u f f i c i e n t condition f o r t h e n o n - t r i v i a l i t y of t h e dual
1.2. Functional Analysis
space 8 ' . Thus, t h e dual space of L . ( U ) ,
U
Rn
C
53
,p
< 1 i s t r i v i a l . The
Using t h e induction argument, one g e t s (1.2.152) f o r Uh
=
and
R:
using Remark 1.2.24, one f i n a l l y e s t a b l i s h e s (1.2.152) f o r each U
I
h'
Remark 1.2.26. In t h e proof of Theorem 1.2.25 we have used an a d a p t a t i o n t o t h e d i s c r e t e s i t u a t i o n of t h e method of prving t h e Morrey's r e s u l t f o r c l a s s i c a l Sobolev space, which i s t o be found i n [Brez. 1 1.2.6.
1
Linear f u n c t i o n a l s and dual spaces
8 be a r e a l ( o r complex) Banach space.
Let
A l i n e a r mapping f ( x )
from 8 i n t o t h e f i e l d of r e a l ( o r complex) number i s c a l l e d a l i n e a r functional. A l i n e a r f u n c t i o n a l f ( x ) defined on t h e whole Banach space
E 8 i f f(x
t o be continuous a t a p o i n t x {xnlnZl C 8 such t h a t
1 Ix
0 -xoI
I
0 for n
+
)
+
+
-.
A l i n e a r f u n c t i o n a l f ( x ) on a Banach space
point x
E 8 i f it
8 is said
f ( x ) f o r any sequence 0
8 i s continuous a t each
i s continuous a t j u s t one p o i n t xo E
A l i n e a r f u n c t i o n a l f ( x ) i s s a i d t o be bounded on
8. 8 i f there exists a
number c 2 0 such t h a t (1.2.161)
If(x)I 5 c / I x I / , V x
E 8.
The l e a s t number c such t h a t (1.2.161) s t i l l holds i s c a l l e d t h e norm of t h e f u n c t i o n a l f ( x ) and i s denoted by (1.2.162)
I If1 I
= sup I f ( x )
xE8
I/j
1x1
1
=
1 If I 1 . SUP
Thus If(x)
I.
11x1 I=1
A l i n e a r f u n c t i o n a l f ( x ) on a Banach space
8 i s continuous i f f f ( x )
i s bounded.
D e f i n i t i o n 1.2.27.
A f a m i l y of l i n e a r f u n c t i o n a l s f ( x ) on a family o f Banach spaces 8
w'
depending upon a parameter w i n some parameter s e t fi 5 R~ i s said t o be equicontinuous (or equiboundedl if there e x i s t s a constant c 2 0 , which
F i f a one-to-one continuous map j of 8 onto a l i n e a r s u b s e t of F i s given, which preserves t h e a l g e b r a i c o p e r a t i o n s . Thus, one has:
space
\ 1 j (x) I I
1
5 C 1x1
IB
i s called natural.
with some c o n s t a n t C > 0 . I f C
=
1 then t h e imbedding
1. Manifolds, Functional Analysis, Distributions
54
I f t h e imeage of
8 i n F i s dense, then t h e imbedding i s c a l l e d dense,
Linear Operators
1.2.7.
Let
x1
and X 2 be two t o p o l o g i c a l vector spaces equipped with t h e
Hausdorff copology. A map A : X,
+
x2 i s said t o be a continuous linear
operator from x1 i n t o x
i f the following conditions are s a t i s f i e d : 2 i s defined on the whole of X1;
lo. A 2O.
A i s l i n e a r , i . e . f o r each couple x and y i n
( r e a l o r complex numbers) A(ax+By)
=
c1
x, and f o r each two scalars
and 5 one has:
aAx + 5Ay.
3O.A i s continuous a t each point x F: x , i n t h e sense of the topologies on x1 and x2, i . e . f o r each neighbourhood u 2 ( h ) of AX i n x2 t h e r e e x i s t s a neighbourhood ul(x) o f x i n x1 such t h a t f o r every x' E ul(x) one has:
Ax' E U2(AX).
A familyog l i n e a r mappings
A
:x1+x2,
respect t o t h e p a r a m e t e r s e t n ) i f f o r :
KCX
1
o E n , i s s a i d t o b e equicontinuous ( w i t h
each given u 0 E G the l i n e a r operator
x 1 + x 2 i s continuousand, moreover, theimage of each bounded s e t under the map A
I w E n ) i s uniformly bounded with respect t o w E n ,
i . e . the union u w E n A w ~belongs t o a bounded s e t i n x2.
81 and B 2 be two Banach spaces. A linear map A : 8, + 8, i s said t o be bounded i f there e x i s t s a constant c > 0 such t h a t 1 / A X ( ( s c ( l x ( 1 , V x E 8,. The l e a s t constant c > 0 Let
x2
x1
w i t h such a property i s called the norm of
I IAl
18 1
2
A :
B,
+
8, and i s denoted by
.
A l i n e a r map A :
8, + 8, i s continuous i f f it i s bounded.
Obviously, one has:
Let {Au}u>O be a sequence of bounded l i n e a r mappings form 8, i n t o 82' The sequence t A v } v > 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
<
r} c 8 ,
B2 11x1
I
< 1) c
B,
Bl
under t h e Zinear mapping A , i . e .
Particularly important for applications is the following consequence of the open mapping theorem: Corollary 1.2.32.
Let A
:
8,
+
8 , be a continuous Zinear one-to-one
mapping o n t o €3.,
Then i t s
1. Manifolds, Functional Analysis, Distributions
56
inverse
A
-1
:
B2
+
B1 is again a continuous l i n e a r mapping onto B
Definition 1.2.33. A family of linear-mappings Aw
: EW +
Fw, w
1'
E 0, i s
said t o be equibounded lor equicontinuous) i f one has:
The following extension of Banach's theorem is useful in applications. Theorem 1.2.34. L e t
: E~
A
F ~ ,w
+
E 0, be linear,one-to-one and equi-
continuous. Assume t h a t one has:
Then -1 AW
: F w + E
W'
w E R
i s equicontinuous. Proof. One -
only has to show that
If the last inequality does not hold, then there exists a sequence
{wk}k21 c R such that
E Fw such that
y,
so that there exists a sequence
k
k
Introduce
-1 x = A , yW , zw Wk k k k
=
xu /IIxw k k
\IE
,
k = 1,2,.,.
w k
so that the last inequality may be rewritten as follows: -1 )lAwkZwkllFw 6 k
,
for
k Let u
E EW,
1 luW/I E
= W
1 and define:
I/zW
k
/IE
=
w k
1,
k = 1,2,...
1.2. Functional Analysis
zw
= u
w
for
w
# wk
and
z
w
=
z
w
for
w
k
=
w k'
Obviously, one has:
the latter being a contradiction with ( 1 . 2 . 1 6 9 ) . Remark 1 . 2 . 3 5 . Aw
:
Ew
+
I
The set L(Ew;Fw) of all linear equicontinuous mappings
Fw, w E R , is a Banach space equipped with the norm:
then L(E ;F ) = L(E ) is a Banach algebra, i.e. besides = F o w the natural linear structure it has a l s o multiplication as a natural
Besides, if Ew
algebraic operation having the property:
Of course, it is readily seen that the set of invertible operators is open in L (Ew;Fw). Let A
o
:
E
w
+
Fw, w E R be a family of bounded linear mappings from
the family of Banach spaces E
into the family of Banach spaces F
.
If gw E FA, i.e. gw is a continuous linear functional on Fw, then the family of functionals fw(Xw) := gw(Awxw), xw E Ew, w E R , is a family of continuous linear functionals on Ew, i.e. f E E ' and, morew w over, one has:
Thus, the correspondence g
w
+
f
w
which is called the adjoint of A Obviously, one has:
so that
defines a linear operator A
.
* 0'
1. Manifolds, Functional Analysis, Disfributions
58
A;
:
F;
E;
+
is again continuous.
It is equicontinuous iff A
: Eu
-f
Fw, w E
R , is equicontinuous.
Let E and F be two Banach spaces and let D(A)
5E
be a vector sub-
space where a linear operator A is defined and takes D(A) into some vector subspace R(A)
5 F. D(A) and R(A) are called respectively, domain
and image (or range) of A . The set N(A)
C E
whose image under the action
of A is zero, is called the kernel of A and sometimes denoted by ker A. Obviously, N(A) is a vector subspace in E. The map A
D(A)
:
R ( A ) is one-
-f
to-one iff N(A) = { O } . In that case the equation Ax = y is uniquely resolvable, V y E R(A). This equation is said correctly resolvable if there exists a constant C > 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 <
m.
Normally
solvable equation (1.2.172) with a closed operator A is called d-normal, if the orthogonal complement coker A to the range R ( A ) of A is finite dimensional: dim coker A
m.
As it has been said hereabove, for a closed operator A the correct
solvability of (1.2.172), (1.2.1731, i.e. the validity of the a p r i o r i
estimates:
with some constant
C >
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
<
(m) =
E q(m-k,E,kc)iE(k), kEZ
( k ) is, of course, the Fourier transform of uE(x) on S
1
.
Now, if B = (bmk)-m<m,k<m is a linear operator in the space l2 of I I = 1lwkl2, then one has the 2 ' k following discrete analogue of Schur's lemma(see Lemma 3.12.8):
~ < ~I Iw the sequences w = { w ~ ] - ~ <with
Indeed, for each w E l2 one has:
Thus, (1.2.184). (1.2.185) yield:
1
S :
1.2. Functioiul Analysis
where C > 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
<
-,
i.e. one has with some
so small that
one gets the conclusion that €QE, E E ( O , E H2(S )
E~
> 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~
<
-.
1.
66
Manifolds, Functional Analysis, Distributions
Therefore, one gets the conclusion that the diagramme
is commutative, V
E
Therefore, A E index
K(A )
ker A E
E
=
= K(A )
0
coker A
E (O,E : H(2,2)
=
),
O
= 0, V
provided that
1 ,E(S ) E
+
H(o,O)
,E
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(<), 5 f [-a,n] be continuous ~II-periodicfunction: a(-n)
=
a(n), and let b(h,<): [O,hol
x
[-n,n]
variables (h,S) and 2n-periodic in 5 : b(h,-s) Further, let c(x,h,c)
: R
X
[O,ho]
X
+ C
be continuous in
! b(h,n),
V h f [O,ho].
[-n,nI be continuous in variables
(x,h,<), twice continuously differentiable with respect to x f R and 2n periodic in 6: c(x,h,-nj
c(x,h,n), V ix,h) E R
Z
x
[O,hol.
Furthermore, assume that one has with some constant K > 0 :
where, as previously, <x>' Let
E$,
=
=
2
l+x
.
hZ be a lattice in IR with meshsize h and 1
(
2 %
discrete analogue of L (IR) on the lattice 2
IR,,,
)
be the
equipped with the family
of inner products and norms:
where, as previously, v*(x) stands for the complex conjugate of v(x). Let F be the Fourier transform on the lattice h
and let Fh' be its inverse:
?-It
67
2.2. F u n c t i u d Analysis For A(x,h,hS) = a(hS) + hb(h,hS) + hc(x,h,hS)
with a(E), b(h,S), c(x,h,S) as defined hereabove, and for each meshfunction u E 12(%
)
%,
which vanishes for x E
1x1 2 r
r > 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\
where
is the Fourier transform of x
+
c(x,h,hrl) on the lattice
%-
Let S(E;,h,hrl) be the integral Fourier transform of c: 6(S,h,hq) = Fc( .,h,hrl)
=
i c(x,h.h~)exp(-ixE)dx. IR
It is readily seen that the following Poisson formula relates (1.2.190) ch(S,h,hn)
=
ch and 6:
&(5+2nkh-',h,hn) ,
C
kEZ
the series on the right hand side of (1.2.190) being convergent uniformly with respect to h E [O,h0 1, hS E [--71,711,hrl E [--TI,TT]. Indeed, for x E
one finds:
and the uniqueness of the Fourier transform on the lattice
%
yields
I . Manifolds, Functional Analysis, Distributions
68
(1.2.190), provided t h a t t h e series on i t s r i g h t hand s i d e converges One f i n d s i n t e g r a t i n g by p a r t : c(S,h,ri) =
J
c(x,h,q)exp(-ixS)dx =
IR =
<5>-2
2
1 c(x,h,n) (l-ax)exp(-ix5)dx
=
IF? =
-2
<<>
J
2 (l-ax)c(x,h,n)exp(-ixS)dx,
IR s o t h a t (1.2.189) y i e l d s : (1.2.191) \;(5,h,rl)
1
<x>-2dx
5 K<5>-2
= ITKic>-*.
IR I n e q u a l i t y (1.2.191) implies convergence of s e r i e s (1.2.190) and, moreover, i t i s r e a d i l y seen t h a t one has with some o t h e r constant K1 > 0:
Hence, one has:
and t h e same argument y i e l d s :
Thus, Lemma 3.12.8
(Schur) i m p l i e s t h a t f o r t h e n o r m o f t h e f a m i l y o f i n t e g r a l
o p e r a t o r s Ch with t h e k e r n e l s ch(S-rl,h,hn),
t h e following i n e q u a l i t y holds:
sup Olh
"hi
'L2(-n/h,n/h)'L2(-n/h,r/h)
0
so t h a t t h e Parceval i d e n t i t y y i e l d s f o r
i.e.
'
'TI
1'
9, introduced
hereabove:
1.2. Functional Analysis
69
is equicontinuous. In the sequel we shall assume that a(S) is continuously differentiable as a 2n-periodic function and satisfies the condition:
Introduce the family of subspaces 12(%,+) functions u € 12(%),
u
:
%
+
Analogously, we denote by 12(lF$,-) c 12(%) functions u € 12(\
),
C
12(\)
of all mesh-
\.
C, which vanish for x < 0, x €
the subspaces of the mesh-
which vanish for x 2 0. Also denote by
multiplication by the characteristic function of the set
x
the
F+ = {x b
0).
Consider the following family of finite difference operators in 12(\,+
) :
(1.2.193) Thu
:=
Since a(n),
X \u,
n
Th
:
12(\,+)
-f
12(%,+).
€ [-n,n] is continuous, 2n-periodic and does not
vanish, the curve [-n,nI 3
n
+
a(q) in the complex plane is closed and
does not pass through the origine. Therefore the so-called winding number m,
is a well-defined integer for a(n) and shows how many times any generic point on the curve [ - n , n ]
3
n
+
a(n) rotates arround the origin when
11
varies from -n to IT. We are going to show that the family of finite difference operators T h defined by (1.2.193), is noetherian, that its index K ( T ) does not depend h on h € (O,hol and that, moreover, (1.2.194)
K
=
m.
First, consider in 12(%,+) difference operators
where
the following family of finite
1 , Manifolds, Functional Analysis, Distributions
70
with a(5) as introduced above. Notice that the winding number for
is zero, so that
is a continuously differentiable periodic function of 5 E [ - n , n I . Introduce 6
J q ( n ) (l-exp(i(5-n)?6))-'dn, 6 > 0. Inl<TI 6 Notice, that q ( 5 ) can be extended as an analytic function in the
(1.2.195) q -+ ( 5 ) = ( 2 n ) - l
[Re 5 [ 5
half-strip
TI,
Im 5 > 0, V 6 > 0 .
6
Analogously, q - ( S ) can be extended as an analytic function in the halfstrip
IRC
<j
s v ,
Im 5 < 0 ,
v
6 > 0 . Moreover, the limits:
exist, are analytic functions in their respective half-strips and are continuous on the closure of these respective half-strips. Indeed, obviously, q , ( S )
are analytic in their respective half-strips
Im 5 > 0 and Im 5 < 0, since the corresponding integrals are uniformly convergent along with their first derivatives with respect to 5 , when Im
f,
t p > 0 and Im 5 S - 0 < 0 , respectively. Now let f, E [--71,n1.
Then, one has
1.2. Functional Analysis
71
The integral on the right hand of the last formula being uniformly convergent (since q ( 5 ) is continuously differentiable), one draws the conclusion that
6
limit (1.2.196) for q+(5) exists and is a Continuous function of 5 E
[-T,T].
6
The same argument yields the same conclusion for q-(5) as 6 4 0. Notice that if u E 12(\,+
then its Fourier transform uh(S) on the lattice
),
be extended analytically to the half-strip ively, if u € 12(%,-),
%
can
IRe 51 5 n , Im 5 > 0. Respect-
then its Fourier transform on the lattice
can be extended analytically to the half-strip
IRe 51 5
71,
IF$,
Im 5 < 0.
Introduce
Then, obviously, a + ( < ) are cotninuous 2r-periodic functions of
-
5 E
[--11,711
and a+(5) are analytic in their respective half-strip -
Im 5 > 0
and Im 5 < 0. Furthermore, one has the following factorization for a
0
(1.2.197) ao(S)
(6):
a+(5)a-(5).
=
Let us rewrite the equation (1.2.198) X+aOu
f,
=
f f 12(\,+)
in the following fashion: (1.2.199) a u = f - v, 0 where v E 1 ( R ) is also unknown. 2 h,After the Fourier transform on Rh equation (1.2.199) becomes: (1.2.200) a0(hS)ih(S) = f ( 5 ) - jh(S), where Gh, ? are analytic for Im 5 > 0 (and 2~rh-l-periodicin Re h analytic for Im 5 < 0 (and 2Th-l- periodic in Re 5 ) .
c),
vh(<) is
Using factorization (1.2.197), one may rewrite (1.2.200) equivalently in the following fashion: (1.2.201) a+(hS)Gh(S) = (a-(hS))-'Fh(5) Since (a-)-'Gh(5)
-
(a-(hS))-l;h(S).
is analytic for Im 6 < 0, and a u is analytic for + h
Im 5 > 0, one has: -1T[+(a-) vh = 0, ~
-1-1 Il+(a+) uh = (a+) u
h'
1 , Manifolds, Functional Analysis, Distributions
12
where iI+ are defined by (1.2.196).
-
Therefore, applying to both sides of (1.2.201) the projection operator
Il+ defined by (1.2.196), one finds:
Obviously, one has:
where
Indeed, (1.2.203), (1.2.204) follows immediately from (1.2.2021, since IT is an orthogonal projection in L (-IT/h,iT/h) and iI+ + iI- = Id. -1
Using the inverse Fourier transform Fh
, and
the Parcevals identity, one
finds for the inverse operator of X+ao:
Now (1.2.206)
we may rewrite
x+%
x+\
+
=
x+(aOOm
in the following fashion:
hQh),
where 0 is the shift operator, (Bu) (x) = u(x+h),
(Ornu)(x)
=
u(x+mh),
and Qh is the finite difference operator defined by b(h,C) andc(x,h,E),i.e. -1
Qh = Fh (b(h,hC) + c(x,h,hS))Fh. Since -1
(x+ao)
:
12(%,+)
+
12(%,+
),
h C (O,hol,
is equicontinuous, we may rewrite (1.2.206) in the following fashion: (1.2.207)
x+%
=
x +ao ( O m + h(x+ao)-1 X+Qh).
Obviously, one has for each u E 12 ( %,+)
:
1.2. Functional Analysis
13
where
Y2
=
SUP
OShSh
0
a s it has been shown hereabove. Furthermore, obviously one has: X+aoBm = m x+aoem = x+B X+ao f o r m 2 0.
x +a o x + O m
f o r m 5 0 and
Since
-1
h(X+ao)
X+Qht
h
E (OthO1
has i t s norm s u f f i c i e n t l y small, V h E (O,hol when ho i s s u f f i c i e n t l y small, one g e t s t h e following conclusion f o r t h e index K ( % ) ,
using t h e
p r o p e r t i e s of t h e index mentioned hereabove: (1.2.198)
K(\)
m
= K ( X a )+K(x+8 ) .=
+ o
m,
given t h a t h,+) x+ao : 12(R
'2(%,+)
i s an equiisomorphism ( a s it was y e t shown) and s i n c e , obviously, m K ( x + G ) = m. Indeed, f o r m 2 0 one has: ker x+Om
=
{6(x-kh) }05k<m
, Coker
x+Om = { O } ,
and f o r m < 0 i t i s v i c e versa; here 6 ( 0 ) = 1 and 6 ( X I x E
..{OI.
5
=
0 for
The same conclusion (1.2.198) is s t i l l t r u e when a depends on x E R , t o o and s a t i s f i e s t h e conditions:
1
. a(x,5)
1 is in C (R )
is 2 ~ - p e r i o d i c i n 5 EX=
Rx
x
2'.
a(x,c)
a s a f u n c t i o n of x with 1 s u f f i c i e n t l y l a r g e , it 1 2 and i s i n C ( R ) as a f u n c t i o n of (x,S) E R2 =
IR
5; =
am(c)+ a'(x,<),
1. Manifolds, Functional Analysis, Distributions
14 where x
a ' ( x , < ) s a t i s f i e s the inequalities:
+
Notice t h a t t h e winding number m i n t h i s case does not depend on x
E
R ,
s i n c e it i s a continuous f u n c t i o n of x E IR, whose values a r e i n t e g e r . Remark 1.2.39.
One may t h i n k of using f i n i t e d i f f e r e n c e approximations f o r
solving numerically t h e Wiener-Hopf i n t e g r a l equations of t h e form:
X+(X) ( u ( x ) +
J
k(X-y)u(y)dy) = f ( x ) ,
x E R+
,
R+ where t h e k e r n e l k ( x ) i s f o r i n s t a n c e , a continuous function of x E R k
E
and
LI(R). Let u s r e c a l l t h a t t h i s equation i s uniquely s o l v a b l e i n L 2 ( R + ) i f f
G ( 5 ) of k ( x ) s a t i s f i e s t h e condition:
t h e Fourier transform
and t h e winding number
K
for 1
+ G ( 5 ),
i s zero: c = 0 . If
K
> 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<) , h
1 . Manifolds, Functional Analysis, Disfributions
76
-I ic = h (0-l), the index
of the corresponding discretized equation 1s h zero for h E (0,lj and -1 for h E ( 1 , 2 ) . Of course, using k ( x ) = k(x), h h * -1 , i<* = h-l(l-e-'), one gets or k (x) such that kh(B) = (l+ic ) Vx E K~
finite difference approximations of the corresponding integral equation whose index is zero for all h > 0 . h Indeed, for k (x) = k(x), x E IR,,
I +
, one
finds:
-1 = ((l+h)e-exp(-h))(e-exp(-h))
ih(e) h
and for the backforward approximation a * of a one has: x,h x 1
+
-h kh(6) = (he)-'
so that in both cases K~ = 0 , '+h > 0 .
I
Now several definitions and results will be given, which are concerned with the linear equations, containing a spectral parameter. Let
A
:
€)(A)
+
R(A)
with domain D(A) and image R(A) in a Banach space E, be linear and closed. is said to be a regular p o i n t of A if the equation:
A complex number (1.2.210)
Ax x
:=
(A-XId)x
=
y, x E D ( A ) ,
y E E,
is correctly and densely resolvable. The set of all regular points is called the r e s o h a n t s e t of A, and the complement of the resolvant set in the complex plane is said to be the spectrum of t h e operator A . The operator A being closed, its spectrum is a closed set in the complex plane, its resolvant set being open. The linear operator RA(A), (1.2.211)
RX(A) = (A-iI)-'
defined for each A in the resolvant set of A, is called the resolvant of A; sometimes it is also denoted by
%.
The resolvant R x is an analytic function of h in the resolvant set, valued in the Banach space L ( E ; E )
of all bounded linear operators in E. Each
connected component of the resolvant set of A is a natural holomorphy domain for its resolvant Rx
and RX can not be extended analytically through
the boundary of this component. For each couple
x , of ~
regular points of A the following Hilbert's (or
1.2. Functional Analysis resolvent) identity is true: RA
-
R
ll
= (A-u)R
A
R
P’
as well as the following consequence of the latter identity: k (d/dA) RA If A
:
E
-f
E
=
k+1 k!RA , V k E
Z+.
is bounded, then its spectrum lies in the disk
1 I A l lE+}.
The l e a s t rA 2 0 such that 1x1 4 rA } still contains the spectrum of A, is called the s p e c t r a l radius of A. The
t
1x1
5
following Gelfand’s formula is valid for rA:
and the limit on the right hand side exists for each linear bounded operator A:E+E. If A.
is an isolated singular point of the resolvent R A , then RA
admits a Laurent serie’s expansion:
where the linear bounded operators An
: E
-t
E commute which each other
and satisfy the relations: €or k >= 0 , m 5 -1,
AkAm = 0 , An
=
A ~ + ’ , for n t I, 0
A-p-q+l
=
for p , q 2 1.
-A A -p -ql
The last relation with p = q = 1 implies that the residue A-1, A-1 : = Res RAIA=A 0
has the following projection property:
Also the following formulae are valied: (A-AOId)A
If A
0
n
=
A n-1‘
v n # 0 and (A-AOId)AO = A-l+Id.
is an m-pole (a pole of order m) of the resolvent RA, then A.
is an
1 , Manifolds, Functional Analysis, Distributions
78
eigenvalue of the operator A and, moreover, one has: = R( (A-hoId)n), V n 2 m. I~/(A-~)
R(A-~)= N ( (A-AOId)"),
Besides, in that case E can be represented as a direct sum: E =
N ( (A-AOId)") fB R ( (A-AOId)"),
V n 2 m,
and for each element RAx, x E E, one has the expansion:
= A.x, ej = Aj-kx, Ae = eo, Ae. = Ae. ,-1 +ej-2, 1 5 j S k 5 m, j i 1-1 the element eo E E being an eigenvector (associated with A 0) and the elements eo,...,ek-l being called the Jourdan chain associated with A
where f
0'
If A
1 A 1-l;
:
E
-+
E is bounded, then its resolvent Rh vanishes at 1
=
as
if A is unbounded, then its resolvent RA has a singularity at
infinity. In the latter case if, moreover, A = point of R A , then A =
m
m
is an isolated singular
is an essential singularity.
The set consisting of the spectrum of an unbounded operator A
: E + E
and of the point A
=
is called the extended spectrum of A.
m,
The extended spectrum is always non-empty. A non-empty part A l of the spectrum A(A) A is said to be a
SpeCtPffl
of a linear closed operator
S e t if A l is at the same time closed and open
in the spectral set A(A) equipped with the natural Hausdorff topology. For each bounded spectral set A1 there exists a closed Jourdan contour r l , which lies in the resolvent set of A, encloses the spectral set A 1 and does not enclose any point of the complement spectral set A 2 = A \ A l . One may always choose the orientation on to the left when one goes along
rl
rl
in such a way, that A
1
stays
arround the spectral set A1.
The operator
P1
:=
-1
-(2i~i)
I
RA(A)dA
rl is the so-called Riesz' projector. The Banach space E can be split in a direct sum: E = E El
=
PIE and E2
=
(Id- P1:E.
8 E2, where
The subspace E l belongs to the domain D(A) of
A and is invariant under the action of A. The spectrum of the restriction A
1
of A to E l coincides with A l .
The restriction A2 of A to the linear Set
(Id- P1)D(A) is a closed linear operator in E 2 , whose spectrum coincides
1.2. Functional Analysis
79
with A2. If Al,...,A are mutually disjoint bounded spectral sets for a linear k closed operator A, then A = A @ ... B 1 $ @ $+1, E = E 1 B... Ek Ek+l' Ej, 1 6 j 6 k, are bounded and Ak+l : where A . : E . Ek+l + Ek+l is I 3 closed.
'
@
-f
The corresponding projectors
P.P.= 1 3
P ,
1 6 j 5 k+l, are mutually orthogonal:
0, for i # j, and the spectrum of A. coincides with Aj, l < j
An important class of operators A
:
E
+
E is the one, for which
the resolvent RX (A) is a compact operator at least for one X
=
X o in the
0
resolvent set. In this case, Hilbert's identity implies that RX is compact for all X in the resolvent set. An operator A
:
E
+
E whose resolvent R A
is a compact operator, is necessarily closed and has a spectrum A(A) which consists of isolated points. Each point in A(A) is a pole of RX(A) of a finite order and, at the same time, an eigenvalue of A. The kernels N( (A-XOId)k, , k = 1.2,. . . with Xo E A (A) have finite dimension and, moreover, there exists no such that N( (A-hoId)no)
=
N( (A-XOId)k), V k 5 n0'
while for k 5 no one has:
* k If the domain D(A) of A is dense in E, then the kernels N(( (A-hoId) ) ) k . have the same dimension as the kernels N((A-X Id) ) , k = 1,2,
...
0
Let X E A(A), i.e. (1.2.210) is not uniquely resolvable. The complex number X is said to belong to the discrete spectrum of A if (1.2.210) has more than one solution for y = 0; X is said to belong to the continuous spectrum if (A-XId)-l is defined on a dense set in E and is unbounded. If A
:
E
-f
E is compact, then its spectrum A(A) consists of at most
m ) , which are poles (Ak -r 0 for k countable set of eigenvalues {X 1 k kZO %, k t 0, of the resolvent RX(A). The kernel -f
of finite multiplicities
N((A-AkId)) is the eigensubspace associated with Xk and the dimension of this kernel is the multiplicity of the eigenvalue X k' The subspace
Nk
:=
N( (A-hkId)mk) is called the root subspace, associated
with the eigenvalue Xk, and its dimension is called the algebraic multiplicity of X k' One may choose in
Nk
restriction of A to
a basis consisting of Jourdan chains so that the
N k will have as its matrix (with respect to the chosen
basis) the normal Jourdan form. We shall end this section by presenting some definitions and results
1. Manifolds, Functional Analysis, Distributions
80
concerning the classical regular perturbation theory. Let 5
+
A(<) be a family of continuous linear operators in a Banach
space, which depends analytically on 5 E C, 5
A(<) =
k
%, 9, E L ( E ; E ) ,
15
1
<
r, i .e.
151 < r,
k>O the series being convergent in the operator norm in L ( E ; E ) , V 5 with 151 < r. Let A
0
E h(A(0)) be an isolated point in the spectrum of A(0) and
assume that the corresponding invariant subspace (associated with A 0 1 (A(0)) has dimension m < m . Let U C C be an open set, such that EAO
U
n
A(A(O)) = {A 1. Then there exists a 6 E (O,r), an integer k 5 m and an 0
6 the set U
integer n such that for 151
n
A(A(5))
A1(5), ...,A ( 5 ) ; moreover, each function A , ( < ) ,
k 1 analitically on the main branch of the function z = = Xo,
condition: A . ( O )
P.(c), 1
1
consists of the points
1 6 j 5 k, depends
c1ln
and satisfies the
1 5 j 6 k; besides, the corresponding projectors
1 5 j 5 k, admit Laurent series' expansions with respect to z = 5
P.(5) 1
=
C
A. 5
s2-N
Is
s/n
1/n
,
where A . E L ( E ; E ) are continuous linear operators (F. Rellich). 1s
Of special interest is the following simplified version of Rellcih's
result. Let A(5) = A0+5A1, 5 E
C,
151 < r, be a family of self-adjoint
A operators in a Hilbert space H. Further, assume that A(0) = . complete set of eigenvectors {e(O)
has a
associated with the eigenvalues
(O) be an being an o:thonormal basis in H. Let A") k ih:o)jn>co' {en lnbO isolated simple eigenvalue of . A (i.e. of multiplicity one). Then the
eigenvalue A ( 5 ) and the associated eigenvector ek(5) of A(<) are analytic k functions of 5 for 151 < 6 5 r, and, moreover, one has:
where
and an analoguous formula may be obtained for eL2). Neither the Rellich result, nor the perturbation formulae in the specific case hereabove are
,
1.2. Functional Analysis
81
true when the perturbation becomes singular (i.e., for instance, when and A1 are unbounded self-adjoint operators in strictly stronger than A
0
H and A1 is in some sense
).
Example 1.2.40. Let us consider here an example in order to illustrate the complexity of spectral problems when the perturbation is singular. Let ( 0 , l ) 3
E
-+
A€ := a-E
AE be the following family of differential operators:
2 3
a , a
=
d/dx
considered as unbounded operators in L (U), U = (0,l). 2 Such a singular perturbation appears in the linear theory of dispersive singular perturbations. The natural domain of A
is the Sobolev space H 3 ( U ) , equipped with
the family of norms:
for each given Sobolev norm
with respect to
1
I . I I (It21,E being equivalent to the with constants which depend on E (i.e. non uniformly
E (0.1) the norms
E
I 1.1 1 E
E (0,l)).
It is readily seen that the family of linear mappings
is equicontinuous. It is also readily seen, that the spectrum of A
with H (U) as its 3
domain, is the whole complex plane. We shall further restrict the domain of A D(A
)
=
{u E H
(1,2), E
(u)
I
by defining it as follows:
u(o) = au(o)
=
~ ( i =)
01
and shall be concerned with the investigation of the spectrum of the family:
Obviously, (1.2.212) is equicontinuous. First, we show that (1.2.213) Ail
:
L2(U)
-+
D(AE)
is equicontinuous too. Indeed, AE1 is a family of integral operators,
I . Manifolds, Functional Analysis, Distributions
82
(Ailf)(x) =
f GE(x,y)f(y)dy, U
where the kernel G (x,y) can be represented in the form: (1.2.214) GE(x,y) = EE(X-y)+EE(Y)+Ea E (y)(exp(-x/-E)+exp(-(l-x)/E)-l)Y E -EE ( 1-y)exp ( - ( 1-x)/ E ) +exp (-l/~) g, (x,y), the kernel EE(x), (1.2.215) EE(x) := (1/2)(l-exp(-lxl/E))sgn x being a fundamental solution of the operator (1.2.216)
( a x--E 2a 3X)E€ (x) =
6(x),
ax-€'a3
x
on R
, i.e.
E IR m - -
with 6(x) the Dirac mass, and gE(x,y) being in C (UxU) and such that (1.2.217)
sup max J E O<E51 (x,y)EUxU
m m-k k ax aYg,(x,y)
I
<
m,
o
5 k
c
m,
v m
2
o
One has: (1.2.218)
k /(€a) EE(x)/ S 1/2, k = 0,1,
k f Ek-1 /axEE(x)/dx 5 1/2, k = 1 , 2 , IR
so that it is readily seen that the following inequalities hold for
GE(x,y) defined by (1.2.214), (1.2.215):
where, of course, (1.2.217) has been used, too. Using Schur's Lemma (Lemma 3.12.8), one gets the conclusion that for u
E'
U
(x)
:=
I U
one has:
2
GE(x,y)f(y)dy = (As'f) (x), f E L (U),
1.2. Functional Analysis where C 1 = max {C,1/2] does not depend on f and
( a x-E 2 d x3 E
F.
(1.2.216) imply:
NOW (1.2.214),
Thus
83
=
f w .
2 3
a u(x) belongs to a bounded set in Lz(U) and, moreover, one has: X
Now, (1.2.219), (1.2.220) yield:
the latter being precisely the claim (1.2.213). Furthermore, the family of integral operators
is compact (uniformly with respect to
E
E
(O,E~]),
2 for each f E L (U) the
(0 < E 5 E 0 ) belongs to a bounded set in H1(U), the latter family A-'f 2 being compactly imbedded into L (U)
.
For each f E H (U), which satisfies the conditions: 3
(1.2.221)
1 f(x)dx
= 0, f(0) = 0,
U
one has the following estimate:
where the constant
c
>
0 does not depend on f and
natural reduced operator for A
(Ai'f) (x)
=
and where Ail is the
E '
1 (1.2.223)
E
-1
1 Go(x,y)f(y)dy
=
0
70 f(y)dy,
the kernel GO(x,y) = H(x-y) = 1 for x > y and H(x-y) = 0 f o r x < y being the point-wise limit of G (x,y) as
E
*
0:
I . Manifolds, Functional Analysis, Distributions
84
G (x,y) = lim G~(x,Y), 0
v
(x,y) E
UxU,
E+O
as a consequence of (1.2.214), (1.2.215), (1.2.218). -1 -1 Indeed, let uE = A f, uo = A f and introduce 0
(1.2.224)
vE (x) = uE (x)-uo (x)-aEexp(-x/E)-bEexp( - (1-x)/ E ) -cE '
where the constants aE,bE,cEare chosen in such a way, that vE E D(AE), i.e. v ( 0 ) =
a XvE ( 0 )
=
~ ~ ( =1 0. )
It is readily seen that one has:
f(y)dy + O(exp(-l/E)), as 6 -t +0, U the latter asymptotic formulae for f satisfying (1.2.221) taking the form: bE = Ef(0) -
(1.2.226)
a
= bE = c
= O(exp(-l/E)),
as
E -t
+O.
For vE(x) defined by (1.2.224), vE E D(AE), one also has:
(a-E 2a 3)vE(X)
=
E
2 3 aXfw,
so that (1.2.213) implies:
Now, (1.2.224) , (1.2.226), (1.2.227) yield (I .2.222)
. If f (x) does not
satisfy (1.2.221), one still has the following estimate:
where again the constant C > 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 <
m
C
U there exist an integer N 5 0 and a constant
such that the inequality
holds for every 4 E DK. Every locally integrable function f in U with complex values is an element in u ' ( U ) , whose action on each $ E D(U) is given by the formula: = J f(x)$(x)dx, U
where dx is the Lebesgue measure on U.
-
Similarly, if p is a complex valued Bore1 measure on U, or if p is a positive measure on U with p ( K ) <
u
for every compact K c U, then
E D'(U) and
If
is a differential operator with coefficients aa E Cm(U),
/(TI <
m, then
1.3. Distribution T h e o y
87
is a continuous linear mapping defined in the following fashion:
< P ( x , a )u,Q>
P*(x,a
where
If f f'D
E
)
=
,
is the formally adjoint of P ( x , a
):
D'(U) is locally intrgrable in U, then its distributional derivatives
are defined as follows:
or, more explicitly:
m
The multiplication by a function @ E C (U) obviously is a continuous linear mapping of A
D'(u)
into itself.
natural (weak*-) topology on
D' (U),
induced by the one on D(U) is
defined as follows: A sequence of distributions
i L v 1v>o
t
D'(U) is said to be convegent
to a distribution L if one has: lim
,$> = < L , $ > ,
V $ E
D(u).
V+-
Furthei, { L v j v Z O c D ' ( U ) is a Cauchy sequence if {
C, V
Each Cauchy sequence { L v } V 2 0 v
-+
-.
C
C
is a
Q E D(U). C
D'(U) has a unique limit L E
D' (U),
as
Moreover, one also has:
and, more generally,
P(x,ax)LV
+
P(x,ax)L,
for each differential operator
as v
P(x,a ) D' (U)
Besides, Cm(U) is dense in on
D' (u)) .
-f
-,
with coefficients in Cm(U). (with respect to the weak*-topology
1. Manifolds, Functional Analysis, Distributions
88
-
Let U' c c u (i.e. U' c
u ) . Two
are said to coincide in U' if
distributions L , and 1, in D'(U) =
v 6E
D(U').
Let {Uk}k20 be a collection of open sets in lRn whose union is U. Then there exists a sequence { $ j ) j 2 0
C
D ( U ) , with j i . 2 0, such that
1 (i) each $ . has its support in some member of the collection {Ukjkz0, 1 (ii) Z J , . ( x ) = 1, V x E U, j 20 1
(iii) to every compact K c U correspond an integer N 5 0 and an open set V
3
K such that
z
v
$.(x) = 1 ,
OSjSN
'
x E
v
Such a collection { J , . ] is called locally finite partition of unity in 3
U , subordinate to the open cover {Uk } ktO Let {Uk}k,o be an open cover of an open set U
5 Rn
and assume that
to each U corresponds a distribution 1, E D'(Uk) such that 1, k
U
k
=
L . in 3
I7 U . whenever this intersection is non-empty. Then there exists a unique l
distribution L E D ' ( U ) such that L = L in Uk for every k = O , l , ... . k Using a locally finite partition of unity { $ . I . in U subordinate to the ] 120 cover {Uk}kzo, one defines L as follows:
=
j20
1
is a member of the collection {U 1 such that U k , contains the k k>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,
<6x ,@
=
$(x,),
0
6x0 = ~ ( x -0x) is the Dirac distribution, V 4 E D(U), and the derivatives a"6 are interpreted in x x 0
the distribution sense. Let L E D ' ( U ) . There exists a collection of continuous functions {fa(x) l a E z ; in U, such that each compact K only finitely many fa and one has
C
U intersects the support of
in the distribution sense:
Moreover, if L has finite order, then the collection {f (x)IaEZ+, can be chosen in such a way, that only finitely many of fa(x) are different from zero. Let U
5
IRn
5
V
x' direct product f ( x )
x
IR" be open sets and let f E D ' ( U ) ,g E D'(U). The Y g(y) is the distribution in D'(UxV) whose action on
each test function x ( x , y ) ,
x
E 2)'(UXV) is given by the formula:
The direct product of distributions is commutative: f x g associative: f X (gx h) = (f x g )
X
h
=
=
g x f, and
f x g x h.
If a distribution f(x,y), f E D'(UxV), is translation invariant in variable x, i.e.
then f!x,y) = lx x g(y), where g E D'(V) and l X is the (regular) distribution in D ' (Rn) defined by the formula
Let fk E D ' ( R n ) , k
=
1,2,
and assume that the support of at least
one of these distributions is compact. Then their convolution f *f 1 2 is defined as follows:
1 . Manifolds, Functional Analysis, Distributions
90
Of course, the convolution is commutative and associative. Moreover, one has:
Each of the following two conditions guarantees the contrinuity of the convolution f*g (with respect to the weak *-topology in ( D ' ( R n ) ) as a
D' (Rn)
function of f E
f v * g + f + g
(i) either supp fv
C
D' ( R n
valued in for
fv+f
if
..., where
K, V v = 1.2.
(ii) or if supp g is compact in
:
K is a compact in E n ,
xn.
The following result (due to L. Schwartz and often called the kernel theorem) is important for applications. Let B ( $ , $ ) be a bilinear functional x D(V), where U 5 m," , V 5 Rn. Y Assume that 8 ( + , $ ) is continuous with respect to each argument 4 and Q.
on D ( U )
Then there exists a distribution f E D'(UxV) =
such that B ( $ J , $ ) =
.
1.3.3. Some other distribution spaces Again let U
5 Rn
m
C -functions in
U
m
be an open set and C
(U)
the vector space of the
with real or complex values. Introduce the following
Fr6chet topology on C m ( U ) : a neighbourhood O ( m , 6 , K ) the set of all functions 4 E
Cm(U)
such that
for given 6 > 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 <<>' u ( E ) E L2 ( a n ) with some s
E R , one can introduce the
The set of all u
norms:
E S ' ( W n ) , whose Fourier transforms are locally integrable
and have the norm
II 1I *
finite is called the Sobolev-Slobodetsky space
of order s E R and is denoted by H (Rn1 . For s = m Z 0 integer Hm (Rn) coincide with the classical Sobolev spaces introduced hereabove. Each Hs(Rn
)
is provided with the Hilbert space structure, the inner product on
Hs(Rn
)
being defined as follows:
-*
where v
=
J
<02' ;(S)<(E)*dS,
stands for the complex conjugate of
6.
The following Paley-Wiener-Schwartz result gives the characterization of distributions u E D ' ( R n ) with compact support. (i) If a distribution f E 0 ' ( R n ) has its support in the ball { 1x1 5 r} (hence, in fact, f E S ' ( n n ) ) and if f has order N, then its Fourier transform
f ( 5 ) can be extended as an entire function of 5 E Cn and, moreover, there exists a constant C > 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: =
-1
I>,
)/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
Tn One has: sup Nli(k)I kEZn
N = 0,1,
m,
... .
These norms define a Frechet space topology on D(Tn), which coincides with the one given by the norms:
where, of course,
$(C1
,...,Cn)
=
...,exp(iCn)).
$(exp(iC1),
One introduces D'(Tn) as the space of continuous lineair functionals on D(Tn). The Fourier coefficients of any f E
'Toeach f E D'(Tn) correspond
D' (Tn) are
defined by
an integer N b 0 and a constant C <
m
such
that
[ i(k) I
5 CN, V k
E Zn.
Conversely, if f ( k ) , k E Zn, satisfies the last inequalities with some
1.3. Distribution Theory N 2 0 and C <
95
then there exists f E D'(Tn), whose discrete Fourier
m,
transform is f(k). The convolution f*g for f and g in
D' (Tn) is
in 0'(Tn), whose Fourier coefficients are f(k)g(k)
defined as the distribution
. Again,
the convolution
is commutative and associative. It is also a continuous function (in each
D' (Tn).
factor) valued in If J 1
...
J2 C
C
are finite sets whose union is Zn, and if f E D'(Tn),
then the partial sums f ,( 5 )
Z f(k)exp(-i)
:=
7
kEJ . 3
converge to f(<) as j
in the weak*-topology on D'(Tn).
+ m
We shall also introduce the space s(Zn) of rapidly decreasing sequences $(k), k E Zn, equipped with the Frechet topology defined by the norms:
The dual Fr6chet space of all continuous linear functionas is called the space of tempered sequences and is denoted by 5" (Z") . Each element of s'(Zn) is a sequence u(k) whose growth for [ k l N most as with some integer N . 1.3.4.
-f
m
is at
Parameter dependent distributions
Now some classes of parameter dependent test functions and corresponding distribution
spaces are introduced, which will be useful in the
following chapters. Let B$ = hZn, h E (0,h0 1, be a one parameter family of grids (lattices) in
mn. A
map u(h,x), u
For meshfunctions $
:
<
: IF$
(1.3.4)
+
Cp is called a (vector valued) meshfunction.
-f
Cp introduce the semi-norms: 1
where, as previously,
h,:a
=
/a:,h$(h,x)I,
k,j = 0,1,
...,
axl,h...axn,h, "1 an axk,h being the forward finite
difference derivative in variable xk for x E Denote (1.3.5)
''
]$Ik
,
:=
sup I$[k,j;h, k,j O
= 0,1,
...
0
Similarly to the Schwartz space S(lRn), introduce the family
sh = S(-nhn)
of test function spaces of meshfunctions u(h,x), whose semi-
1. Manifolds, Functional Analysis, Disfribufions
96
norms (1.3.4) are finite, V k,j
E
Z+, V h E (O,ho]. Denote by s(Zn)the
space of test functions, whose semi-norms (1.3.5) are finite. For each @ E Co (Rn ) the restriction 71 $(x) to 1 s well-defined. h We shall also denote by 1 u(h,x) = v(h,x) an extension of a meshfunction h u :
<‘
<
Proposition 1.3.1. L e t $ E
S(Rn). Then
nh@ E S(Zn), V ho <
m.
Proof. _ _ First, let n = 1 . One has:
1 $,(x)
:=
(ax,,$) (x) =
I
ax@(x+ht)dt
x E 4
0 and the induction argument yields: t l +...+ tk))dtl...dtk.
As a consequence of the last formula one finds: /D:@(~I
I
6
max la:ocx+t) 04t6jh
1,
v x E
The last estimate implies for @ E s(Rn
)
%1 . and each u E Z y , la1
=
j:
Furthermore, using Peetre’s inequality (which is a consequence of the triangle inequality): (1.3.7)
< x > ~6 42<(x+t)>
2
,
one gets using (1.3.6):
where
1 \$I
Ik,
is the semi-norm:
which indices in S (Rn ) the same topology as the norms ( 1 . 3 . 3 ) . Thus,
1.3. Distribiition Theory i.e. A ~ QE s ( z " ) . For 9 E
sh
I
-h as follows: define its discrete Fourier transform 0
-h
9 (5)
97
=
Fx+5,h9 = hn
C
+(XI exp(-i<x,~>),
XE<
the inverse Fourier transform being
One has:
The following auxiliary result will be useful for further applications. Lemma 1.3.2. L e t u and v be meshfunctions and p ( x , ~ D* ) a d i f f e r e n c e x,h' x,h operator, which i s polynomial i n D D* w i t h c o e f f i c i e n t s depending on x,h' x,h x E 5 .
Then t h e f o l l o w i n g L e i b n i t z ' formula holds: (1.3.8) =
P(X,D~,,,D:,~) (uv) =
z -a !1B ! (P(a,B) (XrDx,h,D:,h)@:,h@x,hu(x,h)) -B +
(D:,h(D*
R )v(h.x),
)
x,h
where we have denoted:
P (a,R) (x,c,c*) = a~a~cp(x,crc*),
a and 5
being t h e d e r i v a t i v e s w i t h r e s p e c t t o 5 and 5 * , r e s p e c t i v e z y .
Proof. We write DU, respectively D*
when Dx,h, respectively D* U' x,h: applied to u(h,x) and use the similar notation Dv, respectively D
V'
Dx,h'
respectively DZ,h, is applied to v(h,x). It is readily seen that
is when
1. Manifolds, Functional Analysis, Distributions
98
The last formulae yield: P(X,D~,~,D:,~)(ux) = p(x,DU+(l+ihDU)Dv = = p(x,DU+(l+ihDU)Dv, D*+(l-ihD*)D*) u v (uv). NOW
using Taylor's formula
I
one gets (1.3.8).
Remark 1.3.3. Applying (1.3.8) with P(x,Dx,h,~i,h)= D:,~,
Y E
zy, one
gets the finite difference analogue of Leibniz' formula: (1.3.9)
Y!
Dy (uv) = 1 x ,h u4y
a!(y-a)!
(DY-u00.
U)
x,h x,h
(D:,hV).
: . € a
Now we can give the characterization of the space s(Zn) of the discrete Fourier transforms of elements in s(z").
If u
Proposition 1.3.4. -h transform u ( 5 ) :
€ S(zn) t h e n one has f o r i t s d i s c r e t e Fourier
Proof. It suffices to show __
B
a-h
( i(h,S)aSu
where the constant
C
u,B
One h a s :
Now, ( 1 . 3 . 9 )
yields:
(5) may depend only on its subscripts
1.3. Distribtrtion Theory
99
where we have also used the inequality:
We shall need the following Poisson formula. Lemma 1.3.5. L e t 4 E S ( n n ) and denote by G ( 5 ) and 6h(5) i t s i n t e g r a l and
d i s c r e t e F o u r i e r t r a n s f o r m , r e s p e c t i v e l y . Then one h a s :
t h e s e r i e s on t h e r i g h t hand s i d e of (1.3.11) b e i n g u n i f o i m l y c o n v e r g e n t , V B E , :Z
V h E (O,ho].
Proof. First, we show that the series on the right hand side of (1.3.11) ~
is uniformly convergent. Since 4 E
s (IRn ) , one
also has :
E
s (IRn ) ,
so
that
The last inequality yields with any k t 0: lac@(e+2iTh-la) B1 5 Ckhk (l+lal)-k, V a E Zn \ { O } ,
V
5 E Tn
5,h'
and the latter ensures the uniform convergence of (1.3.11). NOW, using the inverse Fourier transform, one finds, for x E R t :
xB9 ( x )
=
(2ir)-" J
((ia5)'mh(5))exp(i<x,5>:dE
=
Tn =
Note that for x E
J
IF? =
(2ir)-n
<
I
IR*
one has:
exp(i<x,E>)(ia
1
5,h 6((ia5) $(S))exp(i<x,<>)dc.
5
exp(i<x,S>)
Tn
)
8$(S)dE
Z
=
(iac)B6(5+2~h-1a)d5.
aE2"
5 rh The last formula and the uniqueness of the discrete Fourier transform imply (1.3.11). Corollary 1.3.6. L e t (1.3.12)
E S(lRn). Then one has:
1Cy(h,5)ac(9 B -h( 5 ) - $ ( 5 ) ) / 2 Ca,B,khk, V y,6 E Zy, V k t 0 ,
1. Manifolds, Functional Analysis, Distributions
100 where t h e constants
do not depend on 5 E T~ and h E ( 0 ,hol. h,S ~,8,k
C
Indeed, one has:
c Y ag(+ 8 -h- 4 )
=
i: gY(h,S)a~~(S+znh-’a). OfaEZ”
Note that \<(h,5)I 5 151 and
iE
S(Rn),which along with the last identity
implies (1.3.12). it is readily seen that for each $ E S(Rn) and for every
Also,
integer N 2 0 one has for the discrete Fourier transform of 4 (restricted to
<):
Indeed, (1.3.13) is an easy consequence of the Poisson formula (1.3.11) (with B = 0 ) . The same inequality (1.3.13) is also true for any 4
:
<
+
Cp,
4 E S(Zn), as it is readily seen. Definition 1.3.7. DJG t e s t meshfunctions $
k
E
s(<
1, k = 1,2,.
.. ,
said t o beZong t o t h e same equivalence c l a s s if $Jl(h,x)- $2(h,x) t h e topology defined b y t h e norms (1.3.4) , a s h Example 1.3.8. Let 4 E S ( R n ) and let
x
E
+
-+
0 , V k,j = 0,1,.
are 0 in
.. .
1
L ( R n ) be such that x(x) 2 0
a.e., and
Introduce
x
h
(x)
=
h-nx(x/h) and let
where
and, as hereabove, nh is the restriction to
<.
Then 4 (x) and $J (x) belong to the same equivalence class in S(< h h Indeed, this is readily seen by using Poisson formula (1.3.11), estimate (1.3.12) and the fact that F
X+S
x
=
-
n
E C (W 5
),
).
-
~ ( 0 =) 1, the
latter being a consequence of (1.3.14). The same conclusion is true if one considers a non-negative x E L 1( 2n) such that
1.3. Distribution Theory
101
and defines the meshfunctions J, (x) as follows: h $ h ( ~ )= (Xh*Qh)(X) := where, of course, $h
=
<,
,x((x-~)/h)$~(y), x E
C
YE% 7ih$, as hereabove and
xh (x) =
h-nx(x/h).
Definitions 1.3.9. The space S o ( < )
is t h e s e t of a l l continuous l i n e a r f u n c t i o n a h on s ( < ) , V h E (O,ho], w h i l e s ' (Zn) stands f o r t h e space of uniformly continuous l i n e a r f u n c t i o n a l s on S(zn) w i t h r e s p e c t t o h E (O,ho
<,
Remark 1.3.10.It is readily seen that the elements in meshfunctions u(h,x), x E
s' (<
s' (Z")
are tempered
h E (O,ho] (i.e. with at most polynomial
1x1
growth with all their finite difference derivatives as the elements in
)
+ m)
,
while
are precisely those tempered meshfunctions u(h,x)
for which one has: ,$(h,-)>l <
I
SUP
V $
m
E s(Zn),
O
0
i.e. V$ whose norms (1.3.5) are finite. For instance, the discrete Dirac mass 6h(x)
=
h
-n
for x = 0 and
6 (x) = 0 , for x E <\{o>, is an element in S ' (zn), as well as all its h (a;,h)B6h(x), v a , E~ Z ; . However, finite difference derivatives a" X th h-lfih(x) is no longer in s ' (Z") , being still an element of s' (<1 . One can also introduce in an obvious way the discrete versions p ( < z)(Zn) of the test function space D(Rn versions
D' (<
)
, D' (Z")
)
)
,
and the corresponding discrete
of the distribution space
D' ( R n ) .
However, we
shall not need them for further applications. Definition 1.3.11. TWO tempered meshfunctions uk E s ' (< ) , k = 1,2, are said t o belong t o t h e same equivalence cZass if t h e i r d i f f e r e n c e u = ul-u2 converges to z e r o f o r h 0 i n t h e weak*-topology on S'(zn), i . e . -+
+
0 as
h
-+
0,
V $ E s(Zn).
Examples 1.3.12. Let q E (O,l), y 2 0. Consider the meshfunctions u 1 (h,x) = h-'hh(x),
u2(h,x)
=
h-l-'x+(x)
-x/h (1-q)q
I
X
E
51
-
I
where x+(x) is the characteristic function of the set P . + = {x 2 0 ) and, as hereabove 6 (x) h
=
h-l6 ox, 6oo = 1, 6ox = 0 for x E
d\{O}, is the Dirac
.
1. Manifolds, Functional Analysis, Distributions
102
mass on the lattice I+., €st(\
1
s’(Z )
k = 1,2. However uk 1 One finds for u = u1-u2 and V $ E s ( Z ) :
Obviously, uk
),
,$(h,.)> = h-’($(h,O)-(l-q)
E
if y > 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 $(h,y). O
NOW (1.3.4), (1.3.5) with k = 0, j = 1 and the last identity yield:
Therefore, ul(h,x) and u2(h,x) belong to the same equivalence class in ST(<)
if y E [o,I).
Note that v(h,x) = hyu2 (h,x) is the vanishing at infinity solution of the equation:
Thus, up to an error O(h) as h
-f
0, one may replace v(h,x) by 6h(x), the
latter having as its support just the point x = 0. As a second example consider the meshfunction
(1.3.16)
u(h,x)
:=
-1
(l+jC(h,hS)/2)-1exp(ixS)dS, x E
(2n)
\1 ,
IhS j -1 where i(h,hS) = (ih) (exp(ihC)-l), introduced yet hereabove, is the discrete Fourier transform on the lattice operator D
=
-ia
=
%
of the finite difference
(ih)-l (ox,h-l).
x .h x,h Obviously, u(h,x) is the vanishing at infinity solution of the
equation : D* +l)u(h,x) = 6h(x), (Dx,h x,h
x E X$,
1
.
Introducing n = hc and 8 = exp(iq), one can rewrite (1.3.16) in the following fashion: u(h,x) = -h(Zai)-l
i
(02-(2+h2)8+1)-I8x/hdO
iel=i so that the residuum calculus yields:
1.3. Distribution Theory
(1.3.17)
103
u(h,x) = h
where we have denoted 9+ = B+(h)
(1.3.18)
x E
Using (1.3.17) in order to extend u(h,x) to v(x)
=
m , introducing
(1/2)(exp(-lx;)), (1.3.17), (1.3.18) then imply lim (u(h,x)-v(x)) =' 0, V x E I R , h+O
the convergence being uniform with respect to x belonging to any compact subset in R
.
Introduce
It is readily seen that u(h,x) defined by (1.3.16) and v(h,x) defined by 1 (1.3.19) belong to the same equivalence class in ) . Moreover, it
s'(%
can be shown, but will not be done here, that, in fact, h-'u(h,x) h-'v(h,x)
still belong to the same equivalence class in
and
s'(4) , provided
2.
y
We shall also need several spaces of parameter depending distributions valued either in D ' (U) or 1.3.13. With
Definition
s' ( E n )
E
t h e t e s t f u n c t i o n spaces c
E (O,E~],E~ < k
((O,E~];
t h e t e s t f u n c t i o n s @(E;x) of x E iable with respect t o V
6
E (O,E
0
1.
E
for each fixed value of the parameter.
€ (O,E
and U _ C R n a n open s e t , d e f i n e
D(u)), k
u which a r e
1 and
0
m,
2 0
i n t e g e r , u s t h e s e t of
k times continuously d i f f e r e n t -
a ' $ ( c , - ) € D(u), V j = 0,1,...,k,
S i m i l a r l y are d e f i n e d t h e t e s t f u n c t i o n spaces ck ((O,E~I;S(IR~ )),
a s w e l l a s t h e t e s t f u n c t i o n spaces L~((o,E~I ;D(u)) and L ~ ( ( o , E ~ I ; s ( I R ~ ) ) , whose elements a r e t e s t f u n c t i o n s @(E,x), which a r e Lm-functions of E
E
(O,E~]
valued e i t h e r i n D(u) o r i n S(mn
).
The corresponding spaces of parameter depending d i s t r i b u t i o n s a r e denoted by ck ((o,~~l;D'(u)), ck ((O,E~];S'(IR~)), L~((O,E~I;DI(U)) and L ~ ((0 , ~ ~ l ;(mn S'
I ) , r e s p e c t i v e l y . The elements o f t h e f i r s t two t y p e s of spaces are t h e d i s t r i b u t i o n s f(E;x) which a r e k-times continuousZy
differentiable i n
E
w i t h t h e i r v a l u e s i n t h e corresponding d i s t r i b u t i o n
spaces. The elements of t h e l a s t two spaces a r e t h e d i s t r i b u t i o n s f (E;x) such t h a t
1. Manifolds, Functional Analysis, Distributions
104
\~<
sup O<€<E
-
0
f o r any @(E,x) i n t h e corresponding t e s t f u n c t i o n s p a c e . Two d i s t r i b u t i o n s f (E,x), k
Definition 1.3.14.
k
= 1.2,
a r e s a i d t o belong
ts 7-he same equivalence c l a s s ( i n t h e corresponding d i s t r i b u t i o n space) i f one has: lim 1
=
0
E-fO
for any ~ ( E , X )i n
L~((o,E~I;D(u))
or L~((o,E~I;s(IR~ ) ) , respectively.
Examples 1.3.15. First consider a very simple example. m
Let fl
=
6(x), f2 = (2~)-'exp(-jxl/~). For each $ E L ((0,~~1;S(IRR))one
has : I1 6 (i/s)i exp(-jyl) I$(E,o)-$(E,EY)idy 6
w
Hence, E-'~(x)
ElQj
0,l'
and E-Yf2(E,x) for y < 1 belong to the same equivalence
0
class in C ( ( 0 , ~l;S'(R)),the latter being just the reflection of the fact 0
that f2(E,x) converges in S'(W) to the Dirac mass 6(x), as
E +
0. HOW-
-1
ever, fl = 6(x) and f2 = ( 2 ~ ) (exp(-lx\/E))do not belong to the same I
equivalence class in C ( ( 0 ,];S'(R)), ~ since aEf, = 0 and 0 bounded in S ' ( R ) as E + 0. As
a €f 2 is
a second example consider the distributions E(E;x,Y),
y E IR given, which are the vanishing for 1x1
-t
m
E
un-
E (O,E~],
solutions of the
equation: (1.3.20)
2
(8 D
2
q (x)D +1)E = 6(x-y),
Dx = -id/dx,
m
where q(x) is in C ( R ) , q(x) t qo > 0, V x E IE? and q(x) 1x1 b r with r <
m
5
q(m) for
sufficiently large.
Introduce (1.3.21)
F(E;
x,y)
:=
-1 (2~q(y)) exp(-lx-y1/(2sq(y))).
We are going to show that E - ~ Eand E - ~ Fbelong to the same equivalence class in C((O,a ];S'(R ) ) , provided that y < 1. 0 First, note that E(&;x,y) is well-defined and, moreover, for each E
E
1 fixed is an element in the Sobolev space H 1 ( R 1 . 0 Indeed, the integration by parts and the Cauchy-Schwarz inequality
(O,E
imply that the operator
1.3. Distribution Theory
is an isomorphism for each given E E ( o , E o l ,
105
so that E(c;.,y) E H 1 ( R ) ,
.
given that 6 (x-y) E H-l ( R
Moreover, the same argument implies that
where C
>
0 is some constant.
Furthermore, an elementary straightforward computation shows that one has for u = E-F: A u
=
f (E;x,y),
where f(E;x,y) = E$(x,Y) exp(-/x-yl/(eq(y))),
(1.3.23)
with $(x,y) smooth function of x E R such that $ ( * , y ) E D ( R ) . Again it is readily seen that one has for the Fourier transform ?(€;S,y)
=
F
f the following estimate:
X-tS
(1.3.24)
[ 5 l?(~;S,y)
CE
2
-2
<ES>
,
where C > 0 is some constant which depends only on q(x). Thus, (1.3.24) yields:
Combining (1.3.22), (1.3.25), one finds: (1.3.26)
i /U(E;.,Y)I l 1
5 CE
where C > 0 is some constant. As an immediate consequence of (1.2.26), one gets the conclusion
that E-'E
and F'-E
belong to the same equivalence class in
C ( (O,Eol;s'
(Rx ) ) ,
provided that y < 1. This example is also treated with full details in Chapter 4, since it plays an important role in the theory of singular perturbations appearing in the elasticity theory. As the last example, consider the following distributions
1. Manifolds, Functional Analysis, Distributions
106
(1.3.27)
u(E,t;x)
IT)-'
:=
-'sin(
(t/E)<S>)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
m,
{x
:=
I
1x1 < t} and v(E,t;x)
V t > 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 2<~5>s3F (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,<)
=
U) (E,c)
<EE>
-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
= {sis
+S
1
5-3/2,
2-
s
2
+S
3
3
Notice t h a t ( 2 . 1 . 6 ) s a t i s f i e s t h e e q u a t i o n : (-E
where A =
2
A+~)u(E,x)
2
2
2
2
=
6(x),
2
a / a x l + a /ax2+a /ax23
VE > 0 , x
E IR3
i s t h e L a p l a c e o p e r a t o r and 6 ( x ) i s t h e
Dirac's &-function. Remark 2 . 1 . 6 . For e a c h ( € , E l )
E
( 0 . ~ ~X 1IRn-'
f i x e d t h e space H
(s)
(IR) c o i n c i d e s
, € , E l
w i t h t h e u s u a l Sobolev s p a c e Hs +s (IRR) and f o r e a c h E E ( O , E ~ ] f i x e d t h e 2 3 space H ( R n ) is n o t h i n g e l s e b u t H (IRn), but, i n general, there s +s ( s ), E
2
3
i s no e q u i v a l e n c e o f norms o f o r d e r s w i t h p a r a m e t e r s w i t h t h e u s u a l
S o b o l e v norms o f o r d e r s2+s3u n i f o r m l y w i t h r e s p e c t t o t h e p a r a m e t e r s .
2. Sobolev Spaces of Vectorial Order
112 m m n Hence C ( R ) and C (I3 ) 0 0 respectively.
are dense i n H
5 s2+s3t h e n f o r e a c h
I f s;+s;
imbedded i n t o t h e s p a c e s H
E
(s'),E
,€,El
( S )
( R ) and H
(s),E
(Rn),
(Wn) E (00 ,] ~ f i x e d t h e s p a c e s H ( S ) ,E (nn), b u t t h e imbedding D p e r a t o r s are
n o t n e c e s s a r i l y u n i f o r m l y bounded w i t h r e s p e c t t o
E
are
E ( O , E ~ ] . W e are g o i n g
t o p o i n t o u t n e c e s s a r y and s u f f i c i e n t c o n d i t i o n s on s and s ' f o r t h e uniform c o n t i n u i t y of t h e c o r r e s p o n d i n g i m b e d d i n g ' s o p e r a t o r s . D e f i n i t i o n 2.1.7.
The f a m i l y os norms I 1 . 1 i ( s ) , € , E E ( 0 , c01 , a r e s a i d t o be s t r o n g e r than i 1.i I ( S s ) , E l if t h e r e exists a c o n s t a n t C , which may depend only on s, s' and
E
such , that
~
V
1 1 . I 1 (S'),E
Ve w r i t e
,E
I I .j I
E ( 0 . ~ ~ 1V , u E H ( s ) ( R n ) .
>
ll-ll(s),E Or
a r e s t r o n g e r than
I 1.1 I (s)
E
l l - l l ( s # ) , Ei f
~ ~ . ~ ~ ( S ) , E
t h e norms
( S ' ) ,E'
P r o p o s i t i o n 2.1.8.
One has (2.1.8)
l l - l l ( s # ) , E l l . l l ( s ) , E iff s; 5
sl,
si+s; 5 s l + s 2 , s ' + s ' 5 s 2 + s 3 .
P r o_ o f . I t i s immediate t h a t t h e n e c e s s a r y and s u f f i c i e n t c o n d i t i o n f o r _ ( 2 . 1 . 7 ) t o h o l d is 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 , s' and
5
By p u t t i n g i n ( 2 . 1 . 9 ) , f i r s t
5
E +
0' 0 , afterwards
E
=
-1
<5>
,
E +
0 , and,
one g e t s t h e c o n c l u s i o n :hat (2.1.8) is necessary s -si s3-s3 / 2 , f o r ( 2 . 1 . 7 ) t o h o l d , w i t h C 2 max{EO1 ' , 2
finally,
E =
To show t h a t c 0 5 1.
1,
= 0,
E
+ m,
(2.1.8)
i s a l s o s u f f i c i e n t , it i s enough t o c o n s i d e r s'
In t h a t case, -s E
E~
0,
=
(2.1.8) yields:
s 1<5> 2<E5>s3
-s =
(E)
2
If
I
I
s 1<5> s +s
<€<>
s +s
fS
2<E5>s3 t
<5>
2<ES>
s -s 3
1
t
t ~ .
> 1 , t h e n t h e s a m e argument y i e l d s :
-s CE
s l<<> 2 < c p s 3
21
where t h e c o n s t a n t C depends o n l y on s and We u s e t h e n o t a t i o n s'
<
s or s
> s'
E
0'
f o r w r i t i n g (2.1.8)
i n a s h o r t way and
2.1. Spaces on
W"
113
> s') i f t h e r e i s a t l e a s t one s t r i c t i n e q u a l i t y i n
t h e one s ' < s ( o r s (2.1.8).
Equipped w i t h t h e r e l a t i o n s
<, <,
t h e norms d e f i n e d above and t h e c o r r e s -
ponding s p a c e s become p a r t i a l l y o r d e r e d w i t h r e s p e c t t o t h e s e t
i . e . t h e imbeddings H
5 H(S'),E
( s ) ,E
(En)
a r e uniformly continuous with r e s p e c t t o In p a r t i c u l a r , with Hr(Rn),
(wn)
E
E ( 0 , ~I i f f 0
(s'-s) E V .
r E R, t h e u s u a l Sobolev s p a c e , t h e imbeddings
a r e uniformly continuous with r e s p e c t t o
H r ( w n ) E H ( s ' ) ,E iff ( O , r , O ) > s ' , i.e.
i f f si 5 0 ,
si+s;
E
E
(O,E
0
I,
5 r , s'+s' 3 S r .
C o r o l l a r y 2.1.9.
For s H ( S )
>
s'
one h a s : H
( w n )5 H
( ~ , () R n )
c
(IR) 5 H ( s ' ) , E , < ' (IR) H ( s ) , < '(B) H ( s ' ) ,C' (R) and t h e imbeddings operators a r e uniformly bounded I
( S ), C I S '
w i t h r e s p e c t to t h e parameters
( € , < I )
E ( 0 . ~ ~x 1Rn-'.
P r o p o s i t i o n 2.1.10.
Let u E (2.1.10)
m3, s
3
E IR , s < u and l e t
p = s-t(u-s),
t > 0.
P r o o f . O b v i o u s l y , it s u f f i c e s t o p r o v e ( 2 . 1 . 1 1 ) f o r s -
=
0. I n t h a t case
(2.1.11) i s e q u i v a l e n t t o t h e i n e q u a l i t y :
-20 1 5
2
2u 20 2tUl I<<> 2<E<> 3+6-2tE
6
-20
o r , with A = 6
1 s X+A
20
'<€,>
E
-t
, v
20 '<E<>
-2tu
-2tu 2<E<>
',
t o the inequality
A > 0.
The l a t t e r o b v i o u s l y h o l d s , s i n c e t > 0 and inf(A+A-t)
X>O
=
t l ' ( t + l ) + ( , / t ) t ' ( t2+ I, l)
v
t > 0.
3
I
2. Sobolev Spaces of Vectorial Order
114
C o r o l l a r y 2.1.11.
With
s
< u,
g i v e n by (2.1.10) and any 6 > 0 hoZds:
p
2
I lul I ( s ) , E
(2.1.12)
2 5 6
2
-2t
I (u),E+6
IIUI
I Iul I
v
(2.1.13) Vs
*
=
3
{s'ls' E IR
,
s'
<
.
C o r o l l a r y 2.1.12. 3 * L e t s E R and u E i n t v . Then f o r V
T
E
C
u E €3
( 0 ) ,E
(2.1.11) o v e r RnT1.
5
s}, V,* = {s'Is' E R
v
constant
v
w e denote
being c a l l e d reci p ro cal t o
the set V
(p),E
E (O,Eo1,
E
I n f a c t , one g e t s (2.1.12) by i n t e g r a t i n g
For a g i v e n s E IR3
2
S
vS,
which may depend only on S,T,U,S
3
s'
>
s}
V 6 > 0 there e x i s t s a
and
cO
such t h a t
I n f a c t t h e c o n d i t i o n u E I n t v means, t h a t u -sl > 0, u +a -s -s 1 1 2 1 2 > O , a +u - s -s > 0. T h e r e f o r e , t h e r e e x i s t s p = s - t ( u - s ) < T . 0 2 3 2 3 For s E R w e d e n o t e by a V t h e boundary of V which c o n s i s t s o f t h r e e p l a n e components:
(2.1.15)
alvs a3vs
W e define 3 . V
*
= =
I S ~ / ES ~avs, t s l l s i E avs,
S'
= s,?,
si+s'
2
3
=
2
a2vs +S
3
t o b e t h e p l a n e component of
1 s b e l o n g t o t h e same p l a n e .
=
{s'Is~
E 3vs, s ' + s ' = S + S } , 1 2 1 2
1.
av
*
*
such t h a t 2 . v and 3 . v 3 s 1 s
C o r o l l a r y 2.1.13.
3 * L e t s E R and l e t u E I n t 2 . v . Then f o r V
constant
c such t h a t
I S
T
E 2.v
7 5
there e x i s t s a
(2.1.14) h o l d s .
I n f a c t , t h e p r e v i o u s argument c a n be a p p l i e d .
2.2. R e s t r i c t i o n t o h y p e r p l a n e and f u r t h e r imbeddings r e s u l t s D e f i n i t i o n 2.2.1. FOP
(o,E01
x
Rn-'3
(E,E')
--s
= E
+ ~ i ( E , s ' )
E
c let s3
1<<'>s2<E<'>
(l/I(E,cl)
I
2.2. Restriction to Hyperplane
and l e t c
( S ), E , S '
115
be t h e s e t of complex valued f u n c t i o n s $ ( E , S )
where t h e
[ * I ( S ) ,E,S"
d i s t a n c e s are d e f i n e d by Define
[$I
(2.2.2)
S,
= sup
(s), E , S '
EE(O,E ]
0
and l e t c
(s). 5 '
( 0 . ~ ~3 1E
Let e 2 , e
be a f a m i l y of Banach spaces o f t h e f u n c t i o n s
+ $ ( € , < I ) ,
E
whose norms ( 2 . 2 . 2 ) a r e bounded V 5' E mn-'.
R3 b e a s f o l l o w s :
(2.2.3)
e
With n o
: H
=
2
(O,l,O), e
( S ) , E 8 5 '
(IR)
+
= (l,-l,l).
C t h e r e s t r i c t i o n o p e r a t o r t o {xn = 01,
we state
theorem: Theorem 2 . 2 . 2 .
3 L e t s E R , s 2 + s 3> f . Then w i t h some c o n s t a n t C , which may depend o n l y
on
s
and
E~
and f o r V v E H
( s ). E , S "
(s-te2), E , s ' (2.2.4)
5 CI j v ;
I (s)
5 c ( l + / & nE
Proof. Since f o r V E , S ' -
( s ), E , 5 '
i f s2 > 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 <
m,
> 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-'<~c'>t, I(E,c')
becomes : -2s
-2s
'f
I ( E , ~= ' )< E < ' >
3(E22<~c'>-2+t2)-'dt.
R One checks easily that with q
where C
depends only on s
E
s3 > 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-) I 5 ClLn
v
El,
where the constant C may depend only on cO if Hence, with s
2
E
(E,s') E
0
(o,E01
x
> 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<5'>2<~5t>2+t2) 2dt
)
J
<Eel>
R
-2s -2s ' 3dt,
r
-2(s +s dt).
IR
Combining (2.2.5)-(2.2.9) we get (2.2.4).
0
Theorem 2.2.3. 3 Let s E R , s2+s3 > 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 ) ~ <5> 2 < ~ S > / ; ( E , S ' , S , ) I
J
82
dS'dS,,
mn where
I s ( ~ , F , ' , h )= 2 s -1 (2~)-~/<5'>
2s
-2s
3<5>
-2s 3 2 2<~F,> l e x p ( i h F , n ) - l / dF,
R
and
i f s2
7
f
I S ( ~ , F , ' , h )= (2n)
-2
1-2s / E
*<EF,'>
2 s + 2 s -1 2 3
-2s
<<>
-2s 3 2<~<> exp (ih5,) -1
1
R
I n b o t h cases h o l d s : when h
-+
I 'a<,
i f s2
0 f o r each
1
11 ( ~ , t ' , h ) 5 C
(E,S'),
,
$.
V ~ , c ' , hand moreover, I s ( ~ , S ' , h + ) O
so t h a t L e b e s g u e ' s theorem on bounded convergence
i m p l i e s t h e c o n t i n u i t y of t h e f u n c t i o n R
3 xn
+
u(s,x',xn)
E
H ( a ) ,E (IR;,'),
The l a t t e r i s u n i f o r m w i t h r e s p e c t t o E sup I S ( ~ , F , ' , h )= I s ( E ' , h )
+
0 when h
-f
E
V u (O,s0],
E
if
H (s), E 07:).
1 lul I
0 f o r e a c h F,' E Rn-
F) .
<
m,
E
Theorem 2 . 2 . 6 .
If s2+s3> n/2 then each u E f u n c t i o n s of x E R ~ ,
H
(s),E
( m n ) i s a f a m i l y o f continuous
since
2 . Sobolev Spaces of Vectorial Order
118 (O,Eo1 3
(2.2.10)
and moreover, w i t h
E +
U(E,X) E
O
c
n
(IR
I I . I I co (IRn) t h e
)
norm on Co (R:)
one has s
'
-s -s +n/2
1
2
I
2s
CE
= E
(S)
-2s -2s 2<E€,> 3dc <
m,
v
E
E
n/2
,E'
if s
2
= n/2
if s2 < n/2,
&,
where t h e c o n s t a n t c may depend only on s , n and
where
>
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 cll<<>k IL1 ( p ) , and IL1(p) 5 In fact, j / u / 2 where C is the same constant as in the proof of ( s ) ,E' CEI Theorem 2.2.5 with s2 replaced by s2-k.
I bk I;
i;
luI I
Theorem 2 . 2 . 9 . ( I R ~,) s2+s3 = n/2+a w i t h a E (0,l). Then U(E,X)
Let u E H b'
E
( s ) ,E ( 0 , ~1, 0
E
E
Ca(R:),
and, moreover,
(2.2.14) E
n/2+a-s -s 1 2
where t h e c o n s t a n t c may depend o n l y on
s.
n and
E
0'
Proof. We have only to estimate /u(E,x+h)-u(e,x) 1 with lh[
I /u(E,.)
6 1, since
IIco(mn) does satisfy (2.2.14). We find:
12=
lu(E,x+h)-u(E,x)
2s
-2n
5 (2n)
E
1
(27Tr2n / / ; ( € , E l
Rn
(exp-l)dc/ 2 5
-2s -2s 3 2 2<E<> d < /l u / j ( s )
1 Iexp-1I2<<>
IRn
For s2 2 n/2+a write:
which proves the first of the inequalities (2.2.14). NOW consider the case: s2 < n/2. Write
J
" -2s -2s lexp-l/L<<> 2 < ~ < > 3 d ~=
2
= E
5 CE
1 lexp(ihs-lS)-ll Rn 2s -n 2
{E
(E
-s -2s 2 + / < 1 2 ) 2 3d5 5
-1 -2 IEIh' n+l 2 2 -52 h 1 P (E +p ) (l+~~)-'~dp+ 0
,E.
2 . Sobolev Spaces of Vectorial Order
120
0 n-1-2s
J
+
-2s
2
1P
€14-
2 s -n-2a 3dp] 5 C E
(l+p)
2
lh!2n,
which p r o v e s t h e second i n e q u a l i t y i n ( 2 . 2 . 1 4 ) .
0
Remark 2 . 2 . 1 0 . If u
E
H
( s ), E
( R ~ ) ,s2+s3= n/2+a,
s2
=
n/2+8, w i t h 0 < 8 <
Q:
< 1 , then
t h e s a m e argument i s p r e v i o u s l y shows t h a t
v where t h e c o n s t a n t C may depend o n l y on u , B , n and If s2 = n/2
(8
= 0),
s
(O,E01,
E
0' = ( n / 2 ) t a , t h e n t h e l e f t hand s i d e i n
+s
2 3 ( 2 . 2 . 1 5 ) h a s t o b e r e p l a c e d by E
2.3.
E
E
- s l ( l + / h
lu(E,.)
El)-'[
I IcQ:(Rn)
.
Spaces of d i s t r i b u t i o n s s u p p o r t e d by a h a l f - s p a c e
with s
(s), E , C "
(s),S"
o+
H-
(s), E l (5) ( f o r t h e p r o p e r t i e s of t h e corresponding spaces with s c a l a r
E R3
index s
; i
? ;
o+
W e a r e g o i n g t o d e f i n e t h e s p a c e s H-
E R
w e r e f e r t h e r e a d e r t o [Li-MI,[V-E],[E].
D e f i n i t i o n 2.3.1.
A f a m i l y of d i s t r i b u t i o n s ( O , E ~ x] xn-l 3 ( E , S ' ) u + ( E , ~ ' ) E s'(R) i s o + s a i d t o belong t o H ( S ) ( R ) and supp u+ 5 +, if u + H(s) V (E,s') E ( O , E , ] x R"-l; i t i s s a i d t o b e l o n g to H (s), 5 ' i f u + E H ( s ) , t 1 ( R ) -f
, & , E l
and supp u +
~ x +V 5,' E
, € , E l
+
mn-'.
De finition 2.3.2.
A f a m i l y of d i s t r i b u t i o n s (O,E
to H:~,
0
A +(s). E
if u + E
I
3
-f
u+(E,.)
and supp u + 5 R : ;
H(s),E(IRn)
i f u+ E H ( s ) (En) and supp u + 0 -
E
0 -
5
E E ' ( R ~ ) is s a i d to belong
it <s s a i d to belong t o
q. 0-
0 -
H H t h e c o r r e s p o n d i n g s p a c e s of (s), E , C " (s) ,C" H ( s ,)E l (s) ( f a m i l i e s o E ) d i s t r i b u t i o n s whose s u p p o r t l i e s i n 5-and respectively.
W e d e n o t e by H
z!,
Remark 2 . 3 . 3 . For e a c h
E
E ( O , E ~ ] , 5' o+
H ( s ,)E , 5 '
( R ) and H-
( 5 ), E
n-l
E R
o+
f i x e d , H-
( s ), E , S '
a r e c l o s e d s u b s p a c e s of
a r e c l o s e d s u b s p a c e s of H
( 5 ), E
(Rn)
. Besides
2.3. Spaces of Distributions Supported by a Half-Space o+
and Cm(IR:)
C i ( R -+ )
0
o+
and H-
a r e d e n s e i n H-
-
121
( 5 ), E , S '
respectively
( S ) ,E
I n d e e d , t h i s i s a consequence of t h e f a c t , t h a t f o r V E , S ' f i x e d
H
(s)
Hs
,€,el
(IR) c o i n c i d e s w i t h H
2
+
+ s ( 1 ~ " ) . S i n c e H-
s +s
( s ), S '
n
=
+
H;
E>O
(R)and H
coincides with
(IRn) ( 5 ) ,E
3
n
S),E,C"
2 3 conclusions hold f o r these spaces, a s w e l l .
Hf = H+ (s) E>O ( 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
<
+. In
)exp(-xn/E) -i = (icEn+l)
= E-~H(X
fact, ;(E,<,)
On t h e o t h e r hand,
u E H+
E
H+
,
so t h a t
( S ) ,E
if s 1 5 0 , s2 < - f ,
(W)
(S)
compatible with t h e f a c t t h a t l i m u
=
6 ( x ) E H+
E+O
With a > 0 , one h a s :
U(E,X
)
(R), v
52
n
E
(o,E01,
iff
s 2 + s 3< f , which i s
(IR) , V s2 < - f .
= xaH(x ) e x p ( - x n / ~ )E H
n
E
+ ( 5 ) ,E
( R ), V E E ( 0 . ~ ~ 1 ,
i f f s 2 + s 3< a + + . F u r t h e r , u E H I s ) (IR) i f s 1 < l + a , s2 < -?,
s + s < a++. 2 3 Moreover, n Dk-lu = 0 , 1 5 k 5 m, where t h e i n t e g e r m i s s u c h t h a t 0 xn a = m+y, w i t h IyI < + . I n f a c t , one h a s : ;(E,c
)
l+CY =r(a+l)E ( i E C + ~ ) - l and - ~ t h e following asymptotic
f o r m u l a h o l d s i n S ' (R): =
1 +a 2+a 6 ( x n ) + ~ ( E 1,
r(a+i)E
N o t i c e , t h a t i f I$ E H
+ ( o ),E
(En-')
u = $ x D k 6(xn) E H<s),E(IRn)
x,
supp u c {xn=O} and
,
+
0.
with 02+u3 < 0 , t h e n
with s
=
u-(k+t)e2. In fact,
2. Sobolev Spaces of Vectorial Order
122
R 2.4.
S p a c e s on I R ~ -
For s .
I'
j = 2,3,
I lul I
(2.4.1)
where
CY
non-negative
i n t e g e r s d e f i n e t h e norms
I +
=
, E , c I
= (a',an),
6
(6',Bn),
=
= -ia/ax
D~
.
n We d e n o t e by H
(R+) w i t h s .
J' f a m i l y of s p a c e s of a l l d i s t r i b u t i o n s ( S ),
j = 2 , 3 non-negative
-
& , E l
( € , < I )
+
integers the
u E S'(R+) w i t h norms
(2.4.1) f i n i t e . I n t h e s a m e way o n e d e f i n e s t h e norms
t h e s p a c e s of a l l d i s t r i b u t i o n s i n S'(lR:) b e i n g d e n o t e d by H
w i t h norms ( 2 . 4 . 2 ) f i n i t e
($1.
fs) . E
For u E C m ( q ) a n d N > 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,<'
one h a s :
and u .
Using t h e o p e r a t o r Q
d e f i n e d by (2.4.3), (2.4.4) w i t h N 2 s +s N 2 3'
one f i n d s : m -
where C d o e s n o t depend on
E,
5 ' and u.
Therefore, C m ( s + ) being dense i n H
0
Now w e a r e g o i n g t o i n t r o d u c e t h e s p a c e s H (S),E,c'(R+)' 3 V S E R .
" ( S ) , €
for
L e t K + b e t h e F o u r i e r t r a n s f o r m of t h e m u l t i p l i c a t i o n o p e r a t o r w i t h 'n t h e c h a r a c t e r i s t i c function of t h e halfspace
2. Sobolev Spaces of Vectorial Order
124
= I x = (x',xn)
For v E
s(nn),
I
x'
E
En-1,xn
' 01
one has
Indeed, with 6 > 0 one has:
If @,$
E
1
then L (En),
Hence, one finds: F ~ ( H+(x,~ ) exp (-6x ) F-l v) = (27ii)-'I (Sn-nn-i6)-lv(S',nn) dnn. n 5-t~ R
Further,denote: v+(5',cn+i6) and let V+
=
V(
)
-1
=
(27ii)-'J (Cn-nn-i6) V(C',nn)dnn R
+V (2) where
It is obvious that
On the other hand, one finds for V
(2).
.
2.4. Spaces on
One can rewrite
v( ' )
W$
125
as follows:
so that
=
V.P. 1 iSn-~n)-1v(5'.~n)d~n. IR
Denote by Il- the Fourier transform of the multiplication Operator With 'n the characteristic function u f the half-space IR: = { (x',xn) x' E IRn-i,
I
< 0).
x
Then for v E S(Rn) , one has:
and the same argument, as previously, shows that
Hence,
TI+ v + n '
n-
v
=
v,
v
v E S(IRn).
' n
Since the multiplication by H ( f x
z
2
n
)
is an orthogonal projection of -TI
L (XIn) onto the functions in L (Rn)with their support in R+ , + 2 n respectively, TI- are projectors of L ( R ) onto the subspaces of En functions in L 2 ( R n ) which can be extended analytically onto the half
plane: Im 5
En
+
< 0 and Im
Cn
> 0, respectively. For neromorphic functions
v(C',Cn) which have no singularities on IR, one has:
) a closed Jordan curve in the half-plane {Im t where ~ ( 5 ' is
> 0 )
which
2 . Sobolev Spaces of Vectorial Order
126
encloses all the singularitles of the function
nn
w(E',on).
+
Definition 2 .4.3.
A f a m i l y of d i s t r i b u t i o n s (o,Eo]
X
Rn-' 3 (E,'')
*
U(E,<')
E s'(R+)
,'
( R ) i f f o r each ( E ) t h e r e e z s t s an e x t e n s i o n i s s a i d t o belong t o H ( s ) ,E," + E S ' ( R ) of u t o JR such t h a t Lu E H(s),E,c,(R). The nomi i n H
&u
(s), € , < I (IR+)
i s given b y (1)
(2.4-7)
I IUI I (s), E , "
=
inf 9.
1 I L U ~1 (s), E , ' "
where t h e infimum i s taken over a l l e x t e n s i o n s Lu t o R. A f a m i l y (E,S') + u E S ' ( I R + ) i s s a i d t o belong t o
One can introduce on H (s)r E , "
(m
)
H(s),E,
(a+)i f
another norms:
+
These norms do not depend upon the choice of the extension Lu. In fact, if
-
L.u, j 3
=
1,2, are two different extensions of u to R ,then supp(LlU-L2u)~R-
Therefore, the function
being analytic for Im 5 Hence, H
(S)
E,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] <
f , lsz+s3j < f . Then the
extension operator .toi s a continuous l i n e a r mapping uniformZy w i t h respect to
E
from H
( S ), E
ko : H
(m:)into H
,
( s ) ,€
(Rn),
i . e . there e x i s t s a constant c > 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,<'
( € , < I )
E (0,ll
n-1 x R
I I Ts)
,
is defined by (2.4.8), L! 0 is the extension by zero: where I * I ,€,' I 1 u = = for x < O and where the constant C may depend only on s 2 and s 3 ' 0 n ( R ) be an extension of u E H ( s ) ,E," (W+) which Let !Lu E H (S)
re,"
satisfies the inequality (2.4.14)
I /lulI ( s ) ,E,5,
5 CI lul
I;s),E,su
with a constant C > 0 which does not depend on u,E,S'. Then one has:
one finds also:
Using the same argument as in [Ste], we are going to show that
5n where the constant C does not depend on u , s , < ' .
2.4. Spaces on
W;
129
Introducing
one can write: T(!Zu) = Kw, where
After the substitution q n = tSn, one finds: (Kw)( € , S ' , S , )
J kl(E,S',Sn,t)W(E,S',tSn)dt,
=
R
where kl(E,S',Sn,t) = Silk(E,<',Sn,tSn). Further, one finds using the triangle inequality:
Furthermore, one has:
Denote
Therefore, there exists a constant c
0
> 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 <
: Hs
(R:)
B 2
t.
+
HS (Rn) is a continuous linear 2
The following result will be needed, as wel1,in the elliptic theory of pseudodifferential singular perturbations. U
E H(s)
,E,c,
( R + )tvith s2 2 0 , s 3 t 0 and let
$1. Then holds:
where t h e c o n s t a n t
Proof. Im
En
c may depend o n l y on def
Since Lou(S',S
< 0 , one has:
)
=
Fxn+Sn
s2,s3
and 6 .
9. u(S',x ) is analytic for
0
2.5. Dual Spaces
131
where
Using t h e s u b s t i t u t i o n ri
= tlIm
E n / , one g e t s t h e f o l l o w i n g estimate
f o r I(E,S): I(E,s)
I nI
26-2
-2 2 <,-t)) ( t + j I m ~ n l - 2 < < l > 2 ) 6 - f d 5t
I (1+(11mEnl-'Re R
Furthermore ,I one f i n d s :
H e r e , w e have u s e d t h e c o n t i n u i t y o f t h e e x t e n s i o n by z e r o to from
(R+) i n t o H
H(o,t-s,o) ,E,S'
(R) g i v e n t h a t
1 f-61
< f
(see
I
Remark 2 . 4 . 6 ) .
2.5.
(O,t-6,0) , € , S t
Dual s p a c e s Let H
fixed H
*
( R n ) b e t h e d u a l s p a c e of H ( I R n ) . For each E > 0 ( s ), E (s), E ( R n ) b e i n g isomorphic t o t h e H i l b e r t s p a c e w i t h t h e i n n e r
( s ) .E
product
__ -2s
(u,v)
(2.5.1)
( s ), E
=
( 2 ? ~ ) - * ~1 Rn
;(E,~)V(E,~)E
2s
<5> 2 s 2 < ~3d5 p
1
The R i e s z theorem s a y s t h a t any c o n t i n u o u s l i n e a r f u n c t i o n a l @ ( u )on H
(s), E
(2.5.1)
(Rn) and
i s g i v e n by a n e l e m e n t v
I
1
=
ilvli
(s), E
E
H
( s ),E
(IRn)
as an i n n e r product
= (v,v)'
(s), E '
Denote -2s
(2.5.2)
w
= 6
so t h a t w
E
H
2s '2s2<~i;> G,
(-s),E
and
2. Sobolev Spaces of Vectorial Order
132 for V u E H Hence,
(Rn), V w E H (2). (-s) , E (s , E ( 2 5 . 2 ) i s a n isomorphism between H*
u E
H
(mn)
( B n ) and H
( s ) ,E
t h e v a l u e of a f u n c t i o n a l w E H
a n d , moreover
(mn) on
,
(-s) E
(-s).E a function
( R n ) i s g i v e n by t h e f o r m u l a ( 2 . 5 . 3 ) . W e s h a l l u s e H
( s ), E
a s an isomorphic r e a l i z a t i o n o f H
*
(mn)
B e s i d e s , t h e form ( 2 . 5 . 3 )
(El").
( s ),E
(-S) ,E
c a n b e a l s o v i e d a s a c o n t i n u o u s e x t e n s i o n of t h e form -2n
J
(u,$) = ( 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,<).e^f(E,c)d< +
d o e s n o t depend on t h e e x t e n s i o n k .
E H
I n f a c t , i f k,f
( - s ) ,E
(Rn) i s another extension, then ( i f - L l f ) E
H-
(-s), E
and ( u + , ! L f - k l f ) = 0 . A s a consequence of
1 (u+Jf) ( 0 ) , E l
2
( 2 . 5 . 5 ) one g e t s t h e estimate
1 / u + /1 ( S ) , E l
I
IQfI ( - s )
,E
and, s i n c e ( u + , l l f ) d o e s n o t
2.6. Sobolev Spaces of Vectorial Order on Manifolds depend on t h e e x t e n s i o n ,.P
133
one h a s :
T h e r e f o r e , any f E H (Ry) d e f i n e s a c o n t i n u o u s l i n e a r f u n c (-s), E a c c o r d i n g t o t h e formula ( 2 . 5 . 4 ) a n d , moreover, t h e r e i s t i o n a l on H+ (s), E ( R : ) and t h e one t o one c o r r e s p o n d e n c e between t h e e l e m e n t s i n H 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 s on H
( S )
I n f a c t , d i f f e r e n t elements i n H ( s ), E l
i.e.
,E;1
(R ) +
(S),E
+
l i n e a r f u n c t i o n a l s on H
(-s), E
+
t h e e l e m e n t s of
(H
*
L , E )
.
d e f i n e d i f f e r e n t continuous
s i n c e t h e form ( 2 . 5 . 5 ) d o e s n o t depend on
F u r t h e r , l e t @ ( u ) b e a 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 on
t h e e x t e n s i o n s 9,.
( R n ) i s a H i l b e r t s p a c e ( f o r any E f i x e d ) , w i t h (s), E ' (S),E (s), E t h e i n n e r p r o d u c t ( 2 . 5 . 1 ) a n d , a c c o r d i n g t o t h e R i e s z theorem t h e r e
H+
H+
C H
e x i s t s an element @ Let f0(E,c) fo
E
H
= E
E
H+
( 5 ) ,E
s u c h t h a t @ ( u + )= ( u + , $ + )(s)
2s2<E5,2s3
<<>
(Wn),
( - 5 ) ,E
( u + , f o ) ( 0 ) ,E
=
+ -2s 1
and
= f
T+fo
Il4I
=
-
$+(E,S), f0
= F
-
E H(-s),E my),@ ( u + )= ( u + , @ +(s) )
ll$+ll(s),E
=
v
u+EH+ ( s ), E '
-1 5+xfD. Then
l!foll(-s),E
2
=
l l f l l ( - s () 1, E ).
T h e r e f o r e , 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 on H+ can be d e f i n e d (s). E by ( 2 . 5 . 5 ) a n d , moreover, 2 ( '- )S ) , E . On t h e o t h e r hand, a s a (1) SUP consequence of ( 2 . 5 . 6 ) , one h a s : 1101 = 5 ( - s ) , E l (s) (1) sa t h a t = (-S),E'
11@1 I
1If11
I
1 I@ I
I If1 I
I n t h e same way one shows t h a t
(H
h+)I IIflI
I b+I/ *
+
(By)) = H ( S i , E ' same d u a l i t y p r o p e r t i e s have t h e f a m i l i e s of s p a c e s H (S)
H
( - 5 ) ,E,S'
Of c o u r s e , t h e
(-5) ,E
and , € , E l
(R-).
2.6. Sobolev s p a c e s o f v e c t o r i a l o r d e r on m a n i f o l d s We a r e g o i n g t o e x t e n d t h e d e f i n i t i o n of S o b o l e v t y p e s p a c e s o f v e c t o r i a l o r d e r t o t h e f u n c t i o n s d e f i n e d i n a bounded domain U C Wn m
with
m
C -boundary
a U and t o C - m a n i f o l d s w i t h o u t and w i t h boundary.
W e need two a u x i l i a r y r e s u l t s f o r d o i n g t h a t .
Let @
: En+
(2.6.1)
R n b e a Cm diffeomorphism,
and d e f i n e O * b y :
*
0 u ( x ) = (u''@)(x) = u ( @ ( x ) ) .
Assume t h a t 0 i s l i n e a r o u t s i d e some compact s e t i n R n .
Then t h e
2 . Sobolev Spaces of Vectorial Order
134
*
( R n ) with s . j = 2 , 3 non1' n e g a t i v e i n t e g e r , i s a c o n t i n u o u s l i n e a r mapping u n i f o r m l y w i t h r e s p e c t t o chain r u l e implies: @
E
: H
( s ) ,E
E ( O , E ~ ] . The d u a l i t y y i e l d s t h e same r e s u l t f o r s .
j = 2 , 3 non-positive
3'
integer. Lemma 2 . 6 . 1 .
*
( m n ) + H ( ~ , )€ ( I R ~ )d e f i n e d by ( 2 . 6 . 1 ) w i t h O(x) l i n e a r o u t s i d e a compact s e t i n mn, i s a continuous l i n e a r mapping u n i f o r m l y 3 w i t h r e s p e c t t o E E ( 0 . ~ ~f o1r V s E R . The map Q
: H
(s),E
Lemma 2 . 6 . 2 .
Let
C$ E s ( I R ~ ) .Then
i s a continuous l i n e a r map uniformly with r e s p e c t t o
E.
These two l e m m a s w i l l b e p r o v e d ( i n a more g e n e r a l s i t u a t i o n ) i n t h e Chapter 3 ) . Now u s i n g Lemma 2 . 6 . 1 ,
2 . 6 . 2 and t h e p a r t i t i o n of u n i t y t e c h n i q u e , w e c a n
d e f i n e Sobolev s p a c e s of v e c t o r i a l o r d e r f o r f u n c t l o n s on bounded domain on smooth m a n i f o l d s w i t h smooth b o u n d a r i e s . If U
C lRn
i s a bounded domain w i t h Cm-boundary a U one d e f i n e s
(U) a s t h e f a c t o r spaces:
H ( s ),E
0 -
( R n ) such t h a t i s t h e c l o s e d s u b s p a c e of a l l u E H (s),E ( s ) .E SUPP u c n n \u.
where H
The norms
1 . 1 I ( s ), E , U
a r e d e f i n e d by t h e induced t o p o l o g y of a f a c t o r
space :
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 Ru of u t o Rn. One can g i v e an e q u i v a l e n t d e f i n i t i o n o f norms and s p a c e s
H
( s ),E
(U) u s i n g t h e p a r t i t i o n of u n i t y .
Let ) , Q ' ( l < e s r b e a c o v e r i n g of t h e r e e x i s t diffeomorphisms
au
by open ( i n R n )
s e t s , such t h a t
~ ~ : u ~ n u +i sv p .~ z ,r where
Vp. 1s a s u b s e t of R n .
F u r t h e r , it i s assumed t h a t
x a. ( a u n
n-1
U ) c IR
p.
x
{O}.
L e t UO b e an
2.6. Sobolev Spaces of Vectorial Order.on Manifolds
CI
open s u b s e t of U such t h a t Uo C U ,
U
O
'
2
U and l e t
135
( $ 9 . ) o =-
-
<Er
be a
p a r t i t i o n of u n i t y s u b o r d i n a t e t o t h e c o v e r i n g (U.9 ) O59.6r of u . -1 F u r t h e r l e t 9. ( ( u . $ ~ )' X9. ) : I R y + C and L 0 ( u . $ 1 : R n + C be t h e e x t e n s i o n
0
0
by z e r o of t h e f u n c t i o n s (u.$ )
O
X-l and u.Q0 d e f i n e d on t h e s e t s V9. and
9 . 2 ,
Uo,
respectively.
D e f i n i t i o n 2.6.1.
For a f a m i l y of d i s t r i b u t i o n s ( O , E ~ ] 3
E +
u(E,.)
3
E 0'U.J) and f o r s E IR ,
let 2
I IuI I (s)
(2-6.3)
= €1'
and l e t
H ( ~ (u) ) be
u f o r which ( 2 . 6 . 3
1
1srsi
( I / & o ( ( u . $ 9 . ) X i-1 ) I I +( S ) , E ) 2 + I
19.0(~.$o)
I I 2( S ) , E
t h e f a m i l y of spaces of t h e ( f a m i l i e s of) d i s t r i b u t i o n s
is finite, V
E
E
( 0 . ~ ~ 1 .
Introducing
J u i 1 (s), E , U I I. I j
with
( s ) ,E
d e f i n e d by
f a m i l i e s of d i s t r i b u t i o n s u
2 . 6 . 3 ) , we denote by H ( s )
( u ) t h e spaces of t h e
f o r which ( 2 . 6 . 4 ) i s f i n i t e .
Remark 2 . 6 . 2 . Two d i f f e r e n t c o v e r i n g s and c o r r e s p o n d i n g p a r t i t i o n s of u n i t y y i e l d an e q u i v a l e n t f a m i l y of norms H
(U),
(s)
Remark . _
I I . I I (s), E , U
and t h e same spaces H
a s a consequence of Lemmas 2 . 6 . 1 ,
( s ) ,E
(U),
2.6.2.
2.6.3.
D e f i n i t i o n 2 . 6 . 1 e x t e n d s immediately t o f a m i l i e s o f d i s t r i b u t i o n s on a Cm-rnanifold M w i t h a cm-boundary aM. I n f a c t , one h a s t o t a k e s e v e r a l open
sets i n o r d e r t o c o v e r t h e i n t e r i o r of M . L e t rO d e n o t e t h e r e s t r i c t i o n o p e r a t o r t o t h e h y p e r p l a n e R n - l x
{O}.
D e f i n i t i o n 2.6.4.
For a family of distributions ( O , E ~ ] 3
E
+
$(E,.)
E U'(aU) and for s E R 3 ,
let
and l e t H
( s ),€
( a u ) be t h e f a m i l y of spaces of t h e ( f a m i l i e s of) d i s t r i b u -
t i o n s Q which have t h e norms ( 2 . 6 . 5 ) f i n i t e , V Introducing
E
E (0.~~1.
2 . Sobolev Spaces of Vectorial Order
136 we denote by H
(s)
(aU) the Banach space o f the ( f a m i l i e s o f ) d i s t r i b u t i o n s
w i t h the norm (2.6.6) f i n i t e . Again, the norms [. 1 dependonapartitionoftheunity, ( s ) ,E,au' [ - l ( s ),au but are uniformlyequivalent with respect to E E ( 0 , ~1 and define the 0 same topology and the same spaces.
au
In Definition 2.6.4 one can take as
m
the boundary of any C -manifold and
not necessarily the boundary of a domain in IRn Definition 2.6.5.
A family of d i s t r i b u t i o n s ( 0 . ~ ~3 1E belong t o the spaces ( 0 . ~ ~3 1E
supp u
+
c u, v
( s ) ,E
E H
u(E,.)
(S)
+ u(E,.)
( u ) , i f supp u
E
,€
(mn) i s said t o
5 u.A family
(w") i s said t o belong t o the space
(u), i f
( S )
E (O,EO1.
E
All the results in the Sections 2.1 - 2.5 extend in a natural and obvious way to the spaces of vectorial order on a smooth manifold or in a bounded domain U
C
XIn. We shall illustrate this by giving a detailed proof of the
version of Theorem 2.4.5 for spaces H m
( s ) ,E
(U)
with U C R n
a bounded domain
with a C -boundary a U . Theorem 2.6.6.
Let
u
s =
(sl,s2,s3),s a t i s f y the condition:
c
mn be a bounded domain wCth a cm-boundary
(2.6.7)
3,
/s21 <
3u. Let s E R
3
,
Is2+s3\ < f .
Then t h e continuation by zero Lo i s a continuous linear mapping uniformly w i t h respect t o Lo
(2.6.8)
:
H
e
E' ( 0 , l l from
H
( s ) ,E
,E ( s ) ,E (u) + H ( s )
(U) i n t o H
( s ) ,E
(XIn):
(Rn).
x 3. ' lSjSr, orientation x. E Cm(u. n u), 1 5 j 5 r, x.73 ( U3.n aU) C R n-1 { O } .
__ Proof. Let {Uj]lljSr be an open covering of aU and preserving diffeomorphisms, xj where V
1'
n
V. 7' lSjSr, are subsets of mf such that :
U.
U
+
<,
7
3
can always be extended as diffeomorphic mappings of Rn into itself which coincide with the identity outside a ball with sufficiently large diameter (cf. [Pal. 1
I),
so that supp(X.-Id) is compact. Let
set such that {Uj)05j5r
cover
Ti
3
UoCC
U be an open
and let {$j}05jsr be a C"-partition
Of
137
2.6. Sobolev Spaces of Vectorial Order on Manifolds unity subordinate to the covering {Uj}05jsr.
(U) and denote u : By + C the continuation by zero of Let u E H ( s ) ,E j -1 (u.$.) ,,xj : V . C, 1Sj6r. Furthermore, let uo = u$,. 3 7 Definition 2.6.3 yields: -+
L'u O j
Denote by L'u the extension by zero of u . O j I : IRn + c. Using the triangle-inequality in H
( s ) ,E
:
By
-f
C onto IRn
,
i.e.
(U) r times and after that
Lemma 2.6.1, one finds:
where the constant C may depend only on s,r and
x 3.'
15j5r* Thus, by Theorem 2.4.5 with (2.6.7) being valid, one finds using (2.4.12)
(with L ' instead of II ) and (2.6.10): 0 0
where the constant C does not depend on u and Theorem 2.6.6.
E.
This ends the proof of
I
Example 2.6.7. tx E B2 ,1x1 < 1 1 and denote T1 = a B , 8 E T 1 being the angle 1 coordinate. It is readily seen that g E H (T ) iff its Fourier coef-
Let B
=
E
(s) ,E
ficients gE (n),
satisfy the condition:
With I (x) = J (ix) the Bessel function of the imaginary argument, consider the function u (2.6.13)
:
B
+
C
given by the series:
u E ( x ) = (2~)-'E gE(n)I
nEZ
In1
(rc-1 )(I
(c-'))-'exp(in@), In1
r = ixi, e=arg(x +ix ) . 1 2 With gE E H ( s ) ,€(T1),s2+s3 > 0, uE(x) given by (2.6.13) is the (continuous on B)solution of the boundary value problem:
2. Sobolev Spaces of Vectorial Order
138
2 (1-E A)u (x) = 0 , X E B , u (cos 8 ,sin 8 ) = g
I),
8 E aB.
) as X = E-' In1 one can write uE(x) = vE(x)+ wc(x), where
using the asymptotic expansion for I [Grad.-Ryz., 1
(e),
-1 (rE
+ m
(see
vE(x) = X(r) (~n)-' c gE(n) exp(-E-'<En>(l-r)+in8) nEZ
x
with
E Cm([O,l]),
x
:0 for r E
[0,%1,
x
E 1 for r
E [f,11, and where
w (x) has already a better regularity (uniformly with respect to than vE(x). Furthermore, the regularity of v 0
one of vE (p,t)
where
:
B
+
E
E (0,11)
C is the same, as the
R+ x W - t C, given by:
I
0 = vE(p,t)
(2.6.14)
:
9E(S)exp(-E-1p<Eg>+itS)dS, p E R+
,
tER,
R is defined by (2.6.12) with n replaced by 5 E W .
;,(C)
Using (2.4.10) one gets: vo E H
s,
E H(s),E(R1),
gE
H(S),E(T
i.e. u
E
E H
,P+
with s +s > 0 iff 2 3 (s1+t,s2,s3+f) ) ( B ) with s2+s3 > 0 iff
(s1+f,s2,s3+f),E
) -
2.7. Spaces of one parameter families of meshfunctions Discrete analogues of classical Sobolev-Slobodetski spaces H
are introduced
here, whose elements are meshfunctions on one parameter families of grids in Rn or Ry, the parameter h E (O,ho] being the meshsize. These spaces are suitable for the investigation of corresponding classes of difference operators (see ch. 111, 3.11). For meshfunctions on the grid
$=
hZn in Bn , h E (0,h 1 introduce 0
the inner products
and the norms
the family of spaces of the meshfunctions u for which (g) the norms (2.7.2) are finite, h E (O,ho].
Denote by
H
(0),h
V
As a consequence of Parceval's identity, (2.7.2) is equivalent with
the norms (2.7.3) -h where u ( 5
u is the discrete Fourier transform. Fx+S, h Let (sl,s2 E lR2 and let 5 = ( C l , ...,Cn) with 5 k = (ih)-'(exp(ihEk)-l). =
2.7. Spaces of One Parameter Families of Meshfunctions
139
Definition 2.7.1.
For a f a m i l y o f meshfunctions (O,ho] 3 h
-f
u(h,x) C
S'
h
(Rt)
d e f i n e t h e norms of order s
I
(2.7-4)
IUI
s2 I (s),h 1 Ih-sl <5> (Fx+g,hu)(h,hS)11
.
=
L (T
5 rh
)
(37;) if its norms(2.7.4) a r e f i n i t e V h E (O,ho]. We say t h a t u E /f (s),h The meshfunction u € H (2") i f one has ( S )
Obviously,
H ( s ),h inner products:
(IRE)
is a family of Hilbert spaces equipped with the
For s2 non-negative integer one can also introduce the norms
the norms (2.7.4) and (2.7.7) being in this case equivalent uniformly with respect to the parameter
h E (O,ho], as a consequence of Parceval's
identity and the inequalities:
where the constant C depends only on its subscript s2 Proposition 2.7.2. (s)
(2") is a Banach space.
Proof. Indeed, rewriting (2.7.4) in the form ~
n with Tn = Tn 11
n,l
,one gets the conclusion that the operator
u + h
-(s +s +n/2) 1 2
I
(h2+ w ( r i )
1 2)
Fx+ri, 1u
2 n (2") isometrically onto the Banach space B((O,h 1;L (T ) (s) 0 n bounded on f 0.h 1 functions valued in L2(Tn). (I maps H
0
)
of
2 . Sobolev Spaces of Vectorial Order
140 Example 2 . 7 . 3 .
For t h e f a m i l y of m e s h f u n c t i o n s (2.7.10)
u(h,x) = h
-a
exp(-h
-1
1 lxj-/)t lSj5n
one f i n d s e a s i l y i t s d i s c r e t e F o u r i e r t r a n s f o r m :
TI
Ch(h,hC) = h n - a ( l - e - 2 ) n
cos h C . + e - 2 ) - 1 .
(1-2e-1
3
l<jln T h e r e f o r e , t h e meshfunction and b e l o n g s t o h
H
(s ) n - 2 ( o + s +s 1 2
( 2 . 7 . 1 0 ) b e l o n g s t o any
,v h
H (s), h
>
0,
iff
(Z")
JTn ( h 2 + 1 u ( r l )I
)
s2
n
16j5n
(1-2e-1c0srl,+e 3
-2 -1 )
dn
V h E (O,hol, where C
s ,ho
depends o n l y on i t s s u b s c r i p t s .
The l a t t e r h o l d s i f f h
n - 2 ( a + s +s ) 1 2 J
2 ( h + l w ( n ) 1 2 ) s 2 d n 5 C s , h o ~ V h E (O,ho],
Tn
w i t h some c o n s t a n t C
S,ho
.
I t is e a s i l y seen t h a t f o r h
0
-f
one h a s :
O(1)
( h 2 + / w ( q )/ 2 ) s 2 d q
=
Tn 17
{
O ( l n h - l ) , when s 2 n+2s2 O(h ),
so t h a t t h e f u n c t i o n ( 2 . 7 . 1 0 ) b e l o n g s t o
s +s2 5 ( n / 2 ) - o , o r s 1
= -n/2,
F o r a f u n c t i o n u E C o ( Xn )
s
when s 2 > -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<
(h,h<)
,
since S(Rn)
( 1 . 3 . 1 1 ) , t h e Cauchy-Schwarz
V s . The P o i s s o n formula
i n e q u a l i t y and t h e o b v i o u s estimate
141
I
< m if
6
151,
V h E R
,yj.eld:
s-t > n/2.
a Thus, t h e f i r s t of t h e i n e q u a l i t i e s NOW
(2.7.11) i s proved.
we prove t h e second cane. The s a x e argument a s p r e v l c u s l y shows t h a t
where C
depends o n l y on i t s s u b s c r i p t s . S,t
S i n c e t h e l e f t hand s i d e o f t h e l a s t i n e q u a l i t y d o e s n o t depend on h , one f i n d s :
I f one s u b j e c t s a f u n c t i o n u E
ff ( R n ) t o a smoothing o p e r a t i o n by
means of c o n v o l u t i o n w i t h a f a m i l y o f smooth d e l t a - s h a p e d k e r n e l s ( t h e f a m i l y c o n v e r g e s t o 6 ( x ) when h RE
of t h e f u n c t i o n
belong t o
H
op_ osition -P r_
(Z").
(0,s)
U,
+
O), t h e n t h e p r o j e c t i o n s o n t o t h e g r i d s
r e g u l a r i z e d by a s p e c i f i c c h o i c e of k e r n e l s , w i l l More p r e c i s e l y , t h e f o l l o w i n g r e s u l t i s v a l i d .
2.7.5
L e t u E Hs(xn) and l e t s' be any nonnegative numheu. Then t h e r e e x i s t s
2 . Sobolev Spaces of Vectorial Order
142
a f a m i l y of f u n c t i o n s v
,v
1". v E Hs+,(wn)
E yo,s) ( ) " '
20.
71 hv
3 O .
I l ~ - - v IIs-s'
h
=
E
and
v(h,x) s a t i s f y i n g t h e f o l l o w i n g c o n d i t i o n s : t ~ P h ~ < ~ 6 ~Ctlj~lls; ~ 0 'hV/] ( O , s ) 1 ;
/Iv]I~+~
R + , V t 2 0 and s
'
[I
5 Cs,hS'I IUI
Is
IuI Is
'
where ct and cS, depend o n l y on t h e i r s u b s c r i p t s . Proof. -
m
Let 4 ( 5 ) E C ( R n ) satisfy the conditions 0 0 5 (2.7.13)
aa$
5 0
(0) = 6
On
r V a with la1 5 [s'l+l,
where dOa is the Kronecker symbol and [s'] stands for the integral part of s'
-1 2 0. Introduce v(h,x) = FS+x $~~(hg)F~,~u.
Since supp 4
has the diameter 6 ch-',
0
v(h,x) has the property 1";
indeed, h2t
Property 3" is an immediate consequence of (2.7.13):
Il~--vIIs-s' 2
2 2 <5>2(s-s') ll-40(hS)I l u ( 5 ) I d5 6
-
151
2s'
-
2
I u ( E ) I d5
2 2s' 2 ' Cs,h I lulls.
Finally, 2" can be easily derived from Proposition 2.7.4. In fact, for each m
(since $o E C (W") 0 fore, for any t > n/2 the inequality
h > 0 one has v E S(IR:)
I I'hvl
and, thus,
E
S(R9) ) .
There-
t
( 0 , s ) ,h 5
I IVI Is+Cth I IVI I s + t
holds uniformly with respect to h E (O,hO1, V ho > 0. Using 1' and the last inequality one gets 2". Proposition 2.7.6.
L e t u E H ( o , s ) (z"). Then t h e r e e x i s t s an e x t e n s i o n 1 u of t h e meshfunction h
u o n t o IR"
such t h a t for any t b 0 holds:
(2.7*14) SuPO
htl
llhUl /s+t
' 't[
lul
(O,s).
2.7. Spaces of One Parameter Families of Meshfunctions
143
Proof. Introduce v(h,x)
(lhu)(x)
=
-1 (F5-x XTn
=
F
x+S,h
5 .h is the characteristic function of Tn
xn..
where
def
u) (x), x E IRn,
5 rh
'5 ,h
1,
Noticing that 151 5 ( n / 2 ) 15 (h,hS)
V h > 0,
I
={5ERn
v 5
] h S j / < lZjSn}. ~,
E Tn
5 th
,one finds
immediately that h2tl IvI
2 =
In
h
2t
2(s+t) /;h(E)12d5 Z
<5>
T
E, ,h
5 c2
t
2
<5> 2s;1 h ( 5 ) ]2d5 5 Ct[lu/l(o,s). 2
In
I
T
5 th y f R , k E Z+, the difference analogue of the
k n Denote by C ( 2 ) , k n spaces c ( w ) of meshfunctions u(h,x) equipped by the norm (2.7.15)
IUl
sup (kty)= O
Let u E
H
IuI
(kty),h
lul
(kry),h
=h-'
1.
sup 1 /DZ,hu(h,x) xcRn / a / s k
h
k n (Zn) w i t h s > 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>
5 C
yield:
-2 (s-k) d5
2 (s-k) I;h(5)12d5 lta/2<5>
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 (<5>'ln<5>)-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 <
' ' -I
(s), h
iff
s; 2 s l , s'+s' S s +s 1 2 1 2 '
W e s a y t h a t s'
<
s if
(2.7.17)
holds.
The same argument as i n t h e p r o o f o f P r o p o s i t i o n 2.1.10,
c a n b e used i n
o r d e r t o prove t h e following P r o p o s i t i o n 2.7.11. 2 L e t o E R , s E x2 and
let
Then for V 6 > 0 h o l d s :
V h E (O,hol, V V E f f ( u ) , h ( < )
.
For a g i v e n s E R 2 d e n o t e (2.7.20)
vs
=
is'
I s'ER2,
5'
< sl, v;
=
{s'
I
s'E
m2,
s'>
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
x ,h
>' 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
= [
W e s h a l l show t h a t f o r t h e commutant Bh S,@
holds
x rh
>',@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 = <5(h,hS)>2,
one obtains
Using the identity
which along with (2.7.25) yields the estimate:
Indeed,
n2
=
<<(h,hr~)>~and
p =
s/2,
2.7. Spaces of One Parameter Families of Meshfundions
147
Further, since I$ E S(Zn), one can apply(l.3.10)with k = 0, so that (2.7.29)
sup O
Therefore, using (2.7.28), (2.7.29), one gets the following estimate:
Taking j
=
1 s I+2+n and
denoting
one gets immediately the conclusion that the family of the convolution operators Gh with the kernel Gh, (2.7.30)
(GhGh)(5) =
I
-h -h g ( < - n ) v (ri)dri,
Tn
ri ,h
is uniformly bounded with respect to h E (O,ho] in the spaces
ff
(O),h(%) . -1 Indeed, after the inverse Fourier transform F S-fxjh, (2.7.30) becomes -h so that the multiplication operator v + gv, where g(h,x) = F<+x,h '
where the constant C
depends only of n.
The latter implies that with some constant C not depending upon h the inequality
2. Sobolev Spaces of Vectorial Order
148
is s a t i s f i e d .
Using t h e o b v i o u s i n e q u a l i t y
0
one g e t s (2.7.22). c o r o l l a r y 2.7.15.
A s a consequence of
(2.7.22) t h e map u
continuous Linear map from H
(S)
+
$u f o r each '$ E S ( Z n ) is a
(z") i n t o H
(S)
(z"), v
s = (sl,s2)E
R
2
.
W e t u r n now t o t h e s t u d y o f t r a c e s of m e s h f u n c t i o n s on h y p e r p l a n e s of
codimension one,which c a n b e g i v e n i n t h e form Il w i t h some y E Z n , y
=
(yl
, _ . ,.y n ) ,
Y
= {x
E Rn
,
< y , x > = 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 <<(h,hC)> n 1 T1 T Cn,h 2 (1-s2)+l 5 C2<S'(h,hS')>
2 2 -2 (<<'(h,hS')>+En)
dSn 6 2 ( j - s )+1
dEn 6 C3<5' (h,hS')>
1R
En where the constants C . may depend only on s2,and n. -2s
J
(h,h5')>2(s2-'-f),
Multiplying (2.7.35) by h
using (2.7.36) and
integrating over Tn-' , one finds that 5 ' ,h
1 ID;
1j
hu(x~'o)
1
(sl,s2-j-f),h 5 C I ( u ( (s),h
,V
h E [O,holt
n' where the constant C may depend only on s2 and n. We now prove the surjectivity of the map u(x',x ) each k < s2-4. Let a meshfunction a(h,x') E H
+
k D
xn
hu(x',O) for
(B?:T1h) be ( ~ 1 ~ ~ 2 - k,h -f) given. We construct a meshfunction u(h,x',xn) E H(s),h(Rz,h) , such that
k
I
I
'
with s o m e D hu(h,x',O) = a(h,x') and lul (sl,s2-k-f) ,h (s),h xn8 constant which does not depend on a, u and h E (O,hO].For doing that, put
la'
<<'(h,h<')>* and let @ (t) E Cm(R) be such that supp @ 6 0 6 @ (t) 0 for It/ 5 & / 2 , and such that 6
X
=
(2.7.37)
(2n)-'
I @ 6 (t)dt =
1,
R
where 6 > 0 will be chosen later. Let us define
Gh(E',<
)
as follows
C
[-6,61,
2
2.7. Spaces of One Parameter Families of Meshfunctions
151
-1 -h u s a t i s f i e s t h e r e q u i r e m e n t s above W e show t h a t u ( h , x ' , x ) = F n E+x.h I n d e e d , one h a s :
=
(27I
t h e i n t e g r i n d A-l$(h-l
15
2 0
i n e q u a l i t y 6 < rr(h + 4 ( n - 1 ) )', V h E (O,ho], V
5 ' E Tn-l
o:e
1
5 6 h . Choosing 6 t o s a t i s f y t h e f i n d s i m m e d i a t e l y t h a t h6X <
IT,
T h e r e f o r e , i n t h e i n n e r i n t e g r a l one can
E',h' so t h a t t h e s u b s t i t u t i o n n n = A - 1 instead of T i n t e g r a t e o v e r IR1 'n Cnrh' 'n k hu(x',O) = a ( h , x ' ) . and t h e c o n d i t i o n ( 2 . 7 . 3 7 ) y i e l d D xnr F u r t h e r , u s i n g P a r s e v a l ' s i d e n t i t y and t h e o b v i o u s i n e q u a l i t i e s
151 5 151 5 (71/2)1<1,V h L 0 , V 6 E Tn variables
5
A-1
-
nn,
5,hr
and making t h e change o f
one f i n d s
where t h e c o n s t a n t C may depend o n l y on s 2 , n and
I$
6( t ) .
I
C o r o l l a r y 2.7.18.
A s a consequence of Theorem 2.7.17,
integer m
> 0:
(2.7.38)
/ D i , h u ( h . O ) I 5 C1 IuI
one g e t s for n
I (0,m),h8
V j
=
=
1
and any g i v e n
O,l, . . . ,rn-1,
where t h e c o n s t a n t c does not depend on h and u.
2 . Sobolev Spaces of Vectorial Order
152
Remark 2.7.19. I n t h e same manner, one c a n show t h a t t h e f a m i l y o f mappings
(2.7.33) i s
o n t o as w e l l , and c o n s t r u c t e x p l i c i t l y a ( u n i f o r m l y bounded w i t h r e s p e c t t o t h e parameter h
E (O,ho]) family
of l i f t i n g o p e r a t o r s .
W e s h a l l now i n t r o d u c e a n e q u i v a l e n t d e f i n i t i o n of norms i n s p a c e s
'
which i n c e r t a i n c a s e s t u r n s o u t t o b e more c o n v e n i e n t , s i n c e i t s ( s ), h ' d e f i n i t i o n d o e s n o t u s e t h e F o u r i e r t r a n s f o r m of m e s h f u n c t i o n s . For i n t e g r a l s2
>
0 s u c h a d e f i n i t i o n h a s a l r e a d y been g i v e n . T h e r e f o r e , w e
now c o n s i d e r t h e c a s e 0 < s ,
c 1 . F u r t h e r , w i t h o u t 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 t a k e s1
0 , and we s h a l l w r i t e
L
=
1.11
s ,h
i n s t e a d of
I'"'(O,s),h* I n t r o d u c e t h e norms o f o r d e r s = ( s 1 , s 2 ) ,s 2 E ( 0 , l ) f o r m e s h f u n c t i o n s 1 on R x,h
..
where 0 u t
=
u ( x + t ) is the s h i f t operator.
Theorem 2.7.20.
The f a m i l i e s o f n o r m
I . 1 1 s,h ~
and [ . ]
s,h
d e f i n e d by ( 2 . 7 . 3 9 ) a r e m i -
f0mZ.V e q u i V a k 2 t w i t h r e s p e c t t o h E (O,ho], i . e . t h e r e e x i s t s a c o n s t a n t
c
>
0 such t h a t h o l d s :
It s u f f i c e s to c o n s i d e r t h e case s1
= 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' Parseval's identity yields:
1 I (Ot-l)ul 1 '
=
(277)-1
0,h Introduce g(a) =
C
J1 T
5 ,h
k - ( 2 s + 1 ) s i n 2 ka.
k> 1
One h a s w i t h some c o n s t a n t C > 0
I n d e e d , f o r any N 2 0 one h a s :
leitE-l12\;h(E)
I2dE.
2.7. Spaces of One Parameter Families of Meshfunctions
153
m
k - ( 2 s + 1 ) sin2 ka < J t - ( 2 s + 1 ) d t = ( 2 s ) -1 N - 2 s
C
RN+l(a) =
k>N+1
N
one gets:
( 2 s ) -1/ ( 2 s ) l a l - l ,
With t h e l e a s t i n t e g e r N 2 (2.7.42)
.
2s
0 5 RN+l(a) 5 ' a /
.
Now we e s t i m a t e t h e sum
x
k-(2s+1)sin2ka5 15k5N =
0
2
kl-2s
2N
< a / t
dt =
0
16kZ-N
-1 2 2 - 2 s (2(1-s)) a N
s i n c e , a s a consequence of t h e c h o i c e o f N ,
1-2s
5
Cia/ 2 s ,
one h a s : N
-
1al-l.
I n t h e r i g h t hand s i d e o f t h e l a s t i n e q u a l i t y t h e c o n s t a n t C may depend o n l y on s. F u r t h e r , i f M i s a n i n t e g e r s u c h t h a t n/4 5 M i a 1 5 71/2, t h e n
g(a)2
2
2 2 kl-2s X k-(1+2s)sin2ka2 (2/n) a 16k<M 15k5M ~
(2/71)2 a 2
f: t l - 2 s d t
2
2 Cja12s,
1 where a g a i n C may depend o n l y on s . Hence, P a r c e v a l ' s i d e n t i t y a n d ( 2 . 7 . 3 9 ) y i e l d : [uI2
0,h
=
=
(2nI-l
( 2 n I - l J1 T
5th
where t h e e q u i v a l e n c y
5 rh
(1+4h
Jl T
-
1 O
(l+h
k>0
t- ( 2 s + l ) 4sin2
3 2
t,h h - ( 2 ~ + l ) -(2s+l) . 2 h5 k s i n ( k -)) 2
i s u n i f o r m w i t h r e s p e c t t o h E [O,h
0
1.
For n > 1 t h e norms ( 2 . 7 . 3 9 ) a r e d e f i n e d i n a n a t u r a l way:
Also u s e f u l i s t h e f o l l o w i n g d e f i n i t i o n o f t h e norms o f o r d e r s s2 E ( 0 , l ) :
=
(sl,s2),
2 . Sobolev Spaces of Vectorial Order
154
Theorem 2.7.21.
I I . 1 I ( s ),h are
ivomns (2.7.44) and t h e norms
uniformLy e q u i v a l e n t w i t h
respect to h E [ O , h o l . Proof. __ It suffices to consider s1 ~
.1
,h instead of
Putting y
=
=
0, so that we shall write s instead of s 2 and
1 .1
(0,s),h' xtz and using Parseval's identity, one can rewrite (2.7.44)
in the following fashion:
where, as usual, 0 u
=
u(x+z) is the shift operator.
Denote G(n)
=
X
lkl
-(n+2s) i 2 le -11 .
kEzn\{ 0} Again one shows that with some constant C > 0 holds:
the argument being exactly the same as in the case n
=
1.
Using (2.7.45). (2.7.46) one gets immediately the equivalency of the norms
1.
Is,h
and
11.1
Is,h uniformly with respect to h E [O,ho].
Remark 2.7.22. For V nonintegral s = k c e
> 0, k
are defined in a natural way:
E Z + , 8 E ( 0 , l ) the norms (2.7.43), (2.7.44)
2.7. Spaces of One Parameter Families of Meshfunctions
155
and, correspondingly,
The p r o o f o f e q u i v a l e n c y o f norms ( 2 . 7 . 4 7 ) ,
( 2 . 7 . 4 8 ) and t h e norms
I 1 . I Is,h
( u n i f o r m l y w i t h r e s p e c t t o h E [O,h 01 ) d i f f e r s i n no way from t h e case s E ( 0 , l ) . Remark 2 . 7 . 2 3 . One h a s i n t h e s p a c e
'
H
(Z"),
s = (sl,s2),three different equivalent
(S)
norms, which a r e d e f i n e d a s supremum w i t h r e s p e c t t o h E ( 0 . h 01 o f t h e n o r m s
I I.
(s),h' h - s 1 [ . ] s 2 , h
and h - S 1 / .
Is2,h.
<,+ <,
Now w e s h a l l c o n s i d e r s p a c e s o f m e s h f u n c t i o n s o n t h e g r i d s n , + de f Zn-l x z+ = { a = ( a ' , a n ) , = {x E x 2 0 ) . Denote Z
a'
E
zn-l,
an E
Denote by
x+;.
H (o) , h ( < f +
)
t h e c o l l e c t i o n of m e s h f u n c t i o n s u ( h , x ) s u c h
that
and by
H(o)
(Zn'+)
t h e space o f a l l meshfunctions u : ( 0 , h
01
x
Znt+
+
C,
f o r which h o l d s
<,+
a and e a c h m e s h f u n c t i o n u on the f o r w a r d d i f f e r e n c e d e r i v a t i v e s Da a r e w e l l d e f i n e d . Hence, one c a n d e f i n e x,h f o r e a c h i n t e g e r m t 0 t h e norms o f o r d e r ( s l , m ) :
N o t i c e t h a t f o r each multiindex
and r e s p e c t i v e l y ,
W e d e n o t e by
H
( s l , m ), h
(<,+)
t h e s p a c e s o f m e s h f u n c t i o n s u f o r which
2. Sobolev Spaces of Vectorial Order
156
are finite, V h E (O,ho], and by
the norms ( 2 . 7 . 5 1 )
H ( 3 1 ,m) ( Z y )
the space
of all mesh functions u for which the norm (2.7.52) is finite.
proposition 2.7.24.
Let
+
.
E c ~ ( R ~ )Then
where cm depends only on m. Proof Using the Leibnitz formula
one gets immediately (2.7.53). For any meshfunction u extension operator 1 u m
:
where the coefficients C
R : +
P
(2.7.56)
Z
(-1)J
lspsm
c
:
<,++
C and a given integer m > 0 define an
C, as follows :
satisfy the system of equations
pJ = 1,
0 2 j < m,
P
and where H(x) z 1 for x 2 0, H ( 0 )
=
0 for x < 0 is the Heaviside function.
We shall need the following auxiliary assertion concerning any meshfunction on
%,+
:
Lemma 2.7.25.
Let u(t)
:
d,+* c and
let
where lm is defined by (2.7.55), (2.7.56) with xn Then the following formula is valid:
=
t.
157
2.7. Spaces of One Parameter Families of Meshfunctions
where a Qj,p
=
(al,...,a , ) is a muZti-index, / a / =
=
{a
CL
1 1 (al,...,a , ) , 0 5 ak < p, 1 5 k 2 j}.
=
and +...+a. 1'
7
Proof. -
1 It is obvious, that v . (t) 3 u .(t), V t E %,+, V j 2 0. 1 1 We prove (2.7.58) by induction on j. We shall first check it for j = 1. For t < -h, one gets using the definition (2.7.551, (2.7.56) of lm: (2.7.59) v (t) = h-l 1
C C (u(-pt-ph)-u(-pt)) = lspsrn
-
Correspondingly for t = -h, using (2.7.56) (with j (2.7.60) v (-h) = h
-1
1
C
=
01,
(~(0)C C ~ ( p h ) =) - C C C 16pSm '~EQ l
the latter coinciding with (2.7.59) when t
=
C
C
l
ul(-pt-ph+ah) ,p
we find ul(ah),
1 .P
-h, and (2.7.59) being
precisely (2.7.58) with j = 1. Now, let us assume that (2.7.58) holds for j, and we shall prove it for j+l (j+l i m). The formula (2.7.58) being true for j, let us take u to be the polynomial t1/j!, so that using (2.7.571, we obtain the formula: (2.7.61)
1 = 1,
(-111 C C I: lsp<m '~EQ
v
t
E Rt,h, t < O
1 rP hlal?pt+jph
tl/j! one has: u.(t) = H(t) and v.(t) Z 1. 1 Substracting (2.7.61) from (2.7.56) and using the identity
Indeed, for u
=
1, we get the formula: pl = ' a E Q . 18P (2.7.62)
(-1)'
C C C l
1 = 0, v t E
m t,h ' t < O .
3 rP
h/a)
7+1
(t) for t < -h. Putting a = (a',aj+l), we have
v . (t) = h-l(v.(t+h)-v.(t)) = 1+1 1 1 h-l(-l)l+l( I: c c (~.(-pt-jph+la'Ih)-u.(-pt-(J+l)phf/a';h))) = lsp<m ~ ' E Q , , ~1 1
(2.7.63) =
=
(-1)J + l
z c lspsm
h 1a
+
(-1)j+l
x
c
lsp<m
h-l ( u ,(-pt-jph+(a'Ih)-u,(-pt-(j+l)ph+/a'Ih)
C
3
1
cr'EQi rP
1 Lpt+ ( j+1 ) ph
z
h-lu.(-pt-jph+la'/h) =
~ ' E Q ~
tP
h(a'I
1
+
2. Sobolev Spaces of Vectorial Order
158
=
(-1)j+l lspsm
+
(-1)j+l
z
+
c
lspsm
=
(-1) j+l
x
Z u. (-pt-(j+l)ph+(a'lh+a. h)+ l+l 3+1 ~ ' E Q,p , 053 l
1
c
lspsm
c c ~ E Q ~ + ~ , ~
u , (-pt-(j+l)pi>+1 a 1111 + 3+1
hln' (>pt+(j+l)ph + (-1)j+l
c
c
z
z
h 1 a '+ t :1
u.
OSkh<->t-jph+ I a ' 1 h-h 1+1
lsp<m P ~ ' E Q .
(kh).
( j+l)ph
We used here formula (2.7.621, while substracting u . ( O ) in the corresponding 3 sum. We make a change of summation index in the last sum on the right hand
side of (2.7.63), putting kh = -pt- ( j+l) ph+ 1 a
I h+aj+lh,
after which this sum is transformed as fOllOWS:
Substituting this last expression into (2.7.63), we finally obtain for t
-h
which proves formula (2.7.58) for t < -h. We now prove (2.7.58) for t = -h. Using (2.7.61) with t = -h and the fact that u .(t) 3
0 for t < 0, t E lRL,h
,
we have
2.7. Spaces of One Parameter Families of Meshfundions
=
cp c
c
( - 1 ) j+l
lSpSm
a'EQ.
I #P
1011
vj+l
u . (kh). z O
(j-l)p
-jp+la'I+a. and again taking into consideration that 3+1 0 for t < 0, we obtain for the last sum the expression
Putting k u . (t) 3+1
IZ
159
=
=
(-h) = (-1) j+l
=
(-1)j+l
=
(-1)j+l
c
c
c
c
lspzm '~'EQ. p>a 3 rP / a ' It (j-l)p
. (-jph+lalh) = 1 1 u 1+1
jp- a
+
z c u . (-jph+/a/h) = 1:p6mCpa~CQ, O
z cpaEQ. c
u . (-jph+lalh), ,p 3+1
lSp5m
1+1 which proves (2.7.58) for t
=
I
-h.
Although the proof of Lemma 2.7.25 is based on a rather tedious computational work, the idea which is behind is quite transparent: if t 5 -jh then the inward sum on the right hand side of (2.7.58) is over full cube Q .
I tP
and it is easily seen, that in that case (2.7.58) can be
rewritten in the following fashion (2.7.64) v.(t) = c c 3 l
ai,hu(-pt),
v
t 6 -jh.
The same formula (2.7.64) is valid even for all t < 0, if u(kh) k
=
O,l,
...,m-1.
=
0,
Hence, via interpolation procedure (one can use Newton
interpolation polynomials) the general case is reduced to theone of the k polynomials t /k!, k = 0.1, m-1, which have beenusedinorder toderive the
...,
relation (2.7.61). NOW we are in the position to prove the following theorem: Theorem 2.7.26.
For t h e e x t e n s i o n operator im d e f i n e d by (2.7.55), (2.7.56) hoZds (2.7.65)
1 I1,ui 1
(sl,m),h "rnl
lul
I;sl,m) ,h'
V h
E (O,hol,V
s1
E IR,
where t h e c o n s t a n t cm depends o n l g on m. Proof. u(x',xn), one gets for v . = ( 3 1 3 x,.h using Cauchy-Schwarz' inequality, the following estimate
Applying (2.7.58) with u
=
1 u) (x',xn), m
2 . Sobolev Spaces of Vectorial Order
160
Therefore, with some constant C' the following estimate holds: m
'
j ,h'il~O),h' /lVjll(0),h CAjlUjll(0),h = CAjlax Furthermore, for any tangential difference derivative Da' with x',h ja')+j 5 m one gets:
Now.
I
(2.7.65) is an immediate consequence of (2.7.67).
Corollary 2.7.27. For lm d e f i n e d by (2.7.551, (2.7.56) one has uniformly w i t h r e s p e c t t o h E (O,hol, V ho (2.7.68)
' 0:
'
H(sl,m),h(RE,+ ) + H(sl,m),h(I( ,) v s1 E R ,
:
H(sl,m)( 2 : )
m '
and (2.7.69) lm
-f
H
(sl
,m)
(Z"),
v
s1
E R.
Theorem 2.7.28.
The one parameter f a m i l y of mappings Rh' (2.7.70)
H
(sl
,m),h
(R;")
3 u(h,x',xn)
%
i s continuous uniformly w i t h r e s p e c t t o h E (O,ho], i . e . t h e norm of Rh
i s bounded by a c o n s t a n t which may depend only on m and ho,
V h
E !O,hol.
Moreover, t h e map R h i s s u r j e c t i v e , and t h e r e e x i s t s a continuous ( u n i f o r m l y w i t h r e s p e c t t o h E (O,hO1) f a m i l y of l i f t i n g o p e r a t o r s Lh,
Proof. ____ The f i r s t part of the statement above is an immediate consequence of Theorems 2.7.28 and 2.7.17. The same Theorems guarantee the surjectivity of
2.7. Spaces of O n e Parameter Families of Meshfundions
L
i n d i v i d u a l map
h,j
with D j hu(h,x',O) = a ( h , x ' ) . x , I t n r e m a i n s t o show t h e s u r j e c t i v i t y o f
<= j
( 2 . 7 . 7 0 ) and t o c o n s t r u c t a
Lh a s i n ( 2 . 7 . 7 1 ) . Thus, l e t a . ( h , x ' ) E ff
l i f t i n g operator be g i v e n , 0
161
(sl,m-j-?) ,h(m:-l) I m. L e t u s c o n s t r u c t a f u n c t i o n u ( h , x ' , x n ) t H ( s l , m ) , h (iRn h , +1
<
s u c h t h a t D 1 h u ( h , x ' , O ) = a . ( h , x ' ) , 0 5 j < m. W e s h a l l f i n d f i r s t 1 Xnf
,
f u n c t i o n s U ( h , x ' , x ) C ff (<,+) k ( s l ,m) ,h conditions: (2.7.72)
D1
U
(h,x',O)
=
6.a.(h,x'),
xn,h k
ik 1
0 5 k < m, which s a t i s f y t h e
0 i k , j < m.
Then it i s e a s y t o p r o d u c e a f u n c t i o n u ( h , x ' , x ) a s r e q u i r e d , which i s j u s t
u
C U (h,x',xn). k 06kSm 1 Again, a s i n t h e p r o o f o f Theorem 2 . 7 . 1 7 , p u t X = q '( h , h E ' ) > 2 and m 1 d e n o t e by p ( t ) a C - f u n c t i o n on R t , which s a t i s f i e s t h e c o n d i t i o n s : =
Denote x
=
t and i n t r o d u c e
vk ( h , S ' , t )
=
X -k-h ak
-h where ak = F x , + S , , h a k and C r k ,
C C r k ( h , 5 ' ) pr t / h ( h X ) , l5rSm
1 S r 5 m,
0 5 k < m ,
0 5 k < m, a r e t o be chosen t o
satisfy (2.7.72). One g e t s f o r Crk t h e f o l l o w i n g s y s t e m o f e q u a t i o n s : (2.7.74)
C
((pr(hh)-l)/(ihX))'Crk(h,S')
=
Sjk,
0 5 j , k < m,
l
Denote jlr
=
jlr(hh)
=
( i h A ) - ' ( p r ( h A ) - l ) . F o r e a c h k we have i n ( 2 . 7 . 7 4 ) a
subsystem of l i n e a r e q u a t i o n s f o r Crk, determinant
1 5 r 5 m , w i t h Vandermonde
2. Sobolev Spaces of Vectorial Order
162
which d o e s n o t depend on k .
S i n c e A(hh) =
TL lSp
) , the question p g I of a bound f o r A ( h h ) r e d u c e s t o a n e s t i m a t e o f t h e d i f f e r e n c e s i(, -$ q P q > p . S e t t i n g t = hA a n d n o t i n g t h a t 0 5 t 5 C < m, V h E [ O , h O ] ,
V 5'
E
TnT1 , u s i n g t h e f o r m u l a 5 rh
hq-i(,pl= I t - l ( l - p ( t ) ) l J p ( t ) l n ;
c
P 1( t ) !
O
(2.7.75)
0 < C1 5 / $ q ( h h ) - ~ p ( h A ) 5j C2 <
m,
V h,S'.
The l a t t e r i m p l i e s t h a t A ( h h ) a l s o s a t i s f i e s t h e i n e q u a l i t i e s > 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 ~ ' ( 0<)0 , p ( 0 ) = 1 and 0 5 hh 5 C <
m,
V h,S'.
Summation i n j from 0 t o m i n ( 2 . 7 . 7 6 ) and i n t e g r a t i o n o v e r T lead t o the inequalities:
n- 1 5',h
2.7. Spaces of One Parametcr Families of Meshfunctions
1 1 . 1 1 ;m-k-i)
where w e have d e n o t e d by
M u l t i p l y i n g ( 2 . 7 . 7 7 ) by h-'1, (2.7.78)
I lUk/ I (s,,m)
,h
'
, h t h e norm i n
H (0,m-k-t)
163 n-1 ,h(%
) '
one g e t s t h a t
;sl,m-k-f) ,h'
lakl
O S k < r n ,
X Uk, s i n c e t h e c o n s t a n t s C i n ( 2 . 7 . 7 7 1 , 05kcm I ( 2 . 7 . 7 8 ) do n o t depend o n h and a k , 0 5 k < m.
which p r o v e s ( 2 . 7 . 7 1 ) w i t h u
=
Remark 2.7.29. As
a c o r o l l a r y of Theorems 2 . 7 . 2 7 and 2 . 7 . 2 8 a n d o f t h e f a c t t h a t t h e
r e s t r i c t i o n o f a meshfunction i n in
H (sl,m)
(I$+)
,
H (sl,m)
t o Rt,+ i s a m e s h f u n c t i o n
(<)
w e o b t a i n the s u r j e c t i v i t y o f map ( 2 . 7 . 3 3 ) a n d t h e e x i s t e n c e
1 ) bounded l i f t i n g o p e r a t o r from O n (sl,m),h(%) f o r each i n t e g r a l m 2 0.
of a (uniformly w i t h r e s p e c t t o h C ( 0 , h n O - ; j < m H ( s l , m - j - t ) , h ( %n-1)
into
H
Now s e v e r a l e q u i v a l e n t d e f i n i t i o n s of norms i n s p a c e s H (s), h ( < , + ) w i l l b e g i v e n , a l l o f them b e i n g e q u i v a l e n t w i t h t h e norms i n
H
,m),h(<,+)
g i v e n p r e v i o u s l y , when s 2
m > 0 is an i n t e g e r . Further-
=
(sl more, a n e x t e n s i o n o p e r a t o r 1 : H ( ~ , )h ( < , + ) + H ( ~ , )h ( I?;) w i l l b e s2 c o n s t r u c t e d and t h e t r a c e t h e o r e m on h y p e r p l a n e s o f c o d i m e n s i o n o n e w i l l b e p r o v e d f o r e a c h s2 1 0 . We s t a r t w i t h t h e d e f i n i t i o n which u s e s ( 2 . 7 . 4 7 ) . Namely, l e t s 2 = m+€I w i t h m 2 0 a n i n t e g e r and 8
C (C,l).
Define
and
The s p a c e s o f m e s h f u n c t i o n s and t h e norms [ u ] + (S)
f o r which t h e norms [ u ] + V h E (O,hol (s), h ' H (s),h(<,+) and H ( s ) (zl),
a r e f i n i t e , a r e d e n o t e d by
r e s p e c t i v e l y . W e s h a l l c o n s t r u c t f o r each s2 2 0 an e x t e n s i o n o p e r a t o r 1
s2
which w i l l b e u n i f o r m l y bounded i n h E ( 0 , h
0
1.
,
2. Sobolev Spaces of Vectorial Order
164 For s2
m t 0 i n t e g e r it h a s y e t b e e n done.
=
W e s t a r t with t h e case 0 <
S2
?.
<
Theorem 2 . 7 . 3 0 .
Let u E H ( s ) , h ( ~ : , + )
and l e t 0
+ . Then
< s2 <
for t h e e x t e n s i o n operator
uniformly w i t h r e s p e c t t o h E ( O , h o l . Proof. -
Denote v ( x ) = l 0 u ( x ) , ~ € 4 L. e t { e j } l s i s n b e t h e s t a n d a r d b a s i s i n IRn. One h a s t o show (2.7.82)
-2s
- (2s2+1)
1
h
1
O < t .ER I xirh -2s 5 Ch
h
tj -
1 c o
1 XE<
t. 1
(2s2+1) h
2 n Iv(x+t.e.)-v(x)Ih 6 1 1
Z
2 n lu(x+t.e.)-u(x)I h , 15 j s n , 1 1
,h
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 E (O,ho] and u . For 1 5 j < n one h a s e q u a l i t i e s i n ( 2 . 7 . 8 2 ) w i t h C
=
1. Therefore,
f o r showing ( 2 . 7 . 8 2 ) w i t h j = n 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 o f a s i n g l e v a r i a b l e x n , which w i l l b e d e n o t e d by x . Also, i t s u f f i c e s t o c o n s i d e r t h e case s1 = 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 o f s 2 . Thus, one h a s t o show t h a t w i t h some c o n s t a n t C > 0 h o l d s (2.7.83)
t-(2s+l)
Z t ER
h,+
h
I v ( x + t ) - v ( x ) I 2 h 5 C N~ , h ( ~ ) '
Z xER
h
where w e h a v e d e n o t e d
Given t h e d e f i n i t i o n of l o , t h e l a t t e r r e d u c e s t o t h e p r o o f o f t h e inequality
o r , what i s t h e s a m e , t o p r o v e t h e estimate:
2.7. Spaces of One Parameter Families of Meshfunctions
(2.7.84)
165
Z t-(2s+1)h Z lu(x)I2h 5 C N (u). s,h t>G O<x
Changing o r d e r of summation i n ( 2 . 7 . 8 4 ) . w e c a n r e w r i t e i t i n a n e q u i v a l e n t form
:
where w e h a v e d e n o t e d
O b v i o u s l y , one h a s w i t h some c o n s t a n t C: c-l
s
-1
-2s -1 -2s (x+h) 5 P ~ , ~ ( X6 ) CS (x+hl
,V x
2 0 , V s > 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(<,+) u
, with
0 < s < f . Then
( ~ ~ + h I - ~ u
j.
i s a f a m i l y of ( u n i f o r m l y w i t h r e s p e c t to h E (G,ho]) continuous l i n e a r mappings f r o m
H (O,s),h( Rhn, + )
into H(O,O),h(<,+).
Proof o f Lemma 2 . 7 . 3 1 . S i n c e t h e v a r i a b l e s ( x l , ...,x ) p l a y t h e r o l e o f p a r a m e t e r s ( o n e can u s e n- 1 d i s c r e t e F o u r i e r t r a n s f o r m i n x l , ...,x n - l ) , one i s l e f t p r e c i s e l y w i t h proving (2.7.87) with x
=
x.
Introduce (2.7.88)
$(x)
=
-1 (x+h)
Z 05y5x
One h a s
Indeed, put
( u ( x ) - u ( y ) ) h ,$ ( x )
=
1 y-'@(y)h, ydx+h
x L 0.
2 . Sobolev Spaces of Vectorial Order
166
Z
Q(x) =
( u ( x ) - u ( y ) ) h= ( x + h ) $ ( x ) .
ocysx
*
a Y # h and a y t h
The summation by p a r t s f o r m u l a ( w i t h
t h e f o r w a r d and back-
ward d i f f e r e n c e d e r i v a t i v e s , r e s p e c t i v e l y )
gives for $(x) -1 -1 ( y + h ) Q ( y ) h= - C Q(Y)ay,hY h = y>x+h -1 * = (x+h)-lO(x)+ C y-’a* O ( y ) h = $ J ( x ) +1 Y ay,h@(Y)h. yhx+h Y .h ytx+h +(x)
=
Z
y y>x+h
-1
B u t , it i s e a s i l y s e e n t h a t a;,ho(y)
I:
=
05zSy
a
*
Q(y) Y #h
*
=
~ a ~ , ~ u (I n y d)e e. d ,
(u(y)-u(z))- c (u(y-h)-u(z)) 0 5 zSY- h
=
*
z
(u(y)-u(y-h)) 1 = yay,h~(~). Osz5y-h
=
Thus, t h e i d e n t i t y ( 2 . 7 . 8 9 ) i s p r o v e d . Usinq t h e d e f i n i t i o n of $ ( X I , o n e f i n d s a p p l y i n g t h e Cauchy-Schwarz inequality: /$(x)
i2
-1 5 (x+h)
l u ( x ) - u ( y ) I2h,
C
05y5x and c o n s e q u e n t l y
-2s
C (x+h)
(2.7.90)
2 I$(x)I h 5
xto C
=
y>o
h
C
-(2s+l)
(x+h)
-(2s+l)
IU(X)-U(Y)~
Z
h
I: ( x + h ) x>o
IU(X)-U(Y)/
2
h =
O
h =
xty 2
I: h Z ( t + y + h ) - ( 2 s + 1 )l u ( y + t ) - u ( y ) 1 h 5
=
y>o
t>O
I t should be s t r e s s e d t h a t (2.7.90) holds f o r V s.
For $ ( x ) w e s h a l l now p r o v e t h e estimate: (2.7.91)
2
I: ( x + h ) - 2 s 1 $ ( x ) 1 2 h 5 C 1 ( x + h ) - 2 s l $ ( x ) I h . xzo
x>o
Denoting G ( x + h ) = ( x + h ) - ’ $ ( x ) , x 2 0 and g ( x ) = x-’@(x), x > 0 , g ( 0 ) = 0 ,
2.7. Spaces of One Parameter Families of Meshfunctions we c a n r e w r i t e
167
( 2 . 7 . 9 1 ) i n t h e e q u i v a l e n t form:
One c h e c k s e a s i l y t h a t
I n d e e d , one h a s j u s t t o u s e t h e d e f i n i t i o n o f G(x), g(x) and ( 2 . 7 . 8 8 ) f o r $J
(x-h)
.
Therefore, putting x
=
nh, y = k h , G(x) = G
and g ( y )
=
g k , one f i n d s
W e s h a l l d e n o t e by n ( y ) t h e s t e p f u n c t i o n d e f i n e d by t h e f o r m u l a
a(y) =
l g k l , f o r k 5 y < k t l } and by p m
(2.7.92)
p
1
(x) =
1 (li)'
a(y)
X
*, Y
1
(x) the function
x > 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 <
as well.
1.
s
Lemma 2 . 7 . 3 2 .
Let (2.7.943
with
( < I + )
H(O,s),h
-s + u /l ( 0 ) , h
1 1 (Xn+h)
where t h e c o n s t a n t c
s,ho
s
E ( + , I ) and l e t u(0)
’
Cc.,hol
l‘l
j;O,s)
,h‘
=
0.
Then hoZds:
V h E (O.hol,
depends only on its s u b s c r i p t s .
P r o_ of. _ Again, 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 , so t h a t one h a s t o show
With o ( x ) , $(x) d e f i n e d by ( 2 . 7 . 8 8 ) fact that u(0) (2.7.96)
$(O)
one f i n d s , u s i n g ( 2 . 7 . 8 9 )
and t h e
0, Q(0)= 0:
=
def =
-1
Z
y
Q(y)h,
V = 0.
y?x+h Hence, i n t r o d u c i n g
(2.7.96) y i e l d s : $ l ( x ) (2.7.97)
=
- $ ( x ) , so t h a t (2.7.89)
u ( x ) = Q ( X ) + $ ~ ( X )V, x C
A s it h a s b e e n s t r e s s e d i n t h e p r o o f
c a n be r e w r i t t e n a s f o l l o b s :
mi. of Lemma 2 . 7 . 3 1 ,
the inequality
2.7. Spaces of One Parameter Families of Meshfundions
169
( 2 . 7 . 9 0 ) h o l d s f o r V s . T h e r e f o r e , it s u f f i c e s t o p r o v e t h a t
(2.7.99)
g(x)
x-'@(x), G(x) = x
=
-1
C
g(y)h,
x > 0, g(0) = G ( 0 ) = 0,
o
i n a n e q u i v a l e n t way:
S i n c e 0 < 1-s < f , t h e p r o o f o f
( 2 . 7 . 1 0 0 ) i s t h e same a s t h a t o f
(2.7.91)
I Remark 2.7.33. 0
Denoting by
I{
(<'+) t h e s u b s p a c e of a l l m e s h f u n c t i o n s ( s l r s 2 ), h , f < s 2 < 1 , f o r which h o l d s : u ( 0 ) = 0 , one h a s f o r
(qt+) H(s1,s2)
-S
2:
t h e m u l t i p l i c a t i o n o p e r a t o r (x +h)
uniformly with r e s p e c t t o h E ( 0 , h
0
3,
a n d , a s a consequence o f
(2.7.101),
one h a s a l s o --s
(2.7.102)
where
H
:.i
(x +h)
(S1'S2)
n
v
s2 E ( f , l ) ,
i s t h e subspace o f a l l meshfunctions i n H
(Zn'+) ( s l ,s2)
which v a n i s h f o r x
(Z"'+),
(Z"'+), (Sl'S2)
=
0
Theorem 2.7.34.
The e x t e n s i o n operator lo d e f i n e d by ( 2 . 7 . 8 1 ) is a f a m i l y o f uniformly w i t h r e s p e c t to h E ( 0 , h0 1 continuous Linear mappings from H
into H
(s1rS2)
rh
(R:)
s ) ,h(<'+)
(sl' 2 f o r each s2 E ( + , I ] . Moreover, f o r each s 2 E ( f , 1 3
ho l d s :
Proof. For s2
=
1 t h e a s s e r t i o n s above a r e o b v i o u s . T h e i r p r o o f f o r s2 E ( f , l ) i s
2. Sobolev Spaces of Vectorial Order
170
e x a c t l y t h e same a s t h a t o f Theorem 2.7.30 w i t h t h e o n l y d i f f e r e n c e t h a t
I
i n s t e a d o f Lemma 2.7.31 one h a s t o u s e Lemma 2 . 7 . 3 2 . For s2 E R
d e n o t e by m t h e c l o s e s t i n t e g e r and l e t 8 = s 2 - m .
d e n o t e by ff' ( *
,s 1
,
) ,h(<'+)
2
u E H ( s 1 , s 2 ) , h ( q i n , +)
u E H
( s l,s2)
(Zn'+)
s2 2 0 , a l l t h e m e s h f u n c t i o n s
such t h a t u ( x ' , x ) = 0 f o r x
H
R e s p e c t i v e l y , w e d e n o t e by
We shall
(Z"") ( s l ,s2)
such t h a t u ( x ' , x
t h e s u b s p a c e s o f all m e s h f u n c t i o n s
= 0, x
)
= k h , 0 5 k 5 m.
=
k h , 0 5 k 5 m.
Theorem 2 . 7 . 3 5 .
The e x t e n s i o n operator lo d e f i n e d by ( 2 . 7 . 8 1 ) ( e x t e n s i o n by z e r o for x n
< 0)
i s ( u n i f o r m l y w i t h r e s p e c t t o h E ( O , h O ] ) a f a m i l y of continuous l i n e a r (%) f o r aZl s 2 z 0 such n g + ) into H mappings from H ( s l , s 2 ), h ( %
( s l , s 2 ), h
t h a t s2-f i s n o t an i n t e g e r . Moreover, one has under t h e same c o n d i t i o n on s2 t 0:
Proof. ~
Again, 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 For 0 5 s2 < 1 , s 2 - f (see Theorems 2 . 7 . 3 0 , 2.7.35
1.
# 0 , t h e a s s e r t i o n s above have a l r e a d y b e e n proved 2.7.34).
'
Further, for inteyer s
1
i s q u i t e o b v i o u s , s i n c e 1110 u ] ( s l , m ) , h
Now l e t m < s 2 < m + l , 2.7.34
=
=
=
m
0 Theorem
2 +
'uII
( s l , m ) ,h'
s2 # m+f w i t h i n t e g e r m Z 1. Applying Theorems 2.7.30,
t o va - D x , h ~ , la1 = m, one f i n d s :
(2.7'105)
['ova3 ( s l , s 2 - m )
,h 5 c [ v CY I +( s l , s 2 - m )
,h'
where t h e c o n s t a n t C d o e s n o t depend on h and v a r e d e f i n e d by ( 2 . 7 . 3 9 1 ,
T h e r e f o r e , Theorem 2 . 7 . 2 0 , (2.7.105) y i e l d :
V h E
,
(O,hol,
where [ . 1
( s ), h ' [ - ] ; s ) , h
(2.7.79).
d e f i n i t i o n s (2.7.39), ( 2 . 7 . 4 7 ) and i n e q u a l i t y
2.7. Spaces of One Parameter Families of Meshfunctions
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 , so t h a t ( 2 . 7 . 1 0 4 ) immediate consequence o f
171
i s an
I
(2.7.106).
Now we p a s s t o t h e c o n s t r u c t i o n of a n e x t e n s i o n o p e r a t o r from
H(s),h(
into
H(s),h(<)
f o r any s = ( s1 , s 2 ) w i t h s2 2 0 .
For i n t e g e r s2 2 0 it h a s a l r e a d y b e e n done (see Theorem 2 . 7 . 2 6 ) .
we b e g i n w i t h t h e f o l l o w i n g Lemma 2 . 7 . 3 6 .
,
L e t u and v belong t o H ( s ) , h ( % n r + ) s2
+ +. I n
addition, l e t u(x',O)
(2.7.107) w ( x ' , x n )
=
=
s
=
f f ( x n ) u ( x ' , x n ) + [I-H(x
beZongs t o H ( s ) , h ( ~ : )
,
o
( s l , s 2 )with
< s2 < 1 and Let
v ( x ' , O ) . Then t h e meshfunction
)Iv(x',-x
)
and, moreover,
where t h e c o n s t a n t c does n o t depend on u , v and h . 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 o n l y t h e c a s e n
w r i t e x instead of x
and
By v i r t u e o f Theorem 2.7.20
~ t h e norms [ w ] ~ ,and
t h e a s s e r t i o n o f Lemma 2 . 7 . 3 6 , d ef
(wI
(2.7.40) b e i n g v a l i d , i n o r d e r t o prove
C
2 Iw(x+t)-w(x)l h 5
xE Wh
t>O
5 c[N(u)+N(v)],
where w e have d e n o t e d (2.7.110) N(u)
=
C t
-(2s+l)
t>O
Rewrite M ( w ) u s i n g (2.7.107)
h
s,h being equivalent
it s u f f i c e s t o show t h a t
t-(2s+l) h
=
1 , s1 = 0 so t h a t we
i n s t e a d of [ - 1 ~ o , s 2 , , h r[ . 1 ( 0 , s 2 ) , h .
uniformly with respect t o h , i . e .
(2.7.109) M ( w )
=
2
C lu(x+h)-u(x)1 h. x>o
i n t h e following fashion:
2. Sobolev Spaces of Vectorial Order
172
Therefore, for showing (2.7.109) with N(u) defined by (2.7.110), it suffices to show that t-(2s+l)h C ]u(t-x)-v(x)1 2h 5 C[N(u)+N(v)l t>G O<x
Z t-(2s+1)h
(2.7.111)
t>O +
C
I
u(t-x)-u(x)
I 2h i
O<x
2 t-(2s+l)h I: /u(x)-v(x)(h 2 C[N(u)+N(v)l. t>O O<x
Changing the order of summation in the second sum in the left hand side of (2.7.111) (just as in the proof of Theorem 2.7.30), we come to the conclusion that for proving (2.7.111) it suffices to show that (2.7.112) C (x+h)-2slu(x)-v(x)1 2h S C[N(u)+N(v)l. x>o Noticing that u(x)-v(x) = f (x) E H ( o , s ) ,h(IF(r')
,
f (0) = 0 (by virtue of
the condition u ( 0 ) = v(O)), and applying Lemmas 2.7.31, 2.7.32, we immediately obtain that
1
C (x+h)-2S(u(x)-v(x) 2h 5 CN(u-v) 5 C[N(U)+N(V)l.
x>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<x
But obviously, (lu(2t-x)-u(t-x)/2+/u(2t-x)-~(x) \ 2)h = (2.7.113) Q(u) S 2 1 t-(2s+1)h Z t>O O<x
2.7. Spaces of One Parameter Families of Meshfunctions
where P(u) s t a n d s f o r t h e l a s t sum on t h e r i g h t hand s i d e o f
173
(2.7.113).
Making t h e change t + x + t , we o b t a i n t h e f o l l o w i n g e x p r e s s i o n f o r P (u)
P(u) =
-(2s+l)
I: h C ( t + x ) x>o 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~
<
.
r+
Assume, i n a d d i t i o n , t h a t w i t h scme c o n s t a n t
v
(2.7.120) t ( T , y ) 2 C O ( T + Y ) ,
Then fcr any meshfunction f
:
(T,Y)
E
Co > 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<x
t>O
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
Using i d e n t i t y ( 2 . 7 . 1 2 8 ) w i t h j = m-1, on t h e r i g h t hand s i d e o f
Z
(2.7.129) ( - l ) m C C lsp<m The t r a n s f o r m a t i o n
T
w e can r e w r i t e t h e l a s t sum
( 2 . 7 . 1 2 7 ) i n t h e form:
ff - (x)r um-
( x + p t )-u
m- 1 ( 2) 1
aEQm- 1 ,p =
1
I I
x=pz- (m- 1 ) ph+ a h
pt+pz-(m-l)ph+/alh, y = z takes the Set
2 c Moreover, it i s e a s i l y s e e n t h a t t ( T , y ) I- C o ( ~ + y ) , o n t o some D' h,a,p w i t h some c o n s t a n t Co > 0 , which may depend o n l y on m. b' ( T , Y ) E Di,a tP Applying a g a i n L e m m a 2 . 7 . 3 9 , w e o b t a i n t h e r e q u i r e d e s t i m a t e f o r t h a t sum
%,+.
( 2 . 7 . 1 2 9 ) , which t o g e t h e r w i t h ( 2 . 7 . 1 2 4 ) ,
(2.7.126) g i v e s t h e estimate
( 2 . 7 . 1 1 8 ) , and t h a t e n d s t h e p r o o f o f Theorem 2 . 7 . 3 8 .
1
2. Sobolev Spaces of Vectorial Order
178 Remark 2.7.40. Let s
2
=
m+B w i t h i n t e g e r rn t 0 and 0 < 0 < 1. A s a consequence of
Theorem 2.7.20, one h a s u n i f o r m l y w i t h r e s p e c t t o h
E ( 0 , h01 t h e
e q u i v a l e n c y of norms
where, a s p r e v i o u s l y d e f i n e d ,
W e s h a l l d e n o t e by
H(s)
,h(IF()
to
TI+
IF(".
t h e r e s t r i c t i o n o p e r a t o r of meshfunctions i n Since, obviously,
1 1 '+" 1
+ (s
l , O ) ,h
' I Iv'
(s1,0) ,h'
t h e e q u i v a l e n c y (2.7.131) y i e l d s
where t h e c o n s t a n t C d o e s n o t depend on u and h . I n o t h e r words one h a s u n i f o r m l y w i t h r e s p e c t t o h :
and c o n s e q u e n t l y ,
'+
:
(Z")
H(s1,s2)
+
fi
(sl, s 2 )
$ I + ) ,
v (Sl,S2)E
R
x
z,
C o r o l l a r y 2.7.41.
The spaces
H
(sl,s2) ,h
(R:'+)
and H
(z""),
(S1'S2)
s2 L 0 , c o i n c i d e
( a l g e b r a i c a l l y ) with t h e spaces of r e s t r i c t i o n s to
from H(sl,s
)
2
(R:)
and H
(Sl'S2)
< I +
of meshfunctions
( z n ), r e s p e c t i v e l y .
I n d e e d , i t immediately f o l l o w s from Theorem 2.7.38 and Remark 2.7.40 t h a t each meshfunction u i n
H (s),h(IF('+) o r H
(2"")
can be r e p r e s e n t e d
(S)
( R t ) and H (Z"), 1 u with 1 u i n H + m m ( s ), h (S) W e s h a l l i n t r o d u c e a n o t h e r d e f i n i t i o n of norm i n H
i n t h e form u
as follows:
= ~i
respectively. n,+ (sl,s2) ,h(% ) '
2.7. Spaces of One Parameter families of Meshfunctions
179
w h e r e 1 i s a n y e x t e n s i o n operator,
F u r t h e r , w e d e f i n e a new norm i n ti
(zn'+)
a s follows:
(Sl'S2)
Theorem 2 . 7 . 4 2 .
Norms
( 2 . 7 . 1 3 2 ) and ( 2 . 7 . 7 9 )
E
1, s o t h a t norms 0
h
(0,h
are equivaZent uniformZy w i t h r e s p e c t t o
(2.7.133)
and ( 2 . 7 . 8 0 ) are e q u i v a l e n t , V
s1
E R,
Proof. U s i n g t h e e x t e n s i o n operator 1 f r o m Theorem 2 . 7 . 3 8 , m that
one a t once o b t a i n s
+ ( s l r s 2 ,)h ' w h e r e t h e c o n s t a n t C does n o t d e p e n d o n h a n d u , so t h a t
Since
ti
(Z"")
p r o v i d e d w i t h norm ( 2 . 7 . 1 3 3 )
i s a Banach space ( i t i s l e f t
(S)
t o t h e r e a d e r t o show i t ) , ( 2 . 7 . 1 3 5 ) e q u i v a l e n c y o f norms (2.7.133) (2.7.134),
and Banach's theorem y i e l d t h e
and (2.7.80).
shows t h e e q u i v a l e n c y o f
N o w t h e s t a t e m e n t s o f Theorem 2 . 7 . 2 8
H
(<'+)
w i t h a n y s2 >
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
<-'.
Thus, l e t
5 j < s-+,
b e g i v e n . W e have t o e x h i b i t a meshfunction u ( x ' , x n )
E Hs(<'+)
such t h a t
and s u c h t h a t
where t h e c o n s t a n t C d o e s n o t depend on u, a . and h . 7 We a r e g o i n g t o u s e t h e m e s h f u n c t i o n s U ( x ' , x ) , 0 5 k < s - ? , i n t r o d u c e d k i n t h e p r o o f of Theorem 2.7.28, and w e w i l l o n l y have t o show t h a t
where t h e c o n s t a n t C d o e s n o t depend on ak and h , s i n c e , a c c o r d i n g t o t h e d e f i n i t i o n of U .
I'
0 5 j < s-t,
i n t h e p r o o f o f Theorem 2.7.28, one c a n
d e f i n e a m e s h f u n c t i o n u as r e q u i r e d i n Theorem 2 . 7 . 4 3 , i n t h e f o l l o w i n g fashion
L e t s = m+B
w i t h i n t e g e r m t 0 and B
so t h a t it s u f f i c e s t o g e t t h e e s t i m a t e s
E
( 0 , l ) . Then
2.7. Spaces of One Parameter Families of Meshfunctions
181
and
w i t h some c o n s t a n t C , which d o e s n o t depend on h a n d a . k The e s t i m a t e ( 2 . 7 . 1 3 8 ) h a s a l r e a d y b e e n p r o v e d ( s e e ( 2 . 7 . 7 8 ) o f Theorem
i n t h e proof
2.7.28).
We s h a l l v e r i f y ( 2 . 7 . 1 3 9 ) . F i r s t c o n s i d e r t h e e x p r e s s i o n
u 1 ( S ' , x n ) , one and i n t r o d u c i n g V ( S ' , x ) = (F k n x'+fS',h k g e t s u s i n g P a r c e v a l ' s i d e n t i t y , t h e f o l l o w i n g e x p r e s s i o n f o r N (U ) : 1 k Setting a
=
(a',a
)
A s h a s b e e n shown i n t h e p r o o f of Theorem 2.7.20
(see (2.7.41)), the
following i n e q u a l i t i e s hold:
where C e depends o n l y on
e
and, a s u s u a l , t . = r h w i t h r = 1 , 2 , 3
Hence, u s i n g a g a i n t h e i n e q u a l i t y 2 / 5 . 1 / 7 1 6 l C . ( C , ) l ,
1
and r e c a l l i n g t h a t s
m+B,
=
I
1
... .
V hC.E[-n,nI,
I
one g e t s u s i n g t h e l a t t e r e x p r e s s i o n f o r N (U 1 , 1 k
t h e f o l l o w i n g estimate:
where, a s p r e v i o u s l y , w e have d e n o t e d 2
X (h,h5')
=
1 +
C
ICj(h,h5.)I
15j < n Using a g a i n t h e i n e q u a l i t i e s ( 2 . 7 . 7 6 )
I
2
.
i n t h e p r o o f o f Theorem 2 . 7 . 2 8 ,
2. Sobolev Spaces of Vectorial Order
182 a
a p p l i e d t o D nVk (E,',xn), and a l s o t h e P a r c e v a l i d e n t i t y , one g e t s f o r n N (U ) t h e estimate: 1 k (2.7.140)
N
1
(U k
1'
5 CI lakl s - k - f , h '
w i t h a c o n s t a n t C which d o e s n o t depend on a k and h . Now, l e t u s c o n s i d e r
U = F x'-tc',h k' Using t h e f o r m u l a f o r Vk (see t h e p r o o f o f Theorem 2 . 7 . 2 8 ) ,
where, a s p r e v i o u s l y , Vk
we easily
find that
a (2.7.141)
D
x , h
=
( V ( 5 ' , x +t ) - V ( 5 ' , x n ) ) = k n n k
a -k - h X ak(S')
r t /h
CL
C
Crk(hX)$rn(p(hh)
rxn/h -1)p
(hi).
lSr<s+f
W e prove t h e i n e q u a l i t y :
where C e depends o n l y on 8. The l a s t i n e q u a l i t y i s e q u i v a l e n t t o t h e f o l l o w i n g o n e :
where D ( t )
has t h e p r o p e r t i e s (2.7.73).
b s t i m a t i n g q ( t ) i n p r e c i s e l y t h e same way p r o o f o f Theorem 2.7.20
as the function g ( a ) i n the
( s e e ( 2 . 7 . 4 1 ) ) , one g e t s ( 2 . 7 . 1 4 3 ) .
I n d e e d , i n t h e p r o o f o f ( 2 . 7 . 4 1 ) f o r g ( a ) o n e u s e s o n l y two p r o p e r t i e s of 2 2 2 2 2 t h e f u n c t i o n s i n ka : s i n ka $C and s i n ka 5 Ck a 'rhe f u n c t i o n
.
I p ( t )rk-l I
h a s t h e v e r y same p r o p e r t i e s
Ip(t)rk-l12
6 Ck2t2, since p ( 0 )
=
:I p ( t )rk-l l 2
1. Using ( 2 . 7 . 1 4 1 ) ,
i n t h e same way as i n ( 2 . 7 . 7 6 ) , t h a t
5 C and ( 2 . 7 . 1 4 2 ) . we f i n d ,
2.7. Spaces of One Parameter Families of Meshfunctions
183
The l a s t i n e q u a l i t y a l o n g w i t h P a r c e v a l ' s i d e n t i t y y i e l d s t h e r e q u i r e d e s t i m a t e f o r N (U 1 , 2 k (2.7.145)
N (U ) 5 C / l a 2 k k
Combining (2.7.140)
1 I s-k-f,h-
and ( 2 . 7 . 1 4 5 ) , o n e f i n d s t h a t
w i t h some c o n s t a n t C which d o e s n o t depend o n a k and h. Finally, (2.7.146),
(2.7.138)
( e s t a b l i s h e d i n t h e p r o o f o f Theorem 2.7.28)
and
along w i t h t h e formula
I
prove (2.7.137). Remark 2.7.44.
Theorem 2 . 7 . 3 8 and 2 . 7 . 4 3 a l l o w o n e t o a s s e r t t h e s u r j e c t i v i t y o f t h e
t r a c e mapping (2.7.147)
u(x',xn)
+
ID1
X
I
hu(x',O)}osj<s
which i s a c o n t i n u o u s l i n e a r map from
n
(Zn-').
05j<s - + " ( s l f s 2 - j - * ) 2 of l i f t i n g o p e r a t o r s
R
-+,
2
They a l s o
H
) (2") o n t o
ee t h e e x i s t e n c e o f a f a m i l y
h'
which 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 h , s o t h a t o n e a l s o h a s
Obviously,
Rh
i s n o t u n i q u e l y d e f i n e d , s i n c e t h e map (2.7.147)
n o n - t r i v i a l k e r n e l , which c o n t a i n s m e s h f u n c t i o n s u
for
X
=
kh, 0 5 k 5 m .
E H (s
) (2")
has a
vanishing
1' 2
Now w e s h a l l e x t e n d t h e r e s u l t s c o n c e r n i n g t h e t r a c e mapping (2.7.147)
2. Sobolev Spaces of Vectorial Order
184
to the case, when j > 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
uI 1 ' 6 C < l r . h>l lul j (s),h ' 0 (s,,O) .h
(2.7.148)
where t h e c o n s t a n t
C
does n o t depend on u and h.
3ne has
one easily finds that
where the constant C
n,ho
depends only on its subscripts.
Integrating (2.7.149) over T (2.7.148)
.
1
n- 1 and using (2.7.150), one gets S',h
2.7. Spaces of One Parameter Families of Meshfunctions where t h e constant
C
185
does n o t depend on h and u.
Proof. With t h e n o t a t i o n i n t h e p r o o f o f Theorem 2 . 7 . 4 5 , Schwarz
u s i n g a g a i n t h e Cauchy-
i n e q u a l i t y , one f i n d s t h a t
The same argument, a s p r e v i o u s l y , y i e l d s 1-2s2
I1
-2s
<5>
h
1-2s
2d5n 6 C
h
I W n 1 (7
T Snrh
depends o n l y on s
where C
I
s2
2'
2 2(<<'>
2 -s
+\<,I
)
2d5n
=
I
The d e f i n i t i o n o f t h e norms [ . I ( s ),h F o u r i e r t r a n s f o r m and d e f i n i t i o n (2.7.132)
(see ( 2 . 7 . 7 9 ) ) , d o e s n o t u s e t h e
I I1.11 I
is o f t h e norms ( s ) .h n o t c o n s t r u c t i v e , which makes b o t h of ther:i n o t always handy i n a p p l i c a t i o n s .
W e a r e g o i n g t o i n t r o d u c e a n e q u i v a l e n t d e f i n i t i o n o f norms i n t h e s p a c e s
of m e s h f u n c t i o n s , which seems t o b e p a r t i c u l a r l y s u i t a b l e f o r t h e e l l i p t i c t h e o r y 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 .
.
W e s h a l l f i r s t c o n s i d e r meshfunctions u E H (R') (0), h h For e a c h meshfunction u ( t ) i n H ( 0 ) , h ( R k ) t h e d i s c r e t e < - F o u r i e r t r a n s f o r m -h
(5)
I
1 is a w e l l defined function i n L (T 1 , where T S , h = { < E C ll+ihCl = 1 ) 2
I
being, respectively, (2.7.154) u ( t )
=
(27)-'
and
t/hh (l+ihC) u (<)dg,
I T
< gh
2.Sobolev Spaces of Vectorial Order
186
where idrl
=
dc is the Lebesgue measure on Ti,h, <
=
(ih)-'(eih'-l)
being
the corresponding parametrization (angle coordinate) on T 5 ,h. With H(t) = 1 for t L 0, H(t) = 0 for t < 0, the Heaviside function 1 introduced above, each u E H (0),h(Rh) admits the orthogonal decomposition u(t) = u+(t)+u-(t), u+(t) = H(t)u(t) and u-(t)
=
(l-H(t))u(t)
1
where u (t) has its support in Rh,+ and u-(t) is supported by 1
=+{t E lRh , t < 01. R1 h,We denote by :r the corresponding projection operator h (2.7.156) (n+u)(t) = H(t)u(t). Thus, for each h the spaces H (2.7.157)
1 H (0),h(Rh)
=
'TO)
( 0 ) ,h
,h
--
1 decomposes into a direct sum 2 ,h '
'TO)
1 of the meshfunctions supported by R h,+ and lRh , - , respectively, the subspaces H+(0),h being orthogonal to each other with respect to the inner 1
product (2.7.1) in H
( 0 ) ,h* The proof of the following Paley-Wiener type result for the mesh-
functions (in fact, for the Fourier coefficients of periodic functions) is left to the reader, as an easy exercise. Proposition 2.7.47.
A meshfunction u belongs t o H+
iff i t s d i s c r e t e <-Fourier t ra n sf o rm
(0),h -h u (i) e x t e n d s anaZyticaZZy o n t o t h e e x t e r i o r o f T i . e . o n t o t h e domain 5 ,h' T+ < ,h = {iE C1 [ Il+ih 11, ? ( < I + 0 for 5 + m, V h and
(2.7.158)
J lzh(r)/21dgl5 C , V p 2 1, (l+ih
where t h e c o n s t a n t c does n o t depend on
p.
iff its discrete <-Fourier Analogously, a meshfunction v E H(0),h -h transform v ( 5 ) extends analytically onto the disk 1 T= {< E C I Il+iht/ < 1 1 and (2.7.158) holds for all p S 1 with a irh constant C which does not depend on p . With the orthogonal decomposition (2.7.157) one associates the corresponding orthogonal decomposition of the space L ( T ) of periodic 2 5th functions on T which are <-Fourier transforms of the meshfunctions <,h' in H(0),h(mh) ,
2.7. Spaces of One Parameter Families of Meshfunctions
187
( 2 . 7 . 1 5 9 ) L ~ ( T < , ~= )L ~ ( T ~ , L ~ ; )( T ~ , ~ ) . W e s h a l l d e n o t e by
Il;
+
t h e corresponding pr oj ect i on oper at or s onto
L+(T ) ,respectively. 2 C,h One v e r i f i e s e a s i l y t h a t f o r m e s h f u n c t i o n s u
E H
1
(0), h
(IRh)
with
compact s u p p o r t , t h e f o l l o w i n g f o r m u l a i s v a l i d
= ( f ) y h ( 5 ) + i / ( 2 * ) v.p.
where v . p .
-h u (u)(L-u)-'dli,
J T
V 5
E
Tr,h,
li , h
s t a n d s for t h e Cauchy p r i n c i p a l v a l u e o f t h e c o r r e s p o n d i n g
singular integral. A s it h a s been a l r e a d y shown (see Theorem 2 . 7 . 3 0 ) ,
t h e f a m i l y of
o p e r a t o r s n h a c t s c o n t i n u o u s l y ( u n i f o r m l y w i t h r e s p e c t t o h ) from 1 1 I s2 E ( o , t ) . H ( ~ ,s2) l , h ( I R h ) into ( s 1 , s 2 ), h ( R h , + ) P f o r s1
I t is
q u i t e o b v i o u s t h a t i t i s a l s o t r u e f o r s2 = 0 , so t h a t one h a s u n i f o r m l y w i t h r e s p e c t t o h: (2.7.161)
where
Jli
7(s), h ,
:
( s l ,s2),h
+
7'( s l,s2). h ' v
s1
E R,
v
s2
E ro,t,,
iss,
,h are t h e spaces of t h e d i s c r e t e 5-Fourier transforms 1 1 H ( R h ) and H (s), h ( I R h , + ) , r e s p e c t i v e l y . (s),hNh One f i n d s e a s i l y t h a t i f u (5) i s smooth, t h e n I li i s e x p r e s s e d by a
o f t h e meshfunctions i n
f o r m u l a , a n a l o g o u s t o ( 2 . 7 . 1 6 0 ) . Namely
Of c o u r s e , one h a s a l s o t h e f o l l o w i n g f o r m u l a e :
I t i s c l e a r t h a t e v e r y t h i n g a s s e r t e d above i s c a r r i e d o v e r verbatum
:,+,
t o m e s h f u n c t i o n s d e f i n e d on IR X'
=
( x 1 , ...,xn- 1) E IR;-'
since the tangential variables
p l a y t h e r o l e o f p a r a m e t e r s and one c a n p e r f o r m
t h e d i s c r e t e Fourier transform i n x ' . L e t u s now t u r n t o t h e d e f i n i t i o n o f a new norm i n
H(s)
(R:,+)
, with
2. Sobolev Spaces of Vectorial Order
188
s = (sl,s2),s 2 2 0. A s previously, let
Denote by A+(h,S') (respectively by A-(h,C')) the zero of the quadratic equation
A
2
+ (l+ihA)X2 = 0,
which satisfies the condition Il+ihA-l > 1).
Further, put
(2.7.164) 5 -
Cn-A-(h,C'),
=
1 l+ihA+1
< 1
(respectively the condition
5 , = (Cn-A+(h,C'))(l+ihcn)-'.
It is obvious, that for each real s2 the functions
are analytic and non-vanishing. (R:) , s2 2 0 , be H(sl,s2),h an extension of u onto R E . Theorems 2.7.26 and (2.7.38) guarantee the
let lU
For each u E H (sl ,s2)th
existence of such an extension. Let
Introduce (2.7.165) and
Notice, that norms (2.7.165), (2.7.166) do not depend on the extension operator 1. Indeed, if l . u , j = 1,2, are two different extensions of a meshfunction u E ff
since the function
( s ),h
cn
+
3 (RE")
,
then
s2 5 - (l1u-l2U)- l h
is analytic in T-
2.7. Spaces of One Parameter Families of Meshfundions
1 u - 1 u i s s u p p o r t e d by R 1 and 5'' 1 2 h,-
S i n c e t h e norm o f nh
+
. H(0),h(~
189
vs i s a n a l y t i c i n TLnrh' 2' 1 h ) + H ( 0 ) , h ( ~ h , + ) i s equal t o one,
1
t h e same c l a i m i s correct f o r I I ~: L ~ ( T < , , ~ + ) L+(T h ) . Using t h i s 2 5,. f a c t and ( 2 . 7 . 1 6 5 1 , o n e f i n d s a t o n c e t h a t
where t h e c o n s t a n t C d o e s n o t depend o n u and h . I t is 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 space
c o m p l e t e w i t h r e s p e c t t o norm ( 2 . 7 . 1 6 5 1 ,
H
( s l'S2)
( 2 . 7 . 1 6 6 ) . Moreover,
(z"'+)
is
(2.7.167)
yields:
Hence, by v i r t u e of B a n a c h ' s t h e o r e m , one h a s a l s o
where t h e c o n s t a n t C1 d o e s n o t depend on u . Thus, we have p r o v e d Theorem 2.7.48.
The norms ( 2 . 7 . 1 6 5 ) , ( 2 . 7 . 1 6 6 ) and ( 2 . 7 . 7 9 ) , ( 2 . 7 . 8 0 ) are e q u i v a l e n t . I n conclusion, using d e f i n i t i o n (2.7.165),
H(s)
(Z""),
( 2 . 7 . 1 6 6 ) of t h e norms i n
w e s h a l l p r o v e one u s e f u l f a c t ( l o c a l i t y p r o p e r t y ) c o n c e r n i n g
these spaces. Theorem 2.7.49.
Let 4 E
~ ( 2 ' )Then .
where t h e c o n s t a n t c S ( $ ) depends o n l y on 4 and s2. Proof. Without r e s t r i c t i o n o f g e n e r a l i t y , o n e can assume t h a t s1 = 0 . F u r t h e r , it s u f f i c e s t o v e r i f y t h e i n e q u a l i t y
2. Sobolev Spaces of Vectorial Order
190
we find
=
where A-(h,D
X'
)
L
1 1 (Dxn,h-n-(h,Dx,))sl(@u)1 ,:I
,h,
is the difference operator defined by the formula
-1 A-(h,D ,)u = FE; 1
,h
'IFx'+5 ,hU'
A-
I
Just as in the proof of 2.7.14, one shows that
where the difference operator B admits the estimate
where the constant C1 does not depend on h and u, as a consequence of Theorem 2.7.38. Since l u E u for x E R n r + , (2.7.171), (2.7.172) yield h the required inequality (2.7.170). I Example 2.7.50. 1". Consider the meshfunction Eh(. ,y) (2.7.173) Eh(x,y)
=
(?)
:
1 Rh
+
C,
(l+h2/4)-if3+(h) Ix-yl'h,
9+(h)
=
l+h2/2-h(l+h2/ 4 ) f , h
>
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
=
<<(h,hS)>-2eiYS
Hence,
'
2
IEh' (sl,s2) ,h
-2s =
h
1
-4+2s
l1 T
5.h
'dS
-
2.7. Spaces of One Parameter Families of Meshfunctions
-
C
{
191
3-2(sl+s2) h ;If:I
Therefore. Eh E f f ( s
h>
,
if s2
,
if s2
=
3/2
, if s2
<
3/2.
3/2
iff (sl,s2) < (0,3/2).
) (Z)
1’ 2
2”. Consider the meshfunction v (2.7.175) v+(x)
=
e+(h)x/h,
+
:
R1’+
h
x E R;”,
where 0 (h) is given by (2.7.173). An easy computation shows that for any integer m 2 0 holds:
where C depends only on m. m Further, one finds that 2
(2.7.177)
1 lv+(x+t)-v+(x)1 1’ (sl.O) ,h =
-2s h
-2s1 + 1 2 -1 (1-8+(h) ) = h (1-8+(h)t/h)
1 le+(h) (xct)’h-8+(h)x/h/2h
xeo
-
-2s
C h
=
(1-8+(h)t/h) 2
where the constant C does not depend on h. In fact
Therefore, using (2.7.177) and (2.7.79) with n
=
1, m = 0, 2‘
E (O,l),
one gets that
Since lim e+(h)t/h = e-t, h+O one can rewrite (2.7.178) in the form 2 (2.7.179) [v
3’
+ (Slr8),h
-2s1
m
0
t-(2e+i) -t 2 (1-e ) dt
- Clh
-2s
1
given that the integral in (2.7.179) is convergent when 0 < 8 < 1. Combining (2.7.1761, (2.7.179) and using (2.7.79) with n = 1, one gets the conclusion that
2. Sobolev Spaces of Vectorial Order
192 where C
depends only on
S
s2
2'
given by (2.7.175) belongs to all ff
Thus v
(sl , s 2 )
(Z')
with
s 1 6
0.
Now we are going to use norms (2.7.165) for showing that (2.7.180) holds 2 for all s E R . One finds easily that for n = 1 (2.7.164) takes the form
<+
(2.7.181) 5 - = <-A_,
=
<-A+, with A+-
2 f 8 (h) = l+h2/2 3 h(l+h /4) f
=
-1 (ih) (8+(h)-l),
.
Hence, using (2.7.165) one finds for v+ given by (2.7.175)
1 Taking as 1 the extension operator lo which extends v+ onto Rh by putting
it zero for x E R L f -
,
one finds immediately that -1
(2.7.183) (lov+)-'h where
<+
=
-i<+ '
is given by (2.7.181).
Thus, (2.7.182), (2.7.183) yield
If 1s21 <
<52<,1
+, then
operator L2(T1
)
Now assume that
s
5 th
,
2
E L (TS,h) so that
n;
being a projection
one finds immediately that
= m+8
where m
>
0 is integer and 181 <
f . Using the
formula
and observing that s -k-1
(2.7.187) IIh< +
(l+ih$) = 0 , k = 0,l. ...
we reduce the case considered to the previous one, when Now, let s
2
=
-m+8 where m > 0 is integer and 181 <
I s2 I f.
< f
.
2.7. Spaces of One Parametzr Families of Meshfunctions
193
Using t h e formula
I s2I
a g a i n t h e c a s e c o n s i d e r e d i s r e d u c e d t o t h e o n e , when Thus, f o r a l l s
=
so t h a t v
(Z+),
The m e s h f u n c t i o n
<
t.
(sl,s2)E R2 holds
v
Gh(. ,y)
(Sl,S2)
E
:
n2
+
c1
i s t h e d i s c r e t e G r e e n - f u n c t i o n f o r t h e d i f f e r e n c e o p e r a t o r 1+D D* on x,h x,h R1 r + h . S i n c e Eh g i v e n by (2.7.173) b e l o n g s t o a l l H (Z) with s < (0,3/2), (S)
( 2 . 7 . 1 9 0 ) y i e l d s t h e same r e s u l t f o r G, 3". Let q = (3-/5)/2.
:
Gh E H
( S )
(2')
i f f s < (0,3/2).
The m e s h f u n c t i o n
i s t h e d i s c r e t e f u n d a m e n t a l s o l u t i o n f o r t h e d i f f e r e n c e o p e r a t o r 3-0 -0-1 h h ( w i t h O h t h e s h i f t o p e r a t o r ) , which v a n i s h e s as 1x1 + One f i n d s e a s i l y
-.
that Fx+S,hEh = l + / ~ ( h t ) =1 1~ + 4 s i n 2 % 2 ' F u r t h e r , a n e a s y c o m p u t a t i o n shows t h a t
Hence, Eh
E
H(s)(Z) iff s
< (O,-f).
F u r t h e r , c o n s i d e r t h e m e s h f u n c t i o n v+ : RL'+ v+(x)
=
qx'h,
x E
+ C,
nl,+ h .
Again, u s i n g t h e argument s i m i l a r t o t h e one i n 2',
o n e shows t h a t
2. Sobolev Spaces of Vecto&l Order
194
Therefore, for the Green function
-1
of the difference operator 3-0h-Oh
on RL'+
the following inclusion
holds :
2.8. Some spaces of two parameter families of meshfunctions Two parameter families of meshfunctions appear after the approximation
by finite differences (or finite elements) of singular perturbation problems affected by the presence of a small parameter
1 0
parameter h E (0.h
E
E
( 0 . ~ ~ 1the , second small
being the meshsize.
Consider mesfunctions with complex values: (2.8.1)
u ( E , ~ , x ): %(
+
C,
(~,h) E li
EO'ho where we have denoted
and shall write Il instead of I[ when it does not lead to confusion. &OTh0 Sometimes u(~,h,hk),k E Zn, will be interpreted as a function of
(E,h) E TI whose values are in some Banach or topological sequence space. With S'(Zn) the Frechet space of sequences introduced in Chapter 1 for each meshfunction u : Il + s' (Z") , its di-Crete Fourier transform -h u = FX+[ h~ is well defined, and moreover, > I
2.8. Spaces of Two Parameier Families of Meshfunctions where, a s p r e v i o u s l y , 5
IId/
2
Define H u
:
iI
-t
n
(T<,h)
ck
=
195
(exp(ihck)-l)(ih)-' and
=
i s t h e norm in L ~ ( T )~. trh
(mn) t o be t h e c l o s u r e o f t h e s e t of t h e meshfunctions (s).E,h h 1 (zn) w i t h r e s p e c t t o t h e norms (2.8.3) f o r each ( ~ , h ) E 1T. 1
Further, l e t H ( S )
(Zn) be t h e s e t of a l l meshfunctions u E
H ( s ) .E,h(m;)
such t h a t
It is quite obvious that for nonnegative integer s2,s3 the norms (2.8.3) are equivalent (uniformly with respect to (&,h) E Il
)
to the
€0rho
following ones:
where, as previously, a and 6 are multiindices and D
x,h
=
(DX1,hr.. - ,Dx ,h) n
with iD the forward difference derivative in the direction of x k' xk,h = (Ox -l)/h, (0 u) (x) iD xk'h k Xk The following
=
u(x+hek).
Proposition 2.8.2. H(s) (z") i s a Banach space.
is proved in exactly the same way as Proposition 2.7.2. For u E
H
(s)
(Z") introduce the seminorm
and denote by H o
(S)
(Zn) the set of all u E ff
( S )
(Z") such that ( u ( (s) = 0.
Proposition 2.8.3. HYs) (z") i s a c l o s e d subspace of H ( s ) (z"). Proof. -
The statement is an immediate consequence of the triangle inequality for the seminorm (2.8.6). Definition 2.8.4.
Define t h e space H (2.8.7)
(s)
H ( ~ (z") ) =
(z") t o be
(zn)/HYs)(z"),
H (S)
2. Sobolev Spaces of Vectorial Order
196
the latter being provided with the quotient norm induced.by the structure of the quotient-space, i.e.
where u E ff class
u
E H
(Z") is any meshfunction which belongs to the same equivalence (S)
(z").
(S)
Example 2.8.5. With H(x) the Haeviside function, the meshfunctions (2.8.9)
u(h,x) = (l+h)-(X+h)/hH(x), v(x,h) = (l-h)X'hH(x),
x E
1 (Z ) belong to the same equivalence class.
as elements of H (S)
Indeed, an elementary calculation yields
0 (Z'). (0) Notice that u and v are solutions of the difference equations
so that (u-v) E H
(az,h+l)u = 6h (x) and
( a x,h+l)v
= S
h (x+h), respectively, where 6h ( x )
=
the Kronecker symbol, is the difference &-function. = h-16x,0 with 6 x,o Moreover, u and v are the only solutions of the corresponding difference equations above, which vanish at infinity. Using the discrete Fourier transform, one checks easily that the meshfunction w,
is the only solution of the difference equation ($(ax,h+a*x,h)+l)w = 6h (x), 1 + m , x E IR h' Indeed, via discrete Fourier transform one gets the formula
which vanishes as 1x1
w(h,x)
=
-'
i ~~'~(e~+2h+i)-lde,
(Ti)
jBI=l
so that the residuum calculus yields (2.8.10). One has
+
t
when h
-f
0.
Hence, u and w do not belong to the same equivalence class in H
(Zl),
(0)
2.8. Spaces of Two Parameter Families of Meshfunctions
197
although the accuracy in the approximation of the differential operator a +1 by the finite difference operator f ( a x,h+a*x,h)+1 is O(h21 , while the +1 and a* +1 approximate the same differential difference operators a x,h x,h operator with the accuracy O(h). Also the essential difference in the behaviour of u and v, on the one hand, and w on the other, is reflected by their asymptotics when h + 0 . One has u(h,x)
while w(h,x)
=
e-X H(x) (1+0(h)), v(h,x) = e-xH(x) (1+0(h)),
-
e-IX'[ff (~)+(-l)~/~H(-x-h) 1
Example 2.8.6. With p = h/E consider the meshfunctions (2.8.11)
u(E,h,x) = (l+p)-(X+h)/hH(x), v(E,h,x) = (l-p)X/hH(x).
Assuming that p < 2 , one finds that
+
when
p
0 when
E +
0, h
+
0 and
p < 2,
1 H ( 0 ) ,E,h(lRh)
but v $?
t 2.
With (2.8.12)
w(E,h,x)
=
e-X'EH(x)
one finds that
where C
=
sup( (2p+p2)-1+(l-e-2P)-1-2(l+p-e-P)-1)
<
m.
pd0
Hence, u and w belong to the same equivalency class in H
1 (0) (z )
.
Notice, that w = exp(-x/E)H(x) is the solution of the differential equation
(Ea
+l)w
=
E~(x),which vanishes at infinity, while u in (2.8.11)
is the solution of the difference equation
(€a*
+ l ) u = sAh(x). x rh Notice that the meshfunctions E - ~ U and E - ~ Wwith u and w given by
(2.8.11) and (2.8.121, respectively, do not belong to the same equivalence 1 1 nor do they in any H (Z 1 . However, the meshfunctions
class in H ( o ) ( Z
E - ~ Uand (E+h)-'w
(S)
do belong to the same equivalence class in
H
(z')
(S)
2 . Sobolev Spaces of Vectorial Order
198
with any s = (sl,O,O) such that s1 <
-+. Indeed
a straightforward
computation yields:
where ~(p= ) ((l+p)2 (2p+p2)-1+(~-e-2')
-'-2(1+p) (l+p-e-')-')
5 cl, V p t O .
Example 2.8.7. m
Let f E C (IR) and let the meshfunctions u ( E , ~ , x )and v(E,~,x)be given, 0
as follows: -1 -ihS -1 u = FS+x,h~(l+p-e ) Fx+E,hf
(2.8.13)
It is quite obvious that u and v are the solutions of the difference equations with small parameter
which vanish when
x
I
+
E
m.
Since the difference equations (2.8.14) approximate the same differential equation (Ea +l)w = f(x), x E I R , one should expect that both 1 u and v belong to the same equivalence class in some H (s)( Z 1 . Since m
f E C O ( R ) ,it turns out that this holds for V s
=
isl,s2,s3)with s1 < 1.
Indeed, since
I
( l+p-e-ihc)-1- (
(
I-e-P-ihS) -1 1 5 C / ~ - e - ~ ~ ' l *
one finds, using (2.8.3) (and assuming for simplicity
I Iu-vI
2 (s),E,h
Applying (2.7.11) to
'''-v"
-2s 5 C(E+h)
=
-1
1 l?rhf/I (0,s + s +l),h
(s),E,h
E
0 2s 2s 1 2<(E+h)p 3
2-s3 1 5 C(E+h) h(l If
one gets finally the estimate
Is
+s +1 +ChllflJs1+s 2+2 1
1
2
2.8. Spaces of Two Parameter Families of Meshfunctions
SO
199
that u and v belong to the same equivalence class in any H ( s ) ( Z
1
)
with
s1 < 1. Example 2.8.8. Consider the meshfunctions u(E,h,x) = [(e
-AX
-x/E)+~~-x/E -e 1 H(x),
-p+Ah -1 -Ax - X / E -X/E v ( E , ~ , x )= [(l-e-’) (1-e ) (e -e )+@e where A > 0 and 4 are given, p = h/E, h E ( 0 , 1 1 ,
E
1
H(x),
E (0,(2A)-1].
It is easily seen that p-Ah 2 Ah, so that
Hence,
- hx-e-x/ E )H(x), (e -(A+iS)h-e-(p+ihS))(l_e-(~+iE)h-1 -(l+iEc)p -1 h(e ) (1-e ) .
A straightforwardcomputation yields for w
-h wA r E ( 5 )
=
(FX+S,h~A,E) (5)
=
X,E
=
One checks easily that with some constant C > 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'-s
s;-s3
2<(&+h)p
< cS,S'.
2.8. Spaces of Two Parameter Families of Meshfunctions
20 I
Propcsition 2 . 8 . 1 0 .
I I . j I ( s ),E.h are
he norms
stronger than j
1 . 1 I (s'),s,h'
if? s > 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)<<>)- s 1 < p s 1 + s 2 <
? C2'
2 C,,<(E+h)G>
where
C2
depends only on
(E+h)5>s3 2
s2+s3
s -s
(for c O 5 1 , ho 5 1 , one can take C2 = 1 ) . #
E O,ho
The following statement (interpolation inequality:, is proved in exactly the same way, as Proposition 2 . 1 . 1 0 . Proposition 2 . 8 . 1 1 . Let u E lR3 (2.8.17)
,
s E lR3
,
s
iu
and let
p = s-t(~-s), t > 0.
Then with V 6 > 0 holds:
In Chapter IV boundary value problems for one-dimensional difference singular perturbations will be given a special attention, so that in the rest of this section it will be assumed that n = 1 . Of course, the results tc be discussed can be easily extended to the case of or a layer
$-'
x
<,+
= m;-l nh,+ 1 Ih, with Ih = lRhn I, I = (a,b) c R being a finite
interval, or to Tn-' h on the torus Tn-'.
Ih, where TE-l is a one parameter family of meshes
X
Denote by n u the restriction of a meshfunction u E 0 point x = 0 : 71 u = u ( ~ , h , O ) .
H
( s ) ,E,h
0
For a function 4
:
*
I! EO'ho
C,
@ E Co (II
)
€0th0
introduce
(R) to the
2 . Sobolev Spaces of Vectorial Order
202
(2.8.19)
-1 ,h = (E+h) l$(c,h)\,
[$I1,
[$I1
sup [bll,E,h, 1 E R . E,hEli
=
the family of complex planes equipped with 1,Erh and by C1 the Banach space of all functions the family of norms [-I 1,E,h : Jl -f c with the norm [$Il finite. We shall denote by C
Again, one can introduce the seminom 1
[$Il
-
=
lim ,h)+(O,O)
V
["1,E,h'
$
E c1
(E
and, with Co = { $ E C1 1 [$I; = 01, consider the factor space C1/Co 1 with 1 the quotient topology induced by the norm in
5-
Example 2.8.12. Let f E Ci(IR) and let $(E)
= (2111-l
where, as usual,
p =
2 2 -1 I, (ICE E )
f(E)dS,
h/E.
With U(E,X) and v(E,~,x)solutions of the problems:
it is easily seen, that
$(E)
= T ~ U , + ( ~ , h )=
n 0 v.
Further, Cauchy-Schwarz' inequality yields with V s > 3 :
where
I If I I s
is the norm of f in the usual Sobolev space H s ( R )
we have denoted: 2 2 -1- fl-e-P)211-e-P+ihEI-2. @(~,h,c) = ( 1 + ~ 5 ) A
straightforward computation shows that
where C is a constant. The last inequality yields:
and where
2.8. Spaces of Two Parameter Families of Meshfunctions
203
provided that s > 3/2. Hence, $(E) and $(E,h) belong to the same equivalence class in C1/C:
with V 1 < f.
Lemma 2.8.13.
L e t u E ti
( s ) ,E,h
1 (nh) with
s > f , s2+s3 > f . Then 2
i s a continuous l i n e a r mapping u n i f o m Z y w i t h r e s p e c t t o
.
E Il
(E,h)
€0rho Proof. Applying Cauchy-Schwarz' inequality to -1 u(E,h,O) = ( 2 ~ )
one gets
-2s
(E+h)
1
-h u (~,h,5)dS
J1 T
5 rh
lu(E,h,O)1'
Using the inequalities 151 5 151 S
where the constant s2+s3 >
C1
-2s
2
J I I' I I ( s ) ,e,h T1
5 (21r)-~
<5>
-2s
2<(c+h)p
3d5.
5 ,h 1 (1r/2) 151, V h > 0, V 5 E T5,h,
depends only on
s2
, ~ 3 , ~and 0 ho, given that s 2 > 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
-
-'s
1
1
x.--xcs 1
.
i s uniformly continyous w i t h r e s p e c t t o (E,h) E II
k
€0nho
Lemma 2.8.15.
Let u E H
( s ) ,E,h
1 with (mh)
s2 <
f , s2+s3 > f . Then
1
(2.8.22) l f ( s ) , ~ , h ( ~ h )
+ *OU
Csl+s2-f
i s a l i n e a r continuous mapping uniformly w i t h r e s p e c t t o (E,h) E II EO'ho
2. Sobolev Spaces of Vectorial Order
204 Proof. -
Using again the cauchy-Schwarz inequality, one gets
-
1 Again using the equivalency 15 I 151, v h > 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<(E+h)p 3d5 S
5 rh 6 C
2Sl
I(E+h) 2(s
5 Cl(E+h)
-2s -2s <<> '<(E+h)E> 3d5 6
+S
-t) JICj R
-2s2 -2s 2 (s1+s2-f) <<> 3d5 = C (E+h) . 2
I
Remark 2.8.16. I f u E H (s),E,h(4) with s2 =
4,
s +s
2
3
> f , then holds
where the constant C depends only on s3,E0,h0. Indeed, this follows immediately by the previous argument. Remark 2.8.17. Obviously, the statements of Lemmas 2.8.13 - 2.8.15 and Remark 2.8.16 are also true for the meshfunctions in H(s) (Z") and H at x
=
(Z"), and their traces
(s)
0.
4
{x E I x 2 0 ) will be Now spaces of meshfunctions on %,+= 1 with introduced. For s2 5 0, s3 2 0 integer define the norms 1 I . I I + ( s ),E,h subscript s = (sl,s2,s3) in a natural way:
( R 1 ) the family of spaces of the meshfunctions u and denote by H (s),~,h h,+ with norms (2.8.24) finite for each 1 Analogously denote by H(s)(Z+) the space of the meshfunctions u such
that
2.8. Spaces of Two Parameter Families of Meshfunctions + 1 and set, a previously H ( s ) ( ~ + ) =
1 + O 1 H+( s ) (Z+)/H(s) (Z+),
205
+O 1 where H ( S ) ( Z + ) is the
subspace of the meshfunctions u in H+( s ) ( 2 ; ) such that
We are goint to use the extension operator 1 defined by (2.7.55), (2.7.56) m with m
=
s +s
2 3’ The following statement is an immediate consequence of Theorem 2.7.26.
Lemma 2.8.18.
For s2 t 0,
s3
t 0 i n t e g e r the. e x t e n s i o n operator lm d e f i n e d by (2.7.551,
(2.7.56) w i t h m = s 2 + s 3 i s a family of continuous l i n e a r mappings: (2.8.26)
Is2+s 3
’
1 H ( s ,~,h(=h,+) ) -f
uniformZy w i t h r e s p e c t t o (s,h) E Il
1 ff
( s ) , ~ , h ( ~ h’ )
.
EOfho Using the extension operator 1 = lm, one can define spaces H ( ~ ,) E , h ( ~ L , + )f o r v s E w3 . Definition 2.8.19.
With (2.8.27)
I I’I
1;s)
,E,h = inf 1
I IlU/I ( s ),E,h
where t h e infinimum is taken over a l l possibZe e x t e n s i o n s 1 of u t o R 1 h’ 1 1 d e f i n e t h e spaces ff of all meshfunctions u : mh,+ + c w i t h ( R ~ , + ) ( s ) ,E.h norms (2.8.27) f i n i t e f o r each (E,h) E Il EO
A n a l o g o u s l y , t h e spaces H
,h0’
1
( z + ) are d e f i n e d t o c o n s i s t of a l l mesh-
(S)
f u n c t i o n s u such t h a t
1
and t h e spaces H ( s ) ( z + ) t o be t h e f a c t o r - s p a c e s H
HYs)
1
(Z+)
1
1
( Z + ) / H ; ~ ) ( z + ) , where is t h e subspace of a l l meshfunctions u E H ( z + ) such t h a t (s)
( S )
Remark 2.8.20. 1 With H- (S),E,h(%$,)
the subspaces of all meshfunctions in 1
which vanish for x E R h , + , one has the isomorphism
H
1 ( s ) ,~,h(”h)’
2. Sobolev Spaces of Vectorial Order
206
Remark 2.8.21. One can show that with s2 t 0,
s3 2 0
introduced by (2.8.24) and the norms
integer, the norms
1 1 . ! I+
(s),E,h'
1 11.11
,E,h defined by (2.8.27),
are equivalent uniformly with respect to (E,h) E II
EO,ho' It is useful to have a more constructive definition of norms in 1 spaces H (I?$,+) , which would not require the minimization over all ( s )rE,h possible extensions. To that purpose we shall use the discrete <-Fourier transform and carry over constructions introduced by the end of Section 2.7 More precisely, we are going to use the discrete Mellin transform M x+8 ,h defined as follows:
-1 The inverse discrete Mellin transform Ms+x,h has the form
where the upper + means that the unit circle T1 is run counter-clockwise. Parceval's identity for M takes the form: x+8 ,h
(2.8.33)
1
xC4
2 -1 lu(h,x)! h = (2nh) Jl[zh(h,8) 121d8\, T 1
where Id01 = d5 with 5 the angle coordinate on T
:
8 = exp(i5).
Denote
With :n
the multiplication of a meshfunction u
1
: IF$,+
C by Heaviside's
function ff(x), we shall write down the discrete Mellin transform of
I I ' U .
h
Obviously, one has
Substituting (2.8.32) into the right hand side of (2.8.34), summing up the geometrical progression and introducing the notation:
2.8. Spaces of Two Parameter Families of Meshfunctions
207
one obtains, using the Pleijmel formula:
-h (t)u (h,8)+(2i~i)-lv.p.
=
F1 (e-A)-l Gh(h,X)dh,
1 V 8 E T .
T
i by R L and
Denoting the discrete Mellin transform of the operator a using the notation: (2.8.37)
-' lim
(0
-1 ,
(B6-A)
=
~ E T1 , X E 1T ,
6++0
where the limit in the right hand side of (2.8.37) is taken in the sense of distributions on T1, one can finally write down: (2.8.38)
(1;
xh)(h,B)
=
(hi)-' l; T -1
= (2.rri)
where <
,>
(7;)
<(8+o-.)
-1 -h u (h,A)dh
-1 -h ,U
=
>,
is the duality between testfunctions and distributions on T 1 .
Introducing the operator (2.8.39)
(B+o-X)
(h,x)
=
7-
h'
(l-H(x))u(h,x),
and using the same argument, one gets for the discrete Mellin transform Rh -
of rh the formula (2.8.40)
(n,
zh)(h.0)
=
(hi)-'
/1 (8-0-X)
-1 -h
u (h,X)dA =
T =
-(2vi)-1<(a-o--)-1 ,u -h >,
where
and the upper - means that the unit circle T1 is run clockwise. By the Pleijmel formula, one obtains: (2.8.42)
(I[h - -uh ) (h,B) = ( f ) uh (h,8)-(2ni)-l v.p. < ( 8 - A ) - '
Gh(h,X)dh,
T
so that for the meshfunctions u(h,x), which decrease rapidly as 1x1
one has of course (2.8.43)
(Ilizh) (h,B)+(IIh $1
(h,B) = Lh(h,e),
+
-,
2. Sobalev Spaces of Vectorial Order
208 and
% ((n h+-Il h- )u )(h,8)
(2.8.441
=
(Ti)-' v.p. <(O-A)-'
Yh(h,X)dX,
T
V h > O , V8ET1.
yo)
4,-
h, i.e. u E o for x E and Notice that if u E H 1 , then I[h- $ 0. Indeed, Cf"(h,A) is analytic for ] A ] > 1, -h 2 1 since uh z o for x E R and u E L (T ) for each h E W+. The function h,- ' X + (e6-h)-l being also analytic for 1x1 > 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 <,,h
2
2
:
1
where 8+(h) = l+h / 2 7 h(l+h / 4 ) 2 , FO;
V t
T
1 -f
C,
/8+(h)[ < 1 , /8-(h)( > 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 uniformly with respect to ~ , Eh Il when I3 = exp(ihS),one
-
finds, using (2.8.49), that
IuliS)
,E,h 5
where the constant
C
CI
jlul I
( S )
.E,h'
V u E
1 H ( S ) ,E,h(%,+)
does not depend on ( ~ , h ) E Il
5 c inf
IUITs),E,h
1
and the extension 1. EO rho
The last inequality yields:
1 llul I ( s ) ,E,h = c I'I
'ls)
,E,h'
V (E,h) E Jl
1 V u E H (s),~,h(%,+)
EO,ho'
.
One uses the open mapping theorem and its corollary (see, for instance, [Rud, p.49, Corollary 2.12 (d)1 for showing that 1 I V u E H ( s ) , ~ , h ( ~ h , +' ) (Erh) 'Eo,h0
V h
1 jul I (s).E,h +
We emphasize, one more time, that 'lI is an orthogonal projector, 2 1 h 2 1
> 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(<), called also symbol, so that
where F
and F - l
X-tS
are the direct and inverse Fourier transforms defined
S’X
i n Chapter 1.
The splitting of Q ( 5 ) intci a sum of homogeneous polynomials Q ( 5 ) of k degree k,
with m
=
deg Q ( E ; ) the degree of Q , is a natural graduation of Q ( 5 ) where
especially the homogeneous polynomial Q ( 5 ) of highest order,called also m principal symbol, is well defined and plays an important role in the classical elliptic theory of partial differential operators. The main object of the investigation here are differential operators Q(E,D)
whose symbols may depend upon a small parameter
that their degree deg Q depends on deg Q ( E , S ) = m, V
E
E
E
E [O,E
0
1 and such
in the following fashion:
E ( 0 . ~ ~ 1deg , Q ( 0 , S ) = m 1 < m.
A typical example of such an operator is the one appearing in the
linear Elasticity theory:
3 . Singular Perturbations on Smooth Manifolds without Boundary
214
2 2 (3.1.3)
QO(E,D)
2
= E A -A,
f o r which m = 4, m
=
A
=
-
Dk
C
lZk
2.
1 The c o r r e s p o n d i n g symbol Q
5)
(E,
O
f u n c t i o n of d e g r e e 2 i n v a r i a b l e s
(E
=
-1
E
2
15 1 4+1 5 I
,F) E
(3.1.3) can be r e p r e s e n t e d as a product Q
Rn.
R + X
(E,C) 0
i s a homogeneous
Besides, Q ( E , ~ ) i n
2 2 2 151 ( 1 + ~151 ) so t h a t
=
it is n a t u r a l t o a s s o c i a t e w i t h Q i t s v e c t o r i a l o r d e r v = ( 0 , 2 , 2 ) , which i s
t h e l e a s t v e c t o r i a l o r d e r such t h a t Q ( E , D ) 0 uniformly with r e s p e c t t o E .
:
H
(s), E ( R n )
H(S-V) , E
(W")
I f one c o n s i d e r s a s l i g h t l y d i f f e r e n t d i f f e r e n t i a l o p e r a t o r
(3.1.4)
Q(E,D) = QO(€,D)+Q1(D),
then t h e r e p r e s e n t ati o n sum of homogeneous i n
v(l)
=
(E
(3.1.4)
-1
Ql(D)
= =
i s a n a t u r a l s p l i t t i n g of Q ( E , D )
,5) t e r m s
of o r d e r v(O)
> v"),
r e s p e c t i v e l y , b e s i d e s v")
(O,l,O),
1 b D l
(0,2,2) and
=
so t h a t a g a i n one c a n
view Q O ( ~ , S ) a s t h e p r i n c i p a l symbol of Q ( E , ~ ) and s h o u l d e x p e c t Q p l a y thedominant
0 to r o l e i n t h e e l l i p t i c t h e o r y of d i f f e r e n t i a l o p e r a t o r s
a f f e c t e d by t h e p r e s e n c e o f a small p a r a m e t e r .
> i s o n l y p a r t i a l , it i s n o t s u r p r i s i n g t h a t a , 5 ) principalsymboldoesnotalwaysexistforagivensymbol
Since t h e o r d e r i n g geneousin
(E
-1
4
I n f a c t , t h e polynomial Q ( E , ~ ) =
E
s p l i t i n a sum of homogeneous i n
(E
Q,
= E
4
151 -1
5
+E
2
homoQ(E,<)
4
151 + / 5 I 2 + < b . 5 > can b e s t i l l
,5) t e r m s
Q
= E
2
1514+1512
and
1 5 j 5 + < b . < > o f d e g r e e 2 and 1 r e s p e c t i v e l y , b u t t h e i r v e c t o r i a l
o r d e r s v")
= ( 0 , 2 , 2 ) and
n e i t h e r v(O)
> 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(€,<)=
one f i n d s :
[3.1.5),
Qo(l,~g).
E
For d e r i v i n g a p r i o r i estimates f o r e l l i p t i c s i n g u l a r p e r t u r b a t i o n s w e
i
s h a l l u s e symbols of t h e form Q ( E , ~ ; ) w i t h L(E,:',~,)
=
with
<5'>1c'/-l5',
=
<5>1
and even of t h e form
which a r e w e l l d e f i n e d f o r
6 E
En-,
i s a s u b s e t of measure z e r o .
where S c IRn
D e f i n i t i o n 3.1.2.
A function L
:
( 0 . ~ ~x 1m n \ s +
t o be a symbol i n t h e c l a s s Lv
c w i t h s a c o n i c s u b s e t of measure zero, i s s a i d
=
LV(xn) with v
= (vl,v2,v3),
i f
( i ) t h e r e e x i s t s c o n s t a n t s C,m such t h a t
v
-v (3.1.8)
'is>
/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)
<5>
<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
214<5-n>l4,
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
<5>
-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,<',D,)
: H
= L'(D)
i s called
3.1. Singular Perturbations with Constant Symbols
219
Lemma 3 . 1 . 8 .
( i )I f f o r L E lv a reduced symbol L
0
E 1
unique. ( i i )I f L E
Lv w i t h v 3 -2 0 has t h e reduced symbol
Proof. __ ( i ) The i n c l u s i o n ( 3 . 1 . 1 7 ) y i e l d s L
def ined
0
(6) =
e x i s t s then i t i s
(0,v2,O) L
0
then
E L(o,v2,0)
l i m E"L(€,~).
Thus Lo i s w e l l
€4-0
.
( i i )The symbol cV1L-Lo c a n b e r e w r i t t e n a s f o l l o w s :
E
v1
L(E,S)-L
0
(5)
=
-V -v < p v 2 < E p 3 ( E"1 -v 2<€t;> 3L- ~ L o ) + L o ( (~<)E 5 > v 3 - 1 ) .
"
3 E Since < p V 2 < ~ < > V
1 (o,v2,v3),
( 3 . 1 . 1 7 ) and Lemma 3 . 1 . 3
imply t h a t t h e
f i r s t t e r m i n t h e r i g h t hand side of t h e l a s t i d e n t i t y belongs t o For v3 2 0 one h a s : ( < E S > ' ~ - ~ E ) L(-l,l,v - 1 ) , so t h a t +l,v -1). 3 3~ a t h e symbol L ( t ) ( < ~ S > ~ ~ - lb)e l o n g s t o l(-l,v + l , v - 1 ) , a s w e l l . 2 3 D e f i n i t i o n 3.1.9. 5-1,"
The symbol L~
L E
Lv i s c a l l e d e l l i p t i c of order v i f i t s p r i n c i p a l symbol
satisfies the inequality:
(3.1.18)
-vl
IL ( E , S ) ~ 2 C E
0
w i t h some p o s i t i v e c o n s t a n t
"
J '
151 2 < ~ E > ',
V (E,S)
E
1R+
x
(Rn\{O)).
C.
The c l a s s of a l l e l l i p t i c symbols o f order v i s denoted by OPE\,
E ~ w , hile
stands for t h e corresponding c l a s s o f e l l i p t i c s i n g u l a r p e r t u r b a t i o n s
o f order v. def {w
1
w E R n , / w I = 1 ) . Using t h e homogeneity of L 0 Of d e g r e e v +u2, one c a n r e f o r m u l a t e t h e c o n d i t i o n ( 3 . 1 . 1 8 ) e q u i v a l e n t l y a s 1 follows :
Let Q
-v (3.1.19)
ILo(p,w)I
2 Cp
v '
',
V (p,w)
E 0
R+x
I n t r o d u c i n g t h e r e d u c e d p r i n c i p a l symbol L o ( C ) ,
and t h e homogeneous p r i n c i p a l symbol L o o ( c ) ,
nn.
3 . Singular Perturbations on Smooth Manifolds without Bounda y
220
- ( v +v )
(3.1.21)
L
00
(5)
2
= l i m t
3
L0(l,tS), 5 E R n ~ i O l
tJ-m
one c a n r e f o r m u l a t e e q u i v a l e n t l y t h e e l l i p t i c i t y c o n d i t i o n a s f o l l o w s : P r o p o s i t i o n 3.1.10.
0
L
If L E
admits t h e reduced p r i n c i p a l symbol L ~ ( C ) and t h e homogeneous
p r i n c i p a l symbol ~
~
~ d e(f i n5e d ) by ( 3 . 1 . 2 0 1 , ( 3 . 1 . 2 1 ) , then
L
is e l l i p t i c
( L E E )iff t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :
LOO(,)
(i)
( i i )~
~
~
v
# 0, #( 0 w,
) V
# 0, V
( i i i )L O ( p , w )
w E w
E R E R+x an.
(p,w)
Proof. -
L
If L E
i s e l l i p t i c of o r d e r v t h e n (3.1.19) h o l d s f o r V (p,w)
Multip1y':ng
letting
151
+
E
.
n
0 one g e t s ( i ) -(v2+v3) = 1 , m u l t i p l y i n g it by 151 and
( 3 . 1 . 1 9 ) by p V 1 a n d l e t t i n g p
Further, taking i n (3.1.18)
E R + x S2
+
one g e t s ( i i ).
m,
( E , C ) of o r d e r v1+v2 and ( 3 . 1 . 1 2 ) imply (iii). 0 -V ( i ) - ( i i ih )o l d s , t h e n ( i ) y i e l d s / L ( p , o ) /2 C p
The homogeneity of L Conversely, i f
0
E (O,po]
f o r V (p,w)
PO
Rn w i t h s o m e p o s i t i v e c o n s t a n t C which depends 0 0 PO '3-'1 o n l y on p o and y o = min IL (IJI) while (ii) i m p l i e s I L o ( p , w ) 2 C p wERn 0 p1 which depends f o r V (p,wl E [p,,-) X Qn w i t h some p o s i t i v e c o n s t a n t C x
I
1;
o n l y on p 1 a n d y o 0
=
min
IL
00
wERn
((0)
~. The
41
Qn b e i n g compact
s u b s e t [pO,pl]x
R ,
one can f i n d , u s i n g ( i i i ) a c o n s t a n t C > 0 s u c h t h a t n 2 C P - ~ ' < P > " ~which , i s p r e c i s e l y ( 3 . 1 . 1 9 ) and by homogeneity of 0 -1 L ( e , < ) of o r d e r v +v2 i n ( E ,C) is equivalent t o (3.1.18). I 0 1 in R IL
+
X
(p,w)
P r o p o s i t i o n 3.1.11.
Let
L ( E , D ) E O P L , ~ . Then L is e l l i p t i c o f order following e s t i m a t e holds f o r E~ small enough:
(3.1.22)
/ I ~ l l ( S ) , E ~ C ( / I L U / I ( S+ -" l l)u,lE[ ( s , ) , E ) ,v
v where the c o n s t a n t cdoes n o t depend on
alrs
=
is'
v ( L ( E , D ) E OPE")
1
SI
3
E R ,
S I
<
S ,
S;
E E
n3,
v
so E
E
E (O,E01,
a3,
$ 1
iff t h e
v
uEs(Rn),
< s, s' E airs,
and u and where
= s1}.
Proof. Let L(t,D)
be e l l i p t i c of o r d e r v and l e t L 0 ( € , C )
of L ( E , ~ ) .Witn
i
=
< 5 > \ 5 \ -15 t h e e l l i p t i c i t y o f
b e t h e p r i n c i p a l symbol L implies:
3.1. Singular Perturbations with Constant Symbols
22 1
1
C1. Denote L O ( ~ , D )= OpL ( E , E ) . Then, u s i n g t h e P a r c e v a l ' s i d e n t i t y and 0 t h e i n e q u a l i t y C 5 - P 5 1,one g e t s
The l a s t i n e q u a l i t y l e a d s t o (3.1.22) w i t h s'
=
s-e2 i f
E~
< C/(2C ) .
1
Now,
u s i n g t h e i n t e r p o l a t i o n i n e q u a l i t y (2.1.11) f o r Sobolev norms o f v e c t o r i a l o r d e r , one g e t s (3.1.22) w i t h V s'< s , s'
airs.
E
C o n v e r s e l y , assume t h a t (3.1.22) h o l d s w i t h s o m e s and s' = s-e2. Then t h e Parceval's i d e n t i t y implies:
for
v
(E,s)
E
(o,E01
x
(R~,s).
The l a s t i n e q u a l i t y i m p l i e s :
for (E,<)
E ( 0 . ~ ~x 1 ( R n \ S ) ,
positive constant C
16) 2
r w i t h r l a r g e enough, where t h e
depends o n l y on C , v and r .
The homogeneity o f L O ( ~ , S )i n
(E
-1
o f o r d e r v +v2 i m p l i e s t h a t t h e
,<)
1
l a s t i n e q u a l i t y , perhaps with a d i f f e r e n t p o s i t i v e c o n s t a n t , h o l d s f o r
v
(E,S)
E R+
x
I
(W"\IO}).
Example 3.1.12. The s i n g u l a r p e r t u r b a t i o n s
(3.1.3), (3.1.4) a r e e l l i p t i c o f o r d e r (0,2,2),
while t h e s i n g u l a r p e r t u r b a t i o n Q ( E , D ) =
E
2
4
I D 1 -1DI2 i s n o t e l l i p t i c .
Remark 3.1.13. 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 any s i n g u l a r p e r t u r b a t i o n Q ( E , D )
w i t h polynomial symbol Q ( E , ~ )E p a d m i t s t h e r e d u c e d p r i n c i p a l symbol 0 and t h e homogeneous p r i n c i p a l symbol Q o o ( 5 )
Qo(S)
p~o,v2,01
The same i s t r u e f o r symbols which a r e r a t i o n a l f u n c t i o n s of 5 .
p(o,v2+v3,0)
'
3 . Singular Perturbations on Smooth Manifolds without Boundary
222
Remark 3 . 1 . 1 4 . some s ' < s s u c h t h a t s' < s l , t h e n f o r 1 s u f f i c i e n t l y s m a l l t h e s i n g u l a r p e r t u r b a t i o n L ( & , D ) , E € (@,€,,I, If
(3.1.22) h o l d s for
E
-
equiisomorphism, i . e . L -1 -1 L := Op L(E,S) : H
:= L ( s , O )
(s-")
: H
0 is m
(IRn ) and
( s ), E
(IRn),
E
E
(u,E@~,
,E
are b o t h
e q u i c o n t i n u o u s (see D e f i n i t i o n 1 . 2 . 3 3 ) .
D e f i n i t i o n 3.1.15.
An e l l i p t i c s i n g u l a r p e r t u r b a t i o n
L(E,D)
E
OPEv
is caZZed s t r o n g l y e L l i p t i c
of order u if w i t h some c o n s t a n t s 6 E [ 0 , 2 n ) , y E R+ hoZds: -v (3.1.23)
where
Re(eiCLo(s,F)) L
L ~ ( E , ~i )s
YE
u
' / 5 ( 2 < ~ 5 >3v, V
t h e p r i n c i p a l symboZ of
(E,<)
E
(IRn\{O)),
R+x
L.
Example 3 . 1 . 1 6 . The s i n g u l a r p e r t u r b a t i o n s order u
=
(3.1.131,
(3.1.14) a r e s t r o n g l y e l l i p t i c of
( 0 , 2 , 2 ) , t h e c o n s t a n t s i n ( 3 . 1 . 2 3 ) b e i n g 8 = 0 , y = 1; t h e
i D 2 + D 2 ) 4 ( 1 + ~ 2I: D;)" is e l l i p t i c I: k n llksn-1 l
= (
P r o p o s i t i o n 3.1.17.
Let
L(E,D)
E
OPE
be s t r o n g l y e l l i p t i c of order v and Zet 6, y be t h e
corresponding c o n s t a n t s in ( 3 . 1 . 2 3 ) . Then t h e r e e x i s t c o n s t a n t s
Co
and
Cl
such t h a t (3.1.24)
i6 Re<e
L(E.D)U,U> 2
(Y-C 0 E )
1 IuI 1
2 ( v / 2 ) ,E-Cll
V u E H
( v / 2 ) ,E
where < , > 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,<) = LO(~,S)LO(S)
R ( E , ~ )=
anddefining
,
S ( e , S ) = R(E,S)-~
R ( E , ~ ) ,; ( € , < I
= S(E,<),
i
with
=
< < > I t-1/ 5 ,
one
c a n w r i t e down: L(E,D)
(3.1.30) L
where
0
(D)
=
R ( E , D ) L 0 ( D ) + E Q(~E , D )
= ;(E,D)L(E,D)+EQ~(E,D),
Q 1 ( ~ , S )E L v l Q ~ ( E , S )E L ( O ,"*,o ) .
I n f a c t , t h e argument u s e d i n t h e p r o o f of P r o p o s i t i o n 3.1.18 i s 0 a p p l i c a b l e t o t h e symbols p l ( ~ , < ) L ( E , ~ ) - ~ ( E , < ) L( 5 ) and
p2(€,6) = L0 ( < ) - S-( E , < ) L ( E , S ) , s i n c e t h e i r p r i n c i p a l and r e d u c e d symbols 0 v a n i s h i d e n t i c a l l y : P . ( E , S ) 5 0 , ? . ( 5 ) :0 . JO 3 T h i s remark w i l l be u s e f u l l a t e r f o r t h e r e d u c t i o n o f s i n g u l a r
e l l i p t i c p e r t u r b a t i o n s w i t h v a r i a b l e symbols t o t h e r e g u l a r o n e s .
3.2. S i n g u l a r p e r t u r b a t i o n s w i t h homogeneous symbols Let
E 1". Then c l e a r l y t h e f a m i l y ( 0 . ~ ~3 1E
a(E,S)
-f
a ( € , . ) E S'(R;) i s
a c o n t i n u o u s l i n e a r map and one c a n r e w r i t e a ( ~ , D ) ua s t h e f o l l c w i n g
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 :
(3.2.1)
a ( E , D ) u = (AE
where A (x) For V
=
E
F
-1
5-
*
v
U),
a ( E , c ) , ( 0 . ~ ~3 1E
-f
u E S(IRn)
,
Ac E S'(lR:).
E ( O , E ~ ] and V u E S ( R n ) t h e f u n c t i o n
r a p i d l y decreasing as
6
-f
-, so t h a t a ( ~ , D ) E u
<
m
C (IRn)
-f
,
a(E,S);(<)
V
E
E
is
(0,s01.
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 homogeneous f u n c t i o n s o f (E
-1
,6) a r e
of s p e c i a l i n t e r e s t f o r t h e a p p l i c a t i o n s .
D e f i n i t i o n 3.2.1.
A function:
R+X
(R~\[o})3
(E,s)
+
a0(E,<)
E c is said to belong to t h e
3.2. Singular Perturbations with Homogeneous Symbols
symbol c l a s s Hm
V'
(i) . a
E c
i f t h e fo%%owingc o n d i t i o n s a r e s a t i s f i e d :
o i ~ x+ (IR~\IO}))
(ii) t h e r e e x i s t s a c o n s t a n t (3.2.2)
lao(E,~)I5 CE
-1
c
If a
o
-1
ao(t
E,t<)
=
> 0
such t h a t
151 2<E5>v3, v
(iii) ao(E,t) i s homogeneous i n (3.2.3)
225
(E,c)
E
R+X
(IRn\{O}),
( E - ~ , C )of degree u l + u 2 :
u +u 1 2 t ao(E,c), v t E R + , V
(€,
E IR+X(R*\IO}).
E H-u is a polynomial in 5 ,
then a (6,D) is the differential singular perturbation, 0
E
If a .
f , v2
> -n,
then R + 3
E
-f
a0 (E,.) E S'(R;)
i s a family of
regular distributions and the corresponding singular perturbation can be written as the Fourier integral (3.2.5)
aO(E,D)u(x) = (2n)-" 1 ei<x*s'
. a
(E,
5 ) ( 5 )d5,
Rn
v
E
E R + ,v u E S ( R n ) .
v 2 < 0, -(n+l) < v +v < 0. Let a E H: with u E IR3 such that -n 0 2 3 We shall compute the family of distributions IR+ 3 E + A. (E,. 1 E S ' (Rz),
(3.2.6)
-1
A (€,XI = F5+x ao(E,S),
0
which are the kernels of the convolutions a (E,D)u(x)= (A ( E , . ) * u ) (x), 0 0
u E
S(Rn).
The homogeneity of degree v + u 2 of a ( E , S ) implies: 1 0 - ( v +u +n) (3.2.7) AO(€,x) = 6 AO(X/E),
*
3 . Singular Perturbations on Smooth Manifolds without Boundary
226
Let
]5,/
=
E
-1
b0(5)
where S
=
and l e t
R
=
{y 1
-1
a n d , moreover,
v +v 2 3
(3.2.2)
=
N
,E),
E, E
( x O , x ) . Denote
(nfl+',s),
, 5, =_0 o r 5 = 01. E H yields: 0 v
E Wn+'
bO(ty) = t
(C0,5), x
=
a0(1501
/EO1
The homogeneity of a (3.2.9)
5
v -v
N
(3.2.8)
N
N
, 50 E
-
bo(5),
V t E R+, V
y
E (IRn+'\S),
implies:
w i t h some c o n s t a n t C > 0 . one c a n w r i t e
Obviously,
v -v
15,/
(3.2.11)
-1 1A0(1501
-
-1 -1 ,x) = Fg+xbO(So,C) = FX -tS F - - b o ( 5 ) . 0 X0' 5
A s a consequence of t h e assumption n+u2 > 0 , n + l + v 2 + v 3 > 0 , t h e d i s t r i b u -
tion b
0
E
S'
(R:+')
i s a r e g u l a r one a n d , t h u s ,
5
t h e i n t e g r a l i n t h e r i g h t hand s i d e o f
(3.2.13) being convergent f o r
N
x E (E?n+l\{O}) and V 6 > 0 , s i n c e ( 3 . 2 . 1 0 ) h o l d s and n+v2 > 0 , n+l+u +V > 0. 2 3 One f i n d s , u s i n g t h e homogeneity of b ( 5 ) of d e g r e e 0 N
-I-
where z + T(z) i s t h e r - f u n c t i o n , n+l = {w w E IR , = 11.
=
lTl-'T,?
=
V
2
+v 3
lxl-lz and
Therefore,
u -v (3.2.14)
A
0
(E,X)
= E
3
i ( n / 2 ) (u2+u3+n+l) r ( v 2 +v3 + n + l ) 1(2n)-(n+l)e
*
3.2. Singular Perturbations with Homogeneous Symbols
227
-
- ( v +v + n + l ) (<;',;>+is)
A f t e r t h e change of v a r i a b l e : xo
=
Ixlt,
dwdx
0'
(3.2.14) y i e l d s t h e following
formula: (3.2.15) -(n+l),
.
r ( v +u + n + l ) [ x / 2
-1 J
2
3
3
E
- ( v +v + n t l )
-1
.-iE
J
- ( v +v + n ) v3-vl
i ( n / 2 ) (v2+v3+n+l)
= (2n)
A (E,x) 0
dwdt,
'n+1 where y
=
/ x / - l x and t h e d i s t r i b u t i o n ( t + i O ) h
the distributions ( t + i G ) '
6
when
i s t h e l i m i t i n S ' ( R ) of
(see [Sch
J. 0
1
1,
[G-Sh 1
1).
Example 3 . 2 . 2 . Denote by A z ( c , x ) t h e f a m i l y of d i s t r i b u t i o n s E
0
A Z ( ~ , x =) F-l
0
X;(n)
X;(E~),
=
',
where z
E
-f
C,
A'(€,.) 0
E
S'(R:)
,
0 > Re z > - ( n + l ) .
X'S
Then t h e f o r m u l a ( 3 . 2 . 1 5 ) y i e l d s : (3.2.16)
- I
.
- ( n + l ) i ( n / 2 ) ( z + n + l ) r ( z + n + l~) ~ j - ( z + n ) ~ z
A'(E,x) 0
= ( 2 ~ )
-iE-l
J e
1x1 t ( t w + < Y , W > + l O )
0
- ( z + n + ld)r d t .
'n+1
-
The i n t e g r a l 1
z,n
=
( t < t > - l w 0 +-l+iO)
J
-(z+n+l) dw
'n+l -1 and y = 1x1 x E -1 Indeed, i n t r o d u c i n g y = < t > ( t , y ) ,
d o e s n o t depend on t
E
R
-
N
s u c h t h a t y becomes eo
=
(l,O,
...,0) E
an. = 1 and making a r o t a t i o n
one f i n d s :
W e s h a l l n o t compute h e r e t h e c o n s t a n t I and s h a l l g i v e o n l y t h e z,n f i n a l formula f o r A Z ( c , x ) w i t h 0 > R e z > - ( n + l ) :
0
A s a consequence of i t s d e f i n i t i o n ,
the function
3. Singular Perturbations on Smooth Manifolds without Bounda y
228
c a n b e e x t e n d e d a s a n e n t i r e f u n c t i o n of z E C v a l u e d i n S ' ( R n )
v
E
,
for
E R+. Of c o u r s e , one a l w a y s h a s : l i m A'(E,x) 0 €40
= 6(x)where 6(x) is the Dirac's
6-function. For z
=
1-n one c a n compute t h e i n t e g r a l o n t h e r i g h t hand s i d e o f
( 3 . 2 . 1 7 ) so t h a t
For z = 2m-n-lwhere hand s i d e of
m i s a n i n t e g e r , 1s m < ( n + 1 ) / 2 , t h e i n t e g r a l o n t h e r i g h t
( 3 . 2 . 1 7 ) c a n be a g a i n computed, u s i n g t h e r e s i d u u m
CalCUlUS,
so t h a t one f i n d s i n t h i s case:
(3.2.20)
Azm-n-l(~,x) = 21-n 2m-n-1
-
-1 2-E
1x1
.(n-1)/2r((n-1)/2-m) For Rez +n < 0 one c a n e x p r e s s A ' ( E , x )
-
f u n c t i o n s ( s e e [Grad.
If z
=
A;(E~)
0
Ryz.,
1
c 2' (2m- j - 2 ) OSjsm-1 c ' j ! ( m - l - j ) !
i n t e r m s of t h e m o d i f i e d H a n k e l ' s
1)
2m+a w i t h i n t e g e r rn > 0 a n d a E ( - 2 , 0 ) , t h e n , o b v i o u s l y , 2m = Xo ( E ~ ) X ; ( E ~ ) , so t h a t i n t h i s c a s e one g e t s t h e f o l l o w i n g
r e p r e s e n t a t i o n f o r t h e s i n g u l a r p e r t u r b a t i o n X'(ED):
0
(3.2.22)
XE (E,D)u (x) =
where A
= ( ~ - E ~ A In ~ )A; ( E , x - ~ ) u ( y )dy =
R 2 fnA;(~,x-Y) (I-€ A ) m u(y)dy, Y R
v
C a 2 / a x i i s t h e Laplace o p e r a t o r . 16kSn I n p a r t i c u l a r , u s i n g ( 3 . 2 . 1 7 ) one g e t s t h e f o r m u l a e =
A 0- ' ( E , x )
=
(3.2.23)
and f o r n = 3 t h e f o r m u l a ( 3 . 2 . 1 9 ) y i e l d s : (3.2.24)
.A -2
(E,x)
=
2 -1 - E - l ( 4 n ~1x1) e 1x1,
x
E
IR
3
.
u E S(IR?,
3.2. Singular Perturbations with Homogeneous Symbols Notice t h a t t h e d i s t r i b u t i o n s
E
.A
j.
-2
(E,x)
229
are fundamental s o l u t i o n s
2
f o r t h e 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 1-E A : (3.2.25)
2 (1-E A
-2
)Ao ( € , X I
V
6(x),
=
E R + , x E Bn.
E
We s h a l l e x i b i t a f o r m u l a f o r t h e s p h e r i c a l l y symmetric fundamental L
L
s o l u t i o n s f o r t h e 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 E A -Ax,
appearing
i n t h e l i n e a r e l a s t i c i t y theory. Example 3 . 2 . 3 . Denote by
A”’-2
(E,x)
t h e f a m i l y of d i s t r i b u t i o n s F 2
xo- 2 ,
-2 (E,Sl
=
-1 x - 2 , - 2 0
S X ’
(E,S)
where
S i n c e one c a n a l s o w r i t e
I 1-
2
-E
2
<E5>
-2
,
one f i n d s , u s i n g ( 3 . 2 . 1 7 ) and t h e f o r m u l a f o r t h e s p h e r i c a l l y symmetric fundamental s o l u t i o n f o r t h e L a p l a c e o p e r a t o r : (3.2.26)
.A
-2,-2
(€,XI
=
T ( ( n - 2 ) / 2 ) l x 1 2 - n 2 -2 -E A. n/2
(E,X),
E
E R+,
X
E Rn.
4Tl I n p a r t i c u l a r , one g e t s (3.2.27)
-1 A 0 2 f - 2 ( ~ , x )= ( 4 ~ 1 x 1 ) ( l - e - €
-1 E
E R+, x E R
3
.
Example 3 . 2 . 4 . E -f Y ( E D ) , Y (E Dl = O P ( E 5 < E C > - ~ )w i l l be k k k c a l l e d s i n g u l a r l y p e r t u r b e d Riesz t y p e o p e r a t o r . W e s h a l l denote -1 (E E k < ~ F , > - ’ ) ,so t h a t one c a n w r i t e E j. Y (E,D) a s a f a m i l y Y (E,x) = F k S-tX k of s i n g u l a r i n t e g r a l o p e r a t o r s :
The s i n g u l a r p e r t u r b a t i o n
(3.2.28)
Y
k
(E,D)u(x)
I
= l i m
Yk(€,x-y)u(y)dy, V u E S(IRn).
640 I x - y [ > 6
W e s h a l l compute t h e k e r n e l d i s t r i b u t i o n Y
(E,x).
k C o n s i d e r f i r s t t h e c l a s s i c a l R i e s z o p e r a t o r Yk
=
Op(CklCl
-1
I.
One h a s : (Y u )
k
(x)
=
Op(lC
-1
)Dku(x).
1
The k e r n e l 12-l (x) o f t h e i n t e g r a l c o n v o l u t i o n o p e r a t o r Op( 15 - l ) c a n b e
0
e a s i l y computed:
230
3 . Singular Perturbations on Smooth Manifolds without Boundary
where the constant
Tk.e
C
can be found using the formula:
last formula yields:
Therefore, the integration by part
implies:
+
ic lim i 6-("+l) (xk-Yk)u( y "6J-0 lx-yl=6 - (n+l = -ic (n-1)v.p. J (x -y lx-yl Rn k k
+ ic lim i u(6w+x)w dw = k n6J-0 Rn /x-y~-(~+l = -icfi(n-l)v.p.i (x -y =n k k
Further, denoting, as previously,
E
=
/cO1-1 , -5 =
N
(C0,5), x
=
(xO,x), one
gets after the change of variable: xo = lxlt the following formula for the Riesz operator with a small parameter
:
-ix 5 0 On-(n+2)/2T((n+2)/2) v.p. xkiXl-(n+2)dx0 = =-i Je R
The operators Y (ED) can be also expressed in terms ofthe modified Hankel's k functions (see [Grad. Ryz., 1 1 ) :
-
3.2. Singular Perturbations with Homogeneous Symbols
The l a s t f o r m u l a c a n b e a l s o d e r i v e d from ( 3 . 2 . 1 7 ) , and u s i n g t h e r e l a t i o n : K ( r ) = K
231
(3.2.18), s e t t i n g z = 1
( r ), V v > 0 .
-V
Example 3.2.5. The s i n g u l a r p e r t u r b a t i o n A ( E D ) 0
=
O p < ~ pi s c a l l e d t h e
Riesz o p e r a t o r
w i t h s m a l l p a r a m e t e r . One f i n d s :
=
A -1 ( E D ) u ( x ) 0
-
C
i E
Yk(€D)
l5kSn
au ( x ) = axk
w y ayk The f o l l o w i n g s i n g u l a r p e r t u r b a t i o n of t h e one-dimensional
d
t
.
Riesz o p e r a t o r
(which i s t h e H i l b e r t t r a n s f o r m ) a p p e a r s i n t h e d i s l o c a t i o n t h e o r y ( s e e , f o r instance [ Hirth
-
Lothe,
1
1).
Example 3.2.6. Let
oE
= o p ( s g n c ( l + e x p ( - E / t / ) ) ,E,
E
R
. One
finds easily that
E +
OE i s
t h e 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 : OEU(X)
=
i
4v.p.
The m u l t i - d i m e n s i o n a l
(E,x)
u (y) ---dy x-Y
+
a n a l o g u e s of OE a r e t h e o p e r a t o r s :
One f i n d s t h e k e r n e l O k,n
0 k,n
1
= D 0
(E,x)
k n
o f t h e c o n v o l u t i o n o p e r a t o r s OE: k
(E,x),
where
For n
=
2 , e v a l u a t i n g t h e l a s t i n t e g r a l by means of t h e residuum c a l c u l u s ,
3 . Singular Perturbations on Smooth Manifolds without Boundary
232
so t h a t
(3.2.32)
0
k,2
(€,XI
= 2iT
-1
2 -3/2
2
x k ( c +41xI )
,
k = 1.2.
For n 2 2 one h a s : (3.2.33)
a
(E,x)
=
(n-1) ! 7
(2n)
71
J s i nn-2e ( ~ - i / ~ j ec) 1-n ~ sd e , 0
.
i s t h e area o f t h e u n i t s p h e r e i n R n - l where R n- 1 Hence, an e a s y c o m p u t a t i o n y i e l d s :
Of c o u r s e , i n any case, one h a s : (3.2.35)
lim
(E,x-y)u(y)dy
0
€40 Rn
=
2Yku(x),
V u E S(IR~).
krn
W e are g o i n g t o i n d i c a t e one m o r e c l a s s of s i n g u l a r p e r t u r b a t i o n s
with
homogeneous symbols, which i s i n t e r e s t i n g f o r a p p l i c a t i o n s . Example 3.2.7. Denote by AZ(ED',&D ) t h e s i n g u l a r p e r t u r b a t i o n w i t h t h e symbol
where y > 0 and z E C a r e g i v e n . F i r s t assume t h a t R e
2
< 0 . Using t h e formula
ni2/2 (3.2.37)
(-a+iB)'
=
r (-2)
t-z-l
-8t-iat
dt,
Re z < 0 ,
=+
we find that
where H(x ) i s t h e H e a v i s i d e ' s f u n c t i o n One f i n d s e a s i l y t h a t
I n f a c t , u s i n g t h e r e s i d i u m c a l c u l u s ( o r (3.2.37)
w i t h z = -11,
one f i n d s
3.2. Singular Perturbations with Homogeneous Symbols
233
Therefore,
which l e a d s t o ( 3 . 2.39) when x The f o r m u l a e ( 3 . 2 . 3 8 ) .
> 0.
( 3 . 2 . 3 9 ) imply f o r R e z < 0
so t h a t
I n t h e same manner one f i n d s f o r R e z < 0
Let R e z > 0 and l e t z
=
r + <w i t h i n t e g e r r > 0 a n d R e 5 < 0 .
W e c a n w r i t e down:
Hence.
and i n o r d e r t o f i n d t h e k e r n e l of t h e o p e r a t o r s compute t h e i n v e r s e F o u r i e r t r a n s f o r m of f i n d s:
ED' , E D -
)
one h a s t o
15' / X c ( c S ' , ~ C n ) , a s w e l l . One
*
3 . Singular Perturbations on Smooth Manifolds without Boundary
234
Using t h e same argument a s p r e v i o u s l y , one f i n d s :
so t h a t
> 0 , then t h e l a s t formula y i e l d s :
If x
we can again rewrite t h e s i n g u l a r p e r t u r b a t i o n s
Combining ( 3 . 2 . 4 0 ) - ( 3 . 2 . 4 2 ) ,
a s a f a m i l y of i n t e g r o - d i f f e r e n t i a l
,ED,)
A:(ED'
o p e r a t o r s . The s a m e argument
l e a d s t o 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 X-Z ( s D ' , s D n ) . F i n a l l y , i f z = r > 0 i s i n t e g e r , t h e n one u s e s o n l y ( 3 . 2 . 4 1 ) , f o r g e t t i n g a r e p r e s e n t a t i o n of A T ( E D ' , E D
)
-
(3.2.42)
a s a f a m i l y of i n t e g r o - d i f f e r e n -
t i a l operators. Example 3 . 2 . 8 . Denote by S ( E , D ) t h e s i n g u l a r p e r t u r b a t i o n , whose symbol i s t h e f o l l o w i n g
e
matrix
where 0 E ( - n / 2 , n / 2 )
is given.
The s i n g u l a r p e r t u r b a t i o n S ( E , D ) w i t h t h e symbol ( 3 . 2 . 4 3 ) f o l l o w i n g S t o k e ' s problem i n R n (E
(3.2.44)
-2 2 i 8 e -A
where u : R n
,u> -f
)u
+ v p
= f,
= 0,
Cn, p : W n +
Ax i s t h e L a p l a c i a n , B
solves the
:
x
E xtn
x
E
IRn
C a r e t o be d e t e r m i n e d ,
: En+ C is
f
i s t h e g r a d i e n t and = d i v u
=
given,
au/ax.. l<j
'
One c h e c k s e a s i l y u s i n g t h e F o u r i e r t r a n s f o r m t h a t u = S (E,D I f . e x 2 -2i8 2 Since I+E e 151
(3.2.45)
z
O,V
(E,s)
ER+
x
nn,
if
e
E (-n/2,T/2),
it i s
3.3. Singular Perturbations with Variable Symbols q u i t e o b v i o u s t h a t t h e f o r m u l a (3.2.17) h o l d s f o r
235
r e p l a c e d by
E
Ee
-iB
.
I n d e e d , b o t h l e f t and r i g h t hand s i d e i n (3.2.17) a d m i t a n a n a l y t i c extension with r e s p ect t o E
E R
onto the half-plane
larg
€1
< ~/2.
Using t h e second formula (3.2.23) and t h e c l a s s i c a l R i e s z o p e r a t o r s Yk
(Example 3.2.4), one c a n r e p r e s e n t S e ( € , D ) as t h e c o m p o s i t i o n o f two
operators:
(3.2.46) where, a s For n = 3
(3.2.47)
N o t i c e t h a t f o r each 8 E (-n/2,n/2)one h a s :
(3.2.48) l i m
E - ~ S( E , D ) f ( x ) = e - 2 i e ( f ( x ) - ( Y , e
E++O
x
for
v
f
D
Y.)f)(x)), 7
E ( s ' ( R ~ ) ) "= s ~ ( R X ~ . . . X)S ' ( R " )
F u r t h e r m o r e , S ( E , D ) c a n be e x t e n d e d a s
e
(3.2.49) S B ( ~ , D x ):
x (H(s), E ( R n ) I n
+
(H(s-v),e (Rn)) n
w i t h v = (-2,0,-2),univormly w i t h r e s p e c t t o E E ( 0 . ~ ~ 1 ,v E~ < *, i . e . t h e norms of t h e o p e r a t o r s (3.2.49) a r e bounded by a c o n s t a n t which depends o n l y on
and n . I n d e e d , t h i s i s a n immediate consequence o f
E~
(3.2.43) and
Parceval's identity. Using a r e s u l t from [ M i k h . l , Z I , o n e c a n show t h a t (3.2.49) s t i l l h o l d s i n spaces spaces
H
( R n ) w i t h L -norms 1 < p < a, 5 s ),2,E( R n ) b e i n g t h e P ( R n ) ( s e e 3.12 f o r m o r e d e t a i l s ) .
(s)rP,E
H (s),E
3.3. S i n g u l a r p e r t u r b a t i o n s w i t h v a r i a b l e s y m b G W e c o n s i d e r h e r e c l a s s e s of symbols which are € u n c t i o n s of
( x , E , ~ ) and
i n t r o d u c e t h e c o r r e s p o n d 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 i n c l u d e s i n g u l a r l y perturbed d i f f e r e n t i a l o p e r a t o r s with v a r i a b l e c o e f f i c i e n t s ,
as w e l l . D e f i n i t i o n 3.3.1.
Let u be an open s u b s e t of cZass
S'
1,o a(x,E,c) E
R~
and Zet v
(u) i s d e f i n e d t o c o n s i s t
=
(vl,v2,v3)E R
3
.
The syrnboZ
o f a22 f u n c t i o n s
s a t i s f y i n g t h e c o n d i t i o n : f o r any compact K c u such t h a t and any couple of m Z t i - i n d i c e s a , t~h e r e e x i s t s a c o n s t a n t c
c
(UX ( o , E ~ ]x
Ka I3
.
3. Singular Perturbations on Smooth Manifolds without Bounda y
236
D e f i n i t i o n 3.3.2.
The symboZ a E
S'
1,o
(u) belongs t o t h e symbol c l a s s s'(u)
if t h e r e e x i s t s a
representation (3.3.2)
a ( x , ~ , t ; )= a ( x , ~ , < ) + r ( x , ~ , S ) 0
where a O ( x , ~ , E is ) homogeneous i n v +v2
1
(E-',s)
for
E
E ( O , c o l , 151 2 1 , of degree
i.e. - ( v +v )
a (x,t-lE,tS), 0
a0(x,E,S) = t
(3.3.3)
v
t 2 I,
v
E
E (o,EOl,
v 5 E mn, and where t h e remainder r ( x . E . 6 ) on each compact
K c
151
'
u s a t i s f i e s the
inequalities
for a l l x E K ,
E ( O , E ~ ] , E, E B~ and all m u l t i - i n d i c e s
E
a,B
P r o p o s i t i o n 3.3.3.
Let a E
Sv
1,o
(U),
b
E 'S
1,o
.
(U) Then
Proof. The s t a t e m e n t s of t h i s p r o p o s i t i o n are an immediate consequence of t h e chain rule. D e f i n i t i o n 3.3.4.
Let a E sV
op a
=
(u),( r e s p e c t i v e l y , a E sv (u)) . The s i n g u l a r p e r t u r b a t i o n
1,o a ( x , ~ , ~ is)
d e f i n e d as f o l l o v s
1,
231
3.3. Singular Perturbations with Variable Symbols
The s i n g u l a r p e r t u r b a t i o n a ( x , ~ , i~s) s a i d t o belong t o t h e c l a s s ( r e s p e c t i v e l y , t o t h e c l a s s op
op S;,,(U)
Sv(U)).
Theorem 3.3.5.
If
a
E
m
s ~ , ~ ( U ) ,t h e n a ( x , c , D ) : C o ( U ) + C"(Ui
a ( x , ~ , ~: )E ' ( u ) E"
-t
D'(u),
m 1 a(x,E,D) : Co(U)
+
V
E
( 0 , ~I . 0
E
and E"'a(x,E,D)
Cm(U)
and i t can be eztended a s
Moreover, : E'(U)
-t
D'(U)
E
(E
(O,En])
are e q u i c o n t i n u o u s . Proof. If a
rn
E sv
1,o
(U), u E C o ( U ) ,
then t h e function
Evla(x,E,D)u(x) = (2nl-n
e
i < x , S > '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,<)v(x)e dx)dS
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 l a s t i n t e g r a l i s w e l l
E'(U),
s i n c e t h e family o f f u n c t i o n s
i s r a p i d l y d e c r e a s i n g when
5
+
m
uniformly with r e s p e c t t o
E
E (O,E~],
3. Singular Perturbations on Smooth Manifolds without Bounda y
238
f o r any u E E ' ( U )
so t h a t t h e f a m i l y of f u n c t i o n a l s (3.3.11)
bounded w i t h respect t o
E
E (O,E
0
1.
i s uniformly
I
Remark 3.3.6.
(0) (0)
1
L e t a E S v ( " ) ( U ) where i n t e g e r v . t 0, j = 2,3, and l e t E a ( x , E , c ) be 3 polynomial i n ( ~ ~ 5 L)e .t a , ( x , E , S ) b e t h e p r i n c i p a l symbol o f a ,
a-a,
,
(0) ( 0 ) > v(l)+v(l) 1 2 v 1 +v2 I f a t e a c h s t e p o f t h i s p r o c e s s t h e remainder r k = (a-aoa ) E k- 1 (k) ( U ) , t h e n t h e symbol a ( x , E , S ) c a n b e r e p r e s e n t e d a s a f i n i t e sum:
where, of c o u r s e , u(O) > v(l) and moreover,
...-
Sv
a ( x , ~ , C )=
(3.3.13)
C ak(x,E,<) O
where ak have t h e v e c t o r i a l o r d e r v ( ~ ) a, r e homogeneous i n d e g r e e V : ~ ) + V ; ~ ) and, moreover, t h e o r d e r s v ( ~ ) ,0 5 k
=< N,
(E
-1
, 5 ) of
form a
s t r i c t l y d e c r e a s i n g sequence i n t h e f o l l o w i n g s e n s e :
The s p l i t t i n g (3.3.13)
i n t o a sum of homogeneous t e r m s ak, whose o r d e r s
v ( k ) s a t i s f y (3.3.14),
c a n be viewed a s a n a t u r a l g r a d u a t i o n of a ,
a n a l o g o u s t o (3.1.2). Of
c o u r s e , any symbol a
which i s w e l l d e f i n e d f o r
E
h a s a z e r o t e r m a.
Sv(U)
E (O,E~],
E
e x t e n d e d a s a homogeneous f u n c t i o n o f
of graduation,
151 2 1, x E U and can b e u n i q u e l y -1 , 5 ) E IR+x (lRn\{Oj) f o r V x E
(E
U
Example 3.3.7. The symbol a ( x , E , C ) c(x) : U
-f
C,
= E
2
j5j4+1512++c(x), w i t h b ( x ) : U
I t admits a n a t u r a l g r a d u a t i o n :
a (x,E,~) 0
+
Cn,
smooth f u n c t i o n s , b e l o n g s t o S V ( O ) ( U ) w i t h v ( O ) = (0,2,2).
2 = E
a
=
4
a o + a l + a 2 , where
IS/ +l5I2, al(x,E,5)
v ( O ) = (0,2,2), v ( l ) = (O,l,O), v")
= cb(x),C>, 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 <5-n>'
<5>'
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
<5>-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
belongs t o S
Remark 3.4.6. Using t h e s a m e argument a s i n t h e p r o o f of Theorem 3.4.1,
one shows t h a t
m
(U) and g E C o ( U ) , t h e n H (u) 3 u + g O p a u E i f a ( x , E , E ) E S" 1,o ( s ) ,E H f U ) i s a f a m i l y of c o n t i n u o u s l i n e a r mappings, whose norms a r e ( s - v ) ,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 ~ ] . I n o t h e r words, f o r each E~
> 0, s
E
R 3 t h e f o l l o w i n g f a m i l y o f l i n e a r mappings i s e q u i c o n t i n u o u s :
a(x,E,D) where
E +
gu E H
3.5.
H
1oc
H
:
loc
(u)
( s ) ,E
+
H ( s - " ), E
(U),
( U ) i s t h e f a m i l y of s p a c e s c f a l l f u n c t i o n s u s u c h t h a t
( s ) ,E
(U) f o r ( 5 ),E
v
g
E
m
Co(U).
The p s e u d o l o c a l 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
W e s t a r t with t h e following
D e f i n i t i o n 3.5.1.
A f a m i l y of d i s t r i b u t i o n s
on an open s e t
v 5u
(O,E~I3
E + U(E,X)
i f f o r any g i v e n
E
E D' (u) i s s a i d t o be smooth
E ( 0 ,0 I~ it c o i n c i d e s idith some
) f o r any compact s e t f u n c t i o n u ( x ) E c ~ ( v and,moreover, (3.5.1)
max xEK
1
I D ~ ~ ( E , X )5
where t h e c o n s t a n t s
C
K,a
c ~ , ~ v ,a, v
KCC
do n o t depend on
E
KCC
v
one has:
v
-
E ( 0 , ~ 01
Definition 3.5.2.
The complement of t h e maximal open s e t on u h i c h a f a m i l y o f d i s t r i b u t i o n s E + U(E,X)
i s smooth, i s c a l l e d t h e s i n g u l a r support of u and denoted by
s i n g supp u ( E , x )
.
3. Singular Perturbations on Smooth Manifolds without Bounda y
2 42
Example 3.5.3. IR+ 3
The f a m i l y o f d i s t r i b u t i o n s :
singular support t h e o r i g i n x
=
E
+
Eaexp(-/xl
2
/E
2
)
E S ' (Xn) h a s as i t s
0 , V a E IR.
Our p u r p o s e i s t o prove t h e f o l l o w i n g Theorem 3.5.4.
Let ( o , E 3 E~ + I u(x,E) (3.5.2)
E
(u) and l e t
€ 1
(u). Then a ( x , E , ~ )E S" 1,o
s i n g s u p p ( ~ " a ( x , ~ , D ) u ( ~ , x5 ) ) s i n g supp u ( E , x ) .
Proof. W e associate w i t h E"la(x,E,D)
which i s a f a m i l y of
its kernel K(E,x,Y),
distributions
d e f i n e d by t h e r e l a t i o n
According t o Theorem 3.3.5,
the kernel-distribution m
f a m i l y of c o n t i n u o u s l i n e a r maps K : C o ( U ) W e are g o i n g t o show t h a t t h e f a m i l y (O,E
0
+
1
i n (3.5.3)
Cm(U) and
3
E +
K
:
defines a
E' (U)
+
D ' (u).
K ( E , x , ~ ) i s smooth o f f t h e
d i a g o n a l x = y i n UxU. Indeed, u s i n g (3.5.3)
and t h e d e f i n i t i o n of s i n g u l a r p e r t u r b a t i o n s ,
one f i n d s e a s i l y t h a t
Consequently,
The l a s t f o r m u l a f o r l a / > u +u +n c a n b e r e w r i t t e n as a c o n v e r g e n t i n t e g r a l 2 3
a n d , a s a consequence o f t h e u n i f o r m convergence of 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 , one g e t s t h e c o n c l u s i o n t h a t ( x - ~ ) ~ K ( E , x , ~ E) C J ( U W ) i f
/ a / > v 2 +v3 +n+j. Moreover, a l l t h e d e r i v a t i v e s up t o t h e o r d e r j of (~-y)~K(~,xa ,y 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 ( 0 . ~ ~ 1T. h a t p r o v e s t h e claim t h a t
E
+ K(E,x,~)
h a s as i t s s i n g u l a r support a c l o s e d set
243
3.6. Asymptotic Expansions of Symbols belonging t o t h e diagonal x = y i n UXU. I n o r d e r t o end t h e p r o o f of t h e theorem, w e need o n l y t o a p p l y t h e following Lemma 3.5.5.
Let
( O , E ~ ] 3 E + K ( ~ , x , y )E
D ' ( u x u ) . Suppose t h a t t h e famiZy of Zinear
maps m
m
(3.5.7)
K : Co(U)
+
C (U), K : E ' ( U )
is u n i f o r m l y bounded i n t h e diagonal in
E
+
D'(U)
E (O,c0], and, moreover, t h a t K is smooth off
UXU.
Then s i n g supp Ku c - s i n g supp u, V u E E ' ( U )
.
Proof of Lemma 3 . 5 . 5 . Let
v
and W b e open s u b s e t s of U s u c h t h a t W C C V. m
Let $ E
co(v),
Let
u E E ' ( U ) b e such t h a t u i s smooth i n V . Then Ku = K ( $ u ) + K ( ( l - $ ) u )
E -t
$
1 i n a neighborhood of t h e c l o s u r e of W. m
Since E
E +
$u i s smooth on U , and E
E (O,E~],
K(@)
$u E C (U) u n i f o r m l y w i t h r e s p e c t t o
+
0
i s smooth on U . On t h e o t h e r h a n d , w e c a n w r i t e
K ( ( l - $ ) u ) as an i n t e g r a l K((l-$)u) =
1
K(E,x,Y)
(I-$(y))u(y)dy.
U I f x E W , y E s u p p ( 1 - $ ( y ) ) , t h e n ( x , y ) b e l o n g s t o t h e complement o f some s u i t a b l e neighborhood of t h e d i a g o n a l i n U X U , t h a t i s ( x , y ) b e l o n g s t o a m
s u b s e t where K ( E , x , ~ ) i s smooth ( C
in (x,y) with a l l the derivatives
E ( 0 ,0 ~1 ) . Hence, K ( ( 1 - $ ) u ) i s smooth on V a s w e l l . S i n c e V w a s any open s u b s e t of U, one g e t s t h e c o n c l u s i o n
u n i f o r m l y bounded w i t h r e s p e c t t o
t h a t s i n g supp Ku C_ s i n g supp
E
U.
1
3 . 6 . Asymptotic Expansions of symbols A s it h a s been mentioned i n Remark 3 . 3 . 6 ,
admit a n a t u r a l g r a d u a t i o n
(3.3.13),
some p o l y n o m i a l i n (E,E)
symbols
(3.3.14).
One c a n c o n s i d e r f o r m a l i n f i n i t e sums o f symbols of t h e form ( 3 . 3 . 1 3 ) whose o r d e r s s a t i s f y ( 3 . 3 . 1 4 ) . One o f t h e b a s i c m o t i v a t i o n s t o c o n s i d e r such f o r m a l sums i s t h e f a c t t h a t t h e y a p p e a r i f one expands t h e i n v e r s e symbols t o e l l i p t i c polynomial symbols of t h e form ( 3 . 3 . 1 3 1 ,
(3.3.14)
into
a n a s y m p t o t i c series. On t h e o t h e r hand, it i s c o n v e n i e n t t o have a symbol f u n c t i o n f o r which a g i v e n f o r m a l sum of symbols c a n b e c o n s i d e r e d a s an
3. Singular Perturbations on Smooth Manifolds without Boundary
244
a p p r o p r i a t e l y i n t e r p r e t e d asymptotic expansion. Theorem 3 . 6 . 1 .
i)
L e t a . E syl0 ( u ) , v (1")
< v(j), j
7
v (11 ) J.
(3.6.1)
-a
0,1, and l e t
=
for j + ( 01
Then t h e r e e x i s t s a E s V
1.0
a(x,E,c) -
(3.6.2)
such t h a t (N) (U), V N t 0
C a . ( x , ~ , E ;E ) Sv 1r o O=<j
and moreover, f o r each p a i r of m u l t i - i n d i c e s a , @ and every compact s e t K c U
t h e f o l l o w i n g asymptotic r e l a t i o n s hold:
unifomnZy w i t h r e s p e c t t o (x,c;) E
K x
mn.
Proof. L e t + ( t )E
crn(K+) ,
Let K .
C K
K
c
o
E
...,
for t E
[o,tI, 4
z 1 for t t I.
b e a sequence of compact s e t s e x h a u s t i n g U, i . e .
3' 1 2 t h e union o f a l l K . c o i n c i d e s w i t h U. 3 We choose 6, so small t h a t
f o r V j t 1, V
E
E ( 0 . ~ ~ 1V, 5 E mn,
V x E Krr
v
a,B, r w i t h
l a i + l f I + r5 j .
T h i s i s p o s s i b l e s i n c e v!J)
+
-m.
I n f a c t , one h a s
Let
W e choose 6,
> 0 t o b e so s m a l l t h a t
I n t r o d u c i n g t h e numbers m. =
max t +lt
1
(1-1) 1
+ ( t I)
one c a n s a t i s f y t h e i n e q u a l i t y ( 3 . 6 . 6 ) by c h o o s i n g 6 . > 0 s u c h t h a t 7
3.6. Asymptotic Expansions of Symbols
245
Y. w i t h y , = ( v ~ J ) - v ~ J - ' ) ) - <' 0 .
6 . 4 ( 2 J C .m . ) 3
3
3
3
Moreover, w e r e q u i r e :
6.
1
+
j +
for
0
m.
Now we d e f i n e t h e symbol a ( x , ~ , E )t o be (3.6.7)
j
a(x,e,E)
-1
@ ( 6 . ~ ) a .( x , E , S ) ,
C
=
j>o
3
V (x,E,S)
E u
x
( 0 . ~ ~x 1R ~ .
One c h e c k s e a s i l y t h a t f o r a ( x , E , C ) d e f i n e d by ( 3 . 6 . 7 ) w i t h 6 . 3 s a t i s f y i n g (3.6.4), the r e l a t i o n s (3.6.2), (3.6.3) hold.
+
0,
+ m,
I n f a c t , i t i s enough t o check ( 3 . 6 . 3 ) s i n c e ( 3 . 6 . 2 ) i
