Advances in
Deterministic and Stochastic Analysis
EDITORIAL BOARD N g u y h Minh Chuang Phillipe G. Ciarlet Takeyuki Hida Peter Lax David Mumford Duang H6ng Phong Roger Temam Nguyzn VBn Thu Nguy6n Minh Tri Vii Kim Tudn
Advances in
Deterministic and Stochastic
Analysis Editors
N M Chuong Institute of Mathematics, Vietnamese Acad. of Sci. &Tech., Vietnam
P G Ciarlet City University of Hong Kong, Hong Kong
P Lax Courant Institute, USA
D Mumford Brown University, USA
D H Phong Columbia Universitv, USA
N E W JERSEY
*
LONDON
*
K6 World Scientific -
SINGAPORE
*
BElJlNG
*
SHANGHAI
HONG KONG
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TAIPEI
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CHENNAI
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ISBN-13 978-981-270-550-1 ISBN-I0 981-270-550-3
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V
PREFACE
In Junc 4-9, 2005, the Sccond International Conference 011 Abstract and Applied Analysis was held at Quy Nhon, Vietnam. The conference brought together Vietnamese mathematicians working on analysis, as well as many distinguished foreign visitors. It covered all areas of analysis where Vietnamese mathematicians have been playing an active role, including pure and applicd analysis, and dctcrministic and stochastic approaches. The proceedings of the conference are gathered in the present volume. The spccific topics discussed at the confcrcncc rcflcct thc most activc rcsca,rch directions in Vietnam today, and as such, are quite varied. Seventeen papers appear here. For convenience, they have been loosely organized into four different. chapters. Chapter I, “Integral and Pseudodifferential Operators”, includes the papers where these operators play a major role, particularly in the theory of partial differential equations. The paper by Nguyen Minh Tri provides a criterion for the hypoellipticity for a class of operators with sign-changed characteristic form, generalizing earlier works of Beals and Fefferman, The paper by Nguyen Minh Chuong and Dang Anh Tuan deals with generalizations of the oblique derivative problem in several directions, including non-lincar settings. The last two papers in the chapter deal more with the properties of the operators themselves, with the boundedness of commutator integrals of mixed homogeneity studied by Lubomina Softova, and a classification of integral transforms of both convolut.ionand non-convolution types provided by Vu Kim Tuan. Chapter 11, “Partial Differential Equations” , deals of course also with partial differential equations, but with essentially different methods than the pseudodifferential and integral operator methods of Chapter I. The paper by Martin Schechter provides a unified min-max approach to variational problems, through the idea of linking sets separating a given funct.iona1. The papcr by Mikio Tsiji and Peter Wagner studies generalizcd solutions
vi
Preface
and the possible appearance of shock waves to a class of equations generalizing Burgers’ equation. Boundary value problems for the biharmonic equation are studied in the paper by Dang Quang A and Le Tung Son, by a reduction to a boundary value problem for a second order equation, combined with an iterative scheme. A parabolic boundary value problem, arising from an inverse problem in engineering for finding the heating regime for a given desired final temperature, is studied in the paper by Dinh Nho Hao, Nguyen Trung Thanh, and H. Sahli, by the DC (“difference of two convex functions”) method. The paper by Peter Massopust describes explicit solutions, in various coordinate systems, for the Maxwell equations describing magnetic fields generated by a hard ferromagnet. The paper by Ha Tien Ngoan and Nguyen Thi Nga deals with the local solvability of the non-characteristic Cauchy problem for weakly hyperbolic classical MongeAmpere equations. The asymptotic analysis of the solutions of the NavierStokes problem is considered by Makram Hamouda and Roger Temam when the viscosity goes to zero and the flow in a channel of R3 is also considered by them in the case when the boundary is non-characteristic. Chapter 111, “Geometric Analysis”, groups together the papers with a more pronounced geometric flavor. The paper by Le Hong Van provides a construction of obstructions to imbedding a given statistical manifold into another one. A statistical manifold is a Riemannian manifold with an equivalence class of symmetric 3-tensors, a geometric concept which arises from information geometry. The paper by Do Ngoc Diep constructs a 22graded Cech cohomology, building in ideas of Rosenberg and Kontsevich from noncommutative geometry. The paper by Nguyen Minh Chuong and Le Duc Thinh studies Sobolev spaces with weights on Riemannian manifolds. Finally, as its title indicates, Chapter IV, “Stochastic and InfiniteDimensional Analysis”, groups together papers more related to either probability or analysis in infinite-dimensional spaces. They include the paper by Situ Rong, on the existence of solutions to some reflexive stochastic differential equations with jumps and its applications to stochastic population control; the paper by Shigeyoshi Ogawa, which provides a survey of developments related to a notion introduced by the author; the paper by Andrei Khrennikov, which discusses the foundations of quantum mechanics from the point of view of infinite-dimensional spaces, and the paper by Karl Gustafson, which discusses the ideas of the author on Bell’s inequality and contraction semigroups.
The Editors
vii
CONTENTS
Preface
Chapter I
V
Integral and Pseudodifferential Operators
$1.Pseudodifferential Operators of Second Order with Sign-Changed Characteristic Form Nguyen Minh Tri $2. A Semilinear Nonclassical Pseudodifferential Boundary Value Problem in Sobolev Spaces 1 < p < 00
3
15
Nguyen Minh Chuong and Dang Anh Tuan $3. Singular Integral Operators in Functional Spaces of Morrey Type Lubomira Softova
33
$4. Classification of Integral Transforms
43
Vu Kim Tuan Chapter I1
Partial Differential Equations
$5. Unified Minimax Methods
75
Martin Schechter $6. Some Remarks on Single Conservation Laws Mikio Tsuji and Peter Wagner
91
viii
Contents
$7. Iterative Method for Solving a Mixed Boundary Value Problem
for Biharmonic Type Equation Dung Quang A and Le Tung Son
103
$8. Numerical Solution to a Non-Linear Parabolic Boundary Control Problem Dinh Nho Hao, Nguyen Trung Thanh and H. Sahli
115
$9. A Class of Solutions to Maxwell’s Equations in Matter and Associated Special Functions Peter Massopust
131
510. On the Cauchy Problem for a Quasilinear Weakly Hyperbolic System in Two Variables and Applications to that for Weakly Hyperbolic Classical Monge-Ampkre Equations Ha Tien Ngoan and Nguyen Thi Nga 511. Some Singular Perturbation Problems Related to the Navier- Stokes Equations Makram Hamouda and Roger Temam
Chapter I11
177
197
Geometric Analysis
$12. Monotone Invariants and Embeddings of Statistical Manifolds Le Hong Van
231
$13. Graded Cech Cohomology in Noncommutative Geometry Do Ngoc Diep
255
$14. Sobolev Spaces with Weight on Riemannian Manifolds Nguyen Manh Chuong and Le DUC Thinh
269
Chapter IV
Stochastic and Infinite-Dimentional Analysis
$15. Stochastic Population Control and RSDE with Jumps Situ Rong
281
Contents ix
$16. Noncausal Stochastic Calculus Revisited So-called Ogawa Integral Shigeyoshi Ogawa
-
Around the
5 17. Infinite-Dimensional Stochastic Analysis and Foundations Quantum Mechanics Andrei Khrennikov
818. Noncommutative Trigonometry and Quantum Mechanics Karl Gustafson
297
of 321
341
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Chapter I
INTEGRAL AND PSEUDODIFFERENTIAL OPERATORS
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Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 3-13) @ 2007 World Scientific Publishing Co.
3
51. PSEUDODIFFERENTIAL OPERATORS OF SECOND ORDER WITH SIGN-CHANGED CHARACTERISTIC FORM NGUYEN MINH TRI Institute of Mathematics 18 Hoang Quoc Viet, Cau Giay, 10307 Hanoi, Vietnam E-mail: triminhOmath.ac.vn
In this paper we prove hypoellipticity for pseudodifferential operators of second order with sign-changed characteristic form. The obtained results extend the previous ones of Beals-Fefferman and Ganja for differential operators.
In this paper we consider operator of the form k
in a bounded domain R c R$' = IRE x @, where XI,. . . ,xk are proper pseudo differential operators (p. d. 0.) of first order with real principle symbols Xj(x,t,J,~),j = and c ( z , t ) is a smooth function in R. In Ref. 1 Hormander proved that for a differential operator of second order to be hypoelliptic it is necessary that its characteristic form does not change sign at a fked point in 0. However, Kannai's example in Ref. 2 showed that characteristic form of a hypoelliptic operators can change sign from point to point in 52. Necessary conditions for hypoellipticity of operators generalizing the Kannai example were studied in papers of Helffer and Z ~ i l yand , ~ Beals and Fefferman.4 In Ref. 4 it is proved that for the differential operator P in the form (1) to be hypoelliptic it is necessary that XI,. . . ,xk satisfy the following conditions
Xj(~,0,0,1)=0, j = 1 , ..., k. 2000 Mathematics Subject Classification Primary 35H10; Secondary 35A08, 35B65, 35C15. 35D10.
4 N . M. Tri
In Ref. 4 Beals and Fefferman also established some sufficient conditions for the differential operator P in the form (1) to be hypoelliptic. In Refs. 5,6 Ganja obtained further sufficient conditions for the differential operator P in the form (1) to be hypoelliptic. In this note we extend the results of Beah and Fefferman, and Ganja for the pseudodifferential operator P in the form (1).Now we introduce the transformation: %,t)
+ fi;d,t,)
(x,t ) -+ (21,t’); 5’ = 5 , t’
= 6t; 6 E (0,1].
For a function u E C r ( 0 ) we can define the function follows:
E C r ( 0 ) as
ug(z‘, t’) = u ( 5 ,t’6).
Note that when 6 = 1 we can identify the coordinates ( z , t )and ( d , t ’ ) .
we can get easily the desired estimates. Let Ps be the operator in the coordinates (d, t’) defined by the formula:
P.d.0 of Second Order with Sign-changed Characteristic Form
5
Lemma 2. Let the operator Pa have the form ( 2 ) . Then f o r arbitrary s E R and for arbitrary K c R there exists a constant C(6)= C(s, K , 6 ) such that
c k
ll6t’xj,6911,2 6 21(6t’%71g>sl + C(~)lIgl1,2, vg E C,-(K),
(3)
j=1
where
11 . ]Is, ( , ) s
are taken in the (x’,t’) coordinates.
Proof. Put Bj,6 = 6t’Xj,s - (dt’Xj,s)*. Obviously, ord Bj = 0 (since Xj,a has real principle symbol). Note that k
6t‘Ps = C ( 6 t ’ X j , , ) 2
+Qs,
j=1
where
Qs = -t’:
a
k
+ bt’cs(x’,t’)
at
S2t’Xj,s(t’)Xj,6; ord Qs = 1.
-
j=1
Thus, Lemma 2 is proved.
0
Corollary 1. From the inequality (3) it implies that
c k
llt’xj,sgIl: 6 2llt’PsgII,2
+ C’(~)11g11,2~
j=l
Lemma 3. Let X j be the proper homogeneous p. d. symbol Xj(x,t ,El r ) . Moreover assume that axj
-h 37-
t , E,
= t c p j (5, t , E, 7)
0.
with real principal
(4)
6
N . M. Tri
for j = 1,.. . ,Ic; ord
P&=
‘pj = 0.
Then operator Pa can be written in the form
c
k
l d 6 at‘
f
S
k
c
+
Xj,&t‘Xj,& 6t’
j=1
+
‘Pj,&Xj,S
C&(Z’, t’),
j=1
where cpj,g(z’, t’,J’, 7’) = ‘ p j ( ~ ’t’d, , J’, $).
Proof. We have
+
t’X& = Xj,&t’Xj,& [t’,Xj,&]Xj,&. By a theorem on composition of p. d.
0. it
is easy to see that
where o([t’, X j , & ]denotes ) the symbol of the operator [t’,Xj,s].From above we get the desired equality.
Lemma 4. Assume that X j satisfies the condition of Lemma 3. For every s E R, K’ G 52’, there exists a constant C = C(s, K’, X j ) such that llxj,agll: 6 c11g11:+1
+ c(6)llgll:-ll
vg E
GYK‘).
The constant C is independent of 6.
Proof. We have Xj,&(Z’, t’, E‘, 7’) = xj (Z‘,t‘S, J’,
-)S . 7’
ax. O , O , 1) = 0. Since By the assumption of Lemma 3, we have +(z, ..
X j ( ~ 1* ., zn, ti 1
is a homogeneous function of degree one in lowing Euler formula
Putting in Eq. (5)
(1
i
Jnl7)
((1,
. . .,
Jnl
7)
we have the fol-
= . . . = Jn = 0 , =~1 we get
ax.
0 = --3-(2,0,0,1) = X j ( X , o,o, 1).
a7
Consequently
1 Xj,&(Z’,t’, J’, 7’) = -Xj(Z’, t’b, J’b, 7’) 6 1 dX . = 6 [Xj(Z’,0, SJ’, 7’) S t ’at L ( z ’ ,0 , SJ’,
+
7’)
+ O(S2)l.
P.d.0 of Second Order with Sign-changed Characteristic Form
7
Note that if X j satisfy the condition (4) in Lemma 3, then, equivalently, X j can be written in the following form
xj (x,t , t,).
=
tq (x,t ,tl ). + zj (x,t , 6).
Lemma 5 . Let the operator X j have the form x j ( x , t , t , T )= tu,(.,tJ) +Zj(.,t),
n.
where Y,(x,t,O) = 0 , j = Then f o r every s E 1w and a n arbitrary compact K‘ in R’ there exist constants C = C(s, K ) ,60 such that f o r 6 E (0760)
where
11 . / I s ,
(., .)s are taken in the coordinates (d, t’).
Proof. Consider the expression
By Lemma 3, the operator P6 may be rewritten in the form
where ord
(pj,a = 0.
It follows that
8
N . M. Tri
where @a may be estimated by
Let us estimate the term -6
k ,, )o. We have Cj=l (t'Xj,ag, X~JA'~'-'&
It remains to estimate the second term in Eq. (10)
C
-6
j=1 k (
a
t'Xj,sg, -[Xj,a, at'
Dj,3,6= [Xj,hlA'23-1]. Since ord Dj+,g = 2s Eq. (12) will be estimated by
Put
k j=1
It remains to estimate the first term in Eq. (12)
(12) -
1, the second term in
P.d.0 of Second Order with Sign-changed Characteristic Form
9
We have
Consequently
By Lemma 4
where C is the constant appearing in Lemma 4. By choosing 60 = min{ 1,&} and combining (7) - (14) we get (5). 0
Corollary 2. Under the hypothesis of Lemma 5 the following inequality holds
for all g E C r ( K ’ ) .
10 N . M.
Tri
Corollary 3. Under the hypothesis of Lemma 5 the following estimate holds
for
some constant C = C(s, K ) .
Proof. First choose 6 = 2 , g = f6 in Eq. (15), then apply Lemma 1 we immediately get the result. Corollary 4. Let the hypothesis of Lemma 5 be fulfilled and the algebra Lie generated by { XI,. . . ,X k } has rank n 1 in each point of 0. There exists a number u > 0 such that following estimate holds
g,
for some constant C
= C(s,
+
K).
Proof. We can easily deduce the desired estimate from Corollary 3 and the apriori estimate achieved in Ref. 1 (see also Refs. 7-9). 17 Theorem 1. Let P have the f o r m (1). W e impose o n the operators X j the following conditions
X ~ ( I C , ~ , E , ~ ) = ~ Y ~ ( I C , Jw, ~ h e) r+e Z Y ,~( z(,~J , O J ))=, O , j = p .
The algebra Lie generated by {&,XI,. . . ,X,} has rank n + 1 in each point of R.
Then P is a hypoelliptic operator.
Proof. We shall show that iff E D'(R), Pf E Hb,,(Q),then f E H,S:"(R), X j f E H",' (a),j = for some u > 0. First we shall prove this assertion in the case f E E'(R). Since f E E'(R), there exists y E R such that f E HY(Rn+l). We can assume that y 6 s D . Set 77 = inf(s,y). Then f,P f E H". For the reason of convenience we will work in the coordinates ( d t ' ) for 6 < 60. Instead of f we write g in the ( d , t ' ) coordinates. Let
+
P.d.0 of Second Order with Sign-changed Characteristic Form
11
xE*be an averaging kernel. We have *
~ S ( X E 9) =
-j=1
where all TE,6,T:,6,0 < E 6 EO belong to a bounded set in the spase of p. d. 0.of order zero, if EO is sufficiently small. By choosing an appropriate xE from Lemma 5 we obtain
12
N . M. Tri
Therefore
uniformly in E. Passing to the limit as E -+ 0 we obtain That means that g , X j f E H". From (16) it follows that
+IlXE
* fll,
uniformly in E. Passing to the limit as E Schwarz inequality [(XC
g,Xj,ag E
HV.
G Ml 4
0 we obtain
f E Hq+'. By the
* tPf,XE * f),+q I 6 C(llX8 *tw; + llxe * f112,+,).
Hence
2,
uniformly in E . Passing to the limit as E -+ 0 we obtain Xj f E H,+%. We can repeat our argument with inf(s, q+a) instead of q and get a similar . . . At last assertion, the replace inf(s,q n) by inf[s,inf(s,q n) 4,. we obtain the needed assertion for all f E E'(R). Now let fi be a relative compact set in R, and f be the distribution on R, not necessarily with compact support. By the general theory of distributions it follows that the restriction o f f to 0 belongs to HLc(fi)for some y- E R. Consider a sequence of Coo-function {gi} with compact support in R such that gi+l = 1 in a neighbourhood of supp gi, (i = 0 , 1 , . . .,). We have
+
j=1
+ +
j=1
P.d.0 of Second Order with Sign-changed Characteristic Form
13
By a theorem on order of commutators we get
We assume as above that Pf E H,",,(s1). Set 71 = inf(s,y). Then the right E Hq-l. hand side in the ( i + l ) t h inequality belongs t o H q - l , i. e. P(gi+lf) Hence by the previous step we have
Xj(Si+lf)E Hq-l+s. Therefore the right hand in Eq. (i) belongs t o Hq-l+s whence P ( g i f ) E H"-l+%. But by result proved above Xj(gif)E Hq-l+u,gif E Hq-l+%. Repeating this argument again and again we get P(g0f)E Hq. It follows that Xj(g0f)E Hq+q,gofE Hq+'. But we can choose function go E C r ( 6 ) arbitrarily. Therefore
Using this assertion again and again we shall come to t = s and thus we obtain the desired assertion. Our theorem is proved completely. 0
References 1. L. Hormander, Acta Math., 119,147 (1967). 2. Y. Kannai, Zsr. J , Math., 9,306 (1971).
3. B. Helffer, C. Zuily, C. R. Acad. Sci. Paris Sir. A - B , 277,1061 (1973). 4. R. Beals, C. Fefferman, Comm. Part. D i f f . Equat., 1, 73 (1976). 5. €4. E. ramca, Becmuulc M T Y , (1985), pp. 96-99. 6. kl. E. raaza, YMH, 41, 217 (1986). 7. J. J. Kohn, Proc. Symp. Pure Math., 23,61-69 (1973). 8. 0. A. OneCiiHm, E. B. Pameswr, Y p a e u e u u x Bmopozo n o p x d x a c Heompuyamenauto6 xapaxmepucmuueclco6 @ o p ~ 0 6klTorH , Haym. MaT. a ~ a n m . -M.: BMHklTkl, 1971. 9. F. Treves, Pseudo-Differentia1 Operators and Fourier Zntegml Oprators,
(Plenum Press, 1982).
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15
Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 15-32) @ 2007 World Scientific Publishing Co.
$2. A SEMILINEAR NONCLASSICAL PSEUDODIFFERENTIAL BOUNDARY VALUE PROBLEM IN SOBOLEV SPACES He+, 1 < p < 00 NGUYEN MINH CHUONGf and DANG ANH TUAN
Institute of Mathematics 18 Hoang Quoc Viet, Cau Giay, 10307 Hanoi, Vietnam f E-mail:
[email protected] We study a semilinear nonclassical pseudodifferential value problem in Sobolev This paper is thecontinuationof Refs. 7,8,10,12,where spaces H e , p , 1 < p < a. the problem was investigated in He,z.
1. Introduction
In Ref. 5 A. V. Bicatze firstly on the world dropped the classical boundary condition imposed on the oblique derivative D,u = namely, the condition that the vector field D, must be not tangent to the boundary. He investigated such problem, event for Laplace equations in three dimensional region with the assumption that the vector field D, may be tangent to the boundary. Then many mathematicians such as R. Borelli,' L. HOrmander,l4 Yu. V. Egorov and V. A. Kondratiev," Yu. V. Egorov and Nguyen Minh C h ~ o n g Le , ~ Quang Trung1lgNguyen Minh Chuong and Tran Tri Kiet18 Nguyen Minh Chuong and Dang Anh Tuan,' etc., continue to extend this problem in several directions. For instance in Ref. 10 the problem was extended to an elliptic differential operator of second order for a bounded domain R in Rn with a smooth boundary dR, in Ref. 9 the problem was studied for a parabolic differential operator of second order, in Ref. 11 a non-classical boundary value problem for a second order elliptic equation in Sobolev space of variable order, in Ref. 12 a semilinear boundary value problem for an elliptic singular integro - differential equation of order 2m in He,z was investigated, in Ref. 8 a priori estimates were established for nonclassical derivative problems for elliptic and parabolic differential operators of second order in Sobolev spaces He,p, 1 < p < 00. The purpose of this paper is to investigate a semilinear non classical
g,
16
N . M. Chuong and D. A . Tuan
pseudodifferential boundary value problem in Sobolev spaces He,prq,1 < p < 00, q is a complex parameter, (YO 5 arg q 5 ,& This problem was investigated in Ref. 12 by Yu. V. Egorov and by the second author of this paper, with the singular integro - differential opetators of M. S. Agranovich type3 and the Sobolev spaces He,2. In Ref. 3 M. S. Agranovich used the singular integral operators corresponding to spherical functions on a unit sphere Sn-l in R". He used the symbols expanded into sequences of spherical functions converging in Lz(S"-~). Here by means of a new class of singular integral operators it is able to solve the non classical pseudodifferential boundary value problem in Ref. 12 in Sobolev spaces 1 < p < 03. The approach used in this paper is quite different from Ref. 3, even from Ref. 12. It is noticed that, by writing a non classical boundary value problem for a Laplace equation in pseudodifferential operator form, in Ref. 13 by the first author Yu. V. Egorov, an interesting estimate of subelliptic type was obtained. 2. Sobolev Spaces
Let He,p(Rn),C E R, 1 5 p < 00, be the completion of the space C?(Rn) of functions with compact support with respect to the norm
Fu = C(E) is the Fourier transform of function u(x),IEI = Jc; + . . . + t:. Let p , C be in R such that 0 5 C, 1 5 p < 00. We denote by He,p(RI",)the
where
completion of the space C ~ ( I Wwith ~ ) respect to the norm:
Ilulle,p = Ilulle,p,w;
= inf
lIlulle,p,w,
when the infimum is taken over all extension I from "3 to R". Let R be a compact domain in R" with (TI - 1)-dimensional smooth N M boundary d o , { ~ j } ~ a partition , ~ of identity on R and {$j}j=l a partition of identity on dR such that for each support of +j , j = 1,. . . ,M , dR can be locally transformed into a superplane. Let p , e be in R such that 0 5 e, 1 5 p < 00. We denote by He,*(R) the completion of the space C"(G) with respect to the norm
Semilinear Nonclassical P.d. BVP in Sobolev Spaces He,p 17
where C' denotes the sum over all j such that the neighbourhoods Uj do not intersect the boundary dR, C" denotes the sum over all j such that the neighbourhoods Uj intersect the boundary 80. Let p , l be in R such that 0 5 l , 1 5 p < 03. Let He,p(as2)the completion of the space C"(ds2) with respect to the norm
where Il$julle,p,Rn-lis the norm of the space He,,(R"-l) of $ju in the locally coordinate system. Let q be in C , p , l be in R such that 1 5 p < 03. In the spaces He,p,,(Rn), He,p,q(R)or He,p,,(Rn+),He,p,q(8s2) (l 2 0 ) we define a respective norm with parameter q:
We give now some properties of the spaces He,p. Proposition 2.1. Let l , k , p be in R such that 1 5 p following bounded imbeddings hold true
<
03,
k
< t. T h e
Proposition 2.2. Let p , l be in R such that 1 5 p < 03, 0 5 l. T h e restricted operator M f r o m R" to RI;" i s a bounded linear operator f r o m He,p(R") t o There exists extended bounded linear operator L f r o m He,p(Rn+)t o He,p(Rn).
k)
Proposition 2.3. Let p , l be in R such that 1 5 p < 03, (1< l. The operator defined by u H uIzn---o i s a bounded linear operator f r o m He,p(E%n) (or H e , p ( R y ) )t o He-(l-$),,( Rn-1 ). T h e operator defined by u H uI, i s a
bounded linear operator f r o m Ht,,(O) to He-(l-L),p(Xl). P
where C is a positive constant not depending on u.
18 N . M . Chuong and D. A . Than
Note that F’(uIZn=o)(J’,O) = ( 2 7 r - + & F u ( S ) d & for each u E C r ( R n ) and the integral /(l+
w
It‘[+ IEnl)-*dEn
= (1
+ lt’l)-*+l
s,(1+ Itl)-*dt,
(t = (1+ lE’l-l&J,
converges for (1 -
k) < l , so by Holder inequality we have
hence, (1
+ lS/l)(t-(l-~))”l~’(ulzn=o)(~/,
0)l” I Cs,(l
+ lEJ)e*IFu(E)IPdtn,
by integrating both side of the above inequality we get the inequality (1). 3. Pseudodifferential Operators in
R”
Definition 3.1. Let m be in Z+, 71, 7 2 be in R and
Q = (2
E CIyt
I argz 5 7 2 ) .
The following operator is called a pseudodaflerential operator in order m, with parameter q if it is defined by Au(x,4 ) = 4 =
R” of
5 , D , 4 ) 4 x ,4 )
(27-v
[
c
lal+Plm
qp
J’ K a d x ,
5
- Y)qu(Y)dY],
(2)
R;
with 4E
Q,
K a p ( x ,z ) = H a p ( z )
+ k ap( x : ,z ) ,
where (i) H a p ( z ) , kap(x, 2 ) are homogeneous of order (-n) with respect to z , (ii) kap(x, z ) belongs to Cm(Rn) with respect to 2, D,Yk,p(z, z ) --$ 0 when 1x1 + 00, for all y (multi-index)
Semilinear Nonclassical P.d. BVP in Sobolev Spaces He,*
19
(iii)
n:
Izl=l
lz1=1
where o ( z ) is an element area of the sphere { z E IWnllzl = 1).
Remark 3.1. Under conditions (i), (ii), (iii) and (iv), the pseudodifferential operator (2) is defined on C r ( R n , )and has another form
Au(x,Q) = (27rPn/'
/
c,
ei(z'E)aA(z, q ) W E ) & ,
(3)
R[g;
where
= g:p(x,
gEp(x,
E ) +gAa(l),
/
= (2n)-"/2(-1)n
e-+?E)k,p(x,
(5) z)dz,
(6)
R :
gk p(c ) = ( 2 ~ ) - " / ' ( - 1 ) ~
1
e-i(Z>E)Hap(z)dz.
(7)
R:
Theorem 3.1. Let l , p be in IK, m be in Z+ such that 1 < p < 00. T h e pseudodifferential operator A of order m of form ( 2 ) is a bounded linear operator from (C,"(R"), 1 1 1 . Ille,P) t o (He-m,p,q(Rn), 111. l\le-m,p). This OPerator can be extended t o a bounded linear operator from He,p,g(Rn)t o He-m,p,q(IKn) acting as a n operator of order m of form ( 3 ) . Moreover, the estimate
I ll A ~ lle--m,p l
5 CI lbllle,p,
u E He,P,dIK"),
(8)
20
N . M. Chuong and D. A . %an
holds true, where C is a constant not depending
then
OIL
u,q .
Semilinear Nonclassical P.d. BVP in Sobolev Spaces He,p 21
First, we prove (10). We have
Since gAp(r]) is homogeneous of order 0, we have
This implies that
Hence, we have (10). Now, we prove (9). We have
Because of
and D,Y(x,z ) index y,
-+
0, when 1x1 + co,for all multi-index y,so for all multi-
22
N . M. Chuong and D. A . %an
it implies that for all Ic in Z+, we have
where
C5
not depending
vl El q , so
Semilinear Nonclassical P.d. BVP in Sobolev Spaces He,p 23
We now study the pseudodifferential operator which has special form called a homogeneous one with the symbol not depending on x qp<"
aA(<,4) =
bI+P=-m
/
e-i(Zif)K,p(z)dz.
yq
In this case it is not difficult we obtain the following result.
Theorem 3.2. Let l , p be in R, m be in Z+ such that 1 5 p < 00. Suppose that the homogeneous pseudodifferential operator A of order m has symbol O A ( [ , q ) # 0 , when I(l 141 # 0. Then, with q E Q \ {0}, A has the inverse operator A-' which is a pseudodiflerential operator of order (-m) with symbol a;'([, q ) . Moreover, the estimate
+
IIIA-lf
II Il,P 5 CI Ilfllle-m,P,
f E He-m,P,P(Rn)
holds true, where C is a constant not depending on f , q . 4. Pseudodifferential Operator in R y
Definition 4.1. Let m be in Z+, 71,7 2 be in R and
Q = { z E Q'lyl 5 argz 5 7 2 ) .
24
N . M . Chuong and D. A . Than
The operator defined by
A
= MAL,
where
M is the restricted operator from R" to RT, L is the extended operator from RT t o R" (in the Proposition 2.2),
A is the pseudodifferential operator of form (3) of order 'rrt in R" with symbol ~ A ( J :I ,,41, J: E Rn,E E R",4 E Q I is called the pseudodifferential operator of order m in RT with symbol Q. A is said to be homogeneous if the operator A is homoge-
o A ( Z , E , q ) = oA(J:,I,4),
J:
E
Rw"+, IE
Rnl
4E
The operator neous. The operator A is said to be admissible if its symbol has form o A ( x l , 0, E , 4 ) =
gap(x1i0i '?)<"qP la I + B l m m.
k=O
where
OA,
does not depend on
It.
We now consider a boundary value problem in RT. Let s, m l , . . . , ms be in Z+, 71,7 2 be in R and
Q =(2
E CIn
I argz 5 7 2 ) .
We now study the boundary value problem
A 4 J : ,4 ) = A b , D , Q)+I
4) = f(J:,4),
BjU(J:,4)JZn--0 = ~ j ( J : , D , 4 ) z 1 ( 5 , q ) I z , ==gj(J:',q), 0
J:"
> 01 (13)
j = l,...,S,
(14)
where 4 E Q, A , Bj are homogeneous admissible pseudodifferential operators of order (respectively) 2s, m j with symbols (respectively) O A ( J : ,E , q ) , U B j (J:,1 I 4). Put t o = m a x { 2 s , m l + I , . . . , m , I},
+
Semilinear Nonclassical P.d. BVP in Sobolev Spaces He,p 25
with respect to the norm S
IIIV,g)IIIe,P = III(f,gi,...,gs)IIIe,p = I
I I ~ I I I ~C- IIIgsIIIe-~j-(l-~),p. ~~,~+ j=1
It is easy to prove
Proposition 4.1. The operator U is a bounded linear operator f r o m He,P,q(RT)to Hg,p,q(Ry,Rn-'). Moreover, the estimate
I IIUullle,P I CI ll4l e m
u E Ht,P(R?)
holds true, where C is a constant not depending o n u, q . Let us now consider the operators A , Bj possessing symbols CA(Z, <,q ) not depending on x,i.e.
<,q ) ,
U B (x, ~
gA(x,
<,q ) = a A ( < , q ) ,
UBj
(x,<,4 ) = U B j (6,q ) ,
we have the following useful results. The problem (13) - (14) is said to satisfy the condition ShapiroLopatinski if the problem on the halfline t 2 0
d
CA(<', t
CBj(<
iz,q ) v ( t ) = 0, t > 0
'd ,zz,q)v(t)It=o=hj, j = l , . . . , s ~
(15) (16)
+
for [<'I 141 # 0, has a unique solution in the space M of all stable solutions of (15) for arbitrary hj. The problem (13) - (14) is said to be elliptic if g A ( < , 4 ) # 0, for + Iql # 0, 0 the problem satisfies the condition Shapiro - Lopatinski.
Theorem 4.1. Let p , C be in R such that C, 5 C, 1 5 p < 00, and the problem (13) - (14) is elliptic. Then, the following statements hold true.
( i ) If q E Q \ (0) then U has the inverse operator U-' is a bounded linear operator f r o m H E , ~ , , ( RRn-') ~ + , t o H e , p , , ( R ~ )not , depending o n p , C, (aa) If q = 0 , there exists a bounded linear operator R f r o m H ~ , p ~ q ( R T , to He,P,q(RT),not depending o n p , C such that
UR=IfT, where, I is the identity operator o n He,p(R"+RIW"-l) T is a bounded linear operator f r o m He,p,q(RT,Rn-') to H ~ + I , ~ , , ( RRn-'). -~+,
26
N . M . Chuong and D. A . m a n
Proof.
5. Pseudodifferential Operator on a Compact Domain C2 Let R be a compact domain in R",with an ( n - 1)-dimensional, smooth N boundary 82,and { U j , qj}j=l be a partition of identity on R.
Definition 5.1. Let m be in
Z+, ~
1
E R~ and~
2
Q = { z E C l ~I i argz 5 7 2 ) . Let A be the operator of form ( 3 ) . The operator A is said to be a pseudodifferential operator on R of order m if (i) for each p E Cm(R); P A - A9 is an operator of order ( m - 1) on &p,,(R), 2 0, 1 < P < 00, (ii) for each p, II,E Cm(R) whose supports in Uj 0 if U j n r = 0, in the local coordinates
e
PA[$.] = PA, M.1, where A, are pseudodifferential operator of order m in R" with symbols O A , (x, 6 , 4 ) , 5 E R",E E R", 4 E Q 1 0 if U, n r # 0, in the local coordinates
PANJ.1 = PA,M.1, where A j are pseudodifferential operators of order m in R" with symbols b ~( z, , E ,q ) , 5 E Rn+, E E R n l 4 E Q.
Semilinear Nonclassical P.d. BVP in Sobolev Spaces He9p 27
The operator A is said to be admissible if for each 'p, ?1, E Cm(R) whose supports in U j , Uj n r # 0, in the local coordinates
PA[$.]= 'pAj [?1,.1 , Aj are admissible. We consider now the boundary value problem on R. Let s , ml, . . . ,ms be in Z+, 71,7 2 be in R and
Q = { z E Clyi L argz I 7 2 } . We now study the boundary value problem
It is obvious that
Proposition 5.1. Let l , p be in B such that l o 5 l , 1 < p < 00. Then U is a bounded linear operator from He,p,q(R)to He,p,q(R,80).Moreover, the estimate
holds true, where C is a constant not depending on u,q.
28
N . M. Chuong and D. A . m a n
Ellipticity C o n d i t i o n
Condition 5.1. Let xo E R with a neighbourhood U j . In the locall coordinate system of 20, the operator A changes into the pseudodifferential operator Aj of order 2s in R" having the principal part Aoj with the symbol CAoj (2, < I q ) . The problem (17) - (18) is said to satisfy the Condition 5.1 at xo if CAoj
( 0 ,< I 4 ) # O1
for
1"
-tIqI
# O*
Condition 5.2. Let xo E dR with a neighbourhood Uj. In the locall coordinate system of 20, the operator A changes into the admissible pseudodifferential operator Aj of order 2s in RT having the principal part Aoj with symbol C A (x, ~ El ~ q ) , the operator Bk changes into the admissible pseudodifferential operator of order mk in RT having the main part Bokj with symbol CB0k3 ( X I I ,4 ) . The problem (17) - (18) is said t o satisfy Condition 5.2 a t zoif the problem on half-line t 2 0
+
when 5'1 Iq( # 0 , has a unique solution in the space M of all stable solutions of (19) for arbitrary hk. The problem (17) - (18) is said t o be elliptic if it satisfies Condition 1 for every xo E G and Condition 2 for every xo E dR. The following proposition is proved without difficulties
Proposition 5.2. Let I , p be in JR such that l o 5 I , 1 5 p < 00, and the problem (17) - (18) be elliptic. Then, the estimate
holds true, where C is constant not depending on u, q. This implies that, when q having large enough 141, for every (f , g ) E He,,(R, d o ) , if the problem (17)-(18) has a solution, then it is unique. When the problem (17) - (18) is said to be elliptic, we also say the operator
U is elliptic. To obtain the next theorem we need the following easily proved Lemma
Semilinear Nonclassical P.d. BVP in Sobolev Spaces He,p 29
Lemma 5.1. If U is elliptic, then U has a regularizer R satisfying
RU
= I1
+ Ti,
U R = I2 -k Tz,
where I I , 1 2 are the identity operators (respectively) o n He,,,,(R), He,p,q(R,dR);TI i s a bounded linear operator f r o m He,p,q(R)t o He+l,p,q(R);T2 i s a bounded linear operator f r o m He,,,,(R,aR) t o He+i, p , q (a,an). Lemma 5.2. Let Bo, B , B' are Banach spaces with respective n o r m II.110, 11 . 11, 11 . 11'. A s s u m e that B L-) Bo i s a compact imbedding operator and A ; B + B' i s a bounded linear operator. T h e n , the necessary and s u f i c i e n t condition such that d i m ( K e r A ) < +CQ and ImA i s closed B', i s llull 5 C(IIA4' + 11~110)l
21 E
B,
(21)
where C i s a constant not depending o n u. By using Proposition 5.1, 5.2, Lemmas 5.1 and 5.2 we obtain
Theorem 5.1. Let e, p be in R such that lo5 e, 1 < p < 03. If U is elliptic then for large enough IqI the operator U i s a n isomorphism f r o m He3p9q(fl) t o He,p,q(Ql 80). 6. A Linear Non-classical Pseudodifferential Boundary
Value Problem We study now the following non-classical pseudodifferential boundary value problem
Au = f(x) i n R B j u = Bj(D,u) = g j o n d R , j = 1 , 2 , . . . , s
(22) (23)
where A, Bj are of order 2s, mj respectively in the problem (17)-(18), D, is a vector field that can be tangent to dR on a smooth manifold ro c 80 of dimension ( n - 2) and not tangent to ro. We use here the classification of ro given in Ref. 10. For the reader's convenience let us recall shortly this classification. Let p be the inner normal unit vector of the boundary dfl. The real function ( p , v ) is defined on the boundary aR,. The submanifold is said to be of the first class if on some neighborhood of ro, the function ( p , v ) is negative on the negative part (with respect t o the direction v), and positive on the positive one, the second class is positive on the negative part and negative on the positive one, the third class does not change sign when passing ro.
30
N . M . Chuong and D. A . Than
If
ro belongs t o the first class, then we add the following conditions Dhu(x,q) = U~k(x, q), k
= 0,1,..
., s - 1
on Ro.
(24)
The above problem will be discussed in Sobolev spaces He,prq,1 < p < 00. It is assumed that the problem defined by (A, B j ) is elliptic that is A is an elliptic operator and the Shapiro - Lopatinski condition is satisfied. Let r d be the intersection of a d-neighborhood of ro with R, h = hd(x) E C" and
hd(x) =
0, 1,
if x is outside of r
d
if x is in
Denote by IIe,p,q(R)the space with norm
g,
= N = Nd = {x E Gd13y E ro,3 is a normal vector to 6'0, d ( z , ro) I d } , if r o is of the first class and if r0 is not of the first class the term hu in the norm is omitted. Denote by G e , p , q ( d R )the spaces of functions defined on dR with norm
where D,
llullGe,p,q =
Ill4lle,P,an + Illh~llle+l,P,an.
Denote ne,p,q(fl, 80) = 1 3 e - 2 s , p , q ( Q ) x 2).
Cl = max{2s, mj
+
fli=lGe-j-(l-~),p,q(dfl), for C 2
Theorem 6.1. A s s u m e that the operator ( A ,B j ) i s elliptic, l 2 l , . T h e n f o r large enough 141, f o r each (f,gj) E rIe,p,q(R,dR) ifro belongs t o the if ro belongs t o the third class, or additionally uo E He--(l--l/p),p,q(l?O), first class the problem (22)-(23) o r (22)-(23)-(24) has a unique solution 21 E l 3 e , P , m . 7. A Semilinear Non Classical Pseudodifferential Boundary Value Problem Let us now consider the semilinear non classical pseudodifferential boundary value problem D2m-1 u), in R, (25) Au(x7 4 ) = f(x,4, u , . ' . 7 B j u ( x , q )= Bj(D,u(x,q)) = g j ( x , q , u ,. . ., Dm3-'u),on d R , j = 1,.. ., s (26)
Semilinear Nonclassical P.d. BVP in Sobolev Spaces He,p 31
where A, Bj are pseudodifferential operators in (22), (23) in Sec. 6. If belongs to the first class then we add the conditions D:U(S,
q ) = UOk(2, q ) , IC = 0,. . . , s - 1 on ro.
ro
(27)
For this problem we get
Theorem 7.1. Assume that ro is of the first class, !2 !I, ( A ,B j ) is elliptic, U O , A € Hl-k-(l-L),p,q(ro), f,g = (91,.. . ,g8) satisfy these conditions P (i) these maps (2, q , G) H f(z, q , G),where G = ( ~ 1 ,... ,U N ) , (z, q , G) H gj(z, q , Gj), where Gj = ( ~ 1 ,... ,U N , ) , ~= 1 , 2 , . . . , s
satisfy Carathe'odory conditions, i.e. f,gj are respectively continuous at G, Gj f o r almost (2, q ) , measurable at (z, q ) f o r every G, G j , (ii) the map u(z,4 ) (f(z, 4 , G(z, 4)),9j(Z, 4 , G j b , Q ) ) ) , i s f r o m rIl,p,q(R) t o rIg,,,,(R, aR) transforms each bounded set into a relatively compact set, where ,(.'.
4 ) = ( u ( z ,Q ) , . . ., D2s-1u(5,Q)),
(iii)
Proof. According to Theorem 6.1 for 141 large enough, for each w E (R) the linear problem D2s-1
Q) = f ( z , 4 ,w,. . . , w) in 0, Bju(z,q)= g j ( z , q , w,. . ., Dm3-'w), on a R , j = 1 , . . . ,s,
(28)
D;U(~C q ), = u o k ( z ,q ) , IC = 0, I,.. . , s - 1 on ro
(30)
(29)
has a unique solution u E I I g , p , q ( Q ) . So for IqI large enough, we get the map J : w H u from rIl,P,q(R) into itself. By the assumptions ( i ) - (ii) the map J is compact. By the assumption (iii) there exists a positive R such that the map J maps the sphere SR = { w E r I ~ , p , ~ ( R ) ~ ~ = ~R w }~into ~ ~ the ~ , ball ~ , ~BR ( ~ =) { w E
32
N . M . Chuong and D. A . %an
ne,p,~(S1)Illurlln,,,,,(n) I R}. Therefore, t h e m a p J satisfies t h e Fixed Theorem of Rothe so it possesses a fixed point which is a solution of t h e problem (25)-(26)-(27). References 1. R. A. Adams, Sobolev spaces, (Acad. Press, 1975). 2. S. Agmon, A. Dough, L. Nirenberg, Comm. Pure Appl. Math., 624 (1959). 3. M. S. Agranovich, Uspekhi Mat. N a u k 2 0 , 3 (1965). 4. M. S. Agranovich, M. I. Vishik, Uspekhi Mat. Nauk 19,53 (1964). 5. A. V. Bicatze, Dokl. Acad. Nauk. SSSR 148, 749 (1963). 6. R. Borelli, J . Math. Mech. 16,51 (1966). 7. Nguyen Minh Chuong, Dang Anh Tuan, A boundary value problem for singular integro-differential operators in He,p,l < p < 00, Preprint 2002/28, (Institute of Mathematics, Hanoi, 2000). 8. Nguyen Minh Chuong, Tran Tri Kiet, On a priori estimates in L P ( R n ) , p 1 2 for nonclassical oblique derivative problems, Preprint 2000/2 6, (Institute of Mathematics, Hanoi, 2000). 9. Yu. V. Egorov, Nguyen Minh Chuong, Diff. Urav. 24, 197 (1969). 10. Yu. V. Egorov, V. A. Kondratiev, Mat. Sbornik, 78,148 (1969). 11. Yu. V. Egorov, Nguyen Minh Chuong, Di8. Urav. 20, 2163 (1984). 12. Yu. V. Egorov, Nguyen Minh Chuong, Uspekhi Mat. Nauk 53,249 (1998). 13. Yu. V. Egorov, Mat. Sbornik 73,356 (1967). 14. L. Homander, Ann. of Math. 83, 129 (1966). 15. L. Homander, The analysis 0s linear partial differential operators, 111, Pseudodifferential operator, (Springer - Ve,rlag, 1985). 16. J. Leray and J. Schauder, A n n . Sc. Ecole Norm. Sup. 51,45 (1934). 17. E. H. Lieb, M. Loss, Analysis, Vol. 14, (AMS, 1966). 18. G. 0. Okikiolu, Aspects of the theory of bounded integral operators in L p spaces, (Academic Press, 1971). 19. Le Quang Trung, Uspekhi. Mat. Nauk 44, 169 (1989).
Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 33-42) @ 2007 World Scientific Publishing Co.
33
$3. SINGULAR INTEGRAL OPERATORS IN FUNCTIONAL SPACES OF MORREY TYPE LUBOMIRA SOFTOVA Postal address: Politecnico d i Bari Dipartimento d i Matematica Via E. Orabona 4 , 70125 Bari, Italy E-mail: 1ubaQdm.uniba.it
Keywords: Generalized Morrey spaces, Singular integrals, Commutators, Calder6n-Zygmund kernel, B M O , V M O . MR(2000) Subject Classification: 42B20
1. Definitions and Preliminary Results We consider the following integral operator
Kf(.)
Ln
:= P.V.
k ( z ;z - Y).f(Y)dY
and its commutator with essentially bounded functions
C b , kIf(.)
:= P.V.
1L
k(.; .-y)[a(y)-a(.)lf(y)dy
= K:(af)(.)-.(.)Kf(.).
(2)
The generating kernel k ( x ;5) : R" x {R" \ (0)) 4 R is the one studied by Fabes and Rivibre' and is a generalization of the classical kernel of Calder6n and Zygmund. Introducing a new metric p ( x ) as a solution of the equation F(rc,p) = Cy=lz!p-2a' = 1, cri 2 1, they study (1) in LP(R"). The balls with respect to p ( z ) , centered at the origin and of radius r are simply the ellipsoids
with Lebesgue measure I&,.[ = C(n)ra.It is easy to see that E l ( 0 ) = S"-l with respect to the Euclidean metric. This fact allows to impose on the kernel k ( x , [) the Calder6n-Zygmund conditions on the unite sphere, in spite
34 L. Softova
ti,i = 1,.. . ,n. Let we note, that the standard parabolic metric = sup{lzl,&}, z E R",t E ( 0 , ~does ) not permit t o define the mentioned above conditions on kernels having homogeneity of parabolic type. Using the Fourier transform in L2(a") and the Marcinkiewicz interpolation theorem, Fabes and Rivikre obtained that the integral operator (1)is continuous in LP(Rn),p E (1,w). of the lack of "symmetry" of k with respect to the variables
Definition 1.1. The function k ( z ; C ) :R" x (R" variable kernel with mixed homogeneity if
\ (0))
+
R is called a
i) for every fixed x the function k ( z ; .) is a constant kernel satisfying
<)I 2
ii) for every multiindex p : sup l D f k ( z ; EEW-1
C(p) independently of z.
Let us note that in the special case ai = 1 and thus Q = n, Definition 29 gives rise t o the classical Calder6n-Zygmund kernels. One more example is when a1 = . . . = ~ " - 1 = 1, an = 6 > 1. In this case we obtain the kernels studied by B.F. Jones and discussed in Ref. 8. Let w : R" x R+ + R+ and for any ellipsoid E we write w ( z , r ) =: w ( € ) .
Definition 1.2. A function f E Lfbc(Rn),p E (1,m) belongs t o the generalized Morrey space LP+' (R") if the following norm is finite
The space LP+(fi) and the norm l l f \ \ p , U i are ~ defined by taking and & n R instead of & in (3).
f
E
LP(fi)
For w ( z , r ) = 1 we get the Lebesgue space Lkc(Rn)and for w ( z , r ) = r x , X E (0, a ) , Lp>"(Rn)coincides with the Morrey space LP2x(R"). However, there exist weight functions, as w ( x , r ) = T ln(r + 2) for which LPl" (R") does not coincides with any Morrey space.
Singular Integral Operators an Functional Spaces of Morrey Type 35
For a given measurable function f 6 Ltoc(Rn) define the HardyLittlewood maximal operator M f and the sharp maximal operator f !I as
almost everywhere in Rn and the supremum is taken over all ellipsoids E centered at x. Define also the operator M, f (x):= (MI f I"(x))'/" for s 2 1. The next results are weighted variants of the well-known maximal and sharp inequalities (see Refs. 3,5,15). Lemma 1.1 (Maximal inequality"). Assume that there are constants c 1 and such that f o r any xo E Rn and f o r any r > 0
c,
For 1 I s < p < 00, there is a constant Cp,, > 0 such that f o r f E Lpsw(Rn)
IlMJI l P , W I c P , S l l f IlP,W. Lemma 1.2 (Sharp inequality). Let 0 < ~75 1 and E be an ellipsoid centered at xo E Rn of radius r. Suppose that w ( x 0 , r ) satisfies (4)and
Then f o r p E ( 1 , co)and f E L?,,(Rn) there exists a constant C independent o f f such that
Ilf
IlP,W
I cllf811p,w.
Proof. Let X& be the characteristic function of the ellipsoid and denote by 2E an ellipsoid centered at xo and of radius 2r. It is easy to verify that M X E ( X 5 ) r " / ( p ( x - XO) - r)O 5 1 , for all z E Rn. Further, for any x E 2"'& \ 2k&, k = 1 , 2 , . . . the maximal function of X E is comparable with 2-ka for any 2 as above. From certain properties of the maximal
36
L . Softova
and sharp functions (see Ref. 9, p. 410) it follows J :=
s,
If(Y)IPdY=
l"
If(Y)IPX&(Y)dY
The assumptions imposed on the weight function give
Hence
One more background we need is that for spherical harmonics and their properties (see for details Refs. 4,6,8,13). Recall that the restriction to the unit sphere SnP1of the homogeneous polynomial P : R" -i R of degree m is called n-dimensional spherical harmonic of degree m. The space Tm of these harmonics is a finite-dimensional linear one with gm = dim Tm such that go = 1, g1 = n and Sm =
(m+n-I)!- (m+n-3)! < C(n)mnP2,m 2 2. ( n - l)!m! ( n - l)!(m - 2)! -
(6)
The orthonormal base {Ysm(x)}zzl of Tm and the system { Y s m } ~ ~ lis~ = = o a complete orthonormal system in L2(S"-') satisfying
If, for instance, q5 E COo(Sn-') then ~ , , , b s m ~ , ( ~ )is the Fourier series expansion of 4(x) with respect to {Y,,(z)}. It converges uniformly to 4
Singular Integral Operators in Functional Spaces of M o n e y Type
37
(see Ref. 4) and for any integer 1
2. Singular Integral Estimates Let k ( z ;[) be a kernel in the sense of Definition 29. In order to ensure the existence of the operators (1) and (2) in LP(Rn) we restrict our considerations to functions f E LP(Wn), 1 < p < 00 for which the norm (3) is finite. For the sake of convenience we still denote these spaces by Lp9"(Rn). Note that the last restriction is necessary only to obtain global estimates. If we consider a bounded domain R E R" the operators (1) and (2) with a density f E LP?"(R) will be well defined in LP(R) without any additional assumptions. Define the nonsingular operators K, and C,[a, k ] , a E B M O by KEf(.) :=
/
P("-Y)>E
CE[%
k]f(.)
:= L(af)(.)
.
k(z;
-
Y)f(Y)dY,
- a(z)K,f(z).
We are going to prove that ICE and C,[a,k] are bounded and continuous from LP?"(R") into itself uniformly in E . This along with the properties of the kernel will enable to let E + 0 obtaining as limits in the LP+'(Rn)topology the singular integrals (1) and (2). Moreover, we shall show that the last ones are also continuous in LP@(Rn). Let us note that K f exists in LP(R") as a limit of K,f when E + 0 in the LJ'-norm.8 Moreover, the operator K : LP(Rn) -+ LP(Rn) is continuous and this leads also to continuity in LP(Rn) of C [ a , k ] if a is an essentially bounded function. As it concerns to the commutator we are going to derive a result similar to that of Coifman-Rochberg-Weiss (Ref. 7, Theorem l),which asserts: if K is Calder6n-Zygmund operator in LP(Rn), p E (1,m) and a E B M O than the commutator C[a,.] is a well defined linear continuous operator from LP(Rn)into itself. Later, this result has been extended by Bramanti-Cerutti2 in the framework of homogeneous spaces. Based on this background about Calder6n-Zygmund operators, we arc going to obtain continuity in L*+'(R") and boundedness in terms of (lull, for the commutator (2) having a kernel of more general type.
Theorem 2.1. Let k ( z ; [ ) be a variable kernel of mixed homogeneity, f E Lp+'(Rn),p E (I,..), w satisfies (4) and (5), and a E B M O . Then the integrals K f , C [ a ,k]f E Lp?"(R") exist as limits o f K E f and CE[a,k ] f when
38
L. Softova
0 with respect to the LP>"(R")-norm. The operators K and C [ a ,k ] are bounded f r o m Lp+'(W") into itself and
E -+
IlKfllP,W
I Cllfllp,wr
IIC[a, ~ l f l l P , W I Cllall*llfllP,w
where the constants depend on n, p , Q and k through the constant C(p). Proof. Let
2 ,y
E R" and
= Y E S-', P(Y)
k ( z ;z - Y) = P(. - Y)-*
then
Cbsm(x)Ysm(K)s,m
This way, the Definition 29 ii) and (8) imply
for any integer 1 > 1. Replacing the kernel with its expansion, we get
xsm(z
with - y) standing for Ysm(F)y)p(a: - y)-" and being a constant kernel in the sense of Definition 29 i ) . In order t o get series expansions of K,f and C,[a, k ] f ,we let z E R" and y E R" t o be such that p(.: - y) > E . Then (l),(7) and (9) yield
Singular Integral Operators in Functional Spaces of Morrey Type
39
Thus instead of K:f and C [ a ,k ]f we shall study the existence in Lp)"(Rn) of
Csm[a,k]f(x) :=Ksrn(af)(x) - a(x)Xsmf(x).
For what concerns the c i, we dispose of Ref. 8(Theorem 11.1) and this implies, through Ref. 2 (Theorem 2.5), boundedness in LP(Rn) of Csm[a,k ] as well. The cited results however require the kernel to have some "integral continuity" , called the Hormander condition. It turns out that H ' ,(.) satisfies even stronger condition as shows the following lemma.
Lemma 2.1. (Pointwise Hormander condition (see Ref. 13, Lemma 2.2)) Let & and 2& be ellipsoids centered at xo and of radius r and 2r, respectively. Then
for each x E & and y
4 2E.
Remark 2.1. This result ensures that dition
s
:dy)L4ds))
'Hsmsatisfies
also the integral con-
I'Hsm(Y - x) - 'Hsm(Y)l dY
5c
with a constant independent of x (see Ref. 8, (1.1)).
In view of the cited above results there exist K , , f , such that lim
€40
IIxsm,e.f
- xmnf IILp(p) =
Fz
IICsrn,~[a, k]f
Csm[a,k]f E LP(Rn)
- C ~ n ~ [k]f a rI I ~ p p n = ) 0.
Our goal is to show that this convergence is fulfilled also with respect to the LP+'(R")-norm. The proof is broken up into several lemmas.
Lemma 2.2. (see Ref. 13, Lemma 2.4) The singular integrals K,,f Csm[a,klf satisfy (K:smf)W
(Csm[a,k]f)"x)
and
I Cm"/2(M(lf lp)(Z))l/pl 5 CIlall*{ (M(lK:smfIP)(x))11p+mn/2(M(lflp)(.))
l'p}
(11)
for all p E (1,co) and the constant depends on n, p , a but not on f
40 L. Softova
Lemma 2.3. The operators Ksrn and Csm[a,k ] are continuous, acting from Lp,w(Rn),p E (1,oo),into itself and IIKsrnfIIp,w
L Cmn/211fIIp,wr
I I ~ s m [ akIfIIp,w r
5 Cmn/211aII*IIfIIp,w (12)
with constants depending on n, p , and a. Proof. Let p E (1,m). Applying (11) for any q E (1, p ) and the maximal inequality (Lemma 1.1) with s = 1 we get
S, 1
( K s r n f ) n ( x )lpdx
1~mpn/2
S,
l M ( l f l q ) ( x lp/ndx )
-
ICmpn/2w(€)IIIf lql;;,w
I Cmp"/2w(E)llf llg,w.
Dividing of w ( E ) and taking supElwe arrive a t ll(~smf)nllp,w
I Cm"/211f IlP,W
which implies the first inequality in (12) through Lemma 1.2. The Lp,westimate for the commutator follows in the same manner. U
Lemma 2.4. The operators KSrn,€and Cs m, E[a, k] are continuous acting from Lpiw(Rn)into itself and satisfy
IIKsrn,Efllp,w 5 c(n7P , a ) m n / 2 1 1 f I I p , w ,
klf Ilp,w 5 C(n1p1 Q)mn/211all*llfIlp,w.
II~sm,E[al
(13)
Proof. Let €, and ,& I, be ellipsoids centered a t x E R" and of radius and ~ / 2 respectively. , Writing f = fXE, fX(E,)= = f l f 2 we obtain
+
:= 211(x1& / 2 )
+
+
E
12(2,542)
where 11 and 1 2 stand for the terms introduced a t the proof of Lemma 2.2, and the same arguments as therein lead to IKsrn,cf(x)I
L ~ ( nP , a)mn/' , (M(IfIn)(x))l'q
Singular Integral Operators in Functional Spaces of Morrey Type
41
for any q E ( 1 , ~ )It. remains to take the LP)w-normsof the both sides above for 1 < q < p and to apply Lemma 1.1 in order to get (13). The commutator estimate follows analogously. I7 Returning to the series expansions (lo), we are in a position now to complete the proof of Theorem 2.1. First of all, note that
as it follows from (9), (1) and Lemma 16. Choosing 1 > (3n- 2)/4 the series in (10) result totally convergent in Lp+'(Rn), uniformly in E > 0, whence
i CllfllP,w,
IIxEfllP,w
I Cllfllp,w,
Il~fllp,w
llCE[a, kIfll,,w
IlC[a, kIfll,,w
i Cll~ll*llfllP,w.
I Cllall*llfllP,w
through (9), (1) and Lemma 2.3. Finally, the total convergence in Lp?"(Rn) of (lo), uniformly in E > 0, gives
lim CE[a,k]f(z) = C[a,k]f(x).
E+O
It is worth noting that singular integrals like (1) and (2) appear in the representation formulas for the solutions of linear elliptic and parabolic partial differential equations. To make the obtained here results applicable to the study of regularizing properties of these operators we need of some additional local results.
Corollary 2.1. Let R be a bounded domain in Rn and k : R x {Rn\{O}} + R be a variable kernel of mixed homogeneity, a E BMO(R), p E (1,m) and w
satisfies (4) and (5). Then, for any f E LP+('R)
and almost all x E 0,
42
L. Softova
the singular integrals
arc well defined i n Lp+’(R) and
IlXfll*,w;n 5 Cllfllp,w;n, with C = C ( n , p ,a , R, k).
11%
kIfll,,u;n
I Cllall*llfllP,w;n
To obtain t h e above assertion it is sufficient t o extend k ( z ; .) and f(.) as zero outside R. One more necessary extension preserving t h e norm is t h a t of a in BMO(Rn) and we have it according t o the results of Jones’l and Acquistapacel (see Ref. 6 for details). Another consequence of Theorem 2.1 is the “good behavior” of t h e commutator for VMO functions a. Thus using t h e ideas in Ref. 6 (Theorem 2.13) we can prove
Corollary 2.2. Suppose a E V M O with VMO-modulus y a . Then, for each E > 0 there exists ro = TO(&, y a ) > 0 such that f o r any e E (0, ro) and any ellipsoid &, of radius
e holds
IICb, kIfllp,w;E,
I CEllfllp,w;E,.
(14)
References 1. P. Acquistapace,Ann. Mat. Pura Appl. 161,231 (1992). 286,A139 (1978). 2. M. Bramanti, M. C. Cerutti,Boll. Un. Mat. Ital. B 10,843 (1996). 3. W. Burger ,C. R. Acad. Sci. Paris 286, 139 (1978). 4. Amer. J . Math. 79, 901 (1996). 5. F. Chiarenza, M. Frasca, Rend. Mat. Appl. 7,273 (1987). 6. F. Chiarenza, M. Frasca, P. Longo, Ricerche Mat. 40, 149 (1991). 7. R. Coifman, R. Rochberg, G. Weiss,Ann. of Math. 103,611 (1976). 8. E. B. Fabes, N. RiviBre, Studia Math. 27, 19 (1966). 9. J. Garcia-Cuerva, J . L. Rubio De Francia, Weighted Norm Inequalities and Related Topics: North-Holand Math. Studies, Vol. 116, North-Holand, Amsterdam, 1985. 10. F. John, L. Nirenberg,Comm. Pure Appl. Math. 14,415 (1961). 11. P.W. Jones, Indiana Uniu. Math. J . 29,41 (1980). 12. E. Nakai, Math. Nachr. 166,95 (1994). 13. D. Palagachev, L. Softova, Potent. Anal. 20, 237 (2004). 14. D. Sarason, Trans. Amer. Math. SOC. 207,391 (1975). 15. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, (Princeton University Press, Princeton, New Jersey, 1970). 16. L. Softova, C. R. Acad. Sci., Paris, Ser. I, Math. 333,635 (2001).
Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 43-72) @ 2007 World Scientific Publishing Co.
43
54. CLASSIFICATION O F INTEGRAL TRANSFORMS VU KIM TUAN Department of Mathematics, University of West Georgia, Carrollton, GA 30118, USA E-mail:
[email protected]
In this survey paper we give a classification of integral transforms based on their composition structure.
1. Introduction There is no consensus among mathematicians what should be called an integral transform, and what should not. Some assume that every linear integral operator that appears frequently enough to bear some name can be considered as an integral transform. We accept here another concept. We believe that the most distinguish character that enjoy integral transforms among the vast of linear integral operators is their integral inverses. We call therefore integral transform every linear integral operator whose inverse is also an integral operator of almost the same "complexity". To include RiemannLiouville fractional operators and Riesz potentials into integral transforms, we should admit some operators of differentiation of finite order into the inverses. So, when we talk about integral transforms we have in mind pairs of direct and inverse integral transforms. Because of the importance of applications and relations of integral transforms with differential equations it is widely believed that new interesting integral transforms can be constructed as eigenfunction expansions of some differential operators. Many classical integral transforms are in fact arisen from self-adjoint Sturm-Liouville differential operat01-s.~~ We will show here another method to construct new integral transforms, based on few well-known integral transforms. Many nonconvolution (index) transforms, constructed by that way, seem impossible to be obtained by eigenfunction expansions, since their kernels do not satisfy any linear differential equations.
44
V. K. Tuan
The most classical transforms are: The pair of Fourier transforms
The pair of Laplace transforms
rm
The Fourier and Mellin transforms are equivalent subject to some substitution of variables. But each of them has advantages in different areas of applications. If the Fourier transform is considered in the complex domain, then the Laplace transform turns out to be the Fourier transform of functions vanishing on the negative half-line. Otherwise they are distinct. Other well-known transforms include the Hilbert, Hankel, Fourier-sine and -cosine, Y- and H- transforms, Riemann-Liouville and Weyl fractional operators, Kontorovich-Lebedev, Mehler-Fock and Olevskii nonconvolution transforms. We have prepared a draft list of integral transforms coming into our definition. It contained more than 300 integral transforms and was included in final form in Ref. 27. Of course it is impossible to have some classification of all known and unknown integral transforms. In this work we restrict ourselves on classification of just one-dimensional integral transforms listed in Ref. 27. Similar treatment of multidimensional integral transforms is briefly given in Ref. 3 . Our method of classification is based on composition structure of integral transforms. Fortunately (or maybe unfortunately), almost all of these integral transforms have very simple structures. They are build up with the help of very few operators: Fourier transform (equivalent Mellin transform
Classafieataon of Integral Transforms
45
and Laplace transform as a special case are also included), operator of multiplication and substitution of variables. Throughout the paper, if we say that the integral transform
is mapping L,(R; w) into L,I (0’;w’), we understand that i f f E L,(R;w), and 0, is any compact subset of R, over it k ( z , .) is continuous. then the integral
converges to a function gn E Lp~(R’;w’),and moreover, if 01 c 0 2 c . . .Onc . . ., such that u,”==, R, = then gn converges in L,, (0’;w’) to g. All integral transforms can be divided into two classes: -Convolution transforms and -NonconuoZution (index) transforms.
n,
2. Convolution Transforms
A generalized convolution of functions f and k under three operators K , K1, KZ, and with some weight function w is a function, denoted by the symbol k * f , such that the following factorization property holds
If K = K1 = K2, we have the classical convolution. Convolution transforms are called linear integral transforms of the following type
f +g=k*
f.
Depending on the convolution, convolution transforms can be divided into following types:
2.1. Fourier convolution transforms
46
V. K. Tuan
An example is the pair of Hilbert transforms
d z ) = (Hf)(z)=-
fody, Y-2
&dz. Y-X
2.2. Laplace convolution transforms
An example is the pair of Riemann-Liouville fractional integral and differential operators
2.3. Mellin convolution transforms W
dz)=
k(zy)f(y)dy.
An example is the pair of Hankel transforms
f(Y)
=
(Hvg)(Y)
=/
fiJv(zy)s(z)dz.
0
(15)
where J,,(x) is the Bessel function of the first kind.l Similar to the relation between the Fourier and Mellin transforms, transforms of Fourier and Mellin convolution types can be identified by a simple substitution of variables. Transforms of Laplace convolution type are special cases of transforms of Fourier convolution type (if k ( z ) = f(z)= 0 for z < 0 in ( 7 ) ) . Since most of convolution transforms are given in the form of Mellin convolution type27 we identify the class of Fourier and Laplace convolution transforms with the class of transforms of Mellin convolution type. The class
Classification of Integral Transforms 47
of Mellin convolution transforms is the most representative class among known transforms. They have very simple composition structure
(Kf)(z)= (M-lk*(s)(Mf)(l- s))(z),
(16)
where k * ( s ) = ( M k ) ( s )- the Mellin transforms of the kernel k ( z ) . Thus, any Mellin convolution transform is a composition of a Mellin transform, a substitution of variable s 4 1 - s, an operator of multiplication by k * ( s ) , and an inverse Mellin transform. Although the kernels k ( z ) in (13) are very distinct in different transforms, their Mellin transforms k* ( s ) in general behave very simple on the line %(s) = l/2. One can see that in almost cases k*(s) is bounded and not equal to 0 on the line %(s) = 1/2. Based on the asymptotic behavior of k * ( s ) on the line R(s) = 1 / 2 the class of Mellin convolution transforms in the form (13) in its turn can be divided into three subclasses: 2.3.1. Mellin convolution transforms of Watson type These transforms are characterized by the condition4g
c > Ik*(s)l > c > 0,
%(s) = 1/2.
(17)
Since the Mellin transform (5) is an homeomorphism from Lz(0, co) onto Lz(1/2--ico,1/2+ico) and the operator of multiplication by k * ( s ) under the condition (17) is an automorphism on L~(1/2-i00,1/2+i00), from (17) it is easy to see that convolution transforms of Watson type are automorphisms on LZ(0,co).If, moreover,
lk*(s)I = 1, %(s)
=
1/2,
(18)
then convolution transforms of Watson type are unitary on Lz(0, co). In the form of Fourier convolution (7) the condition (17) is replaced by
C > I(Fk)(z)l > c > 0,
zE
R,
(19)
and transforms satisfying condition (19) are automorphisms on Lz( -03, co), and unitary there if I ( F k ) ( z ) (= 1. Examples of unitary convolution transforms of Watson type are the Hilbert and Hankel transforms, the Fouriersine transform
dz)=(Fsf)(z)= g p n z Y f ( Y ) d Y ,
(20) (21)
48
V. K.
Tuan
with the Bessel function of the second kind Yv(z)and the Struve function H,,(x) in the kernels' are automorphisms, but not unitary in &(O, m). Taking a composition of Fourier transforms in the form Feix3/3Ff we obtain the following pair of convolution transforms of Watson type3'l4'
L 00
g ( z ) = (Aif)(z) =
f (Y)
= (&)(Y)
=
/
A+
+Y)f(Y)dY,
(26)
A+
+ Y)g(z)dzl
(27)
00
--oo
with the Airy function Ai(z)in the kernel1 that are unitary on L2(-m, m). Taking a composition of Fourier transforms in the form Fe-aaSinh Ff I a > 0, we obtain another pair of convolution transforms of Watson type41
with the Macdonald function K,(a) in the kernels' that is unitary on L2(-m, m).
Taking a composition of Fourier transforms in the form Feia'Osh Ff,a we obtain one more pair of convolution transforms of Watson type3'
> 0,
with the Hankel functions HL')(z) and HL2)(z)in the kernels' that are unitary on L2(-m, m).
Classification of Integral Transforms
49
2.3.2. Mellin convolution transforms of Riemann-Liouville type The Mellin transform k*(s) of the kernels of these transforms are characterized by power decay on the line (1/2 - zoo, 1/2 zoo) as 3(s) + fw:
+
0 < c < Ik*(S)Sal < c < 03, a > 0,W(s) = 1/2.
(32)
Many transforms of Laplace convolution type (10) such as RiemannLiouville fractional integral (11), and Weyl fractional operators
the Riesz potential
the Prabhakar transform27
with the confluent hypergeometric function 1F1 ( a ;c;x ) in the kernel, the Saigo transform28
V. K. Tuan
50
with the hypergeometric function zFl(a, b; c; z) in the kernel, the Marichev transform2'
Y
X
X
Y
0 < Rc
F3(-a', n - a , -b', n - b; n - c; 1 - -, 1 - -)g(z)&,
< n,
with the Appell hypergeometric function of two variables F3(a,a', b, b'; c; x , y) in the kernel are convolution transforms of RiemannLiouville type. The following pairs of integral transforms27 X
g(x) = / x (x - y)"-l E l ( a , a', b; c; 1- -, A(z - y))f(y)dy, 0 Y
(43)
and
-z z ( - a , -b; n
X
-
c; 1- -, A(x - y))g(")(z)dx, 0
Y
< RC < n,
with the Humbert hypergeometric functions of two variables Z1 and Ez in the kernels' are nonconvolution transforms, but their properties and composition structure containing only Riemann-Liouville fractional integrals and derivatives are very similar to those considered in this subsection. Some other special cases such as transforms with orthogonal polynomials in the kernel are compositions of some Riemann-Liouville or Weyl fractional operators with operators of multiplication. Feller and Riesz potentials are, on the other hand, compositions of two fractional operators of different type (Riemann-Liouville and Weyl). Their inverses contain, in general, operators of differentiation of finite order.
Classaficataon of Integral Transforms 51
2.3.3. Mellin convolution transforms of Laplace type The Mellin transform Ic*(s) of the kernels of these transforms is characterized by exponential-power decay on (1/2 - zoo,1 / 2 zoo) as 3 (s) + f-00:
+
0
< c < IIC*(s)eYlslsal < C < M, y > 0, a
E
R, %(s)
=
1/2.
(47)
Examples are the Laplace transform, the Stieltjes transform
with the modified Bessel function K,(z) and I,(z) in the kernels,' the Meijer K , transform
1
M
ds) =
e-SyKv(sY) f(Y)dY,
0
f(y) =
A 47rva
/
y+im
sesy [(2v
+ l ) ~ - l ( s y )+ 4v1,(sy)
y--ioo
(52)
(53)
+ (2v - l>~,+l(SY)lg(s)ds. All of these transforms (Laplace, Stieltjes, Meijer, Meijer-Bessel,. . . ) map integrable functions into functions analytic in some half plane. Their inverses, if considered strictly only on the real line, include some differential operators of infinite order. For example, the real variable inverse of the Laplace transform has the form
while the real variable inverse of the Stietjes transform has the form
Most of the Mellin convolution transforms are special cases of the following convolution G-transform
52
V. K. R u n
with the Meijer G-function in the kernel of Ref. 8. Kesarwani16-'* has found the condition when this transform is of Watson type
In this case the inverse has the form
One can find conditions when this transform is of Riemann-Liouville type
and the inverse formula in this case has the form
or of Laplace type
c*=p+q-2(m+n)>0.
(61)
with the real variable inverse of the form
(see Ref. 37 for more details).
As it was said, a Mellin convolution transform can be decomposed as composition of direct and inverse Mellin transforms and an operator of multiplication
(Kf)(x) = M-' @ * ( s ) M { f ( y )1;- s ) ; .I.
(63)
As a special case of the Mellin convolution transform, the convolution Gtransform (56) enjoys more interesting composition structure. The Mellin transform of the kernel, the Meijer G-function GE;" (x is a quotient of products of Gamma functions ( m+ n in the numerator and p +q - m - n
):I,
in the denominator). Each Gamma function in the numerator corresponds
Classification of Integral Transforms 53
to a modified Laplace transform, and each Gamma function in the denominator corresponds to a modified inverse Laplace transform.2 Therefore] the convolution G-transform (56) can be decomposed as composition of m n direct Laplace transforms and p q - m - n inverse Laplace transforms. If we do not want to include the inverse Laplace transform] that contains integration in the complex plane, into the composition, then in the composition we have to use some Riemann-Liouville, Weyl fractional operators27 and Hankel transforms (the Mellin transforms of their kernels are quotients of one Gamma function over one Gamma function). The convolution G-transform is of Riemann-Liouville or Watson type if and only if it can be decomposed into composition of only Riemann-Liouville, Weyl fractional operators and the Hankel transforms (without the Laplace and inverse Laplace transforms). contains as special cases many clasMeijer G-function GEhn (x
+
+
:;;) : 1
sical special functions (orthogonal polynomials, Bessel functions] confluent hypergeometric functions] hypergeometric functions] t o name just a few). Therefore it has a very complicated asymptotic behavior near 0 and infinity,' and studying existence and mapping properties of convolution Gtransform (56) in classical L, spaces is a very challenging problem. However, its Mellin transform k * ( s ) is a quotient of products of Gamma functions] and therefore has a very simple asymptotic behavior a t infinity, powerexponential decay. We introduce now a space of functions M;;(L,) based on the power-exponential asymptotic behavior of k * ( s ) on %s = l / 2 . For simplicity we assume p = 2. The general case p is considered in.37 This space turns out to be very convenient in studying convolution G-transform (56).
Definition 2.1. Let 2 signc+signy 2 0. By M,+(L2) we denote the set of functions f on R+ = (Olm)lwhich can be represented as the inverse Mellin transform
f(x) =
/
l/Z+im
f*(s)IC-Sds
1/2--im
+
of f*(s) from L2(1/2 - im, 1/2 im) with weight IsIYeTClsl.If the norm o f f in M;:(L2) is defined through the norm of f*(s), then M;;(Lz) is a Banach space. = 0, then MCA(L2) = L2(R+).If c = 0, y > 0, then f E M;:(L2) if and only if e"/2f(e")belongs t o the space of Bessel potentials L;(R). If c > 0, then f E M;+(L2) if and only if f is infinitely differentiable a.e.,
If c = y
54
V. K. Tuan
and moreover, (2^c)2" ' '
N n
'
2 < 00.
dX
'
L2(R+)
The importance of the space M.C^(LP) can be seen from the following fundamental result Theorem 2. 1.37 All convolution G-transforms, whether they are of Watson, Riemann-Liouville, or Laplace types, are automorphisms from M - i onto A^~
2.4. Fourier cosine and sine convolution transforms There exist some different convolutions related to the Fourier cosine and Fourier sine transforms. The first convolution has the form29 1 f°°
(/**)(*) = -7= / [Ml x - y |) + k(x + y)]f(y)dy, V2.1t Jo
(64)
and it satisfies the convolution property
Fe(f*g)(x) = (Fcf)(x) (Fck)(x).
(65)
Another convolution was first introduced by Churchill7 1 f°° (f*k)(x) = -= \ (k(\ x-y\)-k(x + y)]f(y)dy,
(66)
V27T Jo
and the respective convolution property for (66) has the form
F,(f*k)(x) = (F,f)(x)(Fck)(x).
(67)
Recently, the third convolution for the Fourier cosine and sine transforms has been discovered25 - = V 2?r It was shown there that
- x)k(\ y-x\) + k(y+ x)}f(y)dy.
)(x),
xeR+ .
(68)
(69)
Corresponding to these three convolutions we can construct three classes of integral transforms of convolution type. For example, for the convolution transform of the type (64) it has been proved43 that the condition Jo
*
4
Classification of Integral Transforms
55
is necessary and sufficient for the transform g(x)=(l-^-] f (k1(x + y)+kl(\x-y\))f(y)dy, ax V ) Jo
x e R+ ,
(71)
to be unitary in L2(R+) with the symmetric reciprocal formula / d2 \ f°° t \ f ( x ) =[!- — } I {ki(x + y)+ki(\x-y\)j g(y)dy,
x £ R+ . (72)
If moreover, k± is twice differentiate, and k(x] = k\(x] — fcj (x) is locally bounded on R+, then the integral transform oo (k(x + y) + k(\x-y\))f(y)dy, x & R+ , (73) / satisfies the Plancherel formula \\9\\L,(R+) = ||/IU 2 (fl + ).
(74)
and reciprocally, f(x) = j
(k(x + y)+k(\x-y\))g(y)dy,
x £ R+ .
(75)
As examples we can take k(x] = -(Ai(x) + Ai(-x) + iGi(x) + iGi(-x)),
(76)
and therefore, k(x) = -(Ai(x) + Ai(-x) - iGi(x) - iGi(-x)).
(77)
£t
Since Ai(x) and Gi(x), the Airy functions, are bounded on the whole real line,1 the kernel k(x) is bounded, therefore, transform (73) with the kernel (18) is a bounded operator from Lp(R+), 1 < p < 2, into Lq(R+), p~l + q~l = 1. Moreover, it is unitary on Z/2(-R+) and k ( x ) is the kernel of the inverse operator. Another example is f,fx\ _ —e~^rH^(a) lx 2 '
(78)
i:c >(a} L/'.,.') — _ -e^TfT*'{•><) 9 \ ''
\ <7Q} '
and
56
V. K. Tuan
where H}, (a) and Hi, (a) are the Bessel functions of the third kind. v Transform (73) with kernel (78) is a bounded operator from LP(R+), 1 < p < 2, into Lq(R+), p"1 + q~l = 1, and moreover, it is unitary on L2(R+) and (79) gives the kernel of the inverse operator. Let
k(x) = -V2asinh7ra xia-1/2Kl/2_ia(x),
x & R+ .
(80)
~k(x) = -\/2asinh7ra x'ia-1/2Ki/2+ia(x),
x e R+ .
(81)
Then 7T
Transform (73) with kernel (80) is unitary on L2(R+) and its inverse has (81) as the kernel. For the convolution (66) it was shown that if fci e L2(R+), then the transformation g(x) =
1- —
(k^x - j/|) - ki(x + y)} f(y)dy,
x £ ,R+,
is unitary on L2(R+), and the reciprocal formula _ _ f(x)= (1--J-2 / [ki(\x-y\}-kl(x + y)}g(y)dy, a
x & R+,
O-X
\
X
\
/ JQ
/ JQ
holds almost everywhere, if, and only if, the kernel satisfies the condition (18). Suppose that the kernel k\ is twice differentiable, and k — k\ — k'[ is locally bounded on R+, then the integral transform oo (k(\x-y\~k(x + y \ ) ) f ( y ) d y , x e R+ . (82) //o. satisfies the Plancherel formula (83)
and reciprocally, f(x) = I
(k(\x-y\) - k(x+y)) g(y)dy,
x € R+ .
As examples we can take k ( x ) = -r-r-r - [J«(ia) - J_ ix (ia) + e^/_«(o) - e ii SllHi 7TX
(84)
Classification of Integral Transforms 57
where J p ( z ) is the Anger-Weber function.' For the convolution (68) it was shown that if kl E Lz(R+),then the transformation
is unitary on La(R+), and the reciprocal formula
holds almost everywhere, if, and only if, the kernel satisfies the condition
Similar results are obtained when the kernel Icl is twice differentiable, and Ic = Icl - Icy is locally bounded on R+. As examples we can take 1 k ( z ) = - ( G i ( z ) - Gi(-z) iAi(-z) - i A i ( ~ ) ) ,
+
2
and
3. Nonconvolution Transforms 3.1. W i m p nonconvolution G-transform.
The only class of nonconvolution transforms known until the eighties is the Wimp nonconvolution G - t r ~ n s f o r r n ~ ' ~ ~ ~
Special cases of this class of transforms are the Kontorovich-Lebedev transformlg "f
the Mehler-Fock transform" g(z)
=l"
Piz-I/2(Y)f(Y)dY,
58
V. K. Tuan
with the Legendre function of the first kind P,,(z)l in the kernel, the Olevskii transform26 g(z) = J ;
zB(a
2Fl
(a
+ iz,a - iz;c; -Y)f(Y)dY,
(90)
+ iz,a - ix; c; -y)g(x)dz,
with the Gaussian hypergeometric function 2F1 in the kernel,' and the Lebedev and Lebedev-Skalskaia transform^.'^ The conventional wisdom is that this is the only class of nonconvolution transforms. You will see soon how wrong it is. Analyzing the composition structure of the Wimp nonconvolution G-transform, Y a k u b o ~ i c h has ~ ~ ,found ~ ~ that the Wimp nonconvolution G-transform can be decomposed into the composition of a Kontorovich-Lebedev transform and a convolution G-transform (56). But the Kontorovich-Lebedev transform is a composition of two cosine Fourier transforms and an operator of substitution, and the convolution Gtransform (56) is a composition of Mellin and inverse Mellin transforms and an operator of multiplication, then the Wimp nonconvolution G-transform is a composition of just Fourier transforms (Mellin and Fourier transforms are the same, up to a change of variables), an operator of multiplication and substitution of variables. Using this composition structure we have proved a Plancherel theorem for the Wimp nonconvolution G-transform as well as for its special cases37$45
Theorem 3.1. The Wimp integral transform
where c* = y* = 0 , p # q , and the integral is understood in L2(R+;xsinh2nx) is a homeomorphism from L2(R+) onto L2(R+;x sinh2.rrx) and the inverse transform has the form
If, moreover, p = 2n, q = 2m, m # n; an+i = --ail ai E R, i = 1 , .. . ,n; b,+j = -bj, bj E R, j = 1,.. . , rn, then we have the Parseval identity
1
00
rcsinh2.rrxJg(x)12dx=
If(y)12dy.
(94)
Classification of Integral Transforms
In particular, put m = 1,n = p = 0 , q = 2, bl = a - 1/2, bz = 1/2 obtain the Olevskii transform with c = a. Hence, we have
-
59
a, we
Theorem 3.2. Let c = a . T h e n the pair of Olevskii transforms (go), (91), where the integrals are understood in L2(R+;z2(z and L2(R+;y1-2a) norms, respectively, is a homeomorphism between L2(R+;Y’-~“)and Lz(R+;zz(z l)4a-3). Moreover, the Parseval identity holds 00 00 1 z s i n h 2 m p 2 ( a iz)g(z)12dz. (95) y1-2alf(y)12dy =
+
+
1
1
+
In particular, taking a = 1 / 2 we get the Lz- theorem for the Mehler-Fock transform established before by L e b e d e ~ . ~ ~
3.2. Transforms of Kontorovich-Lebedev type Another generalization of the Kontorovich-Lebedev transform comes directly from its composition structure.37 Let cp(t)be a strictly increasing on R, cp(-00) = -00,cp(00) = 00, so that the improper integral
k ( z , y) = 27r
Jm
e-izt+iYV(t)dt
(96)
converges uniformly on any closed interval of R\{O}. For example, if cp(t)is odd and twice differentiable function, cp’(00) = 00, cp”(t) > c > 0 on [ N ,m) and p”(t)/(cp’(t))’ E L ( N ,cm),then the integral (96) converges uniformly on any closed interval of R\{O}. Then
1, f ( Y ) =A / q-2, 00
s(z)=
k ( z ,Y ) f ( Y ) d Y ,
(97)
m
-y)zg(z)dz, (98) Y -m is a pair of integral transforms. - If cp(t)= a sinht, a > 0, we obtain the following pair of integral transforms
s(z)=;
1
s_, ”
lrz
e~-“’gnYKiz(lyl)f(y)dy,
l f ( y ) =-
TY
J
m
-m
(99)
lrI
e~s’gnYKiz(lyl)zg(z)dz.
(100)
If f is an odd function, (99) and (100) become the pair of Kontorovich Lebedev transforms.
-
60 V. K. Tuan
If cp(t) = sinht transforms
-
-
If cp(t)= t 3 / 3
+ at, a > -1,
we obtain the following pair of integral
+ a t , we obtain the following pair of integral transforms (103)
/
1
f ( y) = -
Y
00
y-1/3Ai(y-1/3(ay - x))zg(z)dx.
(104)
--m
3.3. The Buchholt nonconvolution G-transform The following "odd" transform was introduced by Buchholz4
with the Whittaker function M p , v ( x )in the kernel.' It is a composition of a Mellin transform, a multiplication operator, and a Hankel transform. In its composition structure replace the Hankel transform by a more general convolution G-transform (56) we could construct a nonconvolution Gtransform including the Buchholz transform37
as a special case.
Theorem 3.3. The G- transform (107) is a homeomorphismfrom Lz(R+) onto La(R) and has the inverse -_
f(Y) = (2.rr)rrrY
(
em-i/2y
I
(108)
r 2 ,...,I,f - ( a g f l ) - i x ' 1 - (a,)-iz 2 9
4- ( b ? + ' ) - i x ,
-( b , ) - i x
yi"-
f g (5)dx.
Classification of Integral R a n s f o m s
If, moreover, p = 2 n , q = 2 n , a,+j = -aj E R,j= 1,..., n;bm+j = R , j = 1,...,m, then we have the Parseval formula
-bj
61
E
The G-transform (107) is a composition of a Mellin transform, multiplication by e-iT-xl'r , and a convolution G-transform. Put in formulas (107) and (108) m = 1,n = p = 0 , q = 2, bl = u, b2 = -u, r = a = 1,we obtained the pair of Buchholz integral transforms. Hence, we have
Theorem 3.4. The pair of Buchholz transforms (105) and (106) are home)). omorphisms between L2(R+;y) and L2 ( R ;(x2 l ) x ( v ) e T ( x - ~ z ~Moreover, if u E R then the following Parseval equation holds
+
In particular, if we take the Fourier cosine (sine) transform as the inner convolution G-transform then we obtain the pair of transforms
The Parseval equality
also holds. Here is the composite structure
If we replace the Fourier cosine transform by the Hankel transform we obtain an integral transform with the Legendre function in the
3.4. The cherry nonconvolution G-transform In 1949 Cherry' considered an index transformation with & ( z ) as its kernel and established that, if f E L 1 ( R )is of bounded variation and continuous
62
V. K. man
at x,then
In fact, Cherry's expansion (112) is still valid when the integrability condition on f is replaced by some condition on the asymptotic behavior o f f a t infinity such as
where Q > $, c1 and c2 are constants, which may be different for x --t +m and x .+ -CQ. To derive composition structure and a Plancherel-type theorem for the Cherry transform, we rewrite the Cherry expansion (112) in the following form:
and
respectively. Substituting the integral representation (2.3.15.3) from Ref. 9 that is, for the parabolic cylinder function Dv(z),
and
Classification of Integral Transforms
63
into the formulas (7) and (8), and interchanging the order of integration that is permissible if f ( y ) E S(R), the Schwartz space of all infinitely differentiable functions which rapidly vanish at infinity together with all derivatives, we arrive at the following composite representations for the transforms D+, D-, 0;’ and DI1:
and
Here M and M-’ are the Mellin transform and its inverse, and F and F-’ are the Fourier transform and its inverse. The composite representations (13) and (14) of the transforms D+ and D- are key tools in studying the Cherry expansion (6). By L:(R;e*) (similarly, by L,(R;e*)) we denote a subspace of &(R) consisting of all square-integrable functions f ( y ) such that the Fourier transforms of f(y)e$ vanish on the negative (positive) part of the real axis. These spaces are Hilbert spaces and their elements can be analytically continued onto the whole upper (lower) complex half-plane. Any function f E L2(R) can be uniquely factorized by
f(Y)
= f+(Y)
+ f-h),
where
f+
E
Lz(R;eg)
and
f-
E L,(R;e*).
Applying the Parseval equalities for the Mellin and Fourier transforms to the composite representations (13) and (14), we obtain the following Parseval equalities for the transforms D+ and D-:
V. K. Tuan
64
and
From the composite representations (13) and (14), the Parseval equalities, and the mapping properties of the Fourier and Mellin transforms, by standard procedure we can expand these identities to the whole spaces L t ( R ; e $ )
and L F ( R ; e * ) .
Thus D+ and D-
are isomor-
phisms from L$(R;e* ) and L, (R;e$), respectively, onto the space L2(R;eY(cosh.rrz)-l), and the integrals (7) and (8) converge in the latter Hilbert space. Comparing the composite representations (13), (14), (15) and (123), it is easy to see that (15) is the inverse of (13), and (123) is the inverse of (14). Hence the operator 0;'is the inverse of D+, and DI1 is the inverse operator for D-, and the integrals (9) and (117) converge in L2(R). Thus
DTID+f+ = f +
and
D I I D - f - = f-.
By the definitions of f + and f - , we have
F { f f ( y ) e $ ; -z}
F{f-(y)e$;s} =0
and
=0
(2
> 0).
Consequently, from (13) and (14) it is easy to see that
(D+f-)(x)
=0
and
(D- f+)(x) = 0.
Thus, for any f E L2(R), we have
D+lD+f
+ DI1D- f = D I l D +f + + D I I D - f -
=f+
+f-
= f.
We sum up our results in the following Plancherel-type theorem:
Theorem 3.5. T h e Cherry transforms (7) and ( 8 ) isomorphically m a p the spaces L t ( R ;eP) and L,(R; e s ) onto the space L2(R;ef(cosh7rz)-'), admit the inversion formulas (9) and (117), and satisfy the Parseval equalities (124) and (125), respectively. Furthermore, the expansion formula (6) holds true in L2(R). We will show now that the Cherry transform is related to a singular Sturm-Liouville problem on the whole line. As a consequence, another Plancherel-type theorem for the Cherry transform is established. For more details see.3o Let us consider the following singular Sturm-Liouville problem:
Ly
=
d2Y - q ( 2 ) y = - x y dx2
(-oo < x
< oo),
(126)
Classification of Integral Transforms 65
< 00,
lY(&:.)I
with q(z) being even on R such that the spectrum of the singular SturmLiouville problem (1)is the whole R. Let $(z, A) and O(z,A) be the solutions of the equation (1) such that
d(0, A) = 0, $'(O, A) = -1, 6(0,A) = 1, O'(0, A) = 0.
There exist two functions ml (A) and mz(A) analytic in the upper half-plane such that, as a function of x, $1 (X,
A> = Q(x,A> + ml (A>4J(z,4
is in Lz(-m, 0), and 1cIz(z, A) = O(z, A)
+ mz(A)$(z,A>
is in LZ(0,m), for non-real A. Moreover, since q ( z ) is even, we have ml(A) = -mz(X). In addition, there exist two non-decreasing functions E(A) and C(A) such that lim
6+0+
J*s 0
[
]
2m2(21+26)
du=J(A)
and
where the symbol '3 means the imaginary part of a complex-valued function. It is a known fact31 that if f(z)E Lz(R),then W
JYA) =
Lw
f(x)4(z, 4 dz
belongs to Lz(R,&), and W
belongs to Lz ( R ,@). The following inversion formula
holds true; and in addition, the Parseval equality:
(129)
66
V. K. Tuan
is valid. Now we are ready to consider the following singular Sturm-Liouville problem d2y
x2
d22 + 7 Y = -XY
(--00
< x < a),
(133)
Functions
-5
are solutions of the equation (133). Since q ( x ) = is even, q ( x ) 3 -03 as x -+ f03,and the integral J ' , Iq(x)I-;dx is divergent, there is a continuous spectrum over ( - 0 ; ) , 0 3 ) , ~ ~and the formulas (4), (5), (8) and (132) hold true. It is easy to see that
and
satisfy the initial conditions (2) and (3). As x + 03, e y x has the argument and - e y x has the argument Consequently, the following asymptotic formulas for the parabolic cylinder functiong can be used:
-?.
2,
(-? <
arg(z)
7
< -- . 4
(137)
Classification of Integral Transforms
67
(x-+ cm). If X is in the upper half-plane (A ,ax-+
=T
+ id; b > 0), then
- i7--6--1
--z
2
E
L2(1,00).
Consequently, we observe that
q-z, A) + m2(44(-z,
E
~ ~ ( 0 ~ 0 0 )
for all X in the upper half-plane if and only if the term -z-ix-ie-$ ishes, that is,
and
So, finally, the formulas (4),( 5 ) , (8) and (132) take the forms:
van-
68
V. K. Tuan
respectively. If we put
then the formulas (140), (141), (142) and (143) become
Classzficataon of Integral Transforms
69
and
respectively. Hence we can sum up our results in the following Plancherel-type theorem. Theorem 3.6. The integral transforms (144) and (145) are isomorphisms f r o m L2(R) onto L,(R; e* (cosh AT)-'). Moreover, the inversion formula has the form (146) and the Parseval equality (147) holds true.
One can show that the transforms (144), (145) and (146) are equivalent to the Cherry expansion (112). The composition structure of Cherry's transform gives rise t o a new Cherry nonconvolution G - t r a n ~ f o r m . ~ ~ ~ ~ ~ Let p = 2n, q = 2m, m # n; an+i = -ai, ai E R, i = 1 , .. .,n; bm+j = -bj, b j E R, j = 1 , .. . , m. Then Cherry G-transform
is an isomorphism from L2f(R) onto L2(R+)and the following inverse formula holds
Similarly, Cherry G-transform
is an isomorphism from LT ( R )onto La (R,) and the following inverse formula holds
The following expansion holds in L2 ( R )
70
V. K. Tuan
3.5. Other nonconvolution G-transforms Let A, y @ (-a, 01. Then the integral transform3*
is a bounded operator from L2(0,m) onto L2(-00, m) and has the inverse
If, moreover, m = n , p = q, bj = - u j , j = 1,.. . , p , d j = - c j l j = 1,. . . ,m, X = -y = 26, b E R, then the Plancherel equality holds
1,
l o o -
2n
19t4l2dz=
.Im
If(Y)12dY.
G-transforms (107) and (152) look completely different, but they have very similar composition structure. In fact (152) can be obtained from (107) by replacing the multiplicator e-zTZ1’‘‘ by Playing with only the Fourier transforms, operators of multiplications and substitutions of variables we have found some more new classes of nonconvolution G-transforms and prove L2 theorems for them. As special cases about 60 new transforms with L2 theorems are added t o the list of around 300 known transform^.^^
z.
References 1. M. Abramowitz and I. A. Stegun Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Reprint of the 1972 edition. (Dover Publications, Inc., New York, 1992), pp. xiv+1046 2. Yu. A. Brychkov, H.-J. Glaeske and 0.1. Marichev Itogi Nauki i Tekhniki. Mat. Anal., V I N I T I A N SSSR 21, 3 (1983). 3. Yu. A. Brychkov, H.-J. Glaeske, A.P. Prudnikov and Vu Kim Tuan, Multidimentional Integral Transformations, Gordon and Breach, (New York Philadelphia - London - Paris - Montreux - Tokyo - Melbourne - Singapore, 1992). 4. H. Buchholz, The Confluent Hypergeometric Function, (Springer Verlag, Berlin - New York, 1969), pp. 240 5. I. W. Busbridge, J . London Math. SOC.9, 179 (1934).
Classification of Integral Transforms 71 6. T . M. Cherry, Proc. Edinburgh Math. SOC.(2) 8 , 50-65 (1948). 7. R. V. Churchill, Operational Mathematics. 2nd Ed. (McGraw-Hill Book Co., Inc., New York - Toronto - London, 1958), pp. ix+337 8. A. Erdklyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions, Vol. I. Based, in part, on notes left by Harry Bateman, (McGraw-Hill Book Company, Inc., New York - Toronto - London, 1953), pp. xxvi+302 9. A. Erdklyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions, Vol. 11. Based, in part, on notes left by Harry Bateman, (McGraw-Hill Book Company, Inc., New York - Toronto - London, 1953), pp. xxvi+396 10. A. Fock, Dokl. Akad. Nauk S S S R 39,279 (1943). 11. H.-J. Glaeske and Vu Kim Tuan, Serdica Bulgar. Mat. Publ. 16,143 (1990), MR 91k: 44009. 12. H.-J. Glaeske and Vu Kim Tuan, Boundary Value and Initial Value Problems in Complex Analysis: Studies in Complex Analysis and Its Applications to Partial Diflerential Equations 1 , (Halle, 1988), pp. 209-220, Pitman Res. Notes Math. Ser. N 256,1991, Longman Sci. Tech., Harlow., MR 93h: 42010. 13. H . 4 . Glaeske and Vu Kim Tuan, Math. Nachr. 152,179(1991). 14. G. H. Hardy and E. C. Titchmarsh, Proc. London Math. SOC.35,116 (1933). 15. Jr. Hirschman and D. V. Widder The Convolution Transform, (Princeton Univ. Press, Princeton, 1955). 16. N. Kesarwani, Proc. Amer. Math. SOC.13,950 (1962). 17. N. Kesarwani, Proc. Amer. Math. SOC.14,18 (1963). 18. N. Kesarwani, Proc. Amer. Math. SOC.14,271 (1963). 19. M. I. Kontorovich and N. N. Lebedev, Zhurn. Eksperim. i Teoret. Fis. 8 , 1192 (1938). 20. E. R. Love, Proc. Edinburgh Math. SOC.15,169 (1967). 21. 0. I. Marichev, Soviet Math. Dokl. 13, 703 (1972). 22. 0. I. Marichev and Vu Kim Tuan, Some Volterra equations with the Appell function F1 in the kernel (Russian). Scientific Works of the Jubilee Seminar o n Boundary Value Questions, (Collect. Artic., Minsk, 1985.) pp. 167-172. 23. 0. I. Marichev and Vu Kim Tuan, Reports of the Extended Sessions of a Seminar of the I. N . Vekua Institute of Applied Mathematics (Tbilisi, 1985), (Tbilisi Gos. Univ., Tbilisi, 1985), pp. 139-142. 24. 0. I. Marichev and Vu Kim Tuan, Complex Analysis and Applications '85 (Varna, 1985), pp. 418-433, ( Bulgar. Acad. Sci., Sofia, 1986). 25. Nguyen Xuan Thao, V. A. Kakichev, and Vu Kim Tuan, East- West J. Math. 1,85 (1998). 26. M. N. Olevskii, Dokl. Akad. Nauk S S S R 69, 11 (1949). 27. A. P. Prudnikov, Yu. A. Brychkov and 0. I. Marichev, Integrals and Series. Vol. 5. Inverse Luplace Transforms, (Gordon and Breach Science Publishers, New York, 1992), pp. xx+595. 28. M. Saigo, Math. Rep. Kyushu Univ. 11,135 (1977/78). 29. I. N. Sneddon Fourier Transforms, Reprint of the 1951 original, (Dover Publications, Inc., New York, 1995), pp. xii+542. 30. H. M. Srivastava, Vu Kim Tuan, and S. B. Yakubovich, Quart. J. Math. Oxford Ser. 51,371 (2000).
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31. E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I, Second Edition, (Clarendon Press, Oxford 1962), pp. vi+203. 32. Vu Kim Tuan, Dokl. Akad. Nauk BSSR (Russian) 29, 584 (1985). 33. Vu Kim Tuan, Dokl. Akad. Nauk. Armyan S S R (Russian) 83, 7 (1986). 34. Vu Kim Tuan, Dokl. Akad. Nauk S S S R (Russian) 286,521 (1986), translation in Soviet Math. Dolcl, 33, 103 (1986). 35. Vu Kim Tuan, Dokl. Akad. Nauk B S S R (Russian) 30, 689 (1986). 36. Vu Kim Tuan, Generalized integral transformations of convolution type in some space of functions. Complex Analysis and Applications '85 (Varna, 1985), (Bulgar. Acad. Sci., Sofia, 1986) pp. 720-735. 37. Vu Kim Tuan, Integral Transforms and Their Composition Structure (Russian), Dr. Sci. Dissertation, (Belarusian State University, Minsk, 1987), p. 275. 38. Vu Kim Tuan, Dokl. Akad. Nauk S S S R (Russian) 299, (1988), translation in Soviet Math. Dokl. 37, 317 (1988). 39. Vu Kim Tuan, Dokl. Akad. Nauk S S S R (Russian) 300,521 (1988), translation in Soviet Math. Dokl. 37, 669 (1988). 40. Vu Kim Tuan, Some integral transformations with a Macdonald function in the kernel (Russian) Current Analysis and Its Applications, (Naukova Dumka, Kiev, 1989), pp. 16-22. 41. Vu Kim Tuan, Ukrain. Mat. Zh. (Russian) 4 2 , 9 9 0 (1990), translation in Ukr. Math. J. 4 2 , 880 (1991). 42. Vu Kim Tuan, Airy integral transform and the Paley-Wiener theorem, in Proceedings of the 2nd International Workshop on Transform Methods and Special Functions, held in Varna, Bulgaria, August 24-29, 1996, Editors P. Rusev, I. Dimovski and V. Kiryakova, (Institute of Mathematics, Sophia, 1998), pp. 523-531. 43. Vu Kim Tuan, J . Math. Anal. Appl. 229, 519 (1999). 44. Vu Kim Tuan, Dokl. Akad. Nauk S S S R (Russian) 313, 1299 (1990), translation in Soviet Math. Dokl. 42, 150 (1991), MR 9lk: 44006. 45. Vu Kim Tuan, 0. I. Marichev and S. B. Yakubovich, Dokl. Akad. Nauk SSSR (Russian) 286,786 (1986), translation in Soviet Math. Dokl. 33, 166 (1986). 46. Vu Kim Tuan and Nguyen Thanh Hai, Dokl. Alcad. Nauk S S S R (Russian) 317, 797 (lYYl), translation in Soviet Math. Dokl. 43, 508 (1991). 47. Vu Kim Tuan and S. B. Yakubovich, Dokl. Akad. Nauk BSSR (Russian)29, 11 (1985), translation in Amer. Math. SOC.Transl. 137, 61 (1987). 48. Vu Kim Tuan and S. B. Yakubovich, Ukmin. Math. Zh. (Russian) 44, 697 (1992), translation in Ukrainian Math. J . 44, 630 (1993). 49. G. N. Watson, Proc. London Math. SOC. (2) 35, 156 (1932). 50. D. V. Widder, The Laplace Transform, (Princeton Univ. Press, Princeton, 1946). 51. J. Wimp, Proc. Edinburgh Math. SOC.(2) 14, 33 (1964). 52. S. B. Yakubovich, Izu. Vyssh. Uchevn. Zaved Mat. (6), 77-79 (1986). 53. S. B. Yakubovich, Indez Transforms, (World Scientific Publishing Co., Inc., River Edge, NJ, 1996), pp. xiv+248. 54. S. B. Yakubovich, Vu Kim Tuan., 0. I. Marichev , and S. L. Kalla, Rev. Tecn. Fac. Ingr. Univ. Zulia, Edicion Especial 10, 105 (1987).
Chapter I1
PARTIAL DIFFERENTIAL EQUATIONS
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Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et a2. (pp. 75-89) @ 2007 World Scientific Publishing Co.
75
55. UNIFIED MINIMAX METHODS MARTIN SCHECHTER Department of Mathematics University of California, Irvine Irvine, C A 92697-3875 E-mail: mschechtOmath.uci.edu
The variational approach to solving nonlinear problems eventually leads t o the search for critical points of related functionals. In case of semibounded functionals, one can look for extrema. Otherwise, one is forced t o use minimax methods. There are several approaches t o such methods. In this paper we unify these approaches providing one theory that works for all of them. The usual approach has used Palais Smale sequences. We show that all of them lead t o Cerami sequences as well. Applications are given.
1. Introduction Many problems arising in science and engineering call for the solving of the Euler equations of functionals, i.e., equations of the form G’(u) = 0 ,
where G(u) is a C1 functional (usually representing the energy) arising from the given data. The classical approach was to look for maxima or minima. If one is looking for a minimum, it is not sufficient to know that the functional G(u) is bounded from below. However, if it is bounded from below, one can show that there is a sequence satisfying
G ( u ~-+) a ,
G ’ ( u ~-+ ) 0
(1)
for a = inf G. If the sequence has a convergent subsequence, this will produce a minimum. A functional G is said to satisfy the PS condition if every sequence of the form (3) has a convergent subsequence and consequently produces a critical point. AMS Subject Classification: Primary 35565,58305,49B27. Key words and phrases: Critical point theory, variational methods, saddle point theory, semilinear differential equations.
76
M.Schechter
Actually, one can do better. If the functional G(u) is bounded from below, then there exists a sequence satisfying G(N)
-+
+
a,
(1 l l u k I l ) G ' ( ~ -+ ) 0
(2)
for a. = inf G. Such a sequence is called a Cerami sequence. It will have a convergent subsequence even in cases when a PS sequence does not. However, when the functional is not semi-bounded, there is no organized procedure for producing critical points. One approach is to search for circumstances other than semibound- edness which will produce PS sequences (3). A successful method has been t o find subsets A and B having the property that when they separate a functional, a PS sequence (3) results. Specifically, we want the subsets t o have the property that for every G E C' (E l R) bounded on bounded sets and satisfying
< bo := inf G,
a0 := sup G A
there is a sequence {uk}
cE
I3
(3)
and a constant a such that
bo
(4)
and (3) holds. When this is true, we say that A links B. Fortunately, such pairs exist and help solve many problems. There have been several approaches to linking;2~6~7~13~14 we describe them below. Here we shall present an approach which includes all of them, is more general and is easy t o apply. The idea is as follows. For each A c E one wishes t o find a collection K of sets K with the following properties. 1. If G E C'(E, R) satisfies a0
:= s u p G A
< a := KEK inf supG, K
then there is a sequence {uk} c E such that (3) holds. 2. If CLO
5
SUP K\A
GI
K
E
Ic,
then there is a sequence {uk} c E such that (3) holds. 3. If A , B c E satisfy
BnK=+, K E K
AnB=4, and
a0 := sup G A
5 bo
:= inf G, B
(5)
Unified Minimax Methods
77
then there is a sequence { u k } c E such that (3) holds. 4. If gK = { u E K\A: G ( v )2 ao}
# 4,K E K,
(9)
then there is a sequence { u k } c E such that (3) holds. In the next two sections we determine collections K that have these properties. Moreover, we shall show that all of these collections produce not only PS sequences (3), but also Cerami sequences (4). In Sections 4-6 we describe known methods of linking and find the sets K in each case. In Section 7 we give some applications. 2. Definitions and Theorems
In this section we begin with a Banach space E.
Definition 2.1. We shall say that a map 'p : E + E is of class A, if it is a homeomorphism onto E, and both 'p, 'p-'are bounded on bounded sets. Definition 2.2. For A c E, we define
R ( A ) = {'p
E
A : P(U)
=U,
u E A}.
Definition 2.3. For a nonempty set A c E , we define a nonempty collection K = K ( A ) of subsets K c E t o be a minimax system for A if it has the following property:
p(K)E K,
'p E
A ( A ) , K E K.
Every nonempty set has a minimax system. We have
Theorem 2.1. Let K be a minimax system for a nonempty subset A of E , and let G ( u ) be a C' functional on E . Define a := inf sup G , KEK K
(10)
and assume that a is finite and satisfies a > a0 := supG. A
Let $(t) be a positive, locally Lipschitz continuous function on [0,co) such that
Then there is a sequence { u k } c E such that
M. Schechter
78
Definition 2.4. We shall call a ( t ) E C([O,T]x ElE ) , T > 0, a flow if a ( 0 ) = I and for each t E [O, TI , a ( t ) is a homeomorphism of E onto itself. Theorem 2.2. Let K be a minimax system for a nonempty set A and let G ( u ) be a C’ functional on E such that gK =
{w E K\A: G ( v ) L ao}
# 4,K E K,
and a = infsupG < 03. I C K
Assume that f o r any b > a , K E K and flow g ( t ) satisfying
G(a(t)v)< b, w
E
K,t > 0,
and
G(a(t)u)< a , u E A , t > 0 , there is a K E K: such that there is a K
I? c
E
K such that
U a ( t ) AU a(T)[EbU K ] , tE[O,tl
where
E, = {U E E : G(u) 5 0). Then the conclusions of Theorem 2.1 hold. 3. Linking Subsets We now show how linking can play a major role in finding critical points.
Definition 3.1. We shall say that a set A in E links a set B to a minimax system K for A if
AnB=$
cE
relative
(20)
and
B n K # 4, K
E K.
We shall say that A links B [mm] if there is a minimax system K: for A such that A links B relative to K.
Unified Minimax Methods
79
Theorem 3.1. Let K be a minimax system for a nonempty set A, and assume that there is a subset B of E such that A links B relative to K . Assume that G E C 1 ( E R) , satisfies
a0 := sup G < bo := inf G , A
B
(22)
and that the quantity a given b y (1) is finite. Then for each positive, locally Lipschitz continuous function Q ( t ) on [0,co) satisfying (12), there is a sequence { u k } c E such that (13) holds. Definition 3.2. A subset A of a Banach space E links a subset B of E if for every G E @ ( E lR) bounded on bounded sets and satisfying (22) there is a sequence { u k } c E and a constant a such that
bosa
(23)
and (3) holds.
Definition 3.3. A subset A of a Banach space E links a subset B of E strongly if for every G E C1(E,R) bounded on bounded sets and satisfying (22) and each positive, locally Lipschitz continuous function +(t)on [0, co) satisfying (12), there is a sequence { u k } C E such that (10) and (13) hold. Theorem 3.2.
If A links B [mml, then it links B strongly.
Theorem 3.3. Let K be a minimax system for a nonempty set A, and assume that A links a subset B of E relative to K . Let G be a C1 functional satisfying (14)) (15) and (24)
a0 := sup G 5 bo := inf G. A
B
(24)
Assume that f o r any b > a , K E K and flow a ( t ) satisfying (16), (17) and such that
a ( t ) An B = 4, t E [0,TI, there is a 2.1 hold.
K
E
K such that
(25)
(18) holds. Then the conclusions of Theorem
Theorem 3.4. Let K: be a minimax system for a nonempty set A satsfying
K\A=4,
KEK. (26) Let G E C1 satisfy (14) and (15). Assume that for any K E K and flow a ( t ) satisfying (16), (17), one has S ( K ) E K , where S ( U )= a ( d ( u ,A))u,u E E .
(27)
80
M.Schechter
Assume also that for each K E K there is a p > 0 such that G ( u )is Lipscluitz continuous on
Then the conclusions of Theorem 2.1 hold.
Theorem 3.5. Let K be a minimax system for a nonempty set A, and assume that there is a subset B of E such that A links B relative to Ic. Assume for any K E K andflow a ( t ) satisfying (16)) (17)) one has S ( K ) E K , where S(u) is given by (27). Let G be a C1 functional satisfying (15) and(24). Assume also that for each K E K there is a p > 0 such that G ( u )is Lipschitz continuous on K p given by (28). Then the conclusions of Theorem 2.1 hold. Corollary 3.1. Let K be a minimax system f o r a nonempty set A , and let G E C1 satisfy (14) and (15). Assume that f o r any b > a, K E K and flow a ( t ) satisfying (16),(17)) and (25) for
B = { u E E \ A : G(u)2 ao}, there is a 2.1 hold.
K E Ic satisfying K c a(T)Eb. Then the conclusions
(29)
of Theorem
We present some linking methods which are special cases of linking with respect to minimax systems. 4. A Method Using Homeomorphisms One linking method can be described as follows. Let Ebe a Banach space and let be the set of all continuous maps r = r(t)from E x [0,1] t o E such that
1. r(0)= I , the identity map. 2. For each t E [0,1),I'(t)is a homeomorphism of Eonto Eand r-'(t)E C ( E x [0,1),E). 3. r(1)Eis a single point in Eand J?(t)Aconverges uniformly to r(l)Eas t + 1 for each bounded set A c E. 4. For each t o € [0,1) and each bounded set A c E
Unified Minimax Methods
81
We have
Theorem 4.1. Let G be a C1-functional o n E , and let A be a bounded subset of E . Assume that a0 := supG
< a := inf sup G ( r ( s ) u )< 00. rE+ O < S < l uEA
Let $(t) be a positive, locally Lipschitz continuous function o n [0,m) satisfying (12). Then there is a sequence {uk} c E such that (13) holds. I n particular, there is a Cerami sequence satisfying (4). Proof. In this case, we lct K be the collection of sets of the form
K = {r(tlu :t E
[o, 1 1 , uE ~ , E ra}.
To show that this is a minimax system, let let
'p
r(t)be in a. Define the mapping r,(t)u= 'p r(t) 'p-lu,
Then rl(t)E
(32)
be a mapping in A(A), and E E.
a. Moreover,
rl(+ = 'p r(t)u,
E A.
0
Consequently,
r l ( t ) A= ' p ( { r ( t ) u: u E A } ) , showing that the collection Kis a minimax system. Since (31) now becomes ( 3 ) , the result follows.
Definition 4.1. We say that A links B [ s t ] if A , B are subsets of E such that A n B = 4 and, for each r(t) E i p , there is a t E (0,1] such that
r ( t ) An B # 4. We have
Corollary 4.1. If A links B [ s t ] ,then it links B[mm]. We also have
Theorem 4.2. Let A, B be subsets of E such that A links B [ s t ] ,and let G be a C1 -functional on E , satisfying a0 := supG 5 bo := inf G. A
B
(33)
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M. Schechter
Assume that a : = inf sup G ( r ( s ) u ) 0<5-<1
(34)
uEA
i s finite. Let +(t) be a positive, locally Lipschitz continuous function o n [ O , o o ) satisfying (12). T h e n there is a sequence {uk} c E such that (13) holds. I n particular, there is a Cerami sequence satisfying (4). Proof. The theorem follows from Theorem 3.3. If
~ = { r ( t ) ~[ O : t, ~~ I , ~ E A } for some r E
(35)
a, and a ( t )is any flow, let F(s) = a(2Ts), 0 5 s = a ( q r ( 2 s- I),
It is easily checked that
F E @. Kc
1
I -2 ’ 1 2
- < s I 1.
Morever,
u
a(t)AUa(T)K.
tE[o,Tl
Hence, (2.9) is satisfied. 5. A Method Using Metric Spaces
Another general procedure is described in Mawhin-Willem7 and BrezisNirenberg2 as follows. One finds a compact metric space C and selects a closed subset C* of C such that C* # 4, C* # C. One then picks a map p* E C(C, E ) and defines
A = { p E C ( C , E ): p = p * on C*} a = inf maxG(p(<)). p € d EEC
They assume (A) For each p E A, maxEEc G(p(<)) is attained a t a point in C\C*. They then prove
Theorem 5.1. Under the above hypotheses there i s a sequence satisfying (1.1).
Unijied Minimax Methods
83
We shall prove
Theorem 5.2. Under the same hypotheses, let $(t) be a positive, locally Lipschitz continuous function o n [0,m) satisfying (12). Then there is a sequence {uk} c E such that (13) holds. I n particular, there is a Cerami sequence satisfying (4). Proof. If a > ao, this follows from Theorem 2.1. In fact, we can take A = p * ( C * )and K as the collection of sets
K
= {P(O :E E
c,
P E A}.
If cp E R ( A ) ,then c p ( P ( 0 ) = cp(P*(E)) = P*(E),
E E c*.
Thus, cp(p(<)) E K. Hence, K is a minimax system for A, and we can apply Theorem 2.1 to come t o the desired conclusion. Now assume a0 = a. Since each K E K is compact, the same is true of K,, given by (28) for each p > 0. Moreover, if c(t)E C(R+ x E , E ) is such that ( ~ ( 0= ) I , one has S ( K ) E K ,where
The result follows from Theorem 3.5.
0
6. A Method Using Homotopy Stable Families Another approach is found in Ghoussoub.' The author condiders a closed subset A of E , and a collection K of compact subsets of E having the property that if K E K and $ E C ([0,1] x El E ) satisfies $(O)U
=U,
uE
E,
and
$ ( t ) =~ U , t E [0,1], u E A, then $(l)K E K.He calls such a collection a homotopy-stable family with extended boundary A. The author proves
Theorem 6.1. If K is a homotopy-stable family with extended closed boundary A , B is a closed subset of E satisfying (9) and G is a C1 functional which satisfies sup G 5 a 5 inf G, A
B
M. Schechter
84
where the quantity a is given by ( l ) , then there is a PS sequence satisfying (3).
We shall prove
Theorem 6.2. Under the same assumptions, f o r each f u n c t i o n +(t) satisfying the hypotheses of Theorem 2.1 there is a sequence satisfying (13). In particular, there is a Cerami sequence satisfying (4). Proof. It is easy to show that such a collection K is indeed a minimax system. Indeed, if K E Ic, and cp E A(A), then +(t)U
= tcp(U)
+ (1 - t ) u
satisfies the stipulations above, and consequently cp(K) E K. Since each of the members of K are compact, it follows that every C1 functional is Lipschitz continuous on some set K p defined by (28), provided p > 0 is sufficiently small. Moreover, if o(t)E C(R+ x El E ) is such that a(0) = I , one has S ( K ) E K ,where
S(u) = a ( d ( u ,A))u, u E E. This follows from the fact that S(t)u = a ( t d ( u ,A ) ) u is in E C([O,11 x
E , E ) and satisfies S(O)U= U ,
u E E,
and
S(t)u= U ,
t
E [0,I ] ,
u E A.
Consequently, S ( K ) = S(1)K E Ic by hypothesis. It now follows that the conclusion of Theorem 3.5 holds. 0 Note: It is not required to have a satisfy a 5 inf G. B
If it does, one obtains additional information as described in Ref. 6. Definition 6.1. We say that A links B [ g h ]if A, B are subsets of E satisfying the hypotheses of Theorem 6.1.
Unified Minimax Methods 85
7. Some Applications We now apply the theorems of the preceding sections to various geometries in Banach space. As before, we assume that G E C1(E,W) and that satisfies the hypotheses of Theorem 2.1. Theorem 7.1. G(0) I (Y 5 G ( u ) , u E dBs
(37)
and that there is a cpo E dB1 such that G(Rpo) IT, R > 0.
(38)
Then for each function + ( t ) satisfying the hypotheses of Theorem 2.1, there is a sequence {uk} c E such that
G(U)
--+
c,
cl
Ic IT,
G'(uk)/+(llukll)
0.
(39)
Many elliptic semi-linear problems can be described in the following way. Let R be a domain in Wn,and let A be a self-adjoint operator on L2(R). We assume that A 2 A0 > 0 and that
Cr(R) c D
:= D(A1'2)
c H"'2(R)
(40)
for some m > 0, where Cm(R) denotes the set of test functions in R (i.e., infinitely differentiable functions with compact supports in R), and (0) denotes the Sobolev space. If m is an integer, the norm in H"i2(R) is given by
Here D@represents the generic derivative of order and the norm on the right hand side of (7) is that of L2(R). We shall not assume that m is an integer. Let q be any number satisfying
and let f ( z , t ) be a CarathBodory function on R x R . This means that f ( z ,t ) is continuous in t for a.e. z E R and measurable in z for every t E R. Throught this section we make the following assumptions: (A) The function f (2, t ) satisfies
M. Schechter
86
and
l l ~ l:= l ~llA1’2~11
(46)
and q’ = q / ( q - 1). If R and Vo(z) are bounded, then (44) will hold automatically by the Sobolev inequality. However, there are functions Vo(x) which are unbounded and such that (44) holds even on unbounded regions R. With the norm (46),D becomes a Hilbert space. Define
F ( z ,t ) :=
Jo
t
f(z,s)ds
(47)
and
It follows that G is a continuously differentiable functional on the whole of D (cf., e.g., Ref. 11). For our first result, we assume further that
H ( z ,t ) = 2 F ( x , t ) - tf(x,t)2 -W1(x)
E
L1(R), x
E R,
t
E
R
(49)
and
H ( z , t ) -+ a a.e. as It1
--f
03.
(50)
Moreover, we assume that there are functions V ( x ) W , ( z )E L2(R) such that multiplication by V ( x )is a compact operator from D to L2(Q)and
+
F ( z , t ) I C(v(x)2(t12 V(z)W(z)ltl)
(51)
We wish to obtain a solution of
AU = ~ ( z , u ) u , E D.
(52)
By a solution of ( 5 2 ) we shall mean a function u E D such that (U,V)D
= (f(.,u),v),
E
D.
(53)
Unified Minimax Methods
87
If f ( z , u ) is in L2(R), then a solution of (53) is in D ( A ) and solves (52) in the classical sense. Otherwise we call it a weak (or semi-strong) solution. We have
Theorem 7.2. Assume that Xo is an eigenvalue of A with eigenfunction PO. Assume also
2 ~ ( z , t 5) X O t 2 ,
It1 5 6 f o r some 6 > o
(54)
t > 0 , z E 0,
(55)
and 2 F ( z , t ) L Xot2 - Wl(Z),
where Wl E L1(R). Then (52) has a solution u = 0. Proof. Under the hypotheses of the theorem, it was shown in Theorem 3.2.1 of Ref. 11 that the following alternative holds: Either (a) there is an infinite number of ~ ( xE)D(A)\{O} such that
AY = f(x,Y)
= XOY
or (b) for each p > 0 suficiently small, there is an G(u) 2 &,
(56) E
> 0 such that
llullD = p.
(57)
We may assume that option ( b ) holds, for otherwise we are done. By (55) we have
G(Rv0) 5 R2(11voll; - ~ o l l v o l l )+
1 R
Wl(Z)da:=
s,
Wl(Z)da:= B1.
Thus (38) holds. By Theorem 7.1, there is a sequence satisfying (39). Taking $(T) = l/(r l ) ,we conclude that there is a sequence {uk} c D such that
+
G(%)
-+
C,
In particular, we have
and
E
5 c 5 &,
(1+ ((%((D)G~(WC) ---t 0.
(58)
88
M. Schechter
These imply
If
P k = ( ( u ~ ( (+ D 00, let i i k = u k / p k . Then ( ( i i k ( = ( ~ 1. Consequently there is a renamed subsequence such that i i k 4 ii weakly in D ,strongly in L2(R) a n d a.e. in 0. We have by (51)
Consequently
1 5 2 C L V(x)2iiEdx. This shows that ii $ 0 . Let Ro be the subset of R on which ii I.k(.)I
= PklG&)l
00,
z E Ro.
(63)
# 0. Then (64)
If R1 = R\Ro, then we have
This contradicts (62), and we see that P k = I l u k l ( ~is bounded. Once we know that the P k are bounded we can apply Theorem 3.4.1 of Ref. 11 to obtain the desired conclusion.
Remark 7.1. It should be noted that the crucial element in the proof of Theorem 7.2 was (60). If we had been dealing with an ordinary PalaisSmale sequence, we could only conclude that
- (f(,Uk),Uk) =
11~k112D
which would imply only
H ( z , urc)dz = O(Pk). This would not contradict (65), and the argument would not go through.
References 1. P. Bartolo, V. Benci and D. Fortunato, Nonlinear Analysis T M A , 7,981 (1983). 2. H. Brezis and L. Nirenberg, C o m m . Pure A p p l . Math. 44, 939 (1991). 3. V. Benci and P. H. Rabinowitz, Invent. Math. 5 2 , 241 (1979). 4. K. C. Chang, Infinite dimensional Morse theory and multiple solution problems, (Birkhauser, Boston, 1993).
Unified Minimax Methods 89 5. C. Corduneanu, Principles of Differential and Integral Equations, (Chelsea, N.Y., 1977). 6. N. Ghoussoub, Duality and Perturbation Methods in Critical Point Theory, (Combridge Tracts in Mathemtics, 107,1993). 7. J. Mawhin and M. Willem, Critical Point Theory and Hamiltonian Systems, (Springer-Verlag, 1989). 8. L. Nirenberg, Bull. Amer. Math. Soc. 4 267 (1981). 9. P. H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, Conf. Board of Math. Sci. Reg. Conf. Ser. in Math. No. 65, Amer. Math. SOC.1986. 10. M. Schechter, Pacific J. Math. 171,529 (1995). 11. M. Schechter, Rend. Sem. Mat. Univ. Padova99, 255 (1998). 12. M. Schechter, Linking Methods in Critical Point Theory, (Birkhauser Boston, 1999). 13. E. A. de B. e. Silva, Nonlinear Analysis T M A 16, 455 (1991). 14. M. Schechter and K. Tintarev, Bull. SOC.Math. Belg. 44,249 (1992).[15] K. Tintarev, Isotopic linking and critical points of functionals, in Proceedings of the Second World Congress of Nonlinear Analysts, Part 7 (Athens, 1996), Nonlinear Anal. 30,4145 (1997). 15. M. Willem, Minimax Theorems, (Birkhauser, 1996).
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Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 91-102) @ 2007 World Scientific Publishing Co.
91
56. SOME REMARKS ON SINGLE CONSERVATION LAWS MIKIO TSUJI Department of Mathematics, Kyoto Sangyo University, Kita-Ku, Kyoto 603-8555, Japan mtsujiOcc.kyoto-su.ac.jp
PETER WAGNER Institut fur Technische Mathematik, Geometrie und Bauinformatik, Universitat Inns bruck, Technikerstr. 13, A-6020 Innsbruck, Austria
First we consider the Cauchy problem for Burgers equation whose initial function is x 2 , and study the existence domain of a piecewise smooth weak solution of the Cauchy problem. Then the existence domain of its real-valued solution is different from that of the complex-valued solution. Next we consider the Cauchy problem for a single conservation law which does not satisfy the convexity condition, and study the regularity of weak solutions of the Cauchy problem. Our question is whether the weak solutions become piecewise smooth, and whether “rarefaction waves” and “contact discontinuities” would really appear in the weak solutions. In this talk we solve an example as a tentative trial and construct its piecewise smooth weak solution by the geometric method.
1. Introduction We consider the Cauchy problem for an equation of conservation law in one space dimension as follows:
au
a
- + --f(u) at ax
=0
in {t > 0,x E R1},
on {t = 0, x E R1} (2) where f = f(u)is of class C“. We assume that the initial functions 4 = $(z) are also sufficiently smooth. The global existence of weak solutions of (1)-(2) has been well studied. As there exist too many papers on this subject, we do not mention anything on references. In this talk we will study the regularity of weak solutions of (1)-(2) and their domains of existence. We think that the most typical phenomenon in nonlinear hyperbolic u(0,x) = $(x)
92
M . Tsujz and P. Wagner
equations is the appearance of “shock wave”. Moreover the shock is defined only for the piecewise smooth weak solution. This is the reason why we pay much attention on the existence of “piecewise smooth weak solution”. Before considering the Cauchy problem (1)-(2), we state our program for nonlinear hyperbolic equations and systems. Our first step is to lift the equation into higher dimensional space so that the singularities of solution would disappear. Next we construct a global solution of the lifted equation in the higher dimensional space. We call this as a “geometric solution” of the original problem. Finally we project the geometric solution to the base space and construct a single valued weak solution which satisfies certain additional condition called as the “entropy condition”. For single first order partial differential equations, we have already proved in Refs. 9,lO and 11 that, if the equations are convex, our program is correct. But we do not yet succeed in the realization of our program for non-convex first order partial equations. For second order nonlinear hyperbolic equations and p-systems, we could not construct weak solutions by our method.l4~l5By these investigations, we have now some questions on the mathematical formulation for nonlinear hyperbolic equations and systems. Our final aim is to show that our program is true for all kinds of hyperbolic problems. In this talk we will first consider the Cauchy problem where an initial function is analytic and not bounded, and study an existence domain of solution. In analytic case, the complex method has been effective especially for linear partial differential equations. Therefore we will study, in Sec. 2, what would happen if we might accept a complex-valued solution. for linear partial differential equations. Therefore we will study, in Sec. 2, what would happen if we might accept a complex-valued solution. Next we will construct piecewise smooth weak solutions of (1)-(2). In the case where f = f(u)is convex, we think that we could understand the mechanics of the “propagation of singularities”. See Refs. 9-11, and 12. Therefore we will consider the case where f”(u)changes the sign. This problem has been studied by J. Guckenheimer,’ G. J e n n i n g ~S. , ~Izumiya and G. T. Kossioris,2 etc. In these papers they have used a “rarefaction wave” and a ‘‘contact discontinuity”, because they have reduced the starting problem to the Cauchy problem whose initial functions have jump discontinuities. Our question is on this reduction. If our program written in the above might be accepted, we would not need to introduce the rarefaction waves and contact discontinuities for the construction of weak solutions of (1)-(2), because a family of characteristic curves covers the whole space. As we do not yet arrive at the final conclusion on this problem, we have tried to solve
Some Remarks on Single Conservation Laws 93
a specific problem in an exact form as a tentative trial. This is the subject of the second part of this talk. 2. Existence Domains of Solutions for Burgers Equation
In this section we consider the following Cauchy problem:
and study an existence domain of a piecewise smooth weak solution of (3). What we would like to show in this section is the following: If we may accept only a real-valued solution, then the solution does not exist in the whole space R2 and its existence domain is surrounded by an envelope of a family of characteristic curves. But, if we may accept a complex-valued solution, then the Cauchy problem (3) admits a piecewise smooth weak solution defined in the whole space R2. First we solve the Cauchy problem (3) by the characteristic method. As a system of characteristic differential equations is written by
x=u,
u=o
whose initial condition is 40) =t
I
4 0 ) = E2,
a family of characteristic curves is given by
x = I+ C2t,
u = E2,
E
E R1.
(4)
Here we denote D* = {(t1x);4tx+1 0) and C = {(t,z);4tx+ 1 = 0). Then we can easily see that D+ U C = U E ~ R I { ( ~ , X ) ; = X 5 r 2 t } , and that the curve C is obtained as an envelope of the characteristic curves {(t,2 ) ;x = E e2t} where 5 is a parameter moving in R1.Solving the first equation of (4) with respect to t, we get the solution of (3) by
+
+
~ ( X) t , = 4x2/(1+ d -)'
(5)
in a neighbourhood of the initial line {t = 0). Though the right hand side of (5) is a continuous function defined in the whole space R2, it is not differentiable along the curve C and it takes complex values in the domain D-. Therefore our problem is how to extend the solution (5) beyond the curve C as a real-valued piecewise smooth weak solution of (3).
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M . Tsuji and P. Wagner
Proposition 2.1. Assume that the solution (5), written b y u = u(t,x ) , can be extended as a piecewise smooth real-valued weak solution of (3) beyond the curve C . Then u ( t , x ) does not have a jump discontinuity along the curve C . Proof. Assume that u = u(t,x ) has a jump discontinuity along the curve C. Here we write the curve C as 2 = y ( t ) = -1/4t. Then it holds [u]?= [u2/2]where [u]= u(t,y ( t ) 0 ) - u ( t ,y ( t ) - 0 ) . As u(t,y ( t ) 0 ) = 1/4t2 and ? = 1/4t2, we get u(t,y ( t ) - 0 ) = 1/4t2, that is t o say, u ( t ,y ( t ) 0 ) = u ( t ,y ( t ) - 0 ) . Hence u = u(t,x ) must be continuous along the curve C. CI
+
+
+
Next we consider the following Cauchy problem:
jg +
a 1 &u2)
= 0,
u ( t ,x ) = 4x2,
( t ,x) E Don C
Proposition 2.2. The Cauchy problem (6) can not have a solution in -
C'(D-) n C o ( 5 - ) where D- is the closure of D-. Proof. Assume that (6) has a solution in C'(D-) n C o ( 6 - ) . Take any point P on C and a point Q = (to,X O ) in D- satisfying u(t0,X O ) # 0. Then a characteristic curve passing through the point Q is given by x =20
+ u(t0,m)(t - t o ) .
(7)
As u(t0,xo)# 0, the characteristic line (7) meets with the curve C a t two different points A and B. Obviously it holds U I A # ul~. But, as the solution CI is constant along any characteristic curve, this is a contradiction. We can localize the above proposition. Take any point P on the curve C and any open neighbourhoon U of the point PI and consider the following Cauchy problem:
au
a
1
- + -(-u at ax 2
L
2
) = 0 , ( t , x )E D- n U
u(t,x ) = 4x2,
on C n U
Proposition 2.3. The Cauchy problem (8) can not have a solution in
c ~ ( D -n U ) n
~~( n u). 5-
Some Remarks o n Single Conservation Laws
95
The proof is almost the same as that of Proposition 2.2. Summing up the above results, we get the following
Theorem 2.1. T h e Cauchy problem (3) can not admit a real-valued piecewise smooth weak solution defined in the whole space R2 and the boundary of the existence domain of the solution is the envelope of the family of the characteristic curves coresponding t o the Cauchy problem (3). Remark 2.1. If we may accept a complex-valued solution, the solution (5) is a piecewise smooth and continuous weak solution of (3) defined in the whole space R2. 3. A Single Conservation Law without Convexity Condition
In the following sections we will study the Cauchy problem for a single conservation law which does not satisfy the convexity condition. Assume that the graphs o f f = f(u) and 4 = 4(x) are drawn as in Fig. 1 (i) and Fig. 2 (i) respectively. The graphs of their derivatives are Fig. 1 (ii)-(iii) and Fig. 2 (ii). In the above case, we assume that f ( a i ) = f ’ ( a i ) = 0 (i = 1 , 2 ) , f ” ( b ; ) = 0 (i = 1 , 2 ) and #(c) = @(d) = 0. Moreover we assume max4(x) > b2. For the Cauchy problem (1)-(2), the characteristic curves X are written by
x
=X(t,
Y ) = Y +t f ’ ( W ) ,
7J
= v ( t ,Y) = $ ( Y ) .
(9)
Therefore it follows that
8X 8Y
-(t, Y) = 1 + t f ’ ’ ( W ) 4 J ’ ( y ) .
(10)
We define h ( y ) ?Zf f ” ( 4 ( y ) ) # ( y ) , and assume that the graph of h = h ( y ) is drawn as in Fig. 3. Next, denote Ai = (yi, h ( y i ) ) ( i= 1 , 2 ) where h ( y i ) < 0 and h’(yi) = 0. Then we have a t A1
4’(Yl) > 0 and f ” ( 4 ( Y l ) ) < 0, and a t
A2
~’(Yz)< 0 and f ” ( d ( y 2 ) ) > 0. We now put ti = - l / h ( y i ) and X i = x(ti,y i ) (i = 1 , 2 ) . Then we can see the appearance of a shock whose starting point is (ti,X i ) (i = 1 , 2 ) , and we denote it by Si (i = 1 , 2 ) . We will briefly explain the construction of shocks
96
M . Tsujji and P. Wagner
Fig. 1.
Fig. 2.
Some Remarks on Single Conservation Laws
97
Fig. 3.
at the beginning of Sec. 4. For the detailed explanation.lOill Our problem is how to extend each shock Si (i = 1 , 2 ) for large t. We give some comment on the above example. Let us consider the Riemann problem for (l),that is to say, we assume that +(z) = c+ for z > 0 and +(x) = c- for z < 0 where c* are constant. If f ’ ( c - ) < f’(c+),the region { ( t ,2); f’(c-)t < z < f ’ ( c + ) t } is not covered by the family of characteristic curves (9). Therefore the rarefaction wave was introduced as a solution in this region. On the other hand, as we assume that the initial function (2) is in CF(R1), the family of characteristic curves covers the whole space R2.This is the principal reason why we doubt the necessity to use rarefaction waves in the process of construction of weak solutions. 4. Behavior of the Shock S1
In this section we will extend the shock S 1 for large t. To explain the situation, we repeat briefly how we have constructed the shock 4 . The graph of 2 = z ( t ,y) for t > ti is drawn as in Fig. 4. We explain the meanings of the notations used in Fig. 4. Let y = &(t)(&(t) < < 2 ( t ) , t > t l ) be the solutions of (&/dy)(t,y) = 0 with respect to y in a neighbourhood of y = 91 and denote z i ( t ) = z ( t ,Ei(t)(i= 1 , 2 ) . Then we see that z l ( t ) > z 2 ( t ) for t > t l . Solving the equation z = z(t,y) with respect to y for z E ( ~ ( t z)l (, t ) ) we , get three solutions y = g i ( t , x) (i = 1 , 2 , 3 ) satisfying g l ( t . 2 ) < g 2 ( t , 2) < g 3 ( t , z ) . As v(t,y) = b(y) for all (t,y) E R2,we define
98
M. Tsuji and P. Wagner
ui(t,z) = $(gi(t, z)) (i = 1 , 2 , 3 ) . Then, as qY(y1) > 0, we have
ul(ti x) < U2(tr x) < u3(ti x).
As we are looking for a single-valued weak solution, we jump from the first branch {u = u l ( t , x ) } to the third one {u = u 3 ( t , x ) ) . A shock curve is determined by the Rankine-Hugoniot jump condition as follows:
Though the right hand term is not Lipschitz continuous at the starting point, we have proved in Ref. 12 that the Cauchy problem (4.1) has a unique solution z = n ( t ) .This is the shock curve S1. To extend the shock & , we consider the behavior of u i ( t , n ( t ) )= q5(gi(t,-yl(t>)) (i = 1,3). We put u*(t) = u ( t , y l ( t )h 0 ) . Then u+(t)= U 3 ( t , Y l ( t ) ) and u-(t) = ul(t,-yl(t)).
Fig. 4.
Lemma 4.1. A s long as S1 satisfies the entropy condition, the function g l ( t , -yl(t))is decreasing and g3(t, -yl(t))i s increasing. Proof. By the definition of y = g i ( t , z), we have z = gz(t, z)
+ tf’($(9i(t,.)))
(2
= 1,213).
Some Remarks on Single Conservation Laws
99
Taking the derivatives with respect to t and x, we have
and
Hence it holds that
which leads us to
Then the entropy condition means
Hence the proof of the lemma is complete.
0
The initial function 4 = 4(x) has the properties as drawn in Fig. 2. As long as the entropy condition is satisfied, we see by Lemma 4.1 that 4 ( g l ( t , n ( t ) ) ) = u-(t) is decreasing, and that 4(gs(t,n(t))) = u+(t) is increasing since g3(t,x) < 0. Hence 4(gl(t,rl(t))) advances to 0, and 4(g3(t,-yl(t))) goes to its maximum. We write here P+(t) = ( u + ( t )f,( u + ( t ) ) and ) P-(t) = ( u - ( t ) ,f(u-(t))). The shock S1 satisfies the entropy condition for t > t l where t - tl is small. This means that the graph of f = f(u)lies entirely on the upper side of the chord P+P- joining the two points P+(t) and P-(t). If we extend the shock S1 further, then the chord P+P- may become tangent to the curve {(u,f(u)) : u E R1} in finite time. We assume that P+P- is tangent to the curve f = f(u)at t = T . Since 0 < u - ( t ) < 4(yl) < bz and thus f ” ( u - ( t )< 0, we see that
100
M . Tsuji and P. Wagner
P+P- is tangent t o the curve f = f(u)a t the point P+(T),but not a t the point P-(T). Therefore it holds that d71
-(TI dt d271 Lemma 4.2. i) -(T) dt2
= f’(.+(T)).
> 0,
22)
d -f’(u+(t))lt=T dt
(13) = 0.
Proof. i) Using the definition (4.1) of ( d y l / d t ) ( t ) , we have d2Tl -(t) dt2
=
d dy1 -(-(t)) dt dt
=
[
dt
u+ - u-
Here we recall that the starting point of the shock S1 is the point ( t l ,X I ) . Therefore it follows that ~ ’ ( Y> I )0, u+(t)-u- ( t ) > 0 and ( a g l / a x ) ( t ,x) > 0. Hence we get ( d 2 y l / d t 2 ) ( T )> 0. 0 Part (ii) is easily obtained from (19).
By Lemma 4.2, we have
This implies that the entropy condition is satisfied for t the above results, we have the following
Theorem 4.1. T h e shock fies the entropy condition.
S1
starting f r o m the point
> T . Summing up
( t l ,X
I ) always satis-
Some Remarks on Single Conservation Laws
101
5 . Behavior of the Shock Sz
In this section we extend the shock 5’2 whose starting point is ( t 2 , X 2 ) . The graph of z = z ( t ,y) for t > t 2 can be drawn as in Fig. 4. Though z = z(t,y) in this section is different from z = z ( t ,y ) in the previous section, we use the same notations introduced there. As in Sec. 3, we solve the Cauchy problem for (16) whose initial condition is z(t2) = X 2 , and denote the solution by 5 = 7 2 ( t ) . We write here also u*(t) = u(t,7 2 ( t ) f0 ) ; that is to say, u+(t)= q5(g3(trY 2 ( t ) ) ) and u-(t) = q5(91(tl7 2 ( t ) ) ) . Put P*(t) = (%(t),f (u*(t))). The shock S2 satisfies the entropy condition for t > t 2 where t - t 2 is small. In this case, as f”(q5(y2)) > 0, the entropy condition says that the graph of f = f(u)lies entirely on the lower side of the chord P + R .
Lemma 5.1. As long as the entropy condition f o r g l ( t , 7 2 ( t ) ) i s decreasing and g 3 ( t l Y 2 ( t ) ) is increasing.
S2
i s satisfied,
The proof of this lemma is almost the same as that of Lemma 4.1. When t gets larger, q5(g3(tI72(t)))tends to 0 and q5(g1(t172(t)))tends to the maximum of q5 = q5(z).Therefore we assume that the entropy condition is satisfied for t < T , and that P+P- becomes tangent to the curve {(u, f (u)); u E R1}at t = T . Then it follows that
dY2 -(T) dt
d
i i ) -f’(u+(t))lt=T dt
= f’($(g3(t, 72(t))))lt=T= f ’ ( U + ( T ) ) .
= 0.
The proof is the same as that of Lemma 4.2. Using this lemma, we get
Therefore we get the following
Theorem 5.1. T h e shock S2 starting f r o m the point fies the entropy condition.
( t 2 , X2)
always satis-
It may happen that the shocks 5’1 and 5’2 collide in finite time. Then we see that a new, uniquely defined shock appears. The method of the proof is almost the same as that in Refs. 9 and 10. Summing up these results, we can say that our program is true for the above Cauchy problem.
102
M. Tsuji and P. Wagner
References 1. J. Guckenheimer, Solving a single conservation law, Lecture Notes in Math., Vol. 468, (Springer-Verlag, 1975), pp. 108-134. 2. S. Izumiya and G. T. Kossioris, Bull. Sci. math. 121,619 (1997). 3. G. Jennings, Adv. i n Math. 33,192 (1979). 4. S. N. Kruzhkov, Math. USSR Sb. 1,93 (1967). 5. S. N. Kruzhkov, Math. USSR Sb. 10, 217 (1970). 6. P. D. Lax, Comm. Pure Appl. Math. 10,537 (1957). 7. 0. A. Oleinik, Uspelchi Mat. Naulc 12,3 (1957) (in Russian). English transl. in Amer. Math. SOC.Transl. 26,95 (1963). 8. D. G. Schaeffer, Adv. in Math. 11,358 (1973). 9. Tran D. V., M. Tsuji and Nguyen Duy T. S., The characteristic method and its generalizations for first-order non-linear partial differential equations (Chapman & Hall/CRC, USA, 1999). 10. M. Tsuji, C. R. Acad. Sci. Paris 289,397 (1979). 11. M. Tsuji, J . Math. Kyoto Univ. 26,299 (1986). 12. M. Tsuji, A n n . Inst. H. Poincare' - Analyse nonline'aire 7,505(1990). 13. M. Tsuji, Pitman Research Notes in Math. 381,164 (1998)(Longman). 14. M. Tsuji and T. S. Nguyen Duy, Acta Math. Vietnamica, 27,97 (2002). 15. M. Tsuji, Some remarks on nonlinear hyperbolic equations and systems, in Abstract and Applied Analysisedited by N. M. Chuong, L. Nirenberg and W. Tutschke (World Scientific, 2004), pp. 355-364.
Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 103-113) @ 2007 World Scientific Publishing Co.
103
37. ITERATIVE METHOD FOR SOLVING A MIXED BOUNDARY VALUE PROBLEM FOR
BIHARMONIC TYPE EQUATION* DANG QUANG A$ and LE TUNG SON Institute of Information Technology Vietnam Academy of Science and Technology, 18 Hoang Quoc Viet Road, Cau Giay Dist., 10307 Hanoi, Vietnam E-mail: dangqaQioit.ncst.ac.vn The solution of boundary value problems (BVP) for fourth order differential equations by their reduction to BVP for second order equations with the aim to use the achievements for the latter ones attracts attention from many researchers. In this paper, using the technique developed by ourselves in recent works, we construct iterative method for a mixed BVP for biharmonic type equation. The convergence rate of the method is proved and some numerical experiments are performed for testing it in dependence on the choice of an iterative parameter. Keywords: Iterative method; Mixed boundary value problem; Biharmonic equation
1. Introduction The solution of fourth order differential equations by their reduction to bouridary value problems (BVP) for the second order equations, with the aim of using efficient algorithms for the latter ones, attracts attention from many researchers. Namely, for the biharmonic equation with the Dirichlet boundary condition, there is intensively developed the iterative method, which leads the problem to two problems for the Poisson equation at each iteration (see e.g. Refs. 2,8,9,11). Recently, Abramov and Ulijanova' proposed an iterative method for the Dirichlet problem for the biharmonic type equation, but the convergence of the method is not proved. In our previous work^^-^ with the help of boundary or mixed boundary-domain operators *This work is supported in part by the National Basic Research Program in Natural Sciences, Vietnam
104 Q. A Dang and L. T.Son
appropriately introduced, we constructed iterative methods for biharmonic and biharmonic type equations associated with the Dirichlet boundary condition. It is proved there that the methods are convergent with the rate of geometric progression. In this paper we develop our technique in Refs. 4-7 for a mixed BVP for the biharmonic type equation. Namely, we consider the following problem
A2u
+ bu = f
in R,
(1)
u=g
onr,
(2)
aU
- = g1 on rl,
(3)
AU = g2 on
(4)
all
r2,
where s1 is a bounded domain in Rn(n 2 2) with the Lipschits boundary I? consisting of two smooth pieces rl and I?2, u is the outward normal to I', b is a positive number and A is the Laplace operator. We propose an iterative method for reducing the problem to a sequence BVP for the Poisson equation, study its convergence and finally, present the results of numerical experiments for the illustrating the effectiveness of the method.
2. Reduction of the Problem to Boundary - Domain Operator Equation
As in Refs. 5 and 6 we set AU = U, ~p = -bu
and denote ulr, = vo.
(7)
Then the problem (1)-(4) is reduced to the problems:
where vo and p are temperorily unknown functions. The solution u found from the above problems should satisfy the boundary condition (3) and the relation (6). To find U O , (p, we introduce an operator
Iterative Method for Solving a Mixed BVP f O T Biharmonic Type Equation
105
B as follows: B : w + Bw, where
w=
e)
, Bw
=
(i."Ir1) , 'p+bu
and u and v are the solutions t o the problems
Av = 'p, x E 0, vlr, = 210, ulr, = 0 ,
The operator B primarily defined on couples of smooth functions is extended by continuity on whole space L 2 ( r 1 ) x L2(R) . Its properties will be investigated later. For the reduction of BVP (1)-(4) to an equation with the operator B we set u = u1 u2, v = v1 u2 , where u1, v1 and u2, u2 are the solutions to the following problems
+
+
x E R, ullr = 0.
Aul = 211,
From (13) and (14) we can determine the definition of B,we have
212
and
u2,
and from (15), (16), by
(17)
106
Q. A Dang and L. T. Son
For u t o be the solution of (1)-(4), as mentioned above, the relations (3) and (6) must be satisfied, i.e., there should be conditions
cp+bu=O
inR.
(19)
From these relations we derive
Setting
from (17) we get the equation
BW= F.
(23)
Now, we study properties of B . First, we introduce the space H = L 2 ( r l )x L2(R) with the scalar product (w, V)H = (wo, G ) L z ( r l )
+ (cp, c P ) ~ z ( n )
for the elements
Property 1. B is symmetric in H Proof. We have
since Zi(1.1 = VO and V l r 2 = 0 in the problems (ll),(12) for the definition of BV.
Iterative Method for Solving a Mixed BVP for Bihannonic Type Equation
107
Now, we transform the boundary intergral, taking into account of (11) and (12). We have
Therefore,
Thus, the symmetry of B is proved. Property 2. B i s positive in H . Indeed, from (24) we have
( B w ,W ) =
1
(bv2
+ p 2 ) d x 2 0.
R
The equality appears if and only if w = 0. Property 3. B can be decomposed into the sum of a positive, symmetric and completely continuous operator and a projection operator, namely,
where Bo and
12
are defined as follows
u being defined from (ll),(12). The complete continuity of Bo easily follows from the theory of elliptic problems and embedding theorems of Sobolev spaces (see Ref. 10). Property 4. B i s bounded in H . This fact is a direct corollary of Property 3. Since B = B* > 0 but is not completely continuous in H the use of two-layer iterative scheme13 to the equation (23) does not guarantee its convergence. Hence, in the next section we will disturb this equation and apply the parametric extrapolation technique (see Refs. 4 and 5) for constructing approximate solution for problem (1)-(4).
108
Q. A Dang and L. T. Son
3. Construction of Approximate Solution of the Original
Problem Via a Perturbed Problem
We associate with the original problem (1)-(4) the following perturbed problem
Aualr, = 92, where S is a small positive parameter. Theorem 3.1. Suppose that f E Hn-4(R), g E Hn-1/2(I'), g1 E Hn-3/2(r1),9 2 E Hn-5/2(I'2);n2 4. T h e n f o r the solution of the problem (27)-(30) there holds the following asymptotic expanssion:
where yo = u i s the solution of ( l ) - ( d ) , yi (i = 1 ,2 , ...,N ) are functions independent of 6, yi E H"-i(0), zg E H"-N(s2) and llZ61IH~(S2,
6 c1,
(32)
C1 being independent of 6 . Proof. Under the assumption of the theorem there exists a unique solution u E H " ( 0 ) of the problem (27)-(30).After substituting (31) into (27)-(30) and balancing coefficients of like powers of 6, we see that yi and zg satisfy the following problems:
A 2 yi
+ byi = 0,
yilr dyi
b--Irl dU
XER,
= 0,
= -Ayz-llrl
, i = 1,2, ..., N ,
(33)
AyiIrz = 0.
A2z+ ~ bzs
= 0,
x E 0,
zalr = 0, (34)
Iterative Method for Solving a Mixed B V P for Biharmonic Type Equation
109
It is possible t o establish successively that (33) has a unique solution yi E HnPi(fl) and (34) has a unique solution zg E H"-N(fl). Clearly, yi (i = 1,..., N ) do not depend on 6. It remains to estimate 26. For this purpose, we reduce (34) to a boundary operator equation. Setting
Azs
' ~ s= --b~s, uslrl = us0
= us,
(35)
we obtain
A u= ~ pa, x E 0, 4 r , = uso, U s l r , = 0.
azs= zslr
E
a,
= 0.
Now , denote
then by definition of B , we obtain
Using the second condition of (34), we obtain
(B
+ 6 1 1 ) ~=s h,
where
It follows that (see Lemma 1 in Ref. 3)
( B w s ,ws) 6 ( B w ,w),
(37) where w is the solution of the equation B w = h. This equation has a solution because it is the equation to which the problem (34) with 6 = 0 may be reduced. In Sec. 2, when investigating the properties of B , we have established that
(Bws,ws) In view of (35), we have
=
s,
(bus2
+ 'Ps2)dx. +
(blavsl2 blzs12)dx.
(38)
(39)
110 Q. A Dang and L. T. Son
Since the right-hand side of the above quality defines a norm in the class of functions vanishing on the boundary, which is equivalent t o the norm II.IIHz(n), we have II.zIIHz(n) 6 G , where C1 = GJ-, c 2 being independent of 6. Thus the theorem is proved. 0
As usual, we construct an approximate solution U E of the original problem (1)-(4) by the formula
i=l
where (_l)N+l-iiN+l
"li =
z. ! ( N + 1 - i)! '
(41)
is the solution of (27)-(30) with the parameter 6 / i (i = 1,...,N + 1). Then it is easy t o obtain the following estimate
ug/i
(IUE- UllHz(n) < C2dN+l, where u is the solution of the problem (1)-(4) and pendent of 6.
(42)
C2 is a
constant inde-
4. Iterative Method for Solving the Perturbed Problem First we notice that in the same way as for the original problem (1)-(4), the problem (27)-(30) may be reduced to the operator equation
B 6 ~ g= F
(43)
+
for wg = ( ' ~ g o , ' p g ) ~ ,where ' U ~ O = AugIrl, 'pg = -bug,Bg = B 611, B and F are defined by (10) and (22), respectively. Clearly, Bg is bounded and
Bg = B i 2 6 I ,
(44)
where I is the identity operator. For solving (43) we can apply the general theory of two-layer iterative scheme for equation with symmetric, positive definite operator. l 3 Namely, we consider the iterative scheme
Iterative Method f o r Solving a Mixed B V P for Biharmonic Type Equation
111
where { i - ~ , k + l } is the Chebyshev collection of parameters according to bounds 7:') = 6,$) = 6 llBll (see Ref. 13 for details). In the case of simple iteration
+
we obtain
where
and as above H = L 2 ( r l )x L2(R) . Using estimates for the solution of elliptic problems1' and taking into account (47), we obtain the estimate
Ilup'
< c ( P 6 ) k I I W p )- W ~ I I H ,
- u611~5/2(n)
(48)
where C is a constant independent of 6. The iterative scheme (45) can be realized by the following process: (i) Given a couple (v~o(O),(ps(O)). (ii) Knowing W I S O ( ~ and ) ( p 6 ( k ) , k = 0,1, ... solves successively two problems
Q. A Dang and L. T . Son
112
5 . Numerical Experiments
We perform some limited experiments in MATLAB for testing the convergenve of the approximate solution given by (40) with fixed N = 2 and 6 will be chosen in agreement with the steps of the grid for discretizing differential problems. We consider the computational domain R = (0, with uniform grid and steps of the grid are hl = h2 = 1/M. The part r2 is the top side and rl is the remaining part of the boundary of the square R . For the iterative process we choose by experimental way the parameter r = 2 6 f 0 . 7 5 4 ' The Dirichlet problems (49), (50) are discretized by difference schemes of second order approximation and the normal derivative in (51) is approximated by difference formula of the same order of accuracy. The stopping criterion for the iterative process (49)-(52) is max(IIvg+l) - vg)llool
IIYS (k+l) -
)ol'&
< & = hlh2.
and in view of the estimate (42) we choose 6 = E ~ / For ~ . solving systems of grid equations the method of complete reduction] whose idea is the successive odd-even eliminations1l2is used. We take some exact solutions and for b = 1 construct the right-hand sides and the boundary data in respect with them. The results of computation on PC Pentium 4 with CPU 1.80 GHZ for these examples are presented in the following tables, where Error = llUE - uII, and in the column "Number of iterations " we report the number of iterations for finding the basic solution ugli (i = 1 , 2 , 3 ) , the time is in seconds. Table 1. Case u = (z: - l)(z;
Grid 16 x 16 32 x 32 64 x 64
Number 13 23 42
Table 2. Grid 16 x 16 32 x 32 64 x 64
of iterations 19 24 37 48 70 93
-
1)
Error 0.0023 0.0007 0.0003
Time 0.88 4.95 42.51
Case u = sin(m1) sin(xz2)
Number 10 17 27
of iterations 21 16 26 33 39 47
Error 0.0050 0.0013 0.0003
Time 0.78 3.52 23.59
It is interesting t o notice that for the above examples in the case if 6
=
0 our experiments show that the iterative process (49)-(52) does not
Iterative Method for Solving a Mixed B V P for B i h a m o n i c Type Equation
Table 3. Case u = 0.252: Grid 16 x 16 32 x 32 64 x 64
+ 0.252; + x: + xz
Number of iterations 16 25 32
28 50
46 84
113
61 115
Error 0.0056 0.0022 0.0009
Time 1.08 6.17 51.14
converge. It justifies the reason, why we have to consider the perturbed problem (27)-(30) and extrapolate its solution by the parameter 6. References 1. A. A. Abramov and V. I. Ulijanova, Journal of Comput. Math. and Math. Physics, 32,567 (1992) (Russian). 2. Dang Quang A, Math. Physics and Nonlinear Mechanics, 44, 54 (1988) (Russian) . 3. Dang Quang A, Vietnam Journal of Math. 33,9 (2005). 4. Dang Quang A, Journal of Comput. and Appl. Math., 51, 193 (1994). 5. Dang Quang A, Vietnam Journal of Math. 26,243 (1998) 6. Dang Quang A, Journal of Comp. Sci. and Cyber. No. 4, 66 (1998). 7. Dang Quang A, Journal of Comp. and Appl. Math. 196,634 (2006). 8. A. Dorodnisyn, N. Meller, Journal of Comp. Math. and Math. Physics 8 , 393 (1968) (Russian). 9. R. Glowinski, J-L. Lions and R. Tremoliere, Analyse Numerique des Inequations Variationelles, (Dunod, Paris, 1976). 10. J-L. Lions and E. Magenes, Problemes aux Limites n o n Homogenes et A p plications, Vol. 1, (Dunod, Paris, 1968). 11. B. V. Palsev, Journal of Comput. Math. and Math. Physics 6, 43 (1966) (Russian). 12. A. Samarskii and E. Nikolaev, Numerical Methods for Grid Equations, Vol. 1: Direct Methods, (Birkhauser, Basel, 1989). 13. A. Samarskii and E. Nikolaev, Numerical Methods for Grad Equations, Vol. 2: Iterative Methods, (Birkhauser, Basel, 1989).
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Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 115-129) @ 2007 World Scientific Publishing Co.
115
§8. NUMERICAL SOLUTION TO A NON-LINEAR
PARABOLIC BOUNDARY CONTROL PROBLEM DINH NHO HAO*Z, NGUYEN TRUNG THANH and H. SAHLI Vrije Universiteit B~ussel, Department of Electronics and Informatics, Pleinlaan 2, 1050 Brussel, Belgium E-mail: haoOmath.ac.vn The so-called difference of convex functions algorithm and the continuation technique are applied to solving the nonlinear parabolic boundary control prob-
where a: > 0 is given and y(z, t ) = y(z, t ;u ) is the solution to the non-linear parabolic problem y t ( z , t ) = yzz(z,t), 0 < I < 1 , 0 < t < T, y(z, 0 ) = Y O ( Z ) , 0 < I < 1, yz(O,t)=O, ~z(l,t)=g(y(l,t))+z~(t)O ,
with { u E Lw([O,T])Iumin(t) 5 u(t) 5 umax(t), for a.e. t E [O,T]}.Numerical examples are given t o show the efficiency of the method.
1. Introduction In Ref. 8 Kelley and Sachs proposed a trust region method for the problem of minimizing
where Q > 0 is given and y(z, t ) = y(z, t;u)is the solution to the non-linear parabolic problem
'Also at Institute of Mathematics, 18 Hoang Quoc Viet Road, 10307 Hanoi, Vietnam
116 D. N. Hao, N . T . Thanh and H. Sahli
In (1)-(2) u is constrained to be in the set
U
= {U E
LO”([O,T ] ) I ~ m i n ( t )5 u ( t )5 u m a x ( t ) , for a.e. t E [0, TI}
(3)
and the nonlinear function g is assumed to satisfy g E
C2(R),
g’, g” E
L”(R).
(4)
In order to use some known results by Sachs16 and Schmidt17 of the problem (1)-(2) we will need some monotonicity conditions on g that we will describe in section 2. The problem (1)-(2) can be understood as a variational formulation of the inverse problem where one wishes to find the heating regime u ( t ) at a part of the boundary such that at the final time the temperature of the system reaches a given desired goal z. The particular case g(y) = -y4 corresponding to the Stefan-Boltzmann radiation condition seems to have many applications in engineering.6>7J6 The method of Kelley and Sachs’ is a projected form of the Steihaug trust-region-CG method with a smoothing step added at each iteration to improve performance in the global phase and provide mesh-independent sup-norm convergence in the terminal phase. However, whether their method delivers a global solution to (1)-(2) is not clear. In this paper we approach the control problem (1)-(2) from a quite different point of view. Namely, from that of DC programming. We realize that f is a DC one, i.e. it can be represented by the difference of two convex functions. Therefore, we can consider (1)-(2) as a DC program and apply the DC programming techniques to solve the problem. In fact we apply the difference of convex functions algorithm (DCA).l2>I3 As DCA is a local method, it cannot guarantee to deliver a global solution, in this work we combine it with a so called continuation or cooling1 technique for the search of a global solution. We note that DCA is applicable to finite-dimensional problems only, but in order to solve (1)-(2) we need to disretize it somehow. The convergence of the solutions of the discretized control problems of type (1)-(2) to that of the continuous one has been proved for some cases in Refs. 15 and 10. The problem (1)-(2) has been first considered by Sachs“ and then by Schmidt,17 see also Refs. 3 and 14. In the next section we prove that the functional f is twice Frkchet differentiable and deliver the formulas for the gradients as well as prove that f can be represented by the difference of two convex functionals. In section 3 the difference of convex functions algorithm (DCA) and the combination of DCA with the continuation technique for solving non-convex optimization
Numerical Solution to a Non-Linear Parabolic Boundary Control Problem
117
problems will be described. Finally, in the last section numerical results based on our method will be given.
2. The Control Problem and Its dc Representation In this section we prove that (1)-(2) is a DC program. Further, to make the algorithm more efficient we change the functional f by
F ( u )=
f
1
1
Iy(z, T ;u)- z(z)12dz
+-
I'
( u ( t )- U * ( t ) ) Z d t .
(5)
Here u* plays the role of a guess for the solutions or a selection criterion for the solutions: as (1)-(2) could have many solutions to, we choose that one which is nearest to u*, by minimizing (5). Now we review some results on the solution of (2). Let I be a bounded subinterval of the reals R, say, I = [a,b],--oo < a < b < 00. Let B be a norm space. By writing y E BJ we mean that y E B with ranges in I . We denote II . I l P ( 0 , l ) by II . II. By H1(O,1) we denote the Sobolev space of all elements y E L2(0,1) having weak derivative u, in L2( 0 , l ) . The space H'( Q ) is similarly defined. Here, Q := ( 0 , l ) x (0, TI. Further, let
V''O(Q) := C([O,TI;L2(0,1)) n L 2 ( ( 0 T , ) ;H1(O,1)) with the norm
IYI := vrai max II'u(., t)llL2(0,1) -k OltlT
IIu~IIL2(Q).
On the notion of Sobolev spaces we refer the reader to Ref. 9. Let urnin,umax be given functions with ranges in I belonging to L"([O, TI). To be consistent with the results of Schmidt17 we additionally suppose that g is strictly decreasing in I .
Definition 2.1. Let yo E L;(O, 1). Then y E V;>O(Q)is said t o be a weak solution of (2) if for all 4 E H1((O,1) x ( 0 , T ) )with = 0, then the following integral identity holds
lT
L 1 ( - Y ( 2 , t ) $ t ( x ,t )
=
i'
Yo(z)4(Z, O)dZ +
+ YZ(Z, t)$Z(Z, t ) ) d Z d t T
( d Y ( L t ) )+ u(t))4(1,W t .
It is proved in Ref. 17 that there is a unique weak solution t o (2). And it is proved in Refs. 3 and 14 that y E L"(Q) n C ( G ) for every E > 0. Here Q E , ~ = (0,1) x (6,T).
118 D.
N. Hao, N. T. Thanh and H. Sahli
Now we prove that F is DC and propose a DC decomposition for it, in doing so we note that F E C2(U). Denote by G : L”(0,T) + V1io(Q) the mapping associating the solution of (2) t o every function u.In what follows, we denote the first- and second-order F’rhchet derivatives of G a t a point u in the direction h or ( h ,k ) by G‘(u)h and G”(u)(h,k ) , respectively. Following Refs. 16,17 and 4 it can be proved that G is twice continuously Frkchet differentiable. If u,h E L”(0, T ) ,cp := G’(u)h,then cp = p(z,t; h ) is the solution of the problem
P~(x, t ;h) = (PZZ(Z, t ;h ) , 0 cp(z, 0; h ) = 0, 0 < x < 1,
< x < 1,0 < t < T , (6)
cp,(l,t;h) = g ’ ( y ( l , t ; u ) ) c p ( l , t ; h ) + h , O < t < T.
cp,(O,t;h) = O ,
If h, k E L”(0,T) and v of the problem
=~
( xt ;,h, k ) := G”(u)(h,k ) , then w is the solution
0 < x < 1 , O < t < T, ut = v,,, V ( X , 0; h, k ) = 0, 0 < z < 1, ~ ~ (t; 0h, k, ) = 0, 0 < t < T , U Z ( 1 , t ;h, k ) = g’(y(1, t; u ) ) 4 1 , t; h, k ) +g”(y(l, t ;u))cp(l, t;h)cp(l, t ;k ) ,
(7) 0
< t < T.
From these results we obtain that F is twice continuously Frkchet differentiable and, if h E L“(O,T), then rl
cp(x,T ;h)(y(z,2’; u)- z ( z ) ) d x
F’(u)h = /o
+a
rT
lo
(u- u*)hdt,
(8)
and
F”(u)(h,h ) =
Jo.I’
Icp(x,T ;h)12dz (9)
1
+
4 x 7 T ;h, h ) ( y ( x , T ;u)- z ( z ) ) d J + : aIIh112L2(O,T).
It can be also proved that the gradient of F is
F’(u) = d(1, t ) + a ( u - u*),
(10)
where d ( z ,t ) is the solution of the adjoint problem
-dt(x, t ) = d,,(x, t), 0 < x < 1,0 < t < T , d ( ~T, ) = y ( x , T ;U ) - z(x), 0 < x < 1, d,(O, t ) = 0, &(I, t ) = g‘(y(1, t ;u))d(l, t ) ,0 < t < T.
(11)
Now we estimate 11 F”(u)11. We need the following result by Schmidt in Ref. 17, Corollary A.3.
Numerical Solution to a Non-Linear Parabolic Boundary Control Problem
119
Lemma 2.1. Let w be the weak solution of the problem
0 < x < 1 , O < t 5 TI -wZ(O,t)+al(t)w(O,t)=fi, ,,(l,t)+a2(t)w(l,t)=fi, W ( . , O ) = wo, 0 < x < 1, W t = w,,,
0
(12) are bounded, measurable, non-negative functions defined o n [O, TI, wo E L 2 ( 0 ,1 ) fl L"(0, l ) ,f1, f 2 E L 2 ( 0 ,T ) n L"(0, T ) . Then there exists a positive constant C such that where
01~02
IIWIILm(Q)
5 C m a x { ~ ~ W o ~ ~ L - ( O , l~) ~ f l ~ ~ L - ( o~,~Tf)2~~ ~ L m ( 0 , T ) } ~
Remark 2.1. By inspecting the proof of Schmidt we get 5 C=2T+-. 4
Let
3. DCA and Continuation Techniques 3.1. DCA 3.1.1.
A function f defined in a convex set is called DC, if it can be written in the form f(x) = g ( x ) - h ( x ) .Here, DC means Difference of Convex functions. We suppose that f is defined in a Hilbert space X, although in the future
D. N. Hao, N . T . Thanh and H. Sahli
120
we mainly work with R". Note that the above representation of f is not unique, since [g(z) allzl12]- [h(z) a11z112]for any positive a is also a DC representation for it.
+
+
3.1.2.
We say that a convex function g is proper, if g > -ca and there exists x such that g(z) # +ca. In this case, we write g E I'o(X). 3.1.3. When we work with a convex function g defined in a convex set C we can extend its domain into the whole X by setting its values to +ca outside C so that the new function remains convex. Hence in the future we will work with convex functions defined in the whole X . 3.1.4. Let g be a conwex function. Its subdifferential is defined by
Here, < ., . > is the inner product in X . It is known that g attains its minimum a t z* if and only if 0 E ag(x*). 3.1.5. Let g E I ' o ( X ) and lower semi-continuous (lsc). The Fenchel transform of g is defined by 9*(Y) := SUP{< 2 , Y
> -gb)).
2
It can be proved that g* belongs t o I'o(X) and is lower semi-continuous. Furthermore, g** = g. Note that y E ag(x) if and only if z E ag*(y), if and only if < x , y >= g(z)
+ S*(Y).
3.1.6. Consider the DC program
(PI
a
:= inf{f(z) = g(z) 5
-
h(z)}.
Numerical Solution to a Non-Linear Parabolic Boundary
COntTOl
Problem
121
Here g and h are proper convex functions and lower semi-continuous. With the notation dom g := {xlg(x) < +00} we see that if la1 < 00, then dom g c dom h and dom h* c dom g*. The program (P) has its dual part
(D)
a
= inf{h*(y) - g * ( y ) } . Y
3.1.7. DC algorithm: Since (P) and (D) are not convex programs, we approximate them by convex ones. Namely, the algorithm for solving (P) will use its dual part (D) and their convex approximations. We construct two sequences { X k } and { Y k } as follows. Let xo E dom g.
(D)* ( D k )
mjn{h*(y) - [ g * ( Y k - l ) +
< x k , Y - Yk-1
>I}.
We see that g * ( y k - 1 ) + < X k , y - y k - 1 > is an affine approximation to g * ( y ) at y k - 1 . Hence h*(y) - [ g * ( Y k - l ) + < x k , y - Y k - 1 >] is a convex function. It follows that the minimum of the last function attains a t Y k which is characterized by 0 E a h * ( y k ) - Xk
The last is equivalent t o Xk
E ah*(%)
Yk
E
which is the condition that ah(xk).
so from x k and Y k - 1 we constructed Y k . Again to update X k , with the same idea, we consider the k-th convex approximation t o (P) (pk)
mjn{g(x)- [ h ( Y k ) +
The solution of the last is equivalencies 0E
ag(Xk+l)
- Yk
<
- xk7 Yk
>I}.
Xk+l
which is characterized by the chain of
@ Yk
E
ag(xk+l)
We summarize the algorithm as follows
xo E dom g ,
zk+l
E
ag*(Yk).
122
D. N. Hao. N. T. Thanh and H. Sahla
zk+l = Argmin{g(z) - [h(yk)+< z - zk,Y k
>I}.
The last equality is equivalent to Yk E dh(Zk),
Zk+1
E
%*(Yk).
The convergence properties of DCA have been proved in Refs. 12 and
13. 3.2. The Continuation Technique DCA provides however only critical points of f. In order to find a global minimizer off we consider the regularized function f a ( z ):= f (z)+a/2)1z))2 with Q > 0. When Q is very large, in many cases it is obvious that fa is convex. In these cases we can find a global minimizer of f a . After finding this minimizer, which we denote by x,,, we reduce a to a1 < QO and minimize fal by an iterative method starting from z ,, and so on. This is the continuation method.' As DCA is globally convergent, if we apply DCA to this procedure we can hope that we reach a global minimizer of f. The combination of DCA and the continuation technique is summarized in the following algorithm
Algorithm 3.1. minf(z) X
(1) Initialization: xo E domf, p 2 )If"ll, and a finite decreasing positive sequence { a,}$!O. (2) Choose h ( z ) such that f(z) h ( z ) is convex (e.g., h ( z ) = g11x112) (3) For i = 0 , 1 , 2 , .. ., M
+
+
+
3.1 gz(z) = f(z) 211z112 h ( z ) , Ic = -1 3.2 While stop criterion has not been satisfied
+
3.2.1 k = Ic 1 y k E ah(zk)
3.2.2 3.2.3
z k + l = argmin{gi(z)-
< Y k , z >}
End (While) 3.3 zat = zk+l (minimizer of f(z) ~ ~ ~ z ~ ~ 2 ) 3.4 20 = xat (change the initial guess in the continuation technique)
+
End (for). The sub optimization problems in 3.2.3 of Algorithm 3.1 are convex. In this paper, we used the Fletcher-Reeves nonlinear conjugate gradient method with Amijo-type line search, which ensures global convergence of the algorithm, for solving these sub-problems (see Ref. 5 for more details).
Numerical Solution to a Non-Linear Parabolic Boundary Control Problem
123
4. Numerical Results
To solve (2) we use an iterative technique to deal with the non-linear boundary condition and the finite difference method for the linear boundary value problem. Namely, let yO(z,t ) := yo(z)
and yn(x, t ) ( n= 0 , 1 , 2 , . . .) be the n-th iteration. Solve gyfl (z, t ) = ' : :y (z, t ) , 0 y"+l(z,O) = yo(z), 0 < z
y,"+l(O,t) = O ,
< z < 1 , O < t < T, < 1,
yZf1(1,t) = g ( y n ( l , t ) ) + u ( t ) ,
(17) 0
We stop the iteration procedure when some tolerance between yn and yn+l is achieved. We note that for n > 1, the difference w = yn+l - yn satisfies the simpler system w ~ ( z t, ) = W Z ~ ( Z , t ) l 0 < z < 1 , O < t < T , w(z,O)=O, O < z < l , Wz(O,t)=O, w~(l,t)=g(y"(l,t))-g(y"-l(l,t)),
(18) 0
< t
than y1 and yn, n > 1. For the convergence of this method we refer the reader to Ref. 2. To solve (17) we use the Crank-Nicholson finite difference scheme. We devide [0,1] by N - 1 equal subintervals: z o = 0 , q = h, . . . ,zi = ih, . . . ,X N = 1, where h = 1/N, and [0,TI by M - 1 equal subintervals: t o = 0, tl = 7 , . . . , t j = j r , . . . ,t M = T . In order to approximate the boundary conditions by central finite difference quotients, we extend the interval [0,1] one more layer to each direction and denote the extended grid points by 2-1 and Z N + ~ .For a function y defined on [0,1] x [0, TI, we set $ := y ( z i , t j ) . For simplicity of notion, we denote the solution of (17) by Y and the inhomogeneous boundary condition by V. We discretize (17) as follows
y.j+l- yj
-
7-
yi" = yo,i,
0
yjfil- 2 q j + l + 2h2 05i5
5 i 5 N,
y!+l 2-1 +
-2qj
2h2 N,0 5 j 5 M - 1,
+ qtl 7
(19)
It is well-known that this scheme is absolutely stable and has second order approximation with respect to the both variables. l1 The Crank-Nicholson scheme is also used to solve the adjoint problem (11).
124
D . N. Hao, N. T . Thanh and H. Sahli
4.1. Numerical Examples In this subsection we illustrate DCA and the continuation technique for the control problem (5) subject to (2) with some concrete examples.
Example 4.1. Let a be a number in (0,7r]. Then y(z, t ) = e-a2t cos(az)
is the solution of the problem
< z < 1,0 < t < T, y(z,O) = cos(az), 0 < z < 1, o < t < T. yZ(o,t ) = 0, yZ(1,t ) = -ue-aztsina, y t ( ~ , t= ) yz2(5, t ) , 0
(20)
Let
u ( t ):= e-4aZt cos4 a - ae-aZt sin a. We see that y,(l,t)
=-Y4(W
+4t).
Thus, in this example we set g(y) = -y4. To test our method we set
z ( z ) = y(z, T ) = e-aZT cos(az). In this example, T is set to be 1 and the space and time grid sizes are taken to be 0.01 and 0.025, respectively, resulting in a nonlinear optimization problem of 41 free variables. The control function u is then approximated by linear interpolation from the discrete values for solving the main problem (2) as well as the adjoint problem (11). We tested the algorithm with two different values of the parameter a. In the first case, we set a = 7r, then u(t) = e-4TZt and z(z) = e-K2Tcos(7rz). The lower and upper bounds of the control function are taken to be umin(t)= -1, and umaz(t) = 2, t E [0,1]. In implementing the continuation technique, the regularization parameter a is reduced five times at each iteration from 5 at the first iteration to 5e - 9 at the last one. To analyze the effects of the initial guess and the regularizing function u*(t)on the reconstruction of the control function, the algorithm was run with two different initial guesses
uA(t)= 1 - t , ui(t)= 2, t E [O, 11, and two different regularizing functions u * J ( t )= 0, t E [O, 11,
Numerical Solution to a Non-Linear Parabolic Boundary Control Problem
125
2 9 The results of the four cases are given in Table 1 and Figure 1.
~ * ? ~=(1t-) 8t, t E [ 0 , 0 . 1 ] , ~ * ' ~= ( t )-(1 - t ) , t E (0.1,1].
(c)
(4
Fig. 1. Reconstruction of the control function of example 1, with a = n
As we can see from the figures that the regularizing function u*(t)has certain effects on the results. In the case that the regularizing function u*(t) is taken to be zero, the estimated control function does not approximate very well the exact solution at the points for which the exact solution is far from zero (Figures 1-(a) and 1-(c)). In the cases that the regularizing function u * ( t )is taken to be ~ * ~ ~which ( t )is, near to the exact solution, the results are better (Figures 1-(b) and 1-(d)). The figures also show that the algorithm possibly converges to the exact solution with different values of initial guess, so we hope that it can converge to the global minimum of the regularized objective function. However, we could not test the algorithm with so many different cases of the initial guess because of its
126 D. N. Hao, N . T . Thanh and H. Sahla
time-consuming calculation. Table 1. Result of example 1, with a = 7r Test a
b C
d
Objective function value 3.6840e-008 2.1032e-007 3.4826e-008 3.1505e-007
Obj. func. gradient's norm 8.3236e-007 2.2014e-006 8.3246e-007 3.0112e-006
Number of iterations 2102 1129 2317 971
It is noted from Table 1 that the problem is severely ill-posed. The values of the objective function is very small with different values of the control function u ( t ), particularly, it is much smaller than the discretization error in solving the main and the adjoint problems. It is well-known that for severely ill-posed problems, as in this example, we cannot expect a very good estimation for the control function. However, for the optimal control problem, in which one only need to find a fiinction u(t) such that the temperature of the system at the final state is close enough to a given function, the algorithm works very well. We also tested the algorithm on the previous example for a = 7r/2, then u ( t ) = - ( ~ / 2 ) e - ~ ' ~ / ~z ,( z ) = e-"2T/4~os(7r2/2).The lower and upper bounds of the control function are u,in(t) = -2, and u,,,(t) = 0, t E [0,1].All the setup of the test is similar to that of the previous case. The initial guesses and the regularizing functions in this case are taken t o be
+ 1.4t, t E [0,t ] ,
~ ; ( t=) 0,
u;(t) = -1.58
u*>'(t) = 0,
~*?~ = (-1.58 t)
+ 1.4t, t E [0, t ] .
The results are given in Figure 2 and Table 2. The figures show the estimated control function compared with the exact one. In cases (a) and (b) ((b) and (d) resp.), the algorithm was started at the same initial guess uA(t) (uA(t)resp.) while in cases (a) and (b) ((c) and (d) resp.) it used the same regularizing function u*>'(t) ( ~ * > resp.). ~(t) It is clear that the cases (c) and (d) gives better reconstructions than the other two cases as the effect of the regularizing function u * ( t ) ,while the initial guess does not effect so much the result. However, as we can see from Table 2, the values of the objective function are very close to one another and close to zero. This behavior again shows the severe ill-posedness of the problem.
Numerical Solution to a Non-Linear Parabolic Boundary Control Problem
127
R__mdn.rnWMn
I
"
"
"
"
'
I
(4
(c)
Fig. 2. Reconstruction of the control function of example 1, with a = n/2
Table 2. Result of example 1, with a = n/2
T~~~caSe a b C
d
Objective function value 4.7510e-008 1.1387e-009 2.0380e-009 7.5335e-010
Obj. func. gradient's norm 9.5507e-007 3.7500e-007 1.2505e-007 1.2505e-007
Number of iterations 16848 2866 40230 32760
Example 4.2. In order to compare our results with those of Kelley and Sachs in Ref. 8, we coFsider here the same example as mentioned in Ref. 8, 10 in which g(y) = 1oo~++y4, yo(z) = 0, z(x) = 1, and T = 1. The constraint of the control function is 0 5 u ( t )5 0.1 4t.
+
All the discretization parameters in this example are as in Example 1. Unfortunately, the exact solution is not available in this example so we cannot have any guess about the solution of the control problem. We also tested the algorithm with two different initial guesses given by uA(t) = 0,
U i ( t ) = 4t, t E [O, 11,
128
D. N. Hao, N . T . Thanh and H. Sahli
and two different cases of the regularizing function u*(t)
u * J ( t )= 0 ,
u*12(t> = 4t,
t E [O, 11.
The results are given in Figure 3 and Table 3.
Fig. 3.
Reconstruction of the control function of example 2
Table 3. Result of example 2
Test caSe a b C
d
Objective function value 9.9149e-004 7.2189e-004 9.9563e-004 7.2604e-004
Obj. func. gradient’s norm 4.3683e-004 4.3675e-004 4.5075e-004 4.5646e-004
Number of iterations 2816 3108 2740 2710
It follows from Figure 3 that, for the initial guess uA(t)and regularizing function u*>’(t), (Figure 3-(a)), and a = 0.01, the result is the same as that of Kelley and Sachs. The result is also very similar in the case of the second initial guess u i ( t ) but with the same regularizing function (Figure 3-(b)). However, the results are different for the regularizing function u*)’(t)for
Numerical Solution to a Non-Linear Parabolic Boundary Control Problem 129
a = 0.01 (see the difference between (a) and (b) with (c) and (d)). Those results show the effect of the regularizing function on the results. However, when a is reduced, the four tests converge to almost the same solution. This behavior ensures the efficiency of DCA and the continuation technique in searching for the global solution of the problem. In conclusion, the combination of DCA and the continuation technique has been demonstrated to be possible for obtaining the global solution of the control problem (1)-(2). We also showed that the obtained result by this method is consistent with the result obtained by the trust region method proposed by Kelly and Sachs’ in the case of the same regularization parameter and function, and the same initial guess. However, our technique is apparently proved to behave better because it reaches the same solution with different initial guesses as well as different regularizing functions. References 1. J. C. Alexander and J. A. Yorke, Trans. Amer. Math. SOC.242, 271 (1978). 2. W. F. Ames, Numerical Methods f o r Partial DiJSerential Equations, Third
edition, Computer Science and Scientific Computing, (Academic Press, Inc., Boston, MA, 1992). 3. E. Casas, S I A M J. Control Optim. 35,1297 (1997). 4. E. Casas and F. Troltzsch, Appl. Math. Optim. 39,211 (1999). 5. Yu-hong Dai, Journal of Conputational Mathematics, 9, 539 (2001). 6. Dinh Nho Him: Methods for Inverse Heat Conduction Problems, (Peter Lang, Frankfurt am Main, 1998). 7. H. Jiang, T. H. Nguyen and M. Prud’homme, Control of the boundary heat f l u x during the heating process of a solid material, Manuscript. 8. C . T. Kelley and E. W. Sachs, SIAM J. Optimization 9, 1064 (1999). 9. 0. A. Ladyzhenskaya, The Boundary Value Problems of Mathematical Physics, (Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985). 10. P. Neittaanmki and D. Tiba: Optimal control of nonlinear parabolic systems. Theory, algorithms, and applications, (Marcel Dekker, Inc., New York, 1994). 11. M. Necati Ozisik: Boundary value problems of heat conduction, (Dover Publications, Inc., New York, 1968). 12. T. Pham Dinh and L. T. H. An, S I A M J. Optim. 8 , 476 (1998). 13. T. Pham Dinh and El Bernoussi Souad, Dends in mathematical optimization (Irsee, 1986), pp. 277-293, Internat. Schriftenreihe Numer. Math., (Birkhauser, Basel, 1988). 14. J. P. Raymond and H. Zidani, Appl. Math. Optim. 39, 143 (1999). 15. F. Troltzsch, Appl. Math. Optim. 29, 309 (1994). 16. E. Sachs, ZAMM 58, 443 (1978). 17. E. J. P. G. Schmidt, J . Dig. Eq. 78, 89 (1989).
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Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 131-175) @ 2007 World Scientific Publishing Co.
131
§9. A CLASS OF SOLUTIONS TO MAXWELL’S EQUATIONS
IN MATTER AND ASSOCIATED SPECIAL FUNCTIONS PETER MASSOPUST
GSF
- Institute for
Biomathematics and Biometry Neuherberg, Germany and Centre of Mathematics, M6 Technical University of Munich Garching, Germany massopust @ma.tum.de
The solutions for a class of Maxwell’s Equations in matter are presented. These solutions describe the magnetic fields as generated by a hard ferromagnet of finite length with missing mass and are important in the area of nondestructive evaluation.
Keywords: Maxwell’s equations, ferromagnetism, Poisson equation, Bessel functions, spherical harmonics, Mathieu functions.
1. Introduction and Preliminaries In this section the basic model for a hard ferromagnet is presented. The basic model is subject to modifications that describe the observed physical phenomena more closely but these modifications result in sets of solutions that are no longer expressible in simple closed analytical forms. For the purposes of this paper, however, the basic model contains all the important aspects of ferromagnetic theory. 1.1. Basic Magnetostatics Let 52 c R3 denote a hard ferromagnet* and 852 its piecewise smooth oriented surface. We assume that 52 is a compact subset of R3 and has *A ferromagnet is called hard if its magnetization M is essentially independent of any exterior applied field for moderate field strengths. One may treat such materials as having a fixed specific magnetization M.
132
P. Massopust
nonempty simply-connected interior. Applying an exterior field H to R 0
induces a magnetization vector field M : R R3 in the interior R of R. It is advantageous to assume that M vanishes on the boundary dR: Mian = 0. Note that this introduces a discontinuity of M on the boundary] which is not observed in the physical reality but provides a convenient mathematical idealization. The vector fields H and M produce the magnetic field B in R3 according to the constitutive relationshipt B=H+47rM.
(1)
In addition] there exists also a constitutive relation between the vector fields H and B of the form
B = f (H), where f is a multiple-valued function referred to as the hysteresis function. Figure 1 shows an example of a hysteresis function. In the neighborhood of B
Fig. 1. An hysteresis function.
any point Ho where f is single-valued] one may define the derivative
aB aH
p(H) := -,
called the differential magnetic permeability. For an isotropic ferromagnet and for small values of H and B, there exists a linear relationship between the fields of the form B=pH.
(2)
Maxwell’s equations] which are based on theoretical and experimental results, describe the behavior of electric and magnetic fields in matter. Their differential and integral forms as follows.t(Here ( , ) denotes the canonical tIn what follows, cgs units will be used. $In the cgs system, the integration of the electric field over the surface of a sphere produces the factor 47r (times the electric charge enclosed). This explains the ubiquitous occurrence of 4 ~ .
Solutions t o Maxwell’s Equations
133
inner product on L2(R3).)
4~
VxH=-J+-C
VxE=--,
1~(EE) c at ’
4lr 1 me f ( H, ds) = I + - -, c c at
f
dB dt
( E , ds)
1 awn c at .
--
1
Here the vector function E : IR3 -+ IR3 is called the electric field and E the electric permittivity. The constant c is the speed of light, Q : R3 -+ IR the electric charge density and Q the total electric charge inside the closed surface S. The vector function J : IR3 + IR3 denotes the macroscopic current density and I := Js J - d u the (macroscopic) current encircled by the closed curve y bounding the surface S. Given any closed orientable piecewise smooth curve y bounding a surface S , the quantities CPe, a” : R3 -+ IR are called the electric and magnetic flux, respectively: CPe :=
s,
(E, d u ) ,
CP”
:=
L
(B, do).
(7)
Here the orientation on S is induced by that of y. Eqs. (7) allows the interpretation of B as a magnetic flux density, i.e., as the “number” of magnetic field lines enclosed by the curve y, whereas CPe can be interpreted as the “number” of electric field lines piercing the surface S . For similar interpretations of electrodynamic quantities and their modcrn differentialgeometric interpretations in terms of differential forms, the reader is referred to Refs. 13,17. Eqs. (3) and Eqs. (4) are referred to as Gauss’s law. In particular, Eqs. (4) expresses the fact that there are no magnetic monopoles, i.e. that B is an (axial) vector field or, more precisely, a 2-form: B = B,(z) dy A d z By(z) dz A d z B,(z) d z A dy. The functions B,, B y ,B, : IR3 -+ IR are called the 2, y, and z-component of B, respectively. Eqs. (5) is commonly called Ampkre’s law with Maxwell’s extension and Eqs. (6) Faraday’s Law of Induction. For the purposes of this paper, it is assumed that there are no electric fields present and that the magnetic field B is static, i.e., dB/at = 0. Hence
+
+
134
P. Massopust
the set of Maxwell’s equations reduces to
( V , B) = 0 , i ( H , d s ) = 4-7TI
47T
(V,H)=-J, C
C
In the case when J vanishes, V x H = 0 and thus there exists locally a continuously differentiable scalar function 9 : R3 4 R,called the magnetic scalar potential, so that
H = -V9.
(8)
Using Eq. (8) together with Eq. (11) transforms ( V , B) = 0 into the Poisson equation
A@ = 4~ ( V , M),
(9)
where A denotes the Laplacian. The right-hand side of Eq. (12) can be thought of as a magnetic charge density p : R3 -+ R,z H -( V, M)(z): Choose a point p E 6’0 and apply the Divergence Theorcm to a cylinder of radius E > 0 and height 2 E centered at p making use of the assumption that M vanishes on 6’0 to obtain that the quantity ( M In) equals
and can therefore be thought of as a magnetic surface charge density. (Here n denotes the unit (outward) normal to an.) A solution to (12) is then given by (see for instance in Ref. 5)s
(Here Vz# refers to the gradient with respect to the variable X I E and, for the remainder of this paper, all primed quantities are evaluated at X I E R.) If we assume a linear relationship of the form (2), then one can compute the magnetic field B from the above equation via
B = -V(p@) = -pV@, Assuming that the interchange of V and
is allowed, one obtains
§Here one needs to assume that rk E C2(&)n C(0)
Solutions to Maxwell’s Equations
135
For the above-described idealized situation of magnetizing R, the magnetization field M may, in first approximation, be considered to be uniform and constant: M = MOe z , MO > 0, e, unit vector in the z-direction. Thus, (11) simplifies to
d
< a < b < L, L >> 1, and A(r1,rz) := { ( I c , ~ ): 0 5 TI I m 5 7-2) assume now that R := A(r1~7-2)x [a,b].(For T I > 0 ,
For 0
such a hard ferromagnet is encountered, for instance, in nondestructive evaluations of pipes.) The surface dR of R consists of the annuli A(rl , 7-2) x { a } , A(r1,r2) x { b } , and the two cylindrical surfaces dR1 := {(Ic, y) : z2 y2 = r:} x [ u l b ]r1 , > 0, and dR2 := { ( q y ) : z2 + y 2 = r,”}x [ a l b ] Now . suppose mass, represented by a volume element V , is removed from R in such a way that part of dV belongs to either 6’01 or dRz, but not both. Denote by C that part of the boundary of V which does not belong to dR and assume C is piecewise smooth and orientable. Note that C is also part of the boundary of R \ V . Now if the dimensions of C are big enough and the applied field H is sufficiently large, then the magnetic field lines of B will leave the interior of R, penetrate its exterior, and reenter R. This process is called magnetic ~ T U I Cleakage. As removing mass from R is a local phenomenon in regards to the overall geometry, focus is entirely given to the magnetic field B in the vicinity of C. This field, BE,is called the magnetic flux leakage field and its shape can be explained using AmpZlre’s Dipole Model. One assumes that BE is generated by magnetic dipoles situated on opposite sides of the surface C c R \ V . In other words, the distribution of magnetic dipoles on the surface C, regarded as a Radon measure on C, generates BE. An explicit expression for BE is given by Eq. (11) with only the slight modification of replacing dR by C.
+
Here we also used the fact that the magnetic permeability p is approximately equal to one outside the ferromagnet R. The magnetic dipole distribution is given by the Radon measure p~ := (M, d a c ) whose support is C. The distribution of the magnetic dipoles, i.e., p ~ is, determined by the magnetization M in a neighborhood of C in R and the specific material properties of the ferromagnet R. Precise knowledge of these two properties together with exact information about the geometry of C would uniquely
136 P. Mussopust
-
determine the Radon measure pc. On the other hand, p~ does determine the geometry of C once M and the ferromagnet's material properties are known. In this case, one has the one-to-one correspondence C p x . Unfortunately, M and the material properties of R are only approximately, if at all, known.
1.2. Coordinate Systems Here a summary of the five types of coordinate systems employed in this paper is given and the relevant quantities necessary for the evaluation of the integral in Eq. (13) are presented. Since the integrals in Eq. 13 are additive with respect to volumes and surfaces, the volume elements V associated with these coordinate systems make up the building blocks for more complex removed masses.
1.2.1. Cartesian Coordinates The unit vectors in the Cartesian coordinate system (z, y, z ) are denoted by {T,?, Throughout the remainder of this paper, it is assumed that the y-axis is the symmetry axis for R, that the magnetization M is along the y-axis, and that is perpendicular to X l 2 .
c}.
Definition 1.1. If V has the form
Ku?"= [-w, w] x [--e, 4 x [-d, dl1 where 2w > 0, 2e > 0, and 2d > 0 denotes the width, length, and depth the missing mass element V, then it is called Cartesian.
1.2.2. Circular Cylindrical Coordinates Definition 1.2. V is called cylindrical if it is of the form
V,,l where
rg
= {(r,$, z ) : r = r g , 0 I 4
5 27r, 0 5 z I d } ,
> 0 and d > 0 denote the radius and depth.
1.2.3. Spherical Coordinates Definition 1.3. V is called (hemi)spherical if it is of the form Vsph
with po
= { ( p , 4,O) : p = Po, 0
> 0 denoting the radius/depth.
5 4 I 2.rr, T/2 I 0 I n},
of
Solutions to Maxwell's Equations
137
1.2.4. Parabolic Coordinates The parabolic coordinate system ((, q, 'p) consists of
(1) Confocal paraboloids about the positive z-axis: ( = constant, 0 5 ( < 00.
(2) Confocal paraboloids about the negative z-axis: q = constant, 0
5q<
00.
(3) Half planes through the z-axis: cp = constant, 0 5 'p 5 27r.
As usual, the azimuth angle 'p is measured counter-clockwise from the positive x-axis when viewed from the positive z-axis. The relationship between Cartesian coordinate (x,y,z) and parabolic coordinates ((, v,cp) is given by
{
E=\lJ.2.yz.,2-.
x = Jqcos 'p, y z
and q =
= Jqsincp, = ;(q2 -52)
'p
Jm
= arctan ylx.
It is noted, that the parabolic coordinate system introduced above is a lefthanded system, i.e., i x i j = The unit vectors are denoted by i,6, and +, and the nabla operator V and the Laplacian A are
-+.
and
respectively. The oriented surface element d u for a paraboloid ( = const. is given by
r:")
d u = (07 qsincp
dqdcp,
(0 =
(14)
where, due to the left-handedness of the parabolic coordinate system, n is the inward normal
Definition 1.4. V is called circular parabolic of radius ro and depth d if it is of the form Vpav
= { ( E , 71 c p ) :
E
= 60, 0 I rl I To, 0 I cp I 2 r } ,
138 P. Massopust
where
= r o / a and 90 =
a.
Remark 1.1. (1) The values for and q in the above definition are obtained as follows. When = x 2 + y2 = r; the depth d should equal the difference between z1 = z l , ~ + ~ 2 = ~= ; ( 1 / 2 ) ( r $ / < i- ti) and zo = ZI,=~=O = --<$/2. This requirement then gives the values for and 90 in terms of ro and d. (2) A circular parabolic V is expressed then in cylindrical coordinates (9,cp, by
Note that -ri/4d 5 z 5 d - r;/4d. 1.2.5. Elliptic Cylindrical Coordinates The elliptic cylindrical coordinate system (u, u,z ) consists of (1) Elliptic cylinders u = constant, 0 5 u < 00. (2) Hyperbolic cylinders u =constant, 0 5 u 5 2n. (3) Planes parallel to the zy-plane, z = constant, -00
< z < +00.
The transformation equations between Cartesian coordinates (x,y, z ) and elliptic cylindrical coordinates (u, v , z ) are given by
{
x = acoshucosv, y = a sinh usin 21, z=z
u = Re [arccosh and u = Jm [arccosh
(w)]
I)*(
(15)
Here %e and Jm denotes the real and imaginary part, respectively. The azimuth angle u is again measured in a counterclockwise fashion from the positive x-axis when viewed from the positive z-axis. The parameter a is the focal length of the ellipse given by a2 = (major semi-axis)2 (minor semi-axis)2. The nabla operator V and the Laplacian A in this coordinate system are
and
A=
2 a2(cosh 2u - cos 2v)
Solutions to Maxwell's Equations
139
respectively, with 0,6 , and 2 denoting the unit vectors.
Definition 1.5. V is called elliptic parabolic of width w , length e, and depth d if it is of the form
Vecyl= { ( u ,v , z ) : 0 5 u 5 arctanh ( e / w ) , 0 5 v 5 27r, 0 5 z 5 d } , or, more precisely, KCyl:
+
z = a2(cosh2u cos2v sinh2u sin2 v ) , 0 5 u 5 uo, 0 5 v 5 2n,
where a=
d w ,w > e,
and
uo = arctanh ( t / w ) .
Remark 1.2. The values for a and uo are obtained by requiring that for z = d the boundary of V in this plane is an ellipse with major semi-axis equal to w and minor semi-axis equal to e. The surface element d u for an elliptic parabolic V is given as cos 3~ cash u - cos v cash 3~ sin 3v sinh u - sinv sinh 3u cos 2~ - cash 2~
with the unit normal n being n=
dl +
1 2a2(cosh2u
Remark 1.3. Setting
T
+ cos 2v)
-2a cosh u cos v -2a sinh u sin v
= (a/2)e", Eqs. (15) can be expressed in the form
If the focal length a
-+ 0, v -+ (x, in such a way that ae" remains finite, and the usual cylindrical coordinates are recovered.
2. Cartesian V In this section, the magnetic field BEgenerated by a Cartesian V is explicitly computed. For this purpose, consider a Cartesian V centered at the origin with length 2 4 width 2w, and depth 2d. To compute the magnetic field BE
140
P. Massopust
generated by this type of V, (13) is employed. For a function f : R3 4 R and a triple a := ( a l , a2, a3) E R3, denote by { f , a} the function
{ f,a } ( ~ y , ,Z) := [f(x- ai, y -
+ a2, z -
a3) - f
( x - a l , Y + a27 z
+ a3)I
[f(x+al,y+az,z-a3) -f(x+al,y+a2,zfas)l
- [ f ( x- al, Y
- a21
+ [ f ( x+ a i l y -
- a3) - f ( x -
a2, Z - a3) -
Y - a2, z + a311
f ( X + a l , Y - a2, z + a3)I
Theorem 2.1. Suppose that V,,, i s Cartesian of width 2 w , length 2C, and depth 2d and the magnetization i s of the form M = MY. T h e n the magnetic flux leakage field BE = (BzlBy, B,) generated by V,,, can be explicitly and exactly calculated. T h e exact expressions f o r the components are given by Y , ). = M{741(w, C, d ) ) ( x ,Y , z ) & ( x , y , 2) = M{P, (%C, d ) ) ( X ,Y l z ) ,
Proof. Under the assumption on MI (M’, du’) = M(da’Iyl=-e-da’I,,=e)l and thus (M’, du)’ 9 ( x )=
f
B
=
Ix-x’l
(Lr
dz’dx’
s_dd J ( X
f w
- Ic’)2
d
+ ( y + !)’ + ( 2 dz’dx’
Z’)2
).
+ ( y - C)2 + ( 2 - Z’)2 Since the functions r - ( x , x’) = l / d ( x - x’)2 + ( y + C)2 + ( z - z ’ ) ~and r + ( x ,x’) = l / J ( x - x ’ ) ~+ ( y - C)2 + ( z - z’)2 satisfy all the hypotheses -L w / d
J(X
- X’)2
for the application of the theorem (Interchangeability of Integral and Partial Derivative) in Ref. 15, p. 59, the x-, y , and z-components of BEare obtained as
By(5)= -a9 dY and
= -M
1: s_dd (e 2) -
dz‘dx‘,
Solutions to Maxwell’s Equations
141
respectively. Performing the partial differentiation in the expression for B, yields
-
Setting gives
=x
B,
-
(X - 5 ’ )dz’dx’
1:
+ (y - l ) 2 + ( z -
[(z -
x‘ and integrating over E between [ = x
=M
[l
dz‘
( x - .1)2
+ (y + + e)2
d
-/ d
( 2 - z’)2
dz‘ J(X
+
W)2
+ (y + e)2 + (Z -
d -I
+ w and E = x - w
Z’)2
dz‘
d J(X
- W)2
+ (y e)’ + -
(2
- 2’)2
1.
dz‘ + l d
J(Z
+
211)’
+ (y - e)’ + (Z -
Substituting C = z - z’ in each of the four integrals reduces each one to an integral of the form
dX
= log(x
+ J.2+x2>+ constant
with a different value for u2. Defining a function cp : R3\ (0, 0,O) cp(z, y, z ) = log(-z
+ d x 2 + y2 +
4
Rf by
22),
yields the stated expression for B,. To obtain the y-component By of BE,one proceeds similarly.
By = -M
[/+w -W
+w
ld J(z
d
-L w L d
+ e) dz’dx’ - x’)2 + (y + e ) 2 + ( 2 (y
3
-
z’)2
(y - t )dz’dx’
J ( x - x’)2 + (y - e ) 2
+(z-
+
Setting E = x - z‘ and integrating over E between E = x w and E = x - w and then replacing z’ by C = z - z’ and integrating over C from z + d to
P. Massopust
142
z
-
B,
d yields
= -M
[
(z - w)(y
+t)
Ld
[(y + l ) 2
+P]J(z
d4-4
2
+ (y + l ) 2 +
(2
z-d
-(.
- ul)(y -
l+d
[(y - t ) 2
+ C”](x
- w)2
+ (y
-
t)
+c2
Each of the four integrals is of the form
r
dX
with different values for a2 and b2. Such integrals may be solved employing the substitution X = a t a n u . This substitution yields
I
a2
a sec2 u d u sec2 u db2 a2 tan2 u
du
+
(dv
The latter integral evaluates to
ad-
1
arcsin
sinu)
Introducing the function $ : R3 \ (0, 0,O)
.-+
,
b2 > a2.
R
$(z, y, z ) = sgn (y) arcsin
where sgn (x)denotes the signum function, gives the exact solution for The z-component B, is given by
-
/:
(Z - z’)
[(x - x ’ ) + ~ (y -
By.
dz’dx’
+ ( z - z’)2]3/2
Notice that B, is just B, with x replaced by z . Hence, the conclusion follows. U
Solutions t o Maxwell’s Equations
143
Figure 2 below shows an example of magnetic field components for Cartesian V. The shapes of the depicted surfaces are characteristic of general magnetic flux leakage fields: The field component B, is bipolar and the component B, of quadrupole nature. The field component By is symmetric about a plane perpendicular to the axis of R (for a symmetric defect.) X
2 1
o
Bx
-1 -2
5
Y X
2
1 0
-1 -2 5
Y x
2
-2
5
Y
Fig. 2.
The
I , y, r-component of BE for a Cartesian V.
By
144
P. Massopust
3. Cylindrical V
Suppose that V is cylindrical with radius r o and depth d , i.e., I/cyl = { ( r , $ , z ) : ?- = T o , 0 5 $ 5 27r, 0 5 z 5 d } . Assume again that M is again of the form M = M j , and thus ( M’, da’) = -Mro sin $‘d#dz’. Note that since the magnetic field leaks into the interior of V&, n is the inward normal n = (- cos +‘,- sin$‘, 0). Thus,
In order to evaluate the latter integral, the expansion of the Green’s function 1 5 - z’I-~ in terms of Bessel or cylinder functions is employed. (See, for instance, Ref. 8.)
Lemma 3.1. T h e Green’s function in cylindrical coordinates f o r the equation A@ = 0 is given by
Proof. See Ref. 8 Here I, denotes the modified Bessel function of the first kind of order u and K, the modified Bessel function of the third kind of order u or Macdonald’s function. Exact definitions and properties of these functions may be found in Refs.1,3,5,9. With this expansion of 1 ~ - - 2 ’ 1 - ~ , the magnetic scalar potential becomes
+
00
COS[Y($
u=l
-
$’)]I,(kr<)K,(kr>)
1
d$‘dz‘dk.
Note that the above - and all subsequent - interchanges of the integrals and the infinite sum is justified. The above integral can be further rewritten
Solutions to Manuell's Equations 145 as Q ( r , 4, z) = --4 M r o 7-r
1" .Id1'"
cos[Ic(z - z')]sin$' ( k r < ) K o(Icr,)
1
d4'dz'dk
As
L'"
sin 4'd@ = 0
and
1'"
cos[v(4- 4')]sin +'d+' =
-2 sin[v7-r]sin[v(7-r- 4 ) ] = 0, for v 'v - 1
# fl,
only the v = 1 term in (17) contributes. For this value of v
and therefore,
I'"
cos(4 - 4') sin +'d@ = 7-r sin 4
@(r,4 , z ) = -4Mro sin 4
1" Ld Lm(I
cos[Ic(z - z')]Il(Icr<)Kl(kr>)dz'dk d
=
-4Mro sin4
cos[k(z - z')]dz'
Hence, the next result is established.
)
Il(kr<)K1(Icr>)dIc.
Theorem 3.1. Suppose that V i s cylindrical with radius ro and depth, d . T h e n the magnetic scalar potential generated by V& i s given by Q ( r , 4 , z) = -4Mro sin 4
sin Icz - sin[lc(z - d ) ]
Ic
I1(kr,)K1 (kr,)dk
(18) I n the case that r< # ro # r > , this above representation m a y be expressed in terms of a n infinite series of the f o r m
*I(.,
4, z) = -
TMror, sin 4 r n=O
:
z2n+1- (z - d)'"+l r
E(-V
( 2 n + l)!! (2n)!!
146
P. Massopust
Proof. The first part is obvious from the above calculations; the second part is established as follows. The Taylor series for the function (sin kz sin[k(z - d ) ] ) / kis given by
sin kz - sin[k(z - d ) ]
k
c m
=
-p+l
(-1)n
- ( z - d)2"+1
(2n
n=O
+ l)!
k2"
Substitution of this series into (18) and interchange of infinite sum and integral yields
m
- 4 ~ r osin4
I 22n-1
C (-l)n[z2n+l( 2 n -+ (.l)!-
r-2n-2
< >
n=O
xr(n
+ 1/2) r ( n + 3/2) 2F1(n + 1/2, n + 3/2; 2; ( r < / r > ) 2 > ,
Here ~ F I (b; Uc;,x) denotes Guufl's hypergeometric function:
Note that
&r(n
+
1/2) - 7r ( 2 n - I)!! 2 2 n + q n + 1) 2 3 n + q n 1 ) - 7r ( 2 n - l)!! 22n+l (2n)!! '
-
~
+
where the duplication formula for Gamma functions was used (Ref. 1, 6.1.18). Thus, after simplification,
w r , 4, 2 ) = x
7rMror< sin 4 r
:
2J'1(n
F(-Un
2 n + l - (2
n=O
- d)2"+1 ( a n
7-p
+ l)!!
(2n)!!
+ 1 / 2 , n + 3/2; 2; ( r < / r > 1 2 > ,
which proves the statement.
0
Solutions to Maxwell's Equations
147
In order to obtain formulae for the magnetic field BE, two cases need to be considered.
CASE I: T > T O In this case, rc = min{r, T O } = TO and r> = max{r, T O } = T . Hence,
a(')( T , 4, z ) = -4M'r-o
sin 4
Irn
sin kz
-
sin k(z - d)
k
11 (kro)K1 ( k r)d k .
The radial component B, of BE is the given by
Using the fact that dKl/dr
=
(-1/2)[Ko+Kz] (Ref. 3, p. 401,11.4.6) gives
Jo
The azimuthal component B,#,of BE is obtained in a similar fashion. As B+ = (-1/r)(N/a4), the result is
The z-component B, of Bc is -dG/dz yielding
CASE 11: T < T O In this case, r< = min{r, T O } = r and r , = max{T, T O }
= ro. Hence,
The radial component B, of BE is the given by
Using the fact that d1l/dr = (1/2)[10
+ 121 (Ref. 3, p. 397, 11.116) gives
*m
As above in Case I, the azimuthal component B4 is
148
P. Massopust
and the z-component
BL”)(r, 4 , z ) = 4Mr0 sin4
Lm
{cos k z - cos[k(z - d ) ] }I l ( k r ) K l ( k r o ) d k
The Cartesian components B, and By of BE are obtained via
4. Spherical
V
In this section, an exact solution for the magnetic flux leakage field due to a (hemi)spherical V is derived. To this end, assume that the radius/depth of said V is po. Then V,,, = { ( p , 4 , 0 ) : p = p o l 0 5 4 5 2 ~ ~ , / 52 6 5 T } . The magnetization M is again assumed to be along the y-axis: M = f i . The normal to C is given by ( M’, do’) = -Mpz sin 4’ sin2 O’d4’de’ (inward normal!). Thus, WPl41e)
#
= as
(MI,
l--’l
da’)
pi JT/2 T
J
0
2T
sin2 9‘ sin d‘d4‘de‘ 15-x2’l
To proceed the next lemma is needed. Lemma 4.1. I n spherical coordinates, the Green’s function Ix - x’1-l for the potential equation A* = 0 has a n expansion in terms of spherical harmonics yem of the form.
with p< = min{p, PO} and p> = max(p,po} (Refs. 3,5,6,8,15). Here * denotes complex conjugation. The spherical harmonics Yem(B,4) are expressible as
where Pem(x) are the so-called associated Legendre functions of degree C and order m:
Solutions t o Maxwell’s Equations
149
Proof. See Ref. 5 or Ref. 8, for instance. For definitions and results see Ref. 1 (Ch. 8) or Ref. 3 (Ch. 12). 0 The right-hand side of (19) may be expressed completely in terms of real functions by splitting off the C = 0 and (C, m) = (C, 0) term, combining those terms in the sum over m that correspond to positive and negative indexes, and using the fact that ecim4 eim4 = 2 cos 4. This yields
+
The functions Pi(.) = Pe,o(Z) are the Legendre polynomials of degree e. With the above expression for (z- ~ ’ 1 - lthe ~ magnetic scalar potential now reads 2x
x
li2
sin2 0‘ sin 4‘dqYdO‘
1
sin2 8’ sin $’cos[m(+ - 4’)]Pem(cos8’)d4’d8’ Pt,(cos 8) .
(21) The first and second term inside the bracket vanishes since the integral over 4’ is equal to zero. Moreover,
I””
sin#cos[m(+ - $’)]dq5’
=
ifm>l .rrsin+ if m = 1.
Therefore, in the third term inside the bracket only the m = 1 summand contributes to the sum over m. Thus,
150 P. Massopust
The integral over 8' can be exactly evaluated using the following approach. Let u = cos 8'. Then
/-'
d D P t , l ( ~ ) d ~
sin2 B ' P e , l ( ~ ~ ~ ~ '= ) d-B '
0
=
1'
JsPt,l(-u)du
= ( - ~ ) ~ + l /d'
3 P e , l ( ~ ) d ~ .
0
Here the fact that Pe,(-X) = (-l)'+"P~,(x) was employed (Ref. 3, p. 437, 12.93). Using the definition of the associated Legendre functions Pe, (Ref. 3, p. 435, 12.84),
the function Pe,l can be written as Pg,l(u)= d m d P e / d u . Hence,
11(1
- u2)-du dPe
( - l ) e + l0/ > m P g , l ( u ) d u
= (-l)e+l
du
+
= (-1)'+l [(l- U2)P'(U)IA 2 = (--I)'+'
[-Pi(O)
+2
/'
1 1
uP'(u)du]
uPr(a)du] .
0
Now,
(Ref. 3, p. 424, 12.34 and 12.35) and the moment integral is equal to
11uP*(u)du=
{
if[= 1 ~
~
~
+
~
~
2 i f e+= 2A X)
if l = 2 X
(23)
+ 1,
(Ref. 1,8.14.15).In the equation above, r denotes the real Gamma function, which is the extension of the factorial to real numbers: r(z)= (z+l)!.Using the fact that (Ref. 1, 6.1.17 and 6.1.12)
Solutions to Maxwell’s Equations
151
combining (22) and (23), and simplifying yields X(2X+1)(2X-3)!! i f e = 2 X (X+1)(2X)!! ife=1 if C = 2X
(-l)’+’
S,,
sin2 e’pe,l(cos e’)de’ =
(I
(24)
+ 1.
Therefore one arrives a t the following result.
Theorem 4.1. T h e magnetic scalar potential due to a (hemi)spherical defect of radius po i s explicitly given by
[f
a(p,4 , 0 ) = - 2 r ~ p i +
5 X=l
sin 0 sin 4
1
(-1)X+’(2X - 3)!! p;x 2x+1p2X,1(cos0) sin 4 (2X+2)!! p,
where
Nx =
(-1)x+1(2X - 3)!! (2X + 2)!! .
Proof. Only Pl,l(cos 0) = sin 6 remains to be shown. This, however, follows immediately from (20). 0
152
P. Massopust
In the derivation of Be the following recursion relation of associated Legendre functions was used to replace the derivative of P1,1,respectively, P2x,1 with respect to 0 (Ref. 3, p. 437, 12.90). d
~
ae
~
~
=
( 1~ ~ ~ e ) --[Pe,m+l(cos0 ) - (t m)(t- m
2
+
+ l)P+-,
(cos e)]. (25)
The Cartesian components B,, By, and B, of BE are obtained via
B, = sin Bcos r$ B, + c o d cos4 Be - sin$&, By = sinesin4 Bp + cos Bsin4 Be + C O S B, ~ B, =coseB,-sinBBe. 4.1. Semi-spherical V To complete the discussion of spherical V ’ s , an exact expression for the magnetic flux leakage field BE generated by a semi-spherical V is derived.
Definition 4.1. V is called semi-spherical if it is either of the form V, = { ( ~ , 4 , 0 ): P = PO, 7r 5 4 I 27r, .rr/2 I 8 5 x} or V , = { ( p , 4 , @ ) : P = p0, 0
I 4 5 T , 7r/2 I e I TI.
In the derivation of the exact solution for magnetic flux leakage, the first form is used and the expansion in (21) is employed with the appropriate modifications. (The magnetic flux leakage field for V, is obtained from that of V, via the reflection y H -y.)
Solutions t o Maxwell's Equations
The scalar magnetic potential 9 is comprised of three terms Ti
T i h $,el, T2 = T2(p74, e), and T3 = T3(p,4,e). More precisely, 9 = -MPi with
Ti=
K ( P 7 476) + Tz(P,4 , q + T3(p,4, Q)] .
IT
/2x
x/2
sin' 8' sin 4'd+'dQ' P>
T
where Ke =
-
7r
--
2P> '
l;2
sin26'Pe(cos e')de',
and
(e-
m)! p:
e=i x x/2
/ x
m=l 2x
sin2d'sin 4'cos[m(4
-
4')]Pem(cosO')d$'dO' Pem(cos0)
153 =
154 P. Massopust
where Ke,zP =
1:2
sin2 8' P e , 2 , 8')dO' (~~~
and the fact that
l=
sin 4' cos m(q5 - q5')dq5' =
2cos
i
y
COSrn
;sin4
=
($
- 4)
m2 - 1 form
& 2cos2
0
4
=1
form=2p otherwise
was used. Here [.]I : IR -t Z denotes the greatest integer function. In the above derivations previous results, in particular (24), was used. To obtain the magnetic field components, the partials of TI, T2, and T 3 need to be calculated. To this end, note that
where
Solutions to Maxwell's Equations
and
The derivatives dPe,,/dcos 0 were replaced according to (25).
155
156 P. Massopust
Theorem 4.2. The magnetic flux leakage field BE = ( B p B+, , Be) generated b y a semi-spherical V of radius po is explicitly obtained from the expressions below.
where the derivatives of Ti, i = 1 , 2 , 3 , are given above. The Cartesian components of BE can be obtained via (26). 5. Parabolic V
In this section the magnetic flux leakage field generated by the two types of parabolic V’s introduced in this paper is calculated. 5.1. Circular Parabolic V Assume that V,,, is a circular parabolic with radius ro and depth d : Vpar
= { ( E , ~ I P) , :
E
= Eo, 0
I r~ I 770, 0 I 9 L 2
~ } ,
-&
with 50 = and q o = d%. Using the form of the oriented surface element as derived in (14) and M = M3, one finds for the magnetic scalar potential
Next an expression for the Green’s function l / / x - x’I in parabolic coordinates is derived. To this end, recall that the Dirac delta distribution S in parabolic coordinates is given by
and that a solution to the (distributional) differential equation
Solutions t o Maxwell’s Equations
157
is sought. For (I q,, ’p) # ( [ I , q’, cp’) the above equation can be solved by the ansatz G = E(<)H(q)@(cp).Thus, upon substitution, simplification, and division by SH@ one obtains <2q2
E2+q2
1
[E
a% (@ +--< aE a[) +-
As the variables <, q, and constant m2 so that
1
‘p
are independent of each other, there exists a
and
The former equation has solutions of the form =
The latter equation can be rewritten as
Separation of variables yields
and
for a constant c. (29) is a modified Bessel Equation whose solutions are Bessel functions I , and K , of the third kind, and (30) is just Bessel’s equation with solutions J , and y,. The Green’s function for a magnetic scalar potential concentrated on the surface [ = 6 = const. must involve only the function J, as Y, is singular at the origin. To solve (27), recall that
(this is just the Fourier series for the Dirac delta distribution) and
158 P. Massopust
which is the Hanlcel transform or Fourier Bessel Integral of the Dirac delta distribution (Refs. 8,14). Therefore, a solution of the inhomogenous equation (27) in the form
is sought. Substituting this ansatz into (27) and simplifying gives
b2Jm
+L+ -Jm IC. E2 + v2 77
e*im(v-v‘)Jm(k771)
-
(Here the dots indicate differentiation with respect to the variable upon which the function depends.) Using the fact that
..
k 2 Jm
. ,2 + -k Jm - -Jm = -k2 Jm 77
v2
and (31) and (32), one finally obtains
Hence gm(E, E’) is a Green’s function for the ordinary distributional differential equation
whose solution is known to be (cf. for instance,8)
with E< = min{E, E l } and obtained.
E>
=
max{E, E’}. Finally, the following result is
Solutions to Maxwell’s Equations
159
where
1 ifm=O em={
2
i s the N e u m a n n factor, [< = min{[,
ifm>O
c’}, and E>
= max{[,
[’}.
With this expression for the Green’s function, the magnetic scalar potential becomes
x?7’2Jm(kv)Jm(k?7’)Im(ICE< )Km(kJ>)dv k d k where &
= min{J, Jo}
and [> = max{[, t o } . As
cos[m(cp - p’)]sin cp’dcp’ =
-/rsin+ i f m = 1 otherwise
only the term m = 1 contributes to the sum. Moreover,
and thus the next theorem holds. Theorem 5.2. T h e magnetic scalar potential due t o a circular parabolic V of radius ro and depth d i s given by
where [< = min{f,[o} and [> = max{[,[o}. In the case [< < [>, the above improper integral m a y be expressed as a n infinite series of the form
160
P. Massopust
Proof. Only the second statement needs to be shown. To this end, employ the following series representation of the product of two Bessel functions (Ref. 16, p. 148, ( 2 ) ) : J p( a z )J,, (bz) =
(fa.)"( f b z ) . r(Y 1) M (-1)"(~az)"2F1(-n1 - p - n;v m!I'(p m -t1 ) x n=O c
+
+
+ 1;b2/a2)
Setting 1-1 = 1, u = 2, a = r], and b = 770 in the above equation, using the fact that
yields
(n n=O
+
x 2 ~ 1 ( 2 n, 3
+ n;2; &J:)
Simplification now gives the result.
+ l)!(n + 2)! 2 ~ 1 ( - n , --n -
1; 3; v02/v2). CI
Theorem 5.3. The components of the magnetic field BE = ( B EB,, , Blp) generated b y a circular parabolic V of radius ro and depth d are then given by
Solutions t o Maxwell’s Equations
161
The Cartesian components (Bx, By, B,) of the magnetic field BE are obtained from the parabolic components (Bc,B,, BV)via the transformation
-t 5.2. Elliptic Parabolic
0
rl
V
Now assume that V is elliptic parabolic:
Kpar:
Z‘
=
a2 -(cash 2 ~ + ‘ cos 2 ~ ‘ ) , 0 5 U‘ <_ 2
UO,
05
V’ <_
IT,
with
d
a=
m
and uo = arctanh
(elw).
Assuming that M = Moj, the magnetic scalar potential drical coordinates is given by
-
* in elliptic cylin-
a3Mo LU”l (sin 3v‘ sinh u’ - sin v’sinh 3u’)du’dv’ 2n
2
Iz - 2’1
At this point, the Green’s function G ( z ,2’)= l/lz - z’I must be expressed in elliptic cylindrical coordinates. For this purpose, note that this Green’s function satisfies the (distributional) differential equation
+
a2G a2G +-= a2G A G ( z , d )= ay2
ax2
-4742
- z’),
a22
where 6 ( z - z’)denotes the Dirac delta distribution. In elliptic cylindrical coordinates the above equation is written as
AG(u,u‘)
=
-
2 -8.T
a2(cosh 2u - cos 2v)
+S(u - u’)6(v - v’)S(z - 2 ’ ) .
(35)
For u # u’ a solution of the above equation may be found by the ansatz G ( u ,u’) = V ( u ) V ( v ) Z ( z )Substitution . into ( 3 5 ) and division through UVZ yields
162
P. Massopust
where
G=
2 a2(cosh2u- C O S ~ U )
Since u, u,and z are independent variables, it follows that there exists a constant k2 so that
This equation has solutions of the form Z ( z ) = e f i k r .Again, from the fact that G = lc2 and the independence of u and u there exists a constant p such that 1 d2U a2k2 u aU2 2 cash 2u = p and
1 d2V
a2k2
cos 2u = p . 2 Simplified, these two equations reduce to a Muthieu equation
v
au2
62V
-+ ( p - 2qcos2v)V = 0 dU2
and an associated Mathieu equation
d2U
--
au2
( p - 2qcosh 2u)U = 0,
(37)
with q = -a2k2/4 < 0. (Note that k = 0 does not yield bounded solutions.) It is known that in (36), for each given value of q, there exists a set of eigenvalues p n = p,(q) and an associated set of complete (even or odd) periodic eigenfunctions of period 7r or 27r. (The second integral is nonperiodic and for physical reasons ruled out.) The set of even eigenfunctions of period 27r is denoted by {~e2,+~(v): n = 1 , 2 , . . .} and the set of odd eigenfunctions of period 27r by {se2,+l(u) : n = 1,2,. . .}.q Moreover, cezn+1 and sezn+1 satisfy the orthogonality conditions
and
1'"
se,(u)ce,(u)du = 0,
TThe functions ceZn+l and segn+l do also depend on q. To simplify notation, however, this dependence is suppressed when not explicitly needed.
Solutions to Maxwell’s Equations
163
and may be expressed as infinite series of cosines and sines m
c(-l)n+VAe$’) + < m
seZn+l(v) =
sin(2v
l)v,
q
0.
V=O
where the coefficients A2v+l ( 2 n + l ) = ACA1)(q)and B2”+l depend on q and its sign. Note that
=
B;r+;l)(q)
and m
V=O
Solutions for the modified Mathieu equation (37) for the same values of p and q are of the form cehn,+l(u) associated with ce2,+1(v) and sehz,+l(u) associated with ~ e 2 ~ + 1 ( vNon-periodic ). second integrals of Eq. (37) are given by fekhz,+l(u) and gekh2,+,(u) associated with cehz,+l(u) and sehZn+l( u ) , respectively. It is noted that these modified Mathieu functions also depend on q and its sign. The following behavior of unmodified and modified Mathieu functions applies when q 4 0-. ce,(v)
+ cosnw,
se,(v)
-+
for n 2 1,
sinnv, for n
2 1,
&l,(kr), c, constant, seh,(u) -+ c ~ l , ( k r ) ck , constant,
ceh,(u)
fekh,(u) gekh,(u)
-+
+ c:K,(kr), t
c: constant,
c:K,(kr), c c constant.
For proofs see,” and for the verification of the argument in the modidied Bessel functions I, and K, recall Remark (1.3) and the fact that k2a2 = -49. More details and additional information about Mathieu functions can be found in,1,4 and.1° Set
V ( v )= seZn+l(v, q ) , with q = -a2k2/4 < 0.
164
P. Massopust
Since the system {sezn+l} forms an orthogonal complete set of eigenfunctions, the Dirac delta distribution 6(v - v’) can be written in the form
b ( v - w’) = -
Se2n+l(v)Se2n+l(V/). n=O
(Refs. 3,8,14) In order t o solve (35), the Dirac delta distribution S(Z - 2’) is represented by its Fourier transform as
6(z - z’)
=
21r
Jm -m
cos[k(z - z’)]dk.
The following ansatz for the Green’s function is then made.
where the function gn is determined so that (35) is satisfied. Substituting the above expression for G into (35) yields (As above, the dots indicate differentiation with respect to the variable upon which the function depends.)
+ cos[k(z
-
~’)]~e~~+l(~)~e2~+1(~’)9,]
)
-k2 cos[k(z - z’)]~e2~+l(w)se2~+1(21’)g, dk Employing (36), and simplifying gives
’9Lm( 7T2
n=O
2 cos[k(z - z’)lsezn+l(v)se2,+l(v’) a2(cosh2u - cos2w) x [gn - ( p - q cosh 2u)gn
+4d(u
) -
u’)] dk = 0.
Hence, gn satisfies the (distributional) differential equation
jn - ( p - qcosh2u)gn = - 4 ~ 6 ( u- u’).
(38)
As sezn+l was chosen as a first integral of (36), the functions sehzn+1 and gekh2,+, are the only acceptable solutions of (38). A solution gn(u,u‘) satisfying the symmetry requirement g,(u, u’)= gn(u‘,u)and the finiteness condition a t u = 0 and u + 00 must therefore be of the form gn(u,u’) = csehZ,+l(u<)x gekh,,+,(u,), where u< = min{u,u’}, u> = max{u,u’},
Solutions t o Maxwell's Equations
165
and c is a constant to be determined so that the Wronskian W of sehzn+l and gekhzncl satisfies
= -4T/C.
To determine c, results concerning the expansion of sehzn+1and gekh2,+, in terms of modified Bessel functions and the asymptotic behavior of sehzn+1 and gekh,,+, and their derivatives are employed. Lemma 5.1. The functions sehz,+l and gekh2,+, have the following series representations in terms of modified Bessel functions (for q < 0). (- = d/du) w
and
u=o
The functions sehzn+1 and gekh2n+l and their derivatives satisfy the following asymptotic estimates as u 4 03.
and
Moreover,
and
166
P. Massopust
Proof. See Ref. 4 for the Bessel series expansions and the asymptotic estimates for seh2,+1 and gekh,,+,. To obtain the estimates on the derivatives, consider (39) and (40) together with the following asymptotic estimates for the derivatives of the modified Bessel functions 1, and K , (Refs. 1,3,9).
and
Using the asymptotic estimates provided in the above lemma, the Wronskian of sehzn+1 and gekh,,+, is computed as
Hence,
and therefore
implying the next result.
Solutions to Maxwell’s Equations
167
Theorem 5.4. T h e Green’s function of the equation A@ = 0 i s elliptic cylindrical coordinates i s given by
x ~e~n+l(v’,k)seh2,+l(u<,k)gekh2n+l(u>, k)dk
with (-l)nk2A~fl)B~2n+’)
G ( k )=
(7r/2) S‘e2,+1(0) dezn+1(7r/2)
=;,+I
and u< = min{u, u‘},u, = max{u, u’}. Next the magnetic scalar potential for an elliptic parabolic V is computed. To this end, note that
x [sinhu‘sin3v’ - sinh3u‘sinv‘l se2,+1(v, k ) se~,+l(v’,k )
x seh~,+l(u<, k ) gekhZn+,(u>, k ) du’dv’dk
T x ”l
-~
u
~
MO0 c,(k) ~
(lu12n
1
cos [z- 5U 2( ~ ~ ~ h 2 u ’ - ~ ~ ~ 2 w ’ )
n=O
x [sinhu’sin3v’ - sinh3u’sinw’l se2,+1(2), k ) s e ~ ~ + l ( wk’), x seh~,+l(u’, k ) gekh2n+l(u,k ) du‘dv’
+
luo 12* cos
[Z
x [sinhu’sin3v’
-
-
:(cash 2.11’
- cos 2
4
1
sinh3u’sinv’l se2,+1(vl k)se2,+1(v’, k )
x seh2,+1(u, k ) gekh2n+l(u’,k ) du‘dw‘
168
P. Massopust
To simplify notation, the following definitions are made. Definition 5.1. Let S2n+l(U, z , k ) =
lU12“
7 (-‘In
[ f
cos k z -
(cosh 2u’ - cos 271’)
1
x [sinhu‘sin3v’ - sinh3u’sinv’l ~ e h 2 ~ + 1 ( uk ‘),s e ~ ~ + l ( vk’),du’dv’.
and
1
(cosh 2u‘ - cos 2v’) 2 x [sinhu’ sin 3v’ - sinh3u’sinv’I gekh2n+1(u’,k ) ~ e 2 ~ + 1 ( vk’),du’dv’
G2n+l(U,2, k ; uo)
=
? I -
To further simplify Szn+l and G2n+1, note that
1
1 2 T c o s k [ z - ~ ( c o s h 2 u ’ - c o s 2 v ’ ) sin3v’~e2~+1(v’,k)du’
] 127Tk [
[ ;
z - - cosh 2 ~ ‘
= cos k
cos
1 1”
sink
cos 2v’] sin 3v’ seZn+l(v’, IC) du’
[fcos 2w‘] sin3v‘ ~ e p , + ~ ( w ’Ic),
du’
and, in particular,
I“
cos /c
[
c 03
cos 2v’] sin 3w’ seZn+l(w’, IC) du’ =
(T) 2m
(-1)m
(2m)!
m=O
and
6’” [: sink
00
cos 2v’] sin 3v’ sez,+l(w’, k ) du’ =
00
C m=O
(-1)rn
(2m
27T C O S ~ ~ + ~ (sin3v’ ~ ~ ‘ )sin(2v
u
($)
+ l)!
+ 1)w’dv’.
=o
Using the trigonometric identity sin3v’sin(2v
2mfl
+ 1)v’ = [cos2(v + 2)v’ - cos2(v - I)v’]/2,
Solutions to Maxwell’s Equations
169
one obtains
=
f
C O S 2 y 2 V ’ ) [COS2(”
C O S z~ [COS(Y ~
+ 2)v’ - cos 2(” - l)v’] dv’
+ 2)z - cos(v - 1)z]dz (43)
and
3v’
C O S ~ ~ + ~ sin ( ~ ~ ’ sin(2v )
+ 1)v’dv’
C O S ~ ~ + ~ [cos ( ~ ~2(v ’ )
=
;14=
C0s2m+1
z [cos(v
+ 2).
+ 2)v‘ - cos 2
( -~ l)w’] dv’
(44)
- COS(Y - l)z]dz
To proceed, the following lemma is needed. Lemma 5.2. Let a and m be integers. Then, for m > 0 , 47r
fora=2b andb=O,
..., m.
otherwise. and, f o r m 2 0 ,
10
otherwise.
Thus, applying the above lemma to (43) and (44) and simplifying yields
170 P. Massopust
and
(-1)nT
2m+1
2m+1
-- ~ % ( ~ ~ l ) [ ( m - V - l ) + ( m + Z + v ) j
2m+1 m-u
2m+1
Hence,
1
l r c o s k [z - %(cosh2u1- cos2v’) sin3v’se2,+l(v’, k)du’
)
m- 1
+A%”(m2””)]
* (-1)
m 2m+la4m+2
k
24”+3(2m
-ACA1) [ ?:Ti)
+ (m2m+ li-+ v
+ l)!
)I)).
The integral
[“
cos k [z - %(cosh2u’ - cos 271’)
] l’ k [ ] 12Tk:[ cos
sin
1
sinv’sez,+l(v’, k)du‘
cos 2v’] sin 21’ sean+1(v’,IC)du’ cos 2v’] sin v’~e2~+1(v’, k) du’
is computed in a similar fashion, employing the trigonometric identity
sinv’sin(2v
+ 1)v’ = [cos2(u + 1)v‘ - C O S ~ V V ’ ] / ~ .
Solutions to Maxwell's Equations
171
The result is
m
m-u
m
x ~ ( A ~ ' ) + A [~r m ~ + ) l) )
+ ( 2m+1
m--v
u=o
.)I
m+l+v
For the sake of notational simplicity, define
c O0
CL2)(k)=
2 [(2:':)+ (
(-l)mk2m+la4m+2
m=O
24m+3(2m + I)!
[( )])
2m+1 m-v-1
(A=')
u=O
m2m +l+v +
)+(
2m+1 m+2+v
)]
m
+
c
(2n+l) (A4u-1
-k
u=l
Then, using these four functions, the expressions for Szn+1 and Gzn+1 can be written in a more succinct way. The results are summarized in the following lemma.
172
P. Massopust
Lemma 5.3. S2n+l(U,
z , k)
1
cosh 2u’ sinh u’~eh2~+1(u’, k) du‘
lu
- Ci2)(k)
sin k [ z -
1’
+ CL3)(k) - Ci4)(k)
a2
cos k [ z -
lu
sink [z -
1 1 1
cosh2u’ sinhu’seh2,+l(u’, k ) du’
a2
a2
cosh 2u’ sinh 3u’ ~ e h 2 ~ + l ( uk’),du’ cosh2u’ sinh3u’seh2,+1(u’, k) du‘
and
G2n+l(U, z , k ; 210) = Cil)(k)
luo[
cos k z -
- Eiz)(k) / u o
a2
sink [ z -
1
cosh 2u’ sinh u’gekh2,+, (u’, k) du’
a2
+ EF)(k) J,”” cosk [ z - Ei4)(k)
luo
sink [z -
1 1 1
cosh 2u’ ~ in h u ’ g e k h ~ ,+ ~ (uk ’, ) du’
U
a2 a2
k) du’
C O S ~ ~ U ~inh3u’gekh,,+~(u’, ’
cosh 221’ sinh 3 ~ ’ g e k h ~ , + ~ ( uk ’),du’
Proof. The formula for Gzn+l is essentially given by replaced by gekhzncl.
Szn+1 with
sehzn+1
In summary,
Theorem 5.5. Suppose that V is elliptic parabolic of length 1, width w, and depth d . T h e n the magnetic scalar potential generated by V i s given by
+ G2,+1(u, where
uo = arctanh
z, k;U O )sehz,+l(u)] sean+1(~) dk.
(e/w), a =
d
w and
Solutions to Maxwell's Equations
173
The components (BulB,, B,) of the magnetic flux leakage field BE generated by this defect are obtained as usual via differentiation.
Theorem 5.6. T h e magnetic flux leakage field BE generated by a n elliptic parabolic V of length e, width w,and depth d i s given by BE = (BulB,, B,) where
and
with S2n+l(U, z , k)
1
cosh 2u' sinh u'seh2,+1 (u', Ic) du'
+ EL2)(Ic)
+ Cr)(k) +EL4)(Ic)
Lu
[
cos k z -
1" Au
sink [ z -
a2 a2
1 1 1
cosh 2u' sinh u' seh2,+1
(d, Ic) du'
cosh2u' sinh3u'seh2,+1(u', Ic) du'
a2 cosk [z - - cosh 2u' sinh3u'~eh2~+l(u', Ic) du' 2
174
P. Massopust
and
=k
(Cil)(k) loIc [ z sin
luo luo luo
+ CA2)(k)
-
cosk [ z -
+Cr)(k)
sink [z -
+Ci4)(k)
cosk [z -
a2
a2 a2
a2
1
cosh 2u’ sinhu’ gekh2n+l(u’, k) du’
1 1 1
cosh2u’ sinhu’gekh2n+l(u’,k) du‘ cosh 2u’ sinh3~’gekh~,+~(u’, k) du’ cosh2u‘ ~inh3u’gekh~,+~(u’, k) du’
The Cartesian components (BZ, B,, B,) of Bc are obtained from the elliptic parabolic components (Bu,B,, B,) via the transformation matrix
T=
1 a(cosh 221 - cos 2v)
as
2 sinh u cos u -2 cosh u sin u 0 0 2 cosh u sin u 2 sinh u cos Y 0 a(cosh 2u - cos 271)
(3) (3) =T
7
with the right-hand side completely expressed in Cartesian coordinates.
References 1. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover Publications Inc., (New York, 1972). 2. R. F. Arenstorf, Lectures o n Special Functions (Bessel Functions, Spherical Harmonics, Elliptic Functions), Lecture Notes, (Vanderbilt University, 1973). 3. G. Arfken, Mathematical Methodsfor Physicists, Academic Press, (New York, 1966). 4. R. Campbell, The‘orie Ge‘ne‘rale de L’dquation de Mathieu et de Quelques Autres Equations Diffe‘rentielles de la Me‘canique, Masson et Cie Editeurs, (Paris, 1955). 5. R. Courant and D. Hilbert, Methoden der Mathematischen Physik 1’11, (Springer Verlag, Heidelberg, 1968). 6. E. W. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, (Chelsea Publishing Company, New York, 1955). 7. G. Hammerlin and K. H. Hoffmann, Numerical Mathematics, (Springer Verlag, New York, 1991). 8. J. D. Jackson, Classical Electrodynamics, (John Wiley & Sons, 2nd ed., New York, 1975).
Solutions t o Maxwell’s Equations 175 9. N. N. Lebedev, Special Functions and their Applications, (Dover Publications Inc., New York, 1972). 10. N. W. McLachlan, Theory and Applications of Mathieu Functions, (Oxford Clarendon Press, London, 1947). 11. F. W. J. Olver, Asymptotics and Special Functions, (Academic Press, New York, 1974). 12. J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, 2nd, ed., Texts in Applied Mathematics 12, (Springer Verlag, New York, 1993). 13. B. Schutz, Geometrical Methods of Mathematical Physics, (Cambridge University Press, UK, 1980). 14. A. Terras, Harmonic Analysis o n Symmetric Spaces and Applications I, (Springer Verlag, New York 1985). 15. W. Walter, Potentialtheorie, BI Hochschulskripten, 765/765a, (Bibliographisches Institut, Mannheim, 1971). 16. G. N. Watson, A Treatise o n the Theory of Bessel Functions, (Cambridge University Press, London, 1922). 17. K. Warnick, R. Selfridge and D. Arnold, IEEE Trans. o n Education, 40, 53 (1997).
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177
Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et aI. (pp. 177-196) @ 2007 World Scicntific Publishing Co.
$10. ON THE CAUCHY PROBLEM FOR A QUASILINEAR WEAKLY HYPERBOLIC SYSTEM IN TWO VARIABLES AND APPLICATIONS TO THAT FOR WEAKLY HYPERBOLIC CLASSICAL MONGE-AMPERE EQUATIONS
HA TIEN NGOAN* and NGUYEN THI NGA* Institute of Mathematics 18 Hoang Quo. Viet Road, 10307 Hanoi, Vietnam E-mail: htngoan0math.ac.t.n The Cauchy problem for a normal quasilinear weakly hyperbolic system in two variables is considered. Sufficient conditions for its diagonalization are given. The local solvability of the noncharacteristic Cauchy problem for some weakly hyperbolic classical Monge-Amphe equations is proved.
1. Introduction The classical hyperbolic Monge-AmpBre equation with two variables is that of the form
Ar + Bs + C t t ( ~-ts2 ) - E
F ( 2 1 , 2 2 , z 1 p l , ~ 2 , r , s l= t)
= 0,
(1)
is an unknown function defined for ( 2 1 , 2 2 ) E R2,pl = zxlxl , s = zZlx2and t = zx2x2.The coefficients A , B , C and E are real C2-functions of ( X I , 2 2 , z , p l , p2) and satisfy the condition of hyperbolicity:
where z 2x1 i P2
= z(z1,22)
=
zx2, r
=
A := B2 - 4 ( A C + E ) > 0. In this case the characteristic equation of the Eq. (1)
X2+BX+(AC+E)=0 has two different real roots
*Supported in part by the National Basic Research Program in Natural Science, Vietnam
178
H. T. Ngoan and N. T. Nga
In this paper we consider the case A > 0, i.e. the characteristic roots Ai,A2 may coincide at some poits. In this case the Eq. (1) is said to be weakly hyperbolic and it can be written in the following equivalent form Zxixi + C zXlX2 + AX = 0. un + ^2 2x2*2 + -A
(3)
We are interested in looking for local classical solution to the Eq. (1). So we suppose that we are given C2-functions X°(ai), X§(ai), Z°(ati), P^(QI), P^(a\), ai e /, where / is an interval (0, 5), 6 is some sufficiently small positive number. The Cauchy problem for the Eq. (1) consists in looking for z(x) € C2 which is a solution of (1) such that
where From (4) we have the following necessary condition for the initial Cauchy data
which is assumed to be fulfilled for all a\ £ I. We assume that the Cauchy problem (l)-(4) is not charateristic (see Refs. 6,8), i.e.
0,Vai £ / where the coefficients A,B,C are calculated at Equation (1) was investigated in Refs. 1,2 by G. Darboux and E. Goursat under the assumptions that A > 0 and there exist two independent first integrals for the Eq. (l). We recall that a function tp(xi,X2, z,pi,p 2 ) is said to be a first integral ion the Eq. (1) if it satisfies the following system of equations y £t,^
&,~
Q{n
Q(n
'--\-i-^-- A-~ = 0 . z ap\ ops
Cauchy Problem f o r a Quasilinear Weakly Hyperbolic
179
However, there exist Monge-Ampitre Eq. (1) which do not possess two independent first integrals. For example, as it has been proved in Ref. 7 that the following equation
rt - s2
+ c2(21,
2 2 , 2 , P l , P2) = 0
has two independent first integrals if and only if
In the case A > 0, but without the assumption on existence of two independent first integrals, the Eq. (1) had been also considered in Refs. 3, 5-9 by reducing it to a hyperbolic quasilinear system of first-order partial differential equations with two variables. In this paper when considering the case A 2 0 we do not assume existence of two independent first integrals for the Eq. (1). To avoid this difficulty in Ref. 8 we have proposed another solving method for the Cauchy problem (3)-(4) which reduces it t o that for the following normal quasilinear first-order system of equations in two variables ’
L
ax,
aa2
with the following initial conditions
Here ( X I ,X 2 , Z , P I ,P2) are unknown functions of two variables a1, a2 and the functions X y ( a 1 ) , X z ( a l ) ,Z o ( a l ) ,P f ( a l ) ,Pi(c.1) are given as the same as above, which satisfy the conditions (5), (6).
180
H. T . Ngoan and N. T. Nga
ax, ax, -aal
aa2
det
# 0.
(9)
-__
From the implicit function theorem we can locally solve the following system of equations X l ( Q 1 , Q2)
=21
Xz(a1, a2)
=2 2
t o obtain a1 = $ 1 ( ~ 1 , z 2 ) ,a 2 = &?(zl,2 2 ) . In Ref. 10 we have proved the following theorem.
Theorem 1.1. Suppose that the Cauchy problem (7)-(8), satisfying the conditions (5), (6), possesses a C2-solution (XI,X 2 , Z , PI,P 2 ) . Then the condition (9) is satisfied and a local C2-solutionz(x) to the Cauchy problem (1)-(4) is given by the following f o m u l a r z(51,22)
= Z($l(Zl,
2 2 ) , $2(21, 2 2 ) ) .
Moreever, Z q ( 2 1 , 2 2 ) = Pl(1cll(Tl, 2 2 ) , $ J 2 ( 2 l 7 X 2 ) ) , 222
( 2 1 , 2 2 ) = P2(7f!J1(21,5 2 ) , $ 2 ( 2 1 , 2 2 ) ) .
So, t o study the Cauchy problem (1)-(4) we first concentrate t o investigate the solvability of the Cauchy problem (7)-(8).To do this in Sec. 2 we recall definition on hyperbolicity for systems (7) and some results on solvability of Cauchy problem for them. In Sec. 3 we first reduce the system (7) to an intermediate one (25). Provided some restrictions on coefficients aij and initial Cauchy date, the last system will be in Sec. 4 reduced in Theorem 4.1 to a diagonal one (42), the Cauchy prblem for which is well solved locally according to the Theorem 2.3. In Sec. 5 we describe in Theorem 5.1 a class of weakly hyperbolic classical Monge-AmpBre equations, Cauchy problem for which is well solved locally. In Sec. 6 we give some examples of weakly hyperbolic classical Monge-Ampkre equations that satisfy conditions of Theorem 5.1.
181
Cauchy Problem f o r a Quasilinear Weakly Hyperbolic
2. Hyperbolicity
From (3) we denote C by a l l , A 1 by a12, A2 by a21 and A by a22. From now we consider instead of (7) the following quasilinear system in two variables -
8x1 1)8% 8x1
(a12 -
= -all-
ap2 + a22- ax2 + aa aa
- (a21
aal
a51 + 1)-ak2 aa, - aa1
8x1
= (a12P1 - U I l P 2 ) Z
ak2 - a2 -
+(a22P1 - a21P2)-
- (411022
+
Here (XI, X2,Z, PI,P2) are unknown functions of the variables aij are C2-functions of (XI,Xa, 2, PI, P2). We set
a1
and
(212;
u = (Xl, x2,z,Pl, p 2 y ,
I
0 0 d(U)= a12Pl-allP2 -a21P2 -1 0 -(711a22 a12a21 0 a11a22 - a12a21 0 0 a12 -
-all
1
a22
-a21-l
+
0 -1 -P2 a12
-1
a22
.
Pi
(12)
--a11 -a21 -
1
We write the system (11) in the matrix form
Now we recall some definitions and results on quasilinear hyperbolic systems. To do this we will consider the following more general normal system in two variables
where U and G ( U ) are m-dimensional columm-vectors, H ( U ) is a matrix of size m x m. If the matrix H ( U ) is diagonal, the system (14) is said to be a diagonal one.
182
H. T . Ngoan and N. T. Nga
The Cauchy problem for system (14) consists in looking for U ( a 1 ,a2) E C 1 such that
where
U o ( a l )is
a given m-dimensional vector function.
Definition 2.1. (Ref. 4) The system (14) is said to be hyperbolic if for any U E R" the following conditions hold: 1) All the eigenvalues of the matrix H ( U ) are real; 2) There exists a basis in R" consisting of its corresponding smooth with respect to U left eigenvectors. Theorem 2.1. (Ref. 11) If a12 # then the system (14) i s hyperbolic.
a21
f o r any ( X I ,X 2 , Z , P I ,P2) E
R5,
Theorem 2.2. (Ref. 4) Suppose that the system (14) i s hyperbolic. T h e n it can be reduced t o a diagonal system of 2m quasilinear equations with respect t o 2m unknowns. Theorem 2.3. (Ref. 4) Suppose the system (14) i s diagonal. T h e n there exists locally a unique smooth solution to the Cauchy problem (14)-(15). For the system (11) we do not assume that a12 # a21. This means that these functions may coincide at some points ( X I ,X 2 , Z , P I ,P2). In this case only condition 1) in Definition 2.1 is valid and the system (11) is said to be weakly hyperbolic. We show below in Theorem 5.1 that under some restrictions on coefficients aij ( X I2,P ) , the weakly hyperbolic system (11) can be still reduced to a diagonal one of 7 quasilinear equations with respect to 7 unknowns. From the Theorem 2.3 it follows that there exists locally a unique smooth solution for the Cauchy problem (7)-(8).
3. Reduced System Set
Cauchy Problem for a Quasilinear Weakly Hyperbolic
183
where aij are the same functions of X I ,X z , Z , PI,P2 as in (11). Then it is easy to verify that
C-l(X1, X 2 , Z , PI,P2) =
1;
all
a12
0 000 1 0 0 0 0 1 0 01 . a21 0 1 0 a22 0 0 1
(17)
Proposition 3.1. Suppose that m a t r i x d is defined by (12) and 0
0
0 -1 0 0 0 -1 0 0 0
-1
-1 0
A=
0
0
1 0
Pl
-P2
a12 -a21
-1
0
0
0 a12 - a21 -
Then C-l
I.
(18)
1
AC = A.
Proof. The proof is obvious by matrix multiplication. Set
That means x1
=x1,
x 2
=X2,
z
=z,
PI
= Pl
[p2
+ 2,P ) X 1 + = P2 + a12(X, z, f)X1+ Ull(X,
(20) a21(X,2,P ) X 2 , a 2 2 ( X , 2, P)X2.
We introduce now the following condition for the Cauchy problem (11)-
(8). For the coefficients a i j ( X ,2,P ) and the initial functions X y ( a 1 ) ,X , ” ( a l ) Z , o ( a l ) ,PF(al), f,”(a1)the following inequality holds f o r (Cl):
184 H. T . Ngoan and N. T . Nga
where the derivatives of functions
aij
are calculated at (X,"(al),X~(a.l),
ZO(al),P?(Ql), Pi(Q1)). Proposition 3.2. Suppose that the condition ((71) holds. Then for any (XI2,P ) E R5, which are suficiently closed to (X0((al),Zo(al),Po( al) ) , the following system of equations with respect to P I ,P2
Pl P 2
+ Ull(X, z,P)X1 + a21(X,2,P)X2 + a12(X,2,P ) X l + a22(X,2,PIX2
I
= Pl,
(22) =p2
has a unique smooth solution
where
and
Proof. Set
{
fl
= Pl+ W ( X , 2,P ) X l + a21(X,2,P)X2 - Pl,
f2
= P2 + a12(X12,P)Xl + a22(X,2,P)X2 - P2.
Cauchy Problem f o r a Quasilinear Weakly Hyperbolic
185
We have
of = det DP
Since a i j ( X , 2,P ) and X , 2,P are C2-functions, from (21), (24) and from the implicit function theorem the assertion of the proposition follows. IJ
Theorem 3.1. The vector function U ( a 1 , a z ) satisfies the system (13) i f and only if the vector function d(a1,a2) satisfies the following system
Here d is defined by (18) and
186
H. T . Ngoan and N. T. Nga
I
L
-(a12
-
+
a21 -
-(a12 - a21 -
aa12
1)-
aal
-(a12
t
-
+
a21 -
where in (26) the variables P I ,P2 are replaced respectively by the functions f(X1, X2,Z, fj, &), g(X1, Xz,Z, PI,&,), which are defined in (23). Proof. 1) Suppose U is a solution of (13). Since
0 = C-lU u = c0.
we have
Therefore
dU
-=
aa2
a0 ac c-aa2 + -u, aa2
au a0 a c aal c-aal + -u. aal
-=
By (13), (27) and (28)
a0 + -Ua c -
C-
aa2
ac + -U). aa1 aa1 aG
= A(C-
aa2
Hence,
a0
-- -
aa2
C-lAC-
a0 aa1
ac ac + C-l(A-aal -)u. aa2 -
So we have -1 0 0 0 0 -1 0 -1 o o -1 -p2 0 0 Oa12-a21-l 0 0 0 0
and
1 0
Pl 0 a12 - a21 - 1
-)a c aal aa2
23 = C-l(A(C0)-ac -
Cauchy Problem for a Quasilinear Weakly Hyperbolic
-
-
-&Lz
-&?.22’
8%
aal
h aa1
h
aalL
*P2
-
-
*Pl
EPZ -2
+& + -(a12
(a12 - a21 -
-(a12 - a21 --(a12
187
- a21 - I)%
-
a21
-
P l
11% I)%
000 000 000 .
+ 2o o o + o o 0-
4. Diagonalization It is clear from (18) that the system (25) is still not diagonal. We give now some sufficient conditions under which the system (25) can be reduced t o a diagonal quasilinear one. We introduce now another condition for the system (11). (C2) : Each function a , j ( X , 2,P ) satisfies the conditions = 0, a
aazl
1 1 bat,
a21 ap,
-~a12%
da,,
a22apz
= 0,
(29)
= 0.
We set
Proposition 4.1. Suppose ( X I ,X 2 , Z , P I ,P2) is a solution of (13). Under conditions (C1) and (Cz) we have
1)
2) Suppose ( X , p ) are suficiently closed to ( X o , P o ) . Then there exist smooth functions bij ( X ,p) and cij ( X ,p) such that
188 H. T. Ngoan and N. T. Nga
from (33), (34) and (11),we have (a12
-a21
-
= (a12 - a 2 1 -
ax1
aa,j
- 1)-
ax1
aa 1
aaij
3j7[(-U11(122
-
dUij --[(a11a22
a x 2
aP2
8%
+GI aP1
a x 2
ap1 + (1.12a21)- 8 x 12 + ( a 1 2 - 1)-3% - (1.12021)-
ap2
8x1
aff + 1
ap1 a22-
8%
- (a21
aa..
daij [-a21-
(1.22-
aQ:1 - ((1.21 + 1)-aa1 - 1-aa1
-
-
+
3x1
aaij - -[-alla x 2
-
aaij
aaij
1)- aa1 a a 2 aaij 8x1 aaaj ax2 daij aP1 daij d P 2 - l)(-+ -~ f --+ -8x1 aa1 a x 2 aa1 ap1 aa1 d P 2 aa, 1
8x1
+ a 1 1 2 - (Ull(1.22 -a12(121)--](9x2
aaaj
+
8 x 2
aP2
aaij aaaj - -+all-]-.
[a12-
(35)
+ 1)-]aa1 8x1
dP2
aa,
-
aP1
aa1
dP2
aaaj a x 2
-
aaij
a p 2 all-]
aaaj
+ [a12-8 x 2 a22- ax1 + ( a 1 1 u 2 2 U12U21)-]-ap1 aaij - a22-1-aaij ap1 + [-a21- aaij + dUij
-
aa1
aa1
ap2
ax1
aP1 aa1 It follows from (29) and (35) that (a12
dP2
-a21
-
daij 8x1 8 x 2 1)-aaij - 0.-+o.-+o.-+o.act1
d a 2
aa,
aa,
8P1 aa1
dP2
aa1
=o.
2) From (22) we have ap1 - -_ dull _ aa1 ap2 -
aa,
aa1 x1 -all-
da12
--
x1
aa,
-
a12-
aa21x2 ax1 - a21- 8 x 2 - aal aa1 aa1
8x1 - (1.22-8 aal
x 2
aa1
- -dxa22 2
da1
I
aPl aa1
aP2 +aal .
Applying (29), (36), (37) to (33) we get aaij - daij 8x1
daij +-aal ax, aal ax2aal 13x2
+ ~a a( .-. - x laa,, - u l l - - a ~ l8-x1 aP1 aa1 aal
+ aa.. ~ ( - - xaa12 l - a l a - axl aP2
da1
aa1
8 x 2
aa,
aa,
a22- a x 2
aa21 -x2
aa1
-aaz2x 2
aa1
+ Fl) + F2)
(36)
(37)
Cauchy Problem for a Quasilinear Weakly Hyperbolic
+ ( - =aa11 XI-
M
ax, + 0.-ax2 + ( aal aa,
-
ap1
aa1 = 0.-
+ ( - =X1 aa12
-x2 aa21 + P1)-daij
189
-
dull %X1
N
-
aa21 -x2 + P1)daij aQ.1 ap1
Hence,
(38) Letting i and j vary from 1 to 2 in (38), we obtain the system
This is a linear system with
aa.. . . s, 1 , 2 as unknown functions.
Set
1 d=
Elemeritary calculations yield
Z,J
=
(39)
H. T . Ngoan and N . T . Nga
190
Since (X, p ) are sufficiently closed to (X', that d # 0. The assertion 2) follows now.
Po),from (41) and (21) it follows
We introduce new dependent variables
w = (Xl
" I
x 2I
2,Pl ,P2
7 P 1 I
p2 T .
Theorem 4.1. Assume that the conditions (Cl) and (Cz) hold. Then the system (25) can be diagonalized, i.e. it may be reduced to the following diagonal one:
here
N N
A=
-1 0 0 -1 0 0 0 0 0 0 0 0
0
and
with
0 0 -1 0 0 0 0 0
0 0 0 0
0 0 0 a12 - a 2 1 -
0 0 0
1 a12
-
a21 -
0 0
1
0 0 0 0 0 a12 - a21 -
0
1
0 a12
-
a21 -
1 (*
Cauchy Problem for a Quasilinear Weakly Hyperbolic
191
I n the representations for F I ( W ) ,F z ( W ) the variables P I ,P2 must be replaced by f ( X I ,X 2 , Z , P I , P 2 ) and g ( X 1 , X z , Z , P I , p ~ respectively. )
P2
-
192
H. T. Ngoan and N. T. Nga
and
The following result is a direct consquence of the last theorem, theorem 2.3 and proposition 3.2.
Theorem 4.2. Assume that the conditions (C1) and (C2) hold. T h e n there exists locally a unique C2-solution to the Cuuchy problem (11)-(8).
Cauchy Problem f o r a Quadanear Weakly Hyperbolic
193
5. Application to the Classical Weakly Hyperbolic Monge-AmpGre Equation Applying Theorems 4.1 and 1.1 we obtain the following theorem on solvability of the Cauchy problem (1)-(4) for weakly hyperbolic Monge-Amphre equations.
Theorem 5.1. Suppose that besides the conditions ( 5 ) , ( 6 ) the following conditions also hold: 1) The coeficients A, B , C, E are C2-functions and do not depend o n z ; 2) A 2 0 ; 5’) For all cq E I it holds
where the derivatives of functions A,C,Xl,Xz are calculated at ( x : ( a l ) ,x;(w), Z O ( a l )p, : ( 4 , m a l ) ) ; 4 ) Each function A ( x ,z , p ) , C ( x ,z , p ) ,Xl(x,z , p ) , X 2 ( 2 , z , p ) is a first integral for the Eq. (1); i.e., due to the condition 1 ) above, each of them satisfies the following system XIT
&
A-
=o, =O.
dP2
Then there exists locally a smooth solution z ( x ) to the Cauchy problem (1)-(4). 6. Examples
Example 6.1. (Ref. 5) The coeffiients A , B , C, E are constants with A 2 0. It is easy to see that all the assumptions of Therem 5.1 are satisfied. From (26) it follows that B = 0. From (25) and (22) we have the following quasilinear system
194
H. T.Ngoan and N. T . Nga
aPl + ( 4 x 1-
AX2 - P2)-
aa1
with the following initial conditions
where
po(al)= ( F : ( a l ) , F;(a1)), and
+ CX,"(a1)+ A2X,O(a1) + + AX,O(ai).
P,"(a1)= P,"(a1)
f$(ai) = J'i(ai)
AiX,"(ai)
From the two last independent linear equations of (49) with initial conditions (50) it is easy to solve F l ( a 1 , a ~ &(a,, ), 0 2 ) . Then we substitute them into the first two equations of (49) to obtain independent linear equations with respect to X l ( a 1 , a g ) and X ~ ( a l , a 2 )At . the last, substituting & ( a l , a 2 ) , &(a1,a2),X l ( a 1 , a 2 ) , X 2 ( a 1 , a 2 ) , which have been found, into the third equation of (49) we can solve Z ( a l , a 2 ) . Then by the Theorem 1.1the local solution ~ ( xto ) the Cauchy problem (1)-(4) can be obtained from the functions X l ( a 1 ,ag),X2(a1,a2) and Z(a1,a2).
Example 6.2. Suppose that ~ ( yt,) is a solution of the Burger equation Vt
+ 2121y = 0 ,
which is C2-function in a neighbohood of the point (clP:(O) c#;(O), czX,0(0) + c l X Z ( 0 ) ) and satisfies the condition -v,(clP:(al) c2P,"(a1),c2X?(a1) Clx,O(al))(C2X,O(Ql) c l X ; ( a l ) ) +1 # 0, for all a1 E I and where c1, c2 are some given constants, cf c$ > 0. Then the Cauchy problem for the Monge-AmpBre equation
+
+
rt - s2 + 212(C1ZZ, - C 2 Z Z 2 , c2x1
+
+
ClZ2)
=0
Cauchy Problem for a Quasilinear Weakly Hyperbolic 195
+
with A = B = C = 0, E = -v2(c1p1 - c 2 p 2 , c2z1 c 1 z 2 ) , A = 4 V 2 ( C 1 p 1 c z p 2 , c 2 z i + cizz), XI = -A2 = ~ ( c l p i c 2 p 2 , czz1+ ~ 1 x 2 satisfies ) all conditions of Theorem 5.1.
Example 6.3. Suppose that ~ ( yt,) is a solution of the following equation Vt
+ f(V)Vy
= 0,
which is C2-function in a neighbourhood of the point (-c1@(0) - c 2 @ ( 0 ) , c l x m
+CZX,o(O))
and satisfies the condition -
f'(v(-clP:(al)
x
vy(-ClP:(al)
- C2P,"(C.l), C l X ? ( C . l )
+
CZX20("1))
- C2P,"(C.l),
+
clx;(Ql) C 2 X , o ( a l ) ) ( c l X : ( a l )
+
CZX%l))
+ 1 # 0,
for all a1 E I and where f ( v ) is a given C2-function of one variable, c1, c 2 are some given constants, c; c; > 0. Then the Cauchy problem for the following equation
+
f(~(-clZ,I
+ f2(+C1Z,,
- C2ZZ2, c1z1
+ .2.2))(.
- c2zz2, C l Z l +
+ t ) + (.t
- s2)
c2z2)) = 0
with A = f(v(-clz,, - c 2 z Z 2 , clzl +CZZZ)), C = f ( ~ ( - c l z , ~ -c2zz2, c ~ z ~ f ~ 2 2 2 ) ) ,B = 0, E = - f 2 ( ~ ( - ~ l ~-,c, ~ z , ~~, 1 ~ 1 + ~ 2 ~ A 2 ) ) 0, , X i = A2 = 0 satisfies all conditions of the Theorem 5.1.
References 1. G. Darboux, LeGons sur la the'orie ge'ne'rale des surfaces, tome 3 (GauthierVillars, Paris, 1894). 2. E. Goursat, LeGons sur l'inte'gration des e'quations aux de'rive'es partielles du second ordre, tome 1 (Hermann, Paris, 1896). 3. J. Hadamard, Le problbme de Cauchy et les e'quations aux de'rive'es partielles line'aires hyperboliques (Hermann, Paris, 1932). 4. B. L. Rodgestvenski, N. N. Yanenko, Quasilinear hyperbolic systems (Nauka, Moscow, 1978). 5. M. Tsuji, Bull. Sci. Math., 433 (1995). 6 . M. Tsuji and H. T. Ngoan, Integration of hyperbolic Monge-Ampdre equations, in Proceedings of the Fifth Vietnamese Mathematical Conference (Hanoi, 1997) pp. 205-212. 7. Ha Tien Ngoan, D. Kong and M. Tsuji, Ann. Scuola Norm. Sup. Pisa C1. Sci., 27,309 (1998). 8. M. Tsuji and N. D. Thai Son, Acta Math. Viet., 2 7 , 97 (2002).
196
H. T . Ngoan and N . T.Nga
H. T. Ngoan and N. T. Nga, O n the Cauchy problem f o r hyperbolic MongeAmpdre equations, in Proceedings of the Conference on Partial Digerential Equations and their Applications (Hanoi, December 27-29, 1999) pp. 77-91. 10. Ha Tien Ngoan and Nguyen Thi Nga, Actu Mathematica Vzetnamica, 29, 281 (2005). 11. Ha Tien Ngoan and Nguyen Thi Nga, Vietnam Journal of Mathematics, 34, 109 (2006).
9.
197
Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 197-227) @ 2007 World Scientific Publishing Co.
$11. SOME SINGULAR PERTURBATION PROBLEMS
RELATED TO THE NAVIER-STOKES EQUATIONS MAKRAM HAMOUDA
Laboratoire d 'Analyse Nurnh-ique, Universite' de Paris-Sud, Orsay, France ROGER TEMAMx T h e Institute for Scientific Computing and Applied Mathematics, Indiana University, Bloomington, IN, USA E-mail: [email protected] In this article, we consider the asymptotic analysis of the solutions of the Navier-Stokes problem, when the viscosity goes to zero; we consider the flow in a channel of R3,in the case where the boundary is non-characteristic. More precisely, a complete asymptotic expansion, a t all orders, is given in the linear case. For the full nonlinear Navier-Stokes solution, we give a convergence theorem up to order 1, thus improving and simplifying the results of Ref. 11.
'- EAUE + (U".V) at div u"
'
= 0,
u" = (o,o,
u~
+ v p " = f,
in R,,
in R,,
-u),
on
r,,
uE is periodic in the x and y directions with periods L1, La, \
uEl,,o = u o .
(1)
198
M . Hamouda and R. T e m a m
Throughout this article we assume that f and u g are given functions as regular as necessary in the channel R,, and that U is a given > 0 constant; a t the price of long technicalities, we can also consider the case where U is nonconstant, U > 0 everywhere. By setting uE = u"- (0, 0, -U), we obtain the following problem for u":
'
aVE
-_
at
'
EAU"- UR3v" + (v'.V) ue + Vp" = f,
div u" = 0,
in R,,
u'=o,
roo,
on
in R,,
(2)
uE is periodic in the x and y directions with periods L1, Lz, \
uEl*=O= uo.
Linearizing in u" we find:
I
U" = 0,
on
r,,
(3)
w" is periodic in the z and y directions with periods L1, L z , 0
ItZO
= uo.
In the linear case, the limit problem ( E = 0) corresponding t o (3) is the Euler problem those solutions may not verify the boundary conditions (3)s. More precisely it follows from this work that the limit solution, denoted by u o , is
Singular Perturbation Problems Related to the Navier-Stokes Equations
199
solution (for all time) of the following system:
1
-U
I~30
I
+
D ~ P 0p0 = j ,
div u0 = 0, = 0,
on
in R,,
in R,,
ro, (4)
0
u =O,
on
rh,
vo is periodic in the x and y, directions with periods L1, Lz,
where l?o = R2 x (0) and rh = R2 x { h } . Our aim in this article is to derive asymptotic expansions for u",when the viscosity E goes to zero, for either the linear case (3) or the nonlinear case (2). It is obvious that we can not expect a convergence result of u" to uo in the uniform or H 1 norm spaces since u" and uo do not have the same traces on r0.This question was addressed in Ref. 11 in which the authors gave a representation of the NS solution U" up to the boundary and proved convergence results in several Sobolev spaces. In this article, we continue this work and we first derive the complete asymptotic expansion of the linearized NS solution v" up to the boundary at all orders. For this purpose, we will use the classical tools of the singular perturbation theory based on the use of correctors; see Refs. 4 and 3, or Refs. 12 and 7 for a related point of view. A similar work is performed in the nonlinear case but we then restrict ourselves to orders 0 and 1 (and to limited time), although we believe that higher orders can be treated in the same way. This article is organized as follows. In the next section, we show how to choose and construct the correctors of orders zero and one for the linearized NS equations. The third section deals with the description of the correctors of order N >_ 2 and the fourth section is devoted to the convergence theorem which validates our (empirical) choice for the correctors (linear case). Finally, in the last section (Sec. 5), we treat the nonlinear problem at orders zero and one and give the adequate correctors by proving their validity; this section improves and simplifies the results of Ref. 11.
M . Hamouda and R. T e m a m
200
2. Correctors of Order 0 and 1 Our method to construct the corrector of order zero is based on the observation made in Ref. 11 that, unlike the characteristic case, the Prandtl equation for this problem is very simple, namely
{
-cD;O"
- UD3P = 0,
div 8" = 0,
in R,,
in SZ,, on ro,
(5)
8 " = -U 0 , 8" = 0 , on I'h.
However our method here is quite different than in Ref. 11;the correctors' representation that we give hereafter is simpler than in Ref. 11 and this simplification allows us to continue our analysis at higher orders. We intend to give an asymptotic expansion of u",p" of the following form:
-
To determine the asymptotic expansion above we are guided by the general techniques of boundary layer theory. Hence for the exterior expansion u" &j v j , p" C,"=,E? p l , we would have
c,"=, N
avo
--
at
UD~U+ O Vpo = f,
(7)
and, for j >_ 1:
The interior asymptotic expansion C,"=,& @+ is determined by the condition that @ > € = -d on roo, at all orders j and, for the equation we use the ansatz 0 3 E - ~ which leads to (5) in Ref. 11. Hence, formally
-
-&D$O0~"- UD30°7"= 0 ,
-,cD;(j1,"
- uD3el," =
I &
ae03& at
(9)
and at orders j 2 2
However this general scheme has to be adapted to the present situation because of the intricacies introduced by the incompressibility condition (and the global nature of this equation); hence (7)-(10) will only serve as a general guideline.
Singular Perturbation Problems Related to the Navier-Stokes Equations
201
We begin at order 0 by proposing a corrector solving (approximatively) this system
u ~ ~ e ~on , (0,~h),
--ED;eo$c= 0, div go>"= 0, in R,, = --you 0 , on r01 = 0, on rh,
(11)
dotE
eosc
where -yo (resp. yh) is the trace on z = 0 (respectively on z = h). Firstly, we derive the tangential component @+ of by solving the system composed of the tangential projection of (11)l and (11)3,41and then the normal component (z-direction) O;E will be given by the incompressibility condition (11)~. --O,E Thus, an approximation denoted eT of the tangential component O:+ up to an e.s.t. (exponentially small term) reads:
e0J,
Here and in the sequel an e.s.t. is a function those norm in all classical spaces H m , C' is exponentially small in E. Notice that $" still satisfies equations (11)l and ( 1 1 ) ~ and ~ has an exponentially small value at z = h:
i$!'E = O(e-uh/E), at z
= h.
(13)
Then, thanks to (11)2,we write h
0':
d i V T e E ( Z ly,
and we deduce easily an approximation of
We note that this approximate corrector
QE
c, t)d<, up to an e.s.t.,
e"'" = (@, 82&)verifies the in-
--O& compressibility condition (11)2 and the equation (11)l.However -yoen = O ( E )and y h e E = O(e-Uh/E),these boundary values being different than -o€ the desired ones, namely en' = 0 at z = 0 , h.
202
M . Hamouda and R. T e m a m -0,E
To correct the boundary conditions of 6 , we introduce an additional corrector denoted by p0+ = &(pol which is solution of the following system:
I
-Avo
+ V7ro = 0,
in R,, div po = 0, in R,, (PO, = 0, on rou rh, 1 -0,E 1 p: = - - p O , = -div,(yov:), U p: = o , on rh.
(15) on I'o,
Remark 2.1. The role of the corrector po>'is just to account for an O(E)error in the boundary value of 8" and this "error" in the boundary condition will be actually corrected at the next order. There are also some boundary value "errors" of order O(&-ae-uh/E); they will be corrected all at once later on (at the final order N ) . We now proceed to the first order. The function d appearing in (6), is required to satisfy theses equations and boundary conditions:
UD3v1 div v 1 = 0, {
+ Vpl = --avo + UDypo + Avo, at in R,,
ro,
v31 = 0,
on
v1 = O ,
on F h ,
,v1
in R,,
is periodic in x and y.
on
rourh.
As at order zero, we instead construct an approximation for 0'1' as follows: = e.s.t., with a tangential component @IE
+
Singular Perturbation Problems Related to the Navier-Stokes Equations
203
and, for the normal component,
As before satisfies the desired boundary value exactly at z = 0 and up -1 ,€ to an e.s.t. at z = h, whereas On satisfies the desired boundary value up to order O(e) at z = 0 and up to an e.s.t. at t = h. To correct the O(e) error we introduce an additional corrector pl+ = up1 verifying $IE
I
+
-Aql V d = 0, in a, div cpl = 0, in a, cp: = 0, on rourh, 1 -1,s = = -b;, on Fo, & 1 vn = 0, on I'h.
(20)
Finally let us summarize what we obtained at order 1. We set G1+ = ?)1 + # Y E pl>'. Thanks to (16), and (18)-(20), the functions G1+ and pl' = p 1 7r1 satisfy the following set of equations and boundary conditions
+
+
204
M . Hamouda and R. T e m a m
3. The Corrector of Order N ( N
2
2)
In this section, we will focus on the N t h corrector for all integers N 2 2. Starting with order N = 2, the equations for the vj and Oj?" are different as observed in (8) and (10). More precisely for N 2 2, we propose to define the tangential component 6;' of by O N i E
roUrh,
on
(22) where vN verifies for all N 2 1 the following form of the system (8):
' -dVN -
UD3vN
at -
div uN
-~
= 0,
at
+V p N + UD3pN-l + A v N - ' ,
in R,
(23)
in 52,
on ro, vN = o , on rh, uN is periodic in x and y.
vf = o ,
As for N = 0,1, the p N to be subsequently defined are introduced as correctors of the boundary conditions.
To solve system (22), we need a sequence of lemmas. 3.1. Preliminary results
The first two lemmas are elementary, they will be helpful for computing the correctors.
Lemma 3.1. We have, for any j E
N,
Lemma 3.2. A particular solution of the differential equation
Singular Perturbation Problems Related to the Navier-Stokes Equations
205
is given by
Both lemmas are proved by induction on j. Based on these lemmas, the next lemmas provide the explicit form of the corrector ON?&. We start by describing the tangential component and the normal component is then deduced from the incompressibility condition.
Lemma 3.3. A n approximation up to exponential order of the tangential solution of (22), is given as follows: component OFEof the corrector O N Y E ,
Proof. By induction on N . For N = 2, ( 2 7 ) is verified thanks to Lemma 3.2 and the explicit form of the first order corrector 8;" which was obtained in (18). The tangential component satisfies the system ( 2 2 ) those first equation can be rewritten as follows: O?+llE
Using Lemma 3.2, the resolution of this equation, taking into account the corresponding boundary condition as given by (22)3, yields: - N + ~ , E-
97-
-
+
--You,N+1 e - U ~ / ~ PS,
(30)
M . Hamouda and R. T e m a m
206
where P S denotes a particular solution of (29); based on Lemma 3.2, its expression reads:
d
p s = --(a;) at
c N
j=O
&J+1 ( j
.j+l
+ I)! U 2 N + 1 - j
---UZ/E-
By identification of (30)-(31) and by the general form of the corrector . 8, , a.e.,
-N+~,E
we deduce that
Finally, after some simple transformations of (31) and by identification we obtain (28). This concludes the proof of Lemma 3.3. 0 Thanks t o the incompressibility condition and t o Lemma 3.1, we have
Lemma 3.4. A n approximation of the normal component of the corrector reads:
elv+
where
Remark 3.1. For all N E N and 0 5 i 5 N , the quantities a?, b? are functions o f t , x and y, but they are independent of z and E . Thanks to the -N,E incompressibility condition (22)2, the corrector 8 still satisfies the same
Singular Perturbation Problems Related to the Navier-Stokes Equations
equation as
I
given by (22)l. More precisely we have - UD3gN'"=
div 3"' -N,E
8,
-NiE
07-
207
= 0, N
-N-~,E
-1E {
-
AS^-'^^},
on ( o , h ) ,
in 0, -N,E
8, = --You, 1 - o(&-Ne-Uh/E ) I
=
&
N
ubo, -NIE
0,
on ro, - O(&-N+le-Uh/E -
>,
on rh.
(33)
Finally, to recover the desired boundary conditions, we introduce an additional corrector, pN>e= &pN, rN+= & r N , defined by
+
-ApN V r N= 0 , in 0, div pN = 0, in 0, p,N =o, on rourh, N 1 -N,E pn = - - ~ ~ e , = - b f , on & p,N = o , on rh. Notice that pN is independent of
E.
gN,E
+
We now define GN+ = uN p N + and p" = p N satisfy the following equations and boundary conditions: +
(34)
ro,
+ 7rNiE
which
4. Convergence Result In this section, we will validate our choice for the correctors by proving a convergence result which concludes the study of the linear Navier-Stokes
208
M . Hamouda and R. T e m a m
problem defined in Sec. 1. We start by introducing the following expression:
k=O
(36)
k=O The correctors that we introduced contained exponentially small errors a t the boundary z = h. To correct all these "errors", we now introduce an additional corrector which will account for all the e.s.t. a t z = h. This corrector, denoted by FN?' , satisfies the following system of equations: + v ~ N ~ -E 0 , = 0, in 0,
-& A F N , "
div $1"
FNYE = 0,
on
in 0,
ro,
Note that by the classical results concerning the Stokes problem (see for instance"), all the Sobolev norms of FN>"are exponentially small, namely 0 ( & - 2 N e-Uh/E
).
We then define new quantities, which will allow us t o prove the convergence result and give a complete asymptotic expansion of the linearized NS solution II", namely
,
, i j j N , ~= w N , ~+ & N @ N , E PE
(38)
= p~ + E N ~ N V E .
Thanks t o (3), (23), to our iterative construction for the correctors as described by the system (22) (with N replaced by k), (34) and (37), the equation verified by the pair (GNIE, F") reads:
Singular Perturbation Problems Related t o the Navier-Stokes Equations
209
Using equations (3)2, (22)2, (23)2, (34)2 and (37)2, we deduce the incompressibility condition div G N i E= 0.
(40)
Moreover, our construction leads to the desired boundary conditions for GN+,namely the homogenous ones: from (3)3, (22)3, (23)3,4, (34)3--5 and (37)3-6, we conclude that = 0,
on
rou rh.
(41)
Also the periodicity condition is still satisfied for GN+:
GN3&is periodic in the x and y directions with periods
L1, La.
(42)
Finally, we state and prove the following theorem:
Theorem 4.1. For each N 1 1, the quantity GNvEdefined b y (38) tends t o zero when E -+ 0. More precisely, we have the estimates:
c c
IIGNIEIILm(O,T;LZ(n))I E N + 1 ,
ll~N’EllL2(0,T;H’(s2)) <
(43) (44)
EN+1/2,
where L’(C2) = (L2(R))3,H1(C2)= (H1(R))3, and C denotes a constant which depends o n the data (and N ) but not o n E . Since the (PN,& are exponentially small, the same results hold f o r wNJ. Proof. We multiply (39) by G N } & and integrate over R. Thanks to (40), (41) and (42) we see that the last two terms in the left-hand side of the resulting equation vanish. Also all boundary terms vanish and we find: I d 2 dt where --ll#YE
2 llLz(s2)
+EIIVW
-NE
2 llLz(s2)
=E
~ + ~ +I E? N
N
+
E
N
N
,
(45)
+ AvN + ATgN’& + UD3pN]. GNiEdR,
dPN
at
Now, we proceed to estimate the terms I:, I:, I:. Schwarz inequality, we easily obtain that I&N+lI?I
Using the Cauchy-
I C&2N+2 + ;IIVG”q;2(n).
(46)
210
M . Hamouda and R. Temam
Thanks to the explicit expression of our corrector Lemma 3.3 and Lemma 3.4, we easily see that
e N I E ,
which is given by
pTeN'EIILa(n) 5 cE1/2. Using equation (37) for FN>' and the classical results on the Stokes problem, we deduce that
I E ~ I 5~ CE-' I
e-Uh/E+ II~N%z(s2).
(47)
We recall here that in order to obtain an estimate for the term aFNJ/&, it suffices to differentiate the system (37) in time t , multiply by and integrate over 0. Finally, we deal with the most difficult term I?. For that purpose, we use again the explicit expression of the corrector for all N E N. Thus, we have
e"",
This yields
5 (Thanks to the Hardy inequality)
5 C&2N+2 + 4((VG""lI;z(n). &
(49)
Combining (46), (47), and (49), we arrive at the following energy inequality
Applying the Gronwall inequality to (50), we obtain (43) and (44). This concludes the proof of Theorem 4.1. 5. The Nonlinear Case This section is aimed at deriving a similar asymptotic expansion of the solutions of the full nonlinear Navier-Stokes equations; we believe that all orders can be handled in the same way, but because of the complexity of the calculations we restrict ourselves to order 1 thus recovering and improving
Singular Perturbation Problems Related to the Navier-Stokes Equations
21 1
the results of Ref. 11. Proceeding as in the linear case, we consider v E = u" - (0,0, -U) which is solution of the following system:
Ifi=o, I
EAV"- UD3v" + (v".V) IF + Vp" = f, in
vE=O,
'JI
in R,,
,a,
on ,,?I
is periodic in the x and y directions with periods
L1, L2.
As in Ref. 11 our analysis shows that the leading terms remain the same as in the linear case and consequently the zeroth order corrector that we propose here is the same as in Sec. 2. Of course we should treat the nonlinear term appropriately. For that reason, and besides its importance, the derivation of a convergence rcsult at ordcr zero will bc helpful for the choice of the corrector a t order one.
5.1. Corrector of order zero
Let
&"= v" - vo - p
- Epo -
p+
(52)
where, e"'",pa,p?" are respectively given by (12)-(14), (15), (37) with N = 0, and vo is now solution of the full nonlinear Euler equation
v30 = 0 ,
on
r0,
v0 = 0 ,
on
Fh,
vo is periodic in the x and y, directions with periods L1, L2
(53)
212
M . Hamouda and R. T e m a m
Remark 5.1. The existence of a smooth solution for the system (53) is proved in Ref. 8. Thus, we deduce the following result:
Theorem 5.1. There exists a time T, -0,'
1I.U" -
where
K
> 0 such that we have
IILm(O,T*;Lyn)) 5
V0 - f3
is a constant independent of
&,
E.
Remark 5.2. (1) The complete proof of Theorem 5.1 is similar to that of the corresponding theorem at order one (that is Theorem 5.2), so we skip it. However, it is helpful for the subsequent order to make explicit the form of the nonlinear term. Using (52) we see that (VE.0)
vo + ( u E . 0 ) wO," + (wO+.O) (21" - w y + + (P1'.O) 80'" + (v0.V) + ( 8 O " . 0 ) vo+ + & ( ( p O . 0 ) 80'" + & (eO'".V) (po + & (v0.0)PO+ (56) + & ( ( p O . 0 ) vo + E2 ( ( p O . 0 ) 'PO + (8O".V) p"+
V E = (uO.0)
P"
+ (vO.0) p"+ + ( p ' . V ) uo +
& ((pO.0) ( p E . 0 )
p++ ( p @ . V ) p++
80>' + & ( p . 0 )9 0 .
(2) In order to obtain (54)-(55), it is sufficient to obtain the same estimates for wo+. Indeed, using the equations for (po and p?'which are given respectively by (15) and (37) with N = 0, we have the following estimates:
I
~ ~ ~ ( p O l l L z ( n C, )
Ic
I I V ~ ~ ~ I I L & ~ -(2 ~N -) 1 / 2 e - U h '/ E
(57)
5.2. Corrector of order one
In this section, we are interested in the derivation of an asymptotic representation of v" up to order 1, that is t o find an approximation of 21" in R, and up t o the boundary as follows: vE
p'
N
vo + eo+ + & P O po +&PI.
+ &(d+ el+) + iz1>&,
Singular Perturbation Problems Related t o the Navier-Stokes Equations 213
First, we infer from (56) the equations satisfied by the mode w1 at order one. These are described by the following system:
' -"l -
UD3d
at -
+
~a(Po UD3po - ( w O . 0 ) cpo - ( ( P O . ~ ) WO
at
div w1
+ ( d . V ) + ( w O . 0 ) w1 + Vpl
= 0,
<
+ AwO,
in Cl,,
in R,,
(58) w31 = 0,
on
r0,
w1 = 0,
on
rh,
wl
is periodic in the x and y, directions with periods L1, L2,
Remark 5.3. The existence,uniqueness and regularity of the solutions of system (58) is proved in the Appendix.
5.2.1. Construction of the corrector 8'9' We now present the equations that we propose for by considering the next dominant terms in the nonlinear NS equation (51)i (of course without the source term and the pressure already taken into account). Hence, we obtain
1
4
E
4
-cp!D3e0,'E- -(8 ' .V) 8: &
E
,
in R,,
(59)
div 81iE= 0, in R,, 0I '&= - w 1 , on r O u r h .
Remark 5.4. Contrary to the linear case, it is important here t o emphasize that equation (59)1 is valid only for the tangential component of the corrector The equation satisfied by the normal component 82&is different and it will be derived later on. It will be slightly different than for the tangential corrector equation since the nonlinear term is not divergence free.
214
M . Hamouda and R. T e m a m
We rewrite equation (59)1 using the explicit form of
go'&:
Thus, thanks to Lemma 3.2, we have an approximation, up to an e.s.t., of the exact solution of (59)-(60) which is denoted, as in the linear case, by -1,E OT and given as follows:
Remark 5.5. 1. The approximation "3: of O:+ satisfies exactly the same equation (60) as 2. It is easy to observe, but important to mention, that the boundary values of '3 : satisfy @:YE.
where c > 0 is constant depending on the data but independent of (62) will be useful later on.
E
and z ;
Singular Perturbation Problems Related to the Navier-Stokes Equations
215
Using the incompressibility condition ( 5 9 ) ~and Lemma 3.1, we deduce an explicit approximation g.',; of OkE.Thus, we have
1 -1,E
I roe,
en'
-1e
15 c E ,
on
rol
= O(Ee-UhIE)I on
rh.
(64) Due to our choice of a simpler form for the corrector gl", we observe that, in ( 6 2 ) ~and (64)2,3, we need to introduce an additional corrector in order t o obtain the desired homogenous boundary conditions. For that purpose, we define (Cpl+, Z1iE)as the solutions of the following system:
I
AG1,&+vZ1"=0, in R, - 0, in R, 'p;>" = 0, on ro, -&
div
-
'p, -"; 9
1,E
-
'p;c
T h-o,& e , - yhi$'
- -&-l -
= == -&
-1,s
-1
= O(E), on -0,E
- rh~:E
o ( &e-Uh/E), -~ r0,
=
- O(e-uh/e),
on
on
rh,
(65)
rh.
By the properties of the Stokes operators, all the Sobolev norms of of order E.
FlyEare
216
M . Hamouda and R. T e m a m
5.2.2. Convergeme result: th,e main theorem
Finally, we state and prove the main result of this article which gives the asymptotic expansion at order one of vE:
Theorem 5.2. For v E solution of the Navier-Stokes problem (51), there exist a time T, > 0 and a constant K > 0 depending o n the data but not o n E , such that:
Proof. Let
Before writing the equations satisfied by w ' ~ ~we, start by expanding the nonlinear term:
(v'.V)
WE
+ (v0.V) vo+ + E [(v".a) (Po + ((P0.V)vo + ( V O . 0 ) v' + ( v l . 0 ) + (PE.0) 80" + & ((P0.V)8'"+ (80'&.V)210 + (vO.0) P'+ + E (eo'".v) ((PO + v') + & (e"".v)(VO + & v')+ + E [(vo + 8'' + ap0+ E v1 + ~8""+ E ( P ~ ) ~P1++ ).V] (68) + E [(vO + 80+ + &(PO + & + &P+ &cpl+).V]el,'+ +E + 21' + P ) . V ) e"?'+ & (8l'".O) (PO+ + E2 [(@.V) (Po + ((P0.V) v' + ( V l . 0 ) (Po + (7Jl.V) v' + =
(vE.0)
w1+
+ (W'?V)
(WE
-w'q
4
211
((@'I&
+
E
(cpl>".V)(?I0 + &
w1
+
& (PO).
1
Thanks to (51)1, (53)1, (9), (15)1,(58)~, (60)-(64)1 and (65)1, we deduce the main equation verified by wl,&for which the boundary conditions are derived from (51)3,4, (53)3,4, (15)3--5,(58)3,4, (62)-(64)2,3 and (65)3-6. Notice that both the incompressibility condition and the periodicity property are
Singular Perturbation Problems Related to the Navier-Stokes Equations
'
UD3W1,E + V [ p " - (PO + T O )
--&Awl,Eat
+(v".V) wllE + (wl?".V) (v" - ~ ' 2 " )=
-
&(PI
+
217
?+E)]+
-x$ 2
Jj>",
j=O
{
div w'," = 0,
wl,&= 0 ,
,
wljh
rowh,
on
is periodic in the
2
and y directions with periods L1, L2,
and 5 2 , E = -( E
1
-+ UD3(pl,E) + Av' +
&
+[(vO+8°'E+&(pO+&w1
+[
+ - ( ( p l ? E .V) (wO
at
((PO.0)
Po
+
((pO.0)
+ &P + &(p'q.v] v1
(p'+
+ (vl.V) (Po + ( w l . 0 )
Remark 5.6. The underlined terms in J o ? &and follows before estimating them later on:
+
E
v1 +&YO)
+ &2(p'+D381'"
1
v' .
(72)
J1>&need to be split as
(2L.V) 11 = (7L7.VT) 21 + u3 D3v. Now, multiplying equation (69)' by obtain
~
'
and integrating over 0, we 2
~
M. Hamouda and R. T e m a m
218
where,
Thanks to the incompressibility, boundary and periodicity conditions ((69)2,3,4and (51)2,3,4),we deduce that the following terms vanish:
k3i'
=-
k'" .I
v [ p c - (po
=
b
5
7
&
=
U D ~ W. ~wli"dR ~ ' = 0,
+ TO)
-
+
~(p'
? l " ) ]
. w l i E d R= 0,
(74)
L[(v'.V) w17"dR = 0.
Thanks to the Hardy inequality, we have
1
IJ6, ' l = I 51
s,
(wli".V) (v' - wl,") . wl.EdRl
n
[(w:yc.V,)
-0," 0
+
(wE - w1iE).w13E W ? ~ D ~ (-W w'," "
).~~~~]dRl+
-
w~ED3~'EW1>Edfi +I&
I II v, (vE - w l , & )+ D3(vE- wile - 0 ' ) + D3eEIIL-(n) 11w1'Ell~2(n)+
Icllw
l a 2 llLZ(n)
+ 1I;u 2
Finally, for the term
U
e
J 6 l E
wl,E
IlL-(O,h)IIYO~~IIL~,IIV
2 llLZ(n).
(75)
we impose the following condition: 0
- 1Iz2 e--uz/" &
2 -Uz/e
IIL~(O,~)~~YO~,~IL~~
4E
0
&
I ~ I l ~ n v , I I q ,I -.6
(76)
Since yo$ vanishes at t = 0, IIyov;ll~~,remains initially small and therefore there exists a time T, > 0 such that
Ilro~%&
5
Ue2
24' v t E [O,T*].
(77)
Consequently, we obtain
lJ6,&l 5 CIIW
l a 2
' IILyn)
E
+ ,IlVW
l a 2
IILz(n).
(78)
We now estimate the right hand side of (73). We start by the easier terms included in J2,&. First, by the properties of the solutions of the Stokes
219
Singular PeTtUTbatiOn Problems Related to the Navier-Stokes Equations
problem, the solutions of (65) satisfy: ,,-1,E
IIatIILz(n)
Ic E ,
IIV,-lqILz(n)
Ic E .
(79)
The same estimates are also valid for the tangential derivatives of Hence, thanks to the Cauchy-Schwarz and Holder inequalities, we have: ( p l l E .
1&2
J 2 q I /€E4 + c
(80)
IIW1q&q.
For J 1 ) E , we use the Hardy inequality * combined with the Cauchy-Schwarz and Holder inequalities and we conclude that:
Finally, since v: = v: = 0 on z = 0, we have:
(thanks t o the explicit expression of
5 c E5’2
p w l q L z ( n )
I c &4
&” given by (61) - (63))
+6 &
(82)
-IIvW’.E1l;z(n).
Hence, inserting all the results (74), (78), (SO), (81) and (82) in (73), we deduce the following new energy estimate: ~dtl l w ” E I I E z ( n )
2 + E I I VW l ’e IILZ(n) 5
I 6.E4+ c
IIW
l a 2
’
IILz(n),
(83)
to which we apply the Gronwall inequality to conclude the proof of Theorem 5.2. 0
*Hardy’s inequality states that
W
ll-llL~(n) z
<211Vwll~z(n)provided w ( z = 0) = 0.
220
M . Hamouda and R. Temam
Appendix
In this paragraph, we prove the existence and regularity of the solutions of the linearized Euler systems (4), (16) and (23). For the full nonlinear Euler system (53), we refer the reader to Ref. 8. We first want to apply the Hille-Yosida theorem to prove the existence and uniqueness of the solution of (4). Thus, we start by introducing the adequate function spaces
{
H = u E L2(s2); div u = 0, ullz=o = u11,=L,, u~),=o = W Z ~ , = L and ~,
w3 = 0
en z = 0, h
dU
{
D ( A ) = w E H ; z-d z E L2(n), u = 0 at z aU
3 p E D'(R) such that - U -
dZ
Then for u E D ( A ) , we set Au = -UD3v linear operator A : D ( A ) c H -+ H .
=h
1
,
and
+V pE H
1 .
+ V p , thus defining an unbounded
Remark 0.1. If s o p dR = 0 , then p is unique, otherwise it is unique up to an additive constant. Let us now recall the statement of the Hille-Yosida theorem (for more details see for example Refs. 1,2,5 and 13): Theorem 0.3. (Hille-Yosida Theorem). Let H be a Hilbert space and let B : D ( B ) 4 H be a linear unbounded operator, with domain D ( B ) c H . Assume the following:
( i ) D ( B ) is dense in H and B i s closed, (ii) B i s a positive operator, (iii) 3 po > 0 , such that B p o l i s onto.
+
T h e n ( - B ) is the infinitesimal generator of a semigroup of contractions { S ( t ) } t l o in H . More precisely, the solution u of the system:
satisfies the following properties: (Ho): If uo E H and f E L1(O,T ;H ) then u E C ( [ 0 ,TI;H ) , 'd T
> 0,
Singular Perturbation Problems Related to the Navier-Stokes Equations
221
and,
( H I ) : If vo E D ( B ) and f' E L 1 ( O , T ; H ) then v E C ( [ O , T ] ; Hn ) E L " ( [ O , T ] ; H ) ,'d T > 0. L"([O,T];D ( B ) ) ,and In order t o apply the Hille-Yosida theorem t o the system (4) with B it is sufficient to prove the following proposition:
= A,
Proposition 0.1. The operator A verifies
+
(i) A XI is onto Q' ' X > 0 , (ii) ( A X I ) - ' E B ( H , D ( A ) ) , 'd X # {Ulkl, k E Z2}, (iii) A is closed.
+
Proof. The proof of the property (i) is equivalent t o finding a solution for the following system:
1
+
+
-UD3u V p Xu = f , in R,, div u = 0, in R,, u3 = 0 , on ro, (84) u=O, on r h , u is periodic in the x and y, directions with periods L1, L z ,
where f is a given function of H . Now, we decompose the solution of (84) in the Fourier basis as follows: U =
c
't&(z)eklZ+k2Y,
k=(ki ,kz)EZ2
and let
k=( k i , k z ) EZ2
Hence, the system (84) is equivalent to the following system of equations:
222
M. Hamouda and R. T e m a m
We multiply (85)1 by i k l , (85)2 by ik2 and we differentiate (85)3 with respect to z , then we sum up the equations that we obtained and find: -lkI2pk
;+;
= divf = 0.
Since & ( h ) = 0 (from (85)3 and ( 8 5 ) ~ we ) have
where
(Yk
is a constant to be determined later on.
Remark 0.2. The existence of a k is guaranteed by (85)3. Indeed, we multiply (85)3 by (-6e-6') tion on (0, h);we deduce ak =
for all Ikl
and integrate the resulting equa-
6e-+'f3kdz
1 1 $ [e" k I- 6) z - e- (1 k 1-k 6) Ze21k 1h]dz '
(87)
# 0. If k = (0'0) then the pressure po reads h
pO(z) =
f30(z)dz.
From (87) we deduce the following estimate for a k :
(89)
where X is chosen such that X
6 {Ulkl, k E Z2}.
Lemma 0.1. We have p E L2(R). Proof. By definition of p we have
There exits a constant C such that
Singular Perturbation Problems Related to the Nauier-Stokes Equations 223 Hence, we conclude that
This ends the proof of Lemma 0.1.
0
Lemma 0.2. The solution u of (84) satisfies: u E L2(f2) and
Z&U
Proof. We multiply (85)3 by (-$e-"/') deduce the following expression of U3k:
E
L'(f2).
(94)
and integrate on ( 0 , h ) ;we
Thanks to (86) we have
(thanks to (89)) I
I I Hence u3 E L2(R). Now, since u1 and '112 satisfy similar equations ((85)i and ( 8 5 ) ~ it ) is sufficient t o prove for example that u1 E L2(R). For that purpose, we multiply (85)l by e-"/') and integrate on (0, h ) , and we obtain the expression Of 'Ulk:
(-6
Thus, we infer from (86) that
(98) where
224
M. Hamouda and R. Temam
Thanks to (89) and for llcl large, we deduce the existence of a positive constant C > 0 such that:
Combining (98) and (99), we obtain
We conclude then that u E L2(!2). Now, since we have the explicit expressions of u and p (given respectively by (97)-(95) and (86)), we prove exactly as below that z D 3 ~ 1 , z D g ~Ez L2(Q) and by the incompressibility condition (85)4, we deduce that
zD3u3
E
L2(Q).
This ends the proof of Lemma 0.2. Consequently, we have proved that for every f E H there exists u E D ( A ) such that ( A XI)(u) = f. This proves ( i ) . The property (ii) is obvious. To prove (iii),we consider a sequence (u,), c D ( A ) such that urn+ u and Aum + cp in H . Thanks to (ii)we have
+
u,
=
+
( A X I ) - l ( A + XI)um -+m++m ( A+ XI)-l(cp + Xu).
(101)
By the uniqueness of the limit, we conclude that
( A + XI)(u) = cp
+ Xu * u E D ( A ) and Au = cp.
(102)
This proves that A is closed and consequently ends the proof of the Proposition 0.1.
Theorem 0.4. Under the following hypotheses:
(H2) : v o ~ C ~ ( a , ) n D ( A ) and f ~ C ~ ( [ 0 , T ] x f i , ) n H , the solution ( v , p ) of the s y s t e m
(4) satisfies:
w,p E C,([O,T] x
L).
(103)
Remark 0.3. The hypothesis (H2) on f can be replaced by f C - ( [ O , T ] x Gm). Indeed, we decompose f as follows:
f
=PHf
E
+PH'f,
where PH is the Leray-Hopf projector on H . Thus, only the pressure p is changed if we replace f by P H f. We define a regular function 8 such that PH1f = V8, and then p is just replaced by p - 8.
Singular Perturbation Problems Related to the Navier-Stokes Equations
225
Let us state and prove the following Lemma in order to prove Theorem 0.4:
Lemma 0.3. For
T
E {z, y} and o E
{T,
t } , we have:
(2) I.@ belongs to the space Cm(2,)nH (respectiwelyCCY([0,T]x;2,)nH or C n D ( A ) ) then dQ,/da belongs to the same space, (22) I . @ E Cm([O,T] x1 , ) nL1(O,T;H ) then 8 @ / dtoo. ~
,(am)
Proof. First, we notice that the periodicity is conserved after differentiation with respect to x,y, z or t. To prove (i) it suffices to verify that: if Q, E Cm(fiz,) n H (resp. C"(2,) n D ( A ) ) then 33/80 E H (resp. D ( A ) ) . Hence, let Q, E C"(2,) n H ; by using the definition of H , we have easily 8@/80E H since we differentiate the boundary conditions only in the tangential direction on the boundaries z = 0 and z = h. Now, let Q, E C "(G,)nD(A). Then, in particular Q, E H and consequently aQ,/da E H . Moreover, since Q, = 0 on z = h, and we have = 0 on z = h. From the condition @ E D ( A ) ,we deduce the existence of a regular function p ( V p = ( I - P H ) ( U D ~ V such ) ) that -UD3v V p E H . This implies that -UD3(8@/80) V(8p/80)E H , and thus E D(A). The property (ii) is a consequence of (2). Indeed, if Q, E C"([O, T] x 2,) n L1(0, T ;H ) , then in particular Q, E H for almost all t E [0, TI. Thanks t o (2) in Lemma 0.3, we have 8@/80E H for almost all t E [0,TI. This concludes the proof of Lemma 0.3. 0
+ a@/&
+
Now, we are able to prove Theorem 0.4:
Proof. of Theorem 0.4 We recall that under the hypothesis ( H z ) ,the HilleYosida (Theorem 0.3) and more precisely ( H I )yields:
v E C([O, TI;L2(0)) and
dv
E Lm(O, T ;L 2 ( 0 ) ) , V T
> 0.
(104)
Thanks t o Lemma 0.3, the linearity of the system (4) and after replacing the source function f and the initial data vo by their tangential derivatives, we infer that amV
-E
a?-"
C([O,T];L~(O)),
v T E {z,y), v m E N.
226
M. Hamouda and R. T e m a m
On the other hand, since wo E D ( A ) and f is continuous a t t = 0, then we deduce that (dv/dt)(t = 0) E H . Now, if we differentiate the system (4) with respect to t, we obtain:
av - E C([O, TI;LZ(0)).
at
From the incompressibility condition (4)2,we deduce that D3213 = -01211 - 0 2 2 1 2 E
C([O, TI;L2(s2)).
(107)
Using (106)and (107),the third component of the equation (4)1 yields:
2 E C ( [ O , T ] ;P(s2)). az
(108)
Again, the invariance of the system (4)by differentiation with respect to gives: LtZa7-m
~ ( [ oTI; , L2(a))
E
In particular, we have
g,g
\J 7 E {z, y),
v m E N.
T
(109)
E C ( [ O , T ] ;L2(s2)).
Hence, equation (4)1 projected on the tangential directions z and y implies with (106)that: 03211, 03212 E
C([O,T]; L2(a))*
(111)
Gathering (105),(107)and (1ll), we conclude the following regularity for 21:
IJ
E C([O,T];H1(S1)).
(114
Now, we reiterate the procedure; from (112) and the invariance of (4) by differentiation in time t , we obtain
821 at
- E C([O,T];H1(S1)) i.e.,
w E C1([0,T];H1(0)).
(113)
The invariance of (4)by differentiation with respect to 3: and y implies that: 03213
= -01211 - 0 2 2 1 2 E C1([0,TI;Hl(s2)).
(114)
Then equation (4)l and (113) imply that
9 E C ( [ O , T ] ;Hl(s2)). dz
(115)
Singular Perturbation Problems Related t o the Navier-Stokes Equations 227 Thus,
and consequently,
w E C([O, TI;H2(C2)). Finally, t h e invariance by differentiation in time gives
w E C1([0,T];H2(C2)). This procedure allows us t o prove a higher regularity for w and more precisely (103). 0 This concludes the proof of Theorem 0.4.
Acknowledgements This work was partially supported by the National Science Foundation under t h e grant NSF-DMS-0305110, and by the Research Fund of Indiana University.
References 1. H. BrBzis, Ope'rateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert (North-Holland Publishing Co., Amesterdam, 1973). 2. N. Burq and P. GBrard, Contr6le optimal des e'quations aux de'rive'espartielles (Ecole Polytechnique, Palaiseau, France, 2003). 3. K. 0. F'riedrichs, The mathematical strucure of the boundary layer problem in Fluid Dynamics, eds., R. von Mises and K.O. F'riedrichs, Brown Univ., Providence, RI (reprinted by Springer-Verlag, New York, 1971). pp. 171-174. 4. J. L. Lions,Lectures Notes i n Math, 323 Springer-Verlag, New York, (1973). 5. J . L. Lions, Les Presses de l'universite' de Montre'al (Montreal, Que., 1965). 6. J. L. Lions, Selected work, Vol 1. (EDS Sciences, Paris, 2003). 7. R. E. O'Malley, Communications of the Mathematical Institute, Rijksuniver-
8. 9. 10. 11. 12. 13.
siteit Utrecht, 5. Rijksuniversiteit Utrecht (Mathematical Institute, Utrecht, 1977). M. Petcu, Advances in Differential Equations (to appear). L. Prandtl, Veber Fliissigkeiten bei sehr kleiner Reibung, in Verh. 111 Intern. Math. Kongr. Heidelberg, (Tuber, Leipzig, 1905) pp. 484-491. R. Temam, Navier-Stokes Equations (North-Holland Pub. Company, 1977), Reedition in the AMS-Chelsea Series, AMS (Providence, 2001). R. Temam and X. Wang,J. Differential Equations 179, 647 (2002). M. I. Vishik and L. A. Lyusternik,Uspekki Mat. Nauk, 12,3 (1957). K. Yosida, Functional analysis, (Springer-Verlag, Berlin, 6th edition, 1980).
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Chapter I11
GEOMETRIC ANALYSIS
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Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 231-253) @ 2007 World Scientific Publishing Co.
231
§12. MONOTONE INVARIANTS AND EMBEDDINGS OF STATISTICAL MANIFOLDS LE HONG VAN * Max-Planck-Institute f o r Mathein Sciences InselstraJe 22-26 0-04103Leipzig E-mail: [email protected] In this note we prove certain necessary and sufficient conditions for the existence of an embedding of statistical manifolds. In particular, we prove that any smooth (Cl resp.) statistical manifold can be embedded into the space of probability measures on a finite set. As a result, we get positive answers t o the Lauritzen question on a realization of smooth (C' resp.) statistical manifolds as statistical models. Keywords: Fisher metric, Chentsov-Amari connections, statistical manifolds, statistical models.
1. Introduction
A statistical model is a family A4 of probability measures on a measurable space 0. There are two natural geometrical structures on any statistical model equipped with a differentiable manifold structure. They are the Fisher tensor and the Chentsov-Amari tensor. The Fisher tensor was given by Fisher in 1925 as an information characterization of a statistical model. Rao13 proposed to consider this tensor as a Riemannian metric on the manifold of probability distributions. This Fisher metric has been systematically studied in Refs. 1,4,9 and o t h e r ~ect. , in the field of geometric aspects of statistics and information theory. Chentsov4 and Amari2 independently also discovered a natural structure on statistical models, namely a 1-parameter family of invariant connections, which includes the Levi-Civita connection of the Fisher metric. This family of invariant connections is defined by a 3-symmetric tensor T together with the Levi-Civita connection of the Fisher metric. *Mathematical Institute of ASCR, Zitna 25, 11567 Praha 1, E-mail: hvleOmath.cas.cz
232
L. H. Van
Motivated by the question how much we can describe a statistical model via their Fisher metric and Chentsov-Amari tensor T , in 1987 Lauritzen proposed to call a Riemannian manifold ( M ,g) with a 3-symmetric tensor T a statistical manifold. Since two 3-symmetric tensors T and k . T , k # 0, define the same family of Chentsov-Amari connections, we shall say that two statistical manifolds ( M ,g, T ) and ( M ,g, IcT) are conformal equivalent. A natural and important question in the mathematical statistics is to understand, if a given family M of probability distributions can be considered as a subfamily of another given one N . In the language of statistical manifolds, this question can be formulated as a problem of isostatistical embedding of a statistical manifold ( M ,g , T ) into another one ( N ,g', TI).Here we say that an immersion f : ( M ,g, T ) -+ ij, T ) is called isostatisticul, if f*(?j) = g and f*(T) =T. We shall see in Sec. 2 that the problem of the existence of an isostatistical embedding includes also the Lauritzen question in 1987, if any statistical manifold is a statistical model. It also concerns the following important problem posed by Amari in 1997, if any finite dimensional statistical model can be embedded into the space CapN of probability distributions of the sample space oNof N elementary events for some finite N . We shall construct a class of C" (and C') monotone invariants of statistical manifolds, which present obstructions to embedding of a given C k statistical manifold M into another one N". Here a C kstatistical manifold ( M ,g, T ) is a smooth differentiable manifold with C k sections g E S2T*M and T E S3T*M. These invariants measure certain relations between the metric tensor g and the 3-symmetric tensor T . In particular, using these invariants we show that no statistical manifold which is conformal equivalent to the space CapN can be embedded into the product of m copies of the normal Gaussian manifolds for any N > 3 and any finite m. In the Main Theorem (Sec. 5) we prove that any smooth (C' resp.) statistical manifold M" can be isostatistically embedded to a the space CapN for some N big enough. As a consequence we also get a new proof of Matumoto theorem on the existence of the contrast function for a statistical manifold (see subsection 2.6).
(z,
2. Statistical Models and Statistical Manifolds In this section we recall the definitions of the Fisher metric and the Chentsov-Amari connections on statistical models. We introduce the notion of a weak Fisher metric and a weak potential function. At the end
Monotone Invariants and Embeddings of Statistical Manifolds
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of the section we discuss the problem, if a given statistical manifold is a statistical model. Most of the facts in this section can be found in Ref. 1. Suppose that M is a statistical model - a family of probability measures on a space R. We assume throughout this note that M and R are differentiable manifolds, and R is equipped with a fixed Borel measure dw. We also write
where p ( z , w ) in LHS of 1 is a Borel measure in M and p ( z , w ) in the RHS of (1) is a non negative (density) function on M x 52 which satisfies
l p ( z , w ) d w = l'dz E M . The Fisher metric g F ( x ) is defined on M as follows. For any V,W E we put
g F ( V ,W
) X
=
/
R
(8vlnp(z, W ) ) ( a wlnp(z,W ) ) P ( Z ,w ) .
TxM
(3)
The function under integral in Eq. (3) is well defined, if
Denote by Cap(R) the space of all probability measures on R. Clearly we can consider the density function p ( z , w ) as a mapping M -+ Cup(R). Thus we shall call a function p ( z , w ) a probability potential of the metric g F , if p ( z , w ) satisfies (2), (3), (4). It is known1y4 that for a given Riemannian metric gF on a smooth manifold M there exist many probability potentials f(z, w ) for g F , even if we fix the space (a,dw). Some time it is useful to consider functions p ( z , w ) which satisfy (3) and (4) but not necessary (2). In this case, the Riemannian metric g F will be called weak Fisher metric, and the function p ( z , w ) will be called a weak probability potential of gF.
2.1. Example of a W e a k Fisher M e t r i c
The standard Euclidean metric go on the positive quadrant RT(zi > 0). It is straightforward to check that go admits a weak probability potential {pi(z)= iz:,i = Here R = RN - the sample space of N elementary events.
m.}
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2.2. The Fisher Metric on the Space ( C a p N ) + of all positive probability distributions on ilN (see also Refs. 1,496) By definition we have
Cap?
,pnr)lpi > 0 for i = I , & C p i
:={(PI,.-.
= 1).
We define the embedding map
f :Capy
-+ s N - l ( 2 ) ,
( p l , " ' , p N ) H ( 4 1 = 2 ~ , . ' ., q N = 2 m . It is easy to see that the Fisher metric in the new coordinates ( q i ) is the standard metric of constant positive curvature on the sphere SN-' ( 2 ) .
Divergence Potential (see 1 , l d )
2.3.
A function p on M x M with the following property p(x, y)
2 0 with equality iff x = y
(5)
is called a divergence function. A divergence function p is called a divergence potential for a metric g on M , if
g ( X ,Y ) z = Hf=(P)(il ( X ) ,21 ( Y ) ) .
(6)
where
T(,,,)(M, M ) = (T,M, 0) CB (0,T,M) = ( i l ( T , M ) ) CB ( i 2 ( T z M ) ) . An example of a divergence potential for a Fisher metric is the Jensen function J$"(x, y) of the entropy function H ( x ) on M , or a Kullback relative entropy function K ( z ,y) on M x M . 2.4. Chentsov-Amari connections Let p(x,w ) be a probability potential for a Riemannian metric g . We define a symmetric 3-tensor T on M as follows
T ( X , Y ,2)=
I
(axlnp(~,w))(dyInp(~,w))(~zln~(~,w))p(~,w). (7)
We denote by VF the Levi-Civita connection of the (weak) Fisher metric g F . We define
< V k Y ,2 >:=< VSY,2 > +t . T ( X ,Y,2).
(8)
Monotone Invariants and Embeddings of Statistical Manifolds
235
The connections V t are called the Chentsov-Amari connections.
Remark 2.1. (Refs. 1,8) Any divergence function p ( z , y ) on M x M defines a tensor T on M via the following formula
T ( X ,y, Z)Z = - & * ( z ) H e s s ( P ) ( i l ( X )i ,l ( Y ) ) ( Z , Z ) + & l ( z ) H e s s ( p ) ( i z(XI,iZ(Y))(Z,Z).
If g and T are defined by the same divergence function p ( x , y ) , we shall call p(z, y) a divergence potential f o r the statistical manifold ( M ,g , 7'). It is a known fact that the Kullback relative entropy function is a divergence potential for the associated statistical model. 2.5. Statistical Submanif olds
A submanifold N in a statistical manifold ( M ,g , T ) with the induced Riemannian metric g1N and induced tensor TIN is called statistical submanifold of ( M ,g , T ) . Clearly, if f(z,w ) is a (weak) probability potential for ( M ,g, T ) ,then its restriction to any submanifold N c M is a (weak) probability potential of the induced statistical structure. 2.6. Statistical Models and Statistical Manifolds
Since any probability function p ( x , w ) defines a map M + Cap(R), we shall say that a statistical manifold ( M ,g , T ) is a statistical model, if there probability potential p ( x ,w ) for g and T . By the remark in subsection 2.5, we get that a statistical submanifold of a statistical model is also a statistical model. Furthermore, if a statistical manifold (M, g , T ) is a statistical model, then it must admit a divergence potential. Hence the following theorem of Matsumoto is a consequence of our Main Theorem in Sec. 5 .
Theorem 2.1 (Ref. 8). For any statistical manifold ( M ,g , T ) there exists a divergence potential p for g and for T . 3. Embeddings of Linear Statistical Spaces
An Euclidean space ( R n , g o ) equipped with a 3 -symmetric tensor T will be called a linear statistical spaces. We observe that the equivalence class of linear statistical spaces coincides with the orbit space of 3-symmetric tensors T under the action of the orthogonal group O ( n ) .In this section we discuss certain invariants of these orbits and we show several necessary and sufficient conditions for the existence of embedding of one linear statistical
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space into another linear statistical space by studying these invariants. A class of our necessary conditions consists of monotone invariants A, i.e. we assign to any linear statistical space (R", go, T ) a number A(Rn, go,T ) such that, if (R", go, T ) is a statistical submanifold of (R",go, T ' ) ,then we have
A@", go, T ) 5 X(Rrn,go, T'). Since a tangent space of a statistical manifold is a linear statistical manifold, these invariants play important role in the problem of isostatistical immersion.
3.1. R a c e Type of a Symmetric 3-tensor Let us denote by R" the subspace in S3(Rn) consisting of the following 3-symmetric tensors T V ( z , y , z )=< v , z
>< y , z > + < v , y >< x , z > + < v , z >< x , y >,
where v E R".Using the standard representation theory (see e.g. Ref. 12) we have the decomposition
S3(R")
= R(37r1) @ R".
(9)
To compute the orthogonal projection of a 3-symmetric tensor T on the space R" in the decomposition (9) we can use the following Lemma. We denote by 7r2 the orthogonal projection form S3(Rn) to R".
Lemma 3.1. W e have
Here we identify the 1-form T r ( S ) with a vector in R" by using the Euclideun metric go. We omit the proof of Lemma 3.1 which is straightforward. In view of Lemma 3.1 we shall call any tensor T E R" of trace type. We note that dims3@") = C i + 2 C : + n =
n(n
+ 1)(n + 2) 6
+
Thus the dimension of the quotient S 3 ( R n ) / S O ( n )is at least C i C: n. A direct computation shows that the dimension of the orbit S O ( n ) ( [ C ~ = = l uisi vC: ~ ] )= dimSO(n), if nui # 0. Here {vi} is an orthonormal basis in R". Hence the dimension of S3(Rn)/O(n)= C:+C:+n. This dimension is exactly the number of all complete invariants of pairs
+
Monotone Invariants and Embeddings of Statistical Manifolds
237
consisting of a positive definite bilinear form g and a 3-symmetric tensor
T. Since the dimension of G k ( R n ) = k ( n - k), it follows that generically it is impossible to embed a linear statistical space (Rk, go, T) into a given statistical linear space (R",go, T), unless k ( n - k) 2 Cz Cz k. Clearly the dimension condition is not sufficient as the following proposition shows.
+ +
Proposition 3.1. A linear statistical space (Rk, go, T ) can be embedded into a linear statistical space (RN,gO,T"), if and only i f N 2 k and T i s also a trace type: T = T" with 1wI 5 1211. Proof. The necessary condition follows from the fact that the restriction of T" to Rk equals Tc, where V is the orthogonal projection of v to Rk. Conversely, if IwI 5 IvI we can find an orthogonal transformation, such that w equals the orthogonal projection of v on Rk. 3.2. Commasses as Monotone Invariants Since the metric g extends canonically on the space S3(Rn),we can define the absolute norm
Now we define comasses of a 3-symmetric tensor T as follows
M3(T):=
max Izl=l,Jyl=l,lzl=l
M2(T):=
max
T(z,y , z ) ,
T(z, y , y ) ,
lxl=1,lYl=1
M1(T) := maxT(z, z, z). (xl=1
Clearly we have 0 I M1(T)I M 2 ( T )I M 3 ( T )I IITII.
Proposition 3.2. The comasses are positive functions which vanish at T, i f and and only i f T equals zero. They are monotone invariants of T, since if T is a restriction of 3-symmetric tensor T o n R N , then
IITII 5 IITII, M i ( T )5 M i ( T ) ,V i = 1,2,3.
(11)
L. H. Van
238
Proof. To prove the first statement it suffices to show that M 1 vanishes at only T = 0. To see this we use the identity
+ + + + + + + + +z,z- Y +
- 1 2 T ( x ,Y,2 ) = T ( x Y 2 , ~Y z , 2 Y T ( x Y - Z,Z + Y - Z , T y - Z) T ( x - Y + T(-x + y + Z , -X + y + Z , -Z + y + Z ) - 2(T(s,5, + T(Y,Y,Y) + T ( z ,z , .I).
+
+
2)
Z,%
-y
+
2)
The second statement follows immediately from the definition. Now for a space (R", go,T ) and for 1 5 k
5 n we put
We can easily check that if T is a restriction of T to a subspace I%" then
c R",
Xk(T) 2 Xk(T) 2 0 for all k 5 m. Thus X k ( T )is a monotone invariant of linear statistical manifolds. These invariants are related by the following inequalities
M 1 ( T )= X,(T) 2 X,-l(T) ... 2 Xz(T) 2 X1(T) = 0. The last equality follows from the fact, that the function T ( z , x , z )is anti-symmetric on S"-l(lzl = 1) c R" and S"-' is connected. We observe that if T is of trace type, then Xn-l(T) = . . . = X1(T) = 0. We are going to give a lower bound of the monotone invariant X,-1 of a linear statistical space of certain type. The equality Xn-l(Rn,go, T ) 2 A means that no hyperplane with the norm M 1 strictly less than A can be embedded in (R"-', go, T ) .
Lemma 3.2.
(a) Let T = CZ1(N- ~ i ) ( x a )be ~ a 3-syrnrnetric tensor o n R" with n 2 4, 5 1/4. Then we have N 2 4 and
(b) Let T = N Cy=,( x i ) 3 ,and H be a hyperplane in R" which is orthogonal to ( k n , 1,1,.. . , I), and let n 2 5 , k 2 3. Then we have
N
X"-2(?H)
2 7- 1.
Monotone Invariants and Embeddings of Statistical Manifolds
239
&,
(c) Let x = ((1 - E ) , ... , &) E S"(1) c ItWn+', where n 2 4,k >_ n. W e denote by H the tangential plane TxSn, and by T o the following 3-symmetric tensor on IW"++~:
Then we have
Remark 3.1. The tensor T o in (??) defines on (R", go) a statistical structure with a weak probability potential { i = 1,n}.
ix:,
Proof of Lemma 3.2. The reader shall see that a proof of Lemma 3.2 can be done in the same scheme of the proof of Sublemma 5.4. Therefore we do not repeat this argument here. Remark 3.2. Lemma 3.2.a holds also for n = 3 but not for n = 2, Lemma 3.2.b holds also for n = 4, but not for n = 3, and Lemma 3.2.c holds also for n = 3 but not for n = 2. There are also several obvious monotone invariants of T .
A ~ ( T:= )
max
Ix)=ly~=lzl=1,===0
T ( x ,Y, 2)
is well-defined for n >_ 3.
A ~ ( T:=)
max
1x1=lyl=l,<x,y>=O
T ( x ,Y,Y),
is well-defined for n 2 2. We can check that ker A' = 72". On the other hand we have kerA2 c R(37r1). Thus A' and A2 are different invariants.
0
Lemma 3.3. Let 7r1 be the first component o f T in decomposition (9). Then llTll1 := Ilnl(T)II is a monotone invariant of T . Proof. Let Rk be a subspace of R". We denote by .rr;CnT the restriction of T to R k . Clearly
TE(T)= T , " ( T ~ + T )~ i E ( 7 ~ 2 7 ' ) .
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L. H. Van
We have noticed in Proposition 3.1 that the restriction of the trace form 7r2T to any subspace is also a trace form. Thus 7rc(7r2) is an element in Rkc S3(Rk).Hence we have ~1(
Since all the projections
IlrcTlll
=
4 Y ) = ~1 (r;El(TIT)).
7r1, 7 r c
decrease the norm
(13)
11. I I , we get
ll~l(4Y)ll = llm(7r:(~lT))ll 5 II7rl(T)II = IITII1.
Proposition 3.3. A statistical line (R,g",T) can be embedded into ( R N , g O , T 'if) , and only zf M1(T) 5 M1(T'). Proof. It suffices to show that we can embed ( R , g O , Tinto ) (RN,gO,T'), if we have M1(T) 5 M1(T'). We note that T ' ( w , w , w ) defines an antisymmetric function on the sphere SN-l(IwI = 1) c I R N . Thus there is a point w E SN-l such that T'(w, v , w) = M1(T). Clearly the line w @ IR defines the required embedding. Let us consider the embedding problem for 2-dimensional linear statistical spaces. It is easy to see that S"(IW2) = R2 @
R2.
Thus the quotient S3(R2)/SO(2)equals (R2@R2)/S1.Geometrically there are several ways t o see this. In the first way we denote components of T E S3(R2)via Tiii1Tii2,Ti22,T222.
Lemma 3.4. There exists an oriented orthonormal basic in R2 such that TI11 = M 1 ( T ) > O,T112 = 0 for all non-vanishing T . These numbers (T111,T122, T222) are called canonical coordinates of T. Two tensors T and T' are equivalent, i f and only i f they have the same canonical coordinates. Proof. We choose an oriented orthonormal basis (w1 , 212) by taking as w1 a point on S1((x(= l), where the function T(x, x,x) reaches the maximum. The first variation formula shows that in this case T112 = 0. This shows the existence of the canonical coordinates. Clearly, if two tensors have the same canonical coordinates, then they are equivalent. Next, if two tensors T and T' are equivalent, then their norms M1 are the same. We need to take care the case, when there are several points x a t which T ( x , x , x ) reaches the maximum. In any case, they have the same first coordinates. Next we note that
< T r ( T ) Tr(T) , >= (T111+ 7'122)~
+ 7-2222,
Monotone Invariants and Embeddings of Statistical Manifolds
241
Thus if two tensors are equivalent and have the same first coordinates, they must have the same third coordinate T122, and this third coordinate is uniquely defined up to sign. The condition on the orientation tells us that the sign must be This proves the second statement. 0
+.
Proposition 3.4. W e can always embed the 2-dimensional statistical space (R2,go, 0) into any linear statistical space (R", go, T ) , i f n 2 7. Proof. It suffices to prove for n = 7. We denote by O ( T ) the set of of all unit vectors v E S6 such that T ( v ,v,v) = 0. Clearly O ( T )is a set of dimension 5 in S6.Since T is anti-symmetric, there exists a connected component O o ( T )of O ( T )which is invariant under the anti-symmetry involution. Now we consider the following function f on O O ( T )For . each v E O o ( T )we denote by A" the bilinear symmetric 2-form on the space TzOO(T) considered as a subspace in R": A"(Y, 2 ) = T ( v ,Y, 2 ) . Then we define f(v) equal to det(A"). Since O ( T )has dimension 5, the function f(v) is anti-symmetric on O o ( T ) .Hence the set O;(T) of all E O o ( T ) with f (v) = 0 has dimension 4 and it contains a connected component which is also invariant under the anti-symmetric involution. For the simplicity we denote this connected component also by Og(T).Now we consider the following two possible cases. Case 1. We assume that there is a point E Og(T)such that the nullity of A" is at least 2. Then there are two linear independent vectors y , z E T, such that the restriction of A" on the plane R 2 ( y ,2) vanishes. Since the set O o ( T )is connected and anti-symmetric and of co dimension 1 in S"-', the plane R ( y , z ) has a non-empty intersection with O o ( T )at a point w. Then the restriction of T on the plane R2(v,w) is vanished, because
T ( v ,21, v) = T(w, w,w)= 0 T ( v ,w,w)
= 0 (since
A"(w,w) = O),
~ ( vv, , w)= o (since w E T,o'(T)).
Case 2. We assume that the nullity of A" on @ ( T ) is constantly 1. Using the anti-symmetric property of A" we conclude that the restriction of A" to the plane R4(v) which is orthogonal to the kernel of A" has index
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L. H. Van
constantly 2. Thus there exists a vector z which is orthogonal t o the kernel A" such that A " ( z , z ) = 0. Clearly the restriction of A" t o the plane R2(y,z ) vanishes. Now we can repeat the argument in the case 1 t o get a vector w such that the restriction of T to R2(v, w) vanishes. y of
Theorem 3.1. a) A n y statistical space (Rn,gO,T) can be embedded in the statistical space (Rn("+l),go,T' = 211Tll where xi are the canonical Euclidean coordinates o n IW"("+l). b) The trivial space (Rn,go,O) can be embedded into (R2", go, C;zl(dxi)3) f o r all n.
xzY)x:),
Proof. a) We prove by induction. The statement for n = 1 follows from Proposition 3.3. Suppose that the statement is valid for all n 5 k. Lemma 3.5. Suppose that T E S3(Rk+'). Then there are orthonormal coordinates 21, ' ' . , xk such that k+l i=l
l
Proof of Lemma 3.5. We choose v1 as the unit vector in Sk c Rk+', on which the function T ( v ,w,w) reaches the maximum on the unit sphere S k . The first variation formula shows that T(v1,wl,w ) = 0 for all w which is orthogonal t o w 1 . We denote by Rk the orthogonal complement t o R .w1. Now we consider a bilinear symmetric form A on Wk defined as follows A ( X , Y)
= S ( V l , X , Y).
There is an orthonormal basis on Rk, where we can write A ( z , y ) = six!. Clearly in this orthonormal basis we can write T in the form in (14).
Continuation of the proof of Theorem 3.1.a We shall show explicitly that that any statistical space (IW2,g0,T = U ~ I C ~ ( X can ~ ) ~be ) embedded in ( R ~go, , ~ f = ~ ( y i )if~o )I , la21 I 1/2. We put 1 1 1 1 L(v1) := *(-, -, --, --) (15)
2 2
L(v2):=
2
2
(/F, -/Tl /?-,
-/?).
(16)
Monotone Invariants and Embeddings of Statistical Manifolds
243
Here we take the sign+ in (15), if a2 > 0, and we take the sign-, if a2 < 0. Clearly, L defines the required embedding R2 -+ R4. This together with Proposition 3.3 and the induction assumption complete the proof of Theorem 3.1.a. Proof of Theorem 3.1.b. We decompose the embedding f : (R",go,O) to (Pn, go, z;E1(xi)3) as follows
f ( x l , . . . ,xn) = ( f l ( x ~ ) , *, f*n*( x n ) ) where
f i
(R, ( d ~ z ) ~into , ~ (R2, ) (dx2i-1)
embeds
the
line
+ ( d x 2 i ) 2 ,(dx2i-1) + ( d ~ ' ~ )Clearly, ~). f
is the required embedding.
0
4. Monotone Invariants and Obstructions to Embeddings of Statistical Manifolds Let K ( M ,e ) denote the category of statistical manifolds A4 with morphisms being embeddings. Functors of this category are called monotone invariants of statistical manifolds. Clearly any monotone invariant is an invariant of statistical manifolds. 4.1. Examples There are many monotone invariants which arise from our analysis in Sec. 3. a) Trace type of a statistical manifold. A statistical manifold ( M ,g, T ) will be called of trace type, if for all x E M the form T ( z )is of trace type (see 3.1) It follows from Proposition 3.1 that any statistical submanifold of a statistical manifold of trace type is also of trace type. Thus the trace type is a monotone invariant. In particular we cannot embed the statistical space C a p N and the normal Gaussian space into any statistical space of trace type. On the other hand, unlike the linear case, we cannot embed a statistical manifold of trace type into another one of trace type, even if the norm condition is satisfied. For example, if the trace form is closed (or exact), then the trace form of its submanifolds is also closed (resp. exact). Hence within a class of statistical manifolds of trace type we get a new monotone invariants which can be expressed via the closedness and the cohomology class of the corresponding trace form. b) Decomposability of a statistical manifold. We note that the class of 3-symmetric tensors of trace form is a subclass of all decomposable tensors T 3 which are a symmetric product of 1-forms and symmetric 2-forms. Any
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L. H. Van
statistical submanifold of a statistical manifold with a decomposable tensor
T has also the (induced) decomposable tensor. Thus the decomposability is also a monotone invariant. The Gaussian normal 2-dimensional manifold is an example of decomposable type but not of trace type. c) Rank and comass. We define for any statistical manifold ( M , g , T ) the following number
runk(T) = suprunk(T(z))
M 1 ( T ) o= SUP M 1 ( T ( z ) ) . XEM
Clearly these four numbers are monotone invariants of statistical manifolds. We recall that the normal Gaussian statistical manifold is the two dimensional statistical model which is upper half of the plane R2(p,a ) with the potential
here z E R.
Proposition 4.1. Any statistical manifold which i s conformal equivalent t o the space C u p N cannot be embedded into the direct product of m copies of the normal Gaussian statistical manifold for any N 2 3 and finite m. Proof. It is easy to check that M 1 ( C u p N )= 00. Thus any statistical manifold which is conformal equivalent to C a p N has also the infinite invariant M1. On the other hand, we compute easily that the norm M1 of the Gaussian normal manifold, as well as the norm M1 of a direct product of its finite copies, is finite. Namely the norm M 1 ( p ,a ) is f i for all ( p ,a). [7 4.2. Diameters of Statistical Manifolds For a positive number p
> 0 and
d,(M, g, T ) := sup{l
a statistical manifold ( M ,9, T ) we set
E
Rf U 00 I 3 an embedding of
([O, 11, d z 2 ,P ( d 4 3 ) to ( M ,9, T ) . )
Monotone Invariants and Embeddings of Statistical Manifolds
245
We shall call d,(M, g , T ) the diameter with weight p of ( M ,g , T ) .Clearly d, are monotone invariants for all p. To estimate the diameter with weight p of a given statistical manifold ( M ,g, T ) we can proceed as follows. For each point x E M we denote by D p ( x )the set of all unit tangential vector u E TxM such that T ( v ,v,u ) = p. We denote by D % ( x )the connected components of D L ( x ) . We say that a unite vector u in T x M is p-characteristic with weight c ( x ) , if there exists i such that we have
c(x) = min
< u , w >> 0.
W€D;(X)
We shall say that a point x E M is p-regular, if there is an open neighborhood U,(x)c M such that D,(U,) = U, x D p ( x ) . It is easy to see that the set of all pregular points is open and dense in M for any given p.
Proposition 4.2. The diameter d , of (M", g , T ) is infinite, zf m 2 3 and there exists a number E > 0 such that one of the following 2 conditions holds: a)There exists a ( p &)-regular point x E M such that the convex hull Cou(D$+, ( x ) )of one of connected components DS+,(x) contains the origin point 0 E TxM" as it interior point. b) ( M " , g , T ) has a complete Riemannian submanifold ( N , g ) such that there exists a smooth section x H (D,+,(x) n T N ) ouer N .
+
Proof. The statement under the first condition a) is based on the fundamental Lemma of the convex integration technique of Gromov. Namely Gromov proved that (2.4.1.A, Ref. 5), if the convex hull of some path connected subset A0 c Iwq contains a small neighborhood of the origin, then there exists a map f : S1+ Iwq whose derivative sends S1 into Ao. 0 Lemma 4.1. Under the condition in Proposition 4.2.a there exists a small neighborhood U s ( x ) in M and an embedded oriented curue S1 c U ~ ( xsuch ) that for all point s ( t ) E S1 we have M 1 ( T s ( t ) S 12) p ( ~ / 2 ) .
+
Proof of Lemma 4.1. We denote by E x p the exponential map TxM" + M" and by D E x p the differential of this exponential map restricted to S m - 1 x TxM" c T ( T x M " ) . Here Sm-l is the unit sphere in TxM". The space TxM" is a linear statistical space, so we denote by M : the induced norm-function on S"-l x TxM" as follows:
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L. H. Van
Since DExp is a continuous function, whose restriction to S"-' x 0 is the identity, there exists a ball B(O,6)with center in 0 E T,M such that
Ml(DExp(1)) - M:(1)) <~ / 4
(17)
for all 1 E S"-' x B(6) c T(T,M"). We can assume that b is so small such that DExp is a homeomorphism on S"-l x B(0,S). Now we apply the above mentioned Gromov Lemma (2.4.1.A, Ref. 5) t o get a oriented curve S1(t)in the linear space T,M such that
for all t. Next we observe that for all a > 0 the curve a . S1(t)has the same norm as S ' ( t ) , i.e.
M:(q(a.sll(t))= M ; ( q ( s l ) ( t ) = ) p+~. Thus we can assume that our curve S1(t), which satisfies (18), lies in the ball B ( 0 , b ) . By our choice of S (see (17)), we get from (18) p
+3
ZE
5
I M1(Exp(S1(t))) Ip + ZE,
(19)
for all t. This curve E x p ( S 1 ( t ) )is an immersed curve. To get an embedded curve we perturb the immersed curve such that the condition of Lemma 4.1 is satisfied. This is possible, since rn 2 3. Now let us t o continue the proof of Proposition 4.2.a. We denote by S1( t ) the embedded curve in Lemma 4.1. Next by choosing a tubular neighborhood of S1(t) we can get a (small, thin) oriented embedded solid torus T 3 ( t ,s, r ) = S1( t )x S1( s ) x [0, R] in M" such that our embedded curve is exactly the mean curve S ' ( t ) x (0) x (0) on the solid torus. We can choose this torus T 3 so thin, such that for all s, t , T we have
M1(T;(t,s)) 2 p +
&
4.
(20)
Using (20) we choose a smooth unit vector field V ( t , s ) on the torus T 3 ( t ,s, r ) which is tangential to each torus T:(t, s ) such that T(V,V,V )= p. The integral curve of this vector field is either a circle or an curve of infinite length. If there exists an integral curve of infinite length, then this curve is our desired curve for the Proposition 4.2.a. Assume now that all the integral curves are circles. Then there exist an embedding S1(t) x [0, p] x [0, p] such that for all ( s ,T ) E [0, p] x [0,p] the circle S1(t)x {s} x { T } is an integral curve of V . Now we perturb V in a neighborhood [0, a] x [0, p] x [0, a] with a very small a such that the perturbed unit vector field V' satisfies
Monotone Invariants and Embeddings of Statistical Manifolds
247
T(V',V', V ' ) = p and the integral curve of vector field V' is not any more periodic. This completes the proof of the first part in Proposition 4.2. Using the same argument we can prove the second part b) of Proposition 4.2. First we get the existence of an embedded curve S ' ( t ) of arbitrary length on M such that M ' ( T s l ( t ) )2 p ( 1 / 4 ) ~Now . we consider a torus tubular neighborhood of this curve in M and apply the same argument in the first part, namely we get on each torus T2(t,s) an integral curve whose unit tangential vector V = (a/at)S' (t;s, r ) satisfies the condition:
+
T(V,V,V ) = p. If there exists an infinite integral curve, then we are done. If not, that means all integral curve are circles, then we apply the perturbation method in the proof of the first part and get our desired curve.
5. Existence of Isostatistical Embeddings into CupN
Main Theorem-Any smooth (C' resp.) statistical manifold ( M " , g , T ) can be immersed into the statistical manifold (Cup?, g F , T A V Cf)o r some finite number N . Hence any statistical manifold i s a statistical model. We first deduce our Main Theorem for compact statistical manifolds ( M " , g , T ) from Theorem 5.1 and Theorem 5.4. Theorem 5.1. Let (M", g , T ) be a compact smooth (C' resp.) statistical manifold. T h e n there exist numbers N E N+ and A 2 0 as well as a smooth (C' resp.) embedding f : (M", g, T ) -+ (RN,go, A.To) such that f* (go) = g and f * ( A .TO)= T . Our proof of Theorem 5.1 uses the Nash embedding theorem, the Gromov embedding theorem and an algebraic trick. The existence of monotone invariants prevents us extend Theorem 5.1 for non-compact case (in contrast to the Riemannian case.)
Theorem 5.2 (The Nash embedding theoremloill). A n y smooth (C' resp.) -Riemannian manifold ( M " , g ) can be isometrically embedded into (RN, go) f o r some N depending o n M n . We denote by TOthe "standard" 3-tensor on Rn: n
248
L. H. Van
Theorem 5.3 (The Gromov immersion theorem5). Suppose that M" i s given with a smooth (C' resp.) symmetric 3-form T . T h e n there exists a n embedding f : M" --f WN1(") with N l ( m ) = 3(n (;+')) such that TO) = T .
+ (;+') +
Proof of Theorem 5.1. First we shall take an immersion (M ", g , T)+ (RN1(m),go, To)such that
fl
:
f?(To)= T. The existence of f 1 follows from the Gromov immersion theorem. Then we choose a positive number A-' such that 9 - A-l(f?(go))= 91
is a Riemannian metric on M , i.e. g1 is a positive symmetric bi-linear form. Such a number A exists, since M is compact. Now we shall choose an isometric immersion fi : (M", 91) --t (I@"go). The existence of f2 follows from the Nash isometric immersion theorem. Lemma 5.1. There i s a linear isometric embedding Lm+l : Rm+l such that Ln+1(To)= 0.
4
Proof. We put
L m + l ( x l , . . . ,xm+1) = XI),... f"+'(xrn+l)) 7
where fi embeds the line (R, (dxi)', 0) (It2,(dz2$-l) + ( d x 2 i ) 2 ,(dx2i-1 ) ( d ~ ~ ~ ) ~ ) :
+
into
Clearly, Lm+l is the required embedding. Completion of the proof of Theorem 5.1. Finally we take an embedding f3
: M"
--t
W("+1)("+2)+"
@
W2"+2
as follows. f3(2)
= A - l . f i ( x )@
0f2).
Since f 2 is an embedding, f3 is the required embedding map for Theorem 5.1.
Theorem 5.4. Suppose that C i s a compact subset in Cap?. T h e n any bounded domain D in a linear statistical manifold (Rn,go, A . TO)can be realized as a n embedded statistical submanifold of Cap? \ C .
Monotone Invariants and Embeddings of Statistical Manifolds
249
We denote by Cap:(X) the statistical manifold which is obtained by the restriction of the statistical structure from (R", go, Ef=l (xi)-l(d~i)~) to the positive quadrant S:(X) of the sphere of radius X with center a t the origin of R4.
Proof of Theorem 5.4. We put -
A := m a x { 4 6 , 4 f i . A}.
Let U be an open neighborhood of (A,(2z)-',(2z)-',(2z)-')E S 3 ( 2 / f i ) . Here X is the positive number such that
+
nX2 3 n ( 2 2 ) - 2 = 4.
(21)
We denote by U+ the intersection U n C a p : ( 2 / f i ) . We now choose U so small such that the product U x,times U C S4n-' lies in the complement S4n-1\ C. Since V+ is a statistical submanifold of 4 (R:, go, Ci=l ~ i ' ( d x i ) ~the ) , direct product U+ x n t i m e s U+ is a statis4n x ; l ( d ~ i ) ~ Hence ). U+ X , t i m e s U+ is a tical submanifold of go, statistical submanifold of (Cap4", g F , T A P C ) . We denote by U ( A , r ) the ball of radius r at the point (A, (22)-', (2z)-l, ( 2 z ) - l ) E S 3 ( 2 / J 5 i ) . First we prove
(RP,
Lemma 5.2. For given positive numbers R > 0 and A 2 0 there exists a positive number r such that the bounded domain [0,R] X , times [0,R] c (Rn,go, A * To) can be realized as a n immersed statistical submanifold of u + ( A , r ) Xntimes u + ( A , r ) C (Capyn,gF,TA-C). Proof of Lemma 5.2. It suffices to show that there is a statistical immersion f : ([O, R ] , d x 2 , A .d x 3 ) -+ U + ( A , r ) . On U + ( A , r ) we consider the distribution D ( p ) which is defined by
D,(p) := {w
E T,U+(A,r) : IvI = l,T(w,w,v) = p }
for any given p > 0. Clearly the existence of an immersion f : ([0,R], d x 2 , Ad x 3 ) + U+ ( A ,r ) is equivalent to the existence of an integral curve with the length R of the distribution D ( A ) . The existence of the desired curve is a consequence of the following Lemma
Lemma 5.3. There exists an embedded torus T 2 in U + ( A ,r ) which i s provided with a unit vector field V o n T 2 such that T(V,V,V ) = A. Proof. Let us denote xo := (A, ( 2 2 ) - ' , ( 2 z ) - ' , ( 2 2 ) - ' ) E S 3 ( 2 / f i ) with X satisfying ( 2 1 ) . We shall need the following
250
L. H. Van
Lemma 5.4 (Sublemma). Let H be any %dimensional subspace in T,,U+(A, r ) . Then there exists a unit vector w E H such that T ( w ,w , w ) 2 2A.
Proof of Sublemma 5.4. The subspace H can be defined by two linear equations:
< w , z o >= 0 , +
<w,h>=O. Here w is a vector in H c R4 and iis a unit vector in R4 which is not co-linear with 30 and which is orthogonal t o H . Without loss of generality we can assume that +
h = (0 = h l , h2, h3, h4) and
h: = 1. i
Case 1. Suppose that not all the coordinates hi of sign, so we assume that hl = 0, h2 5 0 , h3 > 0. We put
iare of
the same
w := ( W l , w2 = (1 - &2)k3,w3 = (1 - &2)k2,0= w4).
(24)
The equation (23) for w is obviously satisfied. Now we choose w l , ~2 from the following equations which are equivalent to (22) and the normalization of w:
+
. w1 (1 - E 2 ) . (2A)-1 w:: = (2&2- E ; ) . 0
*
(Ic2
+ k3) = 0 ,
From (25) we get
w1=
(1- E2)(k2 A . 2.4
+ k3)
Substituting this into (26),we get
+
(k2 k3I2 ( (A. 2 4 2
+
1)E;
- (2
+ k3)2 + (-----) k2 + k3 + 2(k2 ( A . 2A)2 A . 2A )&2
2
= 0.
We shall take one of ( 2 possible) solutions ~2 of (28) which is
Monotone Invariants and Embeddings of Statistical Manifolds 0
251
From (29) we get
5
82
I -,Y 16
+
since 0 < k2 Ic3 I 2 (this follows from (24)), and X 2 2n-lI2 (this follows from (21)), so A . 2 2 2 16. Since
T,,(w, w,w)= X-lw;
+ (2Z)(w,3 + w,")
T ( w , w ,W ) 2 8 f i . A ( 1 -
-) . -> 2A.
we get 5 16
1 2Jz-
So in this case 1 Sublenima 5.4 holds. Case 2. Now we shall assume that h l = 0 and h2 2 h3 2 h4 set
+
a > 0 such that a2(a2 2) = 1.
> 0. We
(31)
The Eq. (30) ensures that < w, h >= 0. In order that w is a unite vector and w E H the numbers w1,8 2 must satisfy the following equations - Awl
+ (1
-
&2)(a- 2)" = 0, 2x
w: = (282 - &;). 0
(32)
(33)
From (32) we get (1 - &z)(a- 2)" A271
w1=
Now substituting (34) into (33) we get a solution ~2
= (1
+ 2B2)
-
J(1
(34) EZ
+ 2B2)' - B2(1 + B 2 ) ,
where
B= Since 0
(a-2)" A271
1 X2ii'
<-
< ( a - 2)" < 1 we have E Z < 1/(X221) and ~1 < l/(X2Z). Hence
T(w, w ,w)= X-'W?
+ (2Z)u3(1 - E Z ) ~+(a 3~ ) > 2A.
0
252
L. H. Van
Completion of the proof of Lemma 5.9. First we choose a small embedded torus T 2 in U+( A ,r ) such that for all x E T 2 we have 3 { T ( v , w, w)} 2 TA. (T,TZ)
max
vE S’
(35)
This is possible thank t o Sublemma 5.4. Since T(w,w,w) = -T(v, w,w) and T 2 = R2/Z2 is parallelizable, (35) implies that we can find a smooth vector field V on T 2 sastifying the condition in Lemma 5.3. Completion of the proof of Theorem 5.4. For a given A in Theorem 5.5 we let A’ := A + & for some small positive E and we apply Lemma 5.2 t o ( R ,A’) which is in fact t o apply Lemma 5.3. We can show that the existence of an isostatistical immersion f’ : ([0, R],d x 2 , A’ . dx3) --f U+(A’,r ) implies the existence of an isostatistical embedding f : ([0,R ] ,d x 2 , A . dx3)toU+(A’,r ) by using the same argument in the proof of Proposition 4.4.a. Proof of Main Theorem f o r the non compact case. We can deal with this case by using the compact decomposition of M” as Nash did for the isometric embedding for the smooth case [lo]. Namely we cover M” by disk neighborhoods N,?’,j = 1,m 1 in the following way. For each j let
+
Cj := UiNf.
(36)
Then we require that the union in (36) is a disjoint union, i.e. N: nN l = 8, if i # k. We also require that each N j overlaps only a finite number of other Nf.Now we “compactify” N j via an surjective smooth mapping : qbi : N! -+ S:, where S: is a sphere of the same dimension m. The map 4: can be extended to the whole M”, since it maps the boundary of N i into the north point of the sphere. On the other hand, this map 4: is injective in a large (enough) sub domain c Nf. We can furthermore use the unity partition function to define a C1 statistical structure on each 5’: such that is the given statistical structure on M . In the (sum of) pull back via other words we can consider the C1 statistical structure on M” as induced from (infinitely many) spheres 5’; via the smooth mapping 4:. Now let for each j , 1 5 j 5 m 1, we put
&
+
s j :=
uas;.
Using Theorem 5.1 we can find an isostatistical embedding
inductively, since each 5’: is compact.
Monotone Invariants and Embeddings of Statistical Manifolds 253 Now let us consider the map
as a composition of the m a p 4: : M -+ Finally the product mapping 10 = I1
x
5': and the m a p
* * *
x
$j.
L + l
is the desired isostatistical embedding, since the statistical structure on SN(")-l is induced from the (non-linear) statistical structure on EXN("). We can see this easily by noticing t h a t both symmetric forms go and TS are decomposable w.r.t. the embedding i : Rn + Rn+' for any n, 1 > 0, i.e. go(Rn) = i*(go)(Rn+'), Ts(RY)= i*(Ts)(R";t').
Acknowledgement
I am thankful t o Jiirgen Jost and Nihat Ay for their introduction t o t h e field of information geometry and helpful discussions. References 1. S. Amari and H. Nagaoka, Methods of Information Geometry, Trans. of Math. Monograph (20QO). 2. S. Amari, Contemporary Math. 203, 81 (1997). 3. N. Ay, Ann. of Prob. 30, 416 (2002). 4. N. N. Chentsov, Statistical decision rules and optimal inference, (AMS. 1982), (originally published in Russian, Nauka, Moscow, 1972). 5. M. Gromov, Partial Differential Relations, (Springer-Verlag, 1986). 6. J. Jost, Information geometry, (lecture at MPI MIS 2005). 7. S. Lauritzen, Statistical manifolds, in Differential Geometry in Statistical Inference, IMS Lecture Notes, Monograph Serie lo., (Inst. of Math. Stat. Hayward, California, 1987). 8. T. Matsumoto, Hiroshima Math. J. 23, 327 (1993). 9. E. A. Morozova and N. N. Chentsov, Markou invariant geometry on manifolds of states (in Russian), (Itogi Nauki i Techniki 36, 1990) pp. 69-102. 10. J. Nash, Ann. Math. 60, 383 (1954). 11. J . Nash, Ann. of Math., 63, 20 (1956). 12. E. B. Vinberg, A. L. Onishchik, Seminars on Lie Groups and Algebraic Groups, (in Russian) (Nauka, Moscow, 1988). 13. C. R. Rao, Bull. of the Calcutta Math. SOC. 37, 89 (1945).
14. C. R Rao, Differential metrics in probability spaces, in Differential Geometry in Statistical Inference, IMS Lecture Notes, Mongraph Serie lo., (Inst. of Math. Stat. Hayward, California, 1987).
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Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 255-268) @ 2007 World Scientific Publishing Co.
255
$13. GRADED CECH COHOMOLOGY IN NONCOMMUTATIVE GEOMETRY *t DO NGOC DIEP Institute of Mathematics, Vietnam Academy of Science and Technology, 18 Hoang QUOCViet Road, Cau Giay Dist., 10307 Hanoi, Vietnam E-mail: [email protected]
The Zz-graded Cech cohomology theory is considered in the framework of noncommutative geometry over complex number field and in particular the homotopy invariance and Morita invariance are proven. In some special case we deduce an isomorphism between this noncommutative theory and the classical Zz-graded Cech cohomology theory. Keywords: Cech cohomology of a sheaf, cyclic theory Mathematics Subject Classification 2000 14F20, 19K35, 46L80, 46M20
1. Introduction
Let us fix in this paper the field of complex numbers as the ground field for algebras, modules, etc. The general Cech cohomology presheaf 'l?q for an arbitrary presheaf X on a category C, k Q ( X G) , = 'l?q(G)(X)was introduced in Ref. 7. The idea of Cech cohomology for noncommutative geometry was appeared in Refs.2 and 6. In this paper we use this idea to define the corresponding (periodic) Z2-graded Cech theory. We prove the homotopy invariance and Morita invariance of Cech cohomology in the framework of noncommutative geometry. Our main result is based on a detailed analysis of the structure of C*-algebras. A crucial observation is the fact that for C*-algebras the category of *-representations *The text is presented a t the Second International Conference on Abstract and Applied Analysis, Quinhon, Vietnam, 4-9 June 2005. The author thanks the organizers and especially Professor DSc. Nguyen Minh Chuong for the invitation t o give this lecture. +The work was supported in part by Vietnam National Project of Research in Fundamental Sciences and The Abdus Salam ICTP, UNESCO.
256
D. N. Diep
defines exactly the C*-algebra itself, by the well-known Gelfand-NaimarkSegal Theorem. From this we can deduce the Morita invariance and homotopy invariance of the Cech cohomology, the same properties of periodic cyclic homology of the C*-algebra. Since our result is valid not only for C*-algebras, we work in the general context of a noncommutative algebra over complex numbers. The paper is organized as follows. Taking the Cech cohomology in place of the de Rham theory in the periodic cyclic theory of A. Connes, in Sec. 3 we define the Zz-graded Cech cohomology theory. Then in Sec. 4 we prove two important properties of the theory as the homotopy invariance and Morita invariance. In the last Sec. 5 we deduce also some kind of ConnesHochschild-Kostant-Rosenberg theorem. This lets us see a clear relation with the classical case of commutative algebras and ordinary Cech cohomology theory. Notations: We follow the notations from Refs. 7 and 4 2. Preparation: Differential Systems and the Cyclic Theory
2.1. Differential Systems, Following Kashiwara From M. Kashiwara Ph. D. dissertation:l Linear polynomial differential systems N
R
j=1
p=l
correspond to the V-modules
M
=VN/VN'P
JV= VR/VR'Q
In general if
€
2
3
is a system of partial differential operators from the vector bundle € to 3 and set € = Homo(€, 0), then
V 8 & E { differential operators from € to 0} and we get D-module homomorphism
Graded Cech Cohomology in Noncommutative Geometry
257
Let M be the cokernel, then HomD(27 @ &,0 ) = Homo(&", 0 ) = & and the ?>-module M represents E quence is exact
0
P
, 3, e. i. the following short 5e-
--- - P
E
HomD(M,O)
F
and HomD(&,0 ) appears as the sheaf of solution of P : & In the same way for a given &
and
0
c-
P
M
3
Q
Q , such that Q o P
27@€
P
278.P
.+
F.
=0
Q
27@G
is exact, then the cohomology of the resolution
E-F-Q is isomorphic to ExtD(M, 0 ) .Therefore, E x t & ( M ,0 ) represents the obstruction for an element f E F satisfying the compatibility condition Qf = 0 t o be written as f = Pg for some g E E . Thus the computation of E x t b ( M , 0 ) become an essential problem. Let us recall the first Spencer sequence. Let M be a 27-module with a filtration M = U k M k and we define the D-homomomorphic derivative 6 : 27 80 A P B @ o M
+
(-l)i+jp@ ([vi,vj]A v 1
+ ?>@o A p-16'
80 M
. A Gi A ... A c j
. A vp) 18 u.
I
Theorem 2.1 (D. Quillens). Assume that the diflerential system M is endowed with a good fifiltration M = U k M k . Then locally for suficiently large lc the first Spencer sequence associated to this filtration is exact.
D. N . Diep
258
One of the famous applications of these ideas is the CauchyKovalevskaya Theorem: Let us consider an analytic space X , Y an analytic subspace with defining Ox-ideal Z.For any Ox-modules F . set
H b l ( 7 ) = l ~ E x t ~ x ( O x / l v +7l), . v
There is a canonical homomorphism H b l ( F )-+ H $ ( F ) . In the case where X and Y are complex manifolds, and f : Y 4 X an analytic map, define the sheaf of holomorphic functions onf Y x X , we have
V f = Vy+x = H"[ y l (Q(0'") y x x ) as the sheaf of differential operators of finite order from Y to X. In local coordinate xi of X and yj of Y , the sections of Vy,x
may be written in form
c
P(Y,&)=
aa(Y)a,",
bllm where we use multiindex notation. For any p(x) E OX, we can define ( P P ) ( Y )=
c
a4Y>[~,aP(X)lz=rp(d.
0
This define the product action
vy,x
x
f-bx
-+
Oy.
It is not hard to prove that
vy,x
E!
Oy
Xf-10,
f-lDx.
Define
f * M = Vy-x @f-lvxf - l M
= Oy
@f-10,
f-lM,
then f * O x = Oy,
f*Vx = v y + x .
Theorem 2.2 (Cauchy problem). Let f : Y + X be a smooth holomophic map. For any coherent Vx-modules M and N, the map v : f-'RHomV,(M,N)
4
RHomv,(f*M, f * N )
is an isomorphism. Moreover, for any k, f-l
Extg,(M,N)
-+ Ext&,(f*M,N)
is also an isomorphism. I n the particular case where Y is an submanifold of the complex manifold X , let T * X be the cotangent vector bundle of X
Graded Cech Cohomology in Noncommutative Geometry
259
and P * X = ( T * X \ X ) / C x the cotangent projective bundle of X , TGX the kernel of the canonical projection
T*X xx Y
-+
T * Y , P$X
= (T;X
\ Y ) / CC P*X.
The V = , V x - m o d u l e is non-characteristic if c h a r ( M ) n PGX = 8. If M i s a coherent V)x-module and the submanafold Y i s non-characteristic, then M y = Oy @ox M = V ~ + @vx X M is a coherent V y - m o d u l e and
HomD,(M, O X ) ~ + Y H o m ~ , ( M y OY) , i s an isomorphism. In the next we construct almost the same construction of homological algebra of C*-algebras and projective finitely generated C*-modules.
2.2. Fields of C*-algebras and Sheafification One of the best results of J. M. G. Fell we need is the following proposition. Proposition 2.1. Every C*-algebra of type I can be presented as the C*algebra r ( X ,E ) of cross-sections o n the continuous field of elementary C*algebras &, o n its dual X =
a
It is easy to see that using the Fourier-Gelfand transforms a
H &;
& ( T )= ~
we can define for each open subset U
(a),
c a a closed two-sided ideal
0 kerr.
I ( U ) :=
T€A\U
Proposition 2.2. Let A be a C*-algebra of type I, A its dual object. There i s a bijection between the set of open subsets in A and the set of closed %sided ideals in A . Proposition 2.3. There are naturally localized algebras
A ( U ) = l%A/In(U) n
with respect to the I ( U ) - a d i c topology.
260
D. N . Diep
3. Z2-graded c e c h Cohomology 3.1. Grothendieclc Topos The main purpose of this section is to formulate and define the functor of (periodic cyclic) Zz-graded cech cohomology. The well-known periodic cyclic homology is based on the cyclic homology theory of A. Connes, which is an algebraic framework of the Zz-graded de Rham cohomology theories. In the algebraic context, it was defined by J. Cuntz and D. Quillen in terms of X-complexes and it has become a new chapter of noncommutative algebraic geometry. The most general Gronthendieck algebraic geometry is purely based in terms of categories. In the generic case this turns out to the algebraic version of the Cech cohomology in place of de Rham cohomology theories. We follows the work of Orlov4 in particular to formulate the theory. The main references are Refs. 7 and 4. Many of our results could be obtained in the fields of other characteristics, but we restrict ourselves to the complex case. Let us denote by C a fixed category and Set the category of sets. Any contravariant functor X from C to Set is called a presheaf of sets and the category of all presheaves of sets on C is denoted by c^. The category C can be considered as a subcategory of c^, consisting of representable functors h = hR : C 4 Set. If R is an object of C, then there is a natural isomorphism Honie(hR,X) = X ( R ) . For any object X E C the category over X is the category of pairs ( R ,a), where R is an object of C and E X ( R ) ,and is denoted by C I X . Recall that a sieve in the category C is a full subcategory V C such that any object of C for which there exists a morphism from it to some object in V is contained in Obj(V). A sieve on R is nothing more than a subpresheaf of R in the category c^ A Grothendieclc topology 7 on a category C is defined by giving for each object R in C a set J ( R ) of the so called covering sieves satisfying the following axioms: (Tl) For any object R the maximal sieve C/R is in J ( R ) . (T2) If T E J ( R ) and f : S 4 R a morphism in C, then the induced sieve
f*(T):= (U 01Slfa E T } is in J ( S ) . (T3) If T E J ( R ) is a covering sieve and U is a sieve on R such that f*(U) E J ( S ) for all f : S -+ R in T , then U E J ( R ) .
Graded Cech Cohomology in Noncommutative Geometry
261
A Grothendieck site @ = ( C , I ) is a category C and equipped with a Grothendieck topology 7 . It is reasonable to remind the Jacobson topology on the set of all representations of a group or Zariski topology on algebraic varieties. For the categories with fiber product, a Grothendieck topology can be given by a Grothendieck pretopology which is defined by giving for each object R in C a family Cov(R)of morphism to R such that -+ R } a Ein ~ Cov(R)and S -+ R a morphism of C, the fiber product family R, X R S -+ S is also in Cov(R). (P2) If { R , -+ R } a Eis~in Cov(R) and {Roo -+ R,}P,~J,is in Cov(R,) for each a E I , then the total family { R , 4 R } Y E ~ , E J , I is in Cow(R). (P3) The trivial family { i d R : R -+ R } is in Cow(R).
(Pl) For any family { R ,
Any Grothendieck pretopology P on C generates a Grothendieck topology 7 such that a sieve is covering in 7 if and only if it contains some covering family in P . The topos on the category of functors from a category C to another one
D is defined by the usual rule. 3.2. The Standard Cosimplicial Complex of a Continuous
Functor We define in this subsection the &-graded Cech cohomology theory. Let us recall the definition of the Cech cohomology with coefficients in a sheaf M . Let U = (Ui -+ X ) i E i be a covering sieve of X in the category C. Suppose that the cover has the property that all fiber product and pushout diagrams exist, (see Ref. 7, Exp. 4). Denote A. the associated standard simplicia1 complex:
-
Let us consider again a ringed cite (category) ( C , A ) ,C the topos of presheaves on C, E : C c^ the canonical functor associating to each object X E C the functor h x , presented by X . Define H Q ( h x M , ) = HQ(X,M), (see Ref. 7, Exp IV, 2.3.1),as derived functor of the projective limit functor:
H q ( S ,M ) := Rq 1Lm M J s c/s
262
D. N . Diep
For any A-module M , denote C'(U, M ) := HomA(d., M ) :
One defines Hq(U, M ) := Hq(C'(U, M ) ) . If R is a covering sieve generalized by the family U then
H q ( U ,M ) E H q ( R ,M ) and the functor W ( U ,.) commutes with restriction of scalars (see Ref. 7, Exp. IV, Proposition 2.3.4). One defines, (see Ref. 7, Exp. IV, 2.4)
'FIO(M)(X):= HO(X, M ) = M ( X ) ,
W ( M ) ( X ):= H q ( X , M ) . For an arbitrary presheaf G of A-module G , the groups H q ( R , G ) are called the Cech cohomology with respect to the covering sieve R, with coefficients in G. For a sheaf M of A-modules over C , the group
H Q ( UM , ) := W ( U ,'FIO(M)) is defined as the Cech cohomology group of the sheaf M with respect to he the cover U . One has also
'Fio(G)(X)= l@ G ( R ) R-X
and therefore
H q ( G ) ( X )= l@ H q ( R , G ) R-X
which is called the presheaf of Cech cohomology Define
P ( X , G ) := 'Rq((G)(X), one has also
P ( X ,G ) = l@Hq(U, G). U
For a sheaf M of A-modules, one has
kqx,M ) = kyx,'FI"M)),
3iP(M) = 3iQ(3i0(M)).
The groups H q ( X , M ) are called the Cech cohomology groups of the sheaf
M.
Graded Cech Cohomology in Noncommutative Geometry
263
3.3. The P e r i o d i c Cyclic Bicomplex
Lemma 3.1 (The action of & + I ) . There i s a natural action of the cyclic group Zk+l on the Cech cohomology cochain complex C'(U, M ) associated with a covering U . Proof. The action of the cyclic group Zk+l is defined a cyclic permutation of indices of U's, i.e.
(XM)(Ui, x
*
*
a
x Ui,) := M(Ui, x Ui, x . . . x U i k p 1 ) .
It is not hard also to see that for a covering sieve U = {U, is a natural isomorphism
-i
X} there
M ( U . , ) @ . . . @ M ( U i ,.)
M(Uio x . . . x Ui,)
Therefore the Cech cohomology complex becomes the cyclic complex for lim + M (U). U
Corollary 3.1 (Hochschild differentials and Cyclic operations). The well-known Hochschild difjerentials b' and b and Connes cyclic operators A, N = 1 X . . . A k , s are well-defined on Zz-graded Cech cocycles
+ +
+
Definition 3.1 (Periodic bicomplex). Let ( C , A ) be a ringed U-cite, C the topos of sheaves, M a sheaf of A-modules. Then the bicomplex
is well-defined and is called the (periodic) Cech bicomplex. Definition 3.2 (The total complex and Zs-graded Cech cohomology). The associated total complex of which is defined as Tot C(U,M ) * :=
Ci?j
i+j=f(mod 2 )
264
D. N . Diep
where
Ciij
:= (il
n
M(Ui, x . . . x U i k ) and f = ev (even) or od (odd).
,..., i k ) E l k
The cohomology of this total complex is called the Z2-graded Cech cohomology of M and denoted by Z z H * ( U , M ) and Z 2 H ( X , M ) := lim-; Z2H(U, M ) . It can be also realized as the cohomology of the total complex related with the process of passing through direct limits, i.e. the direct limit bi-complex
For a C*-algebra A, we define it Zz-graded Cech cohomology Z a H ( A ) = Z z H ( d ) as the Z2-graded Cech cohomology of the category of *representations of A. Remark 3.1. In the first periodic bi-complex without direct limits, all the horizontal lines are acylic, but it is in general not the case for the second periodic bi-complex with direct limits. 4. Homotopy Invariance and Morita Invariance
We prove in this section two main properties of the (periodic cyclic) Z2graded Cech cohomology theory: homotopy invariance and Morita invariance, which make the theory easier to compute and being a generalized homology theory.
Definition 4.1 (Chain Homotopy of functors). Let us consider two functors F,G : C -+ 2). Denote the corresponding chain functors between complexes by {F,}, {G,}, where F,, G, : C, -+ D , for complexes 0
-1 -1 -1 co
F~ , G ~
0
__t
Do
c1
F~ , G ~
D1
c 2
...
F~, G ~
__t
D2
...
We say that F and G are chain homotopic i f there exist augmentation
Graded Cech Cohomology in Noncommutative Geometry
functors sn : Cn
-+
265
Dn-l such that for all n
Fn - Gn = S,
0
&-I+
an
o Sn+1,
Lemma 4.1. Two functors F,G : A -+ B are homotopic if and only if for any covering sieve U = (Ui + X ) i E ~there , exists a chain homotopy of chain complexes c
i,jEIxI
iEI
c
and c
Remark 4.1. In the case of smooth manifolds the chain complex homotopy is realized by integration of the so called Cartan homotopy formula for the Lie derivative
Lc = z ( E )
0
d
+d
0
z(()
between de Rham complexes.
Lemma 4.2. Let A and B be C*-algebras, and let A (resp. B) be the category of *-modules . Then the categories A and B are homotopic oneto-another i f and only i f the two algebras A and B are homotopic. Proof. Because of the Gelfand-Naimark-Segal theorem, the C*-algebras are exactly defined by the category of *-representations, the category of *-representations of B 18 C[O, 11 is isomorphic to the category of *representations of B . 0 Lemma 4.3. Let A be a (?*-algebra and A the category of *-representations (i.e. A-modules) of A, then
Z & ( d ) 2 HP*(A). Proof. Let us consider affine covering sieve Ui dual object of A.
-+
Theorem 4.1 (Homotopy Invariance). Let be a homotopy of algebras, then
pt :
X =A
=
SpecA, the 0
Z z H * ( A )2 Z z H * ( B ) .
A
-+
B,t
E
I
=
[0,1]
266
D. N. Diep
Proof. Let A (resp. B) be the category of A-modules (resp., B-modules). Step 1. Change the homotopy by a piecewise-linear homotopy in the space of functors from the category A to the category B. Step 2. A piecewise-linear homotopy gives rise to a chain complex homotopy. Step 3. Two chain complex homotopical functors induces the same isomorphism of Cech cohomology groups. It is an easy consequence from the results of homological algebra: If F, and G, are chain complex homotopic than the induced morphisms satisfies
F* n - G:
=
o S:
+ s:+~ o a;.
The second summand is a zero morphism on cohomology and the first summand is a boundary. The sum on the right is therefore a zero morphism.
Lemma 4.4. Let us denote by Mat,(@) the algebra of all square n x nmatrices with complex entries. Then we have a natural isomorphism ZZH*(Mat,(@))
E ZzH*(@).
Proof. Every complex matrix can be homotopic to a unitary one. Then, every unitary matrix can be by conjugation reduced to a diagonal matrix of complex numbers of module 1. Every elementary block (in this case, diagonal element) [eie]is homotopic to identity [l]by the classical homotopy [ei8t IO
case]
cos 0 - sin 0 -I=[:;]. [sine
Lemma 4.5 (Adjoint functors). There is a natural equivalence of functors Hom and @: Hom(R 8 M,(@), M ) 2 Hom(R, Hom(M,(@), M ) ) .
Proof. This isomorphism of functors is a particular case of the general adjointness between Hom and @ in homological algebra.
Lemma 4.6. There is a natural isomorphism of derived functors Rq Hom(M,(@), M )
Rq Horn(@,M ) .
Proof. It is an easy exercise from homological algebra.
Graded Cech Cohomology an Noncommutative Geometry
267
Theorem 4.2 (Morita Invariance).
Z 2 H * ( A@I Mat,(C))
%
Z2H*(A).
Proof. Let us remark that A A 8 Matn(C) is a fiber bundle. Now apply the Grothendieck's Leray-Serre spectral sequence for this fibration. Following the previous lemmas 4.5 and 4.6, there is a natural isomorphism of functors .--)
RpHom(R 8 Mn(@),R q ( M )
RPHom(R, Rq Hom(Mn(@),M ) ) ,
which are the E2 term of a Leray-Serre spectral sequence converging to the Cech cohomology. 0
Corollary 4.1. The Zz-graded Cech cohomology theory is a generalized cohomology theory. 5. Comparison with the Classical Cech Cohomology Theory In this section we show that a generalization of the Connes-HochschildKostant-Rosenberg Theorem can be obtained easily.
Theorem 5.1. Let A be a stable continuous C*-algebra with spectrum a smooth compact manifold X , in fact A = C ( X , K : ( P ) ) is the algebra of continuous sections of a smooth, locally trivial bundle K ( P ) := P x p u K: on X with fibre the algebra K: of compact operators o n a separable Halbert space associated to a principal P U bundle P on X via the adjoint action of P U o n K:. Let b ( P ) E H 3 ( X ;Z ) be the Dixmier-Douady invariant, that classifies such algebras A and c ( P ) some closed %form o n X , that presents the class 27rid(P) in the real cohomology. Let A be the category of all *representations of the C*-algebra A. Then the Z2-graded Cech cohomology Z,H*(A) is isomorphic to the de Rham cohomology H * ( X ;c ( P ) ) which is isomorphic to the classical Z2 -graded Cech cohomology Z2G* ( X ;c( P ) ) . Proof. In this situation, the Z2-graded Cech cohomology of the category A is isomorphic t o the Connes periodic cyclic homology H P , ( C m ( X , L ' ( P ) ) ) , where C m ( X , L ' ( P ) ) is consisting of all smooth section of the sub-bundle L 1 ( P )= PxpuL' of K ( P ) with fibre the algebra L1of trace class operators on the Hilbert space with the same structure groups P U , see Ref. 3 for a more detailed proof in the language of periodic cyclic homology. 0
268
D. N. Diep
Acknowledgments T h e main part of this work was done while the author was visiting T h e Abdus Salam ICTP in Trieste, Italy. T h e author is grateful t o ICTP and in particular would like to express his sincere thanks t o Professor Dr. Le Dung Trang for invitation and support. T h e text of this paper is presented at the Second International Conference on Abstract and Applied Analysis, Quinhon, Vietnam, 4-9 June 2005. T h e author thanks the organizers and especially Professor Dr. Sc. Nguyen Minh Chuong for the invitation t o give this lecture.
References 1 . M. Kashiwara, Algebraic study of systems of partial differential equations,
MBmoire No. 63, Suplkment Bull. SMF, Tome 123, No. 4, 1995. 2. A. Rosenberg and M. Kontsevich, Noncommutative smooth spaces, The Gelfand mathematical Seminar, 1996-1999, pp. 85-107. 3. V. Mathai and D. Stevenson, O n a generalized Connes-Hochschild-KostantRosenberg theorem, arXiv: math.KT/0404329 v l , 19 April 2004. 4. D. Orlov, Quasicoherent sheaves in commutative and noncommutative geometry, MPI-1999-31 Preprint, 1999. 5. D. Quillen, Formal properties of overdetermined systems of linear differential equations, Thesis (1964). 6 . A. L. Rosenberg,Compositio Math. 112,93 (1998). 7. M. Artin, A. Grothendieck, and J. L. Verdier, The'orie des topos et cohomologie des sche'mas, SGA 4, tome 2, Springer Lecture notes in Malhematics, No. 270, 1972.
Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 269-278) @ 2007 World Scientific Publishing Co.
269
514. SOBOLEV SPACES WITH WEIGHT ON RIEMANNIAN MANIFOLDS NGUYEN MINH CHUONGt and LE DUC THINH Institute of Mathematics 18 Hoang Quoc Vaet, Cau Giay, 10307 Hanoi, Vietnam
tE-mail: [email protected] Sobolev spaces with weight are defined on Riemannian manifolds. The class of such spaces, even in Rn, is very wide and very interesting (see Ref. 10, p. 353). It includes the usual Sobolev spaces of nonnegative integer orders as special cases. Properties of such spaces are studied.
1. Introduction
Many famous geometers, mathematicians have contributed to solve the well known Yamabe’s p r ~ b l e r n . To ~ > solve ~ this geometrical problem one must use various tools, among which mathematical analysis and Sobolev spaces are very powerful. From this geometrical problem arises a very celebrated elliptic problem with critical Sobolev exponent^.^?^ So far, the Sobolev spaces used on Riemannian manifolds are of nonegative interger orders. The aim of this paper is to introduce Sobolev spaces with weighted norms on Riemannian manifolds including usual Sobolev spaces mentioned above as a special case.
2. Sobolev Spaces with Weighted Norm on Riemannian Manifold Let (A&, g) be an n-dimensional smooth Riemannian manifold with the metric tensor g. In local coordinates, g has the form: n iJ=l
Let D be the Levi-Civita connection on Adn. D is extended uniquely to covariant derivative of tensor fields, denoted also by D. For f E C’(M,)(lc 2
N . M. Chuong and L. D. Thinh
270
0, an integer), let us denote by Akf the covariant derivative of f of order k:
AOf
= f,
A1f
=
of, A'flf
=
D(A'f ) , (0 I T 5 k
- 1).
Thus Akf is a tensor field of kth covariance. In local chart (U,'p) of Mn, we denote by Aal...ak f the components of Akf : n
A kf =
C
Aal...ak f dxa'
... @ dxak
@
al,...,ak=l
The metric on the tangent bundle of the Mn generates the metric on all tensor bundles over M,. So, we can define the norm of Akf , denoted by
P"I
:
p k f l z = C g a l B 1 . . p D 1 (Aal...ak,)(np,.../3kf),
where
(gij)
is the inverse matrix of the matrix (gij). For k = 0, ]Aof I
=
If I.
Definition 2.1. For a general Riemannian manifold M , , we call a weight function a positive continuous function on Mn. In the case the Riemannian manifold is R", a positive function w is called a weight function if there exist positive constants C and N such that: w(E
+ 77) 5 (1 + cl[l)Nw(77);
vc,
77 E R"
(These weight functions are introduced by L. Hormander in Ref. 8). We denote by IC(Mn) the set of all weight functions on Mn.
+ 1<12)$,
It is well known that w,([) = (1 If w E K ( R n ) then the functions:
s E
R, belongs to x(Rn).
belong to K(Rn) (see Ref. 8).
Definition 2.2. Let w E IC(Mn), p is a real number, p 2 1 and k is a nonnegative integer. Let .jFp,k,w(Mn)be the vector space of all functions f E Coo(Mn)such that w l V e f l E L p ( M n ) ,for all integers C with 0 5 C 5 k . The Sobolev space with the weight function w , denoted by Hp,k>w(Mn), is the completion of 3p,k,w(Mn)with respect t o the norm: k
e=o
Sobolev Spaces with Weight on Riemannian Manifolds
271
Let us denote by C?(M,) the space of smooth function of M , with compact support, H i , k , w ( M n the ) closure of C,"(M,) with respect t o the above mentioned norm. For k = 0, H,,O,~(M,)= LP,,(M,) is a Banach space equipped with the norm:
Ilfllp,w
= IbJfllP.
As in Ref. 2 we can get Theorems 2.1 and 2.2 below: Theorem 2.1.
cp(Rn)is
dense in Hp,k+,(Wn), that is Hi,k,w(Rn)=
HP,k,W(R").
Theorem 2.2. Let M, be a complete Riemannian manifold. Then C r ( M , ) is dense in HP,1+,(M,). Theorem 2.3. I f w 1 , w2 E K(M,) and there exists a constant C > 0 such that w 2 ( z ) I c w l ( x ) , V z E M,, then Hp,k,wl(Mn)C Hp,k,wz(Mn),and the inclusion operator is continuous. Theorem 2.4. If w1,wg E K(Rn) and Hp,k,wl(Rn)c Hq,e,wz(Rn)f o r some p , q, k,C, then there exists a constant C > 0 such that: w2(z)
I C w 1 ( z ) , V z E R".
Proof. Consider the inclusion operator:
I : Hp,k,wl(Rn)L,H q , e , w z ( R n ) . The operator I has closed graph. Indeed, if u, -+ u in H p , k , w l ( R n ) and Iu, 2, in Hq,e,wz(Rn) then u, --+ u in Lp,wl(Rn)and Iu, --+ Y in Lq,wz(Rn). Consequently there exists a subsequence denoted also by {u,} such that u, -+ u almost everywhere, and u, = lu, Y almost everywhere. So u ( z )=).( almost everywhere in R", and Iu = By the closed graph Theorem, the operator I is continuous. Therefore there exists a constant A > 0 such that: ---f
..
Ilfllq,t,wz
5 AllfI l p , k , w l ,
vf E Hp,k,wl(R")
(1)
Fix a function f E CF(Rn), f $ 0. For each q E R", consider a function f q , defined by:
f,(O
=
f(< - 7 ) .
We have:
D"f,(E) = (D"f>(E- q),V'I..
272
N . M . Chuong and L. D. Thinh
so: Since f E CF(R"), we get On the other hand:
f,
E
CF(R"), thus f, E Hp,k,wl(Rn).
Hence:
Theorem 2.5. Assume that w l , w2 E IC(Rn) and p
<
> n. If wz(') --t 0 as W 1 (El
then the inclusion operator C r ( R n ) n Hp,2,w,(Rn)L-) Lp,wz(RWn) is completely continuous. 4
00
Proof. Let {fi}i?l c C r ( R n ) n Hp,2,wl(Rn)be an arbitrary sequence such that 1 lfil lp,2,wl I 1,V i . We shall prove that there exists a subsequence converging uniformly on every compact subset of W". Let K be an arbitrary compact set. Then there exists an open ball S such that K C S. The closure S of S is a n-dimensional compact Riemannian manifold with smooth boundary in W". Since p > n , applying Sobolev imbedding Theorems to compact manifold with boundary (see Ref. 2), we see that H z ( S ) is imbedded into C1(S). Consequently there exists a positive constant C such that:
Sobolev Spaces with Weight on Riemannian Manifolds
273
Note that:
2
e=o By the continuity of w1, we have:
Hence the sequence {fi} and { IAfil} are uniformly bounded on K , so the sequence {fi} is uniformly bounded and is equicontinuous on K . Since Rn is separable and countable a t infinity, the sequence {fi} possesses a subsequence converging uniformly on every compact subset of Rn. This subsequence, may be denoted by {fi}. Given any E > 0, by wz([)/w1(<) 40 as [ -+ 00,there exists a large enough closed ball S such that:
By Minkowski incquality, we obtain:
where C
= supwz -
> 0.
Since
llfi
-
fjllp,2,wl
I2
and the sequence
{fi}
S
uniformly converges on S we get:
So the sequence {fi} is Cauchy in Lp,wz(Rn)and because LPIW2(Rn) is complete, the sequence {fi} converges in Lp,wz (EXn). 0
N . M . Chuong and L. D. Thinh
274
Remark 2.1. (1) With the assumptions of the Theorem 2.5, it is obvious that the inclusion operator CF(Rn) n Hp,k,wl(Rn)-+ Lp,wz (Rn)is completely continuous, for all integers k 2 2. (2) When Mn is a Riemannian manifold, it is also a metric space that is separable and countable at infinity. But a ball S in M,, is not necessary to be a Riemannian submanifold with boundary, so we cannot apply the Sobolev embedding theorem as above to S. However, the above theorem holds true for the class of Cartan-Hadamard manifold, that is, a simply connected complete Riemannian manifold with non-positive sectional curvature. By the exponential map at any point such a manifold is diffeomorphic to an Euclidean space of the same dimension. Consequently a closed ball in a Cartan-Hadamard manifold is a Riemannian submanifold. So we get:
Theorem 2.6. Assume that M , is a Cartan-Hadamard manifold and
IC(Mn). ~ ~ If p 2 >
and W 2 ( Q ) --+ 0 as d(P,Q ) 4 c o ( ~ i s a fixed wi ( Q ) point of M,), then the inclusion operator f r o m C r ( M n ) Hp,2,wl(Mn)to Lp,w2 (M,) is compact.
~
1
E
TZ
n
Theorem 2.7. Assume that w l , w 2 E K(Rn)and the embedding operator from C?(Rn) to L4,wz(Rn) is completely continuous for reals p , q 2 1 and
for some non-negative Ic. Then w2(') -+ 0 as
E --+ oo.
Wl(E)
Proof. It is sufficient to prove that if{,$,}u21 is an arbitrary sequence such that
Eu
-+ 00
as u
-+
03
w2 ( t u 1 then --+ 0 as v
-+
00.
Wl(EU)
Let us fix a function f E C r ( R n ) ,f $ 0 . Consider sequence {fu} defined by:
From the proof of Theorem 4 we have:
From the first inequality it follows that the sequence { f u } is bounded in H P , k , w i (R") C?(Rn). The assumptions of the theorem give us the relative
n
Sobolev Spaces with Weight on Riemannian Manifolds
275
compactness of the sequence { f,,}in Lq,wz(R,). So there exists a compact set G c R" such that:
where E is any positive number (see Ref. 1, pages 31-33). By the Minkowski inequality, we obtain:
'
Wn\G
G
Since G and supp f are compact sets, for u large enough, we have x - <,, q! supp f , V x E G. So:
-(/w$(x)lf(z-&,)lqdx)' 1 wl(ru)
+Oasu-+cc
G
that implies we obtain:
11 fVllq+,,
-+
0 as u + 00, and by the second above inequality
Similarly to the proof of Proposition 2.11 from [2] (pp. 36-37), we get: Proposition 2.1. If Hqo,l,u(Mn)is imbedded into L,,,w(M,)
1 for - Vn
In [2], Lemma 2.22, T. Aubin proved that: SUP
IVI
I C(~)IIVIIq,l,~V E C,"(Mn),
here q > n and C(q) is a constant, M , is a complete Riemannian manifold having positive injectivity radius and upper bounded sectional curvature ( K 5 b'). If M , is a compact Riemannian, then all these above conditions are satisfied, so we get the desired inequality. However, in the case of a compact manifold M,, another proof will be shown.
Theorem 2.8. Let Mn be a compact Riemannian manifold. Then there exists a positive constant C such that sup I f 1 5 CII f l l q , l , V f E C,oO(Mn).
276
N . M . Chuong and L. D. Thinh
Proof. Since Mn is compact, there exists p > 0 such that V P E M,, the polar coordinates exist on the closed ball:
BP(P)= { Q E MnI4C Q ) I P} (see Ref. 9, p. 22). Because of the compactness of Mn, there exists also the points Pi E Mn(i = 1,..., N ) such that {Bpi P N is an open covery of M , (Bp, P 4 4 P is an open ball of radius with center Pi). 4 be the unity partition corresponding to the above open Let covery. It will be shown that for each i, there exists Ci > 0 such that:
(-)}+,
(-)
{ai}Ll
suplazfl 5 ~ z l l ~ ( ~ z f ) l l qE, CF(Mn). V~ Set f , = a , f , K , = suppa,. Obviously there exists Q, E K , such that suplazfl = Ifi(Qz)l. Consider a closed ball B p of radius p with center 0 in Rn. The exponential map ezpp, a t P, is a diffeomorphism from Bp onto Bp,(p). The function h ( z ) = f,(ezpp,(z)) is defined on B p . Set zo = ezp;%'(Q,). Let ( r ,8 ) be polar coordinate in Rn with center a t 2 0 . For every z E R",satP we have d ( 0 , x) > and d ( 0 , x) < p, so z E Bp and 4 h ( z )= 0. Consequently:
isfying d(z0, z)
=
2P -, 3
h(zo)= - j h h ( r ,8)dr. 0
Hence:
Ifi(Qi)l
= Ih(z0)l 5
f 0
I&h(r, 8)ldr.
Integrating of the two sites of this inequality over S"-l with respect to 8 yields:
1
un-llfi(Qi)l I I&h(r, 8)ldrd8, B
where
B is a closed ball of radius -,23P
with center at z o in R" and u,-1 is
Sobolev Spaces with Weight on Riemannian Manifolds 277
where
dE is the canonical volume element in Rn and
is compact there exists constants A, p with 0
1
1 + - = 1. Since Ki 4 4' -
< X 5 p < co,such that:
X1tI2 I C S i j ( P ) t i t j i plt12,Vt = (tl,...,tTL) E R",V'P E
Ki.
i,j
Consequently:
d~ on K i , where
I X - ? ~ V ,l ~ h I l p f l ~ f i ol ezppi
dV is the Riemannian volume element on Mn. Hence:
We can choose:
Therefore, Vf E C p ( M n )we get: N
i=l
i
N
References 1. R. A. Adams, Sobolev spaces, Academic Press, (New York, 1975). 2. T. Aubin, Nonlinear Analysis o n manifolds, Monge-Ampere Equations, (Springer-Verlag,New York, 1982). 3. T. Aubin, A Course in Differential Geometry, AMS, Providence, Rhode Island, (2000). 4. T. Aubin, Bull. Sci. Math. 2e SCrie, 100, 149 (1976). 5. H. Brezis and L. Nirenberg, C o m m . Pure A p p l . Math. 36,437 (1983).
278
N . M. Chuong and L. D. Thinh
6. N. M. Chuong and T. D. Ke, Existence and nonexistence of solutions f o r a semelinear degenerate elliptic system, J. Abstract and Applied Analysis
(submitted). 7. S. Gallot, D. H u h , and J. LaFontaine, Riemannian geometry, (SpingerVerlag, 1993). 8. L. Hormander, Linear partial diflerential operators, (Springer-Verlag, Berlin, 1976). 9. Jurgen Jost, Riemannian goemetry and geometric analysis, (Springer, 1998). 10. H. Triebel, Theory of functions spaces, (in Russian) Akdemische Verlagsgesellschaft Geest & Portig K. G, (Leipzig Birkhauser Verlag, BaselBoston-Stuttgart, 1986).
Chapter IV
STOCHASTIC AND INFINITE-DIMENTIONAL ANALYSIS
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$15. STOCHASTIC POPULATION CONTROL AND RSDE WITH JUMPS SITU RONG Department of Mathematics, Zhongshan University, Guangzhou 510275, China E-mail: [email protected]. cn We obtain and improve results on the existence, convergence, stability of solutions to reflecting stochastic differential equation (RSDE) with jumps and apply them to discuss the properties of stochastic population solutions and finally solve the optimal stochastic population control problem.
1. A Deterministic Population Dynamical System Consider a population dynamics system. Let us denote by S: the size of the population with age between [i,i l), and T, - the possible largest age of the people. Suppose that the year variable is discrete as t = 1 , 2 , . . .. Then as time t evolves, the size xi,i = 1 , 2 , . . . , r m also changes. Then intuitively, the increment of Ax: caused by the time increment At = 1 can be expressed as
+
{
nx;= [-(1 + v;)z;
+ c:L1-, b:x;pt] nt ,
Ax: = [-(1 + +)xf + S ; - ' ] A t , i = 2 , . . . lrm;
(1)
where ,Ot is the specific fertility rate of females, b: = (1 - p!)kjhi # 0, i = rl, . . ., 1-2; 1 < r1 < T 2 < r,; p: is the death rate of babies, 7: is the forward death rate by ages, kj is the sex rate, hi - the fertility model, and [rl,rZ] is the age interval that women can give babies. The absolute value (1 vt)xi of the first term in the right hand side of the first equation in (1) is the amount of persons who will leave the age interval [1,2)in the next year t 1, and the value b:x$t of the second term is the amount of persons who will arrive a t age interval [l,2) in the next year tt 1. Similar physical meaning can be made for Ax:, 2 5 i 5 r,. If one considers that the time variable changes continuously, and writes the corresponding
+
+
c:&
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ordinary differentialequation in a matrix form, then he will get the following deterministic population control system:
dxt
=
(Atxt
50 =x,
+ Btxt,Bt)dt
(2)
t E [O,T],
where xt = (xi,.. ., xEm),and
Bt=
[
0 .. 0 bL1 0 .... ..
.. b y 0 .. 0 .. .. .... 0 .
1
(4)
Notice that the size of the population should be non-negative, so one should require that
Under appropriate conditons (5) is valid. 2. RSDE for Stochastic Population Dynamical System
In case that the size of the population is disturbed by some stochastic perturbation the situation is quite different. For example, in the simplest case even it is disturbed by a standard Brownian Motion process (BM), caused by the strange properties of the BM (5) still cannot be guaranteed. So, we need to consider the reflecting stochastic differential equations (RSDEs) as the stochastic population dynamic system model. Let us first introduce a more general d-dimensional RSDE with jumps as follows: (Here we write "RCLL" for "right continuous with left limit" .)
r dxt = b ( t ,Z t , w)dt + u(t,~
tw)dwt ,
+ Jz ~ ( ~t ,t ,Z- ,w ) ~ (Id,t ,d ~+) d+t ,
" 0 = z, xt E $,for all t 2 0 , where R$ = { z = (zl,...,zd) E Rd : Z ' >0,l 5 i 5 d , +t is a Rd - valued & - adapted RCLL process with finite variation I+l t on each finite interval [0,t] such that 40 = 0, and
1
Jot I a R p s P 1 4 s +t = 1 ;4 s ) d I& I+lt
=
1
>
,n(t)E Nz,, as ~t
E aR$,
(6)
Stochastic Population Control and RSDE with Jumps
283
where w t is a d-dimensional BM, and f i k ( d t , d ~ is) a Poisson martingale measure generated by a zt-Poisson point process k(.) with a compensator 7r(dz)dt, T ( . ) is a a-finite measure on a measurable space (2,R(2)) such that N k ( d t , d z ) = N k ( d t , d z ) - n(dz)dt, and N, = u,.+oN,,,.,Nx,r= { n E Rd : In1 = 1,B ( x - nr,r ) n 0 = S} , B(xo,E)= {y E Rd : 19 - 201 < E } . Condition = I a R t ( x 3 ) d in (6) obviously means that l$ls inI
s,'
l$lt
creases only when x, c aRd+,and the condition +t = J,"n(s)d141s means that the reflection dq5t happens along the direction of the inner normal vector n ( t ) ,so we may call such a reflection a normal reflection. The geometrical meaning of N, is that it is the set of all inner normal vectors a t x,when x E OR$. Obviously, when x E Rd,,by definition N, = CP (the empty set). Actually, in the case that xt E Rd, no refelction is needed, so we do not need an inner normal vector. More precisely, for ( 6 ) , by definition as x E Rd+, N,= CP (empty set). Notice that
aRd,
= '$=I
d Uil
,ik=l
Ak,
where Ak = { ( 2 1 ~ x 2 ,... ,xd) Xi, = 0, Xj > 0, v j # i,, 1 5 p 5 k } So, if x E Ak, then n = ( 7 2 1 , 1 2 2 , ' . . , nd) E N,, iff, for k 2 3, nil = sin8coscpl coscp2.. .. . . . . . (20s(Pk-21 ni, = sin 8 cos (PI cos cp2 . . .cos (Pk-3 sin ( P k - 2 , nik-, = sin 8 cos cp1, nikPl= sin 0 sin cp1, ni, = cos 8, 8 E [o, ; ] 1 (P11$?2 ' ' ' I (Pk E [o, $1, nj = 0,as j # i l l i z , . . . ,i k ; and for k = 2, nil = cos 8, ni, = sin 8, 8 E [O, $1, n k = 0 , V k # i1,iz; and for k = 1, nil = 1, nj = 0,Vj # il. From this one sees that the inner normal vector a t a boundary point of the domain Rd, is not necessary unique. However, from (6) one knows that there exists a unique n(t)such that as xt E dRd,, n(t) E N,, and the
284
S. Rong
reflection d d t happens along the dircction of the inner normal vector n(t). Since all n, 2 0, i = 1 , 2 , . . . ,d. So we have the following Claims.
&, i = 1 , 2 , . . . ,d, in (6) are increasing, where = (4L 4L.. . ,4 3 .
Claim 2.1. all $t
Claim 2.2. If (xt, d t ) satisfies (6), then t 2 0 , t zp4; = 0, i = 1 , 2 , . . . , d.
so
Claim 2 . 2 is easily obtained by (6) and by the property of N,. From now on we always make the following Hypothesis on the jump term: ( H ~ ) : z + c ( t , z , z , w ) ~f zo r, a l l t 2 0 , z E Z , w E R a n d x E R d , . (Hz) ~ ( d z = ) d z / Izld+', ( 2 = Rd - (0)). The meaning of (HI) is that if the state xt- before the jump lives in then after the jump, the state ~t = xt- c ( t , Z,z , w ) still stays in The Hypothesis (H2) gives a more concrete measure for the compensator of the jump Poisson point process. Notice that under assumption (HI) if ( z t ,4t) satisfies (6), then & should be continuousli Lemma 329. So under assumption (HI) in (6) we may write that
+
nd,.
q5t is a continuous function.
Rd,,
(7)
Now for the solution of RSDE (6) with (7) we introduce the following definition.
Definition 2.1. 1) We say that ( x t , 4 t ) is a strong solution of (6) with ( 7 ) ,if xt is RCLL and $t is continuous and they are 3y1Nk -adapted, where 3W'fi'"
-
U ) ;s I t , u E B(Z)), and (xt,&) satisfies (6) with (7). 2) We say that the pathwise uniqueness for solutions of (6) with (7) holds, if for any two solutions (xi,&), i = 1 , 2 , of (6) with ( 7 ) ,which are defined on the same probability space (fl,3,(St),P ) with the same BM wt and Poisson martingale measure ( d t ,d z ) , P(SUPtL01x2 - x;l = 0) = 1, P(sup,20 14; = 0) = 1. t
= d W s , N k ( ( 0 ,s),
qq
3. Existence of Solutions to RSDE with Jumps
We can have the following general existence theorems of solutions for RSDE with jumps which are some generalization of Theorem 335 and 336 for 0 = R$ in Ref. 1.
Stochastic Population Control and RSDE with Jumps
285
Theorem 3.1. Assume that for t 2 0 , x E @f, z E Z 1" b(t, x),a ( t ,x),and c ( t ,x , z ) are non-random, jointly measurable; moreover, b(t, z), a ( t ,x) are jointly continuous, and as 1% - yl -+0 , It - sI 4 0 Jz Ic(t, 5 ,). - 4%Y , .)I2 .rr(dz) 0, furthermore, b(t, x),o(t,x) and Jz Ic(t, x,. ) I 2 7r(dz) are locally bounded for x; that is, for each r > 0 , as 1x1 5 r , I b ( t , . ) I 2 + Ila(t, %)112 JZ Ic(t, x,z)I2 .rr(dz)I ICT, where kT 2 0 is a constant depending only on r ; 2" there exists a real function V ( t , x ) E C112([0,03) x R d ) ,i.e. it has a first continuous derivative with respect to t and up to second continuous derivatives with respect to x, such that one of the following conditions is satisfied: -+
+
+ J Z [ V ( t2, + c ( t ,5 , z ) )- V ( t ),.
-
EL
Ci(t, x,z)&V(t,
2)17r(dz)
ICl(t), where c l ( t ) 2 0 , s,'cl(t)dt < 03,'v''T < 03; (ii) LV(t,x) 5 c l ( t ) V ( t ,x), where c l ( t ) has the same property as in the above (2); ' 3 there exists a a l ( t ) with al(lt1) 2 0 such that a l ( t ) 00, as t T as 1x1 is large enough, V ( t ),. 2 a1(lzl), where V ( t x) , is the same function occured in 2'; and as xi 2 0 , L i V ( t ,Z) I k o ~ i i, = 1 , 2 , . . . , d;
03;
and
+
- b ( t ,Y ) > Ila(t,). - a(t,dI12 2 c ( t ,Y , z)12~ ( d zI) ~ N , T ( ~ ) P N , T-(YI I~1, as 1x1 IYI I N , t E [O,TI, where for each T < 03, 0 I k N , T ( t ) , l T k N , T ( t ) d t < 03 and P N , T ( U ) i s concave and strictly increasing in u 2 0 such that P N , T ( O ) = 0 , and So+ d u / PN,T ( u )= 03, for each N = 1 , 2 , .... Then (6) with (7) has a pathwise unique strong solution.
4"
2(.
-Y)
. ( b ( 4).
+ Jz I C ( ~2, , z )
-
1
Theorem 3.1 has the following obvious applications.
Corollary 3.1. Under assumptions 1" and 4" in Theorem 3.1 if, furthermore, there exists a constant ko 2 0 such that 2 (x,b ( t , .)) Ila(t,.)1 2 J, Ic(t, 2 , .)I2 T ( d 2 ) 5 ko(1 + 1xI2), then (6) with (7) has a pathwise unique strong solution.
+
+
286
5'. Rong
Corollary 3.1 obviously includes Theorem 335 for 0 = Rd, in Ref. 1. To show this Corollary by Theorem 3.1 one only needs to take V ( t , z )= 1 1212 a l ( t ) = t 2 ,c l ( t ) = Ico. Then one easily sees that under the assumption of Corollary 3.1
+
C V ( t ,).
I Cl(t)V(t,x),
and Theorem 3.1 applies.
Corollary 3.2. Under assumptions 1' and 4" in Theorem 3.1 if, in addition, ( X I b ( t l .)) I C l ( t ) ( l + 1xI2 gk(x))l Il4tl 4 1125 C l ( W + Ixl2nE1 gk(x))l J , I C ( t l 5 1 .)I2 4 d z ) I C l ( t > l where g k ( x )= 1 + l n ( l + l n ( l + . . . l n ( l + 1 x 1 ~ " ~ ) ) ) ~
nlZ"=,
k-times
(m and no are some natural numbers), and c l ( t ) 2 0 is non-random such that for each T < 00, c l ( t ) d t < 00; then (6) with (7) has a pathwise unique strong solution.
s'
Corollary 3.2 obviously includes the statement of the second part of Theorem 336 for 0 = Rd, in Ref. 1. To show this Corollary by Theorem 3.1 one only needs to take V ( t , x )= gm+1(2), a l ( t ) = gm+l(t), c l ( t ) = c l ( t ) . Then one easily sees that under the assumption of Corollary 3.2 q t ,). I k O C l ( t ) I where Ico 2 0 is a constant. So Theorem 3.1 applies. Now let us prove Theorem 3.1.
Proof. Let
where f = b,a;&nd
Then by Theore; 334 in Ref. 1 there exists a pathwise unique strong solution #) satisfying RSDE (6) with (7) but with coefficients b N , oN and c N . Let
(zpl
T
~= inf{t , ~
2 o : Ixrl > k} .
Then by Ito's formula one easily sees that under the assumption (ii) of 2" in Theorem 3.1 for any given T < 00 as 0 I t I T
Stochastic Population Control and RSDE with Jumps
EV(t A
+
tArN*N
L V ( S ,x,N)ds.
xKrN,N) = E V ( 0 , X O ) E J o < kh C l ( S ) E V ( S A T N ' N ,2 f A r N , ~ ) d S . Hence as 0 5 t 5 T EV(t A x E r N , N5) k,eI J 0T C l ( t ) d t rNIN,
+ Ji
287
rNiN,
1
and a ( N ) P ( r N J "< T ) 5 AT, z~N,N,,,T)I~N~N
Therefore, as N
---f
~
(
7
~
1
~
00,
P(rNiN< T ) + 0. For the case that assumption (i) of 2" in Theorem 3.1 holds, the same result is similarly derived and the derivation is even simpler. From this rNvN= co.Now set result one easily sees that P - a s s .limN,, xt = xtN , dt = @, as o _< t < T ~ , ~ . It is easily seen that ( x t ,$ t ) is the pathwise unique strong solution of RSDE (6) with (7). 0
4. Stochastic Population Solutions and Their Properties
Now let
where At and Bt are defined by (3) and (4), and let d = r,, then ( 6 ) with (7) and (8) becomes the stochastic population dynamic system. By Corollary 3.1 we immediately obtain the following result.
Theorem 4.1. Assume that coeficients (T and c satisfy conditions 1" and 4" in Theorem 3.1, and assume that At and Bt are bounded and ,L$(x) is bounded and jointly continuous such that IPt(x) - Pt(y)I I ~ N , T ( ~ ) P N , T ( I XY I ' ) , as 1x1, lyl 5 N , t E [O,T],W'<00; where kN,T(t) and P N , T ( U ) have the same properties as those in the condition 4" of Theorem 3.1. Then the population SDE, i.e. (6) with (7) and with the coeficient (8), has a pathwise unique strong solution.
A simple case for u and c t o satisfy the above conditions in Theorem 4.1 is given as follows:
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S. Rong
Corollary 4.1. Let
i
u(t,x ) = co + ( c p x , .. . , Cid)x) c ( t , x , z ) = (CO G z ) f ( z ) ,
(9)
+
where Co, Cii),i = 1,* . . ,d and 7 7 1 are constant d x d matrices, and 770 is a constant Rd-vector, (i.e. all elements in the matrices or vectors are constants), and , in addition, all elements in 770 and 771 are non-negative, i.e. ?$,i$ 2 O , i , j = 1 , 2 , . . . , d ; where 770 = ,co) -d T , 771 - (- icj ~d ) ~ ,
(.A,..
and f ( z ) 2 0 is such that J R d - { o ) S d z < 00. Then one easily checks that assumption ( H I ) and all conditions for u and c in the above theorem hold. So Theorem 4.1 applies. Let us discuss the properties of the stochastic population solutions 1)Convergence Property. First we can consider the more general RSDEs with jumps. If (z:, @) satisfies (6) with (7) and with coefficients bn(t, x ) , un(t,x ) , and cn(t, x , z ) and with the initial condition x; as well, then we have the following convergence theorem.
Theorem 4.2. Assume that 1' Ibn(t, .)I2 /Inn@,x)1I2 J, I C n ( t , x , z ) I 2 T ( d z )I ko(1 1xI2), n = 0 , 1 , 2 , .., where ko 2 0 is a constant; 2" there exists a real function 0 5 V ( t ,x ) E C1i2([0,00) x R d ) such that
+
+
+
and there exists a a l ( t ) with al(lt1) 2 0 such that a l ( t ) ,t increasing, and
V ( t ,x ) L a1(lxl), moreover,
2 0 is strictly
Stochastic Population Control and RSDE with Jumps
where 1g(y)l 5 k o , and c l ( t ) 2 0 ,
289
SoT cl(t)dt < 00,'v'T < 00;
3"
ao(s~z)II
2
2
l i % + c o ~ ~ [ S U p xIb"(s,z) ~~d - b o ( s , z ) ) + S U P X E R ~ ( ~ ~-~ ~ ( S I ~ ) 2 J , Icn(s,z, z ) - co(s,z, z)I .rr(dz))]ds = 0,f o r all t > 0;
+
z:, as n -, 00, where the initail values xz,n = 0 , 1 , 2 , * - . are all non-random constants. T h e n for all t 2 0 sup,,, V ( t ,zy - z:) = 0. I n particular, for any E > 0 and t 2 0
4"
zt
I n addition, assume that 5" limlQ-xl+OSUPs
sup
IY--zJ+O
s_
s,
= 0,
Ico(s,z, z ) - co(s,y,
.)I
2
~ ( d z=) 0.
Proof. By condition 1" and using Ito's formula one easily obtains that1
EsuplzTl 5 kT, h = 0 , 1 , 2 , * * * , t
where kT 2 0 is a constant depending on T < 00 only. Now applying Ito's formula to V ( t ,zy - zf), one has that as t 5 T
where
+
s,
p ( S ,
z, z ) - c y s , 5 ,
.)I
2 T(dZ)]dS}.
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S. Rong
So, by Gronwall's inequality'!
5 V ( 0 ,J:
- J::)
+ f ( t )+
Lemma 11*
I'
eJt c l ( s ) d s c l ( sf) (s) [
+ V ( 0 ,x; - z:)]ds.
Hence the 1st and 2nd conclusions are easily derived. Notice that
d ( x y - $) = (bn(t,J::,
+ (a"@,
J::,
+s, + 44:
W ) - bo(t,J::, W)
w))dt
- ao(t,J::, w))dwt
(c"(t,J:?-,
2 , ~) Co(t,J:y-,
z , w ) ) f i k ( d t ,d z )
-43.
From this the last conclusion is also easily obtained. Corollary 4.2. Under assumptions 1",3" and 4" in Theorem 4.2 if 6" 2(x - Y ) * (bO(t,J:) - bO(t,Y)) IIflO(t, J:) - aO(t,Y)1I2
+
+J~Ico(t,J:,z)--o(t,Y,Z)12T(dZ) I c 1 ( t ) l J : - y l
2 1
c l ( t ) d t < 00, V T < 00; then f o r all t 2 0 where c l ( t ) 2 0, 2 limn+m sup,^^ (J:: ) = 0. I n addition, i f 5" in Theorem 4 . 2 is also satisfied, then for all > E ) 3 0, as n --f 00. sup,^^ 14; -
J:!(
t,E
2 0,
To see that the first conclusion in Corollary 4.4 is true, one only needs to take V ( t , z ) = (zI2, a l ( t ) = t2 and applies Theorem 4.4.The second conclusion can be proved directly. Corollary 4.3. Under assumptions l o ,3" and 4" in Theorem 4.2 if (x - Y ) * (bO(t,J:) - bO(t,2/11 L C l ( t > 12 - Y I 2 r I L g k ( Z - Y)1 2 ( ( a o ( J:) t , - " O ( ~ , Y ) / \ ~ + /z \c0(t,2 , ~-) co(tly , T(dz) 5 C l ( t ) IJ: - Y12 9k(J: - Y ) , where c l ( t ) has the same property as that in Corollary 4.4, and gk(x) is defined in Corollary 3.2; then for all t 1 0 Esupslt gm(z: - J::) = 0, and for all t , E 2 0 , supslt P ( ~ z : x:( > a) 3 0, as n -+ 00. I n addition, if 5" in Theorem 4.2 is also satisfied, then f o r all t ,E 2 0 , ~(14: > c ) 4 0, as n +. 00.
n:=.=,
-&I
.)I
Stochastic Population Control and RSDE with Jumps
291
To show Corollary 4.4, one only needs to take V ( t ,x ) = g m ( x ) , a l ( t ) = g m ( t ) and applies Theorem 4.4. Applying Corollary 4.2 we immediately have the following result for the convergence of solutions of population SDEs.
Theorem 4.3. Assume that 1" there exists a constant ko 2 0 , for all n = 0 , 1 , 2 , ... lA"(t)l + lB"(t)l + IP,"I I ko, where A"(t), B n ( t ) ,and pz" are non-random such that they do not depend o n x , and P ( t ,x ) and c n ( t ,x , z ) satisfy the condition 1" in Theorem 4.2; 2' oo(t,x ) and co(t,x , z ) satisfy the condition 6" in Corollary 4.2; 3" limn-+mJ:[IA"(t) - A o ( t ) ( ) B n ( t )- B o ( t ) f ) (pp- @l]dt = 0, and the condition 3" in Theorem 4.2 for d ' ( t , x ) , c O ( t , x ) and c"(t, z, z ) ,co(t,x , z ) holds; 4" the condition 4" in Theorem 4.2 holds. Then for all t 2 0 l i m + m E(supslt Iz: - IC, 0 2 ) = 0.
+
1
I n addition, i f 5" in Theorem 4.2 is also satisfied, then f o r all n-+m P(SUP, E ) 0.
t,E
2 0 , as
+
This indicates that if coefficients in the population SDE can be approximatly calculated, then solutions of the population SDE also can be approximatly (in probability) obtained. For more convergence results on the solutions of population SDE one can refer to Ref. 1. 2) Stability Property. For the stability of solutions to RSDE (6) with (7) we have the following theorem. Theorem 4.4. Assume that conditions l a ,3', 4" in Theorem 3.1 hold, and assume that condition 2" in Theorem 3.1 is changed to be 2"' : there exists a real function 0 5 V ( t ,x) E C'v2([0,m) x R d ) such
that C V ( t ,x ) I -.1(t)V(t, x), c l ( t ) d t = 03, and as xi 2 0 , where q ( t ) 2 0 , "ax, V ( t , X) 5 k o ~ i i, = 1 , 2 , . . . ,d. Then (6) with (7) has a pathwise unique strong solution, which satisfies that E V ( t ,x t ) 5 e- Ji~ ~ ( ~ ) ' s E 2v 0()o. , Furthermore, if b ( t , 0 ) = 0 , a ( t ,0 ) = 0 , c ( t , 0 , z ) = 0 , and there exzkts a real function a l ( t ) 2 0 , t 2 0 , which is increasing and a l ( t ) > 0 , as t > 0 ,
292
S. Rong
such that V ( t ),. L a1(Ixl), then (0,O) is the pathwise unique strong solution of (6) with (7) and with the initial condition zo = 0; and f o r any given E > 0 , limt+oo P ( l x t ( > E ) = 0, i.e. the str0n.q solution of (6) with (7) is stable in probability. Proof. First, by Theorem 3.1 (6) with (7) has a pathwise unique strong solution for any initial condition zo. Now applying I t 0 3 formula t o V ( t ,xt), one has that as t 5 T E V ( t ,Z t A T N ) 5 V ( 0 , x o )- J ; C l ( S ) E V ( S ,G A T N ) d S , where TN = infjt 2 0 : lztl > N}. By using Gronwall's inequality and Fatou's lemma the first conclusion is obtained. Secondly, under additional condition one has that for any given E > 0 al(&)p(lxtl> &) 5 E V ( t ,xt) I e- JiQ ( ~ ) ' s E v ( o5 ,0 ) . 0 So the second conclusion is also obtained. An immediate corollary is as follows:
Corollary 4.4. Assume that l o ,4" in Theorem 3.1 hold, and 2 (x,b ( t , x)) + Ila(t,4112 Jz Ic(t, 2, z)I27r(d4 L -c1(t) 1bI27 c l ( t ) d t = 00. where c l ( t ) 2 0 , Then (6) with (7) has a pathwise unique strong solution, which satisfies that
Jr
+
E 1xtI2. < - e-s," cl(s)dsE I x01 .
(10)
Corollary 4.4 obviously follows from Theorem 4.4, if we take V ( t ,x) = 1x12,and a l ( t ) = t2. Now let us apply the above Corollaries to give the stability results on the stochastic population dynamics (6) with (7) and with b ( t , x ) = Atx BtxPt(z).Suppose that a ( t ,0) = 0 and c ( t ,0, z ) = 0 then under conditions of the existence Theorem 4.1 (6) with (7) and with the coefficient (8), has a pathwise unique strong solution (0,O) with 20 = 0. Furthermore, if 2 2 ( A t z , x ) + 2 (B,P,(2)2,2)+lla(t,2)/I2+JZ I c ( ~ , ~ , z ) I~ ( d z F) - c I ( ~ ) where c l ( t ) 2 0, cl(t)dt = 00; then (10) holds, i.e. the solutions of population SDE is exponentially stable in the mean square. More precisely, if a ( t ,x) = (C:'),. . . ,C:'..") z,c ( t ,z,z ) = C 1 z I u ( z ) , where T ( U ) = 1,
+
12121
and Ciz),i = 1,.. . , d ; Let
are defined in Corollary 4.1, and 0
I: a ( x ) 5 PO.
Stochastic Population Control and RSDE with J u m p s 60 =
mintlo(;
+ rll(t),7 7 2 ( t ) , . . . ,r l T m - l ( t ) , ;+
b M = maxt>O(brl( t ) ,’ ’ ’
Then as
7
h-2
77Trn(t)),
(t)).
2
61 = 260 -
293
c:z1IIc,(i)II - 1C11
2
- PObM(r2 - r1+ 1)
> 0,
E 1ztI2I E 1z0l2e-61t, for all t 2 0. That is, the population dynamics of (6) with (7) and with the coefficient (8) is exponentially stable in the mean square under the above assumptions. In particular, for a given constant a > 0 we easily derive from the above inequality that t> - ~61 l n ==+ E 1ztI2I a. This indicates that if the stochastic perturbation is not too large (that
+f
xf=llICii)II
2
is,
+IF1
1’
is very small), and the forward death rate is greater
than a positive constant (that is, 60 > 0), then the stochastic population dynamics (6) with (7) and with the coefficient (8) can be exponentially stable ”in the mean square” if the fertility rate of females is small enough. Furthermore, given a > 0, we can find out when the population size vector ”in the mean square” can be less than this target a. 3) Comparison property. Theorem 4.5. Assume that lo lla(t, .)1 2 J, Ic(4 z, . ) I 2 n ( d z ) I ko(1 where ko 2 0 is a constan,t;
+
+ 1zI2);
2“ a ( t ,z) = (aik(t,x))$=, satisfies the condition that 2
2
5 k N , T ( t ) P N , T ( l z i - yil 1, as IzI,1y1 I N , t E [O,T],vi, Ic = 1 , 2 , . . . , d ; where 0 5 k ~ , ~ ( s,’kN,T(t)dt t), < 00, for each T < 00, and p ~ , ~ ( >u O,as ) u > 0; P N , T ( O ) = 0;and ~ N , T ( u )is strictly increasing ( a i k ( t ,z) - cilc(t,y)(
in u and such that d u / p ~ , ~ (= u 00, ) f o r N = 1,2, .... and T < 00; 3” c ( t ,z, z ) = ( c z ( t X, , z ) d) ~ =satisfies ~ conditions that
so+
z
+ c ( t ,z, 2) E z:,vt 2 0 ,
Ic
E
Ti;, z
+
E 2;
zi 2 yi ===+ zi f C i ( t , z,.) 2 yi C i ( t , y, z ) , vtrO,z,yERd,,ztZ, where x = ( 5 1 , . . . ,z d ) , y = ( y l , . . . ,y d ) ;
4”
(Pt(.) - P t ( Y ) ) I k v , T ( t ) P N , T ( C : l , , , ( z i - y i ) + ) , v z , y E 1x1, IyI I N , t E [O,T],
LZ>yi
as
z:,
where kN,T(t) and ~ N , T ( u )have the same properties as those in 2”; 5” 0 I v ~ , b ~ , P t (I z ko, ) Vz = 1 , 2 , . . . , d ; j = r l , . . . ,r2.
294
S. Rong
If (xt,&) and (&,) , are solutions of the stochastic population dynamics (6) with (7) and with the coeficient (8) and with the initial value X O , the fertility rate of females pt(z) and T o , P t ( z ) , respectively; then x i >?i$,Qi = l , . . ., d ; and A(.) 2 Bt(z),Qt2 0 , Q X E Ed+, implies that P - a.s. X : 2 8,Qt 2 0 , QJi = 1,.. . , d . This indicates that larger initial size and lager fertility rate of female always result larger population size. Theorem 4.5 can be proved by Tanaka's formula as in Ref. 1. (See the proof of Theorem. 346 in Ref. 1. However, now here ,Bt(z)depends on z).
5. The Optimal Stochastic Population Control Now let us discuss the optimal stochastic population control problem for the stochastic population dynamics (6) with (7) and with the coefficient (8). Suppose that the admissible control set is
6={,B : ,O = ,&(x) - is jointly continuous, Po
where 0
I P I Po1
I Do and Pa are constants. Let us minimize the following functional J(P> = .I/'
0
F ( t ,z f ) d t + G ( & ) ] ,
B , dt P ) is the pathwise unique among all ,B E 6, where 0 I T < 00, and (zt strong solution of (6) with (7) and with the coefficient (8) corresponding to the given j3 E where F ( t ,z) and G ( x )are Borel measurable.
6;
Theorem 5.1. Assume that a ( t ,z) and c(tlx , z ) satisfy all conditions in Theorem 4.1 and Theorem 4.5; F ( t , x ) and G ( x ) are jointly Borel measuch that as xi I y i , surable functions defined on t E [O,T]and x E
ad,
1 <_ i 5 d ; x , y
E $,Qt
E [O,T],where we denote d = r,,
G ( z ) I G(Y),F(t1.)
I F ( t ,Y).
Then the smallest constant Po is an admissible optimal control f o r the minimization of the functional J(j3); that is, Po E 6, and J ( & ) = infPGgJ ( P ) . Furthermore, assume that
azrc(t,z) = C Z ; ~ X iC, i ( t , x , z ) = $ X J L r ( Z ) '
Stochastic Population Control and RSDE with Jumps 295
and xfO is the so-called optimal trajectory corresponding to the optimal control Po. Moreover, for a given constant a > 0 we have that
t>&h
E
~ a
+X E
~
~
I ~ ? I/ a. ~
~
This theorem actually tells us the following facts: 1) If a target functional monotonically depends on the population size, for example, the energy, the consumption, and so on, spent by the population, then to minimize this target functional we should control the fertility rate of females to take a value as small as possible. 2) The optimal population size can be exponentially stable in mean square, if we take the fertility rate of females small enough and when the stochastic perturbation is not too large. Furthermore, given a > 0, we can find out when the optimal population size vector ”in the mean square” can be less than this target a. Theorem 5.1 can be derived by applying Theorem 4.1, Theorem 4.5 and Corollary 4.4. To summarize the discussion we have the following conclusion. 6. Conclusion
1) RSDE (6) with (7) and with the coefficient (8) is an appropriate model for the stochastic population dynamic system. 2) The stochastic population size from this model continuously depends on all parameters (i.e. the coefficients in the drift term). If one can ”approximately” evaluate the parameters, then the population solution size can also be ”approximately” obtained. 3) The stochastic population size monotonely depends on the initail size and the fertility rate of females. 4) The stochastic population size can be ”exponentially stable”, if we can take the fertility rate of females small enough, and the stochastic perturbation is not too large. Furthermore, we even can find out when the size vector can be smaller than a given level in the mean square.
296
S. Rong
5) If some target functional, e.g. energy consumption etc., monotonically depends on the population size, then t o minimize this functional one only needs t o take the fertility rate of females as small as possible. References 1. Situ Rong (Rong Situ), Theory of Stochastic Differential Equations with Jumps and Applications. (Springer, 2005). 2. Situ Rong, Reflecting Differential Equations with Jumps and Applications,
Research Notes in Mathematics 408, Chapman & Hall/CRC, (2000). 3. Situ Rong, Reflecting differential equations with jumps and stochastic population control, in Control Theory, Stochastic Analysis and Applications, eds. S . Chen and J. Yong, etc., (World Scientific, Singapore, 1991), pp. 193-202. 4. Situ Rong & W. L. Chen, Stochastic Analysis and Applications 10,45 (1992). 5 . J. Y. Yu, B. Z. Guo, and G. T. Zhu, J . Syst. & Math. Scien. 7,214 (1987) (In Chinese).
Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 297-320) @ 2007 World Scientific Publishing Co.
297
$16. NONCAUSAL STOCHASTIC CALCULUS REVISITED -
AROUND THE SO-CALLED OGAWA INTEGRAL SHIGEYOSHI OGAWA Dept. of Mathematical Sciences, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan E-mail: [email protected]
1. Introduction The causal theory of stochastic calculus originated by K.It8 in 1942 is founded on the Hypothesis of Causality; “In order that the stochastic integral / J ( t , w)d& is well defined the function f ( t , u ) , (t 2 0 , w E a) should be adapted t o t h e increasing f a m i l y of a-fields, generated by the underlying basic process Z t ( w ) , which is a square integrable semi-martingale like the Brownian motion.” The hypothesis seems well fit t o the principle of causality in physics, where the variable ”t”appears as time parameter. Moreover it endows the theory a remarkable feature of being in natural concordance with the notion of martingale which plays indeed an essential role in It6’s Calculus. However it is also true that the Hypothesis of Causality has imposed a significant restriction on the applicability of the causal theory of stochastic calculus. Let us take for example the case of such SDEs driven by general stochastic processes, like the fractional Brownian motion, that do not have the martingale property,s or the case of usual SDEs but under noncausal situations, that is the SDE with noncausal coefficients and/or noncausal initial values. Let us take moreover the case where the parameter ”t” stands for the space parameter, or the case where ”t”is multi-dimensional (ie., ” a stochastic calculus” for the random field14). The notion of Causality looses its sound meaning in such cases because of the lack of natural sense of t i m e direction. Even in the case of physical problems where ”t” appears as t i m e parameter, we can find various situations of noncausal nature, such as the Cauchy problem in the theory of Brownian particle equations, l 3 noncausal
298 S. Ogawa
version of the SDEs, the White noise analysis23 etc. Guided by these motivations the author had presented the noncausal theory of stochastic calculus in 1979,23by introducing the noncausal stochastic integral which is now refereed by author’s name. In this note we will give a unified sketch of that noncausal stochastic calculus. We will show recent development of the theory (Refs. 10 and 13) and we will also refer to some typical applications of the theory to mathematical sciences including mat hematical finance. 2. Noncausal Problems in Stochastic Analysis To motivate our study we like to show in this paragraph some typical examples of noncausal nature in stochastic analysis. Hereafter we fix once for all a probability space (a,F ,P ) on which all random quantities are defined. Among these we understand by W t ( w ) ,t 2 0 the standard Brownian motion and denote by {Ft}the natural filtration generated by the namely the family of o fields Ft = u { W s ; 0 5 s L t } c F. By the random functions we understand those real valued functions f ( t , w ) which are measurable in ( t , w ) with respect to the field B ~ , TxIF and satisfy the condition,
w.,
P{lT
f 2 ( t , z l u ) d t< CO} = 1
We will denote by H the set of all such random functions. A random function f ( t ,w ) E H is called causal (or non-anticipative) if, for each t E [0,T ] it is adapted to the o field Ft, We will denote by M the totality of causal random functions. For the random function f of this class, we have the It6 integral which we will denote by
s
f dWt throughout the paper, while
for the B-differentiable function f E M the integral of symmetric type (or noncausal type) is denoted by the symbol,
s
f d,Wt (see, Paragraph 3).
Example 2.1 (Stochastic conservation law). Suppose given two stochastic processes, X t , Yt (t 2 0 ) which obey a conservation law as follows; E ( X t , Y t , t , a , p , y , ...)= C o n s t , where a , p, y etc are causal constants, that is the deterministic or at least the random variables which are independent of the W. Now suppose that
Noncausal Stochastic Calculus Revisited
299
the process X t is unknown but another one is known to be generated by the following mechanism, d Y , = a(Y,)dt
+ b(Y,)dWt,
Yo= yo E R1.
Given these we are to derive the equation of dynamics for the process X t . W e are already familiar with such problem especially in mathematical physics, but also in the mathematical finance we face to this. For example some authors like M.Schweizer, E.Platen this idea f o r the modelling of the price process in some specific situation, and we, S. Ogawa and M . Mancino," followed the same way to derive the SDE model for price process in such market that admits a feedback of the future information about the delta hedging strategy. For the derivation of the desired SDE, we are only to apply the It6 formula to the conservation equation. But an essential problem can arise when the parameters a , p, y etc. are allowed to arbitrary random variables dependent of the W., for in such case we can no longer suppose that the process X t is causal and we can not apply the It6 theory. Let us take an example again from the mathematical finance.
Example 2.2 (Black-Sholes model in noncausal situation). The SDE in Black-Sholes model is as follows;
dSt = rStdt
+a2StdWt,
(1)
which can be interpreted in the following f o r m of the SDE with the symmetric stochastic integrals, ffz
where the term
s
dSt = ( r - -)Stdt 2
+ uStd*Wt,
d,Wt represents the symmetric integral,l8 that is; for the
B-differentiable causal function f (tlw),
a
is the B-derivative of the f . The symmetric integral is a special case of the noncausal integral that we are to study in the next paragraph. To this simple model there correspond the following two variations of noncausal nature;
-f
awt
(1) Case 1: The case where the constants r, u in the B-S equation are re-
placed by arbitrary random variables.
300
S. Ogawa
(2) Case 2: The case where the Brownian motion W in the equation is replaced b y such random process which does not have a martingale property, for example the Fractional Brownian motion. The first modification can arise when we need to study the B-S model in such situation of admitting the insider trading, and the latter case has been already discussed as the fractional B-S model b y many authors, like A . Shiryaev,6 P. Cheridito’ for example, but without giving a justification of such noncausal type ”SDEs” in terms of the noncausal calculus. For the study of these modified models, first of all we need the introduction of the noncausal stochastic calculus and then we are to give precise meaning to such noncausal SDEs as follows;
+
d X t = a ( t , X t , v ( w ) ) d t b ( t , X t , <(w))dWt, (3)
Xo(w) = t(w). I n view of the application to the SDEs of the cases 1 and 2, we are concerned with the study of the noncausal SDEs, with a special interest o n the validity of a noncausal It6 formula. W e will study these subjects in the paragraphs 3,4 and 5 in a much more general situation.
Example 2.3 (SIE of Fkedholm type). Let us consider the boundary value problem for the second order SDE as follows;
where ( 0 and (1 are arbitrary random variables. where Zt, t E [0,1] is an arbitrary stochastic process defined o n the (Q, F,P ) , with square integrable sample paths. Then as we do for the boundary value problem of ordinary differential equations, by using the Green’s function K ( t ,s, w ) corresponding to the above situation,we may g e t in a very formal way the following integral equation of Fredholm type;
where L ( t , s , w ) = h ( s ) K ( t ,s , w ) and J’d,Z(s)
represents the stochastic
integral of noncausal type with respect to an orthonormal basis {cp,}
in
Nonenusal Stochastic Calculus Revisited
301
LZ(0,l). This is a very typical subject in the noncausal stochastic calculus, since in such situation we can no more suppose that the solution X t is still causal (it i.e. adapted) to the natural filtration generated by the underlying fundamental process Zi,hence the stochastic integral t e r m
s
XtdZt loses its
meaning in the framework of the It6’s theory. I n the article of 198616 the author studied this subject in connection with the boundary value problems of stochastic differential equations and he showed the existence and the uniqueness of solutions (see Theorem 1 in Ref. 16), under some reasonable assumptions on the choice of the fundamental pair (2, { c p n } ) and on the sample regularity of the kernels K and L.
A s a natural extension of such SIE (stochastic integral equation) to the case where the Zt, X , are the random fields, namely the stochastic processes with multi-dimensional parameters t E J = [ O , l l d , we can think of the SIE for the random fields. Imagine for example the case where the driving force process Z,is the Brownian sheet.
+
X(t,u) = f(t,u)
+
L(t,s,w)X(s)ds
K ( t ,s , u ) X ( s ) d , Z ( s ) ,
(6)
For the review of this subject we would refer to the coming a r t i ~ l e which ,~ will appear in another monograph of lecture notes of the same conference. So f a r the examples of noncausal problems are given where the noncausality has entered into the situation through the noncausal quantities, such as the noncausal initial values, the noncausal coeficients in SDE, or the driving stochastic process that does not have the martingale o r semi- martingale property. But the noncausal problem can arise even in the ordinary situation where all the random quantities are supposed to be causal, that is adapted to the natural filtration { 3 t } . Here is another example which is typical in this sense. Example 2.4 (The SPDE called BPE). Given the real Brownian motion Wt and smooth coeficients
we consider the Cauchy problem of a stochastic partial differential equation
302
S. Ogawa
as follows;
a + { a ( t , + b(t,x)-}-u dWt a dt ax
--u
X )
at
dWt dt
= A(t,x)u(t,X , u)-
+ Bu + C ( t ,x )
4 0 ,x ,w ) = u o ( x )
(7) B y the solution of this problem, we understand the random function u(t,x , w ) measurable in ( t ,x , w ) with respect t o the field B p , ~ xl BR1 x 3, especially causal in ( t , w ) for each fixed x E R1,and satisfies the following weak solution equality with probability one for any test function q ( t ,x ) (cf.' 4 ) ;
s
LNVt
+Ll
+ (acplx + A P M t , 2 , w ) + CPldtdX (8)
d x L T ( b P ) 4 t ,x , w)d*Wt +
a where G = [O,T]x R1, pt = -p,
uo(x)Cp(O,x ) d x = 0, L
a pX = -p
1
s
and the t e r m d,Wt stands dX for the symmetric stochastic integral(see the Paragraph 3 below) as it is remarked in the introduction. The equation of this type, called the "Brownian particle equation", was introduced by the author ('4) in the study of wave propagation in random media as a mathematical model for such stochastic propagation phenomenon carried by the Brownian or diffusive particles, namely the propagation along the stochastic trajectory X t governed by the SDE,
at
dXt
= a ( t ,X t ) d t
+ b(t, Xt)d,Wt.
Now the noncausal situation arises in this problem when we try to construct the solution u(t,x , w ) by means of the method of stochastic characteristics which was first studied by the author; B y the formal application of the well known method of characteristics in the theory of the partial differential equation of the first order, or of hyperbolic type, we will get in this case the following system of (symmetric) stochastic integral equations. X ( t i " ) ( s )= x
+
+I
6"
a ( r ,X ( t ' " ) ( r ) ) d r+
6"
b(r, X(t3")(r))d,W,,
t
A~(s,X(~~")(s),w)d,W,.
(9)
Noncausal Stochastic Calculus Revisited
303
Here the first equation gives the characteristic curve X ( t i x ) ( s )( s 5 t ) passing through the fixed point ( t ,x), along which the phenomenon propagates, and the second equation is derived by integrating the PDE along that stochastic calculus. Now remembre that the solution u(t,x,w ) is adopted to the a-field Ft, while the characteristic curve X ( t i x ) ( s ) ,0 6 s 5 t 5 T is measurable with respect to the a-field 3; := a { W ( t ) - W ( r ) s; 5 r 5 t } . The composed function u ( s , X(tix)(s),w) appeared in the second equation above is no more causal with respect to the Brownian motion and hence the term of stochastic integral loses its meaning. Such are the examples of noncausal problems, for the treatment of which we need another theory of stochastic calculus that is free from the restriction of the Causality. The calculus introduced by the author in 1979 is one of such theories and has many advantageous properties compared t o other calculus from the viewpoint of the tool for the modelling and analysis of noncausal random problems. We are going t o give in what follows a resume of that noncausal theory of stochastic calculus, brief but sufficient for us to see that these problems can be treated in very natural way.
3. Review of the Noncausal Stochastic Calculus For the rigorous study of those noncausal problems listed in the previous paragraph, we need to introduce a noncausal stochastic calculus that is free from the restriction of causality and one of such calculus was introduced by the author in 1979 (,23,18,2220 and,” etc.). We are going to give in this paragraph a rapid review of some fundamental results in the theory of noncausal stochastic calculus, mainly following the recent article. l 3 For its special importance of the Brownian motion in the stochastic theory and also for the concreteness of the discussion, we will show mainly the case of the noncausal calculus with respect to the Brownian motion Wt, but as we will see (cf. Remark 3.2) easily the formalism can work even for the ase of more general stochastic processes instead of the Wt. 3.1. Causal functions and the B-diflerentiability Following the,25 we will say that an H-class random function g ( t , w ) is differentiable with respect t o the Brownian motion Wt (or B-differentiable) provided that there exists an M-class random function say & g ( t , w ) such that, for small enough h > 0,
304
S. Ogawa
where the integral
s
dW
stands for the It8’s stochastic integral. The
function &g is called the B-derivative of the g . It is not difficult to see that if the function g ( t , w ) is B-differentiable then its B-derivative is uniquely determined (see25). The B-differentiability of the random function with respect to the multi-dimensional Brownian motion is defined in a similar way.
Remark 3.1. Let g ( t , w ) be a functional of the multi-dimensional Brownian motion, Wt = (W,’, W:, . . . , Wp) where the W i , (1 < i < n) are independent copies of the l-dim. Brownian motion Wt. Then the Bderivative of such function, say V,g, can be defined in the following way: the V,g = ( a t g , w ag , . . . , m ag ) t is a causal random vector such that,
Here we notice that the It8 integral is defined for the causal random functions f ( t , w ) E M and roughly speaking the symmetric integrals (i.e. Z1/2 of O g a ~ and a ~ Stratonovich-Fisk ~ integral) are defined for the causal and B-differentiable functions. That is, the symmetric integral Z 1 / 2 ( f ) of a B-differentiable function was introduced as the limit (in probability) limlal,o Ta(f) of the sequence {Ta(f)} of Riemannian sums,
<
where, A = (0 tl < . . . < t, 5 l} is a partition of the interval [O, 11 and lAl = maxi(ti+l - t i ) . The following result was established by the author in 1970,
Theorem 3.1 (Ref. 25). T h e l i m i t (in p r o b u b i l i t y ) T l p ( f ) = lim lAl-0
exists and is represented in the following f o r m :
z~(f)
Noncausal Stochastic Calculus Revisited
305
3.2. Noncausal stochastic integral
Given a random function f ( t ,w ) E H and an arbitrary complete orthonorma1 system (9,)in L 2 ( [ 0l]), , we consider the formal random series
The stochastic integral of noncausal type introduced by the author in 1979 (23), is given in the following way,
Definition 3.1. A random function f ( t , u )E H is said to be integrable with respect to the basis {pn} (or 9-integrable) when the random series above converges in probability and the sum, denoted by
I'
f ( t , w)d,Wt,
is called the stochastic integral of noncausal type with respect to the basis
{ p n ).
Remark 3.2. The validity of the above definition is not limited to the case of Brownian motion or to other square integrable semi-martingales. Indeed i t can apply even to the case of general square integrable processes say Zt that do not posses the property of semi-martingale, as long as the quantities
are well defined. The simplest example is when we employ the system of Haar functions (Ho,o(t),H,,i(t),O 5 i 5 Z n - l - 1,n E N} as orthonormal basis. Let us remembre that the Hn,i(t) are as follows;
Ho,o(t) = 1,
In this case the quantity above is defined in the natural way as follows;
+
+
2i 1 22 22 2 Jo1Hn,2(t)dZt= 2n'2[{z(-) - Z(-)} - {Z(-) 2" 2n 2"
+
2i 1 Z(-)}I. 2" The noncausal integral with respect to the fractional Brownian motion can be introduced by this way.8 -
306
5'. Ogawa
In general case, the way of convergence of the random series being conditional, the integrability and the sum should depend on the basis, even on the order of the same complete system of orthonormal functions. On the relation between the noncausal integrals with respect to different bases, very few is known except the following, Theorem 3.2 (1984, Ref. 21). If the random function f ( t , u ) E H is
integrable in the L'- sense (ie. convergent in L1(R, P ) sense) with respect to the system of trigonometric functions, (1, JZcos2nnx, JZsin2nnx; n E N}.
Then the f is integrable with respect to the system of Haar functions and the value of two integrals coincide. If the function is integrable with respect to any basis {cp,} and the sum does not depend on the choice of the basis, we will say that the function is universally integrable (or shortly u-integrable). 3.3. Equivalent expressions and variants
Here are some equivalent expressions and a possible variation of the above definition, which are worth to be remarked so that we can have a better understanding of the nature of our noncausal integral. (a) As a limit of the sequence of random Stieltjes integrals: Given the pair (Wt,{ c p n } ) we introduce the sequence of approximation processes W$(t) in the following way.
It is immediate to see that this gives a pathwise smooth approximation of the Brownian motion W ( t , u ) .Moreover, by virtue of the famous theorem due to K.It6 and M.Nishio,2 we know that for any choice of the basis {cp,} the sequence { W z ( t ) } converges uniformly in t E [0,1] as n co with probability one. Now we notice that our noncausal integral can be expressed as the limit (in probability) of the sequence of random Stieltjes integrals;
-
Proposition 3.1. It holds that,
1
1
1
f d p W t := l i m l f d W $ ( t )
(in probability).
Noncausal Stochastic Calculus Revisited
307
(b) Riemannian definition: Let us take the Haar functions { H n , i ( t ) } for basis {cpn}. This is a case of special interest because we have the following,
Lemma 3.1 (1984, Ref. 21). Let us define the approximation process W f ( t )for this case by the following formula, 2k-1
rt
T h e n each W:(t) i s the Cauchy polygonal approximation of the process Wt taken over the set of dyadic points { k / 2 n ; 0 5 k 5 2 n } , that is,
To check this, we introduce the indicator function, xn,i(t) 2 n ’ 2 1 [ 2 - n i , 2 - n ( i + l ) ) ( t ) .It is immediate to see that ( X n , i ,Hm,k) = 0
for ‘ ( m ,k ) with m 2 n
=
+ 1,
here the symbol (., .) denotes the inner product in L2(0,1). Therefore each xn,i should be represented as linear combination of the membres {Hm,k,m 5 n } , say; Xn,i(t) = C ( ~ , JO,O)Ho,o ’;
2(m- 1)- 1
+
c c
l<m
C ( n ,2; m, k ) H m , k ( t ) .
k=O
q n , i;m, k ) = ( X n , i ,H m d . It is also easy to see that we have the following relation;
c
2n- 1
C ( n ,i; m, k ) C ( n ,i; l , j ) = 6m,&j.
i=O
Based on this relation we can get the following equality,
and this is what we want to see.
308 5'. Ogawa
Now applying this result to the expression given in the Proposition 3.1 of (a), we see that in this case the defining formula of the noncausal integral is given as the Riemannian sum,
1'
c .I'
2-"(i+l)
f d H W t = lim
2n-
n-+w i=o
2n
f (s)ds.{ w ( 2 - n (i+l))-w ( 2 - " 2 ) } ,
2-ni
(14) This type of definition can be found in recent publications of some authors. But as we have seen in here ( 2 1 ) 1 this is merely a special case of our integral. (c) Let Dn(t,s ) be the kernel given by, n
Dn(t,).
= C Y k ( t ) P k ( S ) , ( t ,s E
[0,11).
k=l
Then we have the following representation for the noncausal integral,
i'fd,W(t)
= n-co lim
1'
dt
f ( t ,w ) D n ( t , s ) d W s (limit in probabil-
ity).
For the case of trigonometric functions, the kernel Dn(t,s ) is the Dirichlet kernel appearing in the theory of Fourier series. (d) A generalization of the above view: Replace the kernels {D,(t, s)} in the above interpretation by any 6- sequence say { K n ( t , s ) } then , wc will get a generalized formula for the noncausal integral:
3.4. Condition for the integrability - in the framework of
the Homogeneous Chaos theory Let Ho be the totality of all random functions f ( t , w > E H such that, 1
If
( t ,w)12dt < 00. By Wiener-ItB's theory of Homogeneous Chaos, we know that such function f E Ho can be decomposed into the slim of multiple Wiener integrals, that is: There exists a set of kernels, say { k L ( t ;t l , . + ., tn)}rZo, such that
k: E L 2 ( [ 0,In+' , )
with ~ n ! I I ] c ~ l l<~03, +, n
Noncausal Stochastic Calculus Revisited
309
symmetric in n-parameters ( t l ,., tn) E [0,lIn and that, M
n=O
where
11 . [In
stands for the norm in L2([0,1In)space.
We will denote by H1 the totality of all Ho- functions f ( t ,w ) such that, w
~ n n ! l / k ~ l l<~ 00. + lGiven a function f E H1 we introduce its stochastic n=l
derivative V f by the following formula,
n=l Since
El1
i 1 ( V f ( t , s ) ) 2 d t d s= ~ n n ! / l k ~ / l ~we + l notice , that the
stochastic derivative D f (t, s) is well defined for the f E H I . Now we can state the condition for the p-integrability of the HI-class functions in the following theorem that was obtained by the author in 1984.
Theorem 3.3 (Ref. 22). Let f E H1 and let { p n } be an arbitrary orthonormal basis. T h e n the necessary and suficient condition f o r the random
function f t o be cp-integrable is that the lim
n-+m
exists in probability. 3.5. Relation between symmetric and noncausal integrals
We call a random function f ( t , w ) semi martingale when it admits the decomposition, f ( t , w )
= a ( t ,w )
+
s'
f d W s where
f E M and a ( t ) is such
that almost every sample path is of bounded variation in t over [0,1].Notice that if sup E l a ( t ) - a(s)I2 = o(h) then f is B-differentiable. t,slt-sl
The followings are the basic results concerning the relation between the symmetric integrals with the noncausal integral.
310
S. Ogawa
Theorem 3.4 (Ref. 18). Every causal B- differentiable function is integrable in noncausal sense with respect to the system of Haar functions and the sum coincides with that of the symmetric integrals:
We say that an orthonoral basis {cp,} the next condition:
is regular provided that it satisfies
Remark 3.3. Notice that this condition (15) is equivalent to the fact that, 1 w - lim u, = - (in L ~ ) n-cc 2 namely to the fact that, for any f ( t ) E L2(0,1) it holds the following,
Theorem 3.5 (Ref. 18). Every semi martingale (causal or not) becomes cp-integrable, iff the basis (9,) is regular. I n this case the noncausal integral coincides with the symmetric integrals. Related to this result is a natural and interesting question asking whether there can or can not be a basis {cp,} which is not regular. This question is affirmatively answered by P.Mayer and M . M a n ~ i n o .We ~ can go on further. The next result shows that a smoothness in Wt of the integrand ensures the integrability with respect to any orthonormal basis.
Theorem 3.6 (Ref. 18). Every semi martingale that is twice Bdifferentiable, namely the B-derivative f^ is again a semi martingale, is u-integrable. In their article^,^,^ of the authors M.Zakai and D.Nualart, the noncausal integral was referred as the intrinsic Ogawa integral. We like to remark at this stage that, precisely saying what they referred there was our noncausal integral for the case of u-integrable functions, Now let us finish this paragraph by giving another result on the way of convergence of noncausal intgerals for semi-martingales, for its usefulness in appications.
Noncausal Stochastic Calculus Revisited
Proposition 3.2 (1985, Ref. 18). If the B-derivative f of the semi-martingale f , df = fdWt satisfies the following condition,
P{
Then the noncausal integral
I’ I”
311
+ g(t,w)dt,
f 4 ( t ) d t < m} = 1.
fd,Wt with respect to the regular basis con-
verges uniformely in t E [O, 11.
4. Applications to the Noncausal SDEs Now we like to show basic results concerning the Cauchy problem of the noncausal SDEs and the validity of the noncausal It6 formula, which will give us at the same time the appropriate answers to those noncausal problems listed in the Paragraph 2. 4.1. Noncausal Cauchy problem First notice that the SDE in (3) becomes meaningful in the framework of the noncausal stochastic calculus. that is:
d X t = a(t,Xt,77(w))dt+b(t,Xt,ll(w))d,Wt, t E (O,T], (16)
Xo(w) = E ( w ) here the (9,) is a regular basis in L2(0,l),which we will fix throughout the discussion. We notice at this stage that when the parameters [(w), ~ ( ware ) not random and the solution Xt can be supposed to be causal, then by virtue of the Theorem 3.5 the SDE in (16) is reduced to the usual SDE with symmetric integration,
dXt =a(t,Xt,77)dt+b(t,Xt,rl)dWt, t E (O,T], (17)
XO(W) = E.
The Cauchy problem for the noncausal SDE was first studied by the authorlg for such simple case where the parameter 77 is not random or
312
S. Ogawa
does not appear in a ( t lIC), b ( t , z) and only the initial data [ ( w ) arises as a noncausal factor.
d X t = a ( t ,X t ) d t
+ b ( t ,X t ) d p W t , t E (O,T], (18)
Xo(w) = [(w). For this case, the results on the existence and a kind of uniqueness properties of the solution are proved under a milder assumption on the regularity of the coefficients a ( . ) ,b(.) as follows: Assumption 4.1. T h e coeficients a ( t ,IC), b ( t , x) are suficiently regular
in such sense that,
a2
(1) a ( t ,x ) , -b(t, x) are of Cl-class, 8x2 (2) a ( t , x),b ( t , x ) are suficiently regular in the sense that the causal Cauchy problem 17 admits the unique strong solution X ( t ,w ; [) and that the X ( t ,w ; <) i s continuous in ( t ,E ) with probability one. Notice that under such conditions the composit,e
w1w )
= X(t1 w ; E ( w ) )
of the strong solution X ( t ,[, w ) of the (17) and the random variable [ ( w ) is well defined, which we expect to be a solution of the noncausal Cauchy problem (18). In fact we have the following, Theorem 4.1 (1985, Ref. 19). T h e composite X ( t , w ) i s a solution of the noncausal Cauchy problem (18).
We have also found that this solution X ( t , w ) verifies the It6 formula of noncausal type, that is: Proposition 4.1 (1985, Ref. 17). For any function F ( z ) E C4 it holds
the equality, d F ( X t ) = F’(Xt){a(t,Xt)dt
+ b ( t ,X t ) d , W t }
05t51
As an application of this we can show the following result that concerns the uniqueness of the solution for the noncausal problem (18), Corollary 4.1 (Ref. 17). W h e n the b ( x )
# 0 , the composed function
X ( t ,w ) = X ( t ,w ; [(w))i s the unique solution among all random functions verifying the It6 formula 4.1 of noncausal type. The proof of this together with that of the previously presented Proposition 4.1 will be given in the next paragraph for a more general case.
Noncausal Stochastic Calculus Revisited
313
4.2. Discussions f o r the more general cases We are going to give in this paragraph the results on the Cauchy problem for the more general case (16).
Assumption 4.2. W e suppose that the coeficients a ( t , x;r ] ) , b(t, x ; r ] ) are suficiently regular in such sense that, for an arbitrary couple of parameters (6,r ] ) the causal Cauchy problem 1'7 admits the unique strong solution X ( t , w; E, r ] ) and that the X ( t , w; [, v) is continuous in ( t ,E, r ] ) with probability one.
Remark 4.1. The assumption is satisfied when, for example, the a ( t , x;r ] ) , b(t, x ; r ] ) are of the C4-class in x , of C1-class in r] and all derivatives are bounded on [0,1] x R1. We also notice that under the Assumption 4.2 the composite
X ( t , w ) = X ( t , w; E(w),r](w)) of the strong solution X ( t , w ; [ , r ] ) of the (17) and the random variables [ ( w ) , ~ ( wis) well defined, and as in the previous case we expect this composite X ( t , w ) to be a solution of the noncausal Cauchy problem (16). In fact we have the following,
Theorem 4.2. The X gives a noncausal solution of the noncausal Cauchy problem 16. For the verification of this, we need some preparations.
Proposition 4.2. Let f ( t , w; E , v) such that for each fixed ( I ,r ] ) ,
(5, v
E [-A, A ] ) be a semi-martingale
d f ( t ,w ; E , 7 ) = d t ,w; E , r1)dt + h(t1w; 77)dWt
(19)
El
where g(.), h(.) are causal random functions satisfying the following condition,
.rJ_A,
4
1;
dr] 1 1 { g 2 ( t ,w; El r ] )
+ h2(t,w; E , r])W
=1
(i) Then for any regular basis {pn} in L2(0,l), the following equality holds,
(in probability)
314 S. Ogawa
(ii) Moreover i f the coeficient h(t,w ; <,r ] ) in the decomposition (19) again becomes a semi-martingale satisfying the same condition as the f(.), then the equality (20) still holds true for any basis {pn).
Proof. Put f
= f1
+ f2
where,
and
Hence with the help of the Theorem of Nishio-It6 we confirm that,
(in probability) For the term
where,
f2
we have the decomposition,
Noncausal Stochastic Calculus Revisited
and,
zn =
315
1 I
Pn(t)dW(t).
We are t o show that: A
I&(c,r]) = 0 (in probability), (1 5 i 54). Since for the quantities I,,, (i = 1 , 3 , 4 ) this could be easily done by a usual routine, it would suffice to show the result only for the term I z , ~ . By taking the Remark3.3 into account, we see that for each fixed (E, r ] ) we have, 1
h(t,w; E, r]){
n+o3
51 - u,(t)}dt = 0
On the other hand we have,
Now given the unique solution X ( t ,w ; <,'I) of the causal problem 17, we introduce the sequence of random functions in the following way,
X,"(t, w; E , r ] )
= E+
It
a(s, X ( s ,w; E , 'I); r])ds+
It
b(s, X ( s ,w; E , 77); v ) d W Z ( s )
(21) where W$ is the approximate process of the Brownian motion introduced in the previous paragraph. We easily see by the Theorem 3.5 that for each fixed t , (I r], ) , we have (in probablity). Moreover we can see lim X g ( t , w ; <, 77) = X ( t ,w ; E , 'I), n-+m that this convergence is uniform in ( E , r ] ) on every finite set CA = [-A, A] x
[-A, A]. Proposition 4.3. For an arbitrarily large A relation at each fixed t E [ O , 1 ] , lim
n-o3
sup (€,V)ECA
Proof. Put
> 0 it holds the following
I X z ( t ,w ; <,77) - X ( t ,w; <,'I)\ = 0 (in probability)
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S. Ogawa
From equations (17),(21) we obtain the following:
&(t, w ; E , V )
=l
b ( X ( s ;E, 71); rl){dW$(s)
On the other hand we have the following expression,
which implies that,
where
-
We are to show that for each fixed t these J l ( n ) ,&(n),J 3 ( n ) tend to zero 00. Since a t this stage the parameters E , 7 remain in probability as n as deterministic constants, we notice that the X ( t ,w ; E, rl) is causal and derivable in E, 77. In fact under the assumption (4.1) on the regularity of the coefficients a ( . ) ,b(.) it is easy to verify that the derivatives,
Noncausal Stochastic Calculus Revisited
IX 3 ( 0 )
317
=0
This combined with the expression (22) would imply that the quantity A,@, w ; 7) is derivable in <,7 and that the order of the derivation in I ,7 and the integration is exchangeable. For example,
Hence by virtue of the Proposition 4.2 we only need to show that the following quantities,
are semi-martingales satisfying the condition in that Proposition. Since this can be verified by a simple routine work, we see that we are done. 0 Now we are going to give the proof for our Theorem,
S. Ogawa
318
5. Question of Uniqueness - Noncausal It8 Formula
The noncausal solution of the problem (16), a(t,w ) = X ( t ,w ; [ ( w ) , ~ ( w ) ) constructed in the Theorem 4.2, has a remarkable property as stated in the next ,
Theorem 5.1 (Noncausal It6 formula). For any random variable [ ( w ) and any function F ( x ,y ) , which is differentiable in (x,y ) and of C4-class in x with bounded derivatives, it holds the following equality:
d F ( X t , C(w)) = ( W w t ,CbJ))MXt;rl(w))dt+ @t; r l ( w ) ) ~ , W t ) (23) Proof. Let X ( t ,w ; E , 17) be the unique solution of the causal SDE (17) with deterministic parameters 77). Then by the usual It6 formula for causal functions, we have for each fixed deterministic parameters C), the following relation:
(c,
(c,~,
F ( X ( t ;E , r l ) , C) =F(t,
C) +
(24)
I'
( & F ) ( X ( S ,w ; t ,r l ) , C){a(Xs;77)ds + q
Here the stochastic integral
x s ; rl)dWs)
s
dWt stands for the causal symmetric integral.
Given this we introduce the approximation sequence as follows,
F Y t , w ; E , rl, C)
(25)
Following the same argument as in the proof of Proposition 4.2, we would easily verify that for each fixed t E [0,1] the sequence F n ( t , w ; c , ~ , C ) converges to F ( X ( t , w ; E , v ) , C )in probability as n + 00, uniformly in (6,q, C) E C> on any finite set C> = [ - A , AI3. Hence we confirm, again following the same argument as in the proof of the Theorem 4.2, that for each fixed t the sequence F n ( t ,w ; c ( w ) , ~ ( w )<,( w ) ) converges in probability to the F ( X ( t , w ;[ ( w ) , ~ ( w )C(w)) , = F ( X ( t , w ) ,[ ( w ) ) . Now from the equation (25) we see that the following limit, n-cc lim
Lt(W(X(Sw , ; E ( w ) , rl(w)),C(w))b(X(s,w ; E(wL rl(w));rl(w))dW$ (s)
should converge in probability to the limit,
Noncuusal Stochastic Calculus Revisited
319
by definition of the X ( t lw ) and by definition of the noncausal integral with respect to the basis {cp,}. Thus from this fact we get the desired equality (23 ), by letting n 4 00 on 0 both sides of the equality ( 2 5 ) .
As we have mentioned in the previous paragraph, this fact that the solution X ( t , w ) = X ( t , w ; [ ( w ) , ~ ( w ) of ) the noncausal problem (16) satisfies the It6 formula of noncausal type (23) would give us a partial answer to the question of uniqueness of the solution of our noncausal problem. In fact we have the following result that is valid for the case of 1-dimensional SDE.
Corollary 5.1. If the b ( t , z ; q ) does not depend on the t and b ( z ; q )> 0 (or < 0 ) for all ( t ,x,q ) , then the solution X ( t lw ) is unique among the all random functions that werify the noncausal It6 formula (23). We will call such solution the regular solution of the Cauchy problem.
Proof. Without loss of generality we suppose that b(.) > 0. Put Y ( t )= F ( X ( t ) )where F ( z ) is as follows,
Then we have, k ( t , w ) = F - l ( Y ( t , w ) ) . By applying the noncausal It6 formula to the function Y ( t )we get,
Since this is merely a family of ordinary integral equations parametrized by the w , we see the uniqueness of its solution Y ( t )for each fixed and hence the uniqueness of the r?(tlw ) . This completes the proof. 0
References 1. P. Cheridito, Arbitrage in fractional Brownian motion models, (2002). 2. K. It6 and M. Nishio, Osaka J . Math. 5 , 35 (1968). 3. M. Mancino & L. Pratelli, Rendiconti Acad. Nat. delle Scienze detta dei X L , Memorie d i Matematica, 112 Vol. xviii, fasc 1, (1994) pp. 89-101 4. P. Mayer & M. Mancino, Se'minaire de Probabilite's xxxi, 198 (1995). 5. D. Nualart and M. Zakai Ann. Probab. 17,1536 (1989). 6. A . N. Shiryaev, On arbitrage and replication for fractional models, Research Report 20, MaPhySto, (Dept.of Math.Sci., Univ. of Aarhus, 1998) 7. M.Zakai & D. Nualart, Proha. a n d Math. Statist. 3,37 (1987).
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8. S. Ogawa & T. Miura, The fractional B-S model interpreted by the Ogawa integral, presented at the “poster session” of the Congres on RISK Control Theory, Firenze (2005) 9. S. Ogawa, Stochastic integral equations of Fredholm type, (to appear i n ) Harmonic, Wavelet and P-adic Analysis, (2005) 10. S . Ogawa, On noncausal Cauchy problem for the noncausal SDEs, In: S.Watanabe et al. (Eds.): Stochastic Processes and Applications t o Mathematical Finance, (World Scientific, Singapore, pp. 289-304, 2004). 11. S. Ogawa & M. E. Mancino, Nonlinear feedback effects by hedging strategies, In: S.Watanabe et al. (Eds.): Stochastic Processes and Applications to Mathematical Fznance, (World Scientific, Singapore, pp. 255-270, 2004) 12. S. Ogawa & M. Mancino, On a noncausal insider trading equilibrium model of asset pricing (in preparation) 13. S . Ogawa, Rendiconti Acad. Nazionale delle Scienze detta dei XL,119, Vol. xxv (2001) 14. S. Ogawa, On a stochastic integral equation for the random fields, Se‘minaire de Proba. vol.xxv, (Springer Verlag, Berlin, pp. 324-339. 1991) 15. S. Ogawa, Topics in the theory of noncausal stochastic integral equations, Diffusion Processes and Related Problems in Analysis, edt by M.Pinsky, (Birkhauser Boston, 1990). 16. S. Ogawa, On the stochastic integral equation of Fredholm type, W a v e s and Patterns (monograph) (Kinokuniya and North-Holland, pp. 597-605, 1986). 17. S. Ogawa & T. Sekiguchi, On the It& formula of noncausal type, Proc.Japan Acad. Sci, (1985). 18. S . Ogawa, The stochastic integral of noncausal type as an extension of the symmetric integrals, Japan J.Applied Math., (1985). 19. S. Ogawa, Sur la question d’existence de solution de l’equation stochastique differentielle du type noncausal, Kyoto J . of Math., (Kyoto Univ., 1985). 20. S. Ogawa, Quelques propriMs de l’integrale stochastique du type noncausal, Japan J.Applied Math., (1984) 21. S . Ogawa, T 6 h o k u Math. J., 36,41 (1984). 22. S. Ogawa, Noncausal Problems in Stochastic Calculus (in japanese) in Proc. of the Workshop o n Noncausal Calculus and T h e Related Problems held at RIMS Kyoto Univ. in February 1984, (Suuri Kagaku Kokyuuroku (RIMS Report) 527, 1984). 23. S. Ogawa, C.R. Acad. Sci., Paris, t.288, SBrie A, pp.359-362, (1979). 24. S. Ogawa, 2. W.verw. Geb., 28, 53 (1973). 25. S. Ogawa, Proc. Japan Acad. Sci., 70 (1970).
Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 321-339) @ 2007 World Scientific Publishing Go.
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$17. INFINITE-DIMENSIONAL STOCHASTIC ANALYSIS AND FOUNDATIONS OF QUANTUM MECHANICS ANDRE1 KHRENNIKOV
International Center for Mathematical Modeling in Physics and Cognitive Sciences, MSI, University of Vaxjo, S-35195, Sweden E-mail: [email protected] We show t h a t by using an asymptotic expansion of Gaussian integrals of analytic functions on infinite-dimensional spaces it is possible to represent quant u m mechanics as a projection of classical statistical mechanics on infinitedimensional space.
1. Introduction
The problem of reduction of quantum mechanics to classical statistical mechanics has been discussed from the first days of quantum mechanics, see, e.g., Refs. 1-4. Now days this problem is known as the problem of hidden variables or completeness of quantum mechanics, see, e.g., Refs. 5-8 for debates. There is a rather common opinion that quantum mechanics is complete and that it is impossible to introduce “hidden variables” providing more detailed description than quantum mechanics. In Ref. 9 it was demonstrated that in the opposition to this opinion it is possible to represent quantum mechanics as projection of classical statistical mechanics on infinite-dimensional space. In this paper we present this approach (which was called in Ref. 9 Prequantum Classical Statistical Field Theory PCSFT) on the mathematical level of rigorousness; in particular, some functional spaces introduced in Ref. 9 should be modified to obtain the correct results. In the present paper we also find connection of PCSFT with background Gaussian random field on Hilbert space. Finally, we extend the approach of Ref. 9 by considering unbounded operators, see Sec. 6. There are also differences in interpretations of a small parameter of our asymptotic procedure of dequantization. In [9] this parameter was identified with the Planck constant h. In this paper we introduce a new parameter a giving ~
A . Khrennikov
322
the dispersion of prequantum fluctuations, see Ref. 10 for more details on physical interpretation. To simplify considerations, in this paper we consider quantum formalism over the field of real numbers, see Ref. 10 for dequantization of complex quantum mechanics. To exclude possible misunderstanding, we emphasize from the very beginning that our paper is n o t about d e f o r m a t i o n quantization for s y s t e m s w i t h t h e infinite n u m b e r of degrees of freedom, see, e.g., Ref. 11, but about dequantization of conventional quantum mechanics for systems with a finite number of degrees of freedom by using analysis on infinite-dimensional space. 2. Infinite-dimensional analysis Let H be a real Hilbert space and let A : H -+ H be a continuous selfadjoint linear operator. The basic mathematical formula which will be used in this paper is the formula for a Gaussian integral of a quadratic form
f($)
fA("b) =
(A'$,$1.
Let dp(+) be a a-additive Gaussian measure on the a-field F of Bore1 subsets of H , see Ref. 12. This measure is determined by its covariation operator B : H + H and mean value m = mp E H . For example, B and m determines the Fourier transform of p : P(y) = S H e i ( Y ' G ) d p ( $ ) = e ~ ( B y ~ y ) t i ( m ~yY )E, H . In what follows we restrict our considerations to Gaussian measures with zero mean value m = 0, where ( m , y ) = S(,y, + ) d p ( $ ) = 0 for any y E H . Sometimes there will be used the symbol p~ to denote the Gaussian measure with the covariation operator B and m = 0. We recall that the covariation operator B = cov p is defined by (By1,yz) = ~(~~,~CI)(Y~,$)~P(~CI),Y~,Y~ E H , and has the following properties: a). B 0, i.e., ( B y , y ) 0 , y € H ; b). B is a self-adjoint operator, B E C , ( H ) ; c). B is a trace-class operator and Tr B = II$l12dp(+). This is dispersion cr2(p) of the probability p. Thus a 2 ( p ) = Tr B. We pay attention that the list of properties of the covariation operator of a Gaussian measure differs from the list of properties of a von Neumann density operator [4] only by one condition: Tr D = 1, for a density operator
>
>
sH
D. We can easily find the Gaussian integral of the quadratic form
fA($)
:
(1)
The differential calculus for maps f : H + R does not differ so much from the differential calculus in the finite dimensional case, f : R" -+ R. Instead of the norm on R", one should use the norm on H . We con-
Stochastic Analysis and Foundations of Quantum Mechanics
323
sider so called F’rechet differentiability. Here a function f is differentiable if it can be represented as f($o A$) = f($o) f’($o)(A$) o(A$), where l i m ~ ~ ~ Ilo(A‘)tl ~IlA+ll ~ ~ +=o0. Here a t each point $ the derivative f’($) is a continuous linear functional on H ; so it can be identified with the element f’($) E H . Then we can define the second derivative as the derivative of the map $ ---f f’($) and so on. A map f is differentiable n-times iff: f ( $ o A$) = f($o) f’($o)(A$) +f”($o)(A$,A$) ... &f(”)($o)(A$,...)A$) on(A$), where f(”)(+o) is a symmetric continuous n-linear form on H and limlla+ll-o = 0. For us it is important that f”($o) can be represented by a symmetric operator f”($o)(ulu) = ( f ” ( $ ) o ) ~ , ~ ) , E u ,H u (this fact is well know in the finite dimensional case: the matrice representing the second derivative of any two times differentiable function f : R” + R is symmetric). We remark that f($) = f ( 0 ) + f’(O)($) $f”(O)($, $1 ’.. $f‘”’(0)(7b1 . . . I $) on(+). For a real Hilbert space HI denote by the symbol H C its complexification: H C = H @ iH.We recall that a function f : HC + C is analytic if it can be expanded into the Taylor series:
+
+
+
+
+
+
+
+
+
+ +
+
+
which converges uniformly on any ball of H C 3 . Dequantization
3.1. Classical and quantum statistical models We define “classical statistical models” in the following way, see Ref. 9 for more detail (and even philosophic considerations): a). Physical states w are represented by points of some set R (state space). b). Physical variables are represented by functions f : R + R belonging to some functional space V(R).* c). Statistical states are represented by probability measures on R belonging to some class S(R). + * T h e choice of a concrete functional space V(R) depends on various physical and mathematical factors. + I t is assumed (by the Kolmogorov axiomatics) t h a t there is given a fixed u-field of subsets of R denoted by F. Probabilities are defined on F It is, of course, assumed t h a t physical variables are represented by random variables - measurable functions. T h e choice of a concrete space of probability measures S ( R ) depends on various physical and mathematical factors.
324
A . Khrennakov
d). The average of a physical variable (which is represented by a function V(R)) with respect to a statistical state (which is represented by a probability measure p E S(R)) is given by
f
E
A classical statistical model is a pair M = (S(R), V(R)). We recall that classical statistical mechanics on the phase space R2, = R" x Rn gives an example of a classical statistical model. But we shall not be interested in this example in our further considerations. We shall develop a classical statistical model with an infinite-dimensional phase-space. In real Hilbert space H a quantum statistical model is described in the following way: a). Physical observables are represented by operators A : H -+ H belonging to the class of continuous self-adjoint operators C, = C C , ( H ) . b). Statistical states are represented by density operators, see Ref. 4. The class of such operators is denoted by D D ( H ) . d). The average of a physical observable (which is represented by the operator A E L C , ( H )with ) respect to a statistical state (which is represented by the density operator D E D ( H ) )is given by von Neumann's f ~ r m u l a : ~
-
< A >o- Tr DA
(4)
The quantum statistical model is the pair Nquant = (D(H)L , ,(H)). 3.2. Asymptotic Gaussian analysis
Let us consider a classical statistical model in that the state space R = H (in physical applications H = L2(R3)is the space of classical fields on R3) and the space of statistical states consists of Gaussian measures with zero mean value and dispersion
where a > 0 is a small real parameter. Denote such a class of Gaussian measures by the symbol S$(R). For p E S$(O), we have
Tr cov p = 0'
(6)
We remark that any linear transformation (in particular, scaling) preserves the class of Gaussian measures. Let us make the change of variables (scaling) :
Stochastic Analysis and Foundations of Quantum Mechanics
325
(we emphasize that this is a scaling not in the physical space R3, but in the space of fields on it). To find the covariation operator D of the image p~ of the Gaussian measure p ~ we, compute its Fourier transform:
Thus
D=-=%!!?. B a a We shall use this formula later. We remark that by definition:
To make our further considerations mathematically rigorous, we should attract the theory of analytic functions f : RC + C . Here RC = R @ ZR is the complexification of the real Hilbert space R. Let b, : RC x ... x RC 4 C be a continuous n-linear symmetric form. We define its norm by llb,ll = supIIGlII1 I&($, ...,$)I. Thus
Ibn(+r
"'7
$)I 5 Ilbnll11+1In
(9)
Let us consider the space analytic functions of the exponential growth:
If(+)/
5 aebllQll,+ E oC.
(10)
Here constants depend on f : a = a f , b = b f .
Lemma 3.1. l 3 The space of analytic functions of the exponential growth coincides with the space of analytic functions such that: ~ ~ f ( n )5 ( 0c )r n ~ ,~n = 0,1,2, ...
Here constants c = c f and r
=rf
(11)
depend on the function f .
Proof. A). Let f have the exponential growth. For any E RC, we consider the function of the complex variable z E C : g+(z) = f (z+). By the Cauchy integral formula for gG(z) we have: g$)(O) = g+ ( z ) z - ( " f l ) d z ,
+
sJz,=R
where a t the moment R > 0 is a free parameter. Thus: Ig$'(O)l 5 n!RPn If (Reie$)l 5 afn!R-nebfRII+II. By choosing R = n and observing that g$)(O) = f(")(O)($,
11 f(")
...,$) we obtain:
(0)11 5 a;e-nn1/2ebfn.
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A . Khrennikov
Thus the derivatives o f f satisfy the inequalities (11) with r f = e b f . B). Let now derivatives of f satisfy the inequalities (11). Then by the inequalities (9) we have If(+)l 5 C,"==, Ilf(")(0)IIII+lln/n!5 cf C;==,(rf 11 ~ ~ + ~ ~ )5" /cfe'fII+II. n,! Thus f has the exponential growth with bf =r f . 0 We denote by the symbol V(R) the following space of functions f : R 4 R. Each f E V(R) takes the value zero a t the point $ = 0 and it can be extended to the analytic function f : RC 4 C having the exponential growth.
Example 3.1. In particular, any polynomial on R belongs to the space V(R). For example, let A l l ..., AN be continuous linear operators. Then function f ( + ) = C ~ s l ( A n + $, ) n belongs to the space V(R). Any function f E V(R) is integrable with respect to any Gaussian measure on R. Let us consider the family of the classical statistical models Ma
=
(S,%(R),V(R)).
Let a variable f E V(R) and let a statistical state p~ E S,%(R).Our further aim is t o find an asymptotic expansion of the (classical) average < f > p s = f ( $ ) d p ~(+) with respect to the small parameter a.
s,
Lemma 3.2. Let f E V(R) and let p E S,%(R). T h e n the following asymptotic equality holds:
< f >p=
a - Tr D f"(0) 2
+ o(Q),
4
0,
(12)
where the operator D i s given by (8). Here .(a) = a2R(a1f l P I ,
where IR(a,f , p)I
Icf
so erfll+lldpD($).
Proof. In the Gaussian integral
where
(13)
/,
f ( $ ) d p ( + ) we make the scaling (7):
Stochastic Analysis and Foundations of Quantum Mechanics
327
We pay attention that
because the mean value of p (and, hence, of p ~ is) equal to zero. Since p E Sz(R),we have Tr D = 1. The change of variables in (14) can be considered as scaling of the magnitude of statistical (Gaussian) fluctuations. Negligibly small random fluctuations a ( p ) = (where a is a small parameter) are considered in the new scale as standard normal fluctuations. If we use the language of probability theory and consider a Gaussian random variables <(A), then the transformation (7) is nothing else than the standard normalization of this random variable (which is used, for example, in the central limit theorem): E(A)-EE (in our case E J = 0).
6
q(A) =
JE(E(A)-EE)Z
We now estimate the rest term R(a,f , p). By using the inequality (11) we have for a 5 1 :
By using the equality (1) we finally come the asymptotic equality (12). 0 We see that the classical average (computed in the model M a = ( S g (R), V(R)) by using the measure-theoretic approach) is coupled through (12) to the quantum average (computed in the model N q u a n t = (D(R),
&(a))by the von Neumann trace-formula). The equality (12) can be used as the motivation for defining the following classical quantum map T from the classical statistical model M a = (S& V ) onto the quantum statistical model Nquant = (D, L S ): ---f
cov p a (the Gaussian measure p is represented by the density matrix D which is equal to the covariation operator of this measure normalized by a ) ;
T : Sg(R) 4 D(R), D
T : V(Q)
-+
L ( o ) , Aquant
= T ( p )= -
= T(f) =
Our previous considerations can be presented as
1
,f”(O).
(17)
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A . Khrennikov
Theorem 3.1. The map T :
Sz(R) + D(R) i s one-to-one; the m a p T
:
V(s2) + C,(R) is linear surjection and the classical and quantum averages are coupled by the asymptotic equality (12). We emphasize that the correspondence between physical variables f E V(s2) and physical observables A E &(a)is not one-to-one. Let again R = H . We consider a classical statistical model such that the class of statistical states consists of Gaussian measures p on R having zero mean value and unit dispersion
These are Gaussian measures having covariance operators with the unit trace. Denote the class of such probabilities by the symbol SG = SG(R). In this model we choose a class of physical variables consisting of quadratic forms f A ( 2 ) = (Az,z). We denote this class by Vquad. We remark that this is a linear space (over R). We consider the following classical statistical model:
As always in a statistical model, we are interested only in averages of physical variables f E Vquad with respect to statistical states p E &(a).w e emphasize that by (1):
Let us consider the following map T from the classical statistical model (SG,Vquad) to the quantum statistical model Nquant= ( D ,L,) :
Mquad =
T : SG(R) + D ( n ) , T ( p )= cov p
(19)
(the Gaussian measure p~ is represented by the density matrix D which is equal to the covariation operator of this measure), and we define:
(thus a variable f E
Vquad(fl)
is represented by its second derivative).
Theorem 3.2. The map T provides one-to-one correspondence between the classical statistical model M q u a d and the quantum model N q u a n t .
Stochastic Analysis and Foundations of Q u a n t u m Mechanics
329
4. G a u s s i a n u n d e r g r o u n d for pure states
In quantum mechanics a pure quantum state is given by a normalized vector @ E H : llPll = 1. In our model such a state is not pure a t all (in the sense that such a vector Q does not provide a description of an individual system). Such a normalized vector Q is the label of a Gaussian statistical mixture. The corresponding quantum statistical state is represented by the density operator: Dq = 9 @ @. In particular, the von Neumann's traceformula for expectation has the form: Tr DqA = (A@,@). Let us consider the correspondence map T for statistical states for the classical statistical model M a = ( S & V ) , see (16). A pure quantum state 9 (i.e., the state with the density operator Dq)is the image of the Gaussian statistical mixture pq of states $J E H . We use the capital @ to denote a quantum pure state. This is just the special system of labeling of the Gaussian measure pq by the normalized vector @ of Hilbert space. Points of the sample space on that this measure is defined we denote by the low $J. The measure pq has the covariation operator Bq = a&. This means that the measure pq is concentrated on the one-dimensional subspace H a = {x E H : x = s q , s E R}. This is one-dimensional Gaussian distribution.
5. Pure states as one-dimensional projections of spatial white-noise In section 4 we showed that so called pure states of quantum mechanics have the natural classical statistical interpretation as Gaussian measures concentrated on one-dimensional subspaces of the Hilbert space H . On the other hand, it is well known that any Gaussian measure o n H i s determined by i t s one-dimensional projections. To determine a Gaussian random variable [ ( w ) E H , it is sufficient to determine all its one-dimensional projections: [ ~ ( w= ) ( @ , [ ( w ) ) , Q E H . The covariation operator B of [ (having the , = EJ:. We are interested in the zero mean value) is defined by ( B @@) following problem:
Is it possible t o construct a Gaussian distribution o n H such that i t s onedimensional projections will give u s all pure quantum states, @ E H , 11@11 = l?
We recall that in our approach a pure quantum state @ is just the label for a Gaussian random variable [\I, such that E[; = 0111611~ = 01. Thus the answer to our question is positive and pure quantum states can be considered as one-dimensional projections of the &-scaling of the standard Gaussian distribution on H . The standard Gaussian distribution p on H (so
330
A . Khrennikov
the average of p is equal to zero and cov p = I , where I is the unit operator) is nothing else than the white noise on R3 (if one chooses H = L2(R3)), see [12] for details. Thus pure quantum states are simply one-dimensional projections of the spatial white noise. It is well known, see, e.g., [12], that the p is not a-additive on the a-field of Bore1 subsets of H . To escape mathematical difficulties and concentrate on the dequantization of quantum mechanics, we start with consideration of the finitedimensional case.
5.1. The finite-dimensional case We consider the family of Gaussian random variables [Q, 9 E R”,E& = 0 , E [ i = a119112.This family can be realized as [ ~ ( w )= (Q,[(w))where [ ( w ) = &‘q(w) and q(w)E R” is standard Gaussian random variable (so Eq = 0,cov q = I ) . For any $ E R”, we define the projection Pq to this vector: Pqj(k) = (Q, k ) 9 . Denote by the symbol V(R”) the class of functions f : R” -+ R such that f(0) = 0 and f can be continued analytically onto C” and this continuation f(z) has the exponential growth.
Proposition 5.1. Let f E V(R”). Then we have:
+
E f ( W ( w ) )= 7 a”9112j - ( f ” ( 0 ) 9 ,9) o ( a ) ,a
+ 0.
Proof. By using the Taylor expansion o f f we obtain: 1 Ef(Pd(w))= c ( w ) ) 2 ( f ” ( 0 ) Q*), 4a),
p,
+
+
(21)
0.
By setting into this asymptotic equality the dispersion of the random variable P Q [ ( w we ) obtain (21). If 11911 = 1 (a pure quantum state), then we get:
Ef(PqjJ(w))= %(f”(O)Q,
+
9) o ( a ) ,a
-+
0.
(22)
Here A = f”(0) is a symmetric linear operator. We “quantized” the classical variable ~ ( z ) , zE R”, by mapping it to the operator A = $f”(O), see Theorem 3.1. The Gaussian random variable cqj, I 1 9 I I = 1 is “quantize” by mapping it into the pure quantum state 9. Theorem 5.1. There exists a Kolmogorov probability space such that all pure quantum states can be represented by Gaussian random variables on
Stochastic Analysis and Foundations of Quantum Mechanics
this space. The correspondence Q
where XI,
A2
E
4
331
& ( w ) is linear:
R.
Proof. We choose R = R" as the space of elementary events, the a-field of Bore1 subsets is the space of events and the standard Gaussian measure p as the probability measure. Then for @ -+ &(w) = &(Q, w ) , w E R",we have: XIEql(w)+ M i h 2 ( w ) = Al(@i,u) + L ( Q ' z , w ) = (Xi*1 + h Q 2 , w ) = ~Xl*'l+X2Q2
(w).
0
This theorem is rather surprising from the common viewpoint (by that essentially nonclassical probabilistic features of quantum states are consequences of the non-Kolmogorovian structure of the quantum probabilistic model). We pay attention that physical variables &,(w) = P q [ ( w ) , @ E R", (one-dimensional projections of the scaling ( ( w ) of the standard Gaussian random variable ~ ( w E) Rn)cannot be mapped onto nontrivial quantum obserwables. Prequantum classical physical variables &,(w) = (9, w ) are linear functionals of w . Therefore T(&) = tg(0) = 0. Nevertheless, quantum mechanics contains images of c,p given by quantum states @, but only for Q with 11Ql1 = l! We call ( ( w ) a background random field. All pure states could be extracted from the the background random field by projecting it to one dimensional subspaces. PCSFT explains the origin of the scalar product on the set of pure quantum states. We consider the l/a-amplification of the covariation of two Gaussian (prequantum) random variables
Conclusion. The Hilbert space structure of quantum mechanics is induced by the (prequantum) Gaussian random field (the background field [ ( w ) ) through the a 4 0 asymptotic. At the moment we proved this only in the finite-dimensional case. In subsection 5.2 we shall do this in the infinite-dimensional case. Finally, we pay attention to the fact that, for quadratic physical variables f(x) = +(Ax,x), where A : H -+ H is a symmetric operator, the asymptotic equality (22) is reduced t o the precise equality of averages. By considering directly the
332
A . Khrennikov
standard Gaussian random variable q(w) (instead of the background random field t ( ~=)&q(w)) we come to the following classical probabilistic representation of the quantum average: Ef(Pq,q(w))= ( A Q ,Q) =< A >q, . 5.2. Prequantum white noise field To repeat consideration of susection 5.1 for the infinite-dimensional case, we consider measures on the so called rigged Hilbert spaces. We apply some rather abstract mathematical constructions. However, finally we shall consider a simple concrete example which will be then used as the basis of our prequantum classical statistical model. Let R be a nuclear F’rechett topological linear space and R’ its dual space. Suppose that R is densely and continuously embedded into a Hilbert space H , so R c H . Thus the dual space H‘ is densely embedded into R‘. By identifying H and H‘ we obtain the rigged Hilbert space:
RcHcR’ (25) In our final application we shall set R = S ( R 3 ) . This is the space of Schwartz test functions on R3. Here R’ = S‘(R3)is the space of Schwartz distributions. In this case we choose H = L2(R3) and we shall consider the rigged Hilbert space:
S(R3)c L2(R3) c S’(R3)
(26)
Readers who are not so much interested in general theory of topological linear spaces can consider this rigged Hilbert space throughout this section. A Gaussian measure p on 52‘ is determined by its characteristic functional (Fourier transform) which is defined on R : p(9) = e-ib(*>’), where b : R x R --f R is a continuous positively defined quadratic form. By the well known theorem of Minlos-Sazonov, see e.g., [I, p is o-additive on 0’ and its covariation functional is equal to b. Here b ( Q 1 , Q2) = Jo,(q5, @ I ) ( + , 92)dp(4), where * I , E R. This functional defines the covariation operator B : R --f R’ by ( B ~ Q2) I , = b ( * ~ , Q2). This operator is self-adjoint in the following sense. The dual operator B’ : 0’’ 0’. But, since the topological linear space R is a nuclear F’rechet space, it is reflexive. Hence, R” = R. Thus the operator B‘ : R -+ 52‘. Thus it is meaningful to speak about self-adjoint operators in this framework (by extending the ordinary theory of self-adjoint operators in Hilbert space). We also pay attention to the fact that the covariation operator B is positively defined. --f
$so complete metrizable and locally convex
Stochastic Analysis and Foundations of Quantum Mechanics
333
Let us consider the standard Gaussian distribution p on H that is defined by its covariation functional:
b p l , Q2)
= (91,*2).
The corresponding covariation operator B = I : R R' is the canonical embedding operator. Since the embedding R c H is continuous, b : R x R + R is continuous and, hence, the measure p is a-additive on R'. Therefore there is well defined the corresponding Gaussian random variable r ] ( $ ) E R'. In the case of the rigged Hilbert space (26) the Gaussian random field ~ ( 4 E) S'(R3) is nothing else than the spatial white noise. We extend this terminology and we shall call ~ ( 4 white ) noise even in the abstract framework. Let us consider &-scaling of white noise .--)
<($I
=fir]($)
and its one-dimensional projections: &($) = (C($),P),@E R. We have E
*.
Lemma 5.1. Let f : H + R be a polynomial and let f(0) any P E H , the asymptotic equality (21) holds.
= 0.
Then, f o r
Proof. Here the main difference from consideration in section 3 is that the measure p is not concentrated on Hilbert space H on which the function f is defined (and continuous). Therefore even the exponential growth o f f on H would not help so much, because Jn,eal1411dp(+) = 00 (since even ,J, ll$lldp($) = 00). We have for a polynomial f:
334
A . Khrennikov
Since the sum is finite and derivatives of f are continuous forms on H , we obtain (21). 0
9.
We “quantize” f ( u )by mapping it into For quadratic functionals f(u)= $ ( A u , u ) , AE L , ( H ) , we have the precise equality and we can directly use the average with respect to the canonical Gaussian random variable ~ ( w )Here . 1 Ef(P*rl(w))= +f”(O)*,
*I.
6. Unbounded operators
Let f : R -+ R be a smooth function. Then, at any point $0 E R, f”($o) : R + 0’. Therefore f”(0) is in general unbounded operator in H . Moreover, in this way (i.e., starting with PCSFT) we obtain the class of linear operators (quantum observables) that is even essentially larger than in the conventional quantum formalism. In general, A = f”(0) maps R not into H . but into R‘.
Example 6.1. Let us consider the rigged Hilbert space (26). We consider the map f : S(R3)-+ R determined by a fixed point zo E R3 : 1
f($) = 2$2(zo). (For example, the classical field $(z) = e-”’ is mapped into the real number ( f “ ( O ) $ l , 7,h)= $I(~o)$z(~o). Thus
e-”:). Then
1
1
A+(x)= -f”(O)+(z) = -b(z - Z O ) $ ( X ) 2 2 is the operator of multiplication by the &function d(z - 20).Hence
f”(o)(~(~~)) c‘ L ~ ( R ~ ) . For any @ E S(R3),we have 1
E f ( P q q ( ~= ) ) ;*’(Q)
=
(A*, ’@) =< A
However, in general for 9 E L2(R3)the average
>* .
< A > q is not well defined.
Stochastic Analysis and Foundations of Q u a n t u m Mechanics
335
We can consider not only pure states, but general density operators. Let us now consider a Gaussian measure p E S E ( H ) which has the support on the space 52. Thus p can be considered as a measure on R. For such a measure p its covariation operator B : R' 4 R and its Fourier transform p is defined on 52'. We denote this class of statistical states by the symbol SE(52).We remark that SE(R) c S E ( H ) . Let E be a complex locally convex topological linear space. We recall that the topology of E can be determined by a system of semi-norms (the notion of a semi-norm p generalizes the notion of a norm 11 . 11; the only difference is that p ( $ ) can be equal to zero even for a nonzero vector $). Let b, : E x ... x E -+ C be a continuous n-linear symmetric form. There exits a continuous semi-norm p on E such that llbnllp
Ibn($, ..., $41 < 00
= SUP P(*)< 1
(here p zz pb, ) . Thus lbn($,
$>I 5 llbnll P Y $ >
"'7
(27)
An analytic function, see, e.g., [Ill for details, f : E -+ C has the exponential growth if there exits a continuous semi-norm p on E on such that:
If($)l
5 aebp($),$ E E.
Here the constants and the semi-norm depend on f
(28) :a
= a j , b F b j ,p 3 p f .
Lemma 6.1. l 3 The space of analytic functions of the exponential growth coincides with the space of analytic functions such that there exists a continuous semi-norm p = p f : ~ ~ f ( n ) ( 05 )c~rn, ~ pn
Here constants c = c f and r
=rf
= 0 , 1 , 2 , ...
(29)
depend on the function f .
Proof. A). Let f have the exponential growth. For any $ E E l we consider the function of the complex variable z E C : g $ ( z ) = f (zq9). As in Lemma 3.1, we have: IgF'(0)I 5 n!R-nsupole12s I f(Reie$)I 5 a f n ! R P e b f R p ( * ) .We obtain:
~ ~ ~ ( n )5( a;e-nnl/zebfn, o)~~p Thus the derivatives o f f satisfy the inequalities (29) with r f
= ebf
336
A . Khrennikov
B). Let now derivatives of f satisfy the inequalities (29) for some continuous semi-norm p . Then by the inequalities (27) we have
c 03
lf(42
Ilf'"'(o)llpPn(lcl)l~! I CferfP(@).
n=O
Thus f has the exponential growth with b f semi-norm p as in (29).
=rf
and the same continuous
We denote by RC the complexification of R : RC = R e i R . We denote by V(R) the class of functions f : R + R, f(0) = 0, which can be analytically continued onto RC and they have the exponential growth.
Lemma 6.2. L e t p E S$(R). T h e n , for a n y f u n c t i o n f E V(R), t h e following asymptotic equality holds:
< f >,where D
=
s,
f('$)dP('$)
=
;J'
( f " ( O ) h ! b ) d P D ( U ) + 4aL a n
+
0 , (30)
7 .Here .(a) = a2R(a,f l P I ,
where
IR(a,f,P)I 5 C f
1 n
eTfp(')dPD('$).
(31)
(32)
T h e s e m i - n o r m p i s determined by t h e inequality (28). The proof of this Theorem repeats the proof of Lemma 3.2. Instead of Lemma 3.1, we apply its generalization to the case of an arbitrary locally convex topological linear space, see Lemma 6.1. : R + R',so We pay attention that D : R' -+ R, and A = C = D A : R -+ R. In general, this operator can not be extended to a continuous operator in H . We would like to obtain an analogue of the formula (1) for linear continuous operators A : R + R' :
(A'$, '$)dpD (u)= Tr D A
(33)
The main mathematical problem is that in general the operator C = D A is not even continuous in H , so it is not a trace class operator in the Hilbert space H . Nevertheless, we can introduce the notion of trace even in such a framework.
337
Stochastic Analysis and Foundations of Quantum Mechanics
We recall that systems of vectors { e j } Z 1 ,ej E 0, and {eg}j”O=ll eg are called biorthogonal topological bases in R and 0‘ if
E
R’,
j=1
j=1
where the series converge in R and R‘, respectively.
Definition 6.1. A linear continuous operator C : R --+ R is called traceclass operator if, for any pair of biorthogonal topological bases, the series M
j=1
converges and its sum does not depend on bases.
Lemma 6.3. Let p be a Gaussian measure on R and let A : R 4 R’ be a continuous operator. Then the operator C = D A , where D = covp, belongs to the trace class and the equality (5’3) holds. As a consequence of Lemmas 6.2 and 6.3, we obtain:
Theorem 6.1. Let p E SE(R). Then, for any function f E V(R), the following asymptotic equality holds:
< f >,=
f ( $ ) d p ( + ) = Tr Df”(0)/2
+ o ( a ) ,a
+
0,
(34)
where D = y . Thus our prequantum model, PCSFT, provides the motivation to extend the set of quantum observables and consider all continuous operators A : 52 --+ 0’. Operators should be self-adjoint in the ordinary sense: A’ = A . We recall that here A‘ : R“ + R’, but 0‘‘ 3 0, since R is a nuclear Frechet space and hence it is reflexive. Denote the set of such operators by the symbol C,(R,0’).Denote the set of covariation operators of Gaussian measures belonging the space S&(R) by the symbol D(R’, 0).
Definition 6.2. A statistical quantum model corresponding to a rigged Hilbert space 7 given by (25) is the pair N,.,,,,t(7) = (D(R’, R), Cc,(R,a’)). A generalized density operators D E D ( W , 0) represents a statistical state; a linear operator A E C,(R,R’) represents a quantum observable. The average of such an observable with respect to such a statistical state is given by the following generalization of the von Neumann trace-formula:
>D=
Tr DA
(35)
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A . Khrennikov
We choose the state space R - a nuclear F'rechet space. For a rigged Hilbert space 7 given by (25), we consider the classical statistical model M " ( 7 ) = (S$(R),V ( 0 ) ) .Here as always < f >,= f ($)dp($). The equality (34) can be used as the motivation for defining the following classical 4 quantum map T from the classical statistical model M " ( 7 ) = (S,%(R),V ( 0 ) )onto the quantum statistical model Nquant(T) = (D(R', R), LC,(R, 0'))by (16), (17). Our previous considerations can be presented as
s,
Theorem 6.2. The map T : S,%(R) -+ 2)(0',0)i s one-to-one; the map T : V(R) -+ LC,(R,R') is linear surjection and the classical and quantum averages are coupled by the asymptotic equality (34). Example 6.2. The position operators P j , j = 1 , 2 , 3 can be obtained as ?,.-A 'I 3 - 2 fXj(01, where
Here the operator of multiplication Pj : S(R3) 4 S(R3),$ 4 xj$, is continuous. Hence ij : S(R3)4 S'(R3)is also continuous. Thus, for any measure p E S,%(S(R3)), we have
D = cov p/cr (here the trace of the composition DPj is well defined). Example 6.3. Let ICO be a fixed point in R3. Let now A+(%)= 6(x q ) $ ( x ) , $ E S(R3).This operator does not belong to the domain of the conventional quantum formalism. It could not be represented as an unbounded operator in H = L2(R3) with a dense domain of definition. Nevertheless,
and the trace of the composition D A is well defined.
Example 6.4. The momentum operators & , j = 1 , 2 , 3 , can be obtained as fjj = (0), where
if;,
Stochastic Analysis and Foundations of Quantum Mechanics
339
Here t h e operator p j : S(R3)+ S(R3)is continuous. Hence, fij : S(R3)-+ S‘(R3)is also continuous. T h u s for any measure p E S a ( S ( R 3 x) S(R3)), we have (for $(x) = q(x) ip(x)):
< fpj
>p-
-i
/
+
/
% ( x ) m d x d p ( $ ) = cw~rD p j ,
S(R3)xS(R3) dxj
where D = cov p / a . Here Dfij : S(R3)4 S(R3)is t h e trace class operator. Similar considerations can be done for angular momentum operators. This paper was partially supported by EU-network ” Q u a n t u m Probability a n d Applications.”
References 1. D.Hilbert, J. von Neumann, L. Nordheim, Math. Ann., 98,1 (1927). 2. P. A. M. Dirac, T h e Principles of Q u a n t u m Mechanics, (Oxford Univ. Press, 1930). 3. W. Heisenberg, Physical principles of quantum theory, (Chicago Univ. Press, 1930). 4. J. von Neumann, Mathematical foundations of quantum mechanics, (Princeton Univ. Press, Princeton, N. J., 1955). 5. A.Yu. Khrennikov (editor), Foundations of Probability and Physics, Q. Prob. White Noise Anal., Vol. 13, (WSP, Singapore, 2001). 6. A. Yu. Khrennikov (editor), Q u a n t u m Theory: Reconsideration of Foundations, Ser. Math. Modeling, 2 , (Viixjo Univ. Press, 2002). 7. A . Yu. Khrennikov (editor), Foundations of Probability and Physics-2, Ser. Math. Modeling, 5, (Vaxjo Univ. Press, 2003). 8. A. Yu. Khrennikov (editor), Q u a n t u m Theory: Reconsideration of Foundations-2, Ser. Math. Modeling, 10, (Vaxjo Univ. Press, 2004). 9. A. Yu. Khrennikov, A pre-quantum classical statistical model with infinitedimensional phase space. J . Phys. A : Math. Gen., 38,9051 (2005). 10. A. Yu. Khrennikov, Quantum mechanics as an asymptotic projection of statistical mechanics of classical fields: derivation of Schrodinger’s, Heisenberg’s and von Neumann’s equations. http: //www. arxiv.org/abs/quant-ph/O511074 11. A. Yu. Khrennikov, Infinite-dimensional pseudo-differential operators. Duestia Akademii Nauk U S S R , ser.Math., 51,46 (1987). 12. T. Hida, M. Hitsuda Gaussian Processes, Translations of Mathematical Monographs, 120,(American Mathematical Society, 1993). 13. A. Yu. Khrennikov, Equations with infinite-dimensional pseudo-differential operators. Dissertation for the degree of candidate of phys-math. sc., (Dept. Mechanics-Mathematics, Moscow State University, Moscow, 1983).
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Advances in Deterministic and Stochastic Analysis Eds. N. M. Chuong et al. (pp. 341-360) @ 2007 World Scientific Publishing Co.
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$18.NONCOMMUTATIVE TRIGONOMETRY AND QUANTUM MECHANICS Karl Gustafson’ Department of Mathematics, University of Colorado, Boulder, Colorado E-mail: gustafs@euclid. colorado. edu A noncommutativeoperatortrigonometryoriginated by this author in 1966 will be summarized. Some important applications t h a t have taken place during the intervening years will be recalled. Then we will turn t o t h e recent application of this noncommutative operator trigonometry t o Bell’s inequality in quantum mechanics. Our geometrization of this celebrated inequality clarifies t h e proper meaning of several physical issues in the Einstein-Podolsky-Rosen Paradox. Those clarifications lead us t o look more closely at Von Neumann’s formulation of the collapse of the quantum mechanical wave packet. These investigations give us a new clarification of the quantum Zen0 Paradox. We also announce a fundamental new theorem on quantum mechanical reversibility and we take note of some of its physical implications. Keywords: noncommutative operators; antieigenvectors; Bell Inequality; Einstein-Podolsky-Rosen Paradox; Zen0 Paradox; reversibility
1. Introduction, Background, and Summary
Starting from a purely operator-theoretic question which this author had formulated in 1966 within the Hille-Yosida-Phillips-Lumer theory of abstract operator semigroups, this author developed in the period 1966-1972 the essentials of a noncommutative operator trigonometry. In this beginning period, I called this a theory of antieigenualues and antieigenuectors, because intuitively it compares and contrasts in a natural manner to the Rayleigh-Rita variational theory of eigenvalues and eigenvectors. From 1973 *This paper is a summary of the two lectures, ‘Noncommutative Trigonometry’ presented at The Second International Conference on Abstract and Applied Analysis 2005-Quy Nhon, Vietnam, June 4-9, 2005, and ‘Noncommutative Operator Algebra and Bell Inequality’ presented a t the Institute of Mathematics, Vietnamese Academy of Science and Technology, Hanoi, Vietnam, June 15, 2005. In addition, some new results are included.
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K. Gustafson
to 1993, this author was chiefly involved with other research, including scattering theory, mathematical physics, computational fluid dynamics, optical computing systems, neural networks. However, a few further results for the noncommutative trigonometry were obtained. Then from 1994 to the present I returned to the noncommutative trigonometry and successfully applied it to numerical linear algebra, wavelets and control theory, statistical estimation and efficiency, and quantum mechanics. In 1997 I published two books'?' which contain chapters on this noncommutative trigonometry. At that time I decided to call this theory operator trigonometry. This, because I was beginning to realize its importance beyond the original context in which I had viewed it variationally, juxtaposed against the Rayleigh-Ritz variational theory. More recently, I have settled upon the name noncommutative trigonometry. This, because the noncommutative trigonometry now seems to me to convey a spirit much like the theory of A. Connes, which now carries the name noncommutative geometry. Suffice i t t o say that this author has always viewed operator theory as primarily noncommutative. It is time to emphasize that fact with the name noncommutative trigonometry for the noncommutative operator trigonometry I will describe herein. A bibliography of 62 papers written by this author that deal wholely or in part with this noncommutative trigonometry may be found in the recent ~ u r v e yThus, .~ the reader may find more bibliography, facts, and history in Refs. 1-3. Although this noncommutative operator trigonometry has had contact with related contributions by a few other notable mathematicians, including M. G. Krein, H. Wielandt, V. Ptak, C. Davis, B. Mirman, L. Kantorovich, C. R. Rao, for the most part it is my creation. In Section 2 we quickly recall the key essentials of the noncommutative trigonometry. In Section 3 we quickly recount some of its main application to date. More details about these may be found in Ref. 1-3 and especially in the 62 citations of Ref. 3. However, the recent survey3 could not present the recent results of this author in which the noncommutative trigonometry fundamentally clarifies key issues in the celebrated Bell and Zen0 problems of quantum mechanics. So I go beyond3 here and in Sections 4 and 5, respectively, I treat those two problems from the viewpoint of the noncommutative trigonometry. 2. Essentials of the Noncommutative Trigonometry
To fix ideas, let us restrict attention to A a symmetric n x n invertible positive definite (SPD) matrix. The theory extends to arbitrary positive selfadjoint operators in a Hilbert space and to arbitrary invertible matrices
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and Quantum Mechanics
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and even to accretive operators in a Banach space and we refer the reader to Refs. 1-3 for those results. Let A have eigenvalues 0 < A 1 < A:! < * * < A n where again for simplicity we assume them to be all distinct, although the theory does not need that. Then the essential facts of the noncommutative trigonometry depend upon the following fundamental four entities: 9
$(A)
x*
= the maximum turning angle of
=f
A
(3)
(L)1’2zl (4) + (A)1’2z, +An A1 + A n
A1
I named p1 the first antieigenvalue of A. The entity v1 is equally important in applications. I called $(A) as defined from either p1 or vl, the angle of A. I called the two vectors z* the first antieigenvectors of A. They are the vectors (normalized t o norm 1 in (4)) most turned by A. The z1 and z, are normalized eigenvectors for A1 and A, respectively.
A good example is A
=
[:p6].
Then $ ( A )
=
16.2602’ and the two
antieigenvectors are z+ = (4,3)/5 and 2 - = (-4,3)/5. Early on I realized that these entities form the basis of an (angular) spectral theory as rich as the usual (dilation) spectral theory based upon eigenvalues. Moreover I showed that the Euler equation
211Az11211z112(ReA)z- IIzl122Re(Ax,z)A*Az- IIAzlI2Re(Az,z)z = 0 (5) generalizes the usual Rayleigh-Ritz variational theory, and moreover, contains it: for A symmetric or normal, A’s antieigenvectors, and as well all eigenvectors, satisfy (5). I defined the sequence of higher antieigenvalues and corresponding higher antieigenvectors on successive reducing subspaces of A, more or less in analogy with the Rayleigh-Ritz theory. All of that was done in the period 1966-1972. See Refs. 1-3 for more details. A couple of remarks here may be helpful. The two antieigenvectors may not be linearly combined to form an antieigenspace. However, their span is of course the reducing subspace sp{zl,z,} formed by the ‘smallest’ and ‘largest’eigenvectors. Thus the antiegenvectors zk are rather special vectors contained within that spectral subspace. Specifically4 they have relative
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K. Gustafson
angle (x+,x-)= -sin$(A), and thus the angle between them is always
4(A) + 7rP. Second, in applications I often must use the operator trigonometry and the usual trigonometry in conjunction. However, the operator trigonometry generally does not satisfy the identities of the usual trigonometry. Moreover, it has some of its own interesting identities. A useful example is:5 sinqh(A1/2)= sin+(A)/[l + C O S ~ ( A )Here ] . A1/2 denotes the positive square root of the operator A. Generally we have the operator angle inequality:
44AB) 5 $ ( A )+ + ( B ) . Third, although I developed some rudiments of the noncommutative operator trigonometry for operators A on a Banach space, it is better to stay in Hilbcrt space, because the essential identity sin2 4(A) cos2 4(A) = 1 does not hold generally in Banach space.
+
3. Quick Summary of Applications to Date
Applications of this noncommutative trigonometry t o date include, roughly and chronologically: perturbation of operator semigroups, positivity of operator products, Markov processes, Rayleigh-Ritz theory, numerical range, normal operators, convexity theory, minimum residual and conjugate gradient solvers, Richardson and relaxation schemes, wavelets, domain decomposition and multilevel methods, control theory, scattering theory, preconditioning and condition number theory, statistical estimation and efficiency, canonical correlations, Bell’s inequalities, quantum spin systems, and quantum computing. Here is a quick summary. In the initial development period 1966-1989 one could say that the research emphasis was functional-analytic and theoretical. However, I had turned to computational mathematics in the 1980’s, working with several engineering groups. An example is in Ref. 6. In particular, in such research I had used some steepest descent and conjugate gradient scheme computations and had learned of the basic convergence rate bound
Here E ( x ) is the error ((x - z * ) , A ( x - 2*))/2, z*the true solution. This bound also occurs in a key way in some parts of optimization theory. Immediately from my operator trigonometry I knew that what (6) really meant, geometrically, was trigonometric:
Noncommutative P i g o n o m e t y and Quantum Mechanics
345
Although (6) was known as a form of the Kantorovich inequality, nowhere in the numerical nor in the optimization literature did I find this important geometrical result: that gradient and conjugate gradient convergence rates are fundamentally trigonometric. That is, they are a direct reflection of the elemental fact of A’s maximum turning angle 4(A). The known conjugate gradient convergence rate, by the way, becomes, trigonometrically written now,
See Ref. 7 where I published this result (8) for conjugate gradient convergence. And I first announced the steepest descent result (7) at the 1990 Dubrovnik conference.8 Many other important iterative computational linear solvers for large problems Ax = b were shown by the noncommutative trigonometry to have trigonometric convergence rates. I obtained these results in the period 19942004. All were new. It is really quite astounding that these fundamentally trigonometric convergence geometries had lain undiscovered all those years, going back t o Richardson (1910), Kantorovich (1948), other notable computational mathematicians. However, without the cos 4(A) and sin 4(A) of the noncommutative trigonometry a t your fingertips, one cannot see the geometry. See Ref. 9 to come somewhat up-to-date on these results. Another important application of the noncommutative trigonometry has been to statistics. For example, there is the famous Durbin-WatsonBloomfield (1955, 1975) lower bound for least squares relative efficiency
of estimators, namely,
where the A’s are the eigenvalues of the covariance matrix V. In (9) the p* is the best possible estimator and X is an n x p regression design matrix. Coincidentally, I had recently4 extended my operator trigonometry to arbitrary matrices and there (also in earlier papers) I had stressed the alternate definition of higher antieigenvalues and their corresponding higher antieigenvectors as defined combinatorially in accordance with (1)(4) but with successively smaller operator turning angles now defined by corresponding pairs of eigenvectors. It follows readily that the geometric
346 K. Gustafson
meaning of statistical efficiency then becomes clear: l o
n P
RE(&
2
cos2 &(V)
(11)
i=l
It is that of operator turning angles. Other intimate connections between statistical efficiency, statistical estimation, and canonical correlations may be found in Ref. 10. For example, LaGrange multiplier techniques used in statistics lead to equations for the columns xi of the regression design matrices X to satisfy
v2xz (VXi, X i )
+ (V-lxz, Xi
Xi)
= 2vxi,
2 =
1 , .. . , p
I call this equation the Inefficiency Equation. In the antieigenvalue theory, the Euler equation (5) becomes in the case when A is symmetric positive definite,
The combined result is the following. All eigenvectors x j satisfy both the Inefficiency Equation and the Euler Equation. The only other (normalized) vectors satisfying the Inefficiency Equation are the “inefficiency vectors” (my terminology)
The only other vectors satisfying the Euler equation are the antieigenvectors
Further recent results from thc application of the noncommutative trigonometry t o matrix statistics may be found in Refs. 11,12. For other applications of the noncommutative trigonometry, see the citations of Ref. 3. These include applications to quantum computing and to elementary particle theory, which I will not discuss in this paper. In the next two sections I will concentrate on the Bell and Zen0 problems from quantum theory.
4. The Bell Inequalities of Quantum Theory Addressing fundamental issues raised by Einstein, Podolsky, Rosen, l 3 concerning the foundations of quantum mechanics of Von Neumann,14 Bell15
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347
presented his famous inequality
and exhibited quantum spin measurement configurations whose quantum expectation values could not satisfy his inequality. Bell’s analysis assumed that two measuring apparatuses could be regarded as physically totally separated, and free from any effects from the other. Thus his inequality could provide a test which could be failed by measurements performed on correlated quantum systems. It was therefore argued that local realistic hidden variable theories could not hold in quantum mechanics if future physical experiments would violate Bell’s inequality. Later Aspect et al. l6 indeed demonstrated violation of Bell’s inequality in their laboratory experiments. But the controversy about the Bell inequality and related inequalities to be mentioned below and their physical implications, continues to this day. Bell’s arguments in arriving a t his inequality were classical probabilistic correlation arguments. However, it is known and easy to prove that this inequality holds for any real numbers a , b, c in the interval [-1,1]: then ab - bc ac 5 1. Here is a proof. From b2 5 1 and c2 5 1 we have b2(1 - c 2 ) 5 1 - c2 and hence b2 c2 5 1 b2c2. Adding 2bc to both sides and multiplying by a2 5 1 we therefore have a2(b2 c2 2bc) b2 c2 2bc 5 1 b2c2 2bc, that is, a2(b c ) 5~ (1 bc)2. Taking the positive square root yields a ( b c ) 2 la1 Ib cI 5 1 bc. Wigner17 presented his own version, making more clear the issues of locality and socalled realism. Furthermore, Wigner was sure to use quantum mechanical probabilistic correlations. His version of Bell’s theory then becomes the inequality
+
+ +
+
+
+
+
+
+
+
+
+
+ +
where the Oi, are angles between spin directions w i and W k . Another important Bell-type inequality is that of Clauser, Horne, Shimony, Ho1t.l’ Let a,b, c, d be four arbitrary chosen unit vector directions in plane orthogonal to the two beams produced by the source. Let vi(a) and wi(d) be the “hidden” predetermined values f l of the spin components along a and d, respectively, of particle 1 of the ith pair, similarly w i ( b ) and wi(c) for particle 2 values along directions b and c . Then the average correlation value for particle 1 spins measured along a and particle 2 spins measaured along b is E(a,b ) = vi(a)wi(b)/N. Taking into account in the same way the average correlation values E(a,c ) , E ( d ,b), E(d,c )
EL=,
348
K. Gustafson
and adding up all pairs, one arrives a t the CHSH inequality
In a series of papers starting with Ref. 19, see Ref. 3 for citations to later papers, I placed all of the inequalities of the Bell theory, into my noncommutative trigonometry. From my results, it may be argued that many important issues in the Bell theory, about which there are a t times furious arguments about physical and probabilistic meaning among physicists, are really better seen as new mathematical quantum geometry from the noncommutative trigonometry. As a first example, consider Wigner’s version (17) of the Bell inequality discussed above. For the three coplanar directions, our corresponding inequality equality20 becomes in Wigner’s quantum spin setting terminology
sin2
(;el2) + (:ez3) (iell) (;el3)
= 2 cos
sin2
[cos
-
sin2
(ie13)
- cos (:el2)
cos
(:ezj)]
(19)
Violation of the conventionally assumed quantum probability rule I (u, TI)l2 = cos2 9u,vfor unit vectors u and v representing prepared state u t o be measured as state v, is equivalent according to Wigner to the right side of this identity being negative. This is his Bell “violation” test. However, from our point of view, there is no violation, there is just a quantum trigonometric identity, valid for certain formulations of measurement of certain spin systems. As a second example, let us consider the important CHSH inequality (18) given above. Wishing now to preserve equality, we write
Noncommutative Trigonometry and Quantum Mechanics
349
Squaring this expression and writing everything quantum trigonometrically, la. b
+ a .c + d . b - d . cI2 = (2 + 2 cos ebc) cos2 e a , b f c + (2 2 cos 6bc) cos2ed,b-c) + 2(4 4 cos2 &)1/2 cos e a , b + c cos Od,b-c -
-
= 4 cos2(8bc/2) cos2 6a,b+c
+ 4sin2(Obc/2)cos2 ed,b-c + 4 Sin ebc cos ea,b+c cos Od,b-c. In the above we used two standard trigonometric half angle formulas. Now substituting the double angle formula sin 6bc = 2 sin(Obc/2)cos(BbC/2)into the above we arrive at l a . b + a . c + d .b-d.CI2 = 4 [ c o s 8 ~ c / 2 ) C O S ~ a , b + c + S i n ( e b c / 2 ) C O S ~ d , b - c ] 2 and hence we have a new quantum CHSH equality la. b
+ a .c +d .b
-
d . CI = 21 cOs(6)bC/2)COSea,b+c
+ Sin(&/2)
COSed,b-cl
We may also write the righthand side as twice the absolute value of the two-vector inner product U1
’
u2
(cos(ebc/2), Sin(ebc/2)) ’ (cos ea,b+c, cosed,b-c)
to arrive at the quantum trigonometric identity l a . b + a . c f d . b - d . c l = 2 ( c 0 s ~ e ~ , fbc+o~s 2 ed,b-c)1/2/COS8,1,,,1 (20) The right sides of these equalities isolate the “classical limiting probability factor” 2 from the second factor, which may achieve its maximum fi.Fix any directions b and c. Then choose a relative to b+c and choose d relative to b - c so that cos2 Oa,b+c = 1 and cos2 6d,bPc = 1, respectively. Now we may choose the free directions b and c to maximize the third factor to COS~,,,,, = f l . But that means the two-vectors u1 and u2 are colinear and hence u1 = (COS(6bc/2), sin(ebC/2))= 2 ~ l / ~ ( c&,b+c, os cosed,b-c) - 2-1/2(&1, +l)
and thus the important angle 6bc is seen to be f7r/2. More to the point, our identity allows one to exactly trace out the “violation regions” analytically in terms of the trigonometric inner product condition 1 5 Iu1 . u21 5 fi. Thus, and we have only presented two examples here, the noncommutative trigonometry applied to the celebrated inequalities of the Bell theory
350 K. Gustafson
has shown that those inequalities may be made special cases of identities for the noncommutative trigonometry. This enables” much clearer delineations of physical, probabilistic, and geometrical implications of those inequalities. Further analysis” leads to the opinion that one cannot claim quantum locality or nonlocality on the basis of satisfaction or violation of Bell’s inequalities. Rather, one is ledz0 to the assertion that the real issue as concerns nonlocality is the Von Neumann projection rule.14 That rule, under which measurements of quantum probabilities correspond t o projections on Hilbert space subspaces, is considered next, in the last section of this paper.
5. The Zen0 Problem of Quantum Theory The term Zen0 Paradox, i.e., “a watched pot never boils”, was introduced in Misra, Sudarshan’l to highlight certain fundamental issues in quantum mechanical measurement theory. There is a long history to such quantum measurement problems. For example, Von Neumann, l4 when he created his Hilbert space model of quantum mechanics, proved that any given state q5 of a quantum mechanical system can be steered into any other state in the Hilbert space by an appropriate sequence of very frequent measurements. Thus, in particular, you can freeze a quantum evolution in time by “continually watching it”. The allusion t o the greek Zen0 refers t o a debate” which he and Parmenides had with Socrates in Athens approximately 445 B.C.. Parmenides, an elderly philosopher, presented to Socrates an interesting proposition t o start the debate: reality never changes, therefore motion is not possible. Although Parmenides relied on the younger Zen0 for argumentive support, the originating thesis was that of Parmenides. Thus is might be more accurate to call the quantum versions Parmenides Paradox and Parmenides theory. What is a t issue is not a matter of names, it is the fundamental physical question of what happens when a quantum mechanical wave function $(t) is measured by some classical measuring instrument. The act of measuring is seen as a ‘collapse of the wave function,’ which according to the Von Neumann postulates is implemented by an orthogonal projection. This gives rise t o numerous mathematical issues. I was involved, going back to 1974, in some of these questions, and left them unresolved, and hence as concerns my partial results, unpublished. Misra and Sudarshan’l went from the Schrodinger picture to the Heisenberg picture t o partially move beyond these issues, but a close reading of Ref. 21 shows that the issues are not removed, they are only assumed away. Recently I returned to some of these
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351
questions and some of the following results are taken from Ref. 23. Because there is now a large literature on the Zen0 Problem, I must refer the reader to Ref. 23 for further bibliography. Also one should look at several articles in the 2003 volume in which Ref. 20 appeared, a volume dedicated in significant part to the Zen0 problem, and its relation to quantum computing. At issue both physically and mathematically is the nature of the interaction of the measuring instrument or subspace with the evolving wave packet $t = U ~ + O where $0 was the initial state vector and where Ut = eitH is the unitary evolution group generated by the governing self-adjoint Hamiltonian. One also needs t o look a t projected evolutions Zt = PUtP. Here P is an orthogonal projection. It turns out that commutativity or noncommutativity play critical roles in both the mathematical considerations and in the physical outcomes. The following was known early. Let U ( t ) be a n arbitrary unitary evolution with selfadjoint infinitesimal generator H and let P be an arbitrary selfadjoint bounded projection onto closed subspace M in Hilbert space 'H. Let D ( T ) denote the domain of an operator T in 7-l, and R ( T ) its range. We recall that P u t = UtP iff M is a reducing subspace for Ut, and more generally P u t P = Ut P iff M is an invariant subspace for Ut . More general yet, we have (see Refs. 24-27)
Lemma 5.1. The projected evolution Zt = PUtP is a semigroup for all t 2 0 iff M is a proper subspace without regeneration for Ut, i.e., PUsPLULP= O for all t , s 2 0. We remark that when M is a reducing subspace then Zt is a unitary group, when M is an invariant subspace then Zt is a semigroup of partial isometries, and if M fails t o meet the requirement of Lemma 5.1, the Zt loses the semigroup property entirely. The possibility of retaining a unitary, hence reversible, evolution on socalled Zeno subspaces has been investigated in a number of recent b o o k ~ / p a p e r s . ~For ~-~ a ~better understanding of what is involved in such formulations, we wish to take note of certain unbounded operator-theoretic considerations. The following was clear but not published due to the general incompleteness of my study of the Zen0 issues.
Lemma 5.2. Assume D ( H P ) is dense. Then P H P is symmetric in 'H, and P H P is selfadjoint i f PH is closed in 'H. Proof. Since D ( H P )= D ( P H P ) is dense, ( H P ) *and (PHP)' exist and ( P H P ) * = ( H P ) * P 2 ( P H ) P = P H P so P H P is symmetric. If PII
352
K. Gustafson
is closed, then P H previous sentence.
=
( P H ) * * = ( H P ) * , so one obtains equality in the 0
Before looking more closely a t the fundamental underlying issue of D ( H P ) dense, we wish at this point to recall a few operator-theoretic facts which we will use in the remainder of this paper. For more details see the books [32,33]among others. In particular, the notion of (closed) invariant subspace M for an unbounded operator T is better thought of in the more general context as a decomposition of T by a direct sum IFI = M @ N of a pair of subspaces with the requirements that the projection P on A4 map D ( T ) into D ( T ) , T maps M into MI T maps N into N . Here P is the (generally oblique) projection of M along N . Such decomposition of T is equivalent to T commuting with P : PT c T P . Then T P = P T P = PT on D ( T ) . When T is a selfadjoint operator H and P an orthogonal projection and U, = eiHt, then when one says that A4 reduces H one is saying all of the following: P H c H P , P : D ( H ) into D ( H ) , ( H - z I ) - ' P = P ( H - zI)-' for all z with I m z # 0, P E ( s ) = E ( s ) P for all real s and all spectral family projectors of H , and P u t = UtP for all real t. Our point-of-view is that such a simplified situation is not that of the Zen0 issues. A measuring projection already within H's spectral calculus is a far oversimplified situation, although certain proposed spin measurers may have the property. A second set of facts to remember is that T* exists iff D ( T ) is dense, that D ( T * ) need not be dense, but D ( T * ) is dense iff T is closeable, and then its closure satisfies T = T**.For any two densely defined operators A and B and if AB is densely defined, then (AB)' 2 B*A*, with equality if A E B('H). Other conditions for equality when A and B are both unbounded may be found in [34]. We also will use other unbounded operator theory, such as the associativity TI(T2T3)= (TlT2)T3,without comment. We may now sharpen Lemma 5.2 to further clarify the situation.
Theorem 5.1. D ( H P ) is dense iff P H i s closeable. T h e n (HP)' is defined and ( H P ) ' = PH IIP H has domain at least as large as D ( H ) . Thus generally ( P H P ) * = P H P whenever P H is closeable. Furthermore the polar factors satisfy ImI= ( H P 2 H ) ' / ' and lHPl 2 ( P H 2 P ) l 1 2 .
Proof. Because P H is densely defined, its adjoint exists and is ( P H ) * = H P . This operator is densely defined iff P H is a closeable operator. Then H P is a closed densely defined operator and ( H P ) * = ( P H ) * * = PH 3 P H . To obtain the polar factors we form (=)*PH = H P 2 H and ( H P ) * H P = =HP 3 PH2P.
Noncommutative Pigonometry and Quantum Mechanics
353
The key assumption that H P be densely defined is made throughout the recent treatment [35] of quantum Zen0 dynamics. Their approach of quadratic forms follows that of [36] and one is concerned with the form J I H 1 / 2 P ~ 1 1with 2 form domain D ( H 1 l 2 P ) .The operator H p := (H1/2P)*(H1/2P is)associated with this form. It is noted in [35] that H p may not be densely defined but they then assert that it will be a selfadjoint operator in some closed subspace of ‘H. From our analysis here we would like to defer slightly, or a t least point out some ambiguity in such a conclusion. When H 1 / 2 Pis not densely defined in ‘H, one cannot even speak of an operator ( H 1 / 2 P ) * ( H 1 / 2 POf ) . course one can then reduce one’s considerations to the smaller Hilbert space M . But if H 1 l 2 P was not densely defined in ‘HI then H 1 / 2will not be densely defined in M either. Moreover, whatever its domain there, the range R ( H 1 / 21 ~ will ) generally fall a t least partially outside of M . The same reservation applies t o the analysis of Ref. 37. The interesting formulation there combines continuous measurement with a coupling limit t o force the system t o evolve in a set of orthogonal subspaces of the parent Hilbert space. These socalled quantum Zen0 subspaces are the eigenspaces of a Hamiltonian which is supposed to represent the interaction between the evolving quantum dynamical system and the measurement apparatus. The use of a superselection rule and an adiabatic theorem are assumed to determine “the subspaces that the apparatus is able to distinguish”. Thus the physical description is now that of a dynamical evolution allowing changing Zeno subspaces. However, from our point of view, since the modelling and analysis is carried out in the density matrix formulation, its underlying rigorous validity, e.g., the denseness of the domains of the effective Hamiltonians in the individual Zen0 subspace evolutions, has not been considered. A similar comment applies to the Von Neumann Subalgebras Zen0 formulation [38], especially if one does not want the measuring projection E to have t o be within the Hamiltonian’s functional calculus. Finally, I would like to announce here a fundamental new theorem with I believe fundamental implications to quantum mechanics in general. The reversibility of a quantum mechanical Schrodinger evolution corresponds just t o the unitarity of its semigroup eitH. The following result [23] grew out of such thoughts combined with the considerations about domains elsewhere in this paper.
Theorem 5.2. A unitary group Ut necessarily exactly preserves its infinitesimal generator’s domain D ( H ) . That is, Ut maps D ( H ) one to one onto itself, for all -00 < t < 00.
354
K. Gustafson
Proof. For every -oo < t < oo the linear isometry IlUtzll = llzll property and the commutativity property UtH c HUt guarantee that Ut maps D ( H ) one to one into D ( H ) . Is it onto D ( H ) ? Suppose not. Then for some t there exists an z in D ( H ) which is not in the image U t ( D ( H ) ) Apply . to z. By the commutativity property again, we know z = U-tx is in D ( H ) . But then Utz = x must have been in D ( H ) .
Url
The result is quite evident and natural once one sees it. Therefore it may exist elsewhere in the literature. But in a limited search, I did not find it. It importantly distinguishes the special action of Ut on D ( H ) from its one to one onto action on the whole Hilbert space. Remember that D ( H ) ,although dense, is a Baire Category 1 subspace. In particular, a quantum mechanical Schrodinger evolution Ut = eitH where H is the atomic Schrodinger partial differential equation must take this Baire Category 1 subspace D ( H ) of all prepared states exactly onto itself. The rest of the Hilbert space, a Baire Category 2 set which is not a subspace, is separately taken onto itself. Thus Theorem 5.2 expresses a ‘global’ regularity of exact domain preservation. Contrast that with the heat equation evolution Zt = e&*, which instantly takes the whole Hilbert space (e.g., L z ( - m , GO)) into Coo(-oo,oo). See for example the rigorous proof in Ref. 39, pp. 128-131. The Heat evolution operator Zt is an infinitely smoothing operator. Its smoothing action is a ‘local’ regularizing of each initial given heat distribution f E L z ( - m , oo), whether f be in the domain W232(-oo,oo)of the Laplacian -A, or not. Instantly, information in f is spread out and lost. In particular, the domain W2i2is immediately lost. Not so in the Schrodinger evolution: its unitarity 2 Ut at every instant t , requires that W2,2be mapped exactly onto W 2 >by according to Theorem 5.2. To express our interpretation of Theorem 5.2 in dynamical system terms, the Schrodinger partial differential equation initial value problem
$(t) = iH$(t), t > 0 $ ( O ) = $0
given
has two classes of states: $0 in the domain D ( H ) ,and the rest. Only those in D ( H ) are ‘legally’ allowed wave packets from the differential operator theory, in which it is essential that the bounded evolution operator Ut commute with the infinitesimal generator H . Stated in elementary terms: otherwise you cannot substitute the solution back into the differential equation. One can prove Theorem 5.2 in another way which is more useful when one wants to consider also contraction semigroups Zt with infinitesimal $0
Noncommutative %gonometry and Quantum Mechanics
355
generator A. To that end we first establish two lemmas.
Lemma 5.3. Let T be a closed densely defined operator in a Hilbert space ‘FI and let Ut = eitH be a unitary evolution there which commutes with T . Then Ut remains unitary o n the graph-norm Hilbert space ‘FIT.
Proof. ‘HT is the Hilbert space D ( T ) equipped with the inner product (2, y ) r = (2, y) ( T z ,Ty). By the commutativity UtT C TUt we have the isometry property retained,
+
Also the adjoint U,+ considered in ‘HT is the same as the original adjoint, using again the commutativity,
(UtG Y ) T = (Utx,Y) + (UtTx,TY) =
+
(x,U,*Y) ( T x ,TU,*y) = ( 2 ,U,*Y)T
Thus U; = U r l in the original space carries over to ‘HT.
(23) 0
In particular, Lemma 5.3 provides an alternate proof of Theorem 5.2. Let T be H . Then Ut and U,+ are both onto ‘HT = D ( H ) and the rest follows easily from bounded operator theory.
Lemma 5.4. Under the conditions of Lemma 5.3, a contraction semigroup evolution Zt = etA remains a contraction semigroup on ‘ F I T .
Proof. As above. Note that the assumed semigroup commutativity property ZtT C T Z t is essential, as is its subproperty that Z, map D ( T ) into D ( T ) In particular, one may take T = A, the infinitesimal generator of zt 0 Now we can provide a partial converse to Theorem 5.2 in the sense of asking, suppose a contraction semigroup Zt exhibits the regularity preservation property that it map the domain D ( A ) of its infinitesimal generator one to one onto itself. What ‘unitarity’ properties does Zt exhibit for all t 2 O?
Proposition 5.1. Let Zt = etA be a contraction semigroup on a Hilbert space H ‘ such that 2,maps D ( A ) one to one onto D ( A ) . Then Zt on ‘HA can be extended to a group 2-t = 2;‘.
356 K. Gustafson
Proof. By Lemma 5.4 above with A = TI we know 2,remains a contraction semigroup on 'HA. Because Zt maps 'HA1-1 onto itself, we know ZF1 is bounded and also maps 'HA 1-1 onto itself. By known results in semigroup theory (e.g., see Ref. 33, p. 393), we may extend Zt to a group by defining Z-t = Zcl. We mention that 2: remains the same in 'HA as it was in 'HI so that one can relate the adjoint semigroups by (Z,")-' = (Z;')*, pursue further converse statements for the original Hilbert space 'HI etc., which we will not do here. One reason we present these results here is in hopes that they be used elsewhere by those working in Zen0 theory. U What Theorem 5.2 states within the context of unitary quantum mechanical evolutions is that one cannot losc any of the totality of detail embodied in the totality of wave functions $ in D ( H ) . It is quite interesting to think about how much 'mixing around' of D ( H ) the evolution Ut can do. Perhaps one will investigate such questions elsewhere, e.g., from the viewpoint of ergodic theory. However, just by itself, Theorem 5.2 says that to have reversibility in quantum mechanics, you cannot lose a single wave function from D ( H ) as you proceed forward in time. In other words, you must be able to account for all evolving domain probabilities, future and past, all of the time. This finding also gives renewed importance to the role of preparation of states in quantum mechanics. The ensemble ( $ 0 ) of experimental initial states which are prepared to simulate a posteriori in an experiment what you could expect a priori from all possibilities from D ( H ) ,must be sufficiently extensive within D ( H ) in such a way that their trajectories continue to predict properly during evolution. Ideally, you need a dense subset from D ( H ) . Experimentally of course, one must settle for less. To emphasize the above discussion, which to us seems important, we may state, at the risk of sounding repetitive, the following.
Corollary 5.1. A quantum mechanical evolution Ut must and does continuously and simultaneously account f o r all probabilities I$(t)12and all wave functions +(t) in the domain of its Hamiltonian. No probabilities m a y be lost. It is important to note that Theorem 5.2 is of wider interest, i.e., it will apply to any unitary evolution Ut = eitH in any context. For example, even if one has "resorted" to the Heisenberg interaction picture and to unitary evolutions p ( t ) = U,poU;, one knows from Theorem 5.2 that the underlying Hamiltonian must have the domain regularity preservation
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property. Perhaps more interestingly, effective “interaction Hamiltonians” such as those assumed in quantum dynamical d e ~ o u p l i n g ~ must ~ ) ~ ’ also enjoy and respect the domain regularity preservation property of Theorem 5.2. Our point of view is that in such situations, this is a physical property that one should try to understand mathematically, in order to obtain a deeper understanding of the physics. There is another way to place Theorem 5.2 in context. There is a school of thought (see Refs. 41,42) in physics which views irreversibility as “going out of the Hilbert space.’’ This point of view certainly has some merit and goes back in physics to the notion of Rigged Hilbert space, among others. However, as I pointed out in Ref. 43, one is already out of the Hilbert space whcncver one needs to consider weak solutions in partial differential equations, a concept going back to the 1930’s. Also, and in quantum mechanics, the radiation eigenfunctions over the positive (scattering) spectrum of the Schrodinger Hydrogen operator are already non- L2 functions. So we construct L2 wave packets, or work in some Banach function space, or a Rigged Hi1bert space. What Theorem 5.2 says, within this context, is that you do not even need to go out of the Hilbert space to get irreversibility. You need only lose one wave function II,from the domain D ( H ) of the Hamiltonian. Then the unitarity of the dynamical system has been lost. The wave function II, can remain in the Hilbert space. Should Ut immediately reconstitute its unitmity, the lost wave function 1c, would continue its trajectory under the dynamical system Ut but it could not re-enter the ‘eligible’ D ( H ) prepared states until the evolution operator Ut again lost its unitarity for an instant and somehow switched II, back into D ( H ) as another wave function gets thrown out. I have not seen this conceptual picture anywhere in the quantum theory literature. I would assert that “going out of the operator domain” is a much sharper characterization of irreversibility. Elsewhere recently44 I showed that a number of Nobel Prizes depended in a fine analysis on an assumption of detailed balance a t a microscopic physical level. This then leads to a selfadjoint operator and hence to an implicit unitary evolution. The detailed balance was usually a t the level of some, e.g., microscopic reaction operators. I may here interpret Theorem 5.2 as a more precise detailed balance statement. It is at the wave function level and says that Ut must preserve the balance between any two wave functions, and not lose any of them. From the above considerations, which are of course important in themselves, there now enters the question of how to bring the operator trigonom-
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etry to further enrich our understanding of the microscopic dynamics taking place when quantum measurements are performed. The very act of a quantum measurement, except in the case when measuring instrument and the dynamics ‘commute’, involves an interaction which is, intuitively, irreversible. This must be, in my opinion, the key ansatz in the Zen0 problem, and we asserted it in our discussion a t the beginning of this Section. On the other hand, Theorem 5.2 (and the results following) involve only a commuting situation. That gave the important new results about microirreversibility as the loss of a single wave function. Can the noncommutative operator trigonometry now address the measurement (noncommutative) dynamics? It should be remembered that to date, in my opinion, no-one has successfully treated the quantum measurement interaction dynamics in a mathematically rigorous way. In Ref. 45 a beginning may have been made. There I showed that under a multiplicative perturbation H 4B H , although the direction of time is not changed, the speed of evolving dynamics has been altered. Moreover, there always exists a small disturbance such that a unitary evolution U, becomes a completely nonunitary contraction semigroup. In [46] a preliminary theory of interaction antieigenvalues for interacting operators A and B has been worked out. See Refs. 47,48 for some earlier discussion of the Bell and Zen0 problems. Acknowledgements
The invitation to speak at the ICAAA 2005 and the warm hospitality shown me in Quy Nhon by Professors Nguyen Minh Chuong and Nguyen Minh Tri and the others there was greatly appreciated. The invitation to speak at the Institute of Mathematics in Hanoi afterward and the warm hospitality offered me there by Professor Le Tuan Hoa and his group was equally appreciated. References
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Quantum Chemistry 98, 160 (2004). 32. T. Kato, Perturbation Theory for Linear Operators, (Springer, Berlin, 1980). 33. Riesz, F. and Sz.-Nagy, B. Functional Analysis, (Ungar Publishing, New York, 1955). 34. K. Gustafson, A Composition Adjoint Lemma, Stochastic Processes, Physics, and Geometry: New Interplays. I1 (F. Gesztesy, H. Holden, J. Jost, S. Paycha, M. Rockner, S. Scarlotti, eds.), American Mathematical Society, Providence, 2000, pp. 253-258. 35. P. Exner, and T . Ichinose, Product formula related to quantum zeno dynamics, in Proceedings of the X I V International Congress on Mathematical Physics, (Lisbon, World Scientific, 2003). See also arXiv:math-ph/0302060. 36. C. Friedman, Indiana University Mathematics J . 21, 1001 (1972). 37. P. Facchi, S. Pascazio, Quantum zeno subspaces and dynamical superselection rules, in The Physics of Communication (I. Antoniou, V. Sadovnichy, H. Walther, eds.), (World Scientific, 2003), pp. 251-286. 38. A. Schmidt, Journal of Physics A 35, 7817 (2002). 39. K. Gustafson, Partial Differential Equations and Halbert Space Methods, 3rd Edition, Revised, (Dover Publication, N. Y . , 1999). 40. L. Viola, E. Knill, and S. Lloyd, Physical Review Letters 8 2 , 2417 (1999). 41. I. Prigogine, Is Future Given? (World Scientific, Singapore, 2003). 42. I. Antoniou, G. Lumer, (eds.) Generalized Functions, Operator Theory, and Dynamical Systems, Chapman and Hall, CRC Research Notes in Mathematics 399, (London, 1998). 43. K . Gustafson, Rigged Spectral States: A Proclivity for Eigenvalues, in Generalized Functions, Operator Theory and Dynamical Systems (I. Antoniou, G. Lumer, eds.), CRC Research Notes in Mathematics 399, London, 1998, pp. 48-55. 44. K. Gustafson, Discrete Dynamics in Nature and Society 8 , 155 (2004). 45. K. Gustafson, Semigroups and antieigenvalues, in Irreversibility and Causality (A. Bohm, H. Doebner, P. Kielanowski, eds.), Lecture Notes in Physics 504, Springer, Berlin (1998), 379-384. 46. K. Gustafson, J . Math. Anal. A p p l . 299, 174 (2004). 47. K. Gustafson, The quantum zeno paradox and the counter problem, in Foundations of Probability and Physics-2 ( A . Khrennikov, ed.), (Vkixjo University Press, Sweden, 2003), pp. 225-236. 48. K. Gustafson, International Journal of Theoretical Physics 44, 1931 (2005).
