Functional Analytic and Complex Methods, Their Interactions, and Applications to Partial Differential Equations

where

l

R2

P is the formally adoint differential operator.

(1)

25

3

Results

To formulate the main results we will first make more precise a type of the domain D we deal with. Denote by irj:D-yRk,

j = l,2; fc = l,2;

k^j

the projection of D along the vector field lj. Let 7T3 : D - • r be the projection along the vector field 1%. The closure D of the domain D is supposed to satisfy the following hypotheses. (i) For any point p € D its projections

TVJP,

j = 1,2, lie on OAk, j ^ k;

(ii) The set D is /^-convex, j = 1,2. This means that given points p and g on any trajectory 7^ of the vector field Zj all the points r £ jj between p and q belong to D. Furthermore, having dealt with the maps TTJ,J = 1,2,3, we are able to introduce the two maps in T. Cl = 7T3 O 7Ti

and

C2 = 7T3 ° 7T2

which play a crucial role when formulating the main results of the work. These maps generate a noncommutative semigroup $£. The elements of $^ are maps in r of the form O = 0„

0...0CK.

where J = ( j i , . . . , jn) is an arbitrary multi-index with all jh equal 1 or 2. The semigroup $£ generates naturally a dynamic system. In what follows we use the following geometric terminology. 1) Given a map O G $£ , an ordered set (<ji,... ,qn+i) of points in V is called a J- orbit if qk+i=Cjkqk

for

1 < fc < n < 00.

(2)

2) If q-i = qn+1 or equivalently, if G
j = 1,2,

26

where Tq(T) stands for the tangent space of the curve T at the point q. Let We note that 7f is nothing but the set of all characteristic points in V with respect to the operator P(dx, dv). 3) If all the points of a J-orbit belong to the set 7^ , then this orbit is called characteristic. 4) A J-orbit (qi,. •., qn+i) is said to be T^-proper if in (2) Ok = Ci

when

qk <E TCl

and

(jk = (2

when

qk € TC2.

Prom the point of view of a dynamical process generated by the semigroup $£, in moving along any T^-proper orbit we leave each point qj € 7^ along a trajectory transversal to F. Definition We denote by N^ the set of all critical 7^-proper cycles in F. The following result related to the solvability of the First Boundary Problem for any hyperbolic operator P(dx, dy) in question was obtained in [1]. Theorem 1 Assume that at least one of the characteristic sets 7 ^ , 7^2 is finite. Then for any functions G G C(D) and h € C2{dD) there exists a unique generalized solution u(x, y) of problem (1) if and only if the set N^ is empty. The inverse operator (G,h) >—> u is continuous: C(D) x C2(dD) —> C2(D). Here C2(dD) denotes the space of continuous on dD functions whose restrictions to all the sides of the triangle D are twice continuously differentiable functions. However the existence of a classical solution u(x, y) € C3(D) has not been considered in [1]. In the noncharacteristic case (T = 0) and for curves T of a small curvature such a result was obtained in [3]. The following theorem is the main result of this work. Theorem 2 / / at least one of the characteristic sets 7 ^ and 7^2 is finite and the set N^ is empty then for arbitrary functions G £ C1(D) and h € C3(dD) a generalized solution u(x,y) of problem (1) is classical one. 4

Proof of Theorem 2

The proof consists of two parts. In part I we prove Theorem 2 for the case G = 0. After this, in part II, we construct a solution of the problem (1) which belongs to the space C3(D) and vanishes on dD. We will restrict ourselves to a homogeneous operator P. Part I. It is obvious that there exists such a linear change of variables in the space R 2 which reduces the problem (1) (with G = 0) to the problem (hdx + l2dy)dxdyu

= 0 in

D,

u=h

ondD.

27

Here D is a domain in R 2 whose boundary 3D consists of three parts T\ U T 2 U T 3 such that r i = {(z,y)|y = 0 , 0 < z < l } , T2 = {(x,y)\x

=

0,0
T 3 = {(x,y)\x

= x(t),y = y(t);0<

t < 1},

where l\ > 0, l2 > 0 and 1(0) = 0,

i ( l ) = 1,

2/(0) = 1,

3/(1) = 0.

(3)

Let h = hi(x) onTi,h = h2(y) on T2 and /i = /i3(a;,y) on r 3 . The continuity of the function h leads to the natural compatibility conditions

M0) = M0),

/n(i) = M i . o ) ,

/i2(i) =

ft3(o,i).

(4)

Since the domain D = {(x,y)|0 < a; < a;(i),0 < y < y(t);0 < t < 1} is supposed to satisfy conditions (i) and (ii) an arbitrary generalized solution u e C2(D), satisfying the boundary conditions on Ti U T2 is nothing but x

u(x,y)=

y

/ ( F(mis + m2t)dt)ds o o

+ h1(x) + h2{y)-h1{0),

(5)

(x,y) £ D. Here m = (mi,7712) is the unit vector which is orthogonal to the vector I = (h,l2) and mi > 0,m.2 < 0. As for the function F, this is an arbitrary continuous function on the interval / = (m2,mi). The necessity of satisfying the boundary condition u = ft3 on T 3 leads naturally to the following integral equation for the unknown function F £ C(I) x(t)

y(t)

/ ( 0

F(mix + m2y)dy]dx

= H(t),

0 < t < 1,

(6)

0

where H(t) = -h1(x(t))-h2(y{t)]+h3(x(t),y(t)\+h1(0). What is essential here is that the function H(t), generated by an arbitrary function h G C 3 (9D), belongs to the space C§([0,1]) = (C 3 H C 0 )([0,1]V This follows from the compatibility conditions (4). Thus, in order to prove that the function u(x, y) (which is denned by (5)) belongs to the space C3(D) it is sufficient to prove that ifH(t)eC3 ([0,1]V

thenFGC^I). Introduce the new variable z = rnix(t) + m2y(t),

0 < t < 1.

28

Since the domain D satisfies conditions (i) - (ii), we have x'(t) > 0, y'(t) < 0 for any t G [0,1]. Therefore, mix'(t) + m2y'(t) > 0 for all t, and there exists the inverse function t = cr(z),

z G T,

for which o-'(z) > 0. Introduce the functions 5\{z) =m1(xoa)(z),

52{z) = m2{y o a)(z).

where fog denotes the composition of two maps / and g. The following properties of these functions which follow from their definition will be important for us: Si(z) + S2(z) = z,

0 < 5i(z) < mi,

m 2 < 52(z) < 0;

0 < z < 1.

(7)

Substituting a(z) instead of t in the equation (6) we get S1(z)/mi

S2(z)/m2

(

F{m1x + m2y)dy)dx

o

= H(z),

z e I,

(8)

o

where H = Hoa G CQ(I). According to conditions (3) the right and the left hand sides of this equation vanish at the ends of the interval / . This means that differentiating twice this equation we arrive at an equivalent functionalintegral equation of the form F(z) - p1(z)F(s1(z)) z

+6'{(z)

- p2{z)F(52{z))

(9)

z

f F{s)ds + 6'2'(z) f F(s)ds = S2(z)

mim2H"(z),

Siiz)

where Pj(z) = 5?{z),j = 1,2. Since the functions H"(z), 5" and S2(z) belong to the space Cl{I) and F £ C{I), the function z

* = mim2n"

z

- 8'i f F(s)ds-S'2' S2(z)

f

F(s)ds

<5i(z)

belongs to the space C1^). Therefore, what remains to prove is that any solution F G C(I) of the functional equation F(z) - p i ( * ) f ( * i ( z ) ) - p2(z)F(fh(zj) belongs to the space CX{I) if $ G Cl{I).

= $(z)

(10)

29 Note that for the first time functional equations of this type (except when 5j are linear maps in R) have been investigated in above papers 1 and 2 . (in connection with a new problem in Integral Geometry). It turned out that the solvability of the equation is closely connected with dynamic processes generated by the semigroup <E>£. The following proposition is one of the main intermediate results in *. T h e o r e m 3 (ci1) Under the conditions of Theorem 2 functional equation (10) has a unique solution F £ C(I) for an arbitrary function
(11)

where T is a linear operator in the space C(i) given by TF = pxF o5x+ p2F o S2. Making use of inequality (11) we represent the solution of equation (10) in the form oo

oo m— 1

F = y^Tfe$ = y^ y^ Tmp+q§. k=0

p=0 g=0

Now in order to prove the differentiability of F one only need to show that for an arbitrary function $ € C 1 (7) the differentiated series OO 771—1

Y^ J2 dzTmp+q^

(12)

j>=0 q=0

with dz = d/dz converges uniformly on I. Let ||Tm||=7
imi < i. Denote p = max p\ + max pi and p' = max |pi| + max |p'2|. Obviously p < 1. It follows that l|a z T*||< \\dM + (/\\n

(13)

30

and hence for every integer q \\dzV$\\<\\dz$\\+qpf\\$\\

(14)

(Indeed, applying successively the two latter inequalities we find that \\dzTq+1^\\

< \\dzT0$\\ + p ' | | r * * | | < | | 0 , * | | + (q + l)p'\\*\\.

It remains to use the induction on q). On the other hand due to (13) we have | | ^ T m $ | | < HT^T" 1 - 1 *!! + / / | | T m - 1 $ | | < | | T 2 9 z r m - 2 $ | | + 2p'||Tm-2$|| <...<\\TmdM+mp'\\<$>\\. Thus \\dzTm§\\

< 7 | | 0 z $ | | + mp'||*||

Using once again the induction on p we easily arrive at the inequality \\dzTmp\\ < 7 P | | 9 2 $ | | + m / 9 ' p 7 p - 1 | | $ | | .

(15)

Now combining inequality (14) with q < m — 1 and (15) results in \\dzTmp+q§\\

= \\dzTq ( T m p $ ) ||

\\dzTmp\\+qp'\\Tmp$\\

<

<7Pl|9z*||+p'(m + gyP7p"1||*l|, whence m-l

/

mp+

_i_ 1 \

[m

)m

Y^ \\dzT o$\\ < mY\\dzn +

p'p^-'wn-

2

It follows that the functional series (12) is majorized by the convergent series oo

oo

p

m||5z$||^7 +(p'(m+l)m/2)^p7p_1||*llp=0

p=0

Hence series (12) uniformly converges and the differentiability of the function F is established. Together with this Theorem 2 is proved for G — 0. Part II. In this part of the proof we present a C 3 -solution u of the equation (hdx + hdy)dxdyu

= G in

D,

where G is an arbitrary C1-function in D. Changing x on l\X and y on ^2/ and using again the notation u, G and D for the new functions and domain, we reduce our problem to the equation (dx + dy)dxdyu = G.

31

It has an obvious solution x+y

u{x,y) = -

z + x — y z — x + y dz,

H

where x

V

H(x,y) = f ( f G(s,t)dt\ds. 0

0

•<3/

We are going to show that u G C'(D).

It is clear that

x+y

dxu=-

+ -

/ K^

—Hv

y z-x

+y

z+x —y z—x+y

dz,

and x+y

^

3 = 4#X

1 1 /" -^Hy + - / [HXX - 2HXy + Hyy]

_

z+x —y z —x+y dz, 2 ' 2 where Hx = dxH, Hxx = dxH etc. Since G G C 1 ( 5 ) , we have d2xdyH(x, y) G C(D),

dxd2yH(x, y) G C(D),

and therefore the function x+y

HX - ~Hy + ~ / (-2HXy) ( z + x — y z — x + y dz belongs to the space C1(D). Assume for a moment that both functions x+y

$i(x,y)

=

Hxx(

z+x —y z—x +y

dz,

o x+y

*2(l,I/)=

/

flyyf

z + a; — y z — x + y dz,

(16)

32

belong to the space C1(D). Then by virtue of (15) we find that d^u G C1(D) and consequently d^u G C(D). Because of the symmetry between the variables x and y, we verify in just the same way that dyU G C(D), and, finally, u G C3(D). Thus it remains to check that §j{x,y) e C 1 ^ ) , ] ' = 1,2. A direct calculation shows that (z-x+y)/2

o This means that x+y

(z-x+y)/2

*i(x,y)= J (

J

Gx[Z +

X

y

2~

,t)dtyz.

o o Introducing new variables z + x — y = 2s, t = t, we get x

$1(x,y)=2

J (x-v)/2

s—x+y

( J

Gx(s,t)dt\ds.

0

In this form the differentiability of the function $i(x, y) is guaranteed. In just the same way one can check the differentialbility of the function $2(^1 V)This proves Theorem 2. References 1. B. Paneah, On a New Problem in Integral Geometry with Application to Partial Differential Equations, Submitted, 2001. 2. B. Paneah, On a Problem in Integral Geometry Connected to the Dirichlet Problem for Hyperbolic Eqvations. International Mathematical Research Notices (IMRN), 1997, Nr. 5. 3. B. Paneah and B.-W. Schulze, On the Existence of Smooth Solutions of the Dirichlet Problem for Hyperbolic Differential Equations, Preprint 98/5, Universitat Potsdam, Institut fur Mathematik, 1998.

33

ZEILON'S O P E R A T O R A N D L A C U N A E P. WAGNER Institut fur Technische Mathematik, Geometric, und Bauinformatik Universitdt Innsbruck, Technikerstr. 13, A-6020 Innsbruck, Austria E-mail: [email protected]

First, some glimpses at history are taken. Mainly we examine the connexion of Zeilon's operator 9j + 9 | + ^3 with the investigation of lacunae of fundamental solutions of linear partial differential operators with constant coefficients. Second, some essential results from the work of Petrovsky, Atiyah, Bott, and Garding on the lacunae of fundamental solutions of hyperbolic operators are repeated and illustrated by examples. Third, we present a slight generalization of that work to non-hyperbolic operators and we sketch its application to Zeilon's operator.

In this introductory paper, I try to maintain the informal (and hopefully lively) style appropriate to what had originally been prepared for a lecture at the PDE-Workshop in February 2001 at Graz. For more mathematical details, I refer to 16 and 17 .

1

Historical remarks

1) Ivar Fredholm (1866-1927) became famous by establishing the theory of integral equations. Less known are his four articles (of only 18 in all; hence quality and quantity differ at times) on the calculation of fundamental solutions. One of those 4 items was his thesis, which also was his longest paper (42 pages), and which was published in Acta Mathematica in 1900 4 . Fredholm starts with the sentence "On connait le role fondamental que joue I'integrale particuliere £ de l'equation Au = 0 dans la theorie du potentiel." and then sets out to construct fundamental matrices F of elliptic 3 by 3 systems A(d) of linear partial differential operators in three variables with constant coefficients and homogeneous of second order. (We notice, in parentheses, that the notion of fundamental solution could be defined in general only after the advent of L. Schwartz' theory of distributions (cf. 1 5 ) , and therefore Fredholm made reference to the function £, which is, up to the constant factor — 4w, a fundamental solution of A in R 3 .) The respective formula (10) on p. 7 in Fredholm's elegant paper has, in modern language, the following appearance

34

(cf.

13

, 3.2.2, (F')]): F{x)

=

^gn(x2) 2TT

£

3

\a(x)\2

A((k(x)rd

^det(x,Cfc(x),(Vdet^)(C f c (x)))'

ad

where J4(C) denotes the adjoint matrix of A(C), and (,k(x) e C 3 \ {0} are determined up to complex factors by the conditions (Ck(x),x)=0,

deti4(Cfc(aO)=0,

Im (^^-)

> 0, fc = l , 2 , 3 .

(Here and in the following, ( , ) denotes the inner product on R™ or C , respectively.) In his last paper on fundamental solutions 5 , Fredholm calculated a fundamental solution E (actually the only one which is homogeneous and even) of the operator df + d\ + 9 | and obtained (up to the constant factor —
E{x) = -±T,\xJ\r

-r^—

87r

J=i Jt/{2x2j) V4u 3 - u 1 A / . fV2xj\ 1 \ = — x - / xi^ arcsin —. , —= .

Therein t denotes the largest of the three real roots of the cubic

and F denotes the elliptic integral of the first kind (cf. 7 , 3.131.8 and 8.111). 2) Niels Zeilon (1886-1957) was a student and admirer of Fredholm (see 6 ) . He generalized Fredholm's method of construction of fundamental matrices to arbitrary real homogeneous systems with the aim of describing the propagation of light in biaxial crystals. Before reaching this ambitious goal in 1919-1921 through his two big papers (cf. 1 9 ), he considered in 18 , presumably stimulated by Fredholm's last-mentioned work, the operator <93 + d f + <9f. I therefore designated this operator as "Zeilon's operator". Zeilon did not obtain an explicit representation for a fundamental solution as Fredholm did in his case, but he detected that the fundamental solution E which he considers becomes singular only along the surfaces dL U — dL, where dL denotes the boundary of the cone L : = { i e R 3 : i i > 0, x2 > 0, x3 > 0, x/ < X; + xk for all permutations i,j, A; of 1,2,3}.

35

Figure 1. The intersection of L with the plane x\ + X2 + X3 = 1

3) Thereafter Gustav Herglotz, Ivan Petrovsky, and Jean Leray gave representations for the (retarded) fundamental solutions of homogeneous hyperbolic operators with constant coefficients (cf. 8 , 1 4 , 1 0 ) . They also investigated conditions for the appearance of lacunae. Their theory was subsumed and generalized in 1970/73 by Michael Atiyah, Raoul Bott and Lars Garding in the fundamental papers x . Building up on them Johan Fehrman introduced in 1975 the class of hybrid operators, which, by definition, possess fundamental solutions that are real-analytic outside proper cones. As an example, he showed that d\ + 9 | + df is hybrid with respect to the direction N = (1,1,1) (see 2 , p. 223) and, therefore, it possesses a fundamental solution which is real analytic outside the wave front surface dL with respect to N (see 2 , Th. 4, p. 231). He also proved that this fundamental solution of d^ + df + <9f has (except at the origin) sharp fronts everywhere from within L (see 2 , p. 235). However, he did not give an explicit formula for a fundamental solution exhibiting this behaviour. 4) In 1990, Reinhold Meise, B. Alan Taylor, and Dietmar Vogt gave conditions for the existence of a "right inverse" R of a linear partial differential operator P(d) with constant coefficients u . In particular, they showed that there exists a right inverse on R", i.e. a continuous linear operator R : £>'(R n ) —>

36

V'(Rn)

satisfying P(d) o R = id, if and only if

Vr > 0 : 3/9 > 0 : Vx0 G R " with |x 0 | >p:3E€ V'(Rn) : P(d)E = 5 and E\{xeRn.,lx_Xol
= 0,

or, put more vaguely, iff P(d) has enough fundamental solutions with large lacunae. Zeilon's operator df + <9f +
()

=

T7^~si^n

( S v ^ x i ^ z s - x \ - x \ - x\) I J-l

8 V 3 7T 3

=

" VM + 1

/2

33/4

si

S n (3 ^X1X2X?, - x \ - x \ x

'y/2,-l-t\ F[ arccos —= '•'•

x\) ,

\/3 + l •=-

where t is the only simple real root or, if x lies on one of the co-ordinate axes, t is the triple root 0, respectively, of the cubic equation (xf + x%+ x% - 2x\x\ - 2x\x\ - 2xlxl)t2 §\ilx\xlxlt2 / -3v 16a;i^2a:3(a;i + x\ + x%)t - A{x\x\ + x\x\ + x | x | ) = 0. In particular, this implies that the level sets of E are algebraic, and the same also holds true for arbitrary real homogeneous cubic operators in three variables, as follows from the results in 16 . For quartic operators, a systematic analysis comparable to 16 for cubic ones is still missing. 2

Hyperbolic operators (according to Atiyah, Bott &: Garding)

In the sequel, we assume P(f) = °

We set Ppr(0 := E

**?*•

a

a£a, m = d e g P , and i ? e R " \ {0}.

J2 EN

;

'°'"'"

37

Definition The polynomial P(£) (or the operator P(—id) respectively) is called hyperbolic with respect to i?, iff (i) p pr (tf) ^

0 and

(U)

3io e R

. vi < t0 : V£ G R n : P(£ + iitf) ^ 0.

Lemma Suppose that P is homogeneous and that P(i9) ^ 0. Then the following conditions are equivalent: (a) P is hyperbolic with respect to •&, (b) V£ G R™ : the polynomial 11—• P(£ + ii?) has real roots only. In particular, if (a) and (b) hold, then P is—up to a constant factor—a real polynomial. Proof (a) implies (b): If P(£ + stf) = 0 for s G C, then, by the homogeneity of P, we have P( to for all a G R and hence I m s = 0. Thus (b) above is fulfilled. (b) implies (a): If t i—> P(£ 4- t-d) has only real roots, then, in particular, P(£ + it'd) / 0 for t G R \ {0} and therefore condition (ii) in the definition of hyperbolicity is satisfied when setting to = 0. In the sequel, we assume that P is homogeneous of the degree m and that P(tf) ^ 0. We denote by R P n l real projective space, i.e. R P " " 1 := {[£] : £ G R n \ {0}}, where [£] := {c£ : c G R } . (Analogously, complex projective space C P " " 1 is defined.) Then X := {[£] G R P n l : P(£) = 0} is the real projective algebraic variety defined by P. Furthermore, we denote by V that connectivity component of the set {£ G R " : P(£) ^ 0} which contains 79. Visualization of hyperbolicity in projective space If £, -d are linearly independent, then they span a line in projective space, namely g^ := {[r]} : $,,"&, r] linearly dependent}. By the preceding lemma, P is hyperbolic with respect to $ iff, for all £ linearly independent of $, g% has (counted with multiplicity) exactly m projective intersection points with X. In fact, due to P(#) ^ 0, the polynomial P(£ + W) is of the degree m in t. If P(£ + tjft) = 0, j = 1 , . . . , m, are its zeros (counted with multiplicity), which by (b) of the Lemma are all real, then [£ 4- tjfl], j = 1 , . . . , m, are the intersection points of g^ with X, and vice versa. E x a m p l e s 1) Let P(£) = (% - £ | — £f- Then P(d) is the wave operator in R 3 . The variety X is a circle in this case, and Figure 2 below (where £i = 0 represents the line at infinity) shows that P is hyperbolic with respect to -d

38

2) In the case of n = m = 3, X is an elliptic curve. In Hesse's normal form, X is given by the equation P(£) = £? + £f + £3 + 3a£i£ 2 £ 3 = 0. Figure 3 below, where £3 = 1, shows that P can be hyperbolic with respect to some d only if a < — 1. In particular, Zeilon's operator is not hyperbolic with respect to any

a ° -O.Z

-1

-0.8

-0.6

-0.4

-0.2

0 Xi2/xi1

0.2

0.4

0.6

0.8

Figure 2. Hyperbolicity of the wave operator

3

3

r

2

/

2

0

2o

1

-1

2

-2

3

\ .

-3 -2

0 Xi1

2

-2

0 xi1

Figure 3. Real elliptic curves; left: a < —1, right: a > — 1

2

39 Resume of the 1970 paper of Atiyah, Bott fc Garding. We assume that P is homogeneous of the degree m and hyperbolic with respect to -d. (i) 3\E = E(P) € X>'(R") : P{d)E = S and s u p p £ C {x G R " : (0,x) > 0}; (ii) K = K{P) := convex hull (supp£) = T d u a l := {x G R n : V£ G V : (£,x) > 0}; (iii) (J supp£;(P£)Csingsupp(A)E(P)cW:= |J tf(P€), £eR"\{o} £eR"\{o} where P^ is the localization of P at £, i.e., P(£ + r?) = P^T?) + o(M d e g P <) for 7? -> 0. (P^ is likewise hyperbolic with respect to $.) (iv) If \v\ > m — n and x £ K \W,

then dvE{x) = (5^, w^,) = /

where (a) 5^ belongs to the homology group Hn-2(UX), group of cycles modulo boundaries on the space

wXj„,

which is the factor

Ux := {[C] G C P " - 1 : (C,x) = 0,P(C) ^ 0}. Explicitly, ax is given by the map ax : Sx — C P " " 1 : £ — > [ £ - i «(£)],

S x := {£ G R " : |£| = U -1 * } ,

u being a C°° function on Sx which fulfills V C e S , : «(-£) = « ( 0 , «(£) G r ( P £ ) , « ( 0 J- £, V0 < i < 1 : P ( £ - i t o ( 0 ) ^ 0. (b) u)x~u belongs to de Rham's cohomology group Hn~2(Ux), which is the factor group of closed (n — 2)—forms modulo exact forms on Ux. Explicitly, u)x,v is given by !\l-» (27Ti)

(

v

m-- n n ))--ll // £" C x X _ \ k| i|/-| -((m

'

^ ) where r]x is the "Leray form" on Vx := {( G C

n

: ( _L x}, i.e., rjx = Xx\v

and

d((C,a;))AAcl;(C) = ^ ( - i r - 1 0 d C 1 A . . . A d 0 _ 1 A d 0 + i A . . . A d C „ + O((C,x)) 3=1

for (C,^) —> 0, the equation holding in the space £ln~l(Cn) of (n — 1)—forms on C n . We also use j]x to orient the (n — 2)—dimensional sphere Sx on which ax is defined.

40

(v) If x £ K \ W and a^ = 0 (the Petrovsky condition), then E is a polynomial of the degree m — n in the connectivity component of x in K \ W. (The respective components of K \ W are called "Petrovsky lacunae" of E.) E x a m p l e s 3) Let n = 3, P(£) = g - £f - ff, •& = (1,0,0), and E = E(P) as in (i) above. Then T = {£ £ R 3 : & > |£'|}, where £' = ( 6 , 6 ) - The dual cone of T is K = {x £ R 3 : x\ > \x'\} (here x' = (22,23)) and, by (ii), K contains supp E. For £ G R 3 with P(£) ^ 0, the polynomial Pg equals the constant P(£) and hence E(P^) is a constant multiple of the delta function. If P(£) = 0 and £ ^ 0, then

is a linear polynomial and the fundamental solution E(P^) has as support a half-ray on the forward light cone. Hence (iii) yields sing supp E — W = {x£ R 3 : xx = \x'\}. Finally, let us apply the formula in (iv). The homogeneity and the Lorentz invariance of the operator carry over to E, and we thus have E = cY(xi — \x'\)/^/xf — \x'\2. (Y denotes Heaviside's function.) It is thus sufficient to calculate c = E{x) for x = (1,0,0). Then Sx = {(0,£') £ R 3 : |£'| = 1}. For f £ Sx, Pt(r]) is the constant polynomial P(£) = - 1 and T(P^) = R 3 . Therefore we can set v equal to zero. Moreover, Ux = {[0, &, C3] € C P 2 : £f + £ | ^ 0} is homeomorphic to (C\{±i})U{oo) upon using the parametrization z = C2/C3- Since T]x(() = C3CIC2 — C2<3C3 furnishes the orientation on o#, the circle Sx in the [£2, £3]— plane is run through clockwise. Hence, in the variable z, ax corresponds to running twice along the real line oriented in the usual sense. For £ = (0, (2, C3); w e have Vx(C) = C3d(2 - C2dC3 =

P(C) ~

-CI-CI

dz

~ 1 + z2

and we conclude that

1

/"fa*(01

2 f -dz

1

4) Let us consider the wave operator in R 4 , i.e., n = 4, P(£) = £2 — £f~£3 — £i\ •d = (1,0,0,0), and E = E(P). As in Ex. 3, K = {x £ R 4 : xx > \x'\} and sing supp E — W = {x £ R 4 : x\ = \x'\}, where now x' = (x2,X3,X4). Let us check that the Petrovsky condition is satisfied for x = (1,0,0,0). In this the unit sphere in the [£2,£3,£4]— space oriented by Vx(0 = - & d & A d£4 + 6 ( ^ 2 A d£4 - &d& A d£ 3 .

41

Again, v can be chosen as 0. Let us identify Ux with {[C2,C3,C4] G C P 2 : Cl + C| + d 7^ 0}- Then, with ax, we twice cover the real projective space R P 2 contained in Ux. Since r)x{—£) = —r]x{Z,), the homology class a~x vanishes and, by (v) of the above statements, we conclude that E = 0 in K \ W. In fact, as is well known, E = 5{x\ — |a/|)/(47rxi) has such a Petrovsky lacuna inside the forward light cone. 5) Finally, let us consider n = m = 3 and P(£) = £ 3 + ff + £f + 3a£i£2£s (cf. Ex. 2 above) for a < — 1. Then P is hyperbolic with respect to i? = (1,1,1). By 16 , p. 284, l f = { i £ R 3 : A(a;) = 0, xi + x 2 + x 3 > 0}, where A(x) := 3a(a 3 + 4)xfx|a;§+ 4(a 3 + l ) ^ ? ^ + ^ l ^ l + ^ l ) +6a2xia:2^3(a;i + A. + ^i) ~ (^l + x\ +

x 2

l) -

The intersection of W with the plane xi + X2 + X3 = 1 is depicted in Figure 4 below.

x1=x2

Figure 4. The wave surface W for cubic hyperbolic operators

The propagation cone K is the convex hull of the outer, convex part of W. Let us choose x inside the inner, non-convex cone and y in between the two conical surfaces which make up W. Ux is the Riemannian sphere {[<] € C P 2 : (Car) = 0} with three points, the zeros of P(C), excluded. All these zeros belong to the real axis {[£] G C P 2 : (£,x) = 0,£ G R 3 \ {0}} in Ux- The cycle ax runs twice along the real axis turning around the real zeros of P once on both sides (cf. Fig. 5, left side). Therefore ax can be contracted in Ux and thus is homologous to 0, i.e., [ax] = 0. Hence E has a Petrovsky lacuna inside the non-convex part L of W.

42

For y however, one zero belongs to the real axis in Uy and two complex conjugate zeros lie off the real axis. (cf. Fig. 5, right side). Hence the homology class of ay does not vanish in H\(JJy).

< • >

Figure 5. The cycles ax (left) and ay (right)

3

Generalization t o non-hyperbolic operators

Although Zeilon's operator 9j + <9f + <9f is not hyperbolic (cf. II, Ex. 2), the fundamental solution considered by N. Zeilon nevertheless has a lacuna (in the sense of being constant) inside the cone L depicted in Fig. 1. This is implied by the following proposition, which carries over part of the statements of Atiyah, Bott, and Garding to the case where the dimension n is odd and P is of principal type, i.e., VP pr (£) / 0, V£ G R/1 \ {0} (cf. 9 , II, Def. 10.4.11, p. 38). As above, Y denotes Heaviside's function. Moreover, we define the Fourier transform as the extension by duality (or by continuity, which comes up to the same thing) of T : S(Rn) —> «S(Rn) : <j> i—> / R „ ^ e ' 1 ^ ^ d£ to the space 5'(R™) of tempered distributions. Pf denotes the finite part of a meromorphic distribution-valued function. For more details, see 16 , 2., p. 288. Proposition Suppose that n is odd and that P is a real polynomial in n variables, of principal type and homogeneous of the degree m. Define E e <S'(R") by E:=

^PfUHWtaS.)^™^ V SI

(27T) n V Then the following holds:

Ve-0

"

P(f)

(i) P(d)E = S and E(-x) = {-l)mE; (ii) singsupp ( A ) £ = {tVP(0 : £ £ R n ,P(£) =0,te (iii) li\v\>

m-n

n

and x e'R \W1

v

then d E{x) =

R H = W\{ax,ujx,v),

where wXi„ is as in part II and ax = \{PX + /3X), 0X:SX—*

CP""1 : $ ^

[£ - iw(C)].

Sx := « € R " : |C| = 1,Z± x},

43 with

V(0

= e(vP(0-||j)

and e > 0 small enough. (iv) If a; € R™ \ W and cTx — 0 (the Petrovsky condition), then E is a polynomial of the degree m — n in the connectivity component of x in R n \ W. E x a m p l e Let us consider n = m = 3 and P(£) = £ 3 + £ 3 + £f + 3a^i^2^3 (cf. II, Ex. 2) for a > —1. If x is inside the cone L which is the connectivity component of R 3 \ W that contains (1,1,1) (and which is depicted in Fig. 1 in the case of a = 0), then /3X, which equals j3x, has the same shape as ax in Fig. 5, left side. Hence E is constant in the two components of R 3 \ W containing ±(1,1,1), respectively. On the calculation of a fundamental solution of 8f + 9 | + 9 | + 30,818283. a) The polynomial P(£) := £ 3 + £ 3 + £3 + 3a^i^2^3 is of principal type for a G R \ {—1}- For definiteness, we assume a > —1. Let the fundamental solution E of P{8) be defined as in the proposition and let L U — L be the two lacunae of E (cf. the example above). For x £ R 3 \ ( I U ~L), f3x has the same shape as the cycle on the right side of Fig. 5. If px,p^ are the two non-real zeros of P in {[(} € C P 2 : (C,x) = 0}, and if px is chosen properly, then part (iii) in the proposition yields

= -^2-2^(ReSC=P*

= ^Im(Res c = R T [^

(svAO

L

no

Res£=p^ C3r/x(C)

^(0

(0 .

The brackets in these formulae symbolize the passage from one-forms on C 3 to one-forms on C P 2 (cf. \ II, p. 162). b) It is easy to check that Xx(() = \x\~2 det(C,#, d£) verifies the condition d((C, x)) A XX(C) = CidC2 A d( 3 - C2dCi A d( 3 + CsdCi A d( 2 + 0 « C , x))

44

(cf. II, (iv), (b)). Let us parametrize Ux by z = Ci/C3- Then C3>7x(C)

^(0

\P(0J

Cl= z > C2 = -(xiZ+X3)/x2,

l . 2 -det -(xiz \x\ nx{z) \ where Rx(z) := P(z, -{x\z conclude that Resf = P a ;

C3=l

z + x3)/x2 i

+ x3)/x2,1).

xi x2 aj3

If px

dz -xi(dz)/x2 0

dz

x2Rx{zY

[zi(x),z2(x),l],

then we

C3r?x(C)

x2R'x(Zl(x)))'

L ^(0

c) For 0:1,0:2 fixed and x = (xi,x2,xs), let us stress the dependence of R,Zj on the third coordinate by writing RX3,Zj(xz), instead of Rx, Zj(x), j = 1,2, respectively. By integration, the above yields 1

[•X3

E(x) = constant + /

d3E{x\,x2,

fX3 rX3

t) dt = constant — Im 2nx2 —J0Jo

Jo

dt R' RJ\ t(Zl(t))'

Let us substitute z\ = z\ (t) as new variable in the above integral. The identity 0 = Rt{zi{t))

= P{zi{t), -{xlZl{t)

+

t)/x2,1)

•^

--d2P(z1{t),-(x1z1{t)+t)/x2,l)

implies 0=

d

^ ( f l W ) = R't(Zl(t)) dt

di

and hence dt R't(Zl{t))

_

x2 dzi ~~dP/0z2

and

1

n[«*)] ft,

E(x) = constant ± —- Im / 2TT J[0 where fl

— spans the (one-dimensional) space ft1 {Xc) of holomorphic (Jjr j C/2-2

one-forms on the elliptic curve Xc ={[(]£ C P 2 : P(() = 0}, zx = C1/C3, an z2 = C2/C3! d the integral runs over some path in Xc ending in [((x)], which satisfies (((x),x) = 0. Eventually the addition theorem for elliptic integrals is applied in order to evaluate the imaginary part of the integral. For the final result, we refer to the theorem in 16 , p. 286.

45

References

1. Atiyah, M. F., R. Bott, and L. Garding, Lacunas for hyperbolic differential operators with constant coefficients I, II, Acta Math. 124 (1970), 110-189; 131 (1973), 145-206. 2. Fehrman, J., Hybrids between hyperbolic and elliptic differential operators with constant coefficients, Ark. Mat. 13 (1975), 209-235. 3. Franken, U., and R. Meise, Continuous linear right inverses for homogeneous linear partial differential operators on bounded convex open sets and extension of zero-solutions, in: Functional Analysis, Proc. Trier Workshop 1994 (S. Dierolf, S. Dineen, P. Domariski, eds.), de Gruyter, Berlin, 1996, pp. 153-161. 4. Fredholm, I., Sur les equations de Vequilibre d'un corps solide elastique, Acta Math. 23 (1900), 1-42; CEuvres completes: 17-58. 5. Fredholm, L, Sur I'integrale fondamentale d'une equation differentielle elliptique a coefficients constants, Rend. Circ. Mat. Palermo 25 (1908), 346-351; CEuvres completes: 117-122. 6. Garding, L., Mathematics and mathematicians: Mathematics in Sweden before 1950, AMS, Providence, 1998. 7. Gradshteyn, I. S., and I. M. Ryzhik, Tables of series, products, and integrals, Harri Deutsch, Thun, 1981. 8. Herglotz, G., Uber die Integration linearer partieller Differentialgleichungen rait konstanten Koeffizienten I-III, Ber. Sachs. Akademie Wiss. 78 (1926), 93-126, 287-318; 80 (1928), 69-114. 9. Hormander, L., The analysis of linear partial differential operators. I, II, Grundlehren, 256, 257, Springer, Berlin, 1983. 10. Leray, J., Hyperbolic differential equations, Institute of Advanced Study, Princeton, 1952. 11. Meise, R., B. A. Taylor, and D. Vogt, Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse, Ann. Inst. Fourier 40 (1990), 619-655. 12. Meise, R., B. A. Taylor, and D. Vogt, Continuous linear right inverses for partial differential operators of order 2 and fundamental solutions in half spaces, Manuscripta Math. 90 (1996), 449-464. 13. Ortner, N., and P. Wagner, Fundamental matrices of homogeneous hyperbolic systems. Applications to crystal optics, elastodynamics and piezoelectromagnetism, manuscript, 2001. 14. Petrovsky, I. G., On the diffusion of waves and the lacunas for hyperbolic equations, Mat. Sb. 17 (1945), 283-370.

46

15. Schwartz, L., Theorie des distributions, Nouv. ed., Hermann, Paris, 1966. 16. Wagner, P., Fundamental solutions of real homogeneous cubic operators of principal type in three dimensions, Acta Math. 182 (1999), 283-300. 17. Wagner, P., A fundamental solution of N. Zeilon's operator, Math. Scand. 86 (2000), 273-287. 18. Zeilon, N., Sur les integrales fondamentales des equations a characteristique reelle de la Physique Mathematique, Arkiv Mat. Astr. Fys. 9 (1913/14), no. 18, 1-70. 19. Zeilon, N., Sur les equations aux derivees partielles a quatre dimensions et le probleme des milieux birefringents I, II, Nova Acta Reg. Soc. Sci. Upsaliensis, Ser. IV, 5.3 (1919), 1-59; 5.4 (1921), 1-131.

47 R E P R E S E N T A T I O N OF T H E STOKES P O T E N T I A L IN DIVERGENCE FORM

J. A. DUBINSKII Moscow Power Engineering Institute Krasnokazarmennaja Street, 14, Moscow, 111250, Russia E-mail: [email protected] In this article we prove that the classical Stokes potential p(x) 6 Li may be represented in divergence form p{x) = divuj(a:), o

where w(x) = {w\(x),... ,wn(x)) belongs to the Sobolev space W\. It means that the Stokes system has the "symmetric" form div(Vu(a:)) + V(diviu(x)) = h(x)

(*)

with two unknown functions u(x) and w(x). The Stokes system (*) has the simple geometric sense to be the A-image of the o

orthogonal decomposition of W\ in the sum of solenoidal subspace and cosolenoidal subspace.

o

1

Representation

o

o

W^=S^2©VW|

Let R™ be the real Euclidean space and G C R n be a bounded domain with smooth boundary I \ Further, u(x) = (u\(x),... ,un(x))\ G —> R", ° W\= {u(x)\

" \\u\\l = ^2(Vui,Vui)0

< co,

u\r = 0 } ,

»=i

where (•, -)o is t h e inner product in L2, V = {d/dx\,..., d/dxn}. D e f i n i t i o n We introduce t h e subspace of so-called A-solenoidal -51,2 = M * ) &W^:Au(x)

= 0 in

functions

D'(G)}. o

o

It is clear, that 5 ^ therefore

2

is

one

°f

t n e

closed subspaces of the space

W\,

48 in the sense of the inner product n

(u,u)i = (Vu,Vu) 0 = ^ ( V U J , Vui) 0 . i=i

T h e o r e m 1 TTiere is £/ie decomposition

where Wi={p0{x):G->K}\

PaPo|lo
|bo||N E

P o | r = V p 0 | r = 0}.

|a|<2

P r o o f We show that the operator V gives the elliptic isomorphism

V: W*~ (SA,2)X. that is problem o

WPo(x)=q(x),

Po(x)eWi,

(1.1)

o

has the uniqe solution if and only if q(x) £ (S^ 2)^5 moreover \\Po(x)\\2 < M\\q(x)\\-2,

M>0, o

where || • ||_ 2 is the norm in the dual space W2~ =

(W2)*. o

In fact, if q{x) = Vpo(x), where po{x) G D{G), then for any u(x) we have in D'(G) the identity (divAw(x),p0(a;)> = -(u(x),AVp0(x))0

= -(u{x), Vp 0 (z))i-

o

It means that (u(x),S/po(x))

eW2

o

= 0 if u(x) eS2, i.e. Vpo(x) G (S&)2)±o

o

Since the set D{G) is dense in W2, then for any po(x) GW| the function o

q(x) = Vpo(x) is orthogonal to 5 ^ 2 . o

o

Now let q(x) GW2 be orthogonal to the subspace S ^2 of A-solenoidal o

functions. We prove that there exist one and only one function po(x) €W2 such that q(x) = Vpo(x). L e m m a 1.1 The boundary value problem (1.1) is equivalent to Dirichlet problem A2p0(x)=divAq(x),

po(x)€Wi,

(1.2)

49

if and only if q(x) G ( 5 A 2)^ • Proof The implication (1.1)—>(1.2) is obvious. o

On the contrary, let po{x) £]¥$ be the solution of the problem (1.2). This fact includes that divA(Vpo(aO-9(aO)=0

(1-3)

i.e. Vpo(z) - q(x) e Ker(div A) =5 A > 2 °

o

On the over hand for every po(x) e W | the gradient Vpo{x) € (5 o

Therefore, if q(x) e (SXt2)±

^2)'L-

tnen

VP0(x)-

q(x)G(Si,2)X-

(1-4)

Inclusions (1.3) and (1.4) gives Vp0(x) - q{x) = 0. Lemma is proved. To finish the proof of the theorem it is enough to note that for any o

o

1

q(x) GR^ the problem (1.2) has the unique solution po{x) GWf, moreover ||po(a0ll2<M||divAg(a:)||_2,

M > 0. O

R e m a r k It is clear that for any function u(x) £W^ with div Au(x) — div Aq(x) the solution of the problem o

A2p0(x) =divAu(x),

p0(x)£W£,

(1.5)

is the same that solution of the problem (1.2). Therefore, the map u(x) —> Vpo(x) defines the projector

which acts parallel to subspace of A-solenoidal functions. Finally, every funco

tion u(x) €iW\ can be represented as the sum u(x) = UA,S(X) + X

Vp0(x),

where UA S( ) is the A-solenoidal part of u(x), and po(x) is the solution of (1.5).

50 o

Corollary Since the map A:W2<-> W 2 _1 is an isometric isomorphism, then W " 1 = S ' - 1 © V A W22, where S^1 = {h{x) G W^-.divhix)

2

= 0 in

D'(G)}.

Representation W^=S£ © A Q 1 V ( L 2 / R 1 )

Let O

O

Sl= {u{x) eW%:divu(x)

= 0 in D'(G)} o

be the subspace of all solenoidal vector-fields in W2 ; L2/K1 = {po(x) G L2: (p(x), 1) 0 = 0} — the subspace of functions in L 2 , which are orthogonal to unity. The norm in I/2/R 1 will be denoted by ||p(x)|| L , R 1 . Finally, A^"1 means the inverse o

operator to map A: W2—> W^1 • Theorem 2.1 The operator AQ V defines the elliptic isomorphism A^ViLa/R1-*^, i.e. for any p(x) G L2/R 1 the function q(x) = AQ Vp(x) is orthogonal to o

o

subspace S% of all solenoidal functions and, inversely, for any q(x) G (S^) -1 there exists one and only one function p(x) G L2/R-1 such that Ao1Vp(x)^q(x), moreover ||p(z)||WR1<M||divg(aO||0,

M>0.

(2.1)

Proof In fact, since the set of smooth functions is dense in L2, then for o

1

any p(x) G L2 (in particular, p{x) G L2/R ) and for any
i.e.

AQ1VP(X)

= 0, o

J_»S2 in the sense of inner product in W^-

51

Contrary, let q(x) G (S^)1' be the arbitrary function. We show that there exists one and only one function p{x) G L2/H1 such that Ao'Vpix)

= q(x)

( in Wj)

(2.2)

Lemma 2.1 Equation (2.2) is equivalent to equation div (AQ ^ ( a ; ) ) = divq{x)

( in L2/B})

(2.3)

o

if and only if q(x) J-S^The proof this Lemma is similar to the proof of Lemma from § 1 and we omit it. Lemma 2.2 For any h(x) G L2/R 1 there exists the unique solution p{x) G L2/R 1 of the equation div(A-1Vp(x))

= h(x),

(2.4)

moreover, the inequality

IWL 2 / R 1 <M||^)|| L 2 / R 1 is valid. Proof We use the known Galerkin method. Let v\(x),V2(x),... basis in .L2/R 1 . The sequence of approximate solutions

(2.5) is the

N

PN(x) =^2cjNVj{x),

N =

l,2,...,

are found from Galerkins moment equations (div (Ao1\/pN(x),vj(x))0 j=

= (h{x),vj(x))0,

{2A)N

l,...,N,

or, that the same, -(VpAr.VujJ-i = (h(x),vj(x))0

(2.6)

3=

1,...,N, ° _, where (•, -)-i is the inner product in W2 • A n apriori e s t i m a t e Obviously, we have from (2.6) (V P J V ) Vp N )-i

< | | M z ) L 2 / R 1 • ||PAr|| La/R i-

(2-7)

52

Futher, since PN{%) G L2/R1, then using the known inequality (due to Necas, Ladyzhenskaja, Babuska, Brezzi)

\\p(x)\\L2/n^M\\VP(x)\\-v

M > 0

we immediately obtain the estimate (TV = 1,2,...) IKNIWRI

^ M i||Ma:)|| L2/R i, Mi > 0.

This estimate gives the uniqueness and, therefore, solvability of the system Moreover, we can assume that PN{%) —> p(z) weakly in L2, where p(x) £ L2/R 1 is a function. It is clear, that p(x) is the desired solution of the equation (2.4). The uniqueness of solutions is clear too. Lemma is proved. (2.4)JV-

o

Summing up all previous results, we obtain that every function u(x) eW^ may be written in the form u(x) = us{x) + A,71Vp(a;),

(2.8)

1

where divu s (:c) = 0 and p(x) £ L2/R- is the solution of the equation div(Ao 1 Vp(a;)) = divu(x), moreover

l|p( a ; )|lL a /Ri^ M ll d i v U ( a : )llo'

M>0

(2"9)

-

The theorem is proved. o

o

Corollary (inequality "on the contrary"). Let W\ /'S\ be the factoro

o

space, which is identified] with the orthogonal supspace W2 © S\ • Proposition The inequality Hull L, ll \\woysi c < M | |1d1 i v u |no

M > 0,

is hold. Proof Since the decomposition (2.8) is orthogonal then TA

o

o

w}/s\

\U~Us\\l

= llA01VP|ll =

= ||Vp(a;)||_1<M||p(a:)||WRl. Using the inequality (2.9) we obtain now, that o < M||divw(a;)|L, u "wysl "

||M(Z)|| O

11

M > 0.

53

Representation W^ =S£ © A ~ 1 V (H 2 / R 1 ) © V W |

3

In this paragraph we make more precise the previous decomposition. To this o

o

o

°

end we note that S^cS^ 2 , where S^ 2 cWj 1 is the subspace of A-solenoidal functions, considered in §1. According to the main result of this paragraph o

°

o

o

(&A 2)^ — ^ W$, therefore it is necessary to describe the subspace S^ 2© &2Let H2 = {pi{x) £ L2 : APl(x)

= 0 in D'(G)}

be the subspaces of harmonic potentials. Theorem 3.1 The formula Sit2

=k

©Ao ^ ( H a / R 1 )

holds. o

o

Proof Let the function u(x) eS^29

o

^21 that is u(x) -LS2 and at the o

same time divAw(a;) = 0. Due to §2 the orthogonality u(x) _L S 2 implies that u{x) = Aj^VpiCa;), 1

where p\{x) G L2/R is the solution of equation div(Ao 1 Vpi(a;)) =divw(x). (3.1) Taking into account divAw(:r) = 0 we immediately obtain from (3.1), that Api(ar) = 0, i.e. Pl(x) G H2/R1. On the contrary, if u(x) = A^1Vpi(x) where p\{x) G H2/R1 is the o

solution of the equation (3.1), then u(x) G {S2)'LSimulteniously, bearing in mind Api(x) = 0, we find from (3.1) that divAu(a;) = 0. The theorem is proved. o

4

o

Representation L2/R 1 = div(W£ / S£)

Let us prove the decomposition of the space L2, which is dual (in some sense) o

to the decomposition of the space W\ • Theorem 4.1 The orthogonal sum L 2 = R 1 ©div(W 2 1 /5 2 1 ) is true.

(4.1)

54

Proof In fact, if p(x) = c + divw(x), o

where w(x) eW^ (in particular,

o

1

1

w(x) GM^ / S ^), then (c, diviu(a;))o = -(Vc,w(x))0 = 0, that is the sum (4.1) is orthogonal. Contrary. Let p(x) G I/2 be the any function, then p{x) may be written in the form p(x) =c + q(x), where c G R 1 and q(x) G (R 1 )" 1 . We show, that there exist the function (only o

o

one function) w(x) GW^ / S\ such that divtu(x) = p(x)

(4.2)

Lemma 4.1 Equation (4-2) is equvalent to the equation Vdivu>(x) = Vp(x) 1

(4.3)

1

if and only if q(x) G (R )- . The proof is the same as that of Lemma 1.1. Lemma 4.2 For any h(x) G W^1 there exists the unique solution o

o

w(x) GW2 I S\ of the equation Vdivw(x)

= h{x).

(4.3)'

Moreover, the estimate \\w(x)\\ o o <M\\h(x)\\ u wl/s\

.,

M > 0,

(4.4)

is hold. o

o

Proof Let i o i ( x ) , . . . , WN(X), . . . be the basis in W\ / S\. Then we find the approximate solution wN(x) = c^wi(x) + ... + C%WN(X), where unknown coefficients c f , . . . , c$ are determined from the moment equations (VdivwN(x),Wj(x)) j=

= (h(x),Wj(x)) l,...,N O

(Here {•, •) means the duality in the pair (W 2 -1 , W^1)-)

(4.5)

55

Multiplying (4.5) on c^ and summing due to j = 1,... ,N,we obtain the equality (Vdivu/^aO.iu^a;)) =

(h(x),wN{x))

or, that is the same, -(divw; Af (x),divu) W (a;)) =

(h{x),wN{x))

Prom this we immediately receive the inequality \\diywN(x)\\20<\\h(x)\\_1-\\wN(x)\\1 Applying the inequality ||io(x)|| o o < M||divtu(a;)|| n , ii

"

w

i /

s

i

llu

"

M > 0,

(Proposition in the end of paragraph 2) we see that for any N = 1,2,... \\wN(x)\\<M\\h{x)\\-i,

M>0.

This estimates gives the weak compactness of the set {wN(x)}, o

N =

o

1,2,... in the space W\ IS\. It is clear, that the limit-point is the solution of the problem (4.3)', moreover, the estimate (4.4) is true. This apriori estimate gives, obvionsly, the uniqueness of the solution. Lemma is proved. To finish the proof of Theorem 4.1 it necessary to use Lemma 4.2 with h(x) = Vq(x). The theorem is proved. In conclusion we remark, that for n = 3 the formula o

o

1

div(W2 /521)=L2/R1 was proved by Ladyzhenskaja O.A. and Solonnikov V.A. in 3 . o

Let us remark also that the series of more general decompositions of W% , including the fields with values in Clifford algebra, was obtained in the work 4

5

Application to the Stokes s y s t e m

As is known the map o

A: Wo1-* W21

56

is isometrical isomorphysm, therefore the orthogonal decomposition of W2X, proved before, gives automatically the orthogonal decomposition of the dual space W21 = A Si © V ^ / R 1 ) © VA Wl . It means that any function h{x) G W 2 _1 may be decomposed in the sum h(x) = Aus(x) + V(pi(a;) + Apo(aO),

(5-1)

1

where pi(x) G H 2 / R is the harmonic potential, denned as the solution of the equation div(A^ 1 Vpi(a;)) = div(A(71/i(a;))

(5.2)

and potential po(x) (regular part of the Stokes potential) is the solution of the Dirichlet problem A2p0(x)

= div/i(z),

p0|r=0,

Vpo(x)

= 0,

(5.3)

r or, that is same, ,_s_Ji= ..u,-S A,22p„0(x) divh(x),

9

P0(X)

„ /_M = _ n0, po(x)\

dv

= 0 r

{y — normal direction to the boundary T). It is clear, the representation (5.1) is adequate to the Stokes system, which is written in the form Aus(x) + V(pi(ar) + Ap0{x)) = h(x), divus(x)

=0

us(x)\r=0,

p 0 (a;)| r = 0,

Vp0(x)\r=0.

1

Let us notice that both pi(x) G H2/R. and Po(x) are denned from (5.2) and (5.3) independently from the velocity field us(x). Moreover, in accordance with the results of $4 the Stokes potential p{x) = pi(x) + Apo(x) may be written in the divergence form p(x) = div w(x) and, consequently, the Stokes system has the form div{Vus(x))

+ V(divto(x)) = h{x),

us(x)eS12,

w(i)G^/i

(5.4)

57

where w(x) is also unknown pressure field, which my be defined either from equation Vdiv w(x) = Vp(x) or from (6.4) directly, because the (5.4) is equivalent to the following two projection qualities o

(Aus(x),
= (h(x),
Vipa(x) eSl

and (aO,VP(aO> = (h(x),ipP(x)),

Wv(z) &W$/S\

.

References 1. Ju. A. Dubinskii, Doklady RAS, B 373, 727 (2000). 2. Ju. A. Dubinskii, Doklady RAS, B 374, 13 (2000). 3. O. A. Ladyzhenskaja, V. A. Solonnikov, Notices of scientific seminars of Leningrad Branch of Mathem. Steklov Inst. B 59, 81 (1976). 4. H. Begehr and Ju. A. Dubinskii, (Preprint, Freie Univ. Berlin, 1, 2000).

58

O N S P E C T R A OF T H E OPERATOR R O T O R R. S. SAKS Department of mathematics, BSU, Chair of Differential Equations Frunze str. 32, 450074, Ufa, Russia E-mail: [email protected] Boundary value problemes and spectra of the operator rotor are studied The operator rotor is one of the most important first-order differential operators, which appears in various physical models, for example in Maxwell equations of electricity and magnetism. In fluid and plasma physics rotor appears to measure the velocity of various flows. In recent time we observe the extension of investigations dedicated to a basic of mathematical theory of boundary value problems for rotor with low order terms and, in particular, there are investigations of the eigenvalue problem rot u = Xu

in

Gel

3

(1)

with a boundary condition n-u = 0

on

dG

(2)

from a rigorous mathematical point of view. Here u is a vector-function, G is 3-dimensional bounded domain with a smooth or piecewise smooth boundary dG, and n is a unit normal vector on dG. See references in 9 , 1 0 . The eigenfunctions of rotor are called free-decay (or A-force free) fields. Chandrasekhar and Kendall 1 give an explicit calculation of the eigenfunctions for a periodic straight cylinder region. The A-force free fields appear also in Arnold's theorem 2 as an exceptional case. In the theory of fusion plasmas a free-decay field is called Taylor state that is considered to be the ultimate minimum-energy plasma equilibrium 3 . On the other hand, the operator rot u + \u appears in the mathematical theory as example of weakly elliptic operator (in my terminology) in papers of Vainberg and Gruhsin 4 (with A = — 1 and dG = 0 ) . They have considered the more general operator d+ *, where d is the operator of exterior differentiation and * is the well-known operator of zero order transmitting a form uik into another from *wk of degree dim X — k on the compact manifold X without boundary, dimX = 2fc + 1. They proved that this operator is of Predholm type in the Sobolev spaces

d + *:Hs (Xk) -*Hsd-1(xk+1Y

s£R,

59

where the form i k+2

fc+1 7

e Had~x (xk+A

if

fc+1 7

e Hs~l ( x f c + 1 ) and d~/k+1 €

H'- (x y

In papers 5 , 6 I have proved that the operator rot u + A/3 with A e t \ {0} has Predholmian type boundary value problems at the appropriate functional spaces (of Holder or Sobolev type) for every bounded domain G with smooth boundary T. For example Theorem 1 The boundary value problem rot u + \u = h l-u\r

= f,

in G,

h, div h G HS(G),

with A =£ 0,

withfeHs+i(r),

0

0,

(3) (4)

is of Fredholm type if the field I on T satisfies the condition I • n ^ 0. If this problem is solvable, then each solution u belongs to the Sobolev space HS+1(G). When I = n (n is the unit normal) from this theorem it follows Corollary 1 For all eigenvalue A ^ 0 the number of linear independent eigenfunctions of problem (1), (2) is finite. Each of this eigenfunctions from if 1 (G) belongs toC°°(G). We have also considered the case when I -n = 0 on 7 C T. Let I = (0,0,1), then I -n = us, where u = (u\,U2,1/3). Suppose G is the ball B, \x\ < R, T is the sphere \x\ = R, 7 is its equator (where |a;| — R and X3 = 0) and y is any point on 7. Theorem 2 The boundary value problem for equation (3) with conditions U3 | r = / 3 ,

u2 | 7 = /2,

ui{y) = fi(y) = const,

(5)

is of Fredholm type and its index is equal to zero. In the case when A2 does not coincide with any eigenvalue of the Dirichlet problem for scalar Laplace operator -&v = nv

inB,

v\M=R=0,

(6)

the problem (3), (5) has a unique solution. This solution u belongs to C1(G)n C°'a (G) if he C2'<* (G), h e Cl>a(T), h e C 0 ' a (7), 0 < a < 1. In 7 I have studied the general boundary value problems for the operator rot u + a(x), where a(x) is a matrix such that operator div(aV) is elliptic. If a = A/3, this condition reduces to A ^ 0 because div(AV) = AA, where A is the Laplace operator. We distinguish the classes of boundary volume problems which have Predholmian type. Normally solvable and Noetherian (Predholmian) boundary value problems for the system of Maxwell equations in the case of steady-state processes were studied in n by the same method. A basic mathematical theory of eigenvalue problems (1), (2) was given in papers of Berhin 19 , Giga and Yoshida 9 and Picard 10 .

60

Now we consider in detail the eigenvalue problem (1), (2) in the case of G = B, where B is the ball, |x| < R. We proved that the eigenvalues of the problem (1), (2) are the square roots of the eigenvalues of the Dirichlet problem for scalar Laplace operator (6) with condition v(0) = 0, i. e. = X2v

-Av

in

B,

w||x|=fl = 0,

v(0) = 0.

(7)

8

I have proved : Lemma 1 If (A,u) is a solution of the eigenvalue problem (1), (2) with A ^ 0, then the pair (A2, v) is the solution of the eigenvalue problem (7), where v = x • u is scalar product of the vectors x and u. Inversely, if the pair (/z, v) is the solution of the eigenvalue problem (6) and v(0) = 0, then there are two and only two solutions (y/]J,u+) and (^/Jl,u~) of the problem (1), (2) such that u+ ^ u~, but x • u+ = x • u~ = v. The solutions of the eigenvalue problem (6) are well-known 12 . We introduce the spherical system of coordinates (r, 8,
2

m, n G N,

(8)

> 0 are zeros of the function

*.e>^~.e> = <->-(£)"(*r)-

(»)

The multiplicity of the eigenvalues A* n is equal to 2n + 1. Therefore the operator of the problem (1), (2) has pure point spectrum with no accumulation point. Theorem 4 The spherical components ur, UQ, U^ of the eigenfunctions u m n k are 9iven by the formulas («r)n,m,fc=

(uv + iue)^m
^ / " V n l A ^ r ) ^ ^ ^ ,

|*| < „ ,

k, m , n G N ,

=

£M/«<- ( ">*.M-O£

(ab£+«s)*<»•<*

(I0>

o where Y^(8,ip) are spherical functions, c^m

k

are arbitrary real constants, i =

\/—T is the imaginary unit. These functions belong to the class C°° (B \ {0}),

61

where 0 is an axis on which r sin 0 = 0, and they are bounded in B. Moreover the components «i, u2, us of eigenfunctions um n k in the original rectangular Cartesian system of coordinates belong to the class C°° (23). Theorem 5 The eigenvector-functions corresponding to different eigenvalues are orthogonal in the vector space L2(B). I suppose that the set of orthonormal eigenvectors um n k gives a complete orthogonal basis in the subspace of L2(B) which is orthogonal to the kernel of the operator rotor in L2{B). Remark 1 I had the method to solve the eigenvalue problem (1), (2) in 1971 when me and my student A. A. Fursenko have solved problem (3), (4) in the case of G = B and I = n in Holder spaces 13 . This paper was written but it were not published because I didn't know then nothing about an application of the problem (1), (2) in mathematical physics. Remark 2 In papers 1 4 , 1 5 , 1 6 , 1 7 V. I. Arnold used the eigenfunctions of the operator rotor with eigenvalue equal to one in the class of 27r-periodic vector-functions. See also Henon 18 . We give some of my result about spectra of operator rotor on periodic vector-functions. Theorem 6 The eigenvalues of operator rotor in the class of 2-K-periodic vector-functions are equal to ±|k| = ± \ A i + ^ 2 + ^ 3 for a^ ^ = (^11^2,^3) from Z 3 . The multiplicity of non-zero eigenvalue ±|ko| is equal to the number of points k G Z 3 lying on the sphere of radius |ko|. The multiplicity of \ = 0 is infinite. Theorem 7 With every k € Z 3 we can associate three eigenvectors k e j k x , c + e ikx^ c - e ik-x 0j ^ e 0perafor rotor witfi eigenvalues A = 0, A = —|k|, A = |k|, where the vectors k, c£ and c^ are mutually orthogonal. With k = 0 we associate the basic eigenvector-functions e j , e2, e3.

'^i\k2

+

if

k\ = k2 = 0,

if

k\ + k\± 0.

kik^

c£ = I ±i\k1 + k2k3 I , -A2 + k\

Theorem 8 The eigenvectors corresponding to different pairs (A, k) are mutually orthogonal in the space L2{Q), where Q is a cube (0, 2-7r)3. They form a complete basis in this space.

62 Table 1. The multiplicities of eigenvalues

A^ Multiplicity of A A^ Multiplicity of A

1 6 13 24

2 12

3 8

4 6

5 24

6 24

7 0

8 12

14 48

15 0

16 6

17 48

18 36

19 24

A^ Multiplicity of A

25 30

26 72

27 32

28 0

29 72

30 48

X2

37 24

38 72

39 0

40 24

41 96

42 48

Multiplicity of A

20 24

9 30 21 48

10 24 22 24

31 0

32 12

33 48

43 24

44 24

45 72

34 48 46 48

23 0

12 8 24 24

35 48

36 30

47 0

48 8

11 24

The following six complex eigenvectors

±iXl

I 0 j e±«2j /

l

j e±ix3

correspond to the eigenvalue A = 1. One can form six real eigenvectors from them. The values of multiplicities from the above table are calculated by a computer program of my student Yu. N. Polyakov. References 1. Chandrasekhar S., Kendall P. S., On force-free magnetic fields, Astrophys. J. 126 (1957), p. 457-460. 2. Arnold V. L, Sur la topologie des ecoulements stationnaires des fluides parfaits, C. R. Acad. Sci. Paris 261 (1965), no. 1, p. 17-20. 3. Taylor J. B., Relaxation of toroidal plasma and generation of reversed magnetic fields, Phys. Rev. Lett. 33 (1974), p. 1139-1141. 4. Vainberg B. R., Grushin V. V., On uniformly nonelliptic problems. I, Math. Sbornik 72 (1967), no. 4, p. 126-154. 5. Saks R. S., On boundary value problems for the system rot u + Xu = h, Soviet. Math. Dokl. 12 (1971), no. 4, p. 1240-1244. 6. Saks R. S., On boundary value problems for the system rot u + Xu = h, Diff. Equat. 8 (1972), no. 1, p. 126-133. 7. Saks R. S., On boundary value problems of Noether type for some classes of weakly elliptic systems differential equations, Proc. of Inst. Math. Sibirsk. Otdel. Akad. Nauk SSSR, Novosibirsk, Nauka, 1978, p. 237-253.

63

8. Saks R. S., Spectra of operator curl in ball..., Complex analysis, diff. equations and related topics, IV. Applicable mathematics (Proc. of intern. conf.) Ufa, Inst, of math., 2000, p. 61-69. 9. Giga Y., Yoshida Z., Remark on spectra of operator rotor, Math. Z. 204 (1990), p. 235-245. 10. Picard R., On selfadjoint realization of curl and some its applications, preprint Math-AN-02-96, Techn. Univ. Dresden, 1996, 31 p. 11. Saks R. S., Normally solvable and Noetherian boundary value problems for the system of Maxwell equations in the case of steady-state processes, Soviet. Math. Dokl. 28 (1983), no. 2, p. 391-395. 12. Vladimirov V. S., Equations of mathematical physics, 5th edition, Moscow, Mauka, 1988, 512 p. 13. Fursenko A. A., Boundary value problems for one uniformly nonelliptic system, Dissertation of undergraduate work, Novosibirsk, NSU, 1971, 28 p. (not published). 14. Arnold V. I., Hopf asymptotic invariant and its applications, Izbrannoe — 60, (1974), no. 28, Moscow, FAZIS, 1997, p. 215-236. 15. Arnold V. I., Sur la geometrie differentieile des groups de Lie de dimension infinie et ses applications a I'hydrodinamique des fluides parfaits, Ann. Inst. Fourier (Grenoble), 16 (1966), no. 1, p. 319-361. 16. Arnold V. I., Korkina E. I., Growth of the magnetic field in 3-dimensional stationer stream of incompressible fluid, Vestnik MGU. Seria 1. Math. Mech., (1983), no. 3, p. 43-46. 17. Arnold V. I., Some remarks about antidinamo theorem, Vestnik MGU. Seria 1. Math. Mech., (1982), no. 6, p. 50-57. 18. Henon M., Sur la topologie des lignes de courant dans un case particulier, C. R. Acad. Sci. Paris 262 (1966), p. 312-314. 19. Berhin P. E., Selfadjoint boundary value problem for the system *du + Xu = / , Soviet. Math. Dokl. 222 (1975), no. 1, p. 15-17.

64

A L G E B R A I C PROPERTIES OF P O T E N T I A L D I F F E R E N T I A L A N D PSEUDODIFFERENTIAL OPERATORS I.E.PLESHCHINSKAYA Kazan State Technological University, 68 Karl Marx Str., Kazan 420015 Russia E-mail: [email protected], [email protected] The method of potential operators for differential and pseudodifferential operators is considered. The system of equations for axially symmetric electromagnetic field is considered as an example.

1

Introduction

We say that the operator P is a potential for operator S if there exist operators R and Q such that equality SR = QP

(1)

is fulfilled. Let / = Qg. Just from definition it follows that if ip is a solution of equation Pip — g, then u = Rip is a solution of the equation Su = f. We say that the equation Pip = g is a potential for the equation Su = f and solutions of potential equation are potential elements. We say also that the formula u — Rip defines a representation of solutions of the equation Su = f. The main idea of the method of potential operators is to pass from the equation Su — f to an equation Pip — g which has to be simpler or more convenient for research than the assumed one. If the operators Q and R have inverse, then the solutions of the assumed equation and the solutions of the potential equation correspond one-to-one to each other. In this case the method of potential operators can be reduced to a simple replacing u = Rip of the unknown element in the operator equation Su = f and to an equivalent passing from the equation SRip = / to the equation Pip = g. But in general case the replacement of the unknown element and further transformation of the equation will be non-trivial. First of all, not individual potential elements but some classes of such elements coresspond, as a rule, to solutions of the assumed equation (if, for example, the operator R is not invertible). Besides, not all solutions of the assumed equation are obligatory included in the representation u = Rip. Let us consider some classical examples.

65

In mathematical analysis the term "potential" is used in order to describe vector fields. One uses to say that the equation lotp = 0 determines potential field in the domain f2. The field p is a potential iff there exists such scalar function u that p = gradu in il. On the other side, the equation divp = 0 in the domain Q, determines a solenoidal field. The field p is solenoidal iff there exists such vector potential function A that p = rot A in Q,. It is easy to see that in these two cases equality (1) is one of identities for the repeated operations of the field theory rot gradu = 0,

div rot A — 0.

By this Q can be an identical operator and P is a zero operator. For the real Cauchy-Riemann system of equations in a plane domain 0 du dx

dv dy

'

du dy

dv dx

one of the potential equations in O, as it is known, is the Laplace equation d2

dx

d2
=

Q

dy2

If one passes to complex functions (complex-valued functions of complex variable), then any in O analytic function $(z), z = x + iy, will be a potential element for the Cauchy-Riemann system, where u(x, y) = Re &(z), v(x, y) = I m $ ( z ) , and

—-n

o

JL. — 1(—

—\

&z ' ' dz 2 \dx dy) will be a potential equation. The method of potential operators was used for the first time in the works of G.Airy x and J.Maxwell 2 where the system of the main equations of elasticity theory was investigated. In the plane case under absence of space forces tensions and movements are expressed through the second derivatives of Airy's function being a solution of the biharmonic equation. In electrodynamics, for example, seeking an electromagnetic field harmonically dependent on time in a homogeneous isotropic medium all components of the field being a solution of Maxwell's system of equations are expressed through the potential functions being the solutions of Helmholtz's equation. Note that in the theory of nonlinear operator equations 3 it is customary to use the term "potential" for an operator acting from the Banach space X into its conjugated space X* and coinciding with the gradient of some element

66

from X*. This definition does not correspond to the terminology adapted by us. Nevertheless, potential in the sense of the theory of nonlinear equations is a value of the operator S = grad on a functional from X*. So it is possible to consider the term "potential" as "corresponding to some potential element". It is easy to see that in general case the three operators (R, Q, P) for a given operator S are not uniquely determined. The research of algebraic properties of potential operators gives one opportunity to solve a series of questions which arise under choosing the most convenient in that or another sense potential equation and corresponding to it representation of solutions for some given equation. 2

Pseudodifferential and differential operators

Let us use the same denotions as for scalar functions spaces, so for vectorfunctions or functional matrices spaces, every component of which belongs to corresponding function space. Consider pseudodifferential operators (p.d.o.) Su(x) = (2TT)-"/s(x,<£)w(£)e i ( : c ' { ) ^,

R
(27r)-nJq(x,Omeiix'i)d^

P
(2TT)-nfp(x,0mei{x'°dt

with matrix symbols s(x,£), r(x,£), q(x,£), p(x,£) of dimensions I x m, mxp, Ixp, pxp from classes S^(Rn), S^2{Rn), 5o 2 (i?"), S^{Rn) correspondingly. Equality (1) takes place when the condition mi + tri2 = rii + n.2 is fulfilled. We assume that all p.d.o. being considered in the work are eigen operators. Theorem 1 The P.d.o. P is a potential for the p.d.o. S and the p.d.o. R defines the representation of solutions of equation Su = 0 iff for some p.d.o. Q the equality / /

s(x:Or(y,v)ei{x~y^~T')dydC

q(x,Op{y,v)ei{x~v'i"T>)dyd^

Vx£Rn,

Vry G Rn

(3)

67

is fulfilled. This statement follows from the fact that the symbol is uniquely defined by p.d.o.. Besides, if two p.d.o. are equal, then their symbols are equal also. Note that under constructing a potential operator P and an operatorrepresentation R the part of the operator Q is not so significant. One can think that potential operator P is determined to within multiplier Q. Equality (1) defines the right factorisation of p.d.o. P (to within multiplier Q), i.e. representation of operator QP in the form of composition of two operators S and R of lower order. Left factorisation RS = QP means that system of equations Su = f can be transformed after differentiating to a system of equations of higher order. Consider the particular case when S, R, P are differential operators with symbols s(z,£)=

sa(x)£a,

E

r(y,ri)=

I«|<1

E

rp{y)T]p,

|/3|<2

P{X,V) = E P7(X)T?7' M<2 where sa{x), rp(y), p 7 (x) are functional matrices of dimensions Ixm, mxp and pxp correspondingly. In this case Q is a differential operator of no more than 1-st order. Let the operator Q have the symbol N
sar/3=

J2

1<*P&

Va,/3,7, |a| = 1, |/3| = 2, |-y| = 3;

a+/3=7

2Sra+p + sar0 + s0ra = 2Qpa+p + qap0 + q0pa Va, [3, \a\, \/3\ = 1; (4) S r 7 + syr0 = Qpy + q-fPo V7, |-y| = 1; Sr0 = Qp0 are fulfilled.

68

To prove this statement one has to substitute the symbols of the operators S, R, Q, P into the condition (3). Note that the differential operator SR has the 2-nd order iff J2

sar0=O

Va,/3, |a| = 1, |/?| = 2, | 7 | = 3.

(5)

Really, the left-hand sides of conditions (5) are the coefficients of the 3-rd order derivatives in the expression SR. If the conditions (5) are fulfilled, then the operator Q can be of zero order only. In this case conditions (4) have a simpler form. The next particular case can be received by assumption that there are no 2-nd order derivatives of potential functions in the representation R. Then the operator Q has also zero order. The case when the operators S,R,Q,P have orders 1, 1, 0 and 2 correspondingly and all matrices under consideration are square matrices and non-degenerate is considered in details in 8 . Some differential operators and the corresponding representations of solutions for systems of partial differential equations of the 1-st order containing no 2-nd order derivatives of potential functions are investigated in 9 , 10 . 3

Splitting of potential operators

Let I = m= p. Denote by {P}s the set of all operators P which are potential for the operator S. Obviously, every operator has infinitely many potential operators. To simplify the research of the set {P}s, we introduce its partition into classes of elements. Denote by {P}S,R the set of elements from {P}s corresponding to a fixed representation R. If P i , P 2 G {P}s,R, then 3Q\,Qz \ QiPi = Q2P2. So we say that tje potential operators from {P}s,R are Q-equivalent. Consider also a set {R}s of all possible representations for the operator S. One can put in correspondence to every element R from {R}s a set of potential operators {P}s,R- Obviously, {P}S=

|J

{P}S,R.

R€{R}s

Let us consider algebraic properties of sets {P}s, {P}s,R and {R}s- Introduce some restrictions on the set of operators {Q}s permissible for a given operator S. In the simplest non-generate particular case, if all operators are differential operators with constant coefficients and the operators Q have zero

69 order, then {Q}s is algebra of square matrices. We will assume that in more general case the set {Q}s is also an algebra. T h e o r e m 3 If a set of p.d.o. {Q}s is an algebra, then the sets of p.d.o. {P}s, {P}s,R and {R}s are algebras also. Proof. Under fixed S and R for arbitrary elements P\, P2 from the set {P}s,R there can be found such operators Qi,Q2 from {Q}s that Q\P\ = QiP-z- By this to every operator Q there corresponds some operator P (not a unique in general case). This can be easily verified in the particular case which is considered in Theorem 2. If the operator Q is given, then from the last equation (4) the coefficient po of differential operator P will be determined, from the next to the last equation the coefficients p 7 , |-y| = 1 will be determined and etc. A set of operators-representations {R}s of the fixed order will be an algebra if one defines in a special way multiplication and addition operations for elements. We introduce a product of two p.d.o. on the basis of Taylor's series expansion of their symbols. Let p.d.o. R\,R2 have symbols r i ( z , 0 = J2

-,D%n(x,0)C,

L

—' a !

r2{x,0=

£

A^?r2(x,0)^.

—' Or.

We define the product R = R\ R2 as p.d.o. with symbol r(s,0= £ aezi

±D%n(x,0)D%r2(x,0)e-

Multiplication of differential operators in this sense can be reduced to multiplication of the coefficients attached to derivatives of the same order. By this, evidently, the order of a product does not exeed order of each multiplier. At last, a set {P}s can be considered as a set of classes of Q-equivalent potential operators. By this one is to choose one element in every set {P}s which will be a representative of the whole class. Then the operations in the algebra {P}s can be consistent with the operations in the algebra {-R}sIf the matrices of coefficients attached to derivatives of high order of the assumed operator S are degenerate, then equality I = m = p is not fulfilled. In this case number of equations in potential system of equations always is less than number of equations in the assumed system but algebraic properties of sets {Q}s> {-Pis, {P}s,R and {R}s remain the same. By this it is nessesary to research in addition the rank of the coefficients matrices. Further an example will be considered - the system of equations of axially symmetric electromagnetic field.

70

We say that the operator R gives a complete representation of solutions of the equation Su = 0, if the formula u = Rip where ip is a solution of corresponding potential equation Pip = 0 gives all solutions of the equation Su = 0. In this case we say that potential operator P is complete also. In 8 a set of solutions of equation Su = 0 under I = m = p is considered. It is proved there that every operator S has a complete potential operator. This proof is based on the following statement: if the equation Rip = u is solvable for any right-hand side, then formula u = Rip where ip is a solution of equation Pip = 0, P € {P}s,R gives all solutions of the equation Su = 0. We say that a potential operator P is trivial if the set {u}p = {u \ u = Rip, Pip = 0} contains only trivial solution of equation Su = 0. We say that a potential operator P £ {P}s is splittable if a non-trivial operators can be found such that P\,P2 6 {P}s, {u}p = {u}p1 © {u}p2 (subspace of solutions of equation Su = 0 associated with operator P can be non-trivially decomposed into a direct sum of subspaces of solutions of the same equation assosiated with another potential operators). If the complete operator potential for an operator S is splittable, then any solution of the equation Su = 0 can be represented as a sum of solutions just of the same equation belonging to non-intersecting classes. We call this phenomenon a polarisation of the set of solutions of the equation Su = 0. Theorem 4 An operator P e {P}s *s splittable iff it can be represented as a sum of two non-trivial operators P\,Pz € {P}s (in the sense of the usual addition of operators). Proof of the Theorem for differential operators with square matrices of coefficients can be found in 8 . General case can be considered by analogy. Criterion of splittability of a complete potential operator gives Theorem 5 A complete potential operator P for an operator S is splittable iff there exists a non-trivial right ideal in the algebra {P}s4

The system of equations for axially symmetric electromagnetic fields

Let us consider as an example Maxwell's system of equations in cylindrical coordinates for electromagnetic fields with time harmonic dependence in a homogeneous isotropic medium. We will seek the components of the field in the form of Fourier's series expansions with respect to the azimuthal coordinate. For the coefficients of the expansion with just the same number n (for shortness of formulas the index n for the unknown functions is not indicated)

71

we get the following system of equations ^Hz r H^ z

dz dH: dr

-Ha +

8Er

• itJeosEa,

=

—iuJllQllHr,

8EZ

-^

w -TT • ^ —Hr = IUJE0EEZ, r

dHa dr

dEa dz

™EZ r

iujeo£Er,

.

K— =

dz or 1 ,-, dE -Ea + — a r or

-lUJUoflHa,

(6)

in . TT Er = -iwfjL0^H z r

If one groups four above equations of this system into pairs and resolve them for Er,Hz and Ez,Hr, then we get Er —

nEa + nr

2

2 2

kr — n

dEa ,dHai + uu0ur dr dz

Hz

i 2 dEa dHa + Los0erEa + we0er -s— dr k2r2 -- n2 nr dz

Ez

i fc2r2 _ n2

(7)

dK TT 2^Ha nr- dz - wn0nrHa - U)u0fir ——dr

i

Hr

, dHa TT nHa + nr—dr

2 2

kr

uis^Er

2^<* dz

These formulas give the representation of solutions of the system (6), where Ea, Ha are potential functions. To obtain a system of potential equations, one has to substitute right-hand sides of (7) into lower two equations of system (6). It is easy to verify that such potential system breaks up into two independent equations only for n = 0. It is possible to get for the system of equations (6) potential systems for any n. But in this case representations of solutions of the assumed system will contain the 2-nd derivatives of potential functions. Such representations have the form Er -,

Ea

Ez

d2ipv

-iuuou—ipe .

dipe dr

= IUHOU—

drdz ' 1

in dwm — , r dz

d2
rr

'2,„e aV

r

TT

Ha

HT

drdz

=

+ iweoe—u> r

in dwe — r dz

ay2 dz

IUJEQE-

+ k2
dtpm dr

(8)

72

here every potential function ipe, ipm is a solution of scalar Helmholtz's equation d2(p

1 dip

d2ip

/

2

n2\

,9

9

Similar representations of solutions of Maxwell's system of equations just in other coordinate systems are known. By representations (8) many boundary value problems for a system of equations (6) can be transformed into boundary value problems for scalar or vector Helmholtz's equation. In the theory of boundary value problems for Maxwell's equations in axially symmetric mediums, two particular cases of representations (8) are frequently used: the representations which can be obtained by the assumption that one of the potential functions is equal to zero. These two cases correspond to electromagnetic fields of E— or H— polarisations. Solutions of the corresponding boundary value problems for scalar Helmholtz's equation can be obtained by the method of separation of variables 11 . By this method solutions of the classical problem on eigen waves of cylindrical waveguides with metallic walls were obtained, waveguide properties of dielectrical rods of round section (optical fibre) 12 were investigated and some diffraction problems in cylindrical waveguide structures with inhomogenuity were solved. General case of representations (8) is used by researching eigen waves of three-layer dielectrical waveguide in the paper 13 . Investigating axially symmetric electromagnetic fields by the method of potential operators, the following question arises: which variant should be chosen among two ones: representation of solutions with the 1-st order derivatives but with rather complicated systems of potential equations, or simple potential system of two independent equations but with representation of solutions containing 2-nd order derivatives? May be, there exist representations of solutions of system (6) with the 1-st order derivatives of potential functions satisfying independent equations, don't they? The answer to the first question depends on the form of boundary conditions to which the assumed solution of system (6) should satisfy. Let us show that the answer to the second question is "no". Represent the differential operator of system (6) in the form Su — si — + s2 ^

s 0 u,

u = {Er,Ea,Ez,Hr,Ha,Hz).

(9)

73

The coefficients matrices of this operator have block structure 0 s°0

0 s$

si

S2

s? 0

=

IWEQEE

so

-iuj/j,ofj,E

where "00 0 " 00 - 1 01 0

Sn

=

0

"0 - 1 0 " 1 0 0 0 0 0

0 in u.

r

0 - in

0 1

0 0

r

As matrices si,s2 are degenerate, it is sufficient to consider potential operators of the form d2 d2 d2 d d (10) J •Pl21 -P22^-^+Pl-^-+P2^~ +PoI or oroz z oz 2 x 2or. oz the operatorThen with coefficients matrices of dimension representation of the 1-st order ••Pii:

d d r0I ar oz and multiplier Q (matrix) has dimension 6 x 2 . In this case system of equations (4) has the form R

sir i = Qpn,

sir 2 + s 2 ri = Qpi2,

(11)

s2r2 = Qp'22,

drx dri si-rr: + S2-g

s 0 ri + sirQ = Qp1,

dr2 dr2 + s2— si -zor oz

s0r2 + s2rQ = Qp2,

(12)

dr0 dr0 s2— s0r0 = Qpa. si dr oz Assume that the matrices pn,Pi2 are non-degenerate. Then it follows from the 1-st and the 3-rd equations of system (12) that only 2-nd and 5-th lines of the matrix Q are different from zero. As for matrices r-\_,r2, they have zero lines 2-nd and 5-th ones. Therefore the algebra {P}S,R for any possible R is isometrically isomorphic to the algebra of square matrixes of the 2-nd order. Taking into account the structure of the matrix Q, we get from the other equations of system (12) that only four elments of the matrices ri,r2,ro are arbitrary (for example, elements of the 1-st and the 3-rd lines of matrix r-y),

74

and all other are linearly expressed through them. So, algebra {R}s is also isomorphic to the algebra of square matrices of the 2-nd order. One can show that under any selection R G {R}s it is impossible to reduce simultaniously the coefficients matrices of a potential operator to diagonal form by virtue of selecting the matrix Q. Then in algebra {P}s does not exist an incomplete potential for the S operator. Consequently, in the case of representation of solutions with the 1-st order derivatives there does not exist potentials for the system (6) consisting of independent equations. References 1. G. B. Airy, Philos. Trans. Roy. Soc, 153, 49-80 (1863). 2. J. C. Maxwell, Proc. London Math. Soc, 2, 58-60 (1865/69). 3. H. Gajewski, K. Groger, K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen (Akademieverlag, Berlin, 1974). 4. Ju. V. Egorov, Linear Differential Equations of the Main Type (Nauka, Moscow, 1984) [in Russian]. 5. F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators (Plenum Press, New York and London, 1982). 6. E. Hille, R. S. Phillips, Functional Analysis and Semi-Groups (A.M.S., Providence, 1957). 7. N. Dunford and J. T. Schwartz, Linear Operators. Part 2. Spectral Theorie (Wiley-Interscience, 1971). 8. I. E. Pleshchinskaya, N. B. Pleshchiskii in Functional analysis, Approximation Theory and Numerical Analysis, ed. John M.Rassias (Word Scientific, Singapore, 1994). 9. I. E. Pleshchinskaya, N. B. Pleshchiskii, Invest, in Appl. Math. (Kazan), 20, 97-107 (1992) [in Russian]. 10. I. E. Pleshchinskaya in Pitman Research Notes in Mathematics Series, 256 (Longman Scientific & Technical, 1991). 11. H. Honl, A. W. Maue, K. Westpfahl, Theorie der Beugung (SpringerVerlag, 1961). 12. L. Felsen, N. Markuviz, Radiation and propagation of waves (New Jersey, USA, 1973). 13. I. E. Pleshchinskaya, N. B. Pleshchinskii in Mathematical Methods in Electromagnetic Theory. Proc. Int. Conf MMET*98. Kharkov, Ukraine, June 2-5, 1998. Vol.2.

75

T H E M E T H O D OF W E I G H T E D F U N C T I O N SPACES FOR SOLVING INITIAL VALUE A N D B O U N D A R Y VALUE P R O B L E M S

W.TUTSCHKE Graz University of Technology, Department of Mathematics Steyrergasse 30/3, A-8010 Graz, Austria e-mail: [email protected]

D,

In order to apply functional-analytic methods to initial value problems or to boundary value problems, one has to consider a suitably chosen function space, for instance a normed one. In case the desired solution has singularities, these singularities can be compensated by using weight functions. The present paper surveys this method for initial value problems of Cauchy-Kovalevskaya type and for boundary value problems with boundary data piecewise belonging to fractional order spaces. In both cases the method under consideration is based on the combination of functional-analytic methods with methods of Complex Analysis.

1 1.1

Solution of initial value problems in associated spaces Statement of the problem

Consider the initial value problem dtu = F[t,x,u,dXiu\ u(0,x) =
(1) (2)

n

where x = (x\, ...,xn) is a point of M , t means the time, and the right-hand side J 7 of (1) is a function at least continuously depending on its variables. Problem (1), (2) is equivalent to the integro-differential equation t

u(t,x)

- (p(x) + / T\T,x,u,dXju\dT (3) o and, therefore, solutions of (1), (2) can be constructed as fixed points of the operator defined by t

U(t,x) = tp(x) + /

T\T,x,u,dXiu\d,T.

(4)

76

The famous Lewy example 9 shows that there are even infinitely differentiable functions T such that the operator (4) has no fixed points for each choice of the initial function ip. In the paper a sufficient condition on the right-hand side T\t,x,u,dXju) of (1) will be given under which the operator (4) has a fixed-point and, therefore, under which the initial value problem (1), (2) is solvable.

1.2

A weighted Banach space for the construction of fixed points

Let n = <x: x>0>.

Consider the simple initial value problem dtu =

-dxu

u(0,x) = —. x The initial function has a singularity at the point x — 0 which is a boundary point of f2. The solution of this initial value problem is u(x, t) =

showing x—t that a singularity at a boundary point may come into the domain £7 itself in the course of time. This leads to a restriction of the time interval in which the solution exists. The nearer a point x to the boundary, the shorter the time interval may be. Generally speaking, the solution of an initial value problem (1), (2) (i.e., a fixed point of the operator (4)) exits thus in a conical domain over CI only. In order to construct this conical domain, one has to measure the distance of a point x G Q from the boundary d£l. For this purpose consider an exhaustion of Q by a family of subdomains Cts, 0 < s < so, satisfying the following conditions: 1. If s' < s", then fiy is a compact subset of fis». 2. With the exception of a fixedly chosen point i o £ ( l , each point i e f ! belongs to the boundary of a uniquely determined domain fis(x) of the family. 3. There exists a positive constant c\ auch that for any s',s" with s' < s" the distance of f2s< from d£ls» can be estimated by dist (ns,,dns„)

>ci(s"-s').

77

Defining S(XQ) — 0, let M be the conical domain M = Ut,x)

: i e S l , 0 < t < T](s0 ~

s(x))\

whose height is equal to rjso- Then d(t,x) = so — s(x)

(5)

is a weight function which is positive in M, and which vanishes on the lateral surface of M. Using this weight function, it will be possible to consider functions u — u(t,x) having singularities on the lateral surface of M. In order to construct a suitable Banach space of functions defined in the conical domain M in the t, z-space, let B(f2) be any Banach space of functions which are defined in any (bounded) domain fi in the rr-space. Denote its norm by IHIg(jj)- Suppose B(Q) satisfies the following conditions: 1. If ft' C Q", then B(Q") is embedded in B(fi')> i-e-> t n e restriction of a function u G B(£l") to Q,' belongs to B(fi')> and one has

llullB(n') ^ ll u lls(n") • 2. Functions belonging to 6(fi) are bounded, and one has sup|w| < c 2 | | u | | B ( n ) where C2 does not depend neither on u nor on £1. Note that the second condition is satisfied with C2 = 1 for the Holder norm Wu\\B(a) =

max

SU

V n

P l u l.

SU

\u(x') -

u(x")\\

P

x'^tx"

0< A<1. Again let fis, 0 < s < so, be an exhaustion of a given (bounded) domain in the complex plane. Denote B(fls) by Bs, and the norm in Bs will be denoted by ||-|| s . For a fixedly chosen i < JJSO the intersection of M with the plane t = i in the t, x-space is given by < (t,x) : t = i, s(x) < s\

78

where

i

s = s0

rj

.

(6)

Let B*(M) be the set of all (real-valued) function u — u(t,x) satisfying the following conditions: 1. u(t,x) is continuous in M. 2. u(t, x) belongs to Ba^

for fixed i if only s(x) < s where s is given by (6).

3. The norm H , =

sup

\\u(t,-)\\s(x)d(t,x)

(7)

(t,x)eM

is finite. The definition (7) of the norm ||-|| t implies the estimate (8)

for any point (t, x) in M. Proposition 1 B»(M) is a Banach space. Proof Note that the inequality d(t, x) > 5 > 0 defines a closed subset Ms of the conical domain M. Each point of M is contained in such a subset Ms provided 8 is suitably chosen. For points (t,x) in Ms, the definition (7) implies the estimate

Kv)L(x) ^ jIHI. Now consider a fundamental sequence «i, U2, ... with respect to the norm ||-|| t . Then one has \\un(t,-) -um\\s(x)

< - -e

(9)

for points in Ms provided n and m are sufficiently large. This implies also \un - um\ < c2 • - • e

79

for points in Ms- Consequently, a fundamental sequence converges uniformly in each Ms, i.e., the un have a continuous limit function u*(t,x) in M. Similarly, estimate (9) shows that for t — i and s(x) < s the limit function belongs to Ss(a:) because of the completeness of the underlying function spaces B(Q). Carrying out the limiting process m —> oo in the inequality ||u n — u m |L < e i it follows, finally, ||u n — it*^ < e and, therefore, ||u*||„ is finite. 1.3

An estimate of the integro-differential operator

Of course, the operator (4) is defined only for functions u = u(t, x) for which the first order derivatives dXju exist. Suppose such a function u = u(t,x) and its first order derivatives belong to B»{M). We are going to answer the question under which conditions the image U = U(t, x) belongs also to 6*(M). Consider again the Banach spaces 6(£2) introduced above. Definition Suppose £2' is any subdomain of Q," having a positive distance dist (£2', <9£2") from the boundary of £2"'. Then a function u £ B(£l") is called a function with a with a first order interior estimate if dXju belongs to B(Q') and

H^ull» ^ dist(r?,3f2")

l|U B(n )

"

"

(10)

where the constant cj, depends neither on the special choice of u nor on the choice of the pair £2', £2". Applying this estimate to the exhaustion £2S of £2, 0 < s < so, one gets

| | M | , ^ % • p r r 7 • HIprovided s' < s". Now let (t, x) be an arbitrary point of M. Then d(t,x) = s0 — s(x)

> 0.

Define s = s(x) +

-d(t,x)

s < s(x) + - (s0 - s(x)j

= -s(x)

implying + -so < so

(ii)

80

and thus there exists a point x with s(x) = s, i.e., x £ d£l$. One has d(t, x) = s0 — s(x)

= -d(t, x). V * Taking into account the estimate (8), the last relation gives d(t,x)

d(t,x)'

In view of (11) one gets, therefore,

To be brief denote J-(t,x,u,dXju)

by Tu, i.e., in particular one has TO =

J-(t, x, 0 , 0 ] . Next we have to estimate the norm of Fu. For this purpose we assume that the right-hand side T of (1) satisfies the following conditions: 1. J-Q is continuous. 2. The norms ||.F0|| S are bounded and thus H-F0H,,, is finite. 3. Tu satisfies the (global) Lipschitz condition \\Tu - Tv\\s < L0 \\u - v\\s +Y,Lj

\\dXiu -

dXjv\\g.

3

Using (8) and (12), the latter Lipschitz condition implies ||.Fu|| 8(l) < | | ^ e | | s ( x ) + L0 \\u\\s{x) +Y/Lj

\\dXju\\s{x)

j

since d(t,x) < SQ. The definition (5) of the weight function d(t,x) implies /

1

J (P(T,X) o

,

V

d(t,x)

81 and thus it follows t

I

Tu • dr

^ ( ^

e

" *

S 0 + C 4

^ * )

(13)

s(x)

where d = s0Lo H

}

Lj

T h e estimate (13) of the s(x)-norm yields

I

Tu • dr

<

n{\\m*

SQ

+ C4 \\U\\

Suppose, finally, t h a t the norms ||
I^II*
1.4

Solution of initial-value principle

problems

by the

contraction-mapping

T h e Propositions 2 and 3 are t r u e only under the hypothesis t h a t the first order derivatives of u(t, x) exist a n d belong also to S „ ( M ) . In addition, u(t, x) must be a function with a first order interior estimate. This is not the case for an arbitrary element of B*(M). In order to apply t h e above estimations, one has t o find a closed subset of S „ ( M ) such t h a t

82

the assumptions mentioned above are true everywhere in this subset. Such a subset can be defined as kernel of an elliptic operator Q. Define B%(M) = { u € S*(M) : Gu(t, •) = 0 for each fixed t\. Notice that Q has to be an elliptic operator whose coefficients do not depend on t. Condition (10) can be verified using an interior estimate for solutions of elliptic differential equations 2 , whereas 13% (M) is closed in view of an Weierstrass convergence theorem for elliptic equations. In order to apply the construction mapping principle, the operator (4) has to map this subspace Sf (M) into itself. Definition Let T be a first order differential operator depending on t, x, u = u(t,x) and on the spacelike first order derivatives dtu, while T is any differential operator with respect to the space variables Xj whose coefficient do not depend on the time t. Then T, Q is called an associated pair if T transforms solutions of Qu = 0 into solutions of the same equation for fixedly chosen t, i.e., Qu — 0 implies Q{Tu) = 0. Note that Q need not to be of first order 7 . In view of Proposition 3, the corresponding integral operator (4) is contractive in case the height TJSQ of the conical domain M is small enough, and thus the following statement has been proved: Theorem 1 Suppose that T, Q is an associated pair. Suppose, further, that the solutions of Qu = 0 satisfy an interior estimate of first order. Then the initial value problem dtu = J-u w(0, •) = (p

is solvable provided the initial function tp satisfies the side condition Q

83

J- there exist several so-called co-associated 7 operators Q. This leads to decomposition theorems 6 for the solution of initial value problems.

1.5

Examples

The classical starting-point is the classical Cauchy-Kovaleskaya problem. Here the operator T transforms holomorphic functions into holomorphic functions, and the initial functions ip are holomorphic, i.e., the side condition Qu = 0 is the Cauchy-Riemann system. M. Nagumo 13 was the first who used the integral rewriting (3) for solving the classical Cauchy-Kovalevskaya problem. Using this integral rewriting, W. Walter 22 solved the Cauchy-Kovalevskaya problem by the contraction-mapping principle in a Banach space of functions which are holomorphic for each t. In the holomorphic case there is no need to exhaust the underlying domain Q, in the z-plane by a family of subdomains fig, and the weight d(t, z) can be defined by the Euclidian distance d(z) of a point z from the boundary dQ, i.e., W. Walter defines the weight d(t,z) =d(z) - - . Note, further, that W. Walter uses a power da(t, z) as weight where a > 0 (note that one can use a power of the weight (5) can be used also in the general case 1 8 ). Concerning initial value problems with generalized analytic functions as initial functions see 15>17. The construction of associated pairs can be reduced to the inhomogeneous Cauchy-Riemann equation with an algebraic side condition. In case these system is compatible, an associated operator exists and so the initial value problem is solvable (cf. 1 7 ). In case the initial elements are generalized analytic vectors (see B. Bojarski's paper 4 ) , analogous initial value problems are solved in A. Crodel's Thesis 5 . While in the scalar case the underlying partial complex differential equation can be assumed in the canonical form djw = a(z)w + b(z)w, the corresponding normal form (and so the whole theory) is much more complicated in the case of generalized analytic vectors. The above general concept contains also the case that the initial functions are monogenic (cf. the paper 19 jointly written with U. Yiiksel). The same is true for initial functions satisfying differential equations with singular coefficients (see the paper 20 co-authorized by Z. Usmanov).

84

2

2.1

Boundary value problems with boundary data piecewise belonging to fractional order spaces Statement of the problem

Generally speaking, the solution of a boundary value problem has a singularity at a boundary point in case the given boundary data have a singularity at this point. Solving the boundary value problem by a functional-analytic method in a function space, one has to use a weight function in order to compensate the singularity. On the other hand, a very general class of boundary data is given by a fractional order space 1,lA. The solution of the Dirichlet problem for the Laplace equation in Q, belongs, for instance, to the Sobolev space Whfl), p > 1 in case the boundary data belong to the fractional order space Wp(9f2) where s = l-i,see12-14. Unfortunately, even piecewise constant boundary values do not belong to a fractional order space. Therefore, a quite general class of boundary values is given by boundary data piecewise belonging to a fractional order space. Using methods of Complex Analysis, in the sequel it will be outlined how the Dirichlet problem for non-linear partial complex differential equations of type diW = F(z,w,dzw)

(14)

can be solved if the boundary data belong piecewise to a fractional order space. 2.2

The Dirichlet boundary value problem for non-linear first order systems in the plane

Let 0, be a simply connected bounded domain in the complex plane with sufficiently smooth boundary dfl. To be brief, assume that Q is the unit disk {z : \z\ < 1}. The Dirichlet problem for equation (14) is to find a solution w(z) whose real part has prescribed values Re w = g

on

dQ,

(15)

while its imaginary part is prescribed at one point ZQ, Im w(z0) = c.

(16)

85

The well-known TQ- and Iln operators,

(Tnf)[Q = -\\\

{^dxdy, (nn/)[C] = -\fj

Q

j^y^dy

Q

(where z = x + iy) are connected by the relation 3ZTQ=UQ.

Consequently, the boundary value problem (14), (15), (16) is equivalent to the system of integro-differential equations w = y + h = *' +

${w,h)+TnF(-,w,h) *[Wth)+IlaF(;W,h)

where ^ is the holomorphic solution of the boundary value problem (15), (16), ®(w,h) is the holomorpic solution of the homogeneous boundary value problem (15), (16) (i.e., g = 0 and c = 0), and h = dzw. Solution of the boundary value problems of type (14), (15), (16) can thus be constructed as fixed points of the operator W = * + *lWth)+TaF(;w,h)

(17)

H = 9' + $[Wth)+IlaF(.,w,h)-

(18)

Suppose the boundary values g are Holder-continuous with exponent a where 0 < a < 1. Then the holomorphic solution ^ of the boundary value problem (15) belongs to the space C a (fi) of functions which are Holder-continuous with exponent a in CI. Moreover, its derivative VP' belongs to Lp(Cl) if 1

v- 1 , i.e., a>^ . (19) 1—a p Concerning the TQ- and n^-operators one knows 21 that they are bounded operators mapping Lp(Cl) into itself, p > 2. Further, TQ is also a bounded operator mapping Lp(Cl) into C@(Cl) into itself where p<-

P Suppose, finally, that the right-hand side F(z, w, h) of the differential equation (14) satisfies a (global) Lipschitz condition \F(z,wi,h{)

- F(z,w2,h2)\


-w2\

+ L2\wi - w2\

86

and that, in addition, F(z,0,0) belongs to LP(Q). Then the composition F(z,w(z),h(z)) belongs also to Lp(£l) provided w(z) is continuous in Ti and h(z) belongs to Lp(ft) because of F(z,w(z),h(z))|

< \F(z,0,0)|

+ LiKz)| +

L2\h(z)\.

These considerations show that a suitable space for the construction of fixed points of the operator (17), (18) is the pair (C^(Q), Lp(£l)) equipped with the norm \\{w,h)\\ = max( Hlcpjn) > II^ILP(f2) )• The indices can be chosen as follows: Start from a given p > 2 and define /3 = 2r1 ^. According to (19) suppose g G Ca(d£l) where a > 2 ^ . Remark In case of a disk the Th-operator has the remarkable property that the boundary values of TQ coincide with the boundary values of a holomorphic function. Indeed, if fl is the unit disk and ( is any point on its boundary, then one has £ = l/£ and thus

(Ta)[c] =

~l II v=^dxdy=HI

n C Consequently, the holomorphic function

w(0

r^kdxdyn

§//j%**.

(20)

and Tnf coincide on the boundary of the unit disk. That way one gets an explicit representation of the function &(w,h)- The necessary bounds for the norms come from the estimates ||¥X>||c,(n) < Ci \\f\\Lp(n), \\
87

2.3

A weighted Banach space for the construction of fixed point

Now suppose the boundary values g can be decomposed in a sum g\ + g^ where g\ belongs to the fractional order space Wp(dfi), s = 1

, and g^

is piecewise Holder-continuous. Then holomorphic functions whose real part have the boundary values g\ belongs to Wp (0) (see Subsection 2.1). Moreover, the boundary data g2 can be decomposed into a Holder-continuous part and a piecewise linear part. The holomorphic function corresponding to the Holdercontinuous part is Holder-continuous in Q, whereas the holomorphic function belonging to the linear part has logarithmic singularities log(z-Cj) at (finitely many) points Cj of jump discontinuities. Its derivative has singularities of type 1

z-Cj at these jump discontinuities. In accordance with the choice p > 2 (see Subsection 2.2) the derivative does not belong to LP(Q,). However, \z-Cj\P

7- • \z — c7- ' J ' '

is integrable provided 7-p>-2,

i.e.,

7>p-2.

(21)

Then the derivative under consideration belongs to the weighted Lebesgue space Lp(£l, Q"1) where Q is the weight IIf|z — Cj\. Similarly, the holomorphic function itself belongs to a weighted Holder space where the weight is any power of Q with a positive exponent. In most cases the To- and the n^-operators have the same mapping properties in the weighted spaces as they have in the usual spaces 8 . However, the TQoperator has another mapping property as an operator mapping a weighted Lebesgue space into a weighted Holder space. In this case we have n the following statement: Proposition 4 Suppose f belongs to Lp(Q.,gy) where p > 2 and 7 is any positive exponent. Then Tnf belongs to C*(fl, £**) where \x = 7/p and A= m i n ( ^ ^ , 2 + 5 - 1 ] V P V ) provided 5 is any number with 0 < 5 < 1.

88

In order to get A =

= 6, one obtains for 7 the condition P 7 > (p - 2) + (1 - 6)p = (p - 2) + ep

if 1 — <5 will be denoted by e, 0 < e < 1. The latter choice of 7 agrees with the above condition (21). The above considerations can also be applied to the the integral operator in (20) and its derivative with respect to z because these operators are modifications of the TQ- and the II fj-operators. Notice, finally, that the weighted Sobolev space Wp(Q, g) is embedded Cp(n,^-) and, especially, H£(ft) is embedded 1 in C0{Tl).

2.4

Construction 0} fixed points by the contraction-mapping

n

in

principle

Taking into account the relations discussed in the preceding Subsection 2.3, and applying the contraction-mapping principle, one gets the following theorem u concerning the boundary value problem (14), (15), (16): Theorem 2 Suppose the right-hand side of the differential Lipschitz-continuous with sufficiently small Lipschitz constants. given boundary values can piecewise be decomposed into a part a fractional order space Wp(<90) and a Holder-continuous part exponent a where p>2,

s — 1--,

a>\ p

equation is Suppose the belonging to with Holder-

. p

Choose an arbitrary e where 0 < e < 1, and denote the (finitely many) points of jump discontinities of the boundary data by Cj. Then the boundary value problem is solvable in

where g(z) =Uj(z-Cj),

7 = (p-2)+ep

and

fi = j/p.

89 References

1. R. A. Adams, Sobolev spaces. Academic Press 1975. 2. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions I and II. Comm. Pure Appl. Math., vol. 12, 623727, 1959, and vol. 17, 35-92, 1964. 3. B. Bojarski, Generalized solutions of elliptic systems of partial differential equations of first order with non-continuous coefficients, (in Russian). Mat. Sbornik, vol. 43 (85), No.4, pp.451-503, 1957. 4. B. Bojarski, Theory of generalized analytic vectors, (in Russian). Ann. Polon. Math., vol. 17, 281-320, 1966. 5. A. Crodel, Theorems of Cauchy-Kovalevskaya type in classes of generalized analytic vectors, (in German). Thesis (Dissertation A), Halle University, 1986. 6. R. Heersink, W. Tutschke, A Decomposition Theorem for Solving Initial Value Problems in Associated Spaces. Rendic. Circ. Matem. Palermo, Serie II, vol. X L I I I , pp. 419-434, 1994. 7. R. Heersink, W. Tutschke, On Associated and Co-associated Complex Differential Operators. Journ. Analys. Appl., vol. 14, pp.249-257, 1995. 8. B. V. Khvedelidze, Linear discontinuous boundary value problems of the complex function theory, singular integral equations and their applications. (Trudy Tbiliss. Mat. Inst. Akad. Nauk, vol. 23, 1956). 9. H. Lewy, An example of a smooth linear partial differential equation without solution. Ann. of Math., vol. 66, 155-158, 1957. 10. G. F. Manjavidze and W. Tutschke, On some boundary value problems for non-linear first order systems in the plane, (in Russian). Contained in " Boundary value problems in the theory of generalized analytic functions and their applications". Trudy Inst. Appl. Mathem. of University Tbilisi, pp. 79-128, Tbilisi 1983. 11. G. F. Manjavidze, W. Tutschke and H. L. Vasudeva, A complex method for solving boundary value problems with boundary data piecewise belonging to fractional order spaces, (unpublished). 12. A. S. A. Mshimba Construction of the solution to the Dirichlet boundary value problem belonging to WitP(G) for systems of general (linear or nonlinear) partial differential equations in the plane. Math. Nachr., vol. 99, 145-163, 1980. 13. M. Nagumo, Uber das Anfangswertproblem partieller Differentialgleichungen. Japan. Journ. Math., vol. 18, 41-47, 1941.

90 14. H. Triebel, Interpolation Theory, Function Spaces, Differential Operators. Berlin 1978. 15. W. Tutschke, A problem with initial values for generalized analytic functions depending on time (generalizations of the Cauchy-Kowalewski and Holmgren theorems.) (in Russian). Doklady Akad. Nauk 262, No. 5, 1081-1085, 1982. English translation: Soviet Math. Dokl. 25, No. 1, 201-205, 1982. 16. W. Tutschke, Partial Differential Equations. Classical, Functionalanalytic, and Complex Methods, (in German). Teubner-Text zur Mathematik, vol. 27, Leipzig 1983. 17. W. Tutschke, Solution of Initial Value Problems in Classes of Generalized Analytic Functions. Teubner-Texte zur Mathematik, vol. 110, Leipzig 1989. Licensed edition Springer-Verlag 1989. 18. W. Tutschke, Complex methods in the theory of initial value problems. Contained in "Complex methods for Partial Differential Equations". Kluwer Academic Publishers, ISAAC series, vol. 6, 295-311, 1999. 19. W. Tutschke and U. Yiiksel, Generalized monogenic functions satisfying differential equations with anti-monogenic right-hand sides. Contained in " Complex methods for Partial Differential Equations". Kluwer Academic Publishers, ISAAC series, vol. 6, 263-270, 1999. 20. W. Tutschke and Z. Usmanov, Singular Partial Complex Differential Equations as Conditions for Initial Functions. Complex Variables, vol. 42, 375-386, 2000. 21. I. N. Vekua, Generalized analytic functions. 2nd edit. Moscow 1988 (in Russian; Engl, transl. Reading 1962, German transl. Berlin 1963). 22. W. Walter, An elementary proof of the Cauchy-Kowalevsky theorem. Amer. Math. Monthly, vol. 92, 115-125, 1985.

91 OPTIMIZATION OF FIXED-POINT M E T H O D S

S. G R A U B N E R 09228

Wittgensdorf, Untere Hauptstrasse E-mail: [email protected]

86

Subject of the consideration are fixed point equations of the form u = uo + ATu in a Banach space B. Here the operators A. and T are denned in the entire space B mapping B into itself. While A is linear and bounded, T in general is assumed to be nonlinear but Lipschitz continuous on bounded subsets of B. In case the Lipschitz constants for the right-hand side of an operator equation in a Banach space B are not global ones, the contraction mapping principle cannot be applied to the whole Banach space. The paper determines optimal polycylinders in case the growth of the Lipschitz constants is direction-dependent in a sense.

1

Statement of the problem

Let B be an arbitrary Banach space with the norm || • ||. We denote the elements of B by u\, 112, v.3,.... Let B* be the Banach space of pairs (u\, U2) with II (ui,u2) || =max(||wi||,||u 2 ||). In particular, consider Banach spaces B whose elements are functions of a variable x running in a certain set in a Euclidian space. Suppose Aj are linear and bounded operators mapping B into itself where j — 1,2 here and in the sequel. Suppose, further, that for each x under consideration the Tj = Jrj(x,ui,U2) are mappings from B* into B. These mappings are supposed to be Lipschitz continuous in each bounded subset of B*, i.e., one has \\Tj (-,ui,U2) - Tj (•,ui,u 2 ) I) < i j i I K i - ui|| + Lj2\\u1 - U2II with Lipschitz constants Lj\ and Lj? depending on the respective bounded subset of B*. The paper is aimed at solving the system Uj = u0j + AjTj (-, MI, u2), for two desired elements Uj where the UQJ are given elements of B. The right hand sides of this system define an operator mapping B* into itself. We are going to solve the above system by the contraction mapping principle in the poly disc V = {(•ui,M2) € B* : \\UJ - u0j\\ < dj}

92

centred at (uoi!^02) with radii di,d2. Denote the Lipschitz constans of Tj in V by Lji(di,d2) and Lj2(di,d2). In view of the triangle inequality one has \\Fj{;Ui,u2)\\

< ||i^(-,ui,u 2 ) -Fj(;u0i,uo2)\\

+ \\Fj (-, u 0 i,u 0 2) ||

and, consequently, in the polydisc V the estimate ||Fj(-,ui,u 2 ) || < i j i | | u i -uoill + Lj2\\u2 - u02\\ + Mj < Ljidi + Lj2d2 + Mj is true where Mj = \\Fj (•, uoi, M02) ||- Hence the operator under consideration maps V into itself if \\Aj {Ljidi + Lj2d2 + Mj) || < dj. Analogously, the operator is contractive in V if \\Aj(Lj1+Lj2)\\
In other words, one gets the following restrictions of the norms of the operators Aj\ \\A 11

n

11 <

d±

~ Lndi + L12* + M i '

H ^ l l < Mi T + l -^12 r ' l

'" 42 " - L21dx + L22d2 + M2'

11^11 < r

*

•

^21 + ^22

Note that the Lipschitz constants are functions of d\ and d2, i.e., Lij — Lij (d\,d2). For simplification we restrict ourselves to the case L u (di,d2) = L2i (di,d2) = Lx

(di,d2),

L12 (d1,d2) = L22 (di,d2) = L2 (di, d2), and we assume that the Lj are two times continuously differentiable. The restrictions for the norms of the operators are smallest possible if the objective function mm [hi,h2,h3j,

(1)

93

is maximal where kl

{dud2)

h2(dud2)=

=

L1(di,d2)d1+L2(d1,d2)d2

+ M1>

Li(di!d2)di + I/2(dl)d2)d2 +

M2

'

Therefore, one has to solve a non-smooth optimization problem. 2

Optimization under a side condition

We introduce a new parameter K by the substitution d2 = nd\ and consider the ray d2 = nd\ in the first quadrant. Also we require that the functions Li (d\, ndi) and L2 (di,ndi) are strictly monotonously increasing and in special cases strictly convex. At first we will study the properties of the functions hi (di,K,d-\_) and h2 (di,ndi). Without loss of generality we may suppose that Mi < M 2 . Lemma 1 The functions hi (di,ndi) and h2 (di,ndi) have a uniquely determined positive maximum, located at a uniquely determined point (O-(K), KO-(K)) and (T(K),KT(K)) resp. on the ray d2 = ndi. In the case Mi = M2 we have O-(K) = T(K). Proof If we differentiate hj (di,ndi) with respect to di for fixed K > 0 and look for for optimal points we have to solve the equation Mj = 4 (LUi + Lhl2 K + L2di K + L2rl2 K2) , 0 = 1,2). Dividing by d\ > 0 we have with -4- strictly monotonously decreasing functions and if the derivative of {Lid + Lid K + L2d K + L2d K 2 ) is positive, i.e., the second derivative of the Lipschitz constants are positive, we have for two uniquely determined intersection points. Hence we define : hi (di,ndi) has its maximum at the point a (K). Further h2 (di,K,di) has its maximum at the point r (K). In the case Mi = M2 is cr(n) = r (K). Here we define hi (di,ndi) and h2 (di, ndi) has its maximum at the point p (K). This completes the proof. Next we ask whether the two functions hi and h2 can take the same value on the ray d2 = ndi. The answer will be given by the following lemma: Lemma 2 Suppose Mi < M2. If 0 < K < I, then hi(di,ndi) > h2{di,ndi) for each di > 0. Analogously, hi(di,ndi) < h2{di,Kd{) for each di > 0 provided K > M2/Mi. For a fixedly chosen K with 1 < n < M2/Mi the equation hi(di,ndi) = h2(di,ndi) has a uniquely determined solution di =

94

In the case Mi = M2 one has hi{d\,ndi) > h,2(di,ndi) for every K with 0 < K < 1, whereas hi(di,ndi) < h2(di,ndi) whenever K > 1. Finally, in this case the functions hi(di,Kdi) and h.2(<^i, ndi) coincide identically for K = l. Proof For any K with 0 < K < 1 we want to show that the proposition h,2 (di,ndi) < hi (di,K.di) is true. We start from the inequality /J,(K).

0 < (Lidi + L2diK + Mi) (1 - K) + M2 - Mi, which is true for 0 < K < 1 and M2 > Mi and for any di > 0. An elementary calculation gives us: n(Lidi + £ 2 ^ 1 + Mi) < Lidi + L2ndi + M2 and L1d1+L^d1+M2 < L ^ + L I U + M I s i n c e a 0 variables are positive. Further we will show that hi (di,ndi) < h2(di,ndi) for H2- < K < +00 and any di > O.Now we start from the inequality di (1 -

+

K) (LI

L2K)

<

MIK

- M2

2

Since 1 < -jg - < K is the left-hand side negative for all di > 0 and the right-hand side is positive because 0 < MIK — Mi. Now another elementary calculation gives us Lidi + L2Kdi + M2 < K (Lidi + L,2Kdi + Mi). Dividing both sides and multiplying by di > 0, we get the proposition. Further leads the assumption /13 (di, ndi) = fo (di, ndi) o Li (di, nd\)di (1 — K) + M2 = 0 in the case 0 < K < 1 and M2 / O t o a contradiction, also the assumption /i 3 (di,nd{) = hi (di,ndi) 4=» L2 (di,ndi)di (K — 1)+Mi = 0 in the case ^- < K < +00 and Mi ^ 0. Now we want to show that the equation h\ (di,Kdi) = /12 (^i,«) has a uniquely determined solution. From the equation above we get d i ( l - K ) (Li(di,Kd1)

+ L2{d1,Kdi))

= -Mm

+ M2

(2)

The left-hand side of the equation (2) is for 0 < K < 1 nonnegative and we have K < IJ 2 - and the right-hand side is negative, this gives the contradiction. Remark: Since ffi > 1 is K > jjj2- and 0 < K < 1 together impossible. Also in the case ^ < K < +00 is MIK — M2 > 0 and we get a contradiction, since di (1 — K) yL\ (di,K,di)+L,2 (di,ndi) we can change to ( K - 1) (Li(di,Kdi) Obviously

the

function

M

+ L2(di,Kdi)

t-^MiK

(K - 1) (.Li (di,K,di) + L2 (di,ndi)

J < 0. In the last case where 1 < K < If2-

J =

—~——

[s monotonously

decreasing,

while

J is an increasing function. Defining for

95

1 < K < H2- the uniquely determined intersection point with fi — H(K). In the special case Mi = M2 we consider the difference h2 (di, K.d\) — h\ (di, /tdi) = where LKJ^+LI^+M ( -^1 = M2 = M). Now its clear that the difference is positive if K > 1 since di,Li,L2 and M are positive. That means h2(di,ndi) > hi(di,K,di). In the case K < 1 is h2(di,ndi) < hi(di,ndi). Obviously is /i 2 (di, ftdi) = hi (di, /tdi) for any di > 0, if K = 1. From this follows that /13 (di, «di) cannot intersect at least one of the two curves hi (di,ndi) or h2 (di,ndi) in the case 1 < K < H*. So we can write min (hi (di,d2) ,h2 (di,d2) ,/i 3 (di,d 2 ) J = min (hi (di,d 2 ) ,h2 (di,d 2 ) J. Remark: In the special case Mi = M2 is our theorem also true. In this case is A(K) — O-{K) = T(K). Here we have only the two intervals 0 < n < 1 or 1 < K < +00. This was to show. With the end of this proof we can formulate our first theorem. Theorem 1 On each ray characterized by its slope K, the objective function min (hi (di, d 2 ) , h2 (d\, d2), /13 (di,d 2 ) 1 takes its maximum at a uniquely determined point A (K) where A (K) = a (K) or X(K) = r (n) or A (K) = fj, (K). Proof Prom Lemma 2 follows that we only have to consider the functions hi (di, ndi) and h2 (di, ndi). For any K with 0 < K < 1 we have to consider h2(d\,K,di) and so we can optimize by inserting cr(n) in h2{di,ndi). And in this case is A (K) = CT(K). In the interval jg2- < K < +00 we have the function hi (d l5 /cdi) to optimize, this can be done by inserting r (n) and our A (K) is r (K). In the interval 1 < K < ffi we have to distinguish several cases. If we assume that a (K) < /u (K) < r (K) SO we get hi (fi (K) , Kfi (K)) = h2 (fi (K) , K/i (K)) and we receive A (K) = ji (K). Our next assumption is r (K) < fj, (K) and we have to insert r (K) in /i 2 (di, ndi). Our last case is H(K) < a (K) and we get hi (a (K) , na (K)) with A (K) = cr (K). Since this distinction of cases is complete we have got the whole prove. 3

Second step: optimization on the curve of maximum points

Theorem 1 shows that on each ray with the slope K there exists a uniquely determined point ( A(K) ,K\(K) J at which the objective function min ( hi (di, ndi), h2 (di, ndi) 1 is maximal. We define Hj

(K)

- hj (A (K) , KA (K) ) .

(3)

and H(K) = mm(Hi(K),H2(K)).

(4)

96 Thus H (a) is the maximal value of min f hi (di,ndi), h2 (di,K,di) ) on the ray with the slope K. Consequently, it remains to optimize H («) for 0 < K < +00. The points (A (K) , KX (K) J with 0 < K < +00 define a curve in the first quadrant of the di, d2-plane. In view of Theorem 1 the originally stated optimization problem has been reduced to the optimization of H (K) on the curve (A (K) , KX (K) J with 0 < K < +00. 4

The special case M j = M 2

Theorem 2 In the case Mi = M2 there exists a uniquely determined optimal poly cylinder (A(1),A(1) J located on the ray K = 1. The value of the objective function is equals H (1) = Hi (1) = H2 (1)Proof Prom Lemma 1 we get that hi (di,ndi) and /12 (di, ndi) for Mi = M2 have at the point p = p{n) a maximum. Further follows from Theorem 1 h2(di,Kdi) < hi{di,ndi) for 0 < K < 1 and hi(di,ndi) < h2(di,ndi) for 1 < K < +00. So we have to optimize H (K) = H2 (n) = h2 (p (K, ) , up (K) J if 0 < K < 1. Obviously J7 («;)is differentiable for any «; > 0. If we differentiate H (K) with respect to K we get the first derivative as quotient whose numerator is dp(n) K\M-p{n)2 (LUI + Lld2K + L2(1IK + L2d2K2) j dn, +Mp (K) +Lip (K)2 - Lu np (K)3 - L2 n2p (nf , while the denominator is given by {LIP(K)

+ L2KP(K)

+

M)2.

Since p = p(n) is the uniquely determined solution of the equation M = p2 (LUi

+ LU2 K + L2di K + L2d2 K2)

we can insert this in the derivative above and we get dH (K)

P ( K ) 2 ( (Ll-
dn

(LIP(K)

L2

.'i K ) P (M + ^ i )

+ L2KP(K)

+

M)

2

(5)

Obviously the derivative is positive for any K > 0, especially for 0 < n < 1. Analogously we consider H(K) = Hi (K) = hi(p(n) ,KP(K)J if 1 < K < +00

97 and we have in this case the following derivative P (Kf

dH («) da

( iLU2 + L2d2 K) p (K) + L2) (LIP(K)

+ L2KP(K)

+

M)2

<0.

(6)

This completes the proof and now we will come to the general case. 5

Main Theorem

Suppose Mi < M2. From theorem 1 and Lemma 1 we get H (K) = H2 (K) = h2[r (K) , KT (K) J for 0 < K < 1, and the following statememt is true: Lemma 3 H (K) is strictly monotonously increasing if K runs in the interval 0 < K< 1. Proof To prove this, we have only to calculate the numerator of the first derivative of our function H (K) since the denominator is positive. We have: ^

(M 2 - r (Kf (LUi + Lld2 +M2T

K

+ L2di K + L2d2

2 K

) )

(K) - Lld2 T (K) 3 K - L2d2 K2T (nf + LXT {K)2 .

By inserting M 2 = r 2 (K) (Li rfi + L irf2 K + L2di K + L2rf2 K2) we get T

dff(K)

(K)2(Ll + (Llrfl+L2dlK)r(K))

dn

(LtT

(K)

+

L2T (K) K

+ M2)2

for any K > 0. This was to prove. Further, if 1 < K < +oo, then H (K) = Hi (K) = hila(n), KCT (K) J, and the next Lemma can be proved analogously: Lemma 4 H (K) is strictly monotonously decreasing in case K satisfies the inequality 1 < n < +oo. Proof Here we have a

dH (K) ; dn

(Kf

( L 2 +
—

T

(LICT (/t) + L2a

(K) K +

Mi) 2

for any K with 1 < K < +oo. Lemma 3 and Lemma 4 imply

> 0

(8)

98 L e m m a 5 Optimal polycylinders

can only lie on the segment 1 < K < ¥?• of

the curve ( A (K) , n\ (K) J of maximum Note t h a t H(1)

= H2(l)

points.

< Hx (1). Similarly H(^A

= i?i(ff)

<

H2(j^j. Since H\ (K) and H2 (K) are continuously functions, there exist a K* (respectively K**) as the first (respectively the last) solution of the equation Hi (K) = H2 (K). Obviously, 1 < K* < K** < -jg2-. Summarizing the above consideration, the following main theorem has been proved: T h e o r e m 3 The absolute maximum of the objective function and all possible relative maxima, too, are located on the segment K* < K < n** of the curve (A (K) , K\ (K) J of maximum points. T h e main theorem will be illustrated by an example. E x a m p l e Existence of a uniquely determined optimal polycylinder schitz constants are polynomial functions. Suppose .Li we have:

=

k\d\

+ k2d2

h2{dUKdx)

=

k$d\ + k^d\.

i

=

and

d{{kl+k2K2+k3 K+kiK.3)+M2

Like

di(k1+k2^+iU+k4^)+M1-

and L2

if the LipProm here

M<*i./«*i)

in t h e general case we assume also Mx < M2

and we can derive: r (K) = \J

2[kl+k2^+k3K+kiK,3)-

Analogously we have:

CT(K) = ^ / 2 ( f c l + f c 2 K 2 ^ 3 K + f c 4 K 3 ) . Obviously is
=

d

d^kl+k2^+

=

=

kU+k^)+M1

{/(K_l){kl+fcJ?lk3K+kiK:*)-

dl(kx+k2K,+k3\+kiK3)+M2

fOT

M2.

K

^

Wlth

0

< <

K <

1.

2

3

Further we have with H2 (K) == f ( ^ ) monotonously increasing function ,,

j

•

..

^

'

—

aK

val ^

*

=

a strictly <

K <

1 since .

'-

9(M2(fc1+fe2K2+fc3K +

< K < + o o we have M d i , / t d i )

/ii(di,Kdi)

kl+k2^+k3K^

(3k!+2Kk3+k2K2)2i

dH2U)

the derivative

\/

for any K with 0

=

and

di(k1+k2^+k3\+k^)+M2

r fc4K3)2)3

>

0.

In the mter-

d^+k^+t.+k^+M, with

our

<

notation we get:

2

#I(K) dHi(K)

= /ii (*(«)> «*(«))

—4 X - L = aK

=

l ( i r ) 3 yfOi+ktK'+ksK+ktK*

2a(2fc2K+fa+3fc4K2)

-

2

9(Mj(fc 1 +A: 2 K +

^

n

,

, . ,,

and

since

,

- — j - < 0 we have a strictly monotonous 3 2 3

fc3K+fc4K ) )

decreasing function for any n with H2- <

n < +co.

In the interval

99 1 < K < If2- we have to distinguish several cases.
<

/X(K).

Then we get for

jfc with KI = ^m+M;

and

K2

K =

First we assume

the interval 1 < K\ < re < re2 < M

\+M^ •

equals hx (/X(K) , K (K)) = ft2 (M«) >«(«)) =

Here

the

maximal

ue

fci+^l£te£«a'

M^MT V

Denning the following auxiliary function / (re) =

val

k^l^+k^+kl^

with

^ 1 = 0 <^ if (2Mi/c4 + M2fc4 + A^Afc) « 3 + (-3M 2 fc 4 + Mi/c2 + 2AfiJfe3) « 2 + {-2M2k2 + 3Mifei - M2fc3) K - 2M2ki - Miki - M2fc3 = 0. If we denote the left-hand side from the equation above by g (re), then g (re) has the following properties: ^ ^ = 12M1k4K+6Mik2K+6M2k4 (K - l)+2Mi/c 2 +4Mi/c 3 > 0 and g(«i) < 0 and g(re2) > 0. From our next case /z (re)
< re. It is easy to show that y (If 2 -)

< K 2 is

2

true for M\ < M2. So we have in the interval K 2 < re < If - our function hi (cr (re) , no (re)) which is strictly monotonous decreasing. Our last case hi (a (re), KIT (K)) > /i 2 (T (K) , KT (K)) gives us the re interval 1 < re < «i and here is /i 2 (T (K) , KT (re)) strictly monotonously increasing. Summarizing the above considerations, we see that the objective function has a uniquely determined optimum at a point between KI and K2. In this example the optimal polycylinder is uniquely determined, i.e., there are no other polycylinders for which the objective function has a relative maximum. To find sufficient conditions for the uniqueness of the optimal polycylinder is still an open problem. Moreover, it should be mentioned that similar results can be expected for operator equations with an arbitrary number of components. If one applies the above theory to partial complex differential equations (see *), one has to use I. N. Vekua's theory of generalized analytic functions 4 .

References 1. S. Graubner, Uber die Optimierung von Polyzylindern bei Anwendung des Kontraktionsprinzips auf Differentialgleichungen. Thesis (Dr.rer.nat. dissertation) at FU Berlin, 1997.

100

2. S. Graubner, On the optimization of the set of definition of operators in applications of the contraction-mapping principle to differential equations. ISAAC series Volume 6, 1998 Kluwer Academic Publishers, pp. 73-76. 3. W. Tutschke, Optimal balls for solving fixed-point methods in Banachspaces. ISAAC series Volume 6, 1999 Kluwer Academic Publishers, pp. 313-317. (Editors H. Begehr, A. O. Celebi, W. Tutschke). 4. I. N. Vekua, Generalized Analytic Functions. Pergamon Press 1962.

101

P R I N C I P L E OF TELETHOSCOPE S. S A I T O H Department

of Mathematics, Faculty of Engineering, Gunma Kiryu 376-8515, Japan E-mail: [email protected]

University

In this survey article, we shall present a principe of telethoscope which is stating that many solutions of linear partial differential equations of parabolic and hyperbolic types are represented by their local data of space and time of the solutions. Similarly, many solutions of linear partial differential equations of elliptic type are represented by their local data. In particular, we shall refer to some simple representations of analytic and harmonic functions by using their local data.

1.

Introduction

In this survey article, we shall present a principle of telethoscope which is stating that many solutions of linear partial differential equations of parabolic and hyperbolic types are represented, in a constructive procedure, by using their local data of space and time of the solutions of the partial differential equations. At the same time, in those cases we see that the boundary values or initial velocity functions in the differential equations are represented by those local data of the solutions. That is, the inverse source problems in the differential equations may be solved by using the local data of the solutions. Similarly, many solutions of linear partial differential equations of elliptic type are represented by using their local data. In particular, we shall refer to some simple representation formulas of analytic and harmonic functions in terms of their local data, by means of the Riemann mapping function. The principle of telethoscope will have an interesting similarity with the idea of "ergodic theory", in a sense.

2.

Representations of solutions of partial differential equations of parabolic and hyperbolic types by means of time observations

Our results are valid in a very general situation, but for simplicity we shall state the results with a prototype example. Let D be a finitely-connected smoothly bounded domain in Rn. We

102

consider a partial differential equation of parabolic type dxi — = Au = AM - q(x)u (t>0,x&D)

(2.1)

subject to the boundary condition Oil

a{i)u + {1 - a ( 0 } ^

= 0 (i > 0, on dD),

(2.2)

where d/du denotes the outer normal derivative on dD with respect to D. We assume that q{x) is Holder continuous on D = D U dD, a G C2(dD) and 0 < a(£) < 1 on 3D. Let U(t,x,y) be the fundamental solution for the equations (2.1) and (2.2). Then, in particular, recall that for any fixed y G D, U(t,x,y) G (^((O.oo) x D), U(t,x,y) satisfies (2.1) and (2.2). Under the above situations, there exist eigenvalues {Xj}^0 and eigenfunctions {fj}j^o satisfying —oo < Ao ^ Ai si • • • < Xj ^ • • •,

lim A,- = oo,

{(fj}°Z0 forms a complete orthonormal system in L2(D), = e-^tipj(x)

[ U(t,x,y)
on D,

(2-3) (2.4) (2.5)

JD

Aipj(x) — —Xj(pj(x)

on D,

(2.6)

and ipj(j = 0,1, • • •) satisfies the boundary condition (2.2).

(2-7)

Then, OO

U(t,x,y)

= J2e~Xit
(2.8)

3=0

converges uniformly on [8, oo) x D x D for any fixed S > 0. For any / G L2(D) and for oo

f(x)=^2CjVj(x),

(2.9)

3=0

(Utf)(x)

= J2Cie~X,tw(x) J=0

(2-10)

103

converges uniformly on [6, oo) x D for any 5 > 0. Of course, (2.10) represents a "general" solution of (2.1) satisfying the boundary condition (2.2) and the initial condition

limJ(Utf)(x)-f(x)\\L2(D,dx)=0. For these properties, see, for example, 2 . By using the fact that (2.10) converges uniformly on [5, oo) x D for any fixed S > 0, we have . Theorem 2.1 f7) {Cj}£L0 and so, f and (Utf)(x) on {t > 0} x D can be determined and represented by the observation (Utf)(x)

(2.11)

(t>r,xGE)

for any fixed large positive constant r and for a very small set E around any fixed point x* G D. Our representation formula is constructive and for a "small" set E, we can see its exact sense in the following proof. We integrate (2.10) in the form >+T

/

e-M(Utf)(x)dt -(X+\n)T

-(A+A„)s

A + A„

n=0

for A > — AQ.

(2.12)

We first assume that Ao ^ Ai. Then, by A tending to — A0 from A > — A0, we have rs+T

limi

T

e-xt(Utf)(x)dt

A^-A, oo

An —

n=l

AQ

-(A,„-Ao)T

(2.13)

and so, we have rs+T s-t-J

/ =

CoMx)T.

e-x\Utf)(x)dt (2.14)

104

Hence, for a point x such that <po(x) ^ 0, Co is determined by fs+T

1 C0 =

lim

lim

/

e-Xt{Utf)(x)dt.

(2.15)

Further, we shall assume that A0 < Ai < A2. Then, as in (2.12) we consider the integral rs+T

e~xt [{Utf){x) -

= y^,cnipn(x)

1 A + A„

CQe-XatVo{x)\dt -(

lambda+\n)s 1 _ e -(A+A„)T

for A > - A i .

(2.16)

Then, by A tending to - A i from A > —Ai we have r-s+T

lim =

e~Xt [(Utf)(x) - Coe-^M*)]

['/

C1v1{x)T+J2cnipn(x) 1=2

.

dt

-(A„-Ai)s

A„ — Ai

l_e-(A„-Ai)T

(2.17)

Hence, we have, as in (2.14) rs+T

lim

lim

e-Xt[(Utf){x)-C0e-Xotvo{x)]dt

/

s—*oo A—• — A i + 0 /„

(2.18)

= Ci¥)i(a;)r. Hence, for a point a; such that
1

Ci

• -. f

rs+T

. _ lim

ipi{x)T

lim

-At

s-»ooA-»-Ai

(Utf)(x) - C0e-XotVQ{x)} dt.

(2.19)

By induction on n, we can determine all the coefficients {C„}^L 0 from any fixed point x, if all the eigenvalues {A„} are different and all the eigenfunctions {
105

If Ao = Ai < A2, for example, we have in (2.14) rs+T

lim

lim

/

e-Xt{Utf)(x)dt

i—>oo A—>-\0+0 Js

= (CoMx)+Ci
(2.20)

Since (fo and ip\ are linearly independent, by taking two points XQ and x\ in any neighborhood of x* such that ¥>o(zo) ipi{x0) ¥= 0,
(2.21)

we can determine the constants C\ and C\ from (2.20). In Theorem 2.1, we see that the solution (2.1) is analytic on a half complex t plane {t; Re t > r for some T } with respect to time t. By Theorem 3.1, any analytic function on a half plane is represented by using a Taylor expansion around any fixed point of the half plane, in a very simple way. So, in Theorem 2.1, we use the information (2.11) for a large time r, however, by using this analyticity in time t and explicit representation formula, in fact, we need only a local data around any fixed time t. Furthermore, a general corresponding result for the solutions of linear partial differential equations of hyperbolic type is derived by using the Reznitskaya transform. These results may be called the "principle of telethoscope". Of course, for the solutions of partial differential equations of hyperbolic type, we need the data on a reasonable time span in order to get all the information of the solutions. In order to state a physical sense of our result, we stated Theorem 2.1 in connection with the solution u(t,x) of the partial differential equation (2.1) of parabolic type satisfying the boundary condition (2.2). However, the result in Theorem 2.1 may be stated as a general property for the Dirichlet series (2.8), apart from the differential equation (2.1), as follows: We assume that (i) {fn}^Lo are linearly independent functions of class C(D) on a domain D in R m , (ii) {An}^L0 satisfy the monotonicity condition —00 < A0 < Ai < • • •, and (iii) (Utf)(x) defined by (2.10) converges uniformly on [5,00) x D for some fixed S > 0.

106

Then, the coefficients {Cn}^L0 (Utf)(x)

are determined and represented by

(t > r, for a large positive constant r > S)

(2.22)

at any fixed point x g D such that ipn{x) ^ 0 for all n > 0,

(2.23)

by the procedure in the proof of Theorem 2.1. If {^n}^Lo do not satisfy (ii) and satisfy (2.3), and (2.2) is not valid, then we can modify the result, as in Theorem 2.1. Next, we shall consider the differential equation of hyperbolic type _

= Av = Av - q(x)v (t>0,xeD)

(2.24)

subject to the boundary condition v(t,x) = 0 on

3D

for

t>Q

(2.25)

and the initial condition u(0,x)=0

on

D.

(2.26)

For a function / in (2.9), if its Fourier coefficients {C„} satisfy oo

Ys^nfCn)2

< OO

(2.27)

71=0

for

rmi

-] + 1,

A

(2-28)

then (2.9) converges uniformly on D and / is Holder continuous on D. Then, the function oo

^

v(t, x) = J2 -7f= n=0 V

sin

V>*t
(2.29)

X

n

converges uniformly on [0, T] x D for any fixed T > 0, and v(t, x) satisfies (2.25), (2.26) and lim vt(t,x)

= f(x)

on

D,

(2.30)

where the convergence is uniform on D. Here, of course, for the eigenvalues and functions, the boundary condition (2.2) is changed by (2.25) and so, in particular, A„ > 0 for all n is 0.

107

By taking the Laplace transform /•oo

{Lu(-,x))(s)

e-stu(t,x)dt

= / Jo

of (2.1) we have s(Lu(;x))(s)

= A(Lu(-,x))(s)

+ f(x).

(2.31)

Similarly, we have, by the Laplace transform of (2.24) and by using (2.26) and (2.30), s2(Lv(; x))(s) = A(Lv(; x))(s) + f(x).

(2.32)

Hence, (Lv(;x))(s)

= (Lu(;x))(s2).

(2.33)

In terms of the preimages of the Laplace transforms, we have the Reznitskaya transform u(t,x) = ~=J^e^(^-^jv^x)dC

(2.34)

Hence, by the observation v(t,x)

(i>0,for

fixed

i£i)),

(2.35)

we can determine u(t,x)

(t > T for a large fixed r, for fixed x G D)

(2.36)

and so, by Theorem 2.1, / . Hence, we see that for the solution of very general partial differential equations of hyperbolic type, the initial velocity / is determined by the observation (2.35) for a very small set E as observation points x. Note that in the integral kernel in (2.34),

1 m ?

i o« e x p ("9 = 0 '

A m ^ exp (-S) =0 ' and the integral kernel takes a maximum at the point £ = y/2t.

(2.37)

In (2.34), we need the values of u(t,x) for large t (in fact, for any local data, as we saw already) and so, in the observation (2.35), we see that a

108

principal contribution of v(£, x) to (2.34) comes from an interval around \/2t. Note that if the observation point x is outside of the support of the initial velocity / , then v(t, x) = 0 on [0, to] for some positive to > 0. For which general class of linear partial differential equations with general initial boundary and velocity conditions, are Theorem 2.1 and the corresponding versions for hyperbolic types valid? For which general class of linear partial differential equations, can the Reznitskaya transform interpretate the relations between the two partial differential equations of parabolic and hyperbolic types with general initial boundary and velocity functions? It seems that those two problems will propose very delicate and deep research topics. 3.

Representations of analytic functions in terms of local values by means of the Riemann mapping function

Our results and method are valid on any Riemann surface, but in order to simplify our statements we shall state our results on any domain S on the complex Z = X + iY plane. Without loss of generality, we fix a point as 0 € S and we consider any analytic function f(Z) on S. Then, in an e-neighborhood of 0, we have the Taylor expansion oo

f(Z) = Y,a»Zn>

\

Z

\<^

C3-1)

n=0

where

«n = ^ - / 27" 7|C|=e' f(n)(Q)

= t—Pn!

/(OC^C (3.2)

for any 0 < e' < e. Our purpose in this section is to represent f(Z) for any point Z of 5 in terms of {an}^L0 in some "good" way. One fundamental method representing f(Z) in terms of { a n } ^ l 0 is the well-known method by K. Weierstrass which repeats Taylor's expansions by taking a chain by discs, connecting 0 and Z. However, this method is very involved in the sense of practice, because in the first Taylor expansion on the next disc of {\Z\ < e} in the chain, each Taylor coefficient is represented, in general, by using infinitely many Taylor coefficients {an}'^'=0- We discussed this problem by considering the inversion formula of the Cauchy integral representation in ( 3 , 4 ). However, in the method

109 we must, in general, use the eigenvalues and eigenfunctions in the Mercer expansion theorem for positive-definite kernels. Nevertheless, note that the method is applicable widely for the solutions of linear partial differential equations of elliptic type by using integral representations of the solutions. Here, we shall give a new and "good" representation formula, by using some Riemann mapping function. Of course, we know many concrete Riemann mapping functions for special domains. We take a simply-connected domain D C S surrounded by an analytic Jordan curve 3D, containing the two points 0 and Z. We consider a Riemann mapping function
= {|z|

(3.3)

which is analytic on D and a one—to-one mapping from D onto A. The Riemann mapping function z =
~logM

is harmonic on D and 9{Z,0) == 0 o n dD, as follows: ,(Z,0)+ig*(Z,0)}

tp(Z)

where g*(Z,0) is the conjugate harmonic function of g(Z, 0). Then, we set its inversion function which is analytic on A as follows: b

™*m

¥> •\z) = J2

on

A

>

(3-4)

m=l

by the Taylor expansion around 0. Here, we note that { 6 m }i °° ^ = 1 are represented by the Riemann mapping function z = and as fairly simple examples, we showed the formulas in the present situation:

»-(.)-£/ / H | W ^

(3.5)

JdD 1 - Zif(C)

and
fl^l2 dftfr, jD {\-z
(C = £ + ir]).

(3.6)

110

Note that the formula (3.6) is valid for a general simply-connected domain D such that dD need not be a smooth curve and D need not be a bounded domain. Furthermore, we note that in (3.5) and (3.6) we can consider D as a Riemannian domain spread over the complex plane. Then, we have

^-1w = £ (^/ a /^VK)ii^| N

and so, we have the desired representations

bm = ^-j ^

=

J 3D

C^TVtOIKI

^ / / / ^

V

(

C

/

oo

)

|

2

^

(3.7)

Next, from (3.1) and (3.4) we have oo

_1

/(v (*)) = £«n ra=0 oo

\

n

£&mH \m=l

= £cfcZ f c

/

on

A.

(3.8)

fc=0

Here, note the very important fact for our purpose that the Taylor coefficients {cfcj^Q °f / (v~ 1 (- z )) around 0 are represented by the finite numbers of the Taylor coefficients {ao, a\, a,2,..., a^} and {6i, 62, • • •, bk} (Takahasi-Mori Algorithm) as follows: Co = a0 k

Ck = EajWj.fc

(3.9)

where Whk=bk,

k=l,2,... k-j+i

Wj,k=

£

biWj-^k-i;

J =2,3,

, fc; re — 2, 6,...,

(3.10)

111

for the sake of the condition
f(Z) = J2cw(Z)k

onD.

(3.11)

fc=0

In particular, we obtain Theorem 3.1([8j) Let z =
(3.12)

v{Z)=[Z-^dt+*%&*,. (3.13) Jo drj d£ However, on the neighborhood {\Z\ < e} we have the Poisson integral representation

in (3.12). Therefore, by Theorem 3.1 we can obtain the representation f(Z) and so u(Z) by taking the real part on D in terms of {a„}^L 0 and the Riemann mapping function z = with center at the origin and by using the Riemann mapping function z = (Z)| < —*• and so, we can expect that the convergence of the expansion (3.11) will be faster than that of the original one (3.1). By this idea, Takahasi-Mori-Sugihara (see the references

112

in [8]) showed many concrete examples giving faster convergence expansions, by using concrete conformal mapping functions. In [1], we give five concrete examples in Theorem 3.1 by using explicit Riemann mapping functions and furthermore, we give a general principle estimating the speed of convergence of the representation (3.11) in terms of the original expansion (3.1) in the following way: T h e o r e m 3.3(fl]) If Z satisfies \Z\ < p {{\Z\ < p} C D) and (limsup v/jcJ)|y>(Z)| < liminf i/jaJlZI,

(3.15)

then there exists a constant 0 < r < 1 satisfying

\f(z)-j2cnv(zr\
J2 ia"z"i

( 3 - 16 )

n—7n+l

for all sufficiently large m. For r in Theorem 3.3, we can give its precise estimation. See the proof of Theorem 3.3 in [1]. We shall give a typical example of Theorem 3.1 from [1]. Let a be a complex number such that Ima > 0 and let Da be the half plane Da = {Im{Z + a)>0}.

(3.17)

Then, z =
Z

(3.18)

7Mf

Z + (a — a) is a Riemann mapping function of Da onto A. Then, we have
n

0

\

for

a-a for

n=0 n > 1.

(3-19)

(3.20)

In this case we obtain the representation, directly as follows: for any analytic function on Da with Taylor coefficients {an}^L0 around 0

113

J Cfc

=

4.

ao

E; = i^(«-^(;:i)

for for

k = 0 k 1

^-

(3,21)

Representations of meromorphic functions in terms of Laurent expansions

We consider a meromorphic function / on any domain S with isolated singularities. Without loss of generality we fix an isolated singular point of / as 0 G S. Then, for an annulus domain with center 0, we have the Laurent expansion oo

f(Z)=

a

nZn,

Y,

e<\Z\<6

(4.1)

7 l = — OO

where, for any analytic Jordan curve 7 surrounding 0 on the annulus domain a n =

2^/

/ ( C ) C

""

_ l d C

(4 2)

'

Our purpose in this section is to represent f(Z) for any regular point Z of / in terms of {an} in some "good" way. For this purpose we shall use a conformal mapping from a doubly-connected domain on S onto an annulus domain. For any regular point Z of / on 5, we take a doubly-connected domain D(r,7)c5 surrounded by an analytic Jordan curve T ( c {\Z\ ^ 5}c)-c: complement- and 7 and on which / is regular. We consider a conformal mapping tp of Z)(r,7) onto some annulus domain Ar: ip:

D(r,7)-^Ar:={r<\z\
(4.3)

which is analytic and a one-to-one mapping on D ( r , 7 ) . Then, we set its inverse which is analytic on Ar as follows: oo

_1

v ( s ) = 5Z bmzTn onAr m=—oo

by the Laurent expansion on Ar.

(4-4)

114

Now, from (4.1) and (4.4) we have /

oo

n = —oo

\ ™

oo

\m= — oo

which is analytic on Ar. So, by Laurent's expansion of / (ip 1(z)\ on Ar we can set oo

1

ckzk oxxAr.

f(
(4.6)

k= — oo

Our crucial point at this moment is to represent the coefficients {cfc} in terms of {an} and the conformal mapping z = (p(Z) or {bm}. Let Kr(z, u) be the Szego reproducing kernel on Ar and it is given by K

r(Z^) = — J2

1+r2m+l

°nAr-

(4"7)

m—— oo

Then, for any analytic function F(z) on Ar(closure), we have the reproducing property

F(z) = f

F(Z)Kr(£,zM\.

(4.8)

JdAr

By setting F(£) = ^>~1{C)n, we have, on AT

y-\z)n


= ( JdA,.

[

CKr(
/9D(r,7)

=

y f J- /

i > p l 1^)11^ ^.

(4.9)

Hence, from (4.6) and (4.9) we obtain the desired representation oo

f(Z)=

] T ckcp(Z)k,

on£>(r)7)

(4.10)

fc=—oo

for

£ S L ,rT^'^)l|dc|-

7 1 = — OO

In particular, we obtain

(4 n)

-

115

T h e o r e m 4.1 ([9]) Let z = ip(Z) be a conformal mapping from a doublyconnected domain Z?(r,7) on the complex Z-plane surrounded by two analytic Jordan curves T ( c {\Z\ si 5}c) and j(c {\Z\ < 5}) enclosing 0 onto an annulus domain Ar. Then, any analytic function f(Z) on £>(r,7) = -D(r,7) U ({\Z\ ^ 6} etminus {0}) with an isolated singularity at 0 and having the Laurent expansion (4-1) around 0, is represented in the form (4-10) with (4-11) by using the Laurent expansion (4-1) and the conformal mapping z = tp(Z). In Theorem 4.1, let u(Z) = u(X,Y) be a harmonic function on D(T,^f). We form the analytic function f(Z) on £)(r,7) by f(Z)=u(Z)+iv(Z);

(4.12)

Then, f(Z) is represented on the annulus domain e < \Z\ < S as follows: oo

f(Z) = C log Z+ Y,

a n

^ ,

(4.13)

where, C = — / dv (e < r 0 < 6) |Z|=rn 2TT 7|zi =rn 2 " dv

2n J0

de de d ** / udO.

I (4.14) 27r d log r Jo Hence, by Theorem 4.1 we can obtain the representation f(Z) on D(T, 7) and so, u(Z) by taking the real part in terms of {an} and the conformal mapping z —
116

Acknowledgments This research was partially supported by the Japanese Ministry of Education, Science, Sports and Culture; Grant-in-Aid Scientific Research, Kiban Kenkyuu (A)(1), 10304009.

Added in the galley proof One referee pointed out a very valuable paper: L. R. Bragg and J. W. Dettman, Related problems in partial differential equations, Bull. Amer. Math. Soc, 74(1968), 375-378, in connection with the Reznitskaya transform. They also considered similar transforms and problems in the paper.

References 1. K. Amano, M. Asaduzzaman, T. Ooura and S. Saitoh, Representations of analytic functions on typical domains in terms of local values and truncation error estimates. Analytic Extension Formulas and their Applications, (2001), Kluwer Academic Publishers, Chapter 2. 2. S. Ito, Diffusion Equations. Transl. Math. Monographs, Amer. Math. Soc, (1992), 114. 3. S. Saitoh, Some fundamental interpolation problems for analytic and harmonic functions of class L2. Applicable Analysis, 17 (1984), 87-106. 4. S. Saitoh, Theory of Reproducing Kernels and its Applications. Pitman Research Notes in Mathematics Series, 189(1988), Longman Scientific &: Technical, UK. 5. S. Saitoh, Representations of inverse functions. Proc. Amer. Math. Soc, 125 (1997), 3633-3639. 6. S. Saitoh, Integral Transforms, Reproducing Kernels and their Applications. Pitman Research Notes in Mathematics Series, 369 (1997). Addison Wesley Longman, UK.

117

7. S. Saitoh, Representations of the solutions of partial differential equations of parabolic and hyperbolic types by means of time observations. Applicable Analysis, 76 (2000), 283-289. 8. S. Saitoh and M. Mori, Representations of analytic functions in terms of local values by means of the Riemann mapping function. Complex Variables, (to appear). 9. S. Saitoh, Representations of meromorphic functions in terms of Laurent expansions and partial boundary values. Complex Variables (to appear).

118

ISOMETRIC M A P P I N G S A N D T H E P R O B L E M OF A. D . A L E K S A N D R O V FOR CONSERVATIVE D I S T A N C E S THEMISTOCLES M. RASSIAS Department of Mathematics, National Technical University of Athens Zografou Campus, 15780 Athens, Greece E-mail: [email protected] Dedicated to A. D. Aleksandrov in admiration

The theory of isometric mappings had its beginning in the paper by S. Mazur and S. Ulam 10 who proved that every isometry of a normed real vector space onto another normed real vector space is a linear mapping up to translation. A. D. Aleksandrov has posed the problem: Under what conditions is a mapping of a metric space onto itself preserving unit distance an isometry? A generalization of the Mazur-Ulam theorem is given. The A. D. Aleksandrov problem is discussed and a number of open problems are posed.

Introduction Let X,Y be two metric spaces, d\,d2 the distances on X and Y, respectively. A bijection mapping / : X —> Y, of X onto Y, is defined to be an i s o m e t r y if d2(f(x),f(y))

=di(x,y)

for all elements x,y of X. S. Mazur and S. Ulam 10 have proved that every isometry of a normed real vector space onto a normed real vector space is a linear mapping up to translation. The property is not valid for normed complex vector spaces (one may just consider the complex conjugation on C). The hypothesis of surjectivity is essential in general. Without the onto assumption, J. A. Baker 2 proved that every isometry from a normed real vector space into a strictly convex normed real vector space must be a linear isometry up to translation. A mapping / : X —> Y satisfies the distance one preserving property (DOPP) iff for all x, y G X with dx{x, y) = 1, it follows that d2(f{x), f(y)) = 1 (see 1 3 ). A mapping / : X —> Y satisfies the strong distance one preserving property (SDOPP) iff for all x,y £ X with dx(x, y) = 1, follows that d2(f{x), f(y)) = 1 and conversely (see 1 5 ).

119

A. D. Aleksandrov l posed the following problem: Under what conditions is a mapping of a metric space into itself preserving unit distance an isometry? The basic "problem of conservative distances" is whether the existence of a single conservative distance for / implies that / is an isometry of X into Y. F. S. Beckman and D. A. Quarles 3 proved that if / : En -> En for 2 < n < oo satisfies condition (DOPP) then / is an isometry, where En is a finite dimensional real Euclidean space. This property does not hold for E1, the Euclidean line, as well as for E°°, a Hilbert space (cf. 14 , 5 for counterexamples). Independently from Beckman and Quarles, R. L. Bishop 4 , P. Zvengrowski 20 , D. Greenwell and P. D. Johnson 7 have given different proofs of the same result. For non-Euclidean spaces the Beckman-Quarles property has been derived by the Russian school, notably by A. Guc 8 , A. V. Kuz'minyh 9 (See also 1 7 ). We can formulate the following Conjecture If a continuous mapping / : E°° —> E°° satisfies condition (DOPP), then / must be an isometry. B. Mielnik and Th. M. Rassias 12 have proved the following Theorem Every homeomorphism / : En —» En for 2 < n < oo with a non-trivial conservative distance I > 0 is an isometry. Results and open questions In

14

it was proved the following

Theorem. For any integer n > 1, there exists an integer nm such that for N > nm it follows that there exists a mapping f : En —> EN which is distance one preserving but is not an isometry. If one partitions the plane into squares of unit diagonal as follows: Each square contains the bottom edge, the left edge and the bottom left corner but none of the other corners. Next label the nine vertices of the unit 8-simplex in E8 and map each square labeled i to the i-th vertex, then this mapping / : E2 —> Es satisfies condition (DOPP) but is not an isometry. Using hexagons instead of squares one can construct such a mapping / : E2 —* E6. This idea extends easily to higher dimensions. It is an open question whether there is a distance 1-preserving mapping f : E2 —> E3 which is not an isometric mapping. It is not yet known whether there is a

120

continuous mapping / : En —> Ek for k > n which satisfies the (DOPP) but is not an isometry. However, in En three classical metrics induce the same topology: n

dv(x,y)

= ^2\xi-yi\

,

dm(x,y)

= max-fjzi - yt\ : i = 1,2,... ,n}

i=l

and the Euclidean metric dE, where x = (xi,..,,

xn) and y = (J/J, . . . , yn).

Problem. Does the condition (DOPP) suffice for a mapping f : En -> Ek with respect to these metrics to be an isometry if 2 < n < k < +oo? For n = 1 all three metrics are the same. Consider the space E2 with the metric dm. In this case the mapping may satisfy (DOPP) and not be an isometry. For this let / : E2 —> E2 be defined by f(x,y)

=

([x],[y]),

where [x] denotes the integer part of x, in Cartesian coordinates. This mapping, which maps every point to the left-bottom corner of a suitable square with sides of length equal to one, with range equal to Z 2 (Z denotes the set of integers) is not an isometry but it preserves distance one. Now consider the metric d^. Take the mapping

9=(V2-R*/*)ofo(±.R;}iy where / is defined above and R^/A is the rotation:

,

x

fx + y

y-x\

The rotation maps unit balls in metric dm to balls of rarius \/2 with respect to metric d%. It follows that g satisfies (DOPP) but is not an isometry. Problem Examine if the mapping

satisfying (DOPP) is an isometry for n > 3. Th. M. Rassias and P. Semrl

15

proved the following:

Theorem Let X and Y be real normed vector spaces such that one of

121

them has dimension greater than one. Suppose that f : X —> Y is a surjective mapping satisfying (SDOPP). Then f is an infective mapping satisfying \\f{X)-f(y)\\-\\X-y\

<1

for all x,y £ X. Moreover, f preserves distance n in both directions for any positive integer n. The hypothesis that one of the spaces has dimension greater than one cannot be omitted in the theorem. In the theorem (SDOPP) cannot be replaced by (DOPP). The inequality \\f{x)-f{y)\\-\\x-y\\

< 1

for all x, y € X in the theorem is sharp (see 15 for proofs of these statements). Theorem Let X and Y be real normed vector spaces such that one of them has dimension greater than one. Suppose that f : X —> Y is a Lipschitz mapping with k = 1: ||/0»0-/(y)ll
for all

x,yGX.

Assume also that f is a surjective mapping satisfying (SDOPP). Then f is an isometry. Thus f is a linear isometry up to translation. Corollary Let X and Y be real normed vector spaces, dimX > 2, such that one of them is strictly convex. Suppose that f : X —> Y is a homeomorphism satisfying (DOPP). Then f is a linear isometry up to translation. Let X be a sphere in En(2 < n < oo). A natural metric structure is defined on X by the angular distances. The mappings that preserve the angular distance n/2, i.e., orthogonality preserving, have been of particular interest for the mathematical foundations of quantum mechanics. What is their structure? Though the answer has been basically known for a long time, the last brick to the proof has been given by C. S. Sharma and D. F. Almeida 18 (see also 1 6 ). For simplicity, we shall represent X as the sphere of unit vectors in a real Hilbert space H. Now, for any subset Z C X let ZL denote the set of all x £ X orthogonal to all z £ Z. It follows that Z C (Z1-) . The subsets Z for which Z = ( Z x ) are intersections of the sphere X with closed linear subspaces of H. We will refer to them further on as subspaces of X. Two

122

subspaces Y,ZCX are called orthogonal (denoted by Y±Z) iff y±z for every y e Y and z € Z. The orthogonality of the subspaces of X corresponds exactly to the orthogonality of the closed linear subspaces of H. For any two subspaces Y,ZCX one defines the partial ordering relation " < " as the set theoretical inclusion, i.e., Y < Z iff Y C Z. The set L of all subspaces of X with ordering relation " < " and with the mapping Y —> Yy is an orthocomplemented lattice isomorphic to the lattice of all closed linear subspaces of H. Now let / : X —> X be a homeomorphism with ir/2 being a conservative angle. Because of the fact / is orthogonality preserving, it induces a unique authomorphism of the orthocomplemented lattice L, which will be denoted by the same symbol / . Suppose that 3 < dimH < +00. Then, / must conserve not only the orthogonality (±) in L but also the angles between the 1-dimensional subspaces (rays). This fact follows from the existence of some more general numerical invariants on lattices and orthoposets. Indeed, consider the set M of all a- orthoadditive probability measures /i : L —> [0,1]. Assume that for any X £ L there exists at least one measure p, 6 M for which n(X) = 1. This, of course, is true for our lattice of subspaces L. For any pair X, Y € L define the quantity p(X, Y) as the infimum on Y of all probability measures p taking the value 1 on X: p(X,Y)=

inf

n{Y).

As an automorphism of L, f induces an invertible transformation of the aorthoadditive measures \x € M into new such measures \x o f~1, and since the value of p, on X coincides with the value of \i o / _ 1 on f(X), the infimum p(X, Y) must stay invariant i.e.

p(f(X)J(Y))=p(X,Y). If X and Y are two one-dimensional subspaces (rays) spanned by two unit vectors x,y £ H, then by the Gleason theorem 6 there exists exactly one measure fix £ M with fJ,x(X) = 1, and the infimum' p(X, Y) becomes: p(X,Y)

= fix(Y)

=

\<x,y>\2.

Therefore, the quantity | < x,y > | 2 must be invariant too, implying

I 1 = 1 <*.*/> I-

B. Mielnik and Th. M. Rassias

12

have proved the following:

T h e o r e m Let f be a homeomorphism

of the unit sphere X

in a real

123

Hilbert space H(3 < dimH < oo) that conserves the angular distance Then f is an isometry.

n/2.

Proof The mapping / defines an authomorphism of the orthocomplemented lattice L, which for 3 < dimH, implies \\

=

\<x,y>\.

Now by applying Wigner theorem 19 , this conservation formula implies that / differs trivially from a certain linear isometry operator / : X —> X, denned by f(x) = k(x)Ix where I : X —> X is an isometry and k(x) is just a real-valued function on X such that \k(x)\ = 1 for every a; in I . Because of the fact / is continuous and / is a homeomorphism, k(x) must be continuous as well, which leaves only two possibilities, i.e., k(x) = + 1 or k(x) = —1 everywhere on X. Therefore, f(x) = Ix or f{x) = —Ix everywhere on X. Hence / is an isometry. R e m a r k s The proof of the above theorem is based on a very fundamental theorem that was proposed by Eugene Wigner 19 . This theorem asserts that mappings from a Hilbert space to itself which preserve the absolute values of inner products are in a certain sense equivalent to isometries. Absolute values of inner products are related to probabilities of transitions between states of a quantum system and the time evolution of such a system is supposed to preserve these probabilities. Wigner used his theorem to define two linear mappings from a Hilbert space to itself which have played very fundamental roles in the development of quantum theory. These mappings are known to physicists as time reversal and charge conjugation operators. Problem Examine if the above theorem holds when / satisfies a condition weaker than that of a homeomorphism. Some physical questions originated by optics or by quantum mechanics lead to the following concepts of nonexpanding or nonshrinking distances. Definition Let / be a mapping of a metric space X into itself. A nonnegative number I is called a nonexpanding distance of / iff d{x, y) = I implies d(/(x), f{y)) < I for every x,y G X. A nonnegative number I is

124 called a nonshrinking distance of / iff d(x,y) = I implies d(f(x),f(y)) >I for every x, y € X. The distance / is called preserved (or conservative) by / iff for all x, y £ X with d(a;, y) = / i t follows that d(/(a;), /(y)) = /. If X is a sphere of a radius r in a Euclidean space En (2 < n < oo), the existence of a nontrivial nonexpanding (or nonshrinking) distance / (0 < / < 2r) for a homeomorphism / : X —> X implies the existence of a conservative distance, thus proving that / is an isometry (B. Mielnik n ) . For X = En this implication does not hold. For conformal nonisometric transformations defined on En all distances are either nonshrinking or nonexpanding. It is an open and interesting problem to find the structure of a mapping f : En —> En that admits simultaneously a nonshrinking distance d\- and a nonexpanding distance c^- A question remains to be answered as to whether such a mapping must be an isometry. Acknowledgement I wish to express my thanks to Prof. Dr. W. Tutschke for inviting me to participate in the International Workshop in Graz, 12 - 17 February, 2001, and to deliver this lecture.

References 1. A. D. Aleksandrov, Mappings of families of sets, Soviet Math. Dokl. 11 (1970), 116 -120. 2. J. A. Baker, Isometries in normed spaces, Amer. Math. Monthly 78 (1971), 655-658. 3. F. S. Beckman and D. A. Quarles, On isometries of Euclidean spaces, Proc. Amer. Math. Soc. 4 (1953), 810-815. 4. R. L. Bishop, Characterizing motions by unit distance invariance, Math. Mag. 46 (1973), 148-151. 5. K. Ciesielski and Th. M. Rassias, On some properties of isometric mappings, Facta Univ. Ser. Math. Inform. 7 (1992), 107-115. 6. A. M. Gleason, Measures on closed subspaces of a Hilbert space, J. Math. Mech. 6 (1957), 885-893. 7. D. Greenwell and P. D. Johnson, Functions that preserve unit distance, Math. Mag. 49 (1976), 74-79. 8. A. Guc, On mappings that preserve a family of sets in Hilbert and hyperbolic spaces, Candidate's Dissertation, Novosibirsk, 1973. 9. A. V. Kuz'minyh, On a characteristic property of isometric mappings, Soviet Math. Dokl. 17 (1976), 43-45.

125

10. S. Mazur and S. Ulam, Sur les transformations isometriques d'espaces vectoriels normes, C. R. Acad. Sci. Paris 194 (1932), 946-948. 11. B. Mielnik, Phenomenon of mobility in non-linear theories, Commun. Math. Phys. 101 (1985), 323-339. 12. B. Mielnik and Th. M. Rassias, On the Aleksandrov problem of conservative distances, Proc. Amer. Math. Soc. 116 (1992), 1115-1118. 13. Th. M. Rassias, Is a distance one preserving mapping between metric spaces always an isometry?, Amer. Math. Monthly 90 (1983), 200. 14. Th. M. Rassias, Some remarks on isometric mappings, Facta Univ. Ser. Math. Inform. 2 (1987), 49-52. 15. Th. M. Rassias and P. Semrl, On the Mazur-Ulam theorem and the Aleksandrov problem for unit distance preserving mappings, Proc. Amer. Math. Soc. 118 (1993), 919-925. 16. Th. M. Rassias and C. S. Sharma, Properties of isometries, J. Nat. Geometry 3 (1993), 1-38. 17. Th. M. Rassias, Properties of isometric mappings, J. Math. Anal. Appl. 235 (1999), 108-121. 18. C. S. Sharma and D. F. Almeida, A direct proof of Wigner's theorem on maps which preserve transition probabilities between pure states of quantum systems, Ann. Phys. 197 (1990), 300-309. 19. E. Wigner, On unitary representations of the inhomogeneous Lorentz group, Ann. Math. 40 (1939), 149-204. 20. P. Zvengrowski, Appendix to Chapter II of the book by P. S. Modenov and A. S. Parkhomenko, Geometric Transformations, Vol. 1, Academic Press, New York, 1965.

126

N A T U R A L METRICS OF DIFFERENTIAL EQUATIONS (ABSTRACT) Z. D. USMANOV Institute of Mathematics, Tajik Academy of Sciences. pr. Rudaki 33, Dushanbe 734025, Tajikistan E-mail: usmanovQcan. tajik, net An idea of constructing a natural metric of a differential equation was initiated by the problem of time in prediction. The essence of that consists in substantiating a concept of an intrinsic (inherent) time of a process for ranking its states, predicting the process evolution in its specific time and estimating a prognostic quality compared with the forecast based on using the astronomical time. Obviously, we deal with dynamic processes, states of which are changed with time. And what is more, herein dynamic processes described by mathematical models are examined. An aggregate scheme of constructing a natural metric for a certain process is following. Let xi, X2, •••, xn be variables characterizing a state of the process at time t. Its mathematical model is assumed to be and may be resolvable relative to the element dt as a function of a process state and its infinitesimal increment, i.e. dt = f(t,xi,X2,—,xn;dxi,dx2,—dxn),

(1)

in addition the right hand side of the expression is an homogeneous function of the first order with respect to the infinitesimal increments. Formula (1) is interpreted as a formula determining a set of isotropic directions of a Riemann space, *. The following quadratic differential form contains those isotropic directions: ds2 = e{dt2 - / 2 ) ,

(2)

where e is supposed to be +1 or —1 in order to do positive the right side of (2). By the form (2) a Riemann space is defined and called the natural metric of the process. Relative to (2) formula (1) characterizes the class of isotropic directions for which ds2 = 0 takes place. The accompanying formula for the other class of isotropic directions differs from (1) with the minus sign at the right side and corresponds to the process progress contrariwise oriented. 1. Namely in this way, natural metrics of processes described by ordinary differential equations are obtained. In particular, for dynamics of an isolated

127 population, which is characterized by the following mathematical model, 2 :

the Riemann metric has the form ds2 =

N2{a-jN)2dt2-dN2.

2. The natural metric of the gravitational field is ds2 =(?E

+ h\ dt2

_ dx2 _ dy2 __ dz2_

This formula is derived from the equation d2r _

ii

where r = (x, y, z) is the radius vector of the mass point, r is the length of the vector r, t is the time, and \i is the gravitational constant, 3 . It is interesting to note that the metric for the mass point at infinity has the form ds2 = hdt2 - dx2 - dy2 - dz2,

h>0

4

similar to the Minkowski metric, . 3. For the elementary parabolic equation ut = a2uxx,

x G [0,1], t > 0,

in case the difference 9{t) = ux(l,t)

~ux(0,t),

or h{t)

=u(l,t)-u(Q,t),

is given, the natural metric is ds2 = a4(g(t)/l - L02k2^kfdt2

- d&

or respectively ds2 = a4oj2k2(h(t)/l

- ujkr)k)2dt2 - dr]2k

Here 1 fl uk = «*(*) = T / u{x,t)e-lk"xdx, ' Jo u = 2n/l and ^ = Re uk and i / n = I m u f c , k = 0, ± 1 , ±2,....

128

4. For processes described by the elementary hyperbolic equation utt = a2uxx,

x G [0,1],

t > 0,

the natural metrics are presented in the following form ds\ = {(hk)2 + a2kW(gk)2

- a2k2u,2(uk)2]dt2

-

(duk)2,

where uk are Fourier coefficients of the unknown function u(x, t), k = 0, ± 1 , ± 2 , . . . , ui = w/l and the constants hk, gk are computed by initial and boundary conditions imposed on the function u(x, t). Conclusion Perhaps, constructed metrics enable to investigate a specific of processes and their mathematical models on the basis of geometric methods. Later on this investigation is assumed. References 1. Eisenhart L. Pf., Riemannian Geometry. Princeton, 1926. 2. Svirezhev Yu. M., Logofet D. O., Stability of Biological Communities. Mir Publishers. Moscow, 1983. 3. Sommerfeld A., Mechanik. Zweite, Revidierte Auflage, 1944. 4. Minkowski H., Raum und Zeit. Phys. ZS. 10, 104, 1909.

129

E M B E D D I N G T H E O R E M S IN A N I S O T R O P I C F U N C T I O N A L SPACES (ABSTRACT) M. D. RAMAZANOV Chernyshevskiy street 112, 450077 Ufa city, Russia E-mail: [email protected] Functions of n real variables from some Banach space are considered. Precise descriptions of restrictions of functions from these spaces on smooth surfaces are one of the main problems of the theory of embedding of functional spaces. Traditionally this problem was researched for a special case of surfaces being coordinate hyperplanes. It is enough, if the function space is isotropic, videlicet the smooth transformation of variables is an isomorphism of it. However, this is not so in the case of function spaces having diverse degrees of differentiability on diverse directions. In such cases there are not yet final results. The author gives precise descriptions of the restrictions to smooth surfaces of functions from Sobolov spaces W™*1 '™^ provided these spaces have only a small anisotropy, i.e., minrrij > \ maxmj. Moreover, for general reflexive Banach spaces B an operator of the best prolongation of functions on a smooth surface T to the whole space is constructed where ||/||B(r)=

inf HflfllB. 9\r=f

130

D I S T R I B U T I O N A L ANALYSIS OF B O U N D A R Y VALUE P R O B L E M S I N HALF-SPACES-EXEMPLIFIED B Y MELAN'S P R O B L E M OF ELASTOSTATICS (ABSTRACT) G. KIRCHNER Institut fur Technische Mathematik, Geometrie und Bauinformatik Universitdt Innsbruck, Technikerstr. 13, A-6020 Innsbruck, Austria E-mail: Gerhard. KirchnerQuibk. ac. at Using a result on the C°°-dependence (with respect to a distinguished variable) of solutions to partially hypoelliptic operators, we present a definition for Green's function which allows to prove a representation theorem. Afterwards we discuss the construction of a Green's matrix for Melan's problem of elastostatics in this sense using the reflection principle and single layer potentials. This involves the solution of convolution equations as well as the explicit computation of convolutions. The result is the following: Let /

dx dx

dy dy

0

dxx

P(d) =

Ol-Vdl -2(1

0 dy dy

\ , v

G

R

+V)dlydl-Vd2y

be the operator of plane elastic stresses. Then a Green's matrix of P{d) in the half-plane x e R, y > 0 for vanishing stresses on the boundary y = 0, derived by the method sketched above, is [r :— y/x2 + (y — r))2, R := y/x* + (y + T))2): 1

Gv{x, y) = ^ 2

v+1 ' 4irr4

•"

I

lo r -(y-v) vy
x \y ~ V -x y-rj

2

~x(y ~~7?)/2 ^-logr iv-vY

x(x2-(y-r,)2) (y-v)(3x2 + (y-v)2) 2 2 (y - r]){x - (y - r?) ) x((y - r])2 - x2) 2 2 x{3(y - r,) + x ) (y - r?)((y - rj)2 - x2) Sx ».

i *.

x

by + Srj - ^ l o g f l + V+3*?2 + 2yrf •*•

-y + rj

x

iy—n)

-^-logR+V-p-

0N 0 0,

131

+m+J!L\_ 2{y + ri) 2x xr] irR4 2x 2(y + T)) rj(y + 77) u + l

+

+l

W

^4 ™

I

-x

\

- ^-x" ^

—3y — 577 0 X ?Q> y~r)

1/0/-2q) 0 0 \ - x ( y + 77) - ^ y - r ? ) 0,

(l/+1)

^

4

;W J ?S L100 \ -x

y+77

0

This corrects formula (11) in E. Melan's paper "Der Spannungszustand der durch eine Einzelkraft im Innern beanspruchten Halbscheibe", Journal for Applied Mathematics and Mechanics (ZAMM), 12 (1932), p. 343-346 and also the results given in p. 101-105 of A. Foppl, "Drang und Zwang", Vol. 3: Der ebene Spannungszustand, 1969, and p. 158-159, ex. 326 in N.N. Lebedev/I.P. Skalskaya/Ya.S. Ufiyand, "Problems of Mathematical Physics", Prentice-Hall, 1965.

132

BACKLUND TRANSFORMATIONS A N D N O N L I N E A R EVOLUTION EQUATIONS (ABSTRACT) J. PUNGEL Department of Mathematics C, Graz University of Technology Steyrergasse 30/3, 8010 Graz, Austria E-mail: puengelQmatd. tu-graz. ac. at This talk is dedicated to an explicite construction of pairs of differential equations (*)

ut = f(u,ux,uxx)

;

vt=

g(v,vx,vxx)

such that their solutions u(x,t) and v(x,t) are associated by v = -y(u,ux). Such direct differential transformations (DTs) are equivalent to Backlund transformations (BTs) ux = 4>(u,v) ,

ut =

F(u,v,vx)

with integrability conditions (*). Furthermore some generalizations are discussed, for instance a widely applicable method to obtain extensions of scalar DTs and BTs onto systems of evolution equations. References 1. J. Piingel, Transmutations in differential rings, in Generalized Analytic Functions (edited by H. Florian et al.), 55 - 63. Kluwer Academic Publishers, 1998

133

N O N L I N E A R P E R T U R B A T I O N S OF SYSTEMS OF PARTIAL D I F F E R E N T I A L EQUATIONS (ABSTRACT) C. J. VANEGAS Departamento of Matemdticas, Universidad Simon Bolivar Valle de Sarteneja - Baruta, P. O. 89000 Caracas, Venezuela E-mail: [email protected] We show the existence of solutions to boundary-value problems in a bounded region G C R n with smooth boundary : D0w = f(x,w,

-—,..., dxi

Aw = g

-—) axn

inG,

ondG,

(1) (2)

where Do is a linear differential operator of first order with respect to the real variables xi,...,xnThe unknown w and the right-hand side / are vectors of m components, with m > n. g is a given m-dimensional vector-valued function that belongs to the Banach space B1(dG). The operator A is chosen so that (2) leads to a well-posed problem on B1(G) D kerDoFor finding a solution to this nonlinear problem, we use a right inverse of the operator Do and a fix point argument *. As an example, we apply this method to the operator

A D0w=

9 J,—

9x3 9 9x2

9

9

9X3

9X2

9 A -35— 9 9xi

9xi

u>2

= cm\w + \Iw

A

References 1. C. J. Vanegas, Nonlinear perturbations of systems of partial differential equations with constant coefficients, Electronic Journal of Differential Equations, Vol. 2000, No.05, pp.1-10, (2000).

134 PARTIAL D I F F E R E N T I A L EQUATIONS A N D CUBATURE FORMULAS (ABSTRACT) M. D. RAMAZANOV Chernyshevskiy street 112, 450077 Ufa city, Russia E-mail: [email protected] The optimization of cubature formulas is a problem of calculus of variations in a framework of functional analysis theory. So differential equations appears as Euler equations. For example a calculation of optimal error functional of cubature formula is converted into some Wiener — Hopf problem for the polyharmonic operator as it is described in Sobolev's monograph "Introduction into the theory of cubature formulas". Connection between the construction of algorithms for optimal asymptotic formulas and the types of differential operator of these Euler equations are more profound. So if the operator is hypoelliptic we get universal asymptotic optimal cubature formulas for fixed knot lattices. The optimization of the lattice form is possible, too. We intend to give a survey of further results of the theory of cubature formulas and other types of differential operators required in this theory.

C H A P T E R 2:

APPLICATIONS OF FUNCTIONALANALYTIC

AND COMPLEX METHODS

TO MATHEMATICAL PHYSICS

137

CONSERVATION LAWS FOR D I F F E R E N T I A L E Q U A T I O N S V. S. VLADIMIROV Steklov Mathematical Institute Gubkin Street 8, 117966 GSP-1 Moscow, Russia E-mail: [email protected]

1

Introduction

Conservation laws are important for understanding the dynamics of fields described by (linear or nonlinear) partial differential equations (PDE). Usually one uses local conservation laws (currents) which can be obtained by the Nother theorem if we know the symmetry group of the PDE considered. As a rule the symmetry group is " hidden". For the very wide class of PDE we propose a method for construction currents without using the Nother theorem that is without knowing the symmetry group of the PDE 1 - 3 . The method is based on the notion of an adjoint operator to the PDE 4 - 6 , and conserved currents are expressed linearly through solutions of the adjoint PDE. Along the local currents, nonlocal currents exist, they also are important for the dynamics 7 , 8 . The method is applied to the classical PDE of mathematical physics: the nonlinear Dirac equation, the Boltzman equation, anharmonic oscillator, the Navier-Stokes equations, the hydrodynamic equations, the nonlinear Laplace equation, the nonlinear wave equation, the Burgers equation, the nonlinear Schrodinger equation, the Korteweg-de Vries equation, the SineGordon equation, the a-model (chiral field), the Heisenberg spin chain, the Kadomstsev-Petviashvili equation, the Yang-Mills equation, the supersymmetric Yang-Mills equations, and others. The 2-dimensional integrable PDE are considered as compatibility condition for some overdeterminded linear PDE (Frobenius condition). One of the most important features of integrable PDE is the existence of the highest local and nonlocal currents 9 , 1 0 . We apply the small parameter method for constructing an infinite chain of local and nonlocal currents for integrable PDE i- 3 .

138

2

Notations and definitions

Let D be a domain in R n with sufficiently smooth boundary dD, x — (xi,x2, • • • ,xn) e R; let C™xr(D), (m > 1) be the set of all px r -complexvalued matrix-functions of class Cm{D). Let u —> w [u] be a (nonlinear or linear) differential operator from CTXr(D) to C°qXr(D).^ Definition 1 A set of n operators u->Jv[u},

u€C™xr(D),

u=

l,2,....n

is called a conserved current J = (Ji, J 2 , . . . , Jn) in D for the operator UJ [u] if it satisfyies the differential conservation law divJ = dvJv [u] (x)=0,

x GD

(2.1)

on every solution u £ C™r(D) of the equation u> [u] = 0. (Here and below we suppose a summation on repeated indices.) Owing to the Poincare lemma the codition (2.1) is equivalent to the local exactness in D of the differential form (—l)" - 1 Jv [u] (x)dxi A . . . A [dxv] A . . . A dxn, that is Jv [u] (x) = d^^v

[u] (x),

^ ^

= -$!///.

(Here and below [dx„\ denotes that dxv is excluded.) The set of all currents for equation u> [u] = 0 forms a vector space. Definition 2 A current J is called trivial in D if it either vanishes on every solution u of the equation w [u] = 0 or equation (2.1) is fulfilled on every u € C™xr{D). Note that the last condition is fulfilled if Ju [u] (x) = C^M„(:E), $ M J, = Definition 3 Two currents are called equivalent if their difference is trivial. The notion of equivalence possesses the properties of: reflexivity, symmetry and trasitivity. Therefore the set of all currents decompose on disjoint classes of equivalence. We consider trivial currents as inessential, they form zero class of equivalence. Definition 4 A current J is called local in D if every operator Jv[u](x), v = l , . . . , n is a function of finite numbers of arguments x, u(x),.. -, dau{x). The other currents are called nonlocal.

139

Let D be a simply connected bounded domain in R " and u G C™Xr(D) be a solution of equation w[u] = 0. Then from (2.1) it folows the integral conservation law (by the Stokes formula) ( - 1 ) " - 1 / J„ [u] {x)dxi A . . . A \dxv] A . . . A cten = 0. (2.2) JdD If we choose the domain D in the form (0,t) x G, G e R " - 1 , solution u satisfying suitable boundary conditions on dG and denote x\ = t, we obtain from (2.2) the classical integral conservation law (integral of motion) / J\ [u] (t,x2,...,

xn)dx2 • • • dxn =

JG

/ J\ [u] (0, X2, • • •, xn)dx2 • • • dxn = const.

(2.3)

JG

3

Conserved currents for general system

A wide class of (nonlinear in general) system w [u] = 0 of differential equations of order m > 1 can be represented in the following form: u [u] = ( J2

P

" M (x)da)u(x)

= 0,

\a\<m

or equivalently m

P[u]u = J2Pvi...v* M (x)dvi • • • 9Vku(x) = 0,

(3.1)

fc=0

where u —> PVl..,„k [u] are (nonlinear in general) differentional operators from C™xr(D) to Clqxp(D) of order I < m — k; du = j - . (In case when orders I of all operators PVl...Vn are equal to 0 the system (3"l) is quasilinear.) Let us fix u G C™xr(D). We introduce the adjoint to equation (3.1) the linear system

vP*[u] = YtdVl...dv v(x)PVl...Vk [u](x)

(3.2)

fc=0

The adjoint operator v —> vP* [u] is not unique because a representation of the operator LJ [U] in the form (3.1) is not unique.

140

Example 1 The operator w [u] = uux can be represented in the form (3.1) with Pi [u] = u, or PQ [U] = ux, or Pi [u] = l/2ux. The corresponding adjoint operators are: v —> — (uv)x,

v —» uxv,

v —> l/2uvx.

Nevertheless the adjoint operator is unique in some class of equivalence defined by the operator u [u] 4 . Along with the linear equation (3.2) we introduce another linear equation (for fixed u) associated with the equation (3.1) by the following formula: 771

P [u] w = Y^ p^...vh

[«] (x)dVl ... dVkw{x) = 0.

(3.3)

fc=i

For the linear system (3.2)-(3.3) with respect to matrix-functions (v,w), v € C™xq(D), w G C^r(D) the correspodig Lagrangian is Tr(vP [u] w) which is invariant under linear transformatios w —> wC,

v —> C~lv

where C are arbitrary constant (complex-valued) invertible r x r-matrix. By the Nother theorem, the linear system (3.2)-(3.3) has the following conserved currents n : Ju[(v,w)}=

(3.4)

m—1

^2

(-l)kdUl

...dVk v(x)PUVl...VkXl...xi

[u] (x) dXl.-. dxiW(x),

k+i=0

v — l,2,...,n. Putting in (3.4) w = u where u is a solution of equation (3.1) we obtain conserved current for the original equation (3.1) 1 - 3 J„ [«] =

(3.5)

m —1

^2

{-l)kdVl

...dVk v(x)P„l,1...I/kx1...xi[u\(x)

dxi.-.dxMx)'

k+i=0

v = 1,2,..., n. So we have just obtained the following Theorem Every solution of the adjoint linear system (3.2) defines by the formula (3.5) a conserved current for system (3.1). N o t e Components J„ [u] of the current are r x r-matrices, and in fact formula (3.5) defines r 2 conserved currents. In general currents (3.5) are nonlocal.

141

4

Examples

Example 2 The first order equation (3.1) is P [u] dvu +

PQ [U] U = 0.

The adjoint linear system (3.2) is dv(vPv[u])-vPo[u]=0, and the current (3.5) is J„ [u] = vPv [u]u, In particular, for n = 1,

v = 1,2,...,n.

p = r = q = lwe obtain nonlocal current

Po [u] (y) dy J\ [u] = uP\ \u] exp f Po Jo Pi [«](y)

const.

Example 3 The 2-dimentional wave equation utt - uxx = 0,

vtt - vxx = 0.

The solution v — u gives trivial current. The solution v = ut gives known current J\ [u] = v?t - uutt, rb

J2 = uutx -

utux,

fb

/ Ji [u] (t, x)dx = Ja

(uj + u2x)dx = const,

u(t, a) = u(t, b) = 0.

Ja

Example 4 The Burgers equation ut + uux = 0,

vt + (uv)x=0,

J\ [u] = uux,

v = ux.

J2 [u] = u2ux,

fb

/ Ji [u] (t, x)dx = l/2u2(t, b) - l/2u2(t, a)+

Ja

/ [u 2 (r, b)ux(r, b) - w 2 (r, a)ux(r, a)]dr = 0. Jo Example 5 The Korteweg-de Vries equation ut - 6uux + uxxx = 0, J\ [u] = uux,

vt— 6(uv)x + vxxx = 0, J% [u] = -6u2ux

+

v = ux

uuxxx.

142

The integral conversation law for boudary value problem u(t, a) = u(t, b) = 0 is trivial: fe Ji [u] (t, x)dx = l/2u2(t, b) - l/2u2(t, a) = 0. / More complicated problems have been investigated by the method of adjoint operators by V.S.Vladimirov and I.V.Volovich in 1 - 3 , and by G.I.Marchuk, V.I.Agoshkov and V.P.Shutyaev in 5 , 6 . 5

Acknowledgment

The work was supported partly by the RFFI (project no. 00-15-96073.)

References 1. Vladimirov, V. S. and Volovich, I. V., Theor. Math. Physics, v.62, p. 1-20 (1985). 2. Vladimirov, V. S. and Volovich, I. V., Annalen der Physik, v.7, Folge B.47, H.2/3, p.228-238 (1990). 3. Vladimirov, V. S. and Volovich, I. V., Actualnye Problemy Vychislitelnoi Matematiki i Matematicheskogo Modelirovaniya, (Nauka, Novosibirsk, 1985, p.147-162.) (in Russian.) 4. Vladimirov, V. S. and Marchuk, G. I., Dokl. RAS, v.372, p.165-168 (2000). (in Russian.) 5. Marchuk, G. I., Adjoint Equations and Analysis of Complex Sistems, (Nauka, Moscow, 1992.) (in Russian.) 6. Marchuk, G. I., Agoshkov, V. I. and Shutyaev, V. P., Adjoint Equations and Perturbation Algorithms, (CRC Press, Boca Raton, New York, London, Tokyo, 1996.) 7. Bogolubov, N. N., Selected Works, v.3, p.110—173, (Naukova Dumka, Kiev, 1971.) (in Russian.) 8. Liischer, M. and Pohlmeyer, K., Nuclear Physics, v.137, p.46-54, (1978). 9. Zakharov, V. E., Manakov, S. V., Novikov, S. P. and Pitaevskii, L. P., The Soliton Theory, (Plenum Press, New York, 1984). 10. Gardner, C. S., Green, J. M., Kruskal, M. D. and Miura, R. M., Physical Review, v.19, p.1095-1097, (1964). 11. Vladimirov, V. S. and Zharinov, V. V., Differential Equations, v.16, p.534-552, (1980).

143

A P P L I E D Q U A T E R N I O N I C ANALYSIS. MAXWELL'S S Y S T E M A N D DIRAC'S EQUATION V. V. KRAVCHENKO Depto. de Telecomunicaciones, SEPI, Escuela Superior de Ingenierma Mecanica y Elictrica, Instituto Politicnico Nacional, C.P.07738, D.F., MEXICO E-mail: vkravche@maya. esimez. ipn. mx This work is a review of some recent results obtained with the aid of quaternionic analysis for electromagnetic fields and for Dirac's spinors. We consider quaternionic reformulations of Maxwell's equations and Dirac's equation and discuss their advantages for the analysis of boundary value problems as well as for obtaining exact solutions.

1

Introduction

This work is a review of some recent results obtained with the aid of quaternionic analysis for electromagnetic fields and for Dirac's spinors. We consider quaternionic reformulations of Maxwell's equations and Dirac's equation. In the case of Maxwell's equations for a general inhomogeneous medium, the reformulation discussed in this work is new 25 . We consider three special cases. When the medium is homogeneous (chiral or achiral) we show the applications of quaternionic analysis for solving the extendability problem for the electromagnetic field. For static fields in an arbitrary inhomogeneous medium we analyze some possibilities for obtaining fundamental solutions due to the fact that in this case the Maxwell operator written in a quaternionic form together with its conjugate factorize the Schrodinger operator. Finally, for slowly changing media we consider the construction of exact solutions. For the Dirac equation we discuss its quaternionic reformulation obtained in 20 and its advantages compared with the frequently used reformulation due to C. Lanczos. We mention some interesting results obtained for boundary value problems related to the Dirac equation as well as some recent publications where the interested reader can find new exact solutions obtained with the aid of quaternionic analysis. 2

Definitions and notations from quaternionic analysis

The set of complex quaternions (=biquaternions) is denoted by H(C). Each complex quaternion a is of the form a — J2k=o ak^k where {afc} C C, IQ is the

144

unit and {ik\

k = 1,2,3} are the quaternionic imaginary units: *o = -*fc| ioik = «fc«o = ik,

llJ2 =

- « 2 « 1 = 43; ^ 3

k = 1,2, 3;

= - « 3 « 2 = * i ; l&l

= - ^ 3

= «2-

The complex imaginary unit i commutes with ik, k = 0, 3. We will use the vector representation of complex quaternions: a = Sc(a) + Vec(a) , where Sc(a) = ao and Vec(a) = ~a = J2k=i akik- That is each complex quaternion is a sum of its scalar part and its vector part. Complex vectors we identify with complex quaternions whose scalar part is equal to zero. In this way vector calculus is embedded into quaternionic analysis. In vector terms, the multiplication of two arbitrary complex quaternions a and b can be rewritten as follows: a • b = ao&o- < a , b > + a x b + ao b +bo a , where

<~a,b

>:= \\akbk fc=i

GC

and

h "a x b]

12 13

3 a\ a2 a3 e C h 62 b3

The quaternionic conjugation is introduced as follows a = Sc(a) - Vec(a). If a is not a zero divisor then a - 1 = a/(aa). We shall consider H(C) -valued functions defined in a domain G e l 3 . The set of all such functions whose components are continuously differentiable in G we will denote by Cl{G). On this set the well known (see, e.g., 8 , 12 , 30 ) Moisil-Theodoresco operator is defined by the expression D

y^ikdk, fc=i

vhere

dk = -5—• OXk

The action of the operator D on an H(C)— valued function / can be written in the vector form:

145

£>/ = - d i v / + g r a d / 0 + r o t / . That is, Sc(Df) = — div / and Vec(Df) = grad/o+rot / . In a good number of physical applications (see 12 and 30 ) the operators Da = D + Ma and £>_Q = D — Ma are needed, where a is a complex quaternion and Ma denotes the operator of multiplication by a from the right-hand side: Maf = f • a. We start with the case a = SC(GJ). Note that -DaD-a

= A + a2I,

= -D_aDa

(1)

30

2

where / is the identity operator, and moreover (see ): ker(A + a 1) = ker£) a ©ker£>_ Q , for a ^ 0 (here the Helmholtz operator A+a2I is considered in general on the H(C)-valued functions). The factorization equality (1) allows us to obtain the fundamental solutions for the operators Da and D-a. That is, we have that D±aK±a = S, where eia\x\

f

K±a = DTa(

x

x

\

) = \±a + —2 - ia—r 4ir\x\

y

\x\2

\x\J

eia\x\

•

3

,

x =

4TT\X\

}xkik. ^

Consider the following integral operator K±a[f}(x)

= - J K±a{x - y)H{y)f{y)dTyi

iel

3

\r,

where V = dG is a Liapunov boundary and ~n(y) is the quaternionic representation of the outward unit normal to the surface T: ~n{y) = Yl,k=i nk{y)ikLet us stress that all the products in the integrand are quaternionic products. For a bounded domain G we have the following integral representation of the functions from keiD±a(G). Theorem 1 18 , 30 Let f e C1{G) n C(G) and f € ker D±a(G). Then f(x) = K±a[f]{x),

VxeG.

For an unbounded domain G, f must also fulfill the radiation condition (1 ± jHj) • f{x) = o(±), when \x\ -* oo (see 33 and 2 6 ) . Then f{x) = -K±a[f](x), VxGG. The behavior of the integrals K±a[f] near the boundary T is well described by the following analogue of the Plemelj-Sokhotski formulas. Theorem 2 2 9 ; 3 0 Let T be a closed Liapunov surface, G+ be the corresponding interior domain and G~ the exterior one, f be a Holder function defined on T. Then everywhere on Y the following formulas hold

lim G+3x—>rer

K±a\f]ix) = P±a[f}ir),

146

lim

K±a[f](x)

=

-Q±a[f}(r),

G~3x—>rGr

where P±a = § ( / + S ± a ) , S±a[f](x)

Q±a = \{I - S±a) and

:= - 2 J K±a{x - y)Tt{y)f(y)dTv,

x£T

is understood in the sense of the Cauchy principal value. Remark 3 The same results are valid also for the operators D±a, where a is an arbitrary constant complex quaternion (see 30). The corresponding integral operators were introduced in 29 , see also 30 . Their form depends on a. For example, if a = Vec(a) = a then the operator K-g is defined as follows

**[/](*) = -j [w*-y) \\x-y\ (r^s -TT^T) "(y)m+ \ ~y\ I x

+ eu{x-y)-^{y)f{y)S}dTy, where v = v o 3 3.1

2

i£l3\r,

G C is chosen such that Imu > 0.

Maxwell's s y s t e m Quaternionic

reformulation

Let us consider the Maxwell equations for an inhomogeneous medium rotH = £ d t E + j ,

(2)

rotE=-/x9tH,

(3)

div(eE) = p,

(4)

div (/zH) = 0.

(5)

Here e — e(x) and \x = n(x) are absolute permittivity and permeability of the medium respectively. One can verify (see 25 ) that the vectors ~t := v/eE satisfy the following system

and

H := v ^ H

147

and

D-p H = -dtt

+ Vw,

(7)

where c=l/v^

(8)

is the speed of propagation of electromagnetic waves in the medium; _>

gradv/i

__>

grady^Z

and as before D-? := D + Mr, Dj? := D + M K This pair of equations is completely equivalent to the Maxwell system (2)-(5) and represents Maxwell's equations for inhomogeneous media in a quaternionic form. In a time-harmonic case (with the dependence on time as exp(iojt)) the complex amplitudes E and H of the vectors £ and H are solutions of the system 25 D-?~E = -iaH

- -£=

(9)

D^H = iaE + vT^T,

(10)

a := tOy/efl

(11)

and

where

is the wave number. For the sake of simplicity, in what follows we consider the sourceless situation, that is the following pair of equations D-gE = -iaH

(12)

Djtll

(13)

and

3.2

= iaE.

Homogeneous media

When e and /z are constant, the system (12) and (13) has the form DE = -iaH

(14)

DH = iaE

(15)

and

148

and can be diagonalized in the following way 18 (see also 19 and 30 ). Consider functions Tp:=^

+ iH

and

~ip :=~E - iH.

It is easy to see that Tp and ip are solutions of the following equations DaTp = 0

and

£>_a ^ = 0

(16)

if and only if E and H satisfy (14) and (15). Remark 4 Equations (16) mean that Tp and ip are the so called Beltrami fields 1. Thus, using known facts for solutions of the operators D±a, one can obtain corresponding results for the electromagnetic field. For example, under the conditions of Theorem 1 we have the following equalities for "ip and ip tp = KaTp

and

ip = K-a ip,

which rewritten in terms of E and H have the form ~E = ^(Ka(E+iH) + K_a(E-iH))

(17)

and H = ±:{Ka(E+iH)-K-a(E-iH)).

(18)

These two equalities are nothing but the well known 10 Stratton-Chu formulas in a quaternionic form. The quaternionic integral operators introduced above are very appropriate for working with boundary value problems for the electromagnetic field. Let us consider one result which was obtained with the aid of the quaternionic technique 1 8 and which has not been proven by "traditional" methods. For more examples of this kind we refer to 30 . Problem 5 Let G be a bounded (similar results can be obtained for unbounded domains 9) domain in R3 with a Liapunov boundary T. Find a pair of vectors E and H satisfying (14), (15) in G as well as the following conditions on the boundary: E\r — ~e

and

H|r = h ,

where ~e and h are given complex vector functions defined on Y and satisfying a Holder condition there.

149 In other words, to extend the vectors e and h from the boundary onto the domain G in such a manner that their extensions satisfy the Maxwell equations. The problem is ill-posed because the solution does not always exist. A necessary condition for its existence is well known (see, e.g., 3 9 ) : ~e and h must be boundary values of the corresponding Stratton-Chu integrals, but this is not a sufficient condition! The following criterion was obtained in 18 (see also 30 ) and in quaternionic terms has the form Qa{~e+ih)

= 0

and

Q-a{~e - it)

= 0 on T.

(19)

The vector parts of these quaternionic equalities reflect exactly the above mentioned necessary condition and the scalar parts give us the lacking information. We do not write here the explicit vector form of (19) referring the reader, e.g., to 30 . Remaining in the framework of three-dimensional vector calculus it is not easy to find the reason for the necessity of the scalar parts from (19), but it becomes obvious in quaternionic terms. If (19) is fulfilled then the solution of Problem 5 is given by (17), (18) where from the righthand side the capital E and H must be replaced by the corresponding small letters. This result for a domain containing sources of the electromagnetic field can be found in 9 . Remark 6 In the case of a chiral homogeneous medium the diagonalized Maxwell equations have the form 16 Dai ~^P = 0

and

D-a2 ip — 0

(instead of (16) for an achiral case), where at\ := a/(l + a(3),

a
and (3 is the chirality measure of the medium 3 1 . The advantage of the application of quaternionic analysis methods becomes even more appreciable in this case, due to a comparative complexity of vector notations. 3.3

Static fields

For the static electric and magnetic fields in an inhomogeneous medium from (12) and (13) we obtain the following pair of decoupled equations D-gE

=0

DVH

= 0.

and

150

Thus, we are interested in the solutions of the equation (D + M's)J=0,

(20)

where the complex quaternion tt represents "e* or ~jt and has the form ~a = (gia,difi)/tfi. The function w is different from zero. Denote v :=

Aw

,

that is w is a solution of the Schrodinger equation -A

(21)

25

Let ijj be another solution of (21). Then the function ? = ( £ > - -ct)i>

(22)

is a solution of (20). This proposition gives us the possibility to reduce the solution of (20) to the Schrodinger equation (21). Moreover, if ip is a fundamental solution of the Schrodinger operator: ( - A + v)i> = S, then the function / defined by (22) is a fundamental solution of the operator D + M". Remark 8 The proposition 7 is closely related to the factorization of the Schrodinger operator proposed in 5 , 6 . Namely, for a scalar function u we have the equality (D + M*)(D

- M")u

= ( - A + v)u,

if the complex quaternionic function a satisfies the equation D~a + ~ct2 = -v.

(23)

It is easy to check that for ~a = (grad
151 Example 9 2 5 Consider equation (20) in some domain G C K 3 and let Aip/ip = —b2 in G, where b is a complex constant. In this case we are able to construct a fundamental solution for the operator D + Ma . Denote e «6|x|

^W

=

T~T147T | x |

This is a fundamental solution of the operator —A — b2. Then a fundamental solution of D + Ma is constructed as follows -* . f (x) = {D

g r a d e s ) ei6lxl x x = (-T-5 + lbZJ\X\
gradip{x) .„t„\
eibW 47r|x|'

where x = Y,k=i x^kNote that it is not clear how to obtain this result for the Maxwell operators D + M e and D 4- M M using other known methods. 3-4

Slowly changing media

A medium is said to be slowly changing (see, e.g., 2 ) if

JE^£l where c is defined by (8). In this case the system (12) and (13) can be diagonalized also, and we arrive at the following pair of equations 23 , 24 DaTp = 0

and

D-^

= 0,

(24)

are

where a is defined by (11) and the vectors ip and ip linear combinations of vectors of the electromagnetic field. In 24 the following approach for solving equations (24) was proposed. Suppose that the function

)2 = a 2

in G.

(25)

Then the complex quaternion valued functions Q^ := a ± grad^ are zero divisors in G (Q+ • Q~ = 0). Let n = e^. Then the operator Da can be rewritten in the following form Da = n(D + Q+)r)~xI. Consequently, the equation Dau = 0 in G

(26)

is equivalent to the equation (D + Q+)w = 0

inG,

(27)

152

where w = u/r). We look for the solution of (27) in the form w = Q~s, where s is a complex quaternion valued function for which from (27) we have the equation DQ's

= 0.

(28)

In some cases it is possible to find even analytic solutions of this equation. For example, let a depend only on the variable xi. Then, for example, the functions 4>\ = iA, 2 = —iA are solutions of (25). Here A is an antiderivative of a. The corresponding solutions of (26) are obtained 24 in the form Ul(x)

= eiA^G+(x2,x3)

• (1 -

ih)A

and u2(x)=e-iA^G-(x2,x3)-{l

+ ii1)B,

where A and B are arbitrary constant complex quaternions, and the functions G + and G~ have the form G± = G^ + Gfii, where G^ and Gf are complex valued functions satisfying the following Cauchy-Riemann type system d2G0 + d3Gi = 0,

(29)

d3G0 - d2Gi = 0.

(30)

Using the relations given above between the equations (24) and the Maxwell equations we arrive at the following expressions for the electromagnetic field: ^{x)

l!^^((a+eiAMh+(x2,x3)-a-e-iA^h-{x2,X3))(i2-ii3)+

=

+ (b+e-lA^g+(x2lx3)-b-eiA^g-(x2,x3))(i2 3/2 /

\

H(x) = ^-J^l{(a+eiA^h+(x2,x3)

+ ii3)),

+ a~e~iA^h-(x2,x3))(i2

-

ii3)+

(b+e-iA^g+(x2,x3)+b-eiA^g-(x2,x3))(i2+ii3)),

+

where the functions h± = h^ + ihf satisfy (29), (30) and the functions g± = 9o ~ Wi satisfy another Cauchy-Riemann type system: d2G0 - d3Gi = 0, d3G0 + d2Gi = 0; a

±

1

and b^ are arbitrary complex numbers. The explicit analytic solutions are possible not only in the case of a stratified medium but also in some other cases, for example, when the permittivity

153

is analytic with respect to a complex variable z = Xk ± ix\ and depends arbitrarily on xp, where k, I and p are pairwise different and take their values from the set {1,2,3} (see 2 4 ) . For stratified media it is possible to obtain some useful operator estimates justifying the neglection of the relatively small terms (see 2 3 ) . 4

Dirac's equation

We will consider the Dirac operator in its covariant form

D[$] := [7o9t - Y^lkdk + im j [*], where $ is a C4-valued function $ = ($0, $ 1 , $2, $3) T ; fn £ R represents the mass of a particle and the Dirac 7-matrices are defined as follows (see, e.g., 7,37\

70

72 :=

/ 1 0 0 o\ 0 1 0 0 00-1 0 \00 0 - 1 /

/00 0 00-1 7i == 0 1 0 \10 0

'0 0 0 i \ 0 0-20 0-i 00 i 0 00/

73

-l\ 0 0 0/

0 0 - 1 0\ 0 0 01 1 0 00 0-1 0 0/

We will need also the following matrix, usually denoted as 75 /

75 := ^70717273

0 0-1 0\ 0 0 0-1 - 1 0 0 0 \ 0-1 0 0/

Recall some important properties of the 7-matrices: 1- 7o

=

75 = ^4, the identity matrix.

2. 7fc = ~ ^ 4 , A; = 1,2,3. 3- 7j7fc = -Iklj,

j,k = 0,l,2,3,5,

j ^ k.

154

4-1

Historical remarks

Prom the very appearance of Dirac's equation (introduced by P. Dirac in 1928) in quantum mechanics many researchers considered that the form of writing it was unsatisfactory. The reason is that the algebra of matrices used for this purpose has dimension 16, and there does not exist any physically meaningful reason for using an algebra of 16 dimensions if the number of equations under consideration is four. This discrepancy, which sometimes makes the work with the Dirac equation more difficult that it could be, explains the fact that A. Sommerfeld posed the following problem: to rewrite the Dirac equation in a form in which the rank of the algebra of the matrices involved coincides with the number of components of the wavefunction, and apparently he made a first attempt to solve it (see 3 5 ) . Already in 1929 there appeared an article by C. Lanczos 32 in which the Dirac equation

D[$] = 0

(31)

was rewritten in the following form {idt + D-

imZcMi3 )F = 0,

(32)

where F is an H(C)-valued function and Zc is the operator of complex conjugation ZCF = F* — X)fc=o-^fc^- This equation is equivalent to the Dirac equation (see 30 ), and after C. Lanczos it was rediscovered (see n ) by a number of people, e.g., 13 and 14 . Equation (32) played an important role in the development of the Clifford algebras approach to the Dirac equation, but it is not free of some important disadvantages which limited its possible applications. First of all, the simple map transforming the classical Dirac equation into (32) is not even C-linear. But the most serious disadvantage is the presence of the conjugation operator Zc which impedes application of quaternionic analysis methods to (32). Obviously, the simplest possibility to eliminate this difficulty is annihilating the corresponding term making m equal to zero. One year after the publication of the Lanczos work, in 1930 there appeared an article by D. D. Ivanenko and K. V. Nikolski 15 in which the Dirac equation for a massless field was represented as an analyticity condition for a function of a quaternionic variable. This observation was used in a number of works (see, e.g., 3 6 ) . The problem remained, how to rewrite the massive Dirac equation in quaternionic terms without undesirable presence of the complex conjugation operator. An important contribution was made in 4 , where the symmetrical analysis of the Dirac equation for a free particle with spin 1/2 and non-zero rest mass was essentially simplified after using its quaternionic analogue which fulfilled this requirement (see also 3 ) . Nevertheless, there remained a very natural problem: to introduce a transformation

155

which would be able to transform the Dirac equation from its traditional form to an appropriate quaternionic one. Not to introduce one more analogue of the classical Dirac equation but to establish an equivalence between this and the quaternionic equation. A solution of this problem was proposed in 20 where a simple matrix transformation which allows us to rewrite the classical Dirac operator (given in its "traditional" form using 7-matrices) applied to C 4 -vectors as a quaternionic operator applied to H(C)-valued functions (and not containing the conjugation operator) was obtained. We will not discuss here a quite lengthy procedure which led to the matrix transformation referring the reader to 20 or 30 . Finally it is obtained in a very simple form, easy to use. It is C-linear to the difference of the Lanczos transformation. We will show only some necessary properties of this transformation and some of its possible applications.

4-2

Quaternionic

reformulation

Let us introduce an auxiliary notation / := f(t, x\, x2, —X3). The domain G is assumed to be obtained from the domain G C R 4 by the reflection X3 —> —£3. The transformation announced above we denote as A and define it in the following way. A function $ : G C R 4 —> C 4 is transformed into a function F : G C 1 4 - H(C) by the rule F = A[$] := - ( - ( $ 1 - $ 2 )io + i($o - £ 3 ) n - ($0 + $3)«2 + i ( $ i + $2)^3) • The inverse transformation A-1

is defined as follows

$ = A'1 [F] = (-»Fi - F2, -Fa - iF3,F0 - iF3,ih

- F2).

Let us present the introduced transformations in a more explicit matrix form which relates the components of a C4-valued function $ with the components of an H(C)-valued function F:

(

0-11

i F = A[} = -1 \ 0 and

0\

0 0 -i 0 0-1 i i 0/

$2

V53;

156

0-i-l 0\ -1 0 0 -i i1 n 0 n 0 _-i. \ 0 i-1 Oj (

$

A

~ ^

(F0\ p2

\ij

We will need the following properties of the transformation A. L e m m a 10 2 7 (Algebraic properties of the transformation A) 1- -47i727371 ^•~1[F] = -A-(2lzA~l[F]

= ixF ;

2. ^7i727372^" 1 [- F ] = -4717b ^[F] = i2F ; 3. -47i727373-4"1[-Fl = -A-y^A^F]

= -i3F ;

4. -47i72737o-4~ 1[F] = -^7o7i7273^" 1 [-f 1 ] = Fh ; 5. -47i7273^1" 1[F] = -iFi2

•

We introduce the following quaternionic operator R := P?(idt +D) + P{{-idt

+ D) - mMia,

where P * •= _M(1±iifc) fc • 2

Using the algebraic properties of A and . 4 - 1 we obtain the following equality: R=-Am2l3®A-1.

(33)

Thus, the operator R is the Dirac operator for a free massive particle of spin 1/2 in a quaternionic form. 4-3

Given energy

We will consider time-harmonic solutions of the Dirac equation (31) or in other words the solutions with a specified energy w. In this case the solution of (31) $ has the form $(t,x)=q(x)eiut, where w e l and q is a C4-valued function depending only o n i = {x\, x2, X3). Prom (31) we obtain the Dirac equation for q: ®u,mq = 0,

(34)

157

where 3

H\;,m := iwyo - ^7fc9fc + iml. k=i

The corresponding quaternionic reformulation of B W i m will be the operator D-g, where ~a := — (iuii\ + mii)- We have the following equality D-s =-AnmzBwA-1.

(35)

Thus, q is a solution of (34) if and only if / := Aq is a solution of the equation D-ctf = 0.

(36)

This fact allows us to apply systematically quaternionic analysis to the study of Dirac's equation. For example, introducing the operator

we obtain the following result which gives a useful integral representation for solutions of (34) in a bounded domain through their boundary values. Theorem 11 2 0 Let q G C ^ G i C 4 ) n G(G;C 4 ), q G kerD w , m (G). Then q(x) = Ku^mlqlix),

xeG.

A similar fact for unbounded domains was obtained in 26 , where for this purpose an appropriate radiation condition was proposed. Namely, we have the following Theorem 12 Let q G C X (R 3 \G;C 4 ) n G(M 3 \G;C 4 ), q G k e r B u , m ( R 3 \ G ) , and let q satisfy the radiation condition vq(x) - (w7o-m)-^«7(a;) =o ( —r ) , \x\ \\x\J where x^ := J2k=i xklk-

|x| -> oo

(37)

Then

q{x) = -K^m[q\{x),

x G (R 3 \G).

These statements together with the quaternionic Plemelj-Sokhotski formulas are very useful in the analysis of boundary value problems for the Dirac equation. For example, in 2 0 (see also 30 ) a complete solution of the MIT bag model problem was given. We expect to obtain similar results for the mathematical model of the Casimir effect 34 . Methods of quaternionic analysis can be useful not only for solving problems related with free Dirac's particles but also for obtaining solutions of the

158

Dirac equation with different types of potentials. Let us consider formally the following Dirac operator 3

V

D := D + gacEi + igeilo + ^

Ak-yk + #Ps7o75-

k=i

Here gsc is the scalar potential, gei is the electric potential, J2k=1 Ak^k is the magnetic potential and gps is the pseudoscalar potential. Usually these potentials are considered in a separate way. We put them together in order to show their quaternionic counterparts. We have the following equality 24 v R=

v _ -^717273©^ ,

where R:=R-

(igps + ~B)I + A f - ' © * * 1 - 5 " ^ ,

and B has the form B = A\i\ + ^2^2 — ^313. The quaternionic technique appears to be convenient for finding exact solutions of the Dirac equation with potentials. In 2 1 ! 2 2 ) 2 8 new classes of exact solutions were obtained, and in all these cases instead of traditional analytic methods like the separation of variables method, only the algebraic properties of complex quaternions were used. Acknowledgement 13 This work was supported by CONACYT Project 32424-E, Mexico. References 1. C. Athanasiadis, G. Costakis and I. G. Stratis, Reports on Mathematical Physics. 45, 257 (2000). 2. V. M. Babich and V. S. Buldyrev, Short- Wavelength Diffraction Theory. Asymptotic Methods (Springer, Berlin, 1991). 3. A. V. Berezin, Yu. A. Kurochkin and E. A. Tolkachev, Quaternions in relativistic physics. (Nauka y Tekhnika, Minsk, 1989) (in Russian). 4. A. V. Berezin, E. A. Tolkachev and F. I. Fedorov, Doklady Akademii Nauk BSSR. 24, 308 (1980) (in Russian). 5. S. Bernstein, in Proceedings of the symposium Analytical and numerical methods in quaternionic and Clifford analysis, eds. W. Sprossig and K. Giirlebeck (Seiffen, 1996). 6. S. Bernstein and K. Giirlebeck, Complex Variables. 38, 307 (1999). 7. N. N. Bogoliubov and D. V. Shirkov, Quantum fields. (Fizmatlit, Moscow, 1993) (in Russian).

159

8. F. Brackx, R. Delanghe and F. Sommen, Clifford analysis. (Pitman Res. Notes in Math., London, 1982). 9. R. Castillo and V. V. Kravchenko, El problema de la extendibilidad del campo electromagnetico en los dominios no acotados. in Memorias del 5to Congreso Nacional de Ingenieria Electromecanica y de Sistemas (Mexico City, 2000). 10. D. Colton and R. Kress, Integral equations methods in scattering theory. (John Wiley and Sons, N. Y., 1983). 11. A. Gsponer and J.-P. Hurni, Comment on formulating and generalizing Dime's, Proca's, and Maxwell's equations with biquaternions or Clifford numbers. (To appear). 12. K. Giirlebeck and W. Sprofiig, Quaternionic analysis and elliptic boundary value problems. (Akademie-Verlag, Berlin, 1989). 13. F. Giirsey, Phys. Rev. 77, 844 (1950). 14. D. Hestenes, Space-time algebra. (Gordon and Breach, New York, 1966, 1987, 1992). 15. D. D. Ivanenko and K. V. Nikolski, Z. Phys. 63, 129 (1930). 16. K. V. Khmelnytskaya, V. V. Kravchenko and H. Oviedo, Telecommunications and Radio Engineering, to appear (2001). 17. V. G. Kravchenko, V. V. Kravchenko and B. D. Williams, A quaternionic generalisation of the Riccati differential equation. (Available from xxx.lanl.gov.arXiv:math-ph/0101010; submitted for publication). 18. V. V. Kravchenko, On the relation between holomorphic biquaternionic functions and time-harmonic electromagnetic fields. (Deposited in UkrlNTEI, 29.12.1992, #2073 - Uk - 92) (in Russian). 19. V. V. Kravchenko, Russian Academy of Sciences. Doklady Mathematics. 51, 287 (1995). 20. V. V. Kravchenko, Zeitschrift fur Analysis und ihre Anwendungen. 14, 3 (1995). 21. V. V. Kravchenko, J. of Phys. A. 3 1 , 7561 (1998). 22. V. V. Kravchenko, Zeitschrift fur Analysis und ihre Anwendungen. 19, 655 (2000). 23. V. V. Kravchenko, Zeitschrift filr Analysis und ihre Anwendungen.19, 903 (2000). 24. V. V. Kravchenko, Problems and Methods in Mathematical Physics. The Siegfried Prossdorf Memorial Volume, eds. J. Elschner, I. Gohberg and B. Silbermann (Birkhauser Verlag, Operator Theory: Advances and Applications, vol. 121, Basel, 2001).

160

25. V. V. Kravchenko,Quaternionic reformulation of Maxwell's equations for inhomogeneous media and new solutions. (Submitted for publication in Zeitschrift fur Analysis und ihre Anwendungen). 26. V. V. Kravchenko and R. Castillo, An analogue of the Sommerfeld radiation condition for the Dirac operator' (Available from http://arXiv.org/abs/math-ph/0008042, to appear in the Mathematical Methods in the Applied Sciences). 27. V. V. Kravchenko, H. Malonek and G. Santana, Mathematical Methods in the Applied Sciences. 19, 1415 (1996). 28. V. V. Kravchenko and M. Ramirez, Mathematical Methods in the Applied Sciences. 23, 769 (2000). 29. V. V. Kravchenko and M. V. Shapiro, Acta Applicandae Mathematicae. 32, 243 (1993). 30. V. V. Kravchenko and M. V. Shapiro, Integral representations for spatial models of mathematical physics. (Addison-Wesley Longman Ltd., Pitman Res. Notes in Math. Series, v. 351, London, 1996). 31. A. Lakhtakia, V. K. Varadan and V. V. Varadan, Time-harmonic electromagnetic fields in chiral media. (Lecture Notes in Physics 355, Springer, Berlin, 1989). 32. C. Lanczos, Zeits. fur Phys. 57, 447 (1929). 33. A. Mcintosh and M. Mitrea, Mathematical Methods in the Applied Sciences. 22, 1599 (1999). 34. V. M. Mostepanenko and N. N. Trunov, The Casimir effect and its applications. (Oxford Science Publications, 1997). 35. A. Sommerfeld, Atom structure and spectra. (Mir, Moscow, 1956, v. 2) (in Russian). 36. V. Soucek, Variables, Theory and Applications. 1, 327 (1983). 37. B. Thaller, The Dirac equation. (Springer-Verlag, 1992). 38. B. Williams, Una generalizacion cuaternionica de la ecuacion de Riccati. (Master thesis, National Polytechnic Institute, Mexico City, 2001). 39. M. S. Zhdanov, Integral Transforms in Geophysics. (Springer-Verlag, Heidelberg, 1988).

161

INTEGRAL TRANSFORMS METHOD IN T H E C O N J U G A T I O N P R O B L E M S OF ELECTROMAGNETIC FIELDS N. B. PLESHCHINSKII Kazan State University, Department of Applied Mathematics, 18 Kremlevskaya Str., Kazan 420008 Russia E-mail: [email protected], [email protected] Some problems connected with application of method of integral transformations in problems of propagation and diffraction theory of electromagnetic waves are considered.

1

Introduction

As it is known, many problems of electrodynamics in the most general (vectorial) case can be formulated as boundary value problems for Maxwell's system of equations. If a required field in homogeneous and isotropic medium does not depend on one of spatial coordinates (the scalar case), then Maxwell's system breaks up into two independent subsystems. Thus, all components of solutions of everyone subsystem are expressed through solutions of the potential Helmholtz equation. Suppose it is necessary to find an electromagnetic field in complex field, which can be separated into simple subdomains, each of which is filled by homogeneous and isotropic medium. The tangent components of electrical and magnetic vectors should be continuous at the media interfaces. On a boundary with metal (an ideally conductive medium) the tangent components of electrical vector should vanish. Thus, conjugation problems of electromagnetic fields represent by themselves conjugation problems for solutions of the Helmholtz equation or immediately of Maxwell's system of equations. Under solving the conjugation problem of electromagnetic fields the method of partial fields (or method of seaming) is widely used. This method consists in the following. Suppose a solution of Maxwell's system of equations or solution of the potential equation can be represented in each subdomain of complex domain in the form of a series with unknown coefficients or in the form of an integral with an unknown density. Then it is possible to obtain summatorial or integral equations equivalent to the initial problem from the conditions at the media interfaces. If it is possible to express the components of the unknown field in each domain through some analytical functions, then the conjugation problem of fields can be reduced to the conjugation problem

162

for analytic functions. The method of integral transforms is widely applied under solving boundary value problems of mathematical physics. Recall that the basic idea of the method consists in the following: if the required function satisfies some differential equation, then its Fourier transform satisfies an algebraic equation. Further, let us assume that the integral Fourier transform is defined by formula +00

F : f(x) h- F(0 = - ~ J

f(x)e^dx.

—00

For such a transform an inversion formula +00

1

F- : F(0 -> /(*) = ~= J F(Z)e-**d£ —00

and commutation formula FD = MF with operators D : f(x) H- / ' ( * ) ;

M : F(0 -> i£F(t)

take place. The Fourier transform has deep mathematical and physical sense: actually any function can be represented as linear combination of simple functions of the form exp(—i£x). Such functions are used to be called elementary waves or harmonics. In the well known monographies l and 2 solutions of some key problems of an electrodynamics are obtained by classical Fourier transforms method and by methods close to it. But in some cases the justification is not quite strict. The analysis of publications of last years shows that under solving the problems of electrodynamics the application of technique of integral transforms in spaces of generalized functions allows to justify many operations, which were justified earlier more than likely by physical intuition than by mathematical proofs. The statements on existence of solutions of some boundary value problems in the electromagnetic fields theory were obtained also rather recently with the help of modern methods of the partial differential equations

163

theory. For example, complete research of diffraction problems of electromagnetic waves on metal screens, based of the theory of pseudodifferential operators, is given in the monography 3 . In the book 4 the Fourier transforms method is used to reduce many boundary value problems of mathematical physics to boundary value problems for analytical functions. A contraction of classes of given and unknown functions to make the justification more clear has not yielded a possibility to consider solutions such as spreading waves. Nevertheless, expansion of the method on classes of distributions is prepeared: last chapter is devoted to the equations in spaces of generalized functions. It is easy to illustrate that it is advisable to pass to distributions by applying the Fourier transforms method for problems of electrodynamics. Let us consider the Helmholtz equation Au(x, y, z) + k2u(x, y, z) = 0 with real parameter k. The Fourier transforms of unknown functions satisfy an equation

(k2-e-v2-t2)u(t,v,0

= o,

its non-trivial particular solutions in space of distributions of slow growth S' are £/(£, V, 0 = const 5(S - a, rj - (3, C - 7),

a 2 + /?2 +

2 7

= fc2,

and their pre-images are plane waves u{x,y, z) = const e*(°™+/»»+7») where the vector (a, (3,7) determines the direction of the propagation of a wave. Note that it is easy to obtain these solutions by method of separation of variables, but it is difficult to do this by method of classical Fourier transforms. A boundary value problem of electrodynamics is usually called a coordinate problem if boundaries of partial domains are coordinate lines or surfaces. The main key problems of electrodynamics, solutions of which can be obtained in a closed form, are considered in the books x and 2 . Thus, three strict methods are applied: the method of Wiener-Hopf, the method of Jones and the method of residues. Also on the basis of these methods the technique of approximate solution for more complex problems is constructed. The method of Wiener-Hopf in problems of electrodynamics consists in the following: the initial problem is reduced to an integral equation of convolution type (as a rule, by the method of the Green function), which is being reduced to the functional equation by integral Fourier transform. This equation establishes the connection between values of two required analytic functions in some band of a complex plane but not on the common boundary

164

of domains of analyticity, as it is accepted in linear conjugation problems (or Riemann-Hilbert problems). Solving the functional equation, the factorization method is used. In the Jones method 2 , §2.2, 1, chapter 3, §7 the integral transform is applied immediately to initial equation with partial derivatives. As a result, one obtains just the same functional equation of Wiener-Hopf. The method of residues * enables one to find explicit solutions of some infinite systems of linear algebraic equations, evaluating residues of auxiliary functions of complex variable, constructed by special rules. After properties of the Fourier integral with complex values of parameter £ were investigated, it become possible to apply the Fourier transforms method for solving boundary value problems for the differential equations in classes of functions not vanishing at infinity. The theorem of Wiener-Paley 5 has a fundamental meaning. By this theorem, the Fourier transforms of one-sided functions from the space L2 are analytic functions in the corresponding halfplanes. In distributions theory a similar statement takes place 6 : if an analytic function f(z), z = x + iy, has a restricted growth with respect to y in a tubular domain, then its limit, as y —> 0 in the sense of weak convergence in space of distributions, is a distribution on an axis Imz = 0. Further, the converse statement will be extremely important for us: the distribution of slow growth at infinity can be analytically continued from the real axis into upper or lower half-plane, if its carrier belongs to a positive or negative semiaxis correspondingly. Let us consider simple examples. Let a be a complex number, Re a > 0. It is easy to verify that if f(x) = {e~ax, x>0;

0, x < 0} ,

then y/2n£ + ia' Let a be real number. If f(x) - {eiax,

x>0;

0, x < 0} ,

then

This formula can be obtained in the following way: one has to multiply function f(x) by e~ex (or by e+ex), then to calculate the Fourier transform and

165

to pass to a limit as e —» +0, but in the sense of convergence in the space of distributions. Let us consider now the technique of obtaining integral and summatorial representations of solutions of the Helmholtz equation and of Maxwell's system of equations based on a solution of auxiliary overdetermined Cauchy problems. Also let us consider methods of passing from initial boundary value problems of electrodynamics to the equivalent equations. 2

The Cauchy problem for a half-plane

Let us consider the Cauchy problem for the Helmholtz equation in the upper half-plane R2^ d2u

d2u

,9 .

k is a real or complex number, u{x,0) =u0(x),

—(x,0)=ui(x).

(2)

As it is known, the Cauchy problems for elliptical equations with partial derivatives are overdetermined. On the boundary of domain it is enough to give either the value of a required solution, or the value of its normal derivative, but not both these functions simultaneously. Therefore, first of all it is necessary to seek the conditions for boundary functions which make available the existence and uniqueness of a solution. Suppose a required solution u(x, z) 6 Ll°c{R\). But to justify the Fourier transforms metod it is necessary to seek solutions of the Cauchy problem in more wide class, i.e. in the space of distributions of slow growth at infinity. Suppose the distribution u(x,z) is from the Sobolev space H[oc(R2_) (see 3 ); then it and its derivative with respect to z will have correctly defined traces on the line z = 0. Let us define a unique branch of multivalued function

7(0 = \A 2 -£ 2 = y/(k-i){i + k) of complex variable £. Let A; be a real number. Let us carry out a section of the complex plain connecting the branchpoints —k and +k along a segment of the real axis [—k, +k]. Let function 7(£) transform the 1-st quadrant of complex plane to

166

the 4-th quadrant, and the 2-nd quadrant to the 1-st one. Denote by 7 + (£) the limiting value of function 7(£) from the upper half-plane to the real axis,

r g 6 (-co, -k) -HV^F^P;

7 + (0 = < £ € (-*,+*) + V ^ g ; I £ G (+fc, +00) - t ^ 2 - fc2If A; = koezipo, (fo > 0 is a complex number, then assume V ^ 2 - ? = e+i*°^kl-{e-iv°£)2

= e+iipo 7(e- <w ȣ),

where 7(£) is defined above function for real k = ko- For values of new function we will use the same symbol 7(£). Thus, for complex k the one-valued branch of the functions 7(£) is chosen in a plain with a section from —k up to +k along a segment passing through the branchpoints. Theorem 1 The distribution u(x, z) is a solution of the problem (1), (2) for real k iff

MO+il+(OMO=0,

Z>+k,

(3)

and its Fourier transform u(£, C) satisfies an equation

(k2 ~e-c2)««,o

= 4 = M O -KMO]•

(4)

Proof Let u(x, z) be a solution of the problem. Redefine it by zero point up to the whole plain and pass in the equation (1) to a Fourier transform by both variables in view of boundary conditions (2). Then we obtain that the Fourier transform u(£, C) satisfies the equation (4). Extend the equation (4) by variable C to the upper half-plane of the complex plane C, and consider variable £ as a parameter. Under £ < -k and £ > +k a polynomial fc2-£2-C has complex roots ±7 + (£). As the distribution u(£, Q can be analytically continued to the upper half-plane by £, the left-hand side of the proceeded equation vanishes for C, = ± 7 + ( £ ) , laying in the upper half-plane. Then the right-hand side of the equation vanishes at these dots, i.e. the equallities (3) are fulfilled. 0 It is easy to show that if the conditions (3) are satisfied, then a solution of the Cauchy problem is not unique. For a uniqueness of a solution it is necessary to give supplementary the radiation conditions (the conditions at infinity).

167

Let dependence of field on time be of the form exp(iwt). We will distinguish two classes of solutions of the Cauchy problem (1), (2): solutions outgoing to infinity, for which the condition

«i(0 + il+ {0M0 =0,

-k < £ < +k

(5)

is fulfilled, and the solutions coming from infinity, for which « i ( 0 - * 7 + ( 0 « o ( 0 = 0,

-k<£<+k

(6)

is satisfied. If the condition (5) is fulfilled, solution of the Cauchy problem u(x, z) has no harmonics coming from infinity. And if the condition (6) is satisfied, solution of the Cauchy problem has no harmonics outgoing to infinity. In terms of the distribution theory it is possible to give equivalent definitions. For example, for waves outgoing to infinity the following condition sing supp u(£, C) D {C < 0} = 0. should be fulfilled. If k = fcoe*Vo is a complex number, then the Fourier transform of the unknown solution of the Cauchy problem satisfies the equation (4) also. Thus, the distribution u(£, () will be analytically continued by ( to the upper halfplane iff the conditions « i ( O - * 7 + ( O « o ( £ ) = 0,

£<0,

«i(0+*7+(0«o(0=0,

£>0

are carried out. Since 0, we get from this the conditions (3) of the Theorem 1 and supplemental radiation condition (5). Thus, the principle of limiting absorption is valid for the.Cauchy problem in a half-plane. Denote by 7o(0 = { l £ l > *

:

+We

~ k2;

|£|
(this function is not a limiting value of any function, analytical in the upper half-plane), then it is possible to write down the conditions (3), (4) and (5) in the form of one condition MO

- »7o(0 MO

= 0,

- o o < <£ < +oo.

(7)

168

For solutions outgoing to infinity the condition (7) sets the connection between distributions UQ(X) and u\{x) given on a straight line z = 0 in terms of their Fourier transforms. Having applied the inverse Fourier transform, we obtain Ul(x)

= - - L J i 7o (0 «o(0 z~iix <%, +oo

u0[

The right-hand sides of these equalities define the mutually inverse pseudodifferential operators with symbols —i/7o(£) and J7o(0 respectively. Corollary 1 Solution u(x, z) of the equations (1) belongs to the class of solutions outgoing from the straight line z = 0 into a half-plane z > 0 and satisfies the boundary conditions (2) iff +oo

ui(x) = /

U0(T) K0(T,

X)

dr,

— oo

(8) +oo

uQ{x) = I ui(r)

Kx{T,x)dT,

where K0(r,x)

=

K1(T,X) =

^-^—H[2\k\r-x\),

-H^(k\T-x\).

For a lower half-plane all reasonings can be carried out just in the same way. The formulas corresponding to this case can be obtained from the given above if one changes a sign of function 7 + (£)3

The conjugation problems for two half-planes

Let us study two conjugation problems on a straight line by the seaming method. Consider scalar electromagnetic fields in half-planes. Let us use the obtained above representations of solutions of the Helmholtz equation.

169 Assume that field components do not depend on coordinate y. Consider the case of TE-polarization only. Each of problems under consideration can be formulated as a boundary value problem for the Helmholtz equation d2u

d2u

l9,

,

.

.

^ 2 + ai2+ f e2 (*M*>*) = o

0)

+

with piecewise constant coefficient k(z) = {z > 0 : k ; z < 0 : k~~} under different conjugation conditions on the straight line z = 0. Let us restrict ourselves to a case when fc* are real numbers. Let electromagnetic TE-wave coming from infinity fall on a plain z = 0. It is necessary to seek reflected and refracted waves. Such problem is reduced to a jump problem for the Helmholtz equation: one has to seek a solution of the equation (9) for z > 0 and z < 0 in a class of outgoing to infinity solutions satisfying for — oo < x < +oo the conditions u(x, 0 + 0) - u(x, 0 - 0) = a(x), (10) ^(x,0

+ 0)-^{x,0-0)

= b{x).

Theorem 2 The solution of the jump problem (9), (10) has the form

v^ J

7+(0+7~(0

for z > 0 and z < 0, where

7±(0 = {Kl > ** : +»VC2 - (^) 2 ; 1^1 < ^ :-s/^*) 2 - ^2}. Really, from the Theorem 1 follows that boundary distributions 4 ( 0 satisfy the equalities

4 (o - ii+(04 (0 = 0, «r (0+H~ (o«o (o = o. Conditions of the jump problem in the Fourier transforms have the form

4 ( 0 - «o (0 = «(0> 4 ( 0 - «r(0 = KOHaving solved the system of four linear equations, we obtain the Fourier transforms of boundary distributions and by formula (4), we get the Fourier transforms of required solutions of the jump problem.

170 Formulas (11) can be used to prove the evident from physical point of view fact that under fall of a plane wave on the plane media interfaces, the reflected and refracted waves will be plane waves also. Corollary 2 For e± = e solution of the jump problem (9), (10) has the form +oo

—oo

The jump problem for the Helmholtz equation in the plane stratified medium is solved by similar method. Let an ideally conductive infinitely thin metallical strip a < x < /? be placed in the plain z = 0, separating two media z > 0 and z < 0 with dielectric permittivities e + and e~. It is necessary to seek a field which arises under diffraction of the electromagnetic wave, falling from the domain z > 0 on the plain z = 0. Let us seek solutions of the Helmholtz equation for z > 0 and for z < 0 in class of solutions outgoing from the straight line z = 0 to infinity in the form of solutions of the Cauchy problem for the upper and the lower half-planes with boundary distributions u0 (x) and uf(x). Denote by M = (a,/3) and by N = (-co, a) U (/?,+co). At the media interfaces z = 0 the following conditions should be carried out u£(x)

+ u°(x)

u£(x) + UQ(X) = UQ(X),

= 0,

V.Q(X) = 0 ,

x e

M,

uf(x) + it?(x) = Ui(x),

x e N.

Let us consider for simplicity that e± = e. From these conditions and formulas (8) follows Lemma 1 The diffraction problem of electromagnetic wave on a metallic strip is equivalent to the paired equation u£(x) = -UQ(X),

xeM;

wf(x)=0,

x € N

or UQ(X)=0,

x G M;

uj~(x) = u°(x),

x G N.

By this, taking into account the connections between boundary distributions in the overdetermined Cauchy problem, it is possible to obtain the following statement.

171

Theorem 3 The diffraction problem on a strip is equivalent to integral equations J uf (r) Ki (r, t) dr = -u°0 (t),

t £ M,

(12)

x e N,

(13)

M

Ua(t)K0{t,x)

u+{x) =

[U!(T)

-

dt = -u°(x),

f KfaQKofax)

v%(t)K0(t,x)

dt,

dt dr-

(14)

xG M,

M

d£«0(*) = / « o ( O j Ko%T)KX(T,t) - Iu\{T)Kx{T,t)

dr,

dT xeN.

(15)

N

Note that equations (12) and (13) are equations of 1-st kind, and equations (14) and (15) are of the 2-nd kind. The equation (12) has a logarithmic singularity in the kernel, and the equation (13) is a hypersingular one. Under reducing the diffraction problem on a strip to an integral equation, it is possible to use an explicit solution (11) of the jump problem on the straight line. Theorem 4 For e+ ^ £~ the diffraction problem on a strip is equivalent to the integral equation

6(0 e~^ — f- (0+7"(0

x

V2TT J M

d£ = uu0{x)-

7+

+oo

«8(07"(0 i r _u >/2?r J 7+ (0+7"(0

i£x d£+

172

+^ =

/ - ^ L - e - ^ ,

x € M.

(16)

For e^1 = e equation (16) becomes more simple. With the help of similar reasonings it is possible to get the integral equations in the problem of diffraction of the electromagnetic wave on a barrier in the planar waveguide with metallic walls. 4

The Cauchy problem for a quarter of a plane

Let us show how one can use the Fourier transforms metod in more complex case. Let us seek solutions of the Helmholtz equation d2u d2u , o , s , < Q-2 + -Q-1 + k2u{x,z) = 0, x>0, z>0 (17) in a quarter of a plane (k is a real number), satisfying boundary conditions u(0,z) = a0(z),

—(0,z)

= ai(z), (18)

u(x,0)=b0(x),

—(x,Q) = b1(x)

(the overdetermined Cauchy problem for a quadrant). Denote by Roman digits the quadrants of complex planes of variables £ and C- Let bi-half-plane D++ = {Re£ > 0} x {ReC > 0}. Theorem 5 Distribution u(x,z) is a solution of the problem (17), (18) iff the equality is fulfilled ai(()-i£a0(O+b1(i)-i(bo(i)=0 (19)

\/(i,OeD++\k2-£2-(2

= o,

thus, the distribution ii(£, £) satisfies the equation (fc2 _ £2 _ C 2 ) u ( £

c) =

1 rfli(c) _ - £ a o ( 0

+ h { 0

_

i C M 0

] _

(2Q)

V Z7T

Proof. It is easy to see, that the Fourier transform u(£, Q of the required solution satisfies the equation (20). Since the distribution M(£,C) is analytically continued by each component from the reala axis to the upper complex half-plane, i.e. from a skeleton of boundaries of the bi-half-planes in D++, the equation (20) also can be analytically continued to this domain. Then, if k2 - i2 - C2 = 0 for some (£,<) € D++, then the right-hand side of the continued to complex domain equation (20) will vanish. -0"

173

Corollary 3 Distribution u(x,z) the equalities are fulfilled

ai(-i(i))-iia0(-1(£))

is a solution of the problem (17), (18) iff

+ b1(£) + i1(£)b0(i) = o, £el,

ai(7(0)-^oo(7(0) + 6i(0-i7(0fto(0=0,

tell.

(21) (22)

Proof. Really, let C G I. Then 7 (C) G IV and C = -7(C) G / / . If £ € II, then C = 7(C) G / . Therefore the condition (19) is equivalent to the conditions (21) and (22). If C belongs to a positive imaginary semiaxis, i.e £, = ir], r\ £ (0,+co), then 7(C) belongs to the real axis but does not belong to quadrants I and II.

0 By similar way one can obtain the following equalities ai(C)+t7(C)ao(C) + 6 1 ( - 7 ( 0 ) - » C f t o ( - 7 ( 0 ) = 0 , ai(C) - *7(C)ao(C) + 6i(7(0) ~ *CM7(C)) = 0 ,

C G I, C G /J,

(23) (24)

equivalent to (21) and (22). Further operatings are reduced to the following. The left-hand sides of equalities (21) - (24) should be continued on boundaries of quadrants. Then we get relations between the Fourier transforms of boundary functions which make available the existence of a solution of the Cauchy problem. Conditions at infinity (radiation conditions) in the case of the first quadrant can be formulated as follows. In solution of the Cauchy problem the waves outgoing to infinity are absent iff a i ( - 7 + ( 0 ) ~ *Cao(-7 + (0) + M O + H+(C)^o(C) = 0 , 5

C G (-k, 0).

The Cauchy problem for a half-space

Under researching vector-valued boundary problems of electrodynamics, also it is convenient to use the representations of solutions of Maxwell's system of equations obtained under solving supplementary Cauchy problems by the Fourier transforms method. Consider the Cauchy problem for Maxwell's system of equations in the upper half-space i£j_ = {z > 0} rot H = iioeoe E,

rot E = —iw/xoM H,

(x, y, z) G R\

(25)

with real e, fi. Assume the dependence of a field on time has the form exp(iwt).

174

As well as in a scalar case, we consider solutions of the system (25) in space L[oc(R%). We call solution (E,H) of the system (25) as outgoing from a plain z = 0 into a half-space z > 0, if each of its components F(x, y, z) = Ex, Ey, EZ,HX, Hy, Hz 1) is the distribution of slow growth, suppF(x, y, z) C {z > 0} and 2) sing supp F(£, V, 0 n {C < 0} = 0.

(26)

Let us seek in R+ a solution of Maxwell's system of equations by given on a plain z = 0 values of tangent component of vectors E and H, i.e. a solution satisfying the conditions [z0,E](x,y,0)

= e(x,y),

[z0,H}(x,y,0)

= h(x,y),

(27)

where ZQ is a unit vector of an axis z. This problem is overdetermined also. Let A; be a wave number, k2 = ui2/j,ofJ.£o£- Denote by 2

2

7(t,v) = Vk -e-V

£2 + r)2>k2:

+ V2~k2;

+i^e

£2+n2
-y/tf-p-rp.

Theorem 6 Solution (E, H) of the system (25) belongs to the class of solutions outgoing from the plain z = 0 into a half-space z > 0 and satisfies the boundary conditions (27) iff for the Fourier transforms of spurs of its components on the plain z = 0 the equality is fulfilled Loe0s 7 (£, n) e(£, n) + P(£, n) h(£, r,) = 0

(28)

P(Z, V) e(£, v) ~ ^ o M lit

(29)

or V) HZ, r?) = 0,

where

P(^H(/V2_^f). Thus E

ti> V, 0 = , , "lt , A e(£, V) + —B(£, UJ£Q£ C + l&v) i

H& V, 0 = , , "lc ,

V)

h(£, r,)

V B ^ ' ^ e(^'7?) + ^ ^ ' ^

175

where ,

/O 0 0 0

B(£,V)=

Note that P2{^,n) = —k2^2(^,r}) I, I is & unit matrix (it is possible to introduce such normalization that P2{£,T]) = I). The equalities (28) and (29) follow each from another, this is easy to test. From these equalities one can obtain the following statement, defining dependence between given on the boundary of the domain values of tangent components of the field in terms of their Fourier transforms. Corollary 4 If the Fourier transforms of distributions e(x,y), h(x,y) satisfy the equalities (28), (29), then e{x,y) =

/

K(xi,yi;x,y)h(x1,yi)dx1dy1,

(30)

K(xi,yi;x,y)e(xi,yi)dxidyi,

(31)

w£o£ J J

h(x,y) =

/ w^o/i J J

where K{xi,y!;x,y) = _J_ f f 2

I

V*) J J 7&V)

=

p(£ „) ei{xi ~ x)(, + i{yi ~ V)V fjc dn ( V) 4V

*'

'

For solutions outgoing from a plain z — 0 into a half-space z < 0 the condition (26) is replaced. For solutions outgoing into the lower half-space similar statements are valid. It is sufficient to change a sign of the function 7(£, rj) in all above given formulas. 6

The conjugation problem for two half-spaces

Let e(z) = {z > 0 : s+; z < 0 : e _ } . Consider the jump problem: one has to seek solutions (E+,H+) for z > 0 and (E-,H-) for z < 0 of Maxwell's system of equations with piecewise constant coefficient rot H = iujeoe{z) E,

rotE = -iujfi0/j, H,

z ± 0,

(32)

from classes of solutions outgoing from a plain z = 0 to infinity and satisfying o n z = 0 the conditions [z0,E+-E-){x,y,0)=a(x,y), where a(x,y),

[z0,H+ - H_}(x, j/,0) = b(x,y),

b{x,y) are given vectorial functions.

(33)

176

Theorem 7 Solution of the jump problem exists, is unique and E±(Z,T,,0

= , , \ , 7 , L\Ae±(Z,r,) C±7±(?> ?)

+ —^— B(£,r,) weo£±

h±&7,)], -I

ff±£.f, 0 = , • * , \ —B(£,»?) e±(e, 77) + A h±& rj)], C±7±(C,»?)l- w/ioM J where e±(Z,v) =

±etta(Z,n)--^P(Z,T,)b(Z,r,) /> N ^ £ ° 77—c e+7+U,»7)+e-7-U,J?) 1

h±(tv) =

:

P(^VH^V)±1TK^V)

^

The jump problem is equivalent to the problem on the fall of the electromagnetic wave at a plane media of interfaces. Let M be a metallic screen placed in a plain z = 0 and N be its supplement up to the whole plain. Let e± = e, fi± = fj, and EQ(X, y, z) be the electric field strength of a wave falling down on the screen. (It is convenient to consider that the external field is given both from above and from below.) It is necessary to seek the solution of Maxwell's system of equations in the upper and lower half-spaces, outgoing from a plain z = 0 to infinity, satisfying the conditions [z0,E±+E0}(x,y) [zo,E+-E-](x,y)=0,

= 0, (x,y)eM,

[z0,H+-H_)(x,y)

(34)

= 0, (x,y) € N.

(35)

As well as in scalar case, it is possible to obtain by formulas from Corollary 4 integral equations equivalent to the diffraction problem. Theorem 8 The diffraction problem at the plane screen is equivalent to vector integral equations =F / / K(xi,yi;x,y) w£0e J J

h±(x1,y1)

dxx dyx = -e0(x,y),

M

and h±(x,y) +

h±(xi,y1)LN(xi,yi;x,y)dxidy1 M

=

(36)

177

=T

/ / K(xi,ynx,y)e0(xi,yi)dxidyi,

(37)

M

LN(xi,yi\x,y)

=j^ //

K(xi,yi;x2,y2)K(x2,y2;x,y)dx2dy2

N

concerning the spurs e±(x, y), /i±(a;, y) on a plain z = 0 of normal component of solutions of Maxwell's system of equations (E±,H±).

References 1. R. Mittra, S. W. Lee, Analytical Techniques in the Theory of Guided Waves (The Macmillan Company, New York, 1991). 2. B. Noble, Methods Based on the Wiener-Hopf Technique (Pergamon Press, London - New York, 1958). 3. A. S. Ilyinsky, Yu. G. Smirnov. Electromagnetic wave diffraction by conducting screens (Pseudodifferential operators in diffraction problems) (VSP, Zeist, 1998). 4. F . D. Gakhov, Yu. I. Chersky, Convolution type equation (Nauka, Moscow, 1978) [in Russian]. 5. R. Paley, N. Wiener, Fourier Transforms in the Complex Domain (A.M.S., New York, 1934). 6. V. S. Vladimirov, Methods of several complex variables functions theory (Nauka, Moscow, 1964) [in Russian]. 7. N. B. Pleshchinskii, Proc. Lobachevsky Math. Center (Kazan), 7, 153185 (2000) [in Russian].

178

R E M A R K S ON F U N D A M E N T A L MATRICES (GREEN'S TENSORS) IN E L A S T O D Y N A M I C S , PIEZOELECTRICITY A N D CRYSTAL OPTICS N. O R T N E R Institut

fur Technische

Mathematik, Geometrie und Bauinformatik Universitat Innsbruck Technikerstr. 13, A-6020 Innsbruck, Austria E-mail: [email protected]

In Section 1 we outline different formulae for fundamental matrices of hyperbolic systems of differential operators with constant coefficients. In Section 2 we state the analogue of the classical Herglotz-Petrovsky formula for the fundamental matrix of strictly hyperbolic systems. Some hints at the proof and at connections with the formulae of Section 1 are given. In Section 3, the fundamental matrix of the system of transversely isotropic media is computed by means of the HerglotzPetrovsky formula (for systems) - under the hypothesis that the elasticity constants imply the decomposability of the determinant operator. The fundamental matrix of electrodynamics in uniaxial media is a special case thereof (Proposition 2). The proof of the Theorem in Section 2 and more details on the computations in Section 3 will be published in a forthcoming joint paper with Peter Wagner.

1 1.1

Formulae for fundamental matrices of hyperbolic systems of partial differential operators Burridge's formula for the fundamental matrix of anisotropic elastodynamics

The fundamental matrix E G [P'(R 4 )] fulfills

of anisotropic elastodynamics

gI3d?E - K(V)E

= I3S

(1)

where 73 denotes the unit matrix, g the mass density and

3 K

W

= (Yl cpqrsdqds)

(dA

, v = \d2

= (9,),=i,2,3.

The system (1) has the form P(dt, V ) £ = AodfE + A2(V)E

= I35

(2)

where An is a constant 3 by 3 matrix and A2CV) is a 3 by 3 matrix of homogeneous, constant coefficient, second order differential operators in the space

179

variables. The positivity of g and the positive definiteness of the tensor cpqrs of elasticity constants imply the hyperbolicity of P(dt, V) in time direction. Hence, the fundamental matrix E of P ( 9 t , V) with support in t > 0 is uniquely determined. In 1967, Burridge 1, (p. 45, (5.4)) represented E by the following integral over the slowness surface S :

E{t,x) = --±-J2

I AN^)A^f

5'{t - ?*) dg(0, * > 0, a € R3 (3)

Due to the positive definiteness, the matrix K(£) has 3 positive eigenvalues (if £ ^ 0) which define v%(£) by: det(gv2N(0h - K{£)) = 0. The level sets SN = u^ x (l) = {£ G R 3 / U J V ( £ ) = 1} make up the 3 parts of the algebraic surface S given by the equation detfcla - K(0) The vectors Ajv(C)

are

= d e t P ( l , 0 = 0, ^ G R 3 .

normalized eigenvectors of K(£), i.e., they fulfill

(ev2N(0h - K(0)AN(0

= o, \AN(£)\ = i. 3

dS denotes the surface measure on S = \J SN- The appearence of the kernel JV=1

5'(t — £,Tx) implies that the integral in formula (3) is, in fact, the derivative of a "line"-integral over the intersection of the plane t — £,Tx = 0 in R | with the slowness surface S. The integration in (3) can be transformed from S to the unit sphere S2 : In this form formula (3) is presented in 12 (p. 277, (4.17)) and 9 (p. 61, (9.23)). 1.2

"Line"-integral representations for fundamental matrices of piezoelectricity

The equations of piezoelectricity have the form (2) if AQ is replaced by 3 the 4 by 4 matrix and if A
180

1.3

Crystal optics and Griinwald's formula

/1/d 0 0 0 l/e2 0 V 0 0 l/e3, (2) we get the s y s t e m of electromagnetic wave propagation in crystals. In this particular case formula (3) was derived by Griinwald 7 (p. 520 - in a somewhat hidden form) and rediscovered in Garding 6 (p. 378,(47)). Putting A2(V) = V x £>V = rot(£>rot)- with D =

1.4

Generalizations with respect to the size of matrices and the number of space variables

Let us consider the homogeneous system P(dt, V) = Aad2t + 2A1(V)dt + A 2 (V)

(4)

with real, symmetric, I by I matrices Ao,Ai,A2 which are constant, homogeneous of degree 1 and 2, respectively. Suppose furthermore detP(dt,V) strictly hyperbolic in time direction and AQ + A\(£) positive definite on S = {£ e R n / P ( l , £ ) = 0}. Then, if n = 3, the fundamental matrix with support in t > 0 is given by the formula

E(t,x) = -^Y(t)

Je(Oe(0TS'(t s

+

ex)h\(0 (3')

Herein, j-y| denotes the Leray measure on S. The vectors e(£) fulfill (A, + 2A1(£) + A2(0)e(Q

= 0, £ € S,

(5)

e(0 r (4) + ai(0)e(0 = l, £ € &

(6)

Formula (3') is presented in Norris 10 (p. 181, (3.25)). If the number n — 1 of space variables is arbitrary, the analogue of formula (3') is given in Duff 5 (p. 77) - if A0 = Ii, A-L = 0 - : The kernel -—~5'(t has to be replaced by

+ fx)

(in case n - 1 = 3)

2^g!r((* + io + fxrn+2 - (-* + io + fx)-n+2).

181

2

The Herglotz-Petrovsky formula for the fundamental matrix of a strictly hyperbolic, homogeneous system of partial differential operators

Theorem Let P(dt, V) be an I by I system of real differential operators, homogeneous of degree m, with n — 1 space variables. Suppose that det P(dt, V) is strictly hyperbolic in time direction. Denoting by S the slowness surface, S = {£ G R " _ 1 / P ( l , £ ) = 0} and by — | ^[ the Leray measure on S, the fundamental matrix of P(dt, V) with support in t > 0 is given by r p(\ p\&d

E

^=Y^J

hmAt

{dJetp\M

+ ex)m)

-

(7)

s Herein, hm,n = - 2 ( 2 7 r ) 1 - " R e ( i m + 1 ^ ( e r m " 1 ) ) is explicitly known. E.g., for m < n, hm,n = 2(27r) 1 -"(-l) 2 i r 1 a t "~ m _ 1 (upf), if n is odd, and hm,n = ( 2 7 r ) 2 - n ( - l ) 1 + f (n - m - l ) ! ^ " " " 1 " 1 ^ ) , if n is even. Remarks (1) The assumption of strict hyperbolicity (which, e.g., is not fulfilled for the system of crystal optics in 1.3) implies that (dT det P ) ( l , £) does not vanish on the slowness surface. (2) The connection with formula (3') in 1.4: If the system P(dt, V) has the form (4) (hence, m = 2) and if the eigenvectors e(£) of P ( l , f ) on S fulfill the normalization condition (6) then the equation P(l,Oad _l„, ^T = r e ( £ t) we ( 0 J on S (9TdetP)(l,0 2 establishes the connection between (7) and (3') taking into account 1 h2A = - — £2 ' .

4V ' (Putting A0 = gh, A1 = 0, A2 = -K, | 7 | = - ^ -

e(£) = 4 ^ ( 0

and replacing the integration over 5 by a sum of 3 integrals over SV we obtain Burridge's formula (3)).

182

Remarks on the proof The proof is a transfer to systems of the proof of a similar formula for scalar operators given by L. Garding in his article 6 "History of mathematics of double refraction". The starting point is the representation of E as an inverse Fourier transform, E = lim T~x (P(ir + a, i£) _ 1 ) = rmT~^

( P ( T - iO, £ ) - 1 ) •

(8)

The subsequent introduction of polar coordinates with respect to a C°°manifold (which finally becomes the union of the two planes r = ±1) and the integration with respect to the "radial" variable lowers the number of integrations in (8) by one and introduces the distributions /i m ,„. A further integration can be carried out by the introduction of the density 6(detP(l,£)) resulting from the difference -——r — —-—— -. B detP(l-iO,0 d e t P ( l + iO,£) 3

Explicit expressions for transversely isotropic media in a particular case which includes uniaxial crystal optics

Transversely isotropic media are characterized by 5 elasticity constants a» and the system reads as

(9)

m,v)= M 2 --

O i 9 2 —
-a3<9id3

— — a

-a3d!d3

( a l — 04)^102 4 ^ 1 — al92 - 05^3

-a3d2d3

\

-0302(93

S* - a6(8$ + $%) - a2d£

V

/ 2

(see Buchwald , p. 572,(6.1)-(6.4f),(6.7),(6.10)). If the positive constants 01,02,04,05 fulfill the further condition (Payton n , p. 96) a

l = (ai -05X02 —a5),

the determinant of P(dt, V) decomposes into a product of 3 wave operators: d e t P ( 3 t , V ) = W1(dt,V)W2(dt,W)W3(dt,V) with W^duV) = d? - a4A2 W2(dt,V)=d?-aiA2-a2dl, W3(duV) = 8? - a5A3.

a5dl

183

Abbreviating W4 = (ai — a5)Wi + (05 — ai)W2 the adjoint matrix P a d can be written as pad _

'W2W3 + dfW4 d!d2W4 a3did3W-L dxd2W4 W2W3 + d%W j2yyi 4 a3d2d3W\ aadidsWi a3d2d3Wx WXW2 + (a 2 - a 6 )SfWi

The integration over the slowness surface decomposes into 3 integrations over the ellipsoids Wi(l,£) = 0, i = 1,2,3. Since ( 0 T d e t P ) ( U ) = 2[W 2 (1,QW 3 (1,0 + W , 0 ^ 3 ( 1 , 0 + ^ 1 ( 1 , 0 ^ 2 ( 1 , 0 ] we have (0 r detP)(l,0/Wi{u)=0 = 2^(1,0^(1,0The representation of P ( l , £ ) above implies that the expression P(i,0ad / ( a T d e t P ) ( l , 0 / W (i,o=o' is built up from expressions of the type £p£<7

^(1,0

on Wi ( 1 , 0 = 0 .

Hence, by the Theorem in Section 2, the fundamental matrix of P(dt, V) is obtained by evaluation of several integrals

/

W~^J)6'{t

+ e x ) m )

'

^3,P,9

= 1,2,3.

(10)

Wi (1,0=0

The result is Proposition 1 Suppose a\ > a2 > as > 0, a4 > 0, 03 = y/(ai — a5)(a2 — 05), a;' = I * 1 . Then the uniquely determined fundamental matrix E of the system (9) - with support in the half-space t > 0 - is given by

^=^-7I?+lF)+^-v/]?^I)+^-#)+

+H5

Y t

i -^)

-Y(t-J¥^i)

184 where the form factors Hj are given by

Hi

H2

—X1X2 0N

1

-0:1X2

4-Ka4^/o5t\x'\2

0 - a 5 )|x'|2 + ^ ( a 2 - a5)x23]

twa^t[ai(ai

fi(ai - a 5 )xf ^ ( a i - a 5 )xix 2 I f f ( a i ~o-h)x\X2 ^ ( a i - a 5 ) x | a3xix3 a3x2x3

x

H3

xf XiX 2 I X1X2 x;

r/|2

47T05/2i

V 0

0 0 .'12

0

(ai - a 5 )x? (ai - a 5 ) x i x 2 a3xix3 (ai - a 5 ) x i x 2 (ai - a 5 )x2 03X2X3 ! (a-2 a 5 ]x 3 03X1X3 a 3 x 2 X3

1 (ai - a 5 ) | x ' | 2 + (a 2

a 3 xix 3 03X2X3 ^(a2-d5)xlt

a 5 )x§

:

H4

2 - xi - 2 x i x 2 0V -2xiX2 x\ — x\ 0 0

i-Ky/as \x'

(h\\ hi2 /ll3 N h2i h22 h23

1

H5 =

.

=

8^^5(01-02)

^3^3^33

where h11=d1d3X1(R+-R-)/\x'\2, hi3 = h22 =

hi2 = d2d3x1(R+

1

1

-a3d1(R+ -RZ ),

/121 = did3X2(R+ 2

d2d3X2(R+-R-)/\x'\ ,

/123 = - a 3 9 2 ( ^ +

1

1

- -R;

1

hs^-asd^R^-RZ ), h33 = -(02 - o 5 ) 9 3 ( ^

)/!*?, 2 -R- )/k'l ,

- i?_

/132 = - a 3 9 2 ( - R + - -R! 1

1

- RZ )

and R± = V [ V a 5 ( a i - a 2 ) t ± Vai - as X3] + ( 0 2 - 0 5 )

r'|2

185

Remarks

(1) The elements of the fundamental matrix E in Proposition 1 are stated already in Payton n (Section 10, (3.10.54)-(3.10.58),pp. 103,104), though in a less explicit manner. There these elements are derived by the method of Cagniard-de Hoop.

(2) The form factors of the S distributions can also be obtained relativly easily by asymptotic evaluations of the integrals (10) (see, e.g., Burridge 1 , p. 48: ".. .we may obtain for hexagonal media the pulse shape near the edges of the wave fronts..."). The fundamental matrix of e l e c t r o d y n a m i c s in uniaxial crystals is obtained by letting tend first ai\a2, second by observing that the result in Proposition 1 persists if 0 < a\ = a2 < as and, third, by taking the limit a2\0. Renaming the constants (as = — 03 = 1, a\ — d) the result is

Proposition 2 If d > 0 and d 7^ 1, then the uniquely determined fundamental matrix E of the system

'df-ddl-dl ddxd2 dic>3

ddxd2 at2 - dd\ - di d2d3

9i9 3 d2d3 d2t - d\ -

d\

having support in the half-space {(£, x) S R 4 : t > 0} is given by

E = Hts(t

- yjls£.+xl\

+ \htY{t)

+H4 Y(t-y/i^.+4)-Y(t-\x\)

® 5{x) + H3S(t - \x\)+

H5Y(t-\x\),

186

wherein x' =

Hi

H3

and

—X\Xi

0 X\Xi

1 4?rt

X\X2

r'|2

2 ~

4ir \x''14

x •x

0 x

H4

-x\xi 0

Xtj

1 4nd\x'\2t

x

l

-2a;ia;2 £? — 0 0

H5

t 47r|a;|3

3x • x

References 1. R. Burridge, The singularity on the plane lids of the wave surface of elastic media with cubic symmetry. Q u a r t . J. Mech. Apl. M a t h . 20 (1967), 41-56. 2. V. T. Buchwald, Elastic waves in anisotropic media. P r o c . Roy. Soc. London, Ser. A, 253 (1959), 563-583. 3. C. H. Daros, H. Antes, Dynamic fundamental solutions for transversely isotropic piezoelectric materials of crystal class 6 mm. Int. J. Solids Structures 37 (2000), 1639-1658. 4. C. H. Daros, Elastic wave propagation in unbounded piezoelectric media of transversely isotropic symmetry. Thesis, Braunschweig, 1999. 5. G. F . D. Duff, Hyperbolic differential equations and waves, Boundary Value Problems for Linear Evolution Partial Differential Equations, ed. by H. G. Garnir, Reidel, Dordrecht, 1977, 27-155. 6. L. Garding, History of the mathematics of double refraction. Arch. History Exact Sciences 40 (1989), 355-385. 7. J. Griinwald, Uber die Ausbreitung der Wellenbewegungen in optischzweiachsigen elastischen Medien. In: Festschrift Ludwig Boltzmann, J. A. B a r t h Verlag, Leipzig 1904, 518-527. 8. N. M. Khutoryansky, H. Sosa, Dynamic Representation Formulas and Fundamental Solutions for Piezoelectricity. Int. J. Solids Structures 32 (1995), 3307-3325. 9. T . Mura, Micromechanics of defects in solids. 2nd rev. ed., M a r t i n u s Nijhoff Publ., T h e Hague, 1987.

187

10. A.N. Norris, Dynamic Green's Functions in anisotropic, piezoelectric, thermoelastic and poroelastic solids. Proc. Roy. Soc. London A 447 (1994), 175-188. 11. R. G. Payton, Elastic wave propagation in transversely isotropic media. Martinus Nijhoff Publ., The Hague, 1983. 12. C.Y. Wang, J.D. Achenbach, A new method to obtain 3-D Green's functions for anisotropic solids. Wave motion 18 (1993), 273-289.

188

D Y N A M I C A L P R O B L E M S I N T H E (0,0) A N D (1,0) A P P R O X I M A T I O N S OF A MATHEMATICAL MODEL OF CUSPED BARS S. S. K H A R I B E G A S H V I L I , G. V. J A I A N I /. Vekua Institute

of Applied 2 University E-mail:

Mathematics of I. Javakhishvili University St., 380043 Tbilisi, Georgia [email protected]

Tbilisi

State

The dynamical problems in the (0,0) and (1,0) approximations of a mathematical model of cusped bars are studied. Initial boundary value problems are investegated for the hyperbolic systems with order degeneration containing systems of the above (0,0) and (1,0) approximations. The existence and a priori estimates of strong generalized solutions in appropriate weighted spaces are established. T h e obtained results are applied to cusped bars.

Introduction In the fifties of XX century, I.Vekua 1 suggested a new mathematical model of elastic prismatic shells (i.e., of plates of variable thickness) which was based on the expansion of fields of displacement vectors, strain and stress tensors of the three-dimensional theory of linear elasticity into orthogonal Fourier-Legendre series with respect to the variable of plate thickness. Considering only the first N + 1 terms of the expansions, he obtained the TV-th approximation. Each of the approximations N = 0,1,... can be considered as an independent mathematical model of plates, e.g., the approximation N = 1 actually coincides with the classical plate bending theory. In the sixties, I.Vekua 2 offered the analogous mathematical model for thin shallow shells. All his results concerning plates and shells are collected in his monograph 3 . At the same time he recommended to investigate cusped plates, i.e., plates whose thickness vanishes on a part of the plate projection boundary or on the whole one (about investigations in this direction see survey 4 and also I. Vekua's comments in 3 , p.86). Works of I.Babuska, D.Gordeziani, V.Guliaev, I.Khoma, A.Khvoles, T.Meunargia, C.Schwab, T.Vashakmadze, V.Zhgenti, and others (see 5 - 1 3 and references therein) are devoted to the analysis of I.Vekua's models (rigorous estimation of an approximation error, i.e., of the modelling error, numerical solutions, etc.) and their generalizations (for non-shallow shells, anisotropic case, etc.). In 14 , expanding fields of displacements, strains, and stresses of the three-

189

dimensional theory of linear elasticity into double Fourier-Legendre series with respect to the variables of bar thickness and width, an analogous mathematical model of bars with variable rectangular cross-section is constructed. It is assumed that the thickness and width can tend to zero at the ends of the bar, i.e., the case of a cusp is included. More precisely, in Section 1 of 14 some auxiliary formulas are deduced. These formulas give the expressions of the double Fourier-Legendre moments of the first derivatives of functions by means of the double Fourier-Legendre moments of the above functions and of the derivatives of these double moments. Section 2 is devoted to the reformulation of the equilibrium equations, Hooke's law, and of the expression of the strain tensor by means of the displacement vector components in terms of the double moments of the sought fields of displacement vector, strain and stress tensors. In Section 3 the main relations (deduced from the Lame system) of the (iV3, JV2) approximation are derived which are either a system of ordinary differential equations (in the static case) or that of hyperbolic equations (in the dynamical case), in general, with order degeneration for weighted double moments of displacement vector components. As in the case of plates and shells, all these approximations represent independent mathematical models of bars which go into the three-dimensional model when N2,N$ tend to infinity. In Section 4 the initial and boundary value problems are set up. In the nondegenerate case, i.e., in the case when both the bar thickness and width do not vanish anywhere, the well-posedness of problems follows from the well-known theories (see, e.g., 15 , 16 ) of nondegenerate systems of ordinary differential equations (in the static case) and nondegenerate systems of hyperbolic equations (in the dynamical case). The existence of the solutions of the static problems are proved in the (0,0) (see 17 ) and (1,0) (see 18 ) approximations. The relation of the mathematical model of bars under consideration to the three-dimensional theory of elasticity is illustrated in 19 . The present paper deals with the dynamical problems in the (0,0) and (1,0) approximations. In the first part initial boundary value problems are investegated for the hyperbolic systems with order degeneration containing systems of the (0,0) and (1,0) approximations. In the second part, the results of the first part are applied to the cusped bars.

1

Hyperbolic Equations and Systems W i t h Order Degeneration

To the best of our knowledge the theory of degenerate hyperbolic equations mainly deals with the initial boundary value problems in the case of a type

190 degeneration but not in the case of an order degeneration. The mixed problem for hyperbolic equations of the second order with the type degeneration on the lateral part of boundary was investigated in 20 . The case when a type degeneration occurs on the initial hyperplane was studed in 2 1 - 2 5 . 1.1

The mixed problem for a hyperbolic equation of the second order with order degeneration on the lateral part of the boundary

Let us consider the following hyperbolic equation of the second order . . , , . d2u

Lu := p(x)h(x)

w

d (.,

- -

,du\

[h(x)-j

du

du

+ai-+a2-+a3u

_,

,_.

= F,

(1)

where the coefficients and the right hand part of equation (1) are defined in the rectangle D := {(x,t) e R2 : 0 < x < 1, 0 < t < T} , and fc6C[0,l]nC1(0,l];

peC[0,l],

p(x) > pa = const > 0,

a1,a2eC1(D),

hxxa < h(x) < h2xa,

a3GC(D), 0 < x < 1,

(2)

a = const > 0, hi = const > 0, i = 1,2. By virtue of (2) we have /i(0) = 0, i.e., equation (1) has an order degeneration on the left side To : x = 0, 0 < t < T of the rectangle D. Let I^ : x = 1, 0 < t < T, and

_ fr 0 ur 1 ifa
*Q ' \ T i ifa>l. Problem 1.1.1 Find a solution u(x, t) of equation (1) satisfying the initial conditions u(x,0)=0,

^ g ^ l

= 0

,

0<x
(3)

and the boundary condition «lr.„ = 0(4) Parallel with the problem (1), (3), (4) let us consider the problem of finding solutions of L'v : - * ) / . ( * > g

" |

(«*) * ) - | ( « . . ) - l

M

+ «3» = F (5)

0<x
(6)

with the initial and boundary conditions as follows v(x,T)=Q,

^ % ^ = 0 ,

191

«lr.„=0,

(7)

where L* is the operator formally adjoint to the operator L. Below it will be shown that the problems (1), (3), (4) and (5), (6), (7) are adjoint in corresponding function spaces. Let us denote by E and E* function subspaces of the space C2(D), satisfying the initial boundary value conditions (3), (4) and (6), (7), correspondingly. Let W+(W+) be weighted Hilbert spaces obtained after closure of the space E(E*) under the norm \H2w+(w.):=jh(x)[u2

+ uj + ul}dD.

D

Let us denote by W-(W!) by L2(D) and W+(W;) 2 6 , 2 7 . Let

spaces with the negative norms constructed

Mi := sup\h~1(x)ai(x,t)\

< +oo, i = 1,2,

(8)

< +oo, if a > 2,

(9)

D

M3 := sup \a-~1(x)a3{x,t)\ D

where a{x) = xa~2\ln^x\~1~e°, £o =const> 0 is sufficiently small. According to the results of 28 - 30 for any u G W+ and v G W+ by a < 2 we have

I M k w
IMLa(JJ)
(10)

||tr*u|L a(D) < c | | u | | w . ,

(11)

and by a > 2 we have H^ U IL 2 ( D ) < c | M | w + ,

where positive constant c is independent of u and v. Remark 1.1.1 In the space W+(W+), the above - introduced norm H-llw (w*)' °y virtue of (2), is equivalent to

Ml2

fxa[u2+u2t+u2x}dD,

which in turn, in view of (2) and the proved below (see (33)) inequality f h{x)u2dD
D

Vw € W+(WZ),

192

is equivalent to the norms ||u|| 2 := J xa [u2 +u2x] dD and ||M|| 2 := j h{x) [u2 + u2x] dD. D

D

Lemma 1.1.1 Let the conditions (8), (9) be fulfilled. Then the inequalities

||L«||


\L*v\\w_

(12)


(13)

where the positive constants c\ and C2 are independent of u and v, are valid for alluG E, v e E*. Proof According to the definition of the negative norm, for u G E in view of (3), (4) and (6), (7) we have \\Lu\\w,

:= sup \\v\\w\(Lu,v)L2(D) »eiy*

+

sup Hwll^1. [-ph(x)utvt veE* +J

+h(x)uxvx

= sup

\\v\\^.{Lu,v)L2{D)

v€E*

+ aiutv+

+

a2uxv+

a3uv]dD. (14)

By virtue of (8) and of the known inequalities

Jh(x)fgdD < I f h(x)f2dD\

I f h{x)g2dD

D

<(xi + x22)?(yj + yl)

\xiyi+x2y2\ we have / [-p{x)h(x)utvt

< ( 1 + max p(x ~

l

+

h(x)uxvx\dD

J(h(x)(u2 + u2x)dD

0<x
J(h(x)(v2+v2x)dD

<

1 + max p{x) \\u\\

\\v\\w,,

(15)

193

/

/ [a\UtV + a2Uxv]dD < M i

D

+M2

h2(x)v

h(x)u2dD

L2(D)

\D

[h(x)uldD\

hl(x)v

M2))\\u\\w\\v\\w..

(16)

< c 2 s u p | a 3 | | | u | L IM|W.,

(17)

<(M1 +

When a < 2, in view of (10), we have a^uvdD

< sup\as\ \\u\

L2(D)\\V\\L2(D)

D

D

+

and when a > 2, taking into account (9) and (11), we get / cru2dD

/ a^uvdD < M 3 / \a?u\ \piv\dD < M 3 D

D

j2dD

\D

<M3C2|M|W+|ML.

(18)

Prom (14)-(18) easily follows (12). Similarly, taking into account (3)-(7) one can prove (13). Remark 1.1.2 By virtue of inequalities (12), (13) the operator L : W+ —> Wl (L* : W+ —> W-) with the dense domain of definition E{E*) admits closure which is a continuous operator from the space W+(W.£) into the space W1(W-). Leaving for this closure the same notation L(L*), let us note that it is defined on the whole space W+(W^). Lemma 1.1.2 The problems (1), (3), (4) and (5), (6), (7) are mutually conjugate, i.e., for any u G W+ and v £ W+ holds {Lu,v) = (u,L*v).

(19)

Proof By virtue of Remark 1.1.2 it is enough to show the equality (19) in case when u € E and v G E*. In this case obviously (Lu,v) = (LU,V)L2^, and in view of (3)-(7), after integration by parts we get (19). Below we assume Mi :— sup\a1 1(x)as(x,t)\

< +oo,

(20)

D

H ; n I ~ i - a4 -£° if a ^ 1 and o\ := | ^ | a ; | where <7i(a?) := xa -l\ln\x\ £o =const> 0 is an arbitrarily small number

x So

if a = 1,

194

Remark 1.1.3 If a > 2 from (20) there easily follows (9). Let us note that if a < 1, condition (20) is automaticaly fulfilled since by definition a$ G C(D). Let \\v\\2 h(x)v2dD. := f ^

2,ft

Lemma 1.1.3 Let (2), (8) and (20) he fulfilled. Then for any v e W^ the inequality

c|ML2ifc < ||£*«|L_

(21)

with a positive constant c, independing of v is valid. Proof In view of Remark 1.1.2, it is enough to show inequality (21) when v e E*. liveE*, then by A = const > 0 the function t

u(x,t) =

eXTTv(x,T)dr

(22)

o belongs to the space E. By virtue of (22) the equalities ut(x, t) = exttv(x, t),

v(x, t) = e" A 4 _ 1 u t (a;, t)

(23)

are valid. Let (wx, vt) be an outward normal to dD. Taking into account (3)-(7) and (23), we have {L*v,

U)L 2 (U)

= / [-p(x)h(x)vtut

+ h(x)vxux

+ aivut

D

+ ci2vux + asvu]dD

p(x)h(x)exttvvtdD

= —I D

+

I e _At t -1 [/i(a;)u I tWx + o.\u2 + a,2Uxut + a3uut]dD,

(24)

D

-

/p{x)h(x)e xt tvv t dD

p(x)h(x)extt(v2)tdD

= -^ f

D

D

= --

xt

2

f p(x)h(x)e tv vtdS+ldD

= - f p(x)h(x)eXt(l

f p{x)h(x)eXt{l

+Xt)v2dD

D

+ Xt)v2dD = i J

D

+ ]-X J p(x)h(x)e-xtt-1u2dD.

p(x)h(x)eXtv2dD

D

(25)

195

It is easy to see that Vt\x=0 = Vt\x=l = 0,

^t|t=T = l,

-1..2I

t~LUx\t=O = 0.

Therefore, I e-^t^hWuxtUxdD

= ]- j

D

e-^t^h^ulvtdS

dD

+ i f h(x)e-xt[Xt-1

+r2}u2xdD

D

> )- I'h{x)e-xt[\r1

+ t~2)u2xdD.

(26)

D

By virtue of (8) we get / e~xtt~1(aiu2

+

a2uxut)dD

«? + 1 ( « ? + u * ) D

-I-

j e-xtrlh(x)u2dD

e-xH~lh{x)u2xdD,

+ hut j

D

(27)

D

where M = max(M1, M2). Further, using the obvious inequality fe-^t-'fgdD

< I fe-^t-'fdD]

fe-xtr1g2dD

I

by virtue of (2) and (20) when a ^ l w e have e_Att

x

a3uutdD <M4

e-^t^x0-1 Ln—x

I

2

' D/ •

e

xt

t

* |a;2u t | a;

In-x

|u| |tt t |dZ)

dD

2

£>

<M4

je'Xit-1xau2dD

x>

f

J ID

e-^t-1!1

-l-2e0

In-x 2

u 2 dD

<

196 -l-2e0 .a-2 D

ln-x 2

u2dD

D -l-2e0 Xt+-lxa-2

D

m-x

u2dD. (28)

D

Now we use the known inequality [28-30] 1

, 1 In—x 2

-l-2eo

1 , , f | * < a / ^ * *

(29)

where C\ = const > 0 is independent of u € E and t £ [0,T]. Multiplyng both sides of inequality (29) by e _ A ' i _ 1 and integrating the obtained result with limits 0 and T, we get f e-xtrlxa-2

|in2a;r 1 ~ 2 e o u2dD < d / e ^ " 1 / ^ ) ^ * + u*)dl>. (30)

In view of (30) from (28) there follows e

t

< Ml J

a^uutdD

/ •

e-^hix^dD

D

D

At 1

+ ^d

J e" t- /i(a:)(«2 + u\)dD.

(31)

D

By virtue of (20) an estimate analogues to (31) can be obtained when a = 1. Now taking into account that in the domain D t-i

>

T-i5

e\t

>

X)

e -At

>

e -AT

and choosing the number A so large that Ap0-3M-(^1+ci)M4 2i X-M

- M 4 ci 2i

where po := min p{x). 0<x
> -

A / 9 0 - 3 M - ( / i j " 1 + ci)M 4 > ; L 2T - ' A - M - M 4 ci 2T

197

Prom (2) and (24)-(31) there follows (L*v,u)L2{D)

> \ J P{x)h(x)v2dD+l-

Xp(x)h(x)e-xtt-1u2tdD

J

D

D xt

1

2

+t- \u2xdD

+ i f h(x)e- [Xt~

- ^M

f

D

e^H^hix^dD

D

- -M

fe-^t-ihWuldD-^-

e-xtt-xh{x)u2tdD

f

D

D

Mi, -Ci f e-Xtrlh{x)(ul 2

>PiJHx)v<

+ u2)dD

Xp0 - 3M - (/ij 2T

+ ci)M 4

D

+

h(x)e-Mu2dD

/ D

A - M - M4ci • j h(x)e-Xtu2JD 2T~

> y

D

/

h{x)v2dD

D

\D

D

x I jh(x)(u2

)

+ u2x)dD

(32)

\D

In view of w| t=0 and E

2

\h (x)u(x,t)\


2

= \h (x)[u(x,t) - u(ic,0)]|

ut{x,T)dT\

2

h (x)


/

ut{x,r)di

ut(x,T)di

we easily get the inequality f h{x)u2dD
f h{x)[u2 + u2x}dD.

(33)

198

Prom (33) it follows that in the space W+ (W+) the norm Jh(x)[u2+u2t+u2x]dD

IHIw + ( w ; ) •= D

is equivalent to the norm

Ml2 := jh{x)[u2t+ul}dD.

(34)

D

Therefore, leaving for the norm (34) the previous notion I H I ^

(H,.>,

from

(32) we have > Vpoe-XT\\v\\L2J\u\\w+.

\\L*v,u\\Li{D)

(35)

Now, if we apply the generalized Schwartz inequality WL*v>u\\z.a(D)

<\\L*V\\WJ\U\\W+

to the left hand side of (35), after reduction by ||u|| w , we get the inequality (21), where c = \f pae~XT. Thus, Lemma 1.1.3 is fully proved. Now, assume, in addition M5 := sup lo-f 1 ( i ) [ait + a2x - a3] (x,t)\ < +oo.

(36)

D

In view of smoothness of the coefficients aj condition (36) is automatically fulfilled when a < 1. Lemma 1.1.4 Let conditions (2), (8), and (36) be fulfilled. Then for any u 6 W+ the inequality c u

ll IU2,h ^ ll i u llw*

(37)

with a positive constant c independent of u is valid. Proof Similarly to the case of Lemma 1.1.3, in view of Remark 1.1.2, it is enough to show the inequality (37) for u s E. Let u G E. Then it is easy to see that for A = const > 0 the function v{x,t)=

f(T-T)e-XTu(x,T)dT

belongs to the space E*, and vt(x,t) = -(T-t)e-xtu(x,t),

u(x,t) = -(T-t)-1extvt(x,t).

(38)

199 By virtue of (3)-(8) and (38), and of the equalities (r-*)_lux|t=T =0,

"t\t=0 = " I , "t\t=T = 1, ^t|x=0 = Mx=l = ° i we have (Lu,v)L2(D)

= / [-/9(a;)/i(a:)utUt + h(x)uxvx

-

u(aiv)t

D

- u(a,2v)x + a^uv] dD = jp{x)h{x){T

-

t)e-xtutudD

D

-

/ (T - t)~ V

[h(x)vxtvx

- a2wxut

- (au + a2x - a3)vtv - a-^vf] dD,

fp(x)h(x)(T

- t)e-XtuutdD

= ~ f p{x)h(x)(T

(39)

-

t)e~Xtu2vtds

dD

D

+ ]- fp(x)h{x)[l

+ \{T -

t)}e~xtu2dD

+ \{T -

t)}e~Xtu2dD

D

> i fp{x)h(x)[l D

= ]- Ip{x)h(x)e~Xtu2dD

+ ^ f Xp(x)h{x)(T - t^e^dD,

D

f(T - t)-lexth{x)vxtvxdD

-

(40)

D

=--

f (T -

t)-xexth(x)vlvtds

dD

D

+ \ j [(T - ty2

+ \{T - t)" 1 ]

h(x)exth(x)v2xdD

D l

> -j\{T-t)-lh(x)extv2xdD,

(41)

200

f(T - t)-1eXt(a1v2

+

a2vxvt)dD

D

f(T-ty1exth(x) v2 + ^(v2 + v2x) dD

<M D

= ^M f{T - t^e^hix^dD

+ ]-M f (T - t^e^hix^dD,

D

(42)

D

where M := max(M x , M 2 ). By virtue of (36) in a similar way as in obtaining (31) we get f(T - i J - V f a i t + a2x -

a3)vtvdD

D

f(T - t)-lexth{x){v2t

+ vl)dD,

M6 := M6(M5) = const > 0. (43)

D

Now, taking into account that in the domain D (T-t)-1

>T~\

ext>l,

Xt

e-

>e-XT

and choosing the number A so large that \p0 - 3M - 2M 6 2p0 - 3M - 2M 6 2(T -t) ~ 2T A - M - 2M 6 A - M - 2M 6 >1, 2(T - i) 2T from (2) and (39)-(43) there follows (Lu,v)L2{D)

^•e-XTjh(x)u2dD+^J\p(x)h( x)(T-t)-1extvfdD

> + - f\(T

- t)-1h{x)eXtv2xdD

D

- -M f (T D

-iMJiT-tr^hMvldD D

- M6 f(T - t)-lexth{x){v2t

+ v2x)dD >

t)-lexth(x)v2dD

201

P

>

fe-^Jh(xWdD +

"

fh{x)u2dD

2

{Xp0

~ 3g -

+ fh(x){v2

2M6)

Jh(x)e^v2dD

+ vl)dD

1 2

^ V

7

^

3

^ Uh{x)u2dD\

/

ljh(x)(v2+v2x)dD\

,

i

\

2

.

(44)

Prom (44) as in proof of Lemma 1.1.3 there follows the validity of inequality (37). Let us denote by L2th(D) the space of such functions u that h^{x)u G L2(D) under the norm ||U||L 2 A = \\h1/2(x)u\\L2(D). Analogously we introduce the space L 2j /,-i(D). Definition 1.1.1 Let F G L2ih-i(D). The function u will be called a strong generalized solution of the problem (1), (3), (4) of the class W+ if u G W+ and there exists a sequence of the functions un G E such that un —> u in the space W+ and Lun —» F in the space Wl, i.e., lim \\un — u\\w+ — 0, n—>oo

lim ||Zu n — i^Uw* — 0. n—>oo

Theorem 1.1.1 Let conditions (8), (20), and (36) be fulfilled. Then for any F G L2 fl-i{D) there exists a unique strong generalized solution u of the problem (1), (3), (4) in W+, and the estimate c\\u\\L2th < \\F\\W1

(45)

with a constant c independent of F is valid. Proof By virtue of the inequality (21) the functional {F^v)^^ can be considered as a linear continuous functional of L*v, where v G E*, F G L2,h-1(D)- Indeed, using the above-mentioned inequality, we have \{F,v)L2{D)\

<

{{FWL^^WVWL^

< c*||Z,*w||iv_,

c*=const>0.

According to the Hann-Banach Theorem we extend this functional on the whole space W- linearly and continuously. In view of the theorem on the general form of a linear continuous functional over W- there exists a function u G W+ such that (u,L*v)L2(D)

= (F,v)L2(D),

veE*.

(46)

202

The equality (46) means that the function u is a weak generalized solution of the problem (1), (3), (4). Now let us show that this solution is also a strong generalized solution of the problem (1), (3), (4) in W+. Indeed, as far as the space E is dense in the space W+ there exists a sequence un £ E such that lim \\un — u\\w, = 0. +

n—»oo

By virtue of equalities (46), (19), and inequality (12) we have \\Lun-F\\wl

= sup \\v\\^.(Lun-F,v)L2,D) vew* + = sup ||u||j^, [{Lun,v)L2(D) (F,v)L2iD)] UGE*

+

= sup \\v\\^. [(Lun,v) v€E*

= sup ||u||w. {L(un vEE"

(u,L*v)}

+

u),v)

+

< sup ||u||w« \\L{un - u)\\w* \\v\\wx =

vEE* + ||£(U„-M)||W*

+

< ci||u„ -u||vv+-

Hence, lim \\Lun — F\\w* = 0, i.e., u is a strong generalized solution. The uniqueness of a strong generalized solution of the problem (1), (3), (4) in W+ and estimate (45) follows from (37). Thus, Theorem 1.1.1 is proved. Definition 1.1.2 Let F € Wl. The function u will be called a strong generalized solution of the problem (1), (3), (4) in i2,/i, if u G L2,h(D) and there exists a sequence of the functions un e E such that un —» u in L2,h(D) and Lun —> F in Wl, i.e., lim ||wn — n—>oo

U\\L2 h ' '

= 0,

lim \\Lun — F\w* = 0. n—>oo

-

Theorem 1.1.2 Let the conditions (8), (20), and (36) be fulfilled. Then for any F G Wl there exists a strong generalized solution u of the problem (1), (3), (4) in L2,h, and the estimate (45) is valid. Proof As far as L2:h~i{D) is dense in the space Wl for any F € Wl there exists a sequence of the functions Fn e L2th~1{D) such that Mm \\Fn - F\\w. n—>oo

= 0.

(47)

_

According to Theorem 1.1.1 for each function Fn £ L2
203

far as the space E is dense in W+ there exists a sequence un G E such that \\un-

un\\w < - .

(48)

Let us show \\Lun-F\\wl=0.

(49)

Taking into account (12) and (48), we have \\Lun - -FHw* < \\Lun — Lu„||w» + \\Lun - F\\w^ = \\L(un - un)\\wz + UK - F\\W1 <^- + \\Fn -

F\\W1.

Therefore, by virtue of (47), we get (49). Further, from (48) and (37), we have \\un

- Um\\L2:h

< \\un - Un\\L2h

+ \\un - Um\\L2,h

+ \\um ~

Um\\L2,h

< - H h -\\L(un - wm)||iy* n m c = - + - + -\\Fn-Fm\\w. (50) n m c By virtue of (50) the sequence un is fundamental in the complete space L>2,h{D)- Let u e L2,h{D) be the limit of the above sequence. In view of (49) and (37) it is easy to see that the function u is a unique strong generalized solution of the problem (1), (3), (4) in Z/2,/M and the estimate (45) is valid. Remark 1.1.4 In order to make clear in which sense the strong solution u of the problem (1), (3), (4) in W+ satisfies the boundary condition (4), let us note that by virtue of Remark 1.1.1 the metric ll"llw+ • -

[xa{u2+u2t+u2x)dD

(51)

D

of the space W+ according to results of [28-30] ensures in the sense of traces conservation of the zero boundary data (4). When a > 1, the zero boundary data on To : x = 0, 0 < t < T are not preserved. They are preserved only on the lateral side Ti : a; = 1, 0 < t < T of the domain D. Remark 1.1.5 Let a > 1 and h € C^O, 1]. Let us derive a priori estimate for classical solutions u £ C2{D) of (1), satisfying the initial conditions (3) and the boundary condition u{l,t)=0,

0
(52)

204

Indeed, let u € C2(D) satisfy the initial conditions (3) and boundary value condition (52). Then, in view of ^t|t=T = l , Vt\x=0 = Vt\x=l = 0 ,

Vx\t=T = 0,

we have 2(Lu,e-xtut)L2{D)

xt

=

P(x)h(x)e-

(u2t)tdD

D

+2 / e~Xt(aiu2

fe-xt{h(x)ux)xutdD

-2 D

+ a2uxut + a,3Uut)dD, X = const > 0,

(53)

D

p(x)h(x)e-M(u2)tdD

p(x)h(x)e-xtu2vtds

=

D

IXp(x)h(x)e-Mu2dD

+ D

dD 1

= J p(x)h(x)e~XTu2(x,T)dx

\p(x)h(x)e-Xtu2tdD,

+

(54)

D

0

-2 / e~Xt(h(x)ux)xutdD

= -2 / e~xth(x)uxutvxds

D

+2 D

dD xt

2

xt

[e- h(x)(u x)tdD

2

= f e- h{x)u xytds

D D

e~Xth(x)uxuxtdD

Xt

IXe- h(x)u2xdD

+ D

dD 1

J e-Xth{x)u2x{x,T)dx

+

f\e-xth(x)u2xdD.

(55)

Further, by virtue of (8) and (20) carrying out reasonings analogous to them leading us to the inequalities (27) and (31), we get 2/e

{a\ut + a,2Uxut + a^uut)dD

D

< 2

e~Xt(\ai\u2

+ |a 2 u x «t| +

\a3uut\)dD

D

< C2 fe-xth(x)(u2

+ u2x)dD,

C2 = const > 0.

(56)

205

If A is sufficiently large, from (53)-(56) we get l xt

2(Lu,e- ut)L2{D)

fe-XTh(x)[u2t+u2x}{x,T)dx

> o

+ C 3 I"e-xth{x)[u2t + u2x}dD, C3 = const > 0. (57) Further, obviously 2(Lu,e~xtut)

2

(h~^(x)Lu,hi(x)e~xtut)

L2(D)

< 2||LU||L 2 ) I _ 1 |K||z, 2ih < 2||Lu|| I , 2> _ 1 ||u||iv+,

(58)

l

fe-XTh{x)[u2t

+ u2x](x,T)dx + C3 fe-XTh(x)[u2

+ u2x]dD

D ' 1

>CA

fh{x)[u2 + u2x}(x, T)dx + jh{x)[u2

+ u2x]dD

(59)

C± = const > 0. By virtue of (58) and (59) from (57) there follows l

Jh{x){u2

+ u2x}(x,T)dx + jh(x)[u2

+ u2x)dD <

CsWLuWl^^,

(60)

where C5 = const > 0 is independent of u. From a priori estimate (60) there follows the uniqueness theorem for a classical solution u(x,t) € C2(D) of the problem (1), (3), (52). Remark 1.1.6 According to Theorem 1.1.1 for any F £ L2^-1(D) there exists a unique strong generalized solution of the problem (1), (3), (4) in W+ with the weak estimate (45) by the norm of L2,h{D). In order to obtain a strong estimate for this solution in the norm of W+ below we use the Galiorkin method. To this end we define a weak generalized solution of the problem (1), (3), (4) in W+. The function u(x, t) will be called a weak generalized solution of the problem (1), (3), (4) in W+ if u G W+ and the identity 16

206

-p(x){h(x)u)tvt

+ h(x)uxvx

+ aiutv + a2uxv + a3uv) dD

D

FvdD

(61)

/'

D

is valid for any v G V, where F G L2
+ h(x)uxvx

+ a\utv + a2uxv + a^uv] dD

D

^ / [ P ^ K M ^ M t ^ t l + h(x)\uxvx\

+ \aiutv\ + \a2uxv\ + |ci3uu|] dD

D

< C6 [ fh(x)(u2

+ u\ + u2x)dD J

| f h(x){v2 + v2 + v2x)dD

= C
(62)

where Cg = const > 0 is independent of u and v is valid. Prom (62) there follows that the left hand side of the equality (61) has the sense. The right hand side of (61) has also the sense, since F € L2^~^{D) and v G V, and therefore, v G L2,h(D). Further, if v G E*, then v G V and integrating by parts the left hand side of the equality (61), we get (46). Therefore, as it was shown above, there follows that u(x,t) is also a strong generalized solution of the problem (1), (3), (4) in W+. The reverse is true. Now the uniqueness of a weak generalized solution of the problem (1), (3), (4) in the sense of the inequality (61) follows from the a priori estimate (37). For the realization of the Galiorkin method, besides (60) we need the a priori energetic estimate /

h{x){u2+u2t+u2x)dx
-2,h-H

rG[0,T],

(63)

207

for any solution u G E of the problem (1), (3), (4), where CV = const > 0, QT := D fl {t = T } = {(x,t) : 0 < x < I, t = T}, DT := D n {0 < t < T}, 0
2 JLu • utdDT = 2 JFutdDT. DT

(64)

Dr

By virtue of (2), (3), (4) and the equalities Vt\t=T = 1,

^t|x=0 = ^t|s = l = 0 ,

I/x|t=T=0,

after integration by parts, we have 2 / p(x)h(x)uttutdDT

= / p(x)h(x)(u2 )tdDT =

DT

DT

p(x)h(x)u2(x,T)dx,

(65)

fiT

—2 / (h(x)ux)xutdDT

=

h(x)ux(x,r)dx.

(66)

DT nT Further, in view of (8) and (20), similarly to (56), we get

2 / (a\u2 + aiuxut + azuut)dDT


DT

DT T


(67)

where w(a) := / h(x)(u2 + u\ + u2)dx, Cg = const > 0. As far as j FutdDT Or

h~1(x)F2dDT

< DT

/ h(x)u2dx nT from (64)-(69) we have

h(x)u2dDT,

+

(68)

DT

h(x)u2dDT,


(69)

DT

u(r)(v)d*+\\F\\

J

2,h-1

{Dr)

(70)

208

Now, from (70) according to the Gronwall Lemma we obtain (63). o Let {ipn(x)}%Li G C^O, 1] be a basis in W^ity (e.g., i/jn(x) = sinmnr), where fi = QT if r = 0. Applying to the sequence {tpn(x)} the orthogonalization process under the scalar product (u,v)i =

p(x)h(x)uvdQ,

we obtain a complete in W ^ O ) sequence {ipn(x)} for which (
We seek the approximate solution u (x,t)

(71) in the form

n

un{x,t) = Y.Ck(t)Vk{x)

(72)

fc=i

from the relations :

d)22u„n, i i \ p(x)h(x)-—^-,(pej

f +

{h(x)u™
£ = l,...,n,

(73)

and

c,"(o) = o, j t c m

0, k = l,...,n.

(74)

t=o

The equalities (73) represent the system of linear ordinary differential equations of the second order with respect to t for unknown functions C%(t), k = l,...,n. This system is solved with respect to (71) and (72) we have p(x)h(x)-^-,
=

Jp

,

k dt2

, since by virtue of

£=l,...n.

As far as the coefficients of this system belong to the class C[0, T], and in view of F € L2th-i(D) the right hand side fe(t) = (F(x,t),tpe(x)) € L2(0,T), the Cauchy problem (73), (74) is uniquely solvable, moreover —^ € £2(0, T) and un G W+. Let us show that the estimate (63) is valid for un. Indeed, multiplying each of the equalities (73) by the corresponding ^ C " ( i ) and summing

209

up by I from 1 till n, we obtain the equality ^p(x)h(x)^,

+J (h(x)unxu?x + alU?u?

~ j

+ a

«

2

+a3unu?)dx

=

(F,u?)L2{Q).

Therefore, we get (63) for u = un. Thus, the estimate Jh(x)[(un)2

+ ( < ) 2 + K)2}dx

< C 7 ||F||i 2 f c _ l ( 2 , r )

(75)

with a constant C-j > 0, independing of n is valid. From (75) there follows that hn\\w+

= jh{x)[{unf

+«)

2

+ {unxf]dx < C7\\F\\l2h^.

(76)

D

By virtue of (76) and of weak compactness of a closed ball in the Hilbert space W+, from the sequence {«"} one can choose a subsequence (we have the same notation) weakly converging in W+ to some element u G W+. In order to prove the indentity (61) for such u, we multiply each equality (73) by a function dt(t) € C 2 [0,T], de(T) = 0, and sum up the obtained equalities over all t from 1 to n and integrate with respect to t from 0 to T. Further, integrating by parts the first term, we get the identity / {-p(x)(h{x)un)tvt

+ h(x)uxvx

+ a-Litfv + a2uxv + a3unv) dD

D

= JFvdD,

(77)

D n

which is valid for any v of the form ^di(t)(pe(x).

We denote the collection of

£=1

such v by Vn. In (77) passing to the limit over the above chosen subsequence for fixed v from some Vn, we arrive at the identity (61) for the limit function u G W+, which is valid for any v € U Vn. n=l

Now let us show that U Vn is dense in V. Indeed, let v G C2(D) and 71=1

v\x=o = v\x=i = v\t=r — 0. Then there exists a continuation vo of the function v onto the rectangle II : 0 < x < 1, — T < t < T in C 2 (II), such o that uo|an = 0, VQ\D =V [31]. Hence, VQ G W\(II) and as far as the system of

210

functions wm(t + < sirnrkx • sin ^T

T)\°°

(78)

J fc,m=l

is a basis in the space W\ (II) [32], for any e > 0 there exists a linear combi771

nation ^

GJJUJ

of the functions from the system (78) such that

(79)

^ / * i U i || w ^ n ) < e,

«o

7= 1

since ||w||

= HHIw^fn)-

0

Further, by virtue of (2) we have \\w\\v <

V^2lkllw^(D)> Vw G y. Therefore, taking into account (79), we have v-

y^ajUj

S < \Jh2 V-' £jOLi 7=1

7=1

<\fh~2

Wl(D)

< y/fae.

VQ

(80)

7=1

But the function

^OJJUJ

{v e C2{D) : v\x=0

=1

€

U Vn. Therefore, from (80) and as far as

= v\t=T = 0} is dense in V we conclude U Vn

is dense in V. Since u G W+, then in turn it follows that the identity (61) is oo

true for any v G U Vn holds for any v £ V as well. Thus, the limit function Tl=l

u = u(x,t) is a weak generalized solution of the problem (1), (3), (4) in W+. The uniqueness of such a solution was shown above. Remark 1.1.7 As far as the estimate (76) is true for the sequence {un}, and from the weakly convergence of the subsequence {un} to u in W+ it follows that ||u||w + < lisi ||u n ||w + , we conclude that for the solution u of the 77—+OO

problem (1), (3), (4) in W+ from the Theorem 1.1.1 there follows the estimate

\\u\\w+ <

cWFU^

with c = const > 0 independent of F. Remark 1.1.8 Using the known scheme of investigation of smoothness of generalized solutions 16 , one can show that if p, h G C^O, 1], the coefficients

211

au, i = 1,2, satisfy the conditions (8), a3t satisfies the condition of type (20): 2 sup (hai)-^ a3t(x,t)

< +oo,

D

Ft £ L2th-\(D), then the generalized solution of the problem (1), (3), (4) in W+ from the Theorem 1.1.1 will belong to W 2 2 loc (D). 1.2

The mixed problem for a hyperbolic system of the second order with order degeneration

Let us consider the following hyperbolic system of equations [14, 17] a2°>° (A + 2/i)(/i2/i3 « i,i).i +3A(h 2 /i3t'3),i +*i° = Pfohs -—, 0 0

(81)

1,0 0,0 lf0 ,302V3 p,(h2h63 v 3 ,i),i - 3(A + 2p)h2h3 v 3 - Xh2h3 v M + h3Xg = ph2hd3-^-, (82) where A, n = const > 0, p(x) > pa = const > 0, h°xai < hi{x) < h\xai a.i = const > 0, h\= const > 0, i = 2,3; j = 0,1;

keCto.ilnCHo.i], Po, Pi = const > 0,

p e<7(o,i],

7 > 1,

PQX~^ < P{x) < p1x—'\

(83)

7 > 20:3.

o,o i,o 0 0 1)0 X®, X3 are given and vi, v3 are unknown functions. The system (81), (82) will be considered in the rectangle D := {(x,t) G R2 : 0 < x < 1, 0 < t < T} . By virtue of (83), we have /i,(0) = 0, i = 2,3, i.e., the system (81) (82) has the order degeneration on the left side To : x = 0, 0 < £ < T of the rectangle D. Let Ti : x = 1, 0 < t < T and T*a := { ^° U Fl.f " < *' 111 it o. > 1. We rewrite the system (81) (82) in the following form Lu := (Aut)t - (Bux)x

+ (du)x

+ C2ux + C3u = F,

where u = (ui,u 2 ), ui = °'i°, u2 = U3°, F = (Fi,F 2 ), Fj = X ° , F 2 = A =

.' ph2h3 0

0 \ ph2 4)'

fl={(\

+ 2»)h2h3 0 \ V ° M2/13J'

(84) foX°,

212

1

fO -3Xh2h3\ \0 0 )>

°

f 0 0\ \\h2h3o)'

2

3

~ V° 3(A + 2/i)ft 2 /i| _

P r o b l e m 1.2.1 Find a solution u(x,t) of the system (84) satisfying the initial conditions /

„v

du(x,0)

u(z,0)=0,

—^-i=o,

0<x
(85)

and the boundary conditions u

w ilr,«=°> 2|r , x, , = 0 , a = a 2 + a 3 . (86) Parallel with the problem (84)-(86) let us consider the problem of finding solutions of the system

L*v := (Avt)t ~ (Bvx)x

- C[vx - {C'2v)x + C3v = F,

(87)

with the initial and boundary conditions as follows dvix T) —K—r-l = 0 ,

v(x, T) = 0,

Wllr

=°>

«2| r

0 < x < 1,

(88)

=0,

(89)

where C is a transpose of the matrix C, i.e., L* is the formally adjoint to the operator L. Below it will be shown that the problems (84)-(86) and (87)-(89) are adjoint in the corresponding function spaces. Let us denote by E and E* subspaces of C2(D), satisfying the corresponding initial boundary value conditions (85), (86) and (88), (89). Moreover, when 7 > a, we require in addition that these functions vanish in some neighbourhood ofTo:a; = 0 , 0 < i < T ' (own for each function from E or E*). Let W+(W+) be a weighted Hilbert space obtained by closing of the space E{E*) under the norm

Ml 2

.., = / ^2^3 [u\ + pu\t + u\x] dD D

+ f h2h\ [u\ + pu\t + u\x\ dD.

(90)

D

We denote by W-(WZ) the space with the negative norm constructed by L2(D) and W+(W£) 26 .

213

Remark 1.2.1 In the space W+(W+) the above - introduced norm ll-llw (w*)' by virtue of (83) and (90), is equivalent to ||u|| 2 := Jxa

[u\ +pu2lt + u2lx] dD + Jxa>+3a>

D

[u2 + P< + < ] dD,

D

which, in turn, in view of the proved below inequalities / h2h3uldD


D

h2h3 [pu\t + u\x] dD, D

\

h2h\u\dD

D


V(ui,u2)€W+(W;),

(91)

D

is equivalent to the norms

Ml2 :=

J xa [pu2t + u\x] dD + J xa*+3a*

[pu\t + u\x] dD

(92)

f h2h\ [pu\t + u22x) dD.

(93)

D

and IH | 2 := I h2h3 [pu\t + u\x] dD+ D

Remark 1.2.2 Let x6'1 (7&{x) = <

1

\ln-x\

,-i-

,

5^1, (94)

l

x~ \ln-~x\

,

5 = 1,

where eo =const> 0 is as small as desired. Then according to results of [28-30] for any u = (ui,u2) G W+(W+) the estimates J oau\dD

< C J xa [pu\t + u\x] dD < CC* f h2h3 [pu\t + u\x] dD, (95)

D

D

f aa2+3ct3uldD D

D

< C Jxa2+3a*

[pu\t + u22x] dD

D

< CC* / h2h\ \pu\t + u\x\ dD D

hold with positive constant C, independent of u,C* = (/i") - 1 ^ 1 ") - 1 -

(96)

214

Lemma 1.2.1 Let 03 < 1. Then for any u G E, v G E* the inequalities \Lu\\w.


li^llw


(97)

W+ !

(98)

hold, where the positive constants c\ and c2 are independent of u and v. Proof According to the definition of the negative norm for u G E in view of (85), (86) and (88), (89) we have ll-Hlw

: =

SU P \\V\\W'(LU>V)L2(D) v€W^ +

= SUp IMIvy 1 . (Lu,v)L2rD) v&E* +

= sup \\v\\w' / \-ph2h3uuV\t veE' +J

+ (A +

=

2/j,)h2h3Ulxvlx

D

+ 3\h2h3u2vix - ph2h\u2tv2t + ph2h\u2xv2x + Xh2h3Uixv2 + 3(A + 2/j.)h2h3u2v2]dD.

(99)

By virtue of (83), (90)-(93) and of the known inequalities f h(x)fgdD

< ( J h(x)f2dD)

I j h{x)g2dD\

, h(x) > 0, x > 0,

D

< (xj + x%)i(y2 + yl)*,

\xiyi+x2y2\ we have r

/ [~ph2h3Ultvit

+ (A + 2n)h2h3uixvix

- ph2h\u2tv2t

+

\xh2h\u2xv2x\dD

2

< (A + 2/i)

2

J h2hs{pu

lt

j h2h3(pv2t

+ u\x)dD J

JD

+ v2x)dD

ID

2

2 +P j h2h\{pu 2t + u\x)dD

D <(A + 2 / i ) | | « | | w + | H | I v . .

J h2hl(pvl J

+ v\x)dD

ID

(100)

Further, in view of (83) and (94) in the case 0:3 < 1, there exist CQ =const> 0 such that (h2h3)(x)

< c0
0 < x < 1.

(101)

215

In view of (92), (95), (96) and (83), (101) we have

2>\h2hzU2V\xdD

/

< 3A

fh2h3uldD\

< 2>\sfco I / <Ja2+3a3ui dD

< 3 A ^ C (jx^+3a3

<3\yfaC(h$)-i(h°3)-i

I j

h2h3vlxdD

lid

[pu22t + u\x\ dD )

(jh2hl[pu22t

\\v\\w.

+ u22x}dD\

\\v\\w.

<3\yfoC{h°2)-l(h03)-i\\u\\w+\\v\\wV

(102)

Similarly /

Xh2h3uixV2dD


D

/ 3(A + 2fi)h2h3U2V2dD < 3(A + 2^)c 0 I / D

x I

\D

(Ta2+3a3v%dD J


aa2+3a3u\dD ) (103)

Prom (99)-(103) there follows easily (97). The same assertions lead to the inequality (98). Remark 1.2.3 By virtue of (97), (98), the operator L : W+ -> W*_ (L* : W+ —> W-) with the dense domain of definition E(E*) admits closure which is a continuous operator from the space W+(W+) into the space W1{W-). Leaving for this closure the same notation L(L*), let us note that it is defined on the whole space W+(W^jl).

216

Lemma 1.2.2 The problems (84)-(86), and (87)-(89) are mutually conjugate, i.e., for any u £ W+ and v £ W+ we have (Lu,v) = (u,L*v).

(104)

Proof By virtue of Remark 1.2.2 it is enough to show the equality (104) in the case when u £ E and v £ E* .In this case obviously {Lu, v) = {Lu, U)L2(.D)> and in view of (85), (86) and (88), (89), after integration by parts we get (104). Remark 1.2.4 Let us note that parallel with (95) and (96) the stronger estimates 28 ' 29 j oau\dD


D

< CC* j h2h3ujxdD,

D

< C jxa2+3a3u%xdD

j o-a2+3a3uldD D

(105)

D

< CC* I h2h\u\xdD,

D

(106)

D

are valid, where u = {ui,U2) £ W+{W+). In what follows we suppose "3 < \ ,

hlxa-1

< {h2hz)x

(107)

< h\xa-\

(108)

where h\,\, h\ =const> 00. Let ||i;||2

:=

/ ph2h3v\dD+ D

I

ph2h\v\dD.

D

Lemma 1.2.3 Let the conditions (83), (107), (108) be fulfilled. Then one can find a positive number p* = p* (A, p,a2,a3,h%j,T), such that if po > p* or infx1p{x) > p*, then a priori estimate [0,1] t

c\ML2,h<\\L*v\\w_,

Vv£WZ

(109)

is valid with a positive constant c independent of v. Proof By virtue of Remark 1.2.3 it is enough to show that the inequality (109) is true when v £ E*. If v £ E*, then for v = const > 0 the function t

u{x,t)=

I' eVTTv{x,T)dr

(110)

217 belongs to the space E. In view of (110) the equalities ut{x, t) = euttv{x: t),

v(x, t) = e"vtt-lut{x,

t)

(111)

hold. Let (vx, i>t) be an outward normal to dD. Taking into account (85)-(89) and (111), we have (L*v,

U)L 2 (D)

= / [-ph2h3vituit

+ (A + 2p,)h2h3vlxuix

+

Xh2h3v2uix

D

-ph2hlv2tU2t

+ fJ-h2hlv2XU2x - 3\(h,2h3)xviU2 -

3Xh2h3viU2x

/t

+3(A + 2n)h2h3V2U2\dD = - / ph2h3e' tv1v1tdD D +(A + 2/i) / e~vtt~1h,2h3uixtu\xdD

e~utt~1h2hlu2XtU2XdD

+p /

D

D

+ / [Xh2h3V2Ulx - 3\(h2h3)xviu2 D +3(A + 2p)h2h3v2u2}dD, - I ph2h3evttvlvlt}dD

-

3Xh2h3v1u2x (112)

- f ph2h\evHv2v2tdD

D

Iph2hle t{vl)tdD

D vt

= - 1 fph2h3e tv\utdS

D

ph2h\evHv\vtdS

f ph2hlevt(l

+ vt)v\dD

D vt

= \ j ph2h3e {\

+ vt)v\dD + 1 f ph2hlevt(l

D

+ vt)v\dD

D

= \ Iph2h3evtvldD+)D

JPh2h\evtvldD D

f vph2hze-utr1u\tdD

2

f dD

+ vt)v\dD + |

D

\

-]-

D

+ - J ph2h3ev\l

ph2h3eutt{vl)tdD

=- | j

D vt

~\

ph2hlevttv2V2tdD

- / D

+ i f vph2hle-vtr1ultdD.

D

D

As far as "t | „ o = ^l a , = 1 = 0,

i/t | t = T = 1,

* - 1 * 4 | t = 0 = 0,

(113)

218

it is easy to show that (X + 2fjt) / e~tJtt~1h2h3ulxtulxdD

e~vtt~1h2h\u2XtU2xdD

+ /j, /

D

D

= i(A + 2/x) J e-vtrlh2h3u\xvtdS+^

e'vtt-xh2h\u\xvtdS

f

3D

3D 1

2

2

+ i(A + 2M) /"/laAae-"'^- + r ]u lxdD + | D

/' h2hle~vt[vrl

+r2]u2xdD

D ut

1

2

> ^(A + 2^) Jh2h3e- [vt-

2

+ t- ]u lxdD

D

+ f Jhhle-^lvt-1

+r2]u2xdD.

(114)

By virtue of (107), (108), similarly to (101), we have (h2h3)x{x) < c0<Ja2+3a3(x),

0
c0 = const > 0.

(115)

Further / Xh2h3v2uixdD

(h2h3vldD\

I

[h2h3u\xdD

D

< ^Po\h°3)-2

Jph2h\v\dD

+ e Jh2h3u\xdD,

(116)

where e is an arbitrary positive number. By deduction of (116) we used the following inequalities \ab\ < ^-a2 + £b2, 7 > 2a3 and po{h°3)2 < p0(hl)2x2a3-^

< ph\, 0 < x < 1.

By virtue of (83) and (108), we have 7 > 2a3, 7 > 1 and (h2h3)x < h\xa~l < hKhlhD^h^x-ix"!-1

<

p^hUhlhD^ph^,

219

(h2h3f

= (Ph2h3)(h2hl)(p-1h^2)

(ph2h3)(h2h33)(p0(h°3)2)-1^-2a3

<

< Po1(h°3)-2(Ph2h3)(h2hl), l
fsX(k2h3)1

ViU2
<3A I

D

{h2h3)x\viu2\dD

D

< 3A(c0)5

(h2h3)xv2dD

/

aa2+3a3ujdD

< 3A(c0)5 I f po1h14(h02h°3)-1ph2h3v2dD

J

= SXicoipoh0^)-1^)^

/

j ph2h3vldD

< 3\(c0{poh02h°3)-1h14)2{CCt)i 3P coipoh^h^hlCC* 4e

I f ph2h3v2dD

jph2h3v2dD

aa2+3a3u22dD

aa2+3a3u22dD

j

J

h2h\u22xdD

+ 3e f h2h33u2,xdD,

(117)

j 3Xh2h 3viu2xdD D

'o * (ft")"1 ( j ph2h3v2dD J I J h2h2u2xdD <

3A^ Po\h°)~2 4e

fph2h3v\dD

+ 3e f h2h33ulxdD,

(118)

220

/ 3(A + 2p,)h2h3v2u2dD

< 3(A + 2/i)

/ h2h3v%dD J

<3(A + 2^) I J Po1(h°r2ph2hlv22dD\

h2h3u\dD

aa2+3ct3u2dD

I co J

j jph2h\v2dD

< 3(A + 2/i) (poHh^coCC^

I f

J

j /h2h\u\xdD

,<(£>.

(119)

Now, in view of 1

v=

1

^--^-ut[vt-l+r2}

e"*>l,

T2

O<*
t~l > - ,

0
eT 2 '

e~vt>e-1,

from (112), by virtue of (113)-(119) when v = ^ , we have (L*w,u) L2(D) > - / ph2h3v2dD

hjph2h* \u\ dD

2eT 2 7

A ut

r

Po1^)'2

D D

As

' As

D

J ph2hlv22dD - s J D

3A

h2h3u2JD

D

l

c0{Pohlhl)- h\CC*

f ph2h3v2dD

- 3s f

D

Po1^)'2

3{\ +

I ph2h3vldD

2p)2p^1{hl)-2cQCa

h2hlujxdD

D

- 3s I

D

As

ph2h3u2udD

+ X + 22/ i / ph2h3u2xdD + ^ 2 j ph2hlu22xdD 2eT

D

As

ph2h\v22dD + ^ - ^ /

+-

h2h\u\xdD

D

• / ph2hlv2dD D

-3s

I /12/13W dD D

221

+

1 2

3 A 2 ( c o ( ^ ) - 1 / i l C a + (/13T1) 4ep0/i3

J

1 2

A2 + 3(A + 2 / i ) 2 c 0 C a 4 £ / o 0 (^) 2

D

+-L-2 J ph2h3uttdD + ^ (A + 2^) 2eT2

I ph2h\vl dD

J ph 2h\ultdD

ph2h3u\xdD

/

ph2h3v\dD

h2hlulxdD.

2eT 2

(120)

D

D

We see from (120) that if as e we take e = ^min 18eT2 ' (A+2^) and 2eT a suppose that r = po is a sufficiently large number, then for a sufficiently small number 5 = const > 0 we obtain (L*v,u)L2(D)>5

\\v\\

+ \\u\

>25\\v\

from which we get (109) in a well-known way. Remark 1.2.5 Obviously T

= po = po(A, p., T, a2, a3, h)).

Lemma 1.2.4 Let the conditions (83), (107), (108) be fulfilled. Then a positive number p\ = p\{\ n, a2,a3, h^T), will be found, such that when Po > p\ for any u £ W+ the inequality c\\u\Lh

< \\Lu\\w,

(121)

is valid with a positive constant c independent of u. Proof Similarly to the case of Lemma 1.2.3, in view of Remark 1.2.3, it is enough to show the inequality (121) for u e E. Let u G E. Then it is easy to see that for v = const > 0 the function 1

j{T-T)e-VTu{x,T)dT

v{x,t)= belongs to the space E*, and vt(x, t) = - ( T -

tfe-^uix,

t),

u(x, t)

(T-i)-W(z,*)-

(122)

By virtue of (85)-(89) and (122) as well of the equalities vt\t=0

= -l,

vt\t=

1,

vt\

"tL =1 =0,

(T-*)-X|t=T=0,

222 we have (Lu, v)L2{D) = I [-ph2h3ultvlt + (A + 2p)h2h3ulxvlx D +3\h2h3u2vlx - ph2hlu2tV2t + ph2h\u2xv2x -X(h2h3)xu1v2 - Xh2h3u1v2x + 3(A + ^)h2h3U2V2}dD = J ph2h3(T - t)e-vtuxult\dD

+ f ph2hl{T - t)e~vt u2u2tdD

D

D

-(A + 2/x) J{T - trlevth2h3vlxtvlxdD

- (i f (T ~

D

ty^f^hlv^v^dD

D

+ J [3Xh2h3u2vlx - X(h2h3)xUlV2 - Xh2h3u1v2x D

+3(A + 2p)h2h3u2v2}dD, / ph2h3(T - t)e-vtUlult)dD

^3) t)e~vtu2u2tdD

+ j ph2hl{T -

o

i vt

+ i / ph2h3[l + u(T I

t)}e-vtu\dD

+ 2 J ph2hl(T - t)e-vtu22utdS + 1 f ph2h33[l + u{T dD I

t)]e-vtu\dD

= g J Ph2h3{T - t)e- u\vtdS dD

= 2 J P^h3[l + v{T - t)\e~vtu\dD + I fph2h33[l + p{T - t)}e~vtu22dD D

D l/t 2

vt

= 2 / ph2h3e- u 1dD + i D

/ P h2h\e- u\dD D

+lfvph2h3(T-t)-^%*t]dD

+ ±[Vph2hl(T-t)-1el'tvldD,

D

(124)

D

-(A + 2/x) ^ T - tyxevth2h3vlxtvlxdD D

= -\{X + 2/x) / ( T - tyWhihsivufvtdS

-fi f(TD

-

t^e^hahlv^v^dD

223

- | J(T -

ty^^hihUv^fvtdD

dD

+ \{\ + 2M) | [(T -t)~2 + u(T - t)-1} evth2h3v\xdD + | J' [(T - t)~2 D

+ v{T - ty1]

D

evth2h\v\xdD

> i(A + 2y) f [(T - ty2 + v{T - t)'1}

euth2hzv2xdD

D

+ f / [(T - t)~2 + v{T - ty1} evth2h\v2xdD.

(125)

D

In a similar way as by obtaining the inequalities (118), (119), we have / ZXh2h3u2vixdD D

< T - P o 1 ^ ) - 2 / ph2h\u\dD + 3e f h2h3v2xdD,

I

(126)

X(h2h3)xuiv2dD

D

< ^-ca{Poh°2hlylh\CC*

Xh2hzU\v2xdD < ^Pol{h%)~2

f ph2hzu\dD + e I h2h\vlxdD,

f ph2h3u\dD + e f h2h\v\xdD, D

(127)

(128)

D

/3(A + 2/x)/i'i22h3u2v2dD D

< ^ + ^)2Po\h°3y2c0cct

jph2hlu2dD

+ 3£jh2hlv2xdD

(129)

224

Further, taking into account that 1 1 T

+ viT-t)-1]

•.^[(T-tyi 2 e~vt>e-\

>

rp2

(T-i)"1^,

0
from (123), by virtue of (124)-(129) when v = ^, we get (Lu, v)L2(D)

> — / ph2h3u\dD

+ — J ph2h%u\dD

D

D

+ ^f2 J ph2h3vjtdD

J ph2h\v2tdD

+^

D

+ ^ ( A + 2p) J

D

e + ^f2^j

f

D

3A2 2 ph2hlv 2xdD - -—p-\hl)~2

f J ph2h\u\dD

D

f - 3e /

D

2

-—coipohZhD^hlCC, :Pol(lA)~2

h2h3v2xdD

D

2

I ph2h3u dD

- e I h2h3v 2xdD

D

4ef

ph2h3v2xdD

D

J ph2h3ujdD-e

h2h33vlxdD

f D

2

2

3(\ + 2p) pa\h°3)- coCC* 4e

j nh h^2, I ph 2h\uldD

v2xdD h2h\vl

- 3e / D

1 2e

+

2

1

A ((/tip-i^CC. + (hi)- ) 4ep0h%

1_ _ 3A2 + 3(A + 2p)2c0CC* 2e ^epo(hf)2

I

ph2h3u\dD

\Ipk2hlu

2

dD

D

e / ph2h3v2tdD 2T2 D

+~

j

ph2h\v\tdD

D

^ ( A + 2/i) - 3e] J

h2h3v2xdD +

2T2

p — 5e J ph2hlvlxdD.

D

D

Now having taken e—

2

-mm

6^(A

+ 2/i)

'1^2/i.

(130)

225

from (130) by sufficiently large po we obtain (Lu,v)L2{D)

> Si

+IMI2.

Ml? 1

L

" 2,h

"

>2<5i|H| L , J | u | | w , , from which in turn there follows (121). Let us denote by L2!fl(D)(L2,h.-1{D)) (u\,U2) (v = (ui,U2)) with the finite norm

HI2,

= / ph2h3uldD

the space of functions u

+ /

D

MI

S„-i

=

"W*

Si= const > 0 ,

ph2hluldD

D

/P~ l h 2 1 h 3 1 v 2 1 dD + jp-'h^h^vUDj D

-

D

)

Obviously, these spaces are Hilbert spaces. Definition 1.2.1 Let F G L,2th-1(D). The function u will be called a strong generalized solution of the problem (84)-(86) in W+, if u € W+ and there exists a sequence of functions un E E such that un —• u in the space W+ and Lu n —> F in the space Wl, i.e., Hm ||u„ - u\\.1 W i = 0, n—>oo

+

/im ||Lu n - F\\

- 0.

n—*oo

Definition 1.2.2 Let F e W l . The function u will be called a strong generalized solution of the problem (84)-(86) in L,2,h if u € L2,h(D) and there exists a sequence of functions un € E such that un —» u in the space L,2,h(D) and I/Mn —> F in the space WZ, i.e., /im ||u n — u|| L

=0,

lira \\Lun — F\\w.

=0.

n—*oo

Prom the Lemmas 1.2.1-1.2.4 in a well-known way [27] there follows the following Theorem 1.2.1 Let the conditions (83), (107), (108) and p 0 > max(p*,pl) be fulfilled. Then for any F g WZ there exists a unique strong generalized solution u of the problem (84)-(86) in L2th for which the estimate (121) is valid. If F S L2th~1 C WZ, then this solution will be also a strong generalized solution of the problem (84)-(86) in W+. Remark 1.2.6 The condition po > max{pt,,p\) in Theorem 1.2.1 can be replaced by the requirement of T to be sufficient small. E.g., in Lemma 1.2.4

226

we first take e so large that

i

A'ftfegr^cq + Cftg)-1)

Ye

4iMl

> * = const>0,

1

3A2 + 3(A + 2/X) 2 CQCC«

2e

4ep0(^)2

>

'

and then we take T as small that 2jp(\

+ 2n)-3fj.>8,

^fi-5e>5.

Remark 1.2.7 Let us note that according to the well-known embedding theorems 28 ' 29 boundary conditions (86) are satisfied in the following sense. The solution u = (ui,U2) S W+ takes the zero boundary values in the sense of the trace operation. 2 2.1

Cusped Bars Some formulas of a bar model

Let a domain of R3, occupied by an elastic bar, be

V :— {(xi,X2,X3)

: 0 < x\ < L,

2hi(x1):=hi-hi>0,

(-) (+) hi(xi) < Xi < hi(xi), i = 2,3, L = const}, hi e C([0,L}) n C^lO.Lf), » = 2,3,

and let 2/13 and 2/12 be correspondingly the thickness and the width of the bar and their maxima be essentially less than the length L of the bar. Let further f(x1,x2,x3)€C1(V), and at points Xi, where both the thickness and the width of the bar do not vanish, define double moments of the function / and its first derivatives f,j as follows: (+)(+)

f (an) := / (-)(-> /l2 /l3

/ ( * ! , x2, x3)Pn2 (a2x2 - b2)Pn3 (0-3x3 ~ b3)dx2dx3,

(131)

227 (+) (+) i,«2

r hi

f (xi) := / 3

r h3

f,j{xi,X2,X3)Pn2(a2X2-b2)Pn3(a3X3-b3)dx2dx3:

<-) (-> /l2 h-3

rii = 0,l,...,

J = 2,3, j = 1,2,3,

where 1 h { K * Ol V W di := —, bi := —, 2hi := hi + hi, hi hi Pni, i = 2,3, are Legendre polynomials:

• 1 1 t = Z,6,

+i

J Pk(t)W)dt =

y^fa,

-i

i.e., if t = ciiXi — bi, (+> hi

k + - j / Pk(a,iXi - bi)Pi(a,iXi - b^c^dxi = <JW, (-> 5fc; is the Kronecker delta, indices after the comma mean differentiation with respect to the corresponding variable. Let us recall the reformulation of the main relations of the linear theory of elasticity in terms of the double moments of the sought fields of displacements, strain and stress tensors. All double moments at the points x\ £ [0, L], where at least one of /ij = 0, i = 2,3, will be considered as limits of the double moments calculated at points where hi > 0, i = 2,3. To this end let us recall the main relations of the linear theory of elasticity in the isotropic case e

ij := 2^Wi'3

+W

J,i)'

Xi:i := \6ij6 + 2/iey, Xijti+Xj=p-^-,

*>•? = I-' 2 ' 3 !

6 = ekk, d w•

i,j = 1,2,3;

j = 1,2,3;

(132) (133) (134)

228

where we use Einstein's convention, and assume that Latin indices run the values 1, 2, 3; u>i, i = 1, 2, 3, are displacements; e^ and X^, i,j = 1, 2, 3, are strain and stress tensors, correspondingly; p is a density, A and /x are Lame constants; Xj, j = 1,2,3, are components of the volume force. Multiplying both parts of equations (132), (133), (134) by -Pn2(a2^2 — b2)Pn3 {0-3X3 — 63), and then integrating them with respect to x2 and X3 with (-) (+) (-) (+) in /12 and h2, /13 and /13, correspondingly, taking into account (131), after some calculation we obtain the formulas as follows n3,n2 Xij

= Xdij

n3,n2 6

+2/i

(let Wj(xi,X2,%3) & C2(V), series as follows

n3,n2 etj,

™3,"2 n3,n2 6 •= ekk,

i,j

= 1,2,3;

(135)

j = 1,2,3, then we have a uniformly convergent

wj(x1,x2,x3)

=

° 2a3 ( m2 + 2 ) ( m3 + 2 ) m ^ 3

Yl

m2,m3=0 ^ ' ^ x P m 3 (a3a;3 - 6 3 ) ^ 2 ( a 2£2 - 6 2 ),

^

' (136)

where double moments are defined by (131)) "3,"2

"3,"2

3 / 0 0 V~^ / rii ,• ra3,n2 a

e n = wi,i + 2 ^ I n,i=2

"3,"2 ej = -

w

V-*™','

i - 2^

\

»

ai

\ <5i2n3+(5i3S,<5,;2S+<5i3"2 \ Wl

I'

s=m

V - ^ P* 2_^ °isaL s=m+l

2 e23 = - 2_^ 2 ^ "is ai

, __>

t137'

/

*,;2n3+<5i3S,(5i2S+<5i3n2 w i >

3 00 V " ^ V~* 1

„«3ii2

&

. „ „ * = A<*,

/i oo\ (1-5°)

i5i2n3+<Si3S,i5.i2S+5.i3n2 W5 i

-

.

'

^ '

i=1 s = n ; + l

A I 00 ri3,n 2 713,"2 T—"• I «/fc t- " 3 , " 2 V—"> n,fc u i5fc2n3+<5fc3S,<5fc2s+Ofc3n2 w 2 e H = Wi,i + 2_J\ank i - Z^ bs ak w% fc=2 V s=nfc 00

E

™'; bisdi

6i2n3+5i3S,<5i 2 s+(5i3n2 wx

. , 1 = 2,3,

\ I I / ,

.

(140)

229 where

(+)

(-) (-l)s+n'h.i , l

hiA -

(141)

+s

£ : = - ( s + £)[i-(-ir ] (142)

Hi

b \Ou

3 n-,,

13,"2

S ^ Hi,

n

.

.

l5i2«3+<5i3S,<5;2S + <5i3«2

n

.

(5;2«3+<5i3S,<5;2S + <5;37l2

X;

+ a*. t=2 s=0 n3,n2 X

^ a2

J

= P

"3,n2

Q™2 » J' = l , 2 , 3 , n 3 , n 2 = 0 , l , . . . ,

where (+) h;

Q

1+ i=2

(-)

/(+) \ /15-ia

-X <+)

(+) (+) ( x i , <Jj2a;2 + SiSh2,Si2h3 +

5i3x3)

\

hi

+ (-l)"6-^

1+1

/l5_i,l I -X" (_)

\xi,Si2X2

+ 5i3h2,5i2h3 + Si3X3

\ "3 ,"2

x P n i ( a i a ; i - bi)dxi + Xj , j = 1,2,3,

n 3 ,7i 2 = 0 , 1 , . . . ,

/ (±) (±)\ (±) •X(±) . I Xi,6i2h2 + 5,3X2,^2^3 + <5i3ft.3 1 = XkjVik, / -X(±>

(143)

1 = 2 , 3 , j = 1,2,3,

(±) (±)\ ari, fe^ + ^ 3 ^ 2 , ^ 2 ^ 3 + <5j3^3 > J = 1) 2 , 3 , are components of t h e

^ V

/

(±) surface force acting on the surface Xi = hi{x\),i the bar.

= 2 , 3 , from the exterior to

230

Evidently, 1 3 , «2

n;-l

j=2

71.; . (5i2Tl3+5,;3S,5.,;2S+<5i37l2

TU <5,:2rc3+6i3S,(5i 2 S + <5i37l2

s=0 ^

+ / i ^ ^ 3 X? = ph?h$*

J , i = 1,2, 3, n 3 , n 2 = 0,1, • Q,9 dt2

(144)

Now, we assume ™ 3 '™ 2

n

i

oo

(145)

if at least one of the following conditions rii>Ni,

i = 2,3,

is fulfilled. Then from (136), we get Wj(x1,X2,X3) N2

N3 a

~ ^2 E „2=0n3=0 N2 N3

2 f l 3 ( n2 + - ) ( n3 + - ) n3w™2 Pn2 (a2x2 - b2)Pn3 (a3x3 - b3) Z

V ,

S V . , 1 .

j

2+

= E E ^ 2j( n 2 =0n 3 =0

v

y

n3+

^ 1

5j 2 ^ ^n \n \ 3

2

N

3

X

(x3-h3)2-hj\n3

d713

d"2 [(a* - /i 2 ) 2 - /i£

<&£3

dx%2

where "3, " 2 ,

Vj

U)j

.

(n) :=

^

2

+ 1

(n)

(^i>33+

1

(

(^l)'

1

4

6

)

Thus, taking into account (141), (142), (145), (146) from (137)-(140), (135) we obtain: "3,"2

.712 + 1 , H l + l

e n = h2

"3,"2

/ i 3 3 T v\,\

~E E i=2

5=n/

+ l

E (-) = o,

^

/l7;

6^

«i

231 Ni _ P~v

llSi2S+5i3n2 1-2 Hi)

-£

"3,12

=71; + 1

3

w « L&i2S+5i3n2 X—"• < l o

;—••

2 e23 = - ^

i=2

+ luSi2na+Si3s+l "lln3 ~ /li

+ lL,6i2n3+6i3s+l ""<

2^

s=n.i + l

+1

2 e^=^ ^

3+1

„ . 1

ois

hi

3

n3

. . , . n /"* , " ' Oi2ll3+di3S,<>i2S+6i3n2 0i2™3+O«3S,Oi2S+Oi3 ' Ois ^i

N

E

fc = 2 j i Jf ji , * Ok2n3+ok3s,6k2s+Sk3n2

^ ^—\

Si2n3+6i3Si5i2S+Si3n2 Ui

T!^'

/\

, n

i = 2,3;

3 V ~ ^ 713,"2

x«3,n2

i=2

,.

, -

, / , „ , +!,

r

, ,+ i n 3 , T l 2

eu = (A + 2/x) (/i 2 l2+i /i 3 l3+1 u M

JVi Si2S+Si3Ti2+l l u6i2n,3+5i3S+l

E E *S j=2

*

1 = 2,3,

X u = (A + 2/i) e u + A 2 ^ 3

ft'

, - 1 'J* i <Si2n 3 +«;3S,i5»2S+«i3n2 "1

ft."1 6

s=n;+l

-AE 3

/ i ^ s + f a " 2 + i ^ " 3 + f e s + i v 1 6 ; s 5 i 2 n 3 + 6 i 3 ^ i 2 S + a i 3 n 2 , (147)

N;

E

=2 s = i i i + l

l 3

V

2

\ /n3,"2

"3,12

\

, ,.

, -

An3,"2

, / , „ „+ ] ,

n

, + l"3,"2

-Xji = A (^ en + e5_i 5_iJ + (A + 2/z) e* = A (/i2l2+1/i^3+1 wM 3 V ^

JVfc V ^

^fc2S+<Sfc3n2 + l /l <5fc 2 n3+<5fc3S + l / l - l n , f c fc

fc=2 s = n f e +l JV5-i

- £ 5

,

J,<Si2S+l5i3n2 + l L < S i 2 l 3 + < 5 i 3 * + l n , ft ft2 3 L

ni=0,Nh " 33 , " 2 2

w5_i

ft*

S=7lfc + 1

S=Tli + l

X

Oi2l3+di3S,Oi2S+Oi3Tl2

L(5fc 2 S+< 5 fc3n2 + ll ) l5) [ 2n3+l5fc3« + l

k

Xi-E

. , 2 — Z , D,

" ' X 05-s s

< 5 5 - i 2 S + l 5 5 - i 3 n 2 + l1 L < 5 5 - i 2 l l 3 + < 5 5 _ i 3 S + l r — 1 /l25 ft 5-i

+1 l?5-«2»l3+55-i3S,<55-i.2S+l55_i3n2 V5~i

<5

'=2«3+«fc3S^<5 fc 2S + '5fc3n2

232 JVi

-(A + 2M) J2

^ * 2 S + < 5 i 3 n 2 + 1 ^ 2 " 3 + f e s + 1 V 1 £ * a n 3 + 4 «" i * a a + *"* >

t = 2,3,

(148)

"^2 n 3) n 2 A23 = ^M e 23

— ~P

3

JV4

ZJ

Z^

ft

2

ft

3

i

"is

V5_j

,

Z=2 S—Tli + 1

(149)

3

_ y

JVfc

>

y^

h&k2S+Sk3n2

+ lhSk2n3+Sk2s+lh-ln^h

^2n3+Sk33^k2s+6k3n2

k=1 s=nk + l

-

^S+S^

J2

+1

h^^+S^+lh-1g

^n3+Si3sA2S+5i3n2y

^

t = 2,3,

n* =0,JVi,

i = 2,3.

Substituting (147)-(150) into the first (N3 + 1)(7V2 + 1) equations of the system (144) for fixed j = 1,2,3, we will have the following system of 3(N3 + 1)(^2 + 1) equations with respect to 3(A^3 + 1)(A^ + 1) unknown functions vlix^t), r = 0,N3, s = 0,N2, fc = 1,2,3:

A i (/ii n 2 + 1 /iI n 3 + 1 Xi) 1

+ E E E (*£ ^,1+5?, «i) + / « 3 x9(Xl,t) i = l r = 0 s=0 A2hn2 + 1 hn3 + l "3."2

= ^2

2

^3

3

^^

^

H

A

—.

. . _ | A + 2/i, M,

3 = 1,2,3, ^ = 0 , ^ ,

t = 2,3,

j = l, .7 = 2,3,

where fl^xi), S y ^ i ) are expressed by A,/i, /ii(a:i),/ii(xi), i — 2,3; some of them are equal to zero and some can be unbounded.

233

If 3 = 2,3,

/x (hin*+1hin°+1 n%z) / _

3 y ^

Nk V^

\fc=2

S=7lfc + 1

u

fl$k2S+Sk3n2+n.2+l

^6fc2n3+<5fc3S+i3+l^-l

.fc fe

Nj *fc2n3+5fc3S><5fc2S+<Sfc3"2

X

U,-

6j2S+(5j3n2+n2 + ll T (5j2"3+(5j3S+n3 + l ,

+ £ *4

fc

_i

™3' <5j2n3+iSj3S,(5;i2S+6.j3n2

X bis U]S

Vi

3

n.;-l

+ Y,hT-Si2h?-Si>

s=0

%=2

X

5i2S+(5i3ri2 + l i.<Si2n3+i5«3S+l

J2 ^ 3 V"^

iVfc V^

~ Z^

/ ,

6i2"3+(5i3S,6i2S+<5i3n2 V

JM

fc=2 Sk2(Si2n3+6i3s)+Sk3r+l

X

r=(5fc2(<Si2S+<Si3n2)+<5fc3(*i2n3+*i3s) + l l5fc2(l51;2S+l5i37l2)+<Sfe3(<5i2n3+5i3s)

,_1

xti.

i<5fc2r+<5fc3(<5i2S+5i3n2)+l

"2

<5fc2(i5i2n3+i5i3s)+6fc3r, <5fc2J"+t5fc3(<5i2s+<5;3n2) «,Af,-

s+Si3n2)+1 h Sj2r+Sj3(5 2

E

i2

r=Sj2(Si2S+Si3n2)+6j3(Si2n3+Si3s) xflSj2(Si2n3+Si3s)+Sj3r+l

X

+l Sj2(Si2S + Si3n2) + Ojr

,-\

Sj3(Si2n3+Si3s)

8j2(t>i2n3+&i3s)+Sj3r,Sj2r+Sj3(6i2S+8i3n2)

n,3,n2

Vi

= ^"2+1 ^

3 + 1

a"1 v ^ — ^2 - , 9t

n, = Ojh,

i = 2,3,

where 3 ,—.

n

,.

.

\ J - _ n«

Si2n,3+Si3S,Si2S+Si3n.2 Mj

i=2

s=0

, J =2,3,

(151)

234 i.e., 712 — 1

713 — 1 713

712 «3,S

s=0 7l2-l,

h?^ h? Y

s=0 '

/

3

\

k=1

Nk

^ x(hs2+ihr+1 X i - E

s=0

X/l

S,Tl2

<5fc2«+^fc3 7l3

•5fc27i3+<5fc37-+li 1-l l L

rk

K

r=Sk2s+5k3n3+l

Sk2n3+Sk3r,6k2r+8k3S

b:

Vl

- E >i5+lft3+1krl63r rv'3\-(x+2n) r=«3 + l 3

hr2+ih^+ih^ib2r

E

/ 713 — 1

Yhtr+5™s+i

n,3,r V2

r=s+l Ni

713 S=0 i5i27i2+<5i3S

1=2

7-=(5i27l2+l5,:3S+l

<5i2s+<5i3r,15i2r+<5i3n2

712 — 1

«2

H3 = h?-1 h^ E

773-1

71.3,S

b2sX23+h? h^-1 E 712 — 1

1

5=0 ^ n 3

+ S i

3 r

+

l

h

l

Ni

3 772 6

= -»h >- h? E x h

5,712

5=0

8=0

n

773

* 3 . *33 /i ur+at3S+1

^E

E

7=2

T=l5i2S+(5i37l3 + l

* " g » «

2

S,2n3+S,3r^r+Si3s

713 —1

1

+ h? h?- E

^

A(/12 2 + 1 ^ 3 + 1 \

l5fc2772+l5fc3S x/l«k2S+<5/t3r+l/l-l

fefc

^T,l-E fc=2

j;

, r

/l^r+<5fc3"2+1

E

r=6 f c 2n 2 +(5fc3«+l r

,1;

Sk2a+Sk3r,vSk2r+6k3n2

Na

N2

7X2

E 7"=7l2 + l

/ l ^

1

^

1

^

1

*>2r vl

- ( A + 2/i)

E ^ r=s+l

2 + 1

^

+ 1

/ l 3

1

frs/i«3

235

Ifj' = l,

+1

3

+1

N.L

# 2«4-5i3l2+Tl2 + l

(A + 2 A *)(^- ^- Tx) il - E E

i=2 * = o ; + l

yi,^i2»3+^i3-S+n3+1^,-1

(A + 2/z) 6 |

, . ?*

5i2tl3 + l5i3*,5i2« + l5i3Tl2

+A 6 i s

Vj

m~i r 2

' .1 X ft!<5;2»3+6.:3S+l ra

«

/I

Vi

, n

N *i* i

l5

t=2

s=0 I

i2 n 3+l5i3*,'5i2S+6i3Tl2

(A 4- 2^) 6 J

vM

<5i2n3+l5i38,l5i2S+5i3n2

Nk 1 V"" \ "

V"^

2 J )^5 ^ *M ' r+afc 1"''fc3.V««2»'r"i3'iajT-l L;)0(5 f c fc 2 ^2(*i2n3+5i3s)+<5fe 0 i 2 « 3 T - 0 i 3 » ; - r 0 k 3 ' " f -3lr+lj t -) -l 3 (^i2S+<5i3n.2)+l

fc=2 r=<5fc3 (<5<2*+^3 «2)+<5*3 (<5i2 na +6-1,3 s)+l <Sfc2(<5i2S + 5i3"2)+lSfc3(<5i2n3+5,:3*)

(A + 2/i) 6 j <5ka('5i2n3+*i3s)+<S k 3r,6| s 2r+6fc3(6.;2S+(5i3n2) X Wj

/

n; . **2(*«*+*i3n 2 )+i*3(ii2n3+*i3s)

+ ( Ab t

X

bkr

m

5

k2{Sns+Siin2.)+Sk3(Si2n3+Si3s)^

+M 6 i s

&£ JVi

6k2(5i2n.3+6i2s)+8k3r,8k2r+6k3(&i28+Si3ni) Vi

•It

^/i^r+<5i3n2+l r=s+l

X^^+^r+1^-1 ^ £

+ ^ ^

3

*«» S +*3rA2r+*,» a

Xf = p / i ^ » + 1 / # * 3 + 1

a2

"* , m = 0, JV4, t = 2,3.

<%2

(152)

236

2.2

Initial and boundary value problems

(±) We assume that on the surfaces hi, i = 2, 3, of the bar surface forces (which appear in (143) and on the ends of the bar either the components of the displacement vector or the components of the surface forces are given, or both are given in a mixed form. In the (N3, N2) approximation if at an end of the bar 2/ij > 0, i = 2,3, then the boundary conditions (BCs) on the above end have the forms as follows. 1. BCs in displacements: «J- = /J-.

3 = 1,2,3,

r = 07N;,

S = 0T/V2,

(153)

r,s

where constants fj are given at the end. In the dynamical case we have to add the initial conditions as well: r,s V'j |t=0

Xl

rs

d Vj

Vj(zi)>

s]0,L[, j = 1,2,3,

r s

t=o=i>j(xi),

r = 0,W3,

(154)

s = 0,N2,

r s

'

'

where fy, tjjj are given functions, and fj should be, in general, functions oft. 2. BCs in stresses: Xji=r9j,

3 = 1.2,3,

r = 0T/V3, s = Ojh,

(155)

where constants c/j are given at the end. 3. Mixed BCs are called such BCs when either at an one end conditions (153), and at the other one conditions (155), or at both ends for some indices conditions (153) and for the remaining ones conditions (155) are given. If at an end of a bar at least one of 2hi, i = 2,3, vanishes then in conditions (153), (155) the left-hand side terms should be understood as limits from the inside of the bar provided that these limits exist. In some cases conditions (153) should be replaced by conditions of boundedness of Vj. To investigate the above problems one can use the vast literature on ordinary differential equations (see, e.g., 1 5 ), and on hyperbolic partial differential equations (see, e.g., 1 6 ). But to apply them, one needs some efforts especially in case 2hi{x{) > 0, i = 2,3, on [0,L]. If in the static case hi > 0, i = 2,3, on [0, L] then the existence of a regular solution in the classical sense, taking into account that the system

237

(151), (152) can be reduced to the system of first order ordinary differential equations, by virtue of the well-known theorem (see 15 , p. 146), follows from the uniqueness of the solution of the above problems which can be proved by means of the potential energy similarly to [1] (see also 14 Section 8). If in the static case 2hi(xi) > 0, i = 2,3, on [0, L], the following boundary conditions can be set in the (N^,^) approximation: V(0)=

J = 1,2,3 if

7 0 < + o o , ni = 0,Ni,

t = 2,3;

(156)

"3,"2

vj{L)=

iff , j = 1,2,3 if

'3vj2(x1)=0(l), if

IL <+oo,

m=0,Nu

Xl^0+,j

= 1,2,3,

* = 2,3;

= +oo, Hi = 0, Ni, i = 2,3;

7Q

(157)

(158)

i3

Vj2(xi) = 0(1), X! -> L - , j = 1,2,3, if iz = +co, rii = 0, JVj, i = 2,3;

(159)

"3,"2

Zim / i ^ / i ^ X i ^ x i ) 3

Nk

Jun+ A . ^ ^ ^ ^ X i - A . E £ ^<S/fc2S+(<5jt3 + l ) n 2 + l fc=2 s=rifc + l

>,i5t;3S+(<5fc2 + l ) n 3 + l t —1 tfc flo ft. 0 .s fc

^k2n3+Sk3S,6k2S+Sk3ni U,-

Ni LSi2S+(5i3 + l)n2 + l,(Si2

+ l)n3+6i3S+l,-lnLi

ft"1 6,

fljlA/J + <%(<^'2 + Sj3)ll «i2n 3 +6i3«,4i2S+*i3n2 U «3l«+«j2+«j3

S=Ui + l

)

"3,"2

= V°,

n

3,n2

J = 1,2,3, if J 0 < + o o , ni=0,Ni,

t = 2,3;

Km /£ a WJ 3 X y (a;i) = Vf, J = 1,2,3,

x\—+L

if

7L<

+oo, n$ = 0, Ni, i = 2,3;

(160)

(161)

238 "3,712

where

713,7*2

713,712

7*3,712


,7*2

are given constants,

f

7*3,7*2

I™ := J h22n2~1(T)h32n3-1(T)dT,

e = const > 0.

L-e

The boundary value problems (BVPs) (151), (152), (156), (157); (151), (152), (156), (161); (151), (152), (157), (160); (151), (152), (156), (159); (151), (152), (157), (158) are uniquely solvable (here we do not take care to make precise the appropriate classes of solutions). The problem (151), (152), (158), (159) is solvable up to the rigid-body motion. If in a neighbourhood of a cusped end stresses are bounded then at the above end all moments of the stress vector will be equal to zero. Non-zero stress vector moments given at a cusped bar end mean that this end in the three-dimensional case is loaded by a concentrated surface force, and concentrated moments of the corresponding order. In the dynamical case we have to add to the boundary conditions (156)(161) the initial conditions (154) since hi{x\) > 0, i = 2,3, for t = 0, x\ G ]0, L[, and, therefore, the system (151), (152) is not degenerate for such (xi,t). The (0,0) and (1,0) approximations have been considered in Sections 2.3 and 2.4, respectively. 2.3

(0,0)

Approximation

In the (0,0) approximation the system (151), (152) will have the form ( / i 2 M , , i ( i i , t ) ) , i +°Yj = XJlph2h3dVj^t\ 00 t) hi{x\)hz{x{)

°.°

W-(T,

j = 1,2,3,

°'° X? \ + 2fi

°'°

(162)

°'° X? \i

Let us consider the static case of (162). Obviously, X\

T

dr

/

r°>°

Xl

• f

dr

M^/^WM*M+4' „.o

„o

To

(I63)

239 x° = const e]0, L[,

cPa = const, a = 1,2, j = 1,2,3.

Let further t>j(0)=y>S

if

«,-(£) = ¥#

%°<+co,

if

/L<+OO,

j = 1,2,3,

(164)

j = 1,2,3,

(165)

^•(ari) = 0(1),

^i^0+

if I 0 = + o o , j = l , 2 , 3 ,

(166)

w,-(x1)=0(l),

n - L -

if 7 L = +oo, j = 1,2,3,

(167)

X y ( 0 ) = A . - ^ / i s ^ i U ^ o = V-,0 if ^o < +oo,

j = 1,2,3,

(168)

X y ( L ) = Xjhihsvj,!^^ = ipf if 4 < +oo, y!-. V^.
j = 1,2,3,

(169)

0,0

0,0

0,0

Let Yj G L([0,L]), and if IQ = +oo (II = +oo), let it be such that the iterated integral in (163) is bounded by x\ —> 0+ (L-). If - — — is locally summable in ]0,L[, then from (163) it follows that /i2 • A3

for regular solutions (VJ G C 2 (]0,L[)) only the following problems are wellposed: (162), (164), (165) (VJ G C([0,Z,])); (162), (166), (165) («,- G C(]0,L])); (162), (164), (167) (VJ G C([0,i[)); (162), (166), (166) (VJ is bounded); (162), (164), (169) (Vj G C([0,L[), /i2/i3«,M G C(]0,L]); (162), (168), (165) (Vj G C(]0, L}), /i2^3Uj,i € C([0, L[); the mixed problems when on the one end of the bar for some displacement components the Dirichlet and for others either the Neumann or (166), (167) type conditions are given but on the other end for first components either the Neumann or the Dirichlet or (166), (167) type conditions and for the second components the Dirichlet conditions (the Neumann and (166), (167) type conditions are not admissible) are given, are well-posed. In the case (162), (166), (167) we have got the solution up to the rigidbody translation {c\ = 0, c\, j = 1,2,3, are arbitrary). It should be so since in this case the bar under consideration as a three-dimensional body is loaded 0,0

only by surface forces (which are included in Yj) acting on the lateral surfaces of the bar. All other BVPs have unique solutions and constants c£, a = 1,2, j = 1,2,3, can explicitly be calculated.

240

Remark 2.3.1 Let 2hi(x1) = hl0x^L(L - xi)5i,

hl0 = const > 0, «*, 5i = const > 0, i = 2,3.

Then we can give transparent geometric interpretation of setting of above BVPs. To IQ(IL) < +oo there corresponds K2 + K3 < I (82 + 53 < 1), and to IO(IL) = +00 there corresponds K2 + K3 > 1 (82 + 53 > 1). If at least one of the bar profiles and projections has a sharp cusp (see the definition in 4 ) , i.e., K2 + K3 > 1 (82 + 83 > 1), then at this end the components of the displacement vector can not be given. If both the profile and projection have blunt cusps (see the definition in 4 ) and K2 + K3 > 1 (62 + 53 > 1), then the same is true but for K2 + K3 < 1 (62 + S3 < 1) the above components can be given. If

— has nonsummable singularities in ]0, L[ then only arbitrary three /12 • /13

(but one for any fixed j) out of six conditions of (164), (165) can be given (in particular, it contains cases when either only (164) or only (165) can be given). If 2hi := h°x"\

h°, a{ = const > 0, i = 2, 3,

(170)

equations (162) are of the type of (1) for which the conditions (8), (20), (36) are fulfilled since in case of (162) the coefficients a, = 0,i = 1,2,3 in (1), where h := /12/13 and (170) holds. Hence, Theorems 1.1.1 and 1.1.2 are valid for (162). Therefore, the edge x\ = 0 of the bar can be fixed only if a = 012 + ct3 < 1.

More precisely, without loss of generality we can assume L = l, °
j>j(x1)=0,

v>5 = 0, ^ = 0 , 7 = 1,2,3.

Then, if 0,0

YjZWl

or Z, 2ih _i,

there exist unique strong generalized solutions Vj(xi,t)&W+

or L2,h, j = 1,2,3,

of the initial boundary value problems (162), (170), (154) when N2 = iV3 = 0, (164), (165) and (162), (170), (154) when N2 = NZ = 0, (166), (165).

241

2.4

(1,0)

Approximation

Let us consider a symmetric (2/tj = 0, % = 2,3) bar in (1,0) approximation. In view of (151), (152), the corresponding complete system of equations has the following form: ~2°'°

0,0

(A + 2n){h2h3°vhl),1

+3A(/i 2 /i 3 u 0 3),i + i i ° = P^hs-^-,

021'0

1,0 1

X

0

v(h2hl v3A)A-3(\+2iJ,)h2h3 v3-\h2h3 vhl+hsX$

(171)

dt2

= ph2h3—^, 2 ' dt

0, oo ° d2°v°2 p.(h2h3 v 2 ,i),i + X% = ph2h3 ,

(172)

(173) a21'0

10

tf\?-.\ . ^2.hA3 . 1v^ .2 x y 0 _= ph , ,2hi—^-, ,,^3^A*(^2fci * 2,i),i. _- Q 3ph + .h.3X% at2

(174)

0

0,0

g2

^

p.(h2h3 v 34)4 + 3p(h2h3 v i),i + X% = ph2h3—^-, 1,0

(175) Q2^°

(A+2^)(/i 2 /ii 1 A,i),i-At/i2/i3 0 t' 0 3,i-3/i/i2/i3 1 w 1 + /13X? = ph2h33-Qp-,

(176)

In the static case, BVPs for equation (173) have been solved in Section 2.3. Therefore, for this equation, e.g., the Dirichlet problem is well-posed only 0,0

0,0

if Jo < 00, II < 00. For the same reason for (174) the Dirichlet problem is 1,0

i,o

correct only if Jo < 00, II < 00 (see also 15 , 3 4 ) . Hence the approximate value of . . lo,o . . 3a;3i,o . . 102(3:1,0:2, ar3) w - v 2{x1) + - 7 - v 2(2:1) 1,0

,„„„•. (177) 1,0

can be preassigned at the ends of the bar only if Jo < +00, II < +00. Systems (175), (176) and (171), (172) can be correspondingly reduced to the following systems 10 00 _ _ 3fj, v ! = -n v 3,1 - h2 1(x1)h^1(x1)

I 7°'° / X${x1)dx1

V?

I - C\

,

/

C\ = const, x® e]0,L[,

242 0,0

,0,0

(A + 2/j,)(h2hl v X + 2/j,

t^

3,H),II

1,0

= 3X3° + 3(ft3-X-i),i

h2h\ h^hz1 jxl{xr)dXl

-

h^h^G

J,n and 10 00 3Xv3 = -(A+2/x) v i.i-h^ixi^in)

/ f°'° / X^{xl)dx1

0,0

(A + 2n)fi(h2hiv0,0liU),n ( h2h\

- 12(A + p)p(h2h3 v

,'o,o \ X^(x!)dxi

x

h2-\hl

\ - C2 \ , C2 = const,

hl)A

\

\

1

- C2 (ft^AJ )

1

An (A+^ X o + _ ^ x o + S A ^ X i A + 2/i A + 2/x The advantages of these systems consist in the following: the second equations +12

involve only one unknown function and after their solution (see ? ) , v \ and 1,0

v 3 can be readily calculated from the first equations. Now, let us come back to the dynamical case. The system (171), (172) was considered in Section 1.2. Literally in an analogous way can be considered the mixed problem for the system (175), (176). Introducing the notation

=

0,0

1,0

«V«i

0,0

,

F:=

1,0

Xl h3X$

this system can be rewritten in the folllowing form Lu = (Aut)t - (Bux)x

(178)

+ (Ciu)x + C2ux + C3u = F,

where 'ph2h3 0

0 \ ph-ihl)'

5

( ph2h3 0 \ 0 (A + 2n)h2Ki33

>

243

Ci =

3p,h2h3 0

0

C2

r =(° ° 1

°\

3 ^h2h3 OJ' V° 3/i/i 2 /i 3 ;' For the system (178) in the domain JD we consider the mixed problem under initial and boundary value conditions du(x, 0) (179) u{x,0) = 0, = 0, 0 < z < l , dt

0,0

1,0 "1

(180) 0. r . (•>2+ "3) r.„ Below the spaces J5, E*, W+, W+, W_, W l , are the same as for the problem (84)-(86) and in an analogous way we can introduce the notion of the strong generalized solution u of the problem (178)- (180) in W+ and L2^. The folllowing theorem is correct. Theorem 2.4.1 Let the conditions (83), (107), (108) be fulfilled and let either pa > p*, where p* = p*(\,p1,a2,a3,htj,T) be sufficiently large positive number, or let T be sufficiently small, i.e., T < 7b(A,/z,0:2,^3, /i*). Then for any F S Wl there exists a unique strong generalized solution u of the problem (178)-(180) in L^^h for which the estimate 0,

v3

3

c u

< \\F\\W.

u

2,h

holds. If F £ L2,h-1 C Wl, this solution will be also a strong generalized solution of the problem (178)-(180) in W+. Now, under the condition (83) let us consider the split system (173), (174). Assuming v ••

0,0

/ 0,0

1,0

,F=

1,0 \

[X$, h3X$

for the system (173), (174)

in D, we consider the folllowing mixed problem «(a;,0)=0,

dV 0)

^' at

=0,

0
(181)

and 0,0

= 0

v2 r»„

1,0

v2

0.

(182)

r.(«2+3<,3)

There holds Theorem 2.4.2 Let the conditions (83), (107) be fulfilled. Then for any F € Wl there exists a unique strong generalized solution v of the problem (181), (182) in L2,h for which the estimate c\\v\\r

< \\F\ w

244

holds. If F G L2,h~1 C W^, this solution is also a strong generalized solution of the problem (173), (174), (181), (182) in W+. Now, let us consider all the above systems (171)-(176) together as the one system for U = (Ui,U2,u3;v1,V2,v3),

(

Ui = Vi,

0,0

0,0

0,0

1,0

Wj = Uj, 1,0

4=

1,2,3,

1,0 \

X

l, X2 , X3 ; /l3Xj , / l 3 ^ 2 > ^3-^3 J

in the domain D under homogeneous initial boundary value conditions dU( 0) 1,0 g'

0,0I7(x,0) = 0, V i = 0,

vi

r,„

=0,= 0,0
(183) (184)

In this case for the vector-function U the spaces E, E* are introduced in a similar way as. Let W^Wjf;) be a weighted Hilbert space obtained by closure of the space E(E*) under the norm

IMIw+(w., = (hih^ul +

J D

+ pul + uDdD i=i 3

J h2hsY,[vi+Pvit + vl]dDD

i=1

By W- {Wt) we denote the space with the negative norm constructed by L2(D) and W+(W$). Combining the above-obtained results, we conclude Theorem 2.4.3 Let the conditions (83), (107), (108) be fulfilled and either po > p*, where p* — p*(\,p,, 0:2,0:3, ftj,T) is a sufficiently large positive number, orT is sufficiently small, i.e. T < Tb(A,/i, a2,a3,hj). Then for any Fo G Wl there exists a unique strong generalized solution U of the problem (171)-(176), (183), (184) in L2ih for which the estimate c\\U\\L, holds.

<\\F0\\W.

245

IfFo G 1/2 h^1 C W*_ this solution is also a strong generalized solution of this problem in W+. Remark 2.4.1 Let us note that if in the condition (83), the number 7 > max(l,2(3), then the riquirement in conditions of Theorems 1.2.1-1.2.4 Po > P* or T < To can be removed and only the assumptions (83), (107), and (108) can be left. Remark 2.4.2 Repeating Remark 1.2.7 in connection with the last problem, we note once more that according to the embedding theorems the boundary value condition (184) is fulfilled in the following sense. The solution /o,o o,o o,o i,o i,o i,o \ , , , U = I v i, v 2 , v 3 ; v i, v 2, v 3 I G W+ takes zero boundary values in the sense of the trace operation. Remark 2.4.3 Replacing in (177) the index 2 by j,j = 1,2,3, we obtain the approximate expression for the components Wj of the displacement vector. Summing up the above results, we conclude: the displacement vector w := {w\ ,W2,u>3) takes zero boundary value on To in the sense of the trace operation only if (X2 + 3«3 < 1. Thus, the cusped edge x\ — 0 of the bar can be fixed in the (1.0) approximation only if a.2 + 3oj3 < 1 while the same edge can be fixed in the (0.0) approximation only if a.2 + a^ < 1. References 1. Vekua, I. N., On a Way of calculating prismatic shells. Proceedings of A. Razmadze Institute of Mathematics of Georgian Academy of Sciences, 21 (1955), 191-259. (Russian) 2. Vekua, I. N., The theory of thin shallow shells of variable thickness. Proceedings of A. Razmadze Institute of Mathematics of Georgian Academy of Sciences, 30 (1965), 5-103. (Russian) 3. Vekua, I. N., Shell theory: General methods of construction. Pitman Advanced Publishing Program, Boston-London-Melbourne 1985. 4. Jaiani, G. V., Elastic bodies with non-smooth boundaries-cusped plates and shells. ZAMM, 76 (1996) Suppl. 2, 117-120. 5. Babuska, I., Li, L., Hierarchic modelling of plates. Computers and Structures, 40 (1991), 419-430. 6. Gordeziani, D. G., To the exactness of one variant of the theory of thin shells. Soviet. Math. Dokl., 215 (1974) 4, 751-754. 7. Guliaev, V., Baganov, V., Lizunov, P., Nonclassic theory of shells. Vischa Shkola, Lviv 1978. (Russian) 8. Khoma, I., The generalized theory of anisotropic shells. Naukova Dumka, Kiev 1986. (Russian)

246

9. Khvoles, A. R., The general representation for solutions of equilibrium equations of prismatic shell with variable thickness. Seminar of the Institute of Applied Mathematics of Tbilisi State University, Annot. of Reports, 5 (1971), 19-21. (Russian) 10. Meunargia, T. V., On nonlinear and nonshallow shells. Bulletin of TICMI, 2 (1998) 46-49. (electronic version: http://www.viam.hepi.edu.ge/others/TICMI) 11. Schwab, C , A-posteriori modelling error estimation for hierarchik plate models. Numerische Mathematik, 74 (1996), 221-259. 12. Vashakmadze, T. S., The theory of anisotropic plates. Kluwer Academic Publishers, Dordrecht-London-Boston 1999. 13. Zhgenti, V. S., To investigation of stress state of isotropic thick-walled shells of nonhomogeneous structure. Applied Mechanics, 27 (1991) 5, 3744. 14. Jaiani, G., On a Mathematical Model of a Bar with a Variable Rectangular Cros-section. Preprint 98/21, Institute fiir Mathematik, Universitat Potsdam, Potsdam 1998. 15. Kiguradze, I. T., Initial and boundary value Problems for the systems of ordinary differential equations. V. 1. Linear Theory. "Mezniereba", Tbilisi 1997. (see also, Itogi Nauki i Tekhniki, Seria Sovremennje Problemi Matematiki, Noveishje Dostizhenia, V. 30, Moskva 1987, 1-202.) (both in Russian) 16. Ladyzhenskaya, O. A., Boundary value problems of mathematical physics. "Nauka" Press, Moscow 1973. (Russian) 17. Jaiani, G., On a model of a bar with variable thikness. Bulletin of TICMI, v.2 (1998), 36-40. 18. Jaiani G. V., Boundary value problems in (1,0) approximation of a mathematical model of bars. Bulletin of TICMI, v.3 (1997), 7-11. (electronic version: http://www. viam.hepi.edu.ge/others/TICMI) 19. Jaiani G. V., Relation of a mathematical model of bars to the threedimensional theory of elasticity. Workshop in partial differential Equations, University of Potsdam, pp. 15 and 16, 1999. 20. Krasnov M. L., Mixed boundary value problems for degenerate linear hyperbolic differential equations of the second order. Matem. sb., 49(91), 1959, 29-84. (Russian) 21. Baranovskii F. T., A mixed problem for a degenerate hyperbolic equation. Izu. Vyssh. Uch. Zaved., Matematica, 3(16), 1960, 30-42. (Russian)

247

22. Briukhanov V. A., On a mixed problem for a hyperbolic equation degenerating on apart of the domain boundary. Different. Uravnenia, V.8, Nl, 1972, 3-6.(Russian) 23. Vragov V. N., A mixed problem for a class of hyperbolic- parabolic equations of the second order. Different. Uravnenia, V.12, Nl, 1976, 24-31. (Russian) 24. Vragov, V. N., Boundary value problems for nondassical equations of mathematical physics. Novosibirsk: 1983, (Russian) 25. Taniguchi M., Mixed problem for weakly hyperbolic equations of second order with degenerate Neumann boundary condition. Funkc. ekvacioj, 1984, 27, N3, 331-366. 26. Berezanskii, Yu. M., Expansion in proper functions of self-conjugate operators. Naukova Dumka, Kiev, 1965. (Russian) 27. Liashko, I. I., Didenko, V. P., Zitrizkii, O. E., Noise filtration. Naukova Dumka, Kiev. 1979. (Russian) 28. Vishik M. I., Boundary value problems for elliptic differential equations degenerating on a part of the domain boundary. Uspekh. Mat. Nauk, 9, 1(59), 1954, 138-143. (Russian) 29. Vishik M. I., Boundary value problems for elliptic equations degenerating on the domain boundary. Matem. Zb., 35(77), 3, 1954, 513-568. (Russian) 30. Smirnov M. M., Degenerating elliptic and hyperbolic equations, Nauka, Moskva, 1966. (Russian) 31. Fichtengolz G. M., Course of differential and integral calculus. V.l, Nauka, Moskva, 1969. (Russian) 32. Mikhlin S. G., Course of mathematic physics. Nauka, Moskva. 1968. (Russian) 33. Jaiani G. V., The first boundary value problem of cusped prismatic shell theory in zero approximation of Vekua theory. Proceedings of I.Vekua Institute of Applied Mathematics, 29 (1988), 5-38. (Russian, Georgian and English summaries) 34. Jaiani G. V., Bending of an orthotropic cusped plate. Universitat Potsdam, Institut fur Mathematik, Preprint 98/23, 1998, 1-30.

248

VANISHING VISCOSITY LIMIT OF T H E I N C O M P R E S S I B L E NAVIER-STOKES EQUATION (ABSTRACT) K. ASANO Kyoto University, Yoshida College, Inst, of Mathematics Yoshida Nihonmatsucho, Kyoto 606-8501, Japan E-mail: [email protected] We consider the incompressible Navier-Stokes equation in the domain Q C R3 with the boundary S = 9 0 : dtu + u •

VM

— uAu + Vp = 0,

V • u = 0, -yu = u | 0 = 0, u |t=o= wo, V • wo = 0,7U0 = 0, 2

where v = e £ (0,1] is the viscosity coefficient. We assume that S is an analytic surface. The solution u = u(e, t, x) is expected to satisfy the property: u(e, t, x) —> u°(t, x)

as

e —> 0,

if x is away from S. Here M° is the solution of the incompressible Euler equation with the same initial data u°, satisfying the slip boundary condition. In the case Q, = R\ with S = R2, we can prove that u(e, t, x) has the asymptotic form: u = u°(t,x) +eu1(t,x)

+

e2u2(e,t,x)

+u°(t, x) + eu1(e, t, x) + e2u2(e, t, x), p = p°(t,x) + ep1{t,x) +

e2p2{e,t,x)

+ep°(i,x) + e 2 p 1 (e,* ) x) + eV(<:,*1:E)-

249

Here 'u{(e,t,x',x3/e)\ u^(e,i,a;',X3/e) eu3(e,x',x3/e) J

,

x' =

{xi,x2).

From this asymptotic form, we can prove the desired result. We also discuss the stationary problem in the exterior of the ball. The first boundary layer has some interesting properties.

C H A P T E R 3:

PARTIAL COMPLEX DIFFERENTIAL EQUATIONS IN THE PLANE

253

B O U N D A R Y VALUE P R O B L E M S FOR A CLASS OF N O N R E G U L A R ELLIPTIC EQUATIONS N. E. TOVMASYAN State Engineering University of Armenia, 105 Terian Str., Yerevan, 375009, Armenia, E-mail: [email protected] In this paper we consider some boundary value problems for a class of nonregular elliptic equations. These problems we reduce to second kind Fredholm integral equations and calculate their indices

1

Introduction

Let r be a simple closed sufficiently smooth contour, surrounding a bounded simply connected domain D and let D be the closed domain P U T . Without loss of generality we will assume that 0 & D. The positive direction on T is leaving the domain D on the left. Consider the first order differential operators Lj (j = l,...,n) given by d

x

d

(1)

where / is the identity operator, while Xj and a.j (j = 1 , . . . ,n) are complex constants satisfying ImXj

> 0,

j = 1 , . . . , m,

Im Xj < 0, AjT^Afc,

(2)

j = m + 1 , . . . , n, j ^ k,

j,k =

l,...,n.

(3) (4)

For definiteness we will assume that m>n — m. Consider the following boundary value problem: find a n-times continously differentiable solution u(z) z £ D of the equation L\ • • • Lnu(z)

= 0,

z € D,

(5)

satisfying the conditions dm-1u(z) Re Q kQ m-l~k y x Im

dm-1u(z) dykdx m—X~k

fk(z), fm+k(z),

z e l \ fe = 0 , l , . . . , m - l , zGT,

k = 0, l , . . . , n - m - l ,

(6) (7)

254

where the operators Lj (j = 1, ...,ra) are defined by (1) and fk(z) (k = 0 , 1 , . . . , n — 1) are given functions defined on I\ Notice that for m = n we have only the conditions (6). It follows from (2) and (3) that the equation (5) is elliptic. Moreover, if m = n — m, then the equation (5) is regular elliptic, otherwise (5) is nonregular elliptic. For fk(z) = 0 (k = 0 , 1 , . . . , n - 1) the problem (5) - (7) is called homogeneous. We will say that a function u{z) belongs to the class H(D) if it satisfies a Holder condition in the closed domain D. Similarly we define the class H(T) on the contour I\ We will assume that the functions fk(z) (k = 0 , 1 , . . . , n — 1) and their derivatives d3fk{z) dgj

,

, _ k = 0,...,n-l,

,

. , j =

, l,...,n-m-l,

belong to the class H(F), where s is the arc length of the point z £ I\ We look for a solution of problem (5) - (7) in the class of functions, which together with their derivatives up to order n — 1 belong to the class H{D). In the special case where Reafc = 0 (k = 1 , . . . , n — 1) the problem (5) - (7) in the half-plane was considered in [1]. Notice also that for a/t = 0 (k = 1, . . . , n — 1) some Dirichlet type problems for equation (5) in the simply connected domains have been considered in 2 - 4 , where these problems are reduced to normal-type singular integral equations. The purpose of the present article is to reduce the problem (5) - (7) as well as some more general boundary value problems to second kind Fredholm integral equations and to calculate their indices. Recall that the index of problem (5) - (7) is the difference ko — k'0, where k'0 is the number of linearly independent conditions that are necessary and sufficient for solvability of problem (5) - (7), while ko is the number of linearly independent solutions of the corresponding homogeneous problem. Here the linear independence means over the field of real numbers. We prove that the index of problem (5) - (7) is equal to n(2m — n). 2

The general solution of equation (5) and some auxiliary results

Let D and V be as in section 1 and let A be a constant such that ImA ^ 0. Denote by D\ the image of the domain D under the mapping £ = x + Xy {<; = Z + iri€Dx,z = x + iy£D). Definition 2.1 We say that a function
( = x + Xy.

255

Lemma 2.1 The general solution of equation (5) in a simply connected domain D is given by n

u(z) = Y.e~a3yvA* + \y),

(8)

where ipi(x + Xiy),...,
j = 2,...,k-2,

k = 2,...,n,

(9)

and the functions (fj(x + Xjy) (j = 1 , . . . , n) are determined by u(z) uniquely. Proof First we prove the representation (8). To this end we apply induction on n. Let n = 1. In this case the equation (5) takes the form du . du ^ - Xx—+ ay ox

„ alU = 0,

zeD.

„

.,„. (10

By the change of variables u(z) =

e~aiyv(z),

we get from (10) dv , dv - - ^ - = 0,

^ zen.

. . (ii)

The general solution of equation (11) is determined by (see 2 ) v(z) = tpi(x + Xiy),

(12)

where (fi(x + X\y) is an arbitrary analytic function in the argument x + Xiy. From (12) we have u(z) = e-aiytpi(x

+ Xiy),

(13)

yielding (8) for n = 1. Now, assuming that (8) holds for n = p, we prove that it remains valid for n = p + 1. For n — p + 1 the equation (5) can be written in the form L1---,Lp+iv

= 0,

zeD,

(14)

where v = Lm,

zeD.

(15)

256

Since by the induction hypothesis (8) holds for n = p the general solution of equation (14) is determined by v(z) = Yje~~akVMx

+ ^kV),

(16)

k=2

where ip2(x + Xiy),... ,ipp±i(x + \p+\y) are arbitrary analytic functions of arguments x + A2J/,..., x + \p+\y respectively, satisfying the conditions V^ } (0) = 0,

j = 0,...,k-3,

k = 3,...,p+l,

(17)

Substituting v(z) from (16) into (15) we obtain P du , du ^ - A ^ ++ faiul = m ^ e - ^ l ^ ) . dy dx f-^ fe=2

(18)

We set

"* = ? — r >

k = 2,...,P-i.

(19)

We look for a particular solution of the equation //?/

nil

= e-ai°yMx

^-\i^-+alU dy ox

+ ^y).

(20)

in the form u{z) = e-aky
(21)

where ipk{x+\ky) is an analytic function in the argument x+\ky- Substituting u(z) from (21) into (20) and taking into account (19) we obtain

^U0-w(C) = 3^1^(0,

(£DXk.

(22)

k = 2,...,p+l.

(23)

A particular solution of equation (22) is

¥>*(0= , e " \ / e-'-'MQdt, M — M Jo

In (23) we integrate from 0 to £ remaining in the domain D\k. As we have seen above the general solution of equation (10) is determined by (13). It follows from (13), (21) and (23) that the general solution of equation (18) is determined by

u(z) = Y,e~akVMx fc=l

+ *ky),

(24)

257

where y>\{x + X±y) is an arbitrary analytic function of argument x + X\y, while the functions
j = 0,...,k-2,

k=

2,...,p+l,

It folllows from (24) that (8) with functions •

r'tO +E ^ W ^ .

CGDAfc,

(25)

fe=o

where Cfe (k = 0 , . . . , n — 2) are some constants. On the other hand, ¥#°(0) = 0, fc = 0 , . . . , n - 2 .

(26)

A combination of (25) and (26) yields
'

N

A f u(t)dt

(„

2A\ f — / n(t)ds = /(*o),

(27)

where to S T, f(t) is a given and p,(t) is the desired n-dimensional real vectors belonging to the class H(T), ds is the element of arc differential of the contour r , and lo is the length of I\ (Here and below, if otherwise is not specified, all vectors are assumed to be column vectors). In (27) the singular integral is the Cauchy principal value (see 5 , p.50). L e m m a 2.2 Equation (27) is uniquely solvable.

258

Proof The left-hand side of (27) is the boundary value on T of the following function u(z) = Re

A f fi(t)dt

/„

2A\

/" ,

'

(28)

which is harmonic in D. Let fi(t) be a solution of homogeneous problem (27) (for / = 0). Then the harmonic function u(z) vanishes on T, and hence u(z) = 0. Prom (28) for u(z) = 0 we have (see 6 )

A f n{t)dt

/

2.4 \ f

,. ,

where c is some n-dimensional real constant vector. Prom (29) follows (see 5 , p.255) that fi(t) is a constant vector. Therefore from (29) we get loli = ic.

(30)

Since /x and c are real vectors from (30) follows that fi = 0 and c = 0. Therefore the homogeneous equation (27) has only trivial solution. Further, since the index of equation (27) is equal to zero, we conclude that the nonhomogeneous equation (27) is solvable for any left-hand side /(io) from the class H(T) (see 5 , p.510). Lemma 2.2 is proved. Now we study the following equation

r

p

n(to) + / K(t, t0)fi(t) ds + Y] ckojk(t0) = /(to), JT

*o e T,

(31)

fc=i

where /i(io)> f(to) and u>k(to) (k = 1 , . . . ,p) are n-dimensional real vectors, K(t, to) is a real nxn matrix, and c\,..., cp are real constants. Assume that the vectors /x(k{to) (k = 1 , . . . ,p) belong to the class H(F), while the matrix K(t, to) for some constant 6 (0 < 5 < 1) can be represented in the form

*<«•*•>-£%F

(32)

and Ko(t,to) belongs to the class H(T) with respect to both variables t and toWe have to solve equation (31) with respect to a real vector-function ji{t) and real constants c\,... ,c p . For / ( z ) = 0 the equation (31) is called homogeneous.

259 Together with equation (31) we consider the following system of equations x(*o) + / x(t)K(t, t0) ds = 0,

t 0 G T,

(33)

I

= 0, fc = l , . . . , p , (34) X(t)wk(t)ds T where x{t) is unknown n-dimensional real row vector-function from the class H(V). The system of equations (33) - (34) is called associated homogeneous with (31) equation. We have the following lemmas. Lemma 2.3 Equation (31) is solvable if and only if the vector-function f(t) satisfies the condition

J X(t)f(t)ds = 0,

(35)

where x{t) is an arbitrary solution of associated homogeneous equation (33) (34). Lemma 2.4 Let fco and k'0 be the numbers of linearly independent solutions of homogeneous equation (31) and associated homogeneous equation (33) (34), respectively. We have fco —fc0= p. It is clear that under condition (35) we can take only the linearly independent solutions of system of equations (33) - (34). Lemmas 2.3 and 2.4 can be proved by arguments similar to the one given in 5 , (pp.223, 290), so we omit these proofs. The method of resolution of equation (31) with unknown constants Cj, (j = 1 , . . . ,p) is essentially the same as for without these constants (see 5 , p.292). 3

Investigation of problem (5) - (7)

Let Imaj

„

ReA,-Ima,•— ImA,Rea 9

=

,

.

°' ^ fr= Lx3 J = !>-.». (36) Replacing in formula (8) the functions <j>j(x + Xjy) by functions
" ( ) = X ! exP(aix where cpi(x + Xiy),... arguments x + X\y,...,

+ PjV)
(37)

,ipn(x + Xny) also are arbitrary analytic functions of x + Xny respectively, satisfying the conditions

^ f c ) (0) = 0, fc = 0 , . . . , j - 2 ,

j = 2,...,n.

(38)

260

If the partial derivatives up to the order n — 1 of function u(z) belong to the class H(D), then the (n — 1) — th derivatives of functions ipj(x + Xjy) (j = 1 , . . . , n) also belong to this class. First we consider the case where m > n — m. It follows from (38) that the functions

c

Jk{x + *jy)k + ipj(x + Aj-y),

j = l,...,m,

(39)

where Cjfc (/c = j — 1 , . . . , m — 2); (j = 1 , . . . , m — 1) are arbitrary complex constants, and fi(x + Xiy),..., ipn(x + Xny) are arbitrary analytic functions of arguments x + X\y,..., x + Xny respectively, satisfying the conditions ^ f ) ( 0 ) = 0, fc = 0 , . . . , m - 2 ,

j = l,...,m.

(40)

Notice that for j = m the right-hand side of (39) contains only the last term ipn(x + Xny). We put ^•(a; + X5y) =

^

d,-fc(a; + Xjy)k +
k=m—1

where djk = c2m-j,k+m-j\

k = m - 1 , . . . , j - 2;

j = m + 1 , . . . , n,

(42)

and Cjfc are complex constants from (39). It follows from (38) and (41) that ipm+i(x + Xiy),..., ipn(x + Xny) are analytic functions of arguments x + Xm+iy,..., x + Xny respectively, satisfying the conditions ^fc)(0)=0, It is If m m> from

k = 0,.,.,m-2,

j = m + l,...,n.

(43)

clear that (39) contains (1/2) • m(m — 1) arbitrary complex constants. = n - m + 1, then (42) contains all the constants Cjk from (39). If n — m + 2, then (42) contains only (1/2) • (n — m)(n - m + 1) constants (39). From (38) and (41) we have ^ f e ) (0) djk = - ^ T J — .

k = m-l,...,j-2,

j = m+l,...,n.

(44)

In view of (44) the equalities (41) and (42) can be written as follows j 2 fc) j(x + \jV)

-

~ v- J (o) J2

k=m~ 1

- -j^-(x

+ xjy)k'

(45)

261

j=m + C2m-j,k+m-j

l,...,n, =

k = m — 1,... ,j — 2,

^

'

(^)

Jfc! j = m + 1 , . . . ,n.

We put ^ m - 1 ) ( s + A,-!/) = ^ ( a : + A.y),

j = 1 , . . . , n,

(47)

Prom (40), (43) and (47) we obtain

^(x + XjV) = —z^y f

'\x + XjV - 0m-2
(48)

j= l,...,n. The function <j>j{x + Aj-y) we represent in the form (see[5], p.254)

«^=H£%?-+*X+id'-

(49)

j = l,...,n where £ = ^ +177 is an integration point, Hj{t) (j = 1 , . . . ,n) are real functions from the class H(D), dj (j = 1 , . . . , n) are real constants, and these functions and constants are determined by j(x + Xjy) (j = 1 , . . . , n) uniquely. In (49) we integrate along T in positive direction. In (49) we make the substitution ^•(*) = Mj(*) exp(a j C + / ^ ) -

(50)

Let to = Co + ^ 0 be a point on boundary T. Denote by j(Co + XJTJO) the boundary value of integral (49) when the point z = x + iy tends to the point to = Co + iVo inside the domain D. Applying Sokhotzky-Plemelj formula and taking into account (2) and (3) we obtain (see 5 , p.66) m JY C + Xji] -t,o-

XjT]0

j = l,...,m

0j(& + Vto) = -Mj(*o) + ^ / / f f V ^ t ,

+

^''

(52)

^ J r C + Aj?7 - Co - AjJ7o j = m + l,_.-.,n Let t = C + Aj-r; G T, t 0 = Co + XJTJQ € T, £ = £ — irj and io = Co — «%• We

lj(t) = £ + *jV,

put

262

hm -^-i ^^, T->t, r e r T- t Kj(t0,t)

v*w

=

Kj(to,t)

t = 1

lim , -r->t, r e r T - t j = l,...,m,

7J(*)-7J(*O)

*-*>'

7j(*)-7j(*o)

*-*o'

=

j = m + l, . . . , n .

It is known (see 5 , p.573) t h a t the functions Kj(t0,t) (j = 1, . . . , n ) can be represented in the form (32). T h e formulas (51) and (52) can be written as follows

(Oo) = H (to) + — [ ^Mp- + i - j Kj(t0, t)iij(t) dt + i dj, •KI

JT

t — t0

m

Jr

(53)

j = l,...,m

,(0o)

-lijfo)

+ — / W ^ . + — / Kjito,t)m(t)dt TTl Jr t - to 7" J r

+ idj,

(54)

where £j 0 = £o + Aj-r/o- It follows from (46) t h a t (39) and (49) contain totally q = n(2m — n) arbitrary real constants, which we denote by dj (j = 1 , . . . , q). T h e b o u n d a r y conditions (6), (7) can be written as follows (to € F) lim Re z—>to

K>

=fk(to),

dykdxm~1~k . dm-1u(z) "dykdxm-i-k\

lim Re Z —*to

k = 0,l,...,m-l,

-fm+k(to),k

(55)

= 0,l,...,n-m-l,

(56)

In (55) and (56) the limits are calculated inside the domain D. Let A(t) be some function denned on T and to € I \ Since fij(t) is a real-valued function we have

Re

,

,

j = m +

l,...,n,

/ r t — to Consider t h e matrices Ak and Bk defined by

A,=

1 Ai

1 A2

A*" 1 A*" 1

... 1 ... X„ \k-i

Bk =

1 1 ... 1 Xm+i A m +2 • • • Ara \fc-i

A

m+1

\fe-i

A

m+2

•••

\k-i

A

n

(57)

263

and the block-matrix AQ :

J3„

Am

A0

(58)

iBn-

where Bk is the complex conjugate to BkIt follows from (4) that det A0 = 2i det Am • det B n _ m ^ 0.

(59)

Substituting the general solution (37) into the boundary conditions (55) and (56), and taking into account (39), (45) - (50), (53), (54) and (57), for determination of functions vi(t),..., vn(t) and constants dj (j — 1 , . . . , q, q — n(2m — m)) we get the equation Re

Ao {to) +

^

v{t)d,e

hit-t0

+ y^ckUk(to)

= f(t0),

fM{t,t0)v(t) ds t0 € T,

(60)

it=i

where v{t) = fa{t),...,vn(t)), f(t) = (/i(*),...,/„(*)); M{t,t0) and LJi(to),.. .uiq(to) are well-defined nx n real matrix and n-dimensional real vector-functions, respectively. We assume that the vectors u)i(to),...u)q(to) belong to the class H(T), and the matrix M(t,to) admits a representation of the form (32). Consider the operator Qg = Re

f9«o)

Bo_ r g{t)dt 2m Jr t

m-(°»-*)L^

where Bo is the inverse of the matrix Ao, g(t) is an n-dimensional real vector, and En and lo are as in (27). According to Lemma 2.2 the vector v(to) can be uniquely represented in the form K*o) = Qg.

(61)

Substituting v(to) from (61) into (60) and applying the formulae of multiplication of singular integrals (see 5 , p.508), we obtain the following Predholm integral equation g{t0) + / K(to, t)g{t) ds + Y] cfcwfc(i0) = /(to), to € I\

(62)

where K(to,t) is well-defined real nx n matrix, that admits a representation of the form (32).

264

Thus, the problem (5) - (7) is reduced to Fredholm integral equation (62) with respect to unknown real-valued n-dimensional vector-function g(t) and real constants a,.. •, cq (q — n(2m —n)). A similar equation we have considered in the previous section (equation (31)). The above unique representations show that the number of linearly independent solutions of homogeneous problem (5) - (7) is equal to the number of linearly independent solutions of homogeneous equation (62) (for / = 0). It follows from Lemma 2.4 that the index of problem (5) - (7) is equal to n(2m — n). Now let m = n — m. In this case the problem (5) - (7) can be reduced to Predholm integro-differential equation of the form (62) without unknown real constants. Consider now more general boundary conditions

= fk(z),

Re p+Km-1

zGT,

k = 0 , 1 , . . . , n - 1,(63)

"

where the coefficients akpi(z) are complex constants for p+l = m—1 (which we denote by a,kpi), and are sufficiently smooth complex-valued functions defined on r for the remaining values of p and I. We put d

kj =

]C

a

M^>

J = 0 , . . . , m - 1, k = 0 , . . . , n - 1,

p+l=m—1 d

kj =

^2

°-kPi>?j, j = m, . . . , n - 1, k= 0, l , . . . , n - 1.

p+l=m— 1

Let Co be the matrix with elements dkj (k,j= 0 , . . . , n — 1). The conditions (6) - (7) is a special case of general condition (63). In this special case the matrix Co coincide with the matrix AQ defined by (58), and hence by (59) det Co ^ 0. This shows that there exists an infinite number of conditions of the form (63) satisfying det Co ^ 0. We have the following result. Theorem 3.1 If det Co ^ 0, then the problem (5), (63) can be reduced to a Fredholm integral equation of the form (62) with q — n(2m — n), and the index of this problem is equal to n(2m — n). Proof Under the boundary conditions (6) - (7) the proof is given above. Under the general boundary condition (63) the proof is similar to before.

265

References 1. N. E. Tovmasyan, Non-regular differential equations and calculations of Electromagnetic fields. (World Scientific, Singapore, New Jersey, London, Hong-Kong, 1998). 2. A. V. Bitsadze, Boundary value problems for second order elliptic equations. (Nauka, Moscow, 1966) (in Russian). 3. N. E. Tovmasyan, Boundary value problems for partial differential equations and applications in Electrodynamics. (World Scientific, Singapore, New Jersey, London, Hong-Kong, 1994). 4. P. A. Soldatov, Function theory methods in boundary value problems on the plane, Izvestia AN SSSR, Mathematika. 55, no 5, (1991), pp. 56-61 (Russian). 5. N. J. Muskhelishvili, Singular integral equations. (Nauka, Moscow, 1962) (in Russian). 6. M. A. Lavrentiev and B. V. Shabat, Methods of theory of complex variable functions. (Nauka, Moscow, 1973) (in Russian).

266

O N T H E SOLUTIONS OF T H E ITERATED B E R S - V E K U A EQUATION P. BERGLEZ Department of Mathematics, Graz University of Technology Steyrergasse 30, 8010 Graz, Austria E-mail: berglez@weyl. math, tu-graz. ac.at For the solutions of the iterated Bers—Vekua equation Dnw = 0, n e N,n>

Dw := —— + a(z, z) w + b(z,z)tii Oz we give a representation theorem using the solutions of the equation D w = 0. From this result several further representations of the solutions considered are deduced.

1

1,

with

Introduction

The Bers-Vekua equation characterizing the pseudo-analytic functions is given by Dw = Q with

Dw := —— + a(z,z) w + b(z,z)w (1) oz where the functions a and b are assumed to be analytic in a domain V of the complex plane (cf. e.g. L. Bers 6 ) . For the representation of these functions several methods have been developed such as the use of integral operators by I.N. Vekua 9 ' 1 0 or the use of differential operators by K.W. Bauer 3 and the author 4 ' 5 . In this paper the iterated Bers-Vekua equation Dnw = 0, neN,n>l,

with

Dnw = DiD"-1

w)

(2)

is considered. D. Pascali 8 gives a representation of the solutions of (2) by iterated application of a certain integral operator. Here we show that the solutions of (2) can be determined in a simple way using pseudo-analytic functions. 2

Relations to pseudo-analytic functions

With the transformation w = eA w

where

Ag = —a

267

we get Dw = D(eA w) = eA Dw with the operator D given as „

dw

_

Dw := -— + cw with oz Thus equation (1) can be transformed into Dw = -^r+cw oz

,

c = oe

i_A

=

ti.

(3)

On the other hand we get Dn w = Dn-k

(eA Dk w)

for

k =

0,l,...,n

using D°w := w which leads to Dnw =

eADnw.

Hence instead of (2) we can consider the reduced iterated equation Dnw = 0 with

Dw:=-^

+ cw

(4)

oz without loss of generality. Now we deduce a representation for the solutions of (4) by means of a set of pseudo-analytic functions which are solutions of (3). In the following let T> denote a fundamental domain for the differential equation (3) in the sense of I.N. Vekua 10 . Using the function iui := Dn~lw

Dn~2w , 77 = z + z

V equation (4) can be written in the form Dw1 + -wi=0.

(5)

77

Following this method we set wk := Dn~kw - - D " - ' 0 - 1 ^ , k = 2, 3 , . . . , n - 1

(6)

77

and get for the functions u;* ,fc= 2, 3 , . . . , n — 2, the relations

With k = n-l

wk = Dwk+i + -Wfc+i.

(7)

_. Dw

(8)

in (6) n—1 w — 7«n_i

268

follows. Setting wn := w we obtain from (5), (7) and (8) a system of n differential equations of first order for the functions wi,..., wn , which may be written in matrix notation as Wz + AW + BW = 0

(9)

for the vector valued function W = t{w\,... ,wn) with the matrices B = cE (where E denotes the (n x n) unit matrix) and A = (ajk) with a,jk given by I/77 (1 - n)/r) _ ! for 0

_ a Jk-\

for 1 < j = k < n - 1 for j = k = n j = fc + l , l < f c < n - l else

'

Using the transformation W = pV with the nonsingular square matrix p = (pjfe) with

Pjk = <

^0kY 0

j
l
-fc(n-i-fc)i J=n> rf1 1 j =k = n

l
system (9) turns over into the non-coupled system Vg + c& = 0 t

for the vector V — (vi, • • •, vn). Therefore the functions i>k,k = 1 , . . . , n, are solutions of the differential equation Dvk = 0 which means that they are pseudo-analytic functions. The solution W of (9) now can be calculated from W = pV. To get the solution w of equation (4) we have to take the last component wn of the vector W: W = Wn = Y^ PnkVk fe=l

1 1 -T]n-2V1 x (n-2)!'

- —.

^ 7 f ? n _ 3 W 2z

2(n-3)!'

1 n - 1TVn-1

+ T? n _ 1 U n

Since with Vk the function Vk = avk,a G R, is a solution of Dvk = 0 also the function w can be represented in the final form as n-l

w = Y, fc=0

Vk Vk

(10)

269 with Dvk = 0,k = 0, . . . , n — 1, and we have the following Theorem 1 1) Let the functions Vk, k = 0 , . . . , n—1, be solutions of the BersVekua equation Dvk = 0 in the fundamental domain V. Then the function w according to (10) represents a solution of the iterated Bers-Vekua equation Dnw=0 inV. 2) For each solution w of the iterated Bers-Vekua equation defined in V there exist functions VQ,- • • ,vn-i, pseudo-analytic in T> such that w is given by equation (10). To obtain a relation between a solution w of (4) in the form (10) and the pseudo-analytic functions Vk we consider "-1

i.|

Dlw = y2-—-Vk-lVk,l

= 0,l,...,n-l

(11)

which follows from equation (10). The expressions in (11) represent n equations for the functions VQ, ... ,vn-i, from which we prove by induction that the Vk , k = 0 , 1 , . . . , n — 1, are given by n—l — k

»* = k\* ^E

^D^u,. l\

(12)

1=0

Lemma 1 For each solution w of (4) in the form (10) the functions Vk are determined uniquely by (12). R e m a r k In the particular case c = 0 the iterated Bers-Vekua equation (4) reduces to dnw dzn ~ which characterizes the polyanalytic functions (cf. e.g. M.B. Balk 1). Here the functions Vk,k = 0,.. ,,n — 1, considered in Theorem 1 are holomorphic functions and the expression in (10) can be converted into the form w = 2_\

Z>C v

k i

v

k holomorphic

fe=o

the well known representation for polyanalytic functions. Therefore the solutions of (4) are called poly-pseudo-analytic functions. For pseudo-analytic functions there holds a similarity principle proved by L. Bers 6 as a consequence of which a close relationship between the analytic

270

functions and the pseudo-analytic functions can be proved. For example the zeros of a pseudo-analytic function are isolated as the zeros of an analytic function are. For poly-pseudo-analytic functions there cannot exist such a principle which would show that the function theoretic properties of a polypseudo-analytic function are in close connection to the properies of an analytic function. We can see this from the following example: Let w = Y^kZo rlk vk according to Theorem 1 be a poly-pseudo-analytic function with VQ =0. Then with z = x + iy the function w may be written as n-1

w = r]'^2r]kvk+i

n-2

=2x^2vkvk+i

k=0

•

fc=0

This function vanishes on the whole imaginary axis (x = 0), therefore the zeros of a poly-pseudo-analytic function are not isolated in general. 3

Integral representations of solutions

After I.N. Vekua 10 for every solution u of (3) analytic in the simply connected domain T> there exists a function such that u can be represented in the form T1(z,z,t,Co)!P(t)dt+

u(z) = (p(z)+

r2(z,z,z0,T)cp*(T)dT

Jzi

JCi

with ZQ, 21, CO, CI € T> arbitrary fixed points. Here (p* denotes the conjugate function to ip and the resolvents Ti and T 2 can be determined from T1(z,C,t,T)

= W1(z,C,t,T)

+

r 2 (*,<,t,T) = Wl(3,C, t,T)

iW2(z,C,t,T) -iW2(z,Ct,T)

where W\ and W2 are two-parametric solutions of Dw = 0 which satisfy the following conditions Wi | C = T =

-C(Z,T),

W2

|C=T=

-ic(z,r).

For particular equations the corresponding resolvents were determined by K.W. Bauer 2 and K. Nusser 7 . Starting with this result we can give the following integral representation for the solutions of (4).

271

Theorem 2 For every solution w of (4) defined and analytic in the domain T> there exist functions such that w can be represented in the form n—l

-z k

w{z) = Y,V Vk(z)+

n—1

T1(z,z,t,Ca)Y^Vk
/ Zl

fc=0

fe=0

+ f r 2 ( z , z, z0, T) J2 Vk
(13)

fc=n

Conversely for all functions Pk(z) holomorphic in T> the expression in (13) represents a solution of (4)For a pseudo-analytic function u there exists the following generalized Cauchy integral formula (cf. I.N. Vekua 1 0 ). Let Q be a finite, multiplyconnected domain bounded by a finite number of rectificable closed curves. Let C denote the boundary of Q and ZQ = XQ 4- iyo be a fixed point inside Q . Then the value of u in ZQ is given by u z

( o) = y~-• /

(u{z)U(z,z0)dz-u(z)V(z,z0)dz)

with the kernels U(z,z0) = Ui(z,z0) V(z,z0) =V1{z,z0) Ui{z,z0)

=

U2(Z,ZQ)

(14)

-V2(z,z0)

r1(z0,^,z,z)\og((z

- z0)(z

-z^))

z — zo +

{^i(zo,zo,z,z)-T1{t,z^,z,z))Jza

+ / Jzo

Vi(z,z 0 ) f* +

t-ZQ ~, l

/

~ z0

ri(£,zti,z,z)c(£,z5)d£

Jzn

-r*2{zE,z,z,z)log{{z-z0){z-z^)) _ _ (T^(Z0,Z0,Z,Z)-V^{T,Z0,Z,Z))

JTE

+ /

=z /

Ai(z,z,z0,v)dr] Jo

—

r - Z0 T2{r],zo,z,z)c*(r),zo)dri

rzo

U2{z,z0)=

dr

+

272

+

drj

Jo rZQ

+

rZQ

dV

Jo

r1(z0,zE,C,v)Ai(z,z,^r))d^

Jo

r1(z0,z0-,t,r])A*2(z,z,r],Z)dZ

Jo

rzo

V2{z,zo)=j

Al{z,z,z$,£)d£

Jo rZQ

rZQ

d

+

€

Jo

r*1(z0-,z0,T),t)A?i(z,z,n,t)dr) Jo

rzo

pzo d

+

£

Jo

mzo,z0,V^)Ai(z,z,^,ri)dr] Jo

where the functions At(z, C, z0, Co), i = 1,2, are the analytic continuations of the functions . . . dUi(z,z0) A1{z,z0) = ~ \-c(zo, z0)Vi(z, z0) azo dVi{z,z0) A2{z,z0) = — ^ z z — +c(z0,z0)U1(z,z0) (which are analytic in the real variables x,y,XQ,y$) into the complex plane. They are analytic functions of z, C, .zo, Co- The resolvents Ti, T2 are defined as above. The kernels U, V depend on the coefficient c of the Bers-Vekua equation only and do not depend on the domain considered. With this we can give the following representation for a solution of (4) which is a generalized Cauchy integral formula for a poly-pseudo-analytic function. T h e o r e m 3 Let C denote the boundary of a (multiply-connected) domain Q. Then (*°) = i E ( ^ + ^

/

( i f J2

Ti7f(2 + ^

({-~(z -1)'

+

[Dk+lw(z)]

U(z,z0)dz

z)l[Dk+lw(z)}v(z,zo)dz

1=0

gives the value of a solution w of (4) in zo G Q if for w and the expressions Dkw, k — 1 , . . . , n — 1, the values are prescibed on the contour C. The kernels U, V are given by (14).

273

L. Bers 6 introduced a generating pair of a pseudo-analytic function in the following way: The pair (F, G) of particular solutions of (3) is called generating pair if the following conditions are satisfied: • $s[F{z) • G(z)} > 0 in the domain D, • Fz,Fz,Gz,Gz

exist and are Holder continuous in V.

Then Bers showed that for every pseudo-analytic function v there exist two real-valued functions $(x,y),'$>(x,y) such that v can be represented in the form v = $F + VG. The functions $ and ^ are determined uniquely by vG-vG =rrz, FG-FG'

$ = ^ s

vF-vF FG- FG

T

*

Starting from these results we can use for the pseudo-analytic functions vk in the representation of a poly-pseudo-analytic function w according to Theorem 1 the form vk = $ fc F + * fc G , k = 0 , . . . , n - 1 with suitable real-valued functions Qk,tyk,k n—1

= 0,...,n

— 1. This leads to

n— 1

w^Ys^^kF + YsV^kG. k=0

(15)

fc=0

Now the functions
vkG-vkG FG-FG

=

'

k

vkF-vkF FG-FG'

K

'

To determine the functions v^ from the poly-pseudo-analytic function w one can use the relation (12). Thus we have the following Theorem 4 Let (F, G) be a generating pair of (3) in the sense of L. Bers. Then for every poly-pseudo-analytic function w there exist real-valued functions $£, tyk, k = 0 , . . . , n — 1, such that for w the representation (15) holds. The functions $k, $k are given by the relations (16) whereas vk can be calculated from w using (12).

274

4

Differential operators for the solutions

Differential operators have been introduced successfully by K.W. Bauer 3 for the representation of solutions of partial differential equations. The author 5 proved a necessary and sufficient condition on the coefficient c in equation (3) under which such operators exist, which give all the solutions v of (3) by means of a holomorphic function / in the form

v = Y,aj(z,z)g-f(z)

+£

j=0

b^z^—fiz)

=: Kmf + K*m_J,

mGN.

j=0

The coefficients a,j and bj can be calculated from the factor c in a simple way. Now we use for the solution w of (4) in the representation (10) for the pseudoanalytic functions Vk the expressions Vk = Kmfk + K^_1fk,fk holomorphic in T>, k = 0 , . . . , n — 1. We get n-l fc=0 which can be rearranged to the form m

TO—1

w = yjaj(-z,2)(5 J M+ \ J bj(z,z)5:>u j=o j=o where u denotes the polyanalytic function u = Y^l=o Vkfk and S is a differential operator defined as Su :— uz — w2 with 8^+lu = S^u), 5° := u. This result means that in the case of the existence of differential operators Km and K^n_1 of Bauer-type for the differential equation (3) the poly-pseudoanalytic functions can be calculated from the polyanalytic functions by means of differential operators. References 1. M.B. Balk, Polyanalytic Functions, Akademie Verlag, Berlin, 1991. 2. K.W. Bauer, Bestimmung und Anwendung von Vekua-Resolventen, Monatsh. Math. 85 (1977), 89 - 97. 3. K.W. Bauer and St. Ruscheweyh, Differential Operators for Partial Differential Equations and Function Theoretic Applicatons, Lecture Notes in Math., 791 (1980). 4. P. Berglez, Characterization of certain differential operators in the representation of pseudo-analytic functions, Rend. 1st. Mat. Univ. Trieste 14 (1982), 2 7 - 3 1 .

275

5. P. Berglez, Differentialoperatoren bei partiellen Differentialgleichungen, Ann. Mat. Pura Appl., 143 (1986), 155-185. 6. L. Bers, Theory of Pseudo-Analytic Functions, New York University, 1953. 7. K. Nusser, Uber die Darstellung von pseudoanalytischen Funktionen, Diplomarbeit, Graz, 1994. 8. D. Pascali, The structure of n-th generalized analytic functions, in: G. Anger (ed.): Elliptische Differentialgleichungen, Vol. 2, Akademie Verlag, Berlin, 1971, 197 - 201. 9. I. N. Vekua, Verallgemeinerte analytische Funktionen, Akademie Verlag, Berlin, 1963. 10. I. N. Vekua, New Methods for Solving Elliptic Equations, North Holland Publ., Amsterdam, 1967.

276 THE BAUER-PESCHL EQUATION — DERIVATION A N D SOLUTION OF A PARTIAL DIFFERENTIAL EQUATION B Y LAPLACE'S M E T H O D

K. W. TOMANTSCHGER Department of Mathematics A, Graz University of Technology Steyrergasse 30, 8010 Graz, Austria E-mail: tomanQweyl. math, tu-graz. ac. at The present paper is concerned with the Bauer-Peschl equation (1 ±zQ2wz(

+ Xw = 0.

The problem, for which A-values the solutions of the equation can be represented by differential operators, can be solved by the method of Laplace. Assuming the solution in form of a differential operator w(z,0

= A0(zX)X(z)

+ Ai(z,OX'(z)

+---An{z,C)X(n){z),

X{z) being arbitrary holomorphic functions, we obtain an overdetermined system of partial differential equations for the functions Aj(z,Cj. Since the coefficients Aj(z, £) are polynomials in zC,/{\ ± z ( ) , the solution is a double sum. This is quite another kind of representation of the solution as we know it from K. W. Bauer, who solved the equation in a completely other way.

1

Introduction

In this paper we shall derive a general solution of the equation (1 + ezC,fwzi:

+ \w

= 0,

AeC,

e = ±l

(1)

with differential operators. During the past two decades, this elliptic resp. formal-hyperbolic differential equation has often been referred to as the Bauer-Peschl equation. 1966 introduced K.W. Bauer the differential operators with this equation ( 1 ). K.W. Bauer had derived the following solution for A = en(n + 1), n e IN, .

.

- A (2n - u)\

(-e)"- 1 '

/^

„ dv

n_v

dv

\

^o = EA(^i (1 + e , 0 n-,(c" j?f(*) + *" or9(0), f(z), g(() beeing arbitrary holomorphic functions. But when K. W. Bauer started to solve equation (1) he didn't know that special differential equations possess solutions, which can be represented by arbitrary holomorphic functions and a finite number of their derivatives. So his solving way was

277

completely another one than we use it for calculating differential operators today. In this paper Gi and Gi are simply connected domains of the complex z resp. £ plane; z £ Gi, ( £ Gi. G\ and G2 is assumed to be such that ez( y^ — 1 in Gi x Gi- We look for a general solution of (1), represented by two arbitrary holomorphic functions X(z) and Y(() and a finite number of their derivatives, that is,

w(zX) = 52M*X)—£J- + '£Di(z,0—j£l. 3=0

j=0

^

Aj(zX) and Dj(zX) are determinate coefficients and have to be calculated. Since not every partial differential equation possesses such a kind of solution, it is clear, that we first determine all these constants A, for which such solutions exist. This can be made by the method of Laplace a (see 5 , pp.68). In Chapter 2 this method is sketched shortly. Applying it on the Bauer-Peschl equation we obtain A = e n (n + 1 ) , n e IN. In Chapter 3 we set Y = 0, i.e., we take the part of the solution with X(z) and its derivatives. Inserting this integral into the equation (1) with A = e n ( n + 1), and using the fact that the functions X, X', X", • • • , X^n+1^ are independent, we get an overdetermined system of n + 2 equations for n + 1 coefficients Ao, • • • , An (see (25) - (28)). Only if the last equation is identically fulfilled for the already calculated coefficient Ao, the equation possesses an integral in such a form (see (28)). In our case the last equation will be satisfied, because we already have used the method of Laplace. There we demanded that the invariant hn must be equal zero. We treat our system (25) - (28) with n = 1,2, 3. That means, that we look for solutions which go till to the first, to the second and to the third derivative of X(z). Looking at the three calculated solutions we see, that we need a double sum for representing our general integral (see (44)). Applying the symmetry of equation (1) in z and £, we easily obtain the coefficient Dj. Remark 1 Equation (1) is of importance, for instance in the case A = — n(n+l), e = —1. This Bauer-Peschl equation is closely related to several basic equations which are of mathematical and physical importance ( x , 2 ). It is related to the twodimensional wave equation WX1X1 + WX2X2 = WX3X3 a T h i s method given by Laplace 1777 was developed and extended by Darboux. Today it is upon his exposition that the present account of the method is based.

278

from which it results by separation of variables and suitable projection. It may also be obtained from the ultrahyperbolic equation WXlXl + WX2X2 = WX3X3 + WX4X4 as well as from the parabolic equation WXlXl + WX2X2 = WX3X3 + WXi . 2

Laplace's method

The differential equation (E) :

wzC + a(z, ()wz + b(z, C)wc + c(z, ()w = 0 ,

w = w(z, Q , (2)

can always be written d (dw

\

+aw

d-z{-dC

)

, fdw + b

\ +aw

{-OC

,

..

hw

)=

{3)

'

where h = az + ah — c. If h = 0, equation (3) reduces to a linear equation uz + bu = 0, where we have taken u = w^ + aw as an auxiliary unknown. The general integral of this first order equation leads us to an intermediate integral of equation (2) dw - _ fbdz — + aw = Ye J where Y is an arbitrary function of £. Integrating this differential equation, we obtain the general solution of equation (2) w

-fadi

X(z)+ [Y{()efad<:-fbdzdt

,

X{z) being an arbitrary function. We see that this integral is of the form w = aX(z) + JpY(C)dC,

(4)

a, 0 beeing determinate functions of z and (. If h

T£

0, we set dw wi = — +aw.

... (5)

279 Substituting this into (3), eliminating w from the resulting equation and inserting w into (5) yields (Ei) :

witZ{ + aiwi
(6)

where ai = a- (log h)^,

bi = b,

a = c - az + b^ — b (log h)^ .

If we would know ioi, we could calculate w, because from (3) and (5) we get W

=

u>i,z + bwi —h •

It is clear that the integration of equation (2) and of equation (6) constitutes two equivalent problems. Making the substitution Wi = wz + bw

(7)

and defining k — &£ + ab — c, we obtain equation WiX + aWi = kw .

(8)

If k = 0, the general integral of (2) resp. (8) can be represented in a formula analogous to formula (4), which would be derived from it by permuting z and £, X and Y. Providing k ^ 0, equations (7) and (8) will lead by elimination of w to the new equation, analogous to (6). Remark 2 The two expressions h and k are called the invariants of equation (2), because h and k don't change if we make a change of the unknown function such that w = M(z, ()w for any function M.

Denoting the invariants of (6) by hi and ki, an easy calculation shows hi = aiyZ + aibi — ci = 2h — k — (log h)z^ , ki = &i,£ + ai&i — c\ = h .

280

k\ cannot be zero since we have supposed h ^ 0. But it may happen that h\ is zero without either of the invariants h and k vanishes. If this is so, the general solution of equation (6) can be derived. According to formula (4) we obtain w = A0X(z)

+ A1X'(z)

+ f BY(()d(

,

(9)

A 0 , Ai, B being definite functions of z and (, and X(z), Y(£) being arbitrary functions. If we set Y — 0, the solution (9) consists of an arbitrary function X(z) and of its first derivative. The repeated application of Laplace's method will lead to a sequence of equations (E), (E1), (E2), . . . . We also get a sequence of h, hi, h2, •••, and k, k\, k2, • • •, for which generally holds hn = 2hn_i - kn-i - (log/i n _i) z C , kn = hn-i.

(10) (11)

If we know the invariants and the nth transformation wn we can write down the dependent variable w(z,(). For the transformation holds wm+i = wm£ + amwm .

(12)

Then equation {Em), already written as a first order equation with the invariant hm, becomes hmwm

= ium+i,a +bwm+i.

(13)

Consequently, fbdz WmeJ

1

d /

=

h^d-z rm+ieJ

fbdz\

)'

and therefore _d d__ d ( Jbdz\ (14) hdz hidz hn-idz \ ) If we come, after n successive transformations, to an equation En for which the invariant hn = a n , 2 + anb — cn = 0, this equation is integrable. Going back step by step, we can derive from it the general integral of the equations (En-i), {En-2), • • •, up to equation (E). Inserting wm+i from (12) into (13) and solving the resulting equation for m = n and with hn = 0 , we have wJbdz

=

wn = a{X + I YfidQ)

with

281

X(z) and y(C) are arbitrary holomorphic functions, cx(z,() and (3(z,() are determinate functions. Now, according to (14) the relation between w and wn is Jbdz me

3 d hdz h\dz

d hn-\dz

°* (x(z) + JY(()e-td(

(15)

and therefore, effecting the differential operations, we have the value of w expressed in the form w =

(16)

A0 X + jY(3dc\+AAx' + jYpzdC +.-- + An\x^ + J t-d(^ where Ao, A\, ••• , An are determinate functions of z and ^. X^' represents the j t h derivative of X with respect to its argument z. Since the functions X and Y are arbitrary, (15) is a general solution. Taking Y(£) = 0, we get the specialised integral (particular solution) w = A0(z, C) X(z) + Ar(z, Q X'(z) • • + An(z, 0 I w ( z ) ,

(17)

so that an integral exists of an arbitrary function X(z) and of a finite number of its derivatives. The assumption for the existence of such a solution is that one of the invariants vanishes in the succession of equations, which are constructed by repeated transformations of Laplace. 2.1

The invariants of the Bauer-Peschl equation

Let us apply this method to the Bauer-Peschl equation (1). We ask for which coefficient resp. for which constants A our differential equation W +

(l+Xez(rW

«

= 0

'

e = ±1

'

possesses an integral of the form (17). In our case we have a = 0,

6 = 0,

c = — (1 +

and therefore -A h = k = -c = (1 + £ZC) 2 '

—J, ezQ2

282

Eliminating A;„_i from (10) by (11), we get the recursion relation ft„ — 2/i„_i - hn-2 - (log/i„_i)^,

h0 =

ft,

n > 2.

(18)

Starting with n = 1, we use the recursion (10) und set k$ = k. Then h

2e 1

(1+£*C)

_ 2

- A + 2e

~~ (1 + ^ C ) 2 '

In the next place, we take n = 2 in (18). This yields h2

2e 4 /, , £ 2 = hi + (/i! - ft) + 7;—• TvT = fti + (1 + szQ (1 +ff^C) 2

- A + 6e (1 + e z < ) 2 '

Making one step more, we obtain from (18) with n = 3 , ,

2e , 6e - r 2, = ft2 + (1 + £zC) (1 + e ^ C ) 2

"3 = "2 + (ft2 - ftl) + 7-—

- A + 12e (1 + ezC)2'

Consequently, ,

,

hn = « n - l + (1 +

2ne ezC,f

and -A + £ n(n + l) (1+ezC)2 ' That means, that the invariant ft„ only can vanish at any stage of the successive transformations, if the constant A is an integer of the form A = en(n

+ 1),

(20)

where n is a positive integer. Only in this case the solutions of the BauerPeschl equation can be represented by an arbitrary function X(z) and its first n derivatives X'(z), • • • , X ' " ' ( 4 Remark 3 Similarly, as we have constructed the solution (17), we can get a solution of the form w = D0(z,QY(0

+ D1(z,C)Y'(C)

+ ••• + Dm{zX)Y^m\Q,

(21)

where Y(() is an arbitrary holomorphic function and Do, D\, • • • , Dm are determinate functions. Here we used the transformation (7).

283

Lemma (1) When the differential equation (2) WZQ + a(z,()wz

+ b(z,C)w<; + c(z,C)w = 0

possesses an integral in the form w = A0X{z) + --- + resp.

AnX^{z)

w = D0Y(Q + ••• + DmY^

((),

then the transformation w^ + aw = w\

resp.

wz + bw = W±

was effected. X and Y are arbitrary holomorphic functions. (2) Let us suppose that both sets of transformations are finite in the sense that, in each set, only a finite number of operations is needed to produce a vanishing invariant. Then the general integral of equation (2) is

w j=0

j=0

After we have determined the invariants, we use formula (15) for getting the solution of our differential equation. Taking Y = 0 and inserting 6 = 0 yields d

d

hdz

h\dz

d

hn-\dz

,- J an dCX(z)

(22)

Now we have two possibilities. The first possibility is to calculate an and to substitute it into formula (22). By effecting the differential operations, we obtain the solution of the form (17). The second way is to substitute the solution (17) with unknown coefficients AQ(Z, £),• • •, An(z,() into our differential equation (1) with A = en(n + 1). Since the functions X(z), X'(z), X"(z), • • • , are linear independent, the equation is only satisfied, if the factors of X^n>(z), n = 0,1,2, • • •, are equal zero. In this way we get a system of differential equations for the coefficients.

284

3

Calculation of the solution of the Bauer-Peschl equation by a system

Now we want to determine the solution (17) in G\ x G2 of the differential equation enln 4- 1) = 0, £ = (l + ezC)2W The integral is represented by a differential operator WzC+

, > ±1

(23)

"

fc=0

where X{z) is an arbitrary holomorphic function. Then by directly substituting (24) into (23) we obtain the recursive system X(n+1)(*) = ^ X(n)(z).

= 0

(25)

a ^ n - i _ e n ( n + l)

X<*)(z): _2^± = p^ oC

X(z):

^

.

ozo(

+

+

l^±A^

k = n-1,.,1

(1 + e z Q

(27)

^±AAO=0.

(28)

azaC (1 + ezQY Since this system contains n + 2 equations but only n + 1 undetermined functions Ak, the system is overdetermined. So the last equation must be satisfied. In our case this surely will be done, because Laplace's method guarantees us the existence of a solution in form of a differential operator (24) of the partial differential equation (23). Taking the case where h\ = 0, e.g. n = 1, then a solution of the form w = A0(z,C)X(z)

+

A1(z,0X'(z)

exists. Integrating relation (25) and (26) with n = 1, we have 2

A\ = Ai(z)

arbitr.

and

A0 = —7— AAz). v y v ; z{\ + ezQ Inserting AQ into (28) with n = l w e get by comparison of coefficients the two equations " 1 " : -zA[ + 2Ai = 0 ,

285

C : -z2A[

+ 2zA± = 0.

A\ = z2 represents a particular solution of these two dependent equations. Hence AQ = 2z(l + ez()~ . The equations (27) drop in our case. We thus have Proposition 1 In G\ x G2 the differential equation 2e Wz +

<

(1 +

szCy

w = 0

possesses the particular solution 2z w = , "~ , X + z2X'. 1 + ezC,

(29) '

v

In our next step the invariant h2 vanishes. That means that n = 2. But let us ask, what happens, if we don't set n = 2 in equations (26), (27) and (28). Will we get more solutions or not? Now the integral possesses the form w = A0(z,()X(z)

+ A1(z,C)X'(z)

+

A2(z,C)X"(z).

For this solution the system (25) — (28) reduces to A A 1 \ i-x. A2 = A2(z) arbitr.,

.

/ .

en(n

+ 1) . \

1

V

sn(n + 1) ^ A2 (1 + ezC)

A

AltC =

C + " 0

.

/

en(n

+ 1) ,

(1 +

ezQ

From the first equation we obtain n(n + 1) A

i

= —FT

. , .

,

TTA2Z.

. 30

w K z{\ + ezQ ' By inserting this into the second equation, we get by integration with respect to C

A0 = -n(n + l) z(l + ezQ

, 2

en(n

+

l)(s

Z*{1+EZQ2

+

2zQ

n2 (n + if 2z2(l+£zC)

286

Substituting this into the last equation and making comparison of coefficients, we get • l V ^ -

j B

( „

+

l ) . ^

+

n(„+.)(=fiLtii+2)M,-M»)

C : z2A'2' - [2 + n{n + l)]zA'2 + 3 n ( n + l)A2 = 0, C2 : z2A'2' - 4zA'2 + 6A2 = 0.

(33) (34)

The Euler-Cauchy equation (34) possesses the solutions A2 = z2 and A2 = z3. By substituting A2 = z2 into (32) and (33), we see that these equations are only fulfilled for the positive integer n = 1. Now we can calculate the two other coefficients. With (31) and (30) we get A0 = 0 and Ai = 2z(l + ez £ ) _ 1 . Proposition 2 In G% x G2 the differential equation ezQ2

(1 + possesses the particular solution

2z X' + z2X". 1 +ez(

(35)

But this is quite the same result (29) as we had before. We only have to introduce a new arbitrary holomorphic function X\{z) = X'{z). Now we consider the case A2 = z3. Substuting this in equation (33), we see that it is identically satisfied for all n. Giving A2 = z3 in (32), we get the relation [n(n + 1) - 2][n(n + 1) - 6] = 0, which implies the positive integers n = 1 and n = 2. From (31) and (30) we have with A2 = z3, n = 1 and n = 2 -2-2

2z2

,

A0 = — 7, M 1 + ez(' . . 6z(l-ez() . and A0 = —±- r f , Al = (1 + ezQ2

1 + ez( 6z2 -— 1 + ezC,

Now we can say: Proposition 3 In G\ x G2 the differential equation (1 +

ezQ2

respectively.

287

possesses the particular solution w = - r ^ — . X

+ J^—JX*

1 + ezC,

+ z3X".

(36)

1 + e^C

Proposition 4 In G\ x Gi the differential equation (1 + e^C) possesses the particular solution

w

= w^Ax

+

(1 + ezQ1

rf~rx'+ z*x"- (3?)

1 4- ezC,

Starting with a solution of the form w = A0(z,()X(z)

+ A1(z,QX'(z)

+ A2(z,()X"(z)

+

A3(z,OX'"(z),

we obtain in a similar way the following results: Proposition 5 In G\ x Gi the differential equation 2e ezQ2

(1 + possesses the particular solution 97

1 +

ez(

X' + z2X".

(38)

This corresponds to (29) and (35). Proposition 6 In G\ x Gi the differential equation 2e (1 +

ezQ2

possesses the particular solution 2z „, 2z2 X' + 2 ,X" 1 + ezC, 1 + ezC, Defining X\(z) = X'(z),

we have Proposition 3.

+ Z*X'" •

(39)

288

Proposition 7 In G\ x G2 the differential equation 6e 'JzC

(1 +

ez()2

w = 0

possesses the particular solution p ^ O x > (1 + ezQ2 Defining X\{z) = X'(z),

+

- ^ - X » 1 + ez(

+ Z*X»>.

(40) v

we have Proposition 4.

Proposition 8 In G\ x G2 the differential equation (1 + e^C) possesses the particular solution A-72

A-7

w =

?73

= ^ — X' + X" + z4X'" . 1 + ezC, 1 + ezC,

.X 1 + ezC,

(41)

Proposition 9 In G\ x G2 the differential equation Wzd + 7 — (1 + possesses the particular solution w

12z(l - ezQ = -W-^X (1 + ezC)2

jr^w ezQ2

1 - ,, l 2 £ Z \22 X' (1 + ez() "

+

= 0

r^-rX" ' 1 + ez(

+ ,4X'".

(42)

Proposition 10 In Gi x G2 the differential equation (1 +

ezQ2

possesses the particular solution A2\ 2 4 z ( l - 3 e < + zV 2C )

(i+^o

3

y

+,

1 0 ,22 / o _ n 0 , r t 12z (3-2ezQ

(i+^o

2

v

, ,

+

- | Q „ 33

12*

m^cx

y

„ ,

4 +zX

• (43)

289

Taking the integrals of Proposition 1, 4 and 10, we see that they also can be written in the following form £z(

w = 2z I 1

w =

2z2e

3ezC 1 + ezC 6ez( s 1 + £Z(

w = Gz [ 1 24z\l +I2zz

3

+12z d

1 -

(1 + szCy l0z2C2 s 2 + (1 + EZ()

1 + e^C £ZC

l +

z2X',

X +

1 + ezC,

X + 6z2

1 -

ez( X' + z33Xv-n 1 + ezC

$£ZA3^3 C

X +

(1 + £^C) 3

' (1 + e ^ O 2

\ X" + zAX'". ez(J

Now the building law can be seen easily. Therefore we set n—k

n

w(z,0

,

= ^>fc+1 k=0

v

j=0

-ezQ V 4- ezC,

X^k\z

(44)

where £?£,., are constant coefficients which have to be determined. Inserting solution (44) into the differential equation (23), all factors of the independent functions X^k\ k = n + 1, n, • • •, 0, must vanish. This implies the system n—k

Y^(-eyjBkJ((k

+ j + l)

J=0

C

C

I ) -(1 + £2C)^' +2 (1 + ezCy+1 T - C wJ +" *'

+

n-fc+1

E (-e^B*-u(1 + n—k

+ en(n

+ 1) ^ ( _ e ) J " B f c j .

i?n+i,-i = 0 ,

(45)

ezC)i+i

^•+i ^

(1 +

ezQi+2

0,

A; = n + 1, • • •, 1,0.

Starting with k = n + 1 we get 0 • Bni0 = 0. This coefficient cannot be calculated because its factor is j = 0. From our solutions (29), (37) and (43)

290

we can assume that Bn,a = 1, because it is the constant factor of the highest derivative of X{z) multiplied by zn+1. Remark 4 If it would hold Bn$ ^ 1, we could devide the solution (44) by the constant Bnfi for obtaining the term 1 • zn+1X^n\ The three integrals of Proposition 1, 4 and 10 also allow us the conjecture £n-i,o = -Bn-i,i- This relation can be proved by writing the factor of X^n~^ in the following form: const. zn

ez( 1+ezC

= const. zn

Bn-1,0 + Bn-\\— 1+C 1 + ezC Taking (45) with k = n and deviding the resulting equation by (—ez)/(l + ezQ2 we have the relation Bn-i,i

- n(n + l)Bnio

= 0,

which implies J 5 „ _ u = B n _i, 0 = n ( n + 1).

(46)

In our next step we set in (45) k = n — 1. Furthermore, we make a comparison of coefficients with respect to (e J 'z- 7 ^ J_1 )/(l + ez(y+1 for j = 1,2. The two resulting equations are - B n _ 2 , i + n(n + l)B n _i,o - (n + l ) B „ - i , i = 0 and

2B n _ 2 ,2 + [2 - n (n + l)]Bn-hl

= 0.

Applying the results before we have Bn-2,i

= ( n - l ) n ( n + l)2 ,

Bn-2,2 = \ (n - 1) n (n + 1) (n + 2).

(47) (48)

Entirely in the same way as before, (45) yields for k = n — 2 - n B „ _ 2 , i +n(n

+ 1) B„_2,o - # n _ 3 , i = 0,

[2 - n ( n + l)]B„-2,i + 2(n + l ) B n _ 2 > 2 + 2 5 n _ 3 , 2 = 0, [ 6 - n ( n + l p n - 2 , 2 + 3B„-3,3-0.

(49) (50) (51)

291 Using (47) and (48), we deduce from equation (50) and (51) that Bn-3,2 = \ (n - 2) (n - 1) n (n + l ) 2 (n + 2),

(52)

B„-3,3 = ^ ( n - 2)(n - l ) n ( n + 1) (n + 2) (n + 3).

(53)

Bn-3,i cannot be calculated with equation (49) because £?n_2,o is undetermined. We don't get an equation for a coefficient JB/^o because in the recursion equation (45) it possesses the factor j = 0. First we consider the coefficients (48), (53) and -B n -i,i in (46). Writing them with faculties, we obtain n_1,1

(n + 1)! ~~ 1! (n - 1)! '

(n + 2)! " 2! (n - 2)! '

n 2,2

"

n

(n + 3)! ~ 3 ' 3 " 3! (n - 3)!

which implies n p p

" '

_ (n + j»)l " p! (n - p)\ •

We next turn to the coefficients Bn-\fi senting them with faculties, we have

B

-'<° = 0 ! ( ^ T ) ! ( " +

X)

•

, (

j

in (46), then (47) and (52). Repre-

B

-"

' i!^TT25! <" + !) •

Now we see, that

B

—- = ^w^i ( n + 1)'

(55)

Looking to (54) and (55), let us suppose that -Bn_PiP_2 possesses the form (n + p - 2)! S p p 2 (56) - ' - ~ (p-2)!(n-p)! C with the unknown constant C, which holds for all p = 2,3, • • •, n. Putting (56) with p = 2 into (49), we see that C=

|n(n+l).

292

This can be seen as a binomial coefficient. Furthermore, we write the factor 1 of the fraction in (54) and the factor n + 1 of the fraction in (55) as binomial coefficients n + 1\

(n + 1 resp

o j

-

v i

Taking the relations (54), (55) and (56) with binomial coefficients, we obtain the conjecture

In our next step we make the index transformation k = n — p and j = p — s resp. p = n — k and s = n — k — j . Now the coefficient (57) becomes

B

" = ^^JiF(kl+j 11)> °- fc - n ' ° ^ » - * -

(58)

Inserting (58) into (45), we see that this equation is identically satisfied, which proves the correctness of our relation. According to (44) and (58) we can say: T h e o r e m 1 J n f t x G2 the Bauer-Peschl equation (23) en (n+1) zC + 7T~, 7v2

w

(1 +

w

=

„ °'

ezQ1

possesses the particular solution f v r)i(n+j)\(

n + 1 \

n—k

n

mT

n e JN0 ,

w(zX) =fc=0 5>* +1

z^j (59)

the function X{z) beeing arbitrary holomorphic. Since we treat a self adjoint differential equation, it holds for the invariants h = k. So it easily can be shown, that for the integrals (17) and (21) holds m = n. The symmetry of z and £ in (23) implies Dj(z, C) = Aj((, z), 0 < j < n (see Lemma). Now we can say: T h e o r e m 2 In G\ x G2 the Bauer-Peschl equation (23) en (n+1) < + 77~i 7vi w (1 + ezC) 2

w

n =

°>

_T n e JN 0 ,

293 possesses the particular solution n fc=0

n—k

^

}

y(fc)( C ) t

j\k\

\k + j + l)(l

+ e zC)i (60)

the function Y(() being an arbitrary holomorphic function. Theorem 3 The general solution of (23) is obtained by adding the integrals (59) and (60). This easily can be seen, because the sum includes two arbitrary functions (see also Lemma, (2)). It should be mentioned once more, that G\ and Gi are simply connected domains of the complex z resp. £ plane. Furthermore, G\ and G-z are assumed to be such that 1 + ez( does not vanish in G i x G2. Remark 5 Taking the solutions of Proposition 1, 6 and 8, we obtain another building law. It yields a solution for the differential equation (1 +

EZQ2

References 1. K. W. Bauer, Uber die Losungen der elliptischen Differentialgleichung (1 ± zz)2 + Xw = 0, Journal f. d. reine u. angewandte Mathematik, 221, Teil I, 48 - 84; Teil II, 176 - 196 (1966). 2. K. W. Bauer, Uber eine der Differentialgleichung (1 ± zz)2 ±n(n+l)w = 0 zugeordnete Funktionentheorie, Bonn. Math. Schr., 23 (1956). 3. K. W. Bauer, St. Ruscheweyh, Differential Operators for Partial Differential Equations and Function Theoretic Applications, Lecture Notes in Mathematics, 791, Springer-Verlag, Berlin • Heidelberg • New York (1980). 4. R. P. Gilbert, Constructive Methods for Elliptic Equations, Lecture Notes in Mathematics, 365, Springer-Verlag, Berlin • Heidelberg • New York (1974). 5. M. Kracht, E. Kreyszig, Methods of Complex Analysis in Partial Differential Equations with Applications. A Wiley-Interscience Publication, New York • Chichester • Brinsbane • Toronto • Singapore (1988).

294 KOMPLEXE DIFFERENZENGLEICHUNGEN M. CANAK Poljoprivredni fokultet (Landwirtschaftliche Fakulf'at), Katedra za matematiku Nemanjina 6, 11000 Zemun, Beograd, Yugoslavia

1

Einfuhrung

Definition 1.1 Es sei eine komplexe Gleichung der Form z = g(z)

(1.1)

gegeben, wobei g{z) eine beliebige analytische Funktion ist. Die Gleichung (1) kann eine geschlossene oder nichtgeschlossene Kontur, wie auch eine Menge isolierter Punkte bestimmen. Im weiteren werden wir die Menge der Punkte in der komplexen Ebene, die durch die Gleichung der Form (1.1) bestimmnt sind, eine K - Kontur nennen. Eine grosse Zahl fur die Praxis wichtiger Konturen (Gerade, Kreis usw.) lasst sich in der Form (1-1) darstellen. In seiner Arbeit x hat der Verfasser eine Abbildung ag(z) der Menge der stetigen komplexen Funktionen w = w(z, z) auf die Menge der analytischen Funktionen auf folgende Art und Weise eingefuhrt: Definition 1.2 Es sei g{z) eine analytische Funktion und w = w(z,z) eine stetige, komplexe Funktion, die eine konvergente Potenzentwicklung nach z und z besitzt. Dann ist die zusammengesetzte Funktion w(z, g(z)) analytisch und wir bezeichnen sic mit ag^w. Der geometrische Sinn dieses Operators ist folgender: Wenn z = g(z) die Gleichung einer geschlossenen Kontur ist, so besitzen die Funktionen w — w(z, z) und ag(z)W den gleichen Randwert auf dieser Kontur. Darstellung der Kontur in der Form z = g(z) und die Anwendung der Abbildung ag^ stellen die Hauptidee fur die sogenannte KRK - Methode (KRK = Komplexe Representation der Kontur) dar. Diese Methode hat der Verfasser in der Arbeit 2 eingefuhrt und niitzte dieselbe in der Theorie der verallgemeinerten analytischen Funktionen und Randwertaufgaben. Definition 1.3 Es seien LQ : z = go{z), L\ : z = gi(z), • • •, Ln : z = gn(z) die gegebenen Konturen, die die Gebiete Go, Gi, • • • ,Gn begrenzen, und es sei Go C Gi C • • • C G„. In seiner Arbeit 3 hat der Verfasser zuerst die

295 sogenannten areoldren V? - Differenzen a9lW-agoW=q/igogi) 91

aa„w gn -

-go

aa,._,w -V (gn-i,

gn-i

^(gl,g2)-^(fl0,gl) gi

fr(gl,g2,---

*(go,gi,g2)

-go

* ( g n - l , g n ) ~ fr(gn-2,gn-l) 9n - gn-2

,9n) - ^ ( g Q . g l , - - - , g n - l )

gn - g o

gn)

_ ,T, , , x — * \gn-2>gn-l,gn,)

, T ,,„

„

„

/, 9x U-4)

„ x

= W v g O , g l , ' •• i g n - l , g n )

Auf Grund dieser Differenzen konstruierte er weiterhin folgende Funktionenfolge

go-z ^o,g,)-m9o) 9i-

=

z

: ^(gQ,--- , g n ) - ^ ( z , g Q , - - - ,gn-l)

(i.3) lTf,_

x

gn-^

Diese Differenzen spielen in der Interpolationstheorie der nichtanalytischen, komplexen Funktionen eine wichtige Rolle. Die Gleichungen in denen, neben den unbekannten komplexen Funktionen auch ihre Differenzen der Form (1.3) fiir das System von Konturen z = gi(z), (i = 0 , 1 , . . . , n) erscheinen, nennen wir weiterhin komplexe, areolare * Differenzengleichungen. Es herscht eine starke Analogie bei der Form und Struktur der allgemeinen Losungen von der komplexen Differential - und Differenzengleichungen (siehe

296 3

). So enthalt die allgemeine Losung der komplexen Differenzengleichung erster Ordnung eine beliebige analytische Funktion und die allgemeine Losung der komplexen Differenzengleichung n-ter Ordnung enthalt n beliebigen, analytischen Funktionen. 2

Einige lineare komplexe Differenzengleichungen

In seiner Arbeit 3 hat der Verfasser einige lineare, komplexe Differenzengleichungen gelost. Dabei hat er folgende Satze bewiesen. Satz 2.1 Es sei die Gleichung einer einfachen, glatten, geschlossenen oder nichtgeschlossenen Kontur L : z = g(z) gegeben und es sei Vgw(z,z)

= V(z,g) =

-2——. 9 z

Es seien weiterhin a{z, z) und b(z, z) gegebene, stetige, komplexe Funktionen. Die allgemeine Losung der linearen, nichthomogenen Differenzengleichung $>w = a(z,z)w + b(z,z)

(2.1)

wird durch die folgende Formel v

;

V

l + (g-z)-a(z,z)

'

gegeben, wobei £(z) eine beliebige analytische Funktion ist. Satz 2.2 Es seien LQ : z = go(z), L\ : z = gi(z), • • • Ln : z = gn{z) die gegebenen Konturen, die die Gebieten GQ,GI, • • • ,Gn begrenzen, und es sei Go C G\ C • • • C Gn. Die allgemeine Losung der komplexen Polydifferenzengleichung y(z,9o,9i,---

,9n-i,gn)=0

(2.3)

wird durch die Formel w(z, z) = (z- g0){z - g{) • • • (z - gn-i)
(2.4) z

^ ( ~ 9o)
gegeben, wobei
297 Differenzengleichung n-ter Ordnung ist eine Gleichung der Form *(£,So, ••• ,9n-i) + ai(z)^(z,g0,--,gn-2) + a2{z)^{z,g0r-,g„-3) H \r an-2(z)$(z,go,gi) + a n _i(2:)*(z,g 0 ) + an(z)w = 0 (2.5) wobei ai(z),a2(z), • • • ,an(z) gegebene, analytische Funktionen sind. Das Auflosen dieser Gleichung ist ahnlich wie das Auflosen der entsprechenden komplexen Differentialgleichung. J. Keckic 4 untersuchte die komplexe Gleichung der Form Dnw + ai(z)Dn~1w

H

+ o„_iDw + an(z)w = 0

(2.6)

mit den analytischen Koeffizienten a,k(z), (k = 1,2,..., n) wobei Dw = {ux - v'y) + i(uy + v'x) = 2w'g (Dnw =

D{Dn-lw))

der bekannte Operator von Kolossov ist. Keckic zeigte, dass die allgemeine Losung der Gleichung (2.6) die Form w(z, z) = (piiz^^*

+ ip2{z)er2^'s

+ ••• + ipn{z)elhi^li

(2.7)

besitzt, wobei
^rn{z)

die Losungen der charakteristischen algebraischen Gleichung r" + a^zy71-1

+ ••• + an-i(z)r

+ an(z) = 0

(2.8)

darstellen. Hier betrachten wir zuerst die sogenannte Modell-Differenzengleichung Vw(z,g0)

= w(z,z)

(2.9)

wobei Lo : z = go(z) gegebene Kontur darstellt. Ihre allgemeine Losung besitzt die Form

m(z, z) =

*y—-.

(2.io)

298

Suchen wir auf Grund des Wertes (2.10) die Losung der Differenzengleichung (2.5) in der Form

m

»(*•*)-ITS^Fi'

= 1)

(2 U)

-

wobei b(z) eine unbekannte analytische Funktion ist. Nach einer langeren Rechnung erhalt man die folgende Funktionenfolge w(z,z) = v

;

l +

—— b(z)-z

(l + b(z)-g0)(l ^w(z,go,9i)

+

b(z)-z)

(1 + b(z) - g0)(l + b(z) -

ffi)(l

+ b(z) - z) (2.12)

<M*,S0,5i, • • • ,9n-x) -

( 1 + h{z)

_go)...{l

1 + b{z)

_ gn_i){l

+ b{zy_

-z) •

Durch Substitution der Werte (2.12) in (2.5) erhalten wir die charakteristische, algebraische Gleichung n-ter Ordnung 1 + (1 + b -

9 n _i)oi(z)

+ (1 + b - s„_i)(l + b - gn-2)a2{z)

+ {l + b- ff„_i)(l + b - <7„_2) • • • (1 + b - g0)an(z) = 0

+ ••• + (2.13)

im Bezug auf b = b(z). Nehmen wir an, dass b\{z) ^ b2{z) ^ ••• ^ bn(z). Dann existiert n linear-unabhangigen partikularen Losungen der Differenzengleichung (2.5) in der Form wk(z,z)

= ———— k = l,2,---,n. (2.14) z; 1 + bk(z) — z Es seien C\{z) und C2(z) zwei beliebige, analytische Funktionen. Auf Grund der Eigenschaft ag (Ci(z)wi + C2(z)w2) = Ci(z)agwi

+

C2(z)agw2

folgt der Satz 2.3 Wenn w\{z, z), • • • ,wn(z,z) linear-unabhangigen, partikularen Losungen der Differenzengleichung (2.5) sind, so besitzt die allgemeine Losung die Form W(z, z) = C1(z)w1 + C2(z)w2 + ••• + Cn(z)wn wobei Ci(z), • • • ,Cn(z) beliebige, analytische Funktionen sind.

(2.15)

299 3 3.1

Einige nichtlineare komplexe Differenzengleichungen Komplexe Differenzengleichung vom Riccatischen Typus

Es sei L : z — g(z) die Gleichung einer gegebenen, einfachen, glatten, geschlossenen Kontur und es seien a(z, z), b(z, z) und c(z, z) gegebene, stetige Funktionen. Komplexe Differenzengleichung vom Riccatischen Typus ist die Gleichung der Form ^w{z, g) = a(z, z)w2 + b(z, z)w + c(z, z).

(3.1)

Durch Substitution des Wertes T

^w{z,g)

=

a„w-w ——— 9-z

in (3.1) erhalt man die Relation a(g - z)w2 + [b(g - z) + l]w + (g - z)c - agw = 0.

(3.2)

Da die allgemeine Losung der Gleichung (3.1) eine beliebige analytische Funktion
=0

(3.3)

geniigt und dass heisst, dass sich die unbekannte analytische Funktion a s u; auf die beliebige analytische Funktion ip{z) zuriickgefuhrt wird. Die Losungen der quadratischen Gleichung (3.3) sind + (g- z)bf ~ M A ~ z)\J9 ~ *Y ~
_ - [ 1 + (g - z)b] ±K°[l ~

2->g(z)

unsere Losung auch auf L definieren. Dass ist aber nur fur den ersten Wert moglich. In seiner Arbeit 1 hat der Verfasser den Satz iiber die Entwicklung einer nichtanalytischen, differenzierbaren, komplexen Funktion in eine konvergente Potenzreihe w(z, z) = h{z) + (z- g{z))h{z)

+ (z- g(z))2f2(z)

+ •••

(3.5)

in einer Umgebung der Kontur z = g(z) (z. B. in einem Kreisring die diese Kontur enthalt) bewiesen. Die Koefnzienten der Entwicklung lassen sich durch

300 die Formel a

9i*)Dkw

/ (y\ _

2kk\

bestimmen. In unserem Fall erhalt man y/1+ {g-z)F(z,z)

lim

« 1 + -(
-[1 + (g - *)<>] + Vll + (g - *)6]2 - 4a(ff - z)[(g - f)c - yfo)] _

2^ 9 (z)

2(g - f)a 2 2

lim

z^g(z)

i( f f - *) 6 - 2a(g - zfc + 2a(g - z)ip(z) — 3= 2{g - z)a

=
(3.6)

So besitzt die allgemeine Losung der Riccatischen Differenzengleichung (3.1) die Form (3.4) mit dem positiven Vorzeichen und lasst sich auf der singularen Linie mit dem Wert (3.6) definieren. 3.2

Komplexe Differenzengleichung vom Vekuatypus

In seiner Monographie 5 hat Vekua ausfuhrlich diejenige elliptische Gleichungssysteme untersucht, die sich in der folgenden komplexen Form iuf = b(z, z)w + c(z, z)

(3.7)

darstellen lassen, wobei die KoefRzienten b(z, z) und c(z, z) gegebene, stetige Funktionen sind. Er hat gezeigt dass die allgemeine Losung von (3.7) die Form w(z, z) = $(z) + ff r i ( z , *)*(*) dT + ff T2(z, *)$(*) dTT

T

(3.8)

- - II nfatywar - - II\i2{z,ty(f)dT mit

ri(2,t) = J2K2j(^t),

r2(z,t) =

Kx(z,t) = - A ,

Kn(z,t) = JJ K&^K^faQdT,,

^2K2j+1(z,t),

301

besitzt, wobei
= b(z,z)w + c{z,z)

(3.9)

wobei z = g(z) die Gleichung einer einfachen, glatten, geschlossenen Kontur ist, wahrend b(z,z) und c(z,z) gegebene, stetige, komplexe Funktionen darstellen. Unmittelbar, aus (3.9) folgt . a„w —w T , Vw(z,g) = — — = bw + c

=>

w + (g — z)bw = agw — (g — z)c.

(3.10)

Parallel betrachten wir auch die konjugierte Gleichung w + (g — z)bw = a^w — (g — z)c.

(3-H)

Durch Auflosen des Systems von zwei Gleichungen (3.10) und (3.11) erhalt man =

agW-(g-z)_[c + byJ-(g-z)bc)^ 1 - bb(g - z)(g - z)

Da die allgemeine Losung von (3.9) eine beliebige analytische Funktion ip(z) enthalt, so nehmen wir auf Grund (3.12) an, dass sie die Form w{z,z) =

-=—— . (3.13) 1 - bb{g - z){g - z) hat und das heisst, dass die unbekannte analytische Funktion agw in die beliebige analytische Funktion ip(z) iibergefiihrt wird. Da der Wert (3.13) der Gleichung (3.9) geniigt, so stellt er ihre allgemeine Losung dar. Bemerkung Die Untersuchung der komplexen Differenzengleichungen ist teilweise ahnlich wie bei den reellen, mit folgenden Analogien: a) Anstatt der Punktenfolge XQ,X\, ... ,xn auf der x-Achse erscheinen die Konturen L0 : z = g0(z), Lx: z = gx{z), ••• Ln: z = gn(z). b) Anstatt der Veranderliche x erscheint die Veranderliche z. c) Die Rolle der beliebigen Konstante c ubernimmt die beliebige analytische Funktion
302

e) Die allgemeine Losung der komplexen Differenzengleichung n-ter Ordnung enthalt n beliebigen, analytischen Funktionen. Es besteht auch eine Ahnlichkeit zwischen den linearen komplexen Differenzen- und Differentialgleichungen. Die allgemeine Losung der Polydifferenzengleichung (n + l)-ter Ordnung stellt ein Polynom n-ter Ordnung im Bezug auf z, mit den analytischen Koemzienten dar. Das Auflosen der linearen Differenzengleichung n-ter Ordnung reduziert sich auf eine algebraische Gleichung n-ter Ordnung. Die allgemeine Losung enthalt n linear unabhangige partikulare Losungen. Interessante Einzelheiten erscheinen bei Untersuchung der nichtlinearen, komplexen Differenzengleichungen. Wahrend die allgemeine Losung (3.8) der Vekuaschen Differentialgleichung (3.7) praktisch unanwendbar ist, da sie singulare Doppelintegrale vom Cauchyschen Typus enthalt, so lasst sich die allgemeine Losung der entsprechenden Differenzengleichung leicht bestimmen. Ahnliches gilt auch fur die Riccatische Differenzengleichung. Dabei spielt die erwahnte KRK - Methode eine wichtige Rolle. Sie ermoglicht das Zuruckfiihren der unbekannten analytischen Funktion agw(z, z) auf die beliebige analytische Funktion tp(z) und erleichtert dadurch das Losungsverfahren. 4

Anwendungen

Es sind verschiedene Anwendungen der komplexen Differenzengleichungen moglich. In seiner Arbeit 3 hat der Verfasser das folgende Interpolationsproblem untersucht: Problem 4.1 Es seien LQ : z = go(z), L\ : z — 9i(z),-- • Ln : z — gn{z) die gegebenen Konturen, wobei alle analytischen Funktionen gi(z), i — 0,l,...,n verschiedene sind. Es sei weiterhin W(z, z) eine gegebene, stetige, nichtanalytische Funktion und agoW = SQ(Z), agiW — si(z),---, a97lW = sn(z). Man soil das Interpolationspolynom Pn(z,z) n-ter Ordnung bestimmen, das den Interpolationsbedingungen (Xgo(z)Pn = S0(z),

Otgi(z)Pn

= Si(z),

••• , agn(z)Pn

= Sn(z),

(4.1)

geniigt. Mit Hilfe der allgemeinen Losung (2.4) von der Polydifferenzengleichung (2.3) hat er auch die Losung des Problems (4.1) in der Form Pn(z, z) = ago{z)W + {z- g0)^w(go,gi) + {z - gQ)(z gi)^w(.9o,9\,92)+ + • • • +(z - g0)(z - g{) • • • (z - gn-i)^w(9o, 9i,---,9n) (4.2)

303

gefunden. In dieser Arbeit betrachten wir eine andere Anwendung bei der Methode vom Eulerschen Typus zum Naherungsauflosen einer komplexen Differentialgleichung erster Ordnung. Problem 4.2 Es sei W(z,z) eine gegebene komplexe Funktion, die beliebig oft im Kreisring P : K°a — 5/2 < \z\ < K°a + 5/2 nach z differenzierbar ist. Nehmen wir an, dass die positive Zahl b = sup bk, k = n + l,n + 2,... existiert, wobei bk Majoranten von Q „ / Z f | ' sind. Aus der Formel (3.5) folgt die Approximation W(z, z) « f0(z) + (z- a/z)h{z)

+ ... + (z- a/z)nfn(z).

(4.3)

Die Koefnzienten fk{z) lassen sich durch die Formel a i W^

ausrechnen. Fiir die Abschatzung des Approximationsfehlers konnen wir die Ungleichung

niitzen und \R\ —+ 0 wenn n —> oo. Im speziellen Fall n = 1 haben wir die Approximation W(z,z)*aa/zW+(z-a/z)aa/zW's

(4.4)

mit dem Approximationsfehler

\R\ < ^-f—• Die Formel (4.4) lasst sich auch in der Form ,

W(z,z)-aa/zW v ' J - £ ^ _ (4.5) w z v z-a/z ' schreiben und man ersieht, dass die ^/-Differenz der Funktion W(z, z) eine Approximation fiir die Ableitung dieser Funktion in einer Umgebung des Kreises z = a/z (zB. in einem Kreisring) darstellt. Es sei die komplexe Differentialgleichung W's = F(z,z,W) (4.6) «

mit der Anfangsbedingung aa/zW

= w0(z)

(4.7)

304

gegeben, wobei w0(z) eine gegebene analytische Funktion ist. Nehmen wir an, dass die Funktion F(z, z, W) im Gebiet

'

K°a-f-<\z\
II: <

2

K\W-w\
stetig ist und dass sie der Bedingung \F(z, I, W2) - F{z, z, W{) |< L |W2 - Wi.\\ fiir beliebige Werte W\,W% geniigt. Es sei ein endliches Intervall [a, A], (a, A € R) gegeben. Teilen wir dieses Intervall auf n gleicher Abschnitten a = xo, X\ = XQ + h, x2 = xo + 2h,..., A = xn — xo + nh (h = (A — a)/n) und konstruieren wir ein System von Kreisen Ki : z = Xi/z (i = 0 , 1 , . . . , n ) . Eine nichtanalytische, komplexe Funktion kann einerseits in der expliziten Form W = W(z, z) erscheinen. Andererseits lasst sie sich auch in der Form einer Tabelle, wenn ihre Grenzwerte auf den gegebenen Kreisen W(z,z)\K. bekannt sind, darstellen. Durch Ausniitzung des Operators ag ist moglich, eine Folge der analytischen Funktionen wo(z), w\(z),..., wn(z) zu konstruieren, dere Grenzwerte auf Ki mit den entsprechenden Grenzwerten von W(z, z) identisch sind. Darstellung einer nichtanalytischen Funktion W{z, z) in folgender tabellaren Form

Ki Wi(z)

K0 : z = x0/z w0(z)

K\\

z = x\/z w1(z)

•**• n ' % — %n 1 %

wn(z)

nennen wir a - R e p r a s e n t a t i o n dieser Funktion auf dem System der Kreisen Ki. Wenn aXi/zW(z,z) = Wi(z) (i = 0 , 1 , . . . , n ) , so ist die Representation (4.8) eindentig und jedem Wert z = Xi/z entspricht eine und nur eine analytische Funktion Wi(z). Das Interpolationspolynom (4.2) das die Funktion W(z, z) approximiert, entspricht der Tabelle (4.8). Suchen wir die Naherungslosung der Aufgabe (4.6)-(4.7) in der Form (4.8). Dabei suchen wir solche Folge der analytischen Funktionen Wi(z), (i — 0 , 1 , . . . , n) dere Grenzenwerte auf den Kreisen z = Xi/z (ZJ+I —Xi = h = {A — a)/n) mit den Grenzwerten der exakten Losung naherungsweise gleich sind. Die Formel (4.4) geht auf Grund (4.6) in W(z, z) « aa/zW iiber.

+ [z- a/z)- aa/zF(z,

z, W)

(4.9)

305

Den ersten Naherungswert w\ erhalten wir durch Anwendung des Operators aXl/z auf (4.9), d.h. 3/1 — d

aXl/zW

= w\ = aa/zW

+

aa/zF(z,

z, W)

oder * w ( o , xi) = aa/zF{z,

z, W)

oder Wl

=

z

w0-F(z,x0/z,w0).

Auf dem Kreis z = x\/z geht die Formel (4.9) in W(z, z) w aXl/zW

+ (z- x1/z)aXl/zF(z,

z, W)

(4.10)

iiber. Den zweiten Naherungswert W2 erhalten wir durch Anwendung des Operators aX2/z auf (4.10) d.h. < W 2 W = w2 = w1 +

-F(z,xi/z,wi). z Durch Fortsetzung dieses Verfahrens erhalten wir auf eine gleiche Art und Weise die allgemeine rekurrente Formel wn+i =wn + -F(z, xn/z, wn) (4-11) z die uns die Naherungslosung der Aufgabe (4.6)-(4.7) in der tabellaren Form (4.8) gibt. Bemerkung Die Methode vom Eulerschen Typus (4.11) fur die Aufgabe (4.6)-(4.7) lasst sich verallgemeinern. Anstatt (4.11) konnen wir die allgemeine lineare Methode der Form

Y^ oiiWn+i = -^2 i=0

(3iFn+i

(4.12)

i=0

untersuchen. Wegen Fn+i = F(z,xn+i/z,wn+i) stellt die Relation (4.12) im allgemeinen Fall eine nichtlineare Differenzengleichung dar. Diese Gleichung lasst sich oft mit Hilfe unserer Theorie der <3>-Differenzengleichungen losen.

306

Literatur 1. Canak M., " Uber die a-Approximation einer nichtanalytischen Funktion durch ein areolares Polynom", ZAMM-Z.angew.Math.Mech., 69 (1989) 4, s. 71 - 73. 2. Canak M., " Anwendung der KRK-Methode auf eine inverse Randwertaufgabe", ZAMM-Z.angew.Math.Mech., 70 (1990) 6, s. 573-574. 3. Canak M., " Uber die komplexen Differenzengleichungen", Proceedings of the eighth symposium of mathematics and its applications, Timisoara 1999, s. 31-38. 4. Keckic J., " 0 jednoj klasi parcijalnih jednacina", Mat. Vesnik, Beograd 6 (21), 1969, s. 71-73. 5. Vekua I., "Systeme von Differentialgleichungen erster Ordnung vora elliptischen Typus und Randwertaufgaben", Veb Deutscher Verlag der Wissenschaften, Berlin, 1956.

307

STABILITY OF SOLUTIONS OF N O N U N I F O R M L Y ELLIPTIC SYSTEMS A N D B O U N D A R Y VALUE P R O B L E M S A. MAMOURIAN Department of Mathematics and Computer Sciences Faculty of Sciences, University of Tehran Enghelab Avenue, 14174 Tehran, Iran E-mail: [email protected] This work is aimed at the investigation of systems of nonlinear equations with degeneration of ellipticity in two-dimensional domains. Introduction It is well-known that methods of complex function theory have influenced the theory of partial differential equations for at least fifty years. The possibility and importance of employing complex function theory in partial differetial equations is so wide that it presents a real difficulty to give a survey of them. For a great many special references one may consult for instance 3 or 8 . For the investigation of systems of differential equations with non-uniform ellipticity, we confine ourselves to the nonlinear system of first order of type w-z = H(z,wz)=G{wz)

+ F{z)

(1)

in D, where w = w(z) = u(x,y) + iv(x,y), z = x + iy,wz = §f = (|jjjf + i^)/2, Wz = <£ = (?%- i | e ) / 2 ; G(0(£ = wz) and F(z) are complex valued functions. We assume that D is a multiply-connected Liapounoff region with boundary L = LQ + L\ H + Lm. Clearly, equation (1) contains the complex form of the Cauchy-Riemann system wz = 0. We assume that (1) fulfils the following condition Condition C\ (I) H satisfies the following inequality

|tf(z,6)-ff(*,6)l
(2)

9(*,6,6) < 9(16-61) < i

(3) s

(II) the function q as a function of \i (fi = | 6 — 61) i continuous in [0, co]; q(n) < 1 for /x € (0, oo); the function /xg2(/x) is increasing and concave.

308

Concerning the nonlinear system (1), we study the following boundary values problem: Problem A Re R t M * ) ] = 7 (*)

(4)

on L, where the coefficients a, 7 are given functions on the boundary L. Similar to the Riemann-Hilbert problem for a uniformly elliptic system of equations, the index corresponding to the boundary value problem A is defined as follows l

-- f d(loga(t)). (5) 2m J Li Remark 1 Let us recall that, in the classical Riemann-Hilbert boundary value problem of the type (l)-(4), relative to the uniform ellipticity of the nonlinear system of equations of Lavrentiev type, the solution w is sought in the Sobolev space Wp(D), for some p > 2. For the equation (1) in the case of non-uniform ellipticity (2)-(3), the L p -theory has not been applied directly for the proof of existence of the solution of (l)-(4). Hence the formulation of the boundary value problem A involves the weak boundary condition (see 2 ). We shall consider the problem A with usual assumptions on the coefficients: Condition Ci The complex function a(t) and the real function 7(£) are Holder continuous on the boundary L, with the exponent v, where 0 < v < 1 (01,7 G HV(L), 0 < v < 1). The complex function is F(z) assumed to be measurable and to belong to the class LP(D), for some p > 2. The solution w will be sought in the Sobolev space Wp(D),p > 2. If G = 0 in (1), the non-homogeneous boundary value problem (l)-(4) will be called problem A$. Lemma Under the Hypothesis (Ci),(C2), if n < 0, the non-homogeneous boundary value problem AQ is solvable, if and only if n = inda

/ a{t)X(th(t)dt-2iRe\ JL

f

1

X(z)F(z)da:

= 0

(6)

JD

where x IS a n arbitrary solution of the homogeneous boundary value problem adjoint to the problem AQ (see for instance 3 , pp. 98-101). Remark 2 Making use of representation formula for the solution w of the problem A: w = T(C){z)+iP(z)

(7)

309

(see for instance J ) , where T(() = (T(Q)D,C^ Lp(D),p > 2, and ip{z) is the solution of the boundary values problem AQ, we observe that T, depending on the index n, satisfies the homogeneous boundary condition corresponding to (4) on L, when z -> t (z G D, t G L). Moreover ^f= ((z). Defining S(() = dgP we conclude that the L 2 -norm of S is equal to one, also S is a bounded operator from Lp(D),p > 1 into itself, and the continuity of || S||p with respect to p > 1 can be proved through the well-known RieszThorin convexity theorem. Remark 3 The norm || || is defined by

IKIlL2(i5) = ( ^ / z ) I C W I 2 ^ ) 2 In view of (1), (4), (7), we obtain the following equation for £:

C = G(S(C)+V')-

(8)

If we assume that C G -^(-D), the integral equation (8) can be solved through a successive approximation method, i.e.

0+i=G(s(0)+V')

(9)

£0 = 0, j = 0,1, • • •. According to the concavity property of C the well-known Jenssen inequality, we obtain an inequality of the form

IIC 1 -C 2 II<9(II< 1 -C 2 II)IK 1 -C 2 II,

an

d

(10)

and the existence, also uniquencess of the solution of the integral equation in 1/2(-D) can be proved, i.e. lim Cj = C ( s e e 6 )j—»oo We shall prove that the solution w G WjiD) of the problem (l)-(4) actually belongs to Wp{D) for some p > 2. The number q0 = lim sup(g(0) < 1

(11)

C-»+oo

is called the ellipticity coefficient corresponding to the boundary values problem (l)-(4). This go shows how fast the gradient may approach infinity, and consequently qo will influence the exponent p > 2 of the integrability of the gradient (see also 2 ) .

310

Let p > 2 be such that *>||S||p
(12)

Proposition Under the Hypothesis {C\),{C2), condition (6) and inequality (12), the solution w of the problem (1)~(4) belongs to the Sobolev space W}(D). Proof Since 0 converges to C in L2, it is sufficient to show uniform estimates

for ||0|| P . In view of the relation (9), Q (j = 0,1, • • •) is in LP(D). Co G

Clearly

LP(D),

then, if we write

0 + i = G(S(G) + V') - GW) + G(V/),

(13)

|0+iI < 9(|5(0)|)|5(C-)| + \G(i>%

(14)

we obtain

Taking into account the properties of G, q, and inequality (14), we have

|0+1|<|S(0)l+ 1^1^00), which is derived by induction hypothesis and the Lp - continuity of S. Moreover, because of continuity of ||5|| p , relative to p, a number (5 € (qo, 1) can be chosen such that P\\S\\P<1.

Now, according to (11), there exists N > 0 for which g(C)
for

|C|>JV.

If we write q(() = q~i(Q + 92(C)) where fg(C)if qi J

^

\0

|C|
otherwise

and ( )

fO if |CI < iV \ 9(C) otherwise

(15)

311

It is clear that
< 9 i ( | S ( 0 ) | ) | S ( 0 ) | +P\S(Q)\ + W\-

(16)

By integration of both sides of (16) it follows \\Q+I\\LT(D)< P\\S\\P

•

(17)

(qi{\S(Q\))\S(Ci)\pdaz)

+ ( [

\\Q\LP(D)

"+

\W\\LV(D)

but since |5(Cj)| < N, we have \\Q+ihp(D)

< P\\S\\P.\\tj\\Lp(D) +N(mes

{D))' + \W\\LP{D)-

(18)

The following inequality holds \\CJ+I\\LP{D)

<

1

_ ^

s

^ (^V(mes D ) i + HV'IU,^)).

We observe that the right-hand side of the latter inequality does not depend on j , that indicate the uniform estimates of ||CJ+I||L P (D)- Therefore we conclude that C £ LP{D), and we obtain the upper bound

IICIIL

N{mssD)

"^ - i-l\\s\\p{

'

+

mL

"^)-

Since if, e W*(D), T : LP(D) -> W ^ D ) , the solution w = T(C) + V belongs to W^(D). References 1. H. Begehr, G. C. Hsiao, The Hilbert boundary value problem for nonlinear elliptic systems, Proc. of the Royal Society of Edinburgh, 94A, (1983) PP. 97-112. 2. T. Iwaniec, A. Mamourian, On the first order nonlinear differential systems with degeneration of ellipticity, Proc of Sec. Finish-Polish Summer school in Complex Analysis, Jyvaskyla, Edited by J. Lawrynowicz and O. Martio, (1984), PP. 40-52. 3. E. Lanckau, W. Tutschke, Complex Analysis, Methods, Trends and Applications. North Oxf. Acad. (1985). This work is partially supported by grant V-CiRA, Univ. of Tehran

312

4. A. Mamourian, Boundary Value problems and general systems of nonlinear equations elliptic in the sense of Lavrentiev, Demonstratio Math. 17, (1984), PP. 633-645. 5. A. Mamourian, On a mixed boundary values problem for Lavrentiev type equations,Ann. Polon. Math. 45, (1985), PP. 149-156. 6. A. Mamourian, First-order nonlinear system of Lavrentiev type equations and Hilbert BVP, Proc. of Asia Vibr, Conf., Edited by W. Bangchun and T. Iwatsubo, Shenzhen, Northeast Univ. of Tech. (1989), PP. 735-738. 7. A. Mamourian, BVP of a non-uniformly elliptic system of partial differential equations, Demon. Math. X X V I , No. 3-4, (1993), PP. 735-741. 8. G. C. Wen, H. Begehr, Boundary value problems for Elliptic Equations and Systems, Pitman New York(1990).

313

A N EXPLICIT SOLUTION TO A CLASS OF A S E C O N D K I N D COMPLEX I N T E G R A L EQUATION WITH SINGULAR A N D SUPER-SINGULAR KERNEL N. RAJABOV Tajik State National University Dushanbe 734025, Tajikistan E-mail: [email protected] In this paper we have proved for a class of two-dimensional complex integral equations of second kind with singular and super-singular kernels on the boundary of a domain that the homogeneous equations have an infinite number of linearly independent solutions and the non-homogeneous integral equations have always a solution provided the kernels and the right-hand sides satisfy certain conditions. It is proved that the solution of the non-homogeneous equation contains an analytic function, which was found in an explicit form. The corresponding solutions of the integral equations are used for formulation, investigation and t h e construction of solutions of boundary value problems of linear conjugation.

Let D+ be the disc \z\ < R, complementary to D+. 1

T be circle \z\ = R, and D

the domain

Explicit solution of the complex model integral equation for the disc

In D+ we shall consider the following integral equation

U{Z)

-~J J (R-P)4-*)=f{z)>

(1)

where A - is complex parameter, a = constant > 0, ip = arg£, £ = £ + in, p2 = £2 + n2, z — x + iy, and f{z) is a given function. Prom equation (1) we can see that in the case, when f(z) € C{D+), equation (1) has solution in the class function U(z) having zero order 7 > a— 1 in r . The kernel of equation (1) has on T a weak singularity for a < 1, a singularity for a = 1 and a super-singularity for a > 1. Therefore the equation (1) is called complex model integral equation with weak singularity of kernel on the boundary V when a < 1, model complex integral equation with singular kernel on V when a = 1 and complex model integral equation with super-singular kernel on V when a > 1.

314

Problems of finding continuous solutions of first and second order elliptic systems with singularity and super-singularity on V are reduced to the consideration of the complex integral equation (1) ? ' ? ? The problem of finding the general solution of the differential equation for generalized analytic functions with regular coefficients ? s reduced to the consideration of regular integral equation of the type (1) (Vekua type integral equation). In the case a < 1 the solution of equation (1) can be found by succession approximations, see ' In this case equation (1) has a unique solution, which is given by formula

uM = /(.) + X-j j -

^

f -J*'-"' f(Q
P)

D+

where w„(r) = [(a - 1)(R - r ) " - 1 ] " 1 . In the case a > 1 we have the following statement. Theorem 1 Suppose a > 1, ReA > 0, f(z) € C(D+) neighbourhood ofT and

and has a zero in a

- r) 71 ]

f(z) = o[exp(-2ReXwa(r)){R

when r —> R where 71 > a — 1. Then the homogenous equation (1) has an infinite number of linearly independent solutions of type Uk{z) = zk exp(-2Au; a (r)),

k = 0,1,2,...

The non-homogenous equation (1) has always a solution and its general continuous solution contains one analytic function and is given by the following formula U(z) =

A f

+

f(z)+exp(-2\cja(r))$(z)+

f exy[2\(LJa(p) - ua(r)) + itp]f(Q
W J

(R-pnc-z)

'

... (3)

D+

where <&(z) is an arbitrary analytic function in D+. In the case a = 1 the following theorem is true: Theorem 2. Let in equation (1) a = 1, ReA > 0, f(z) £ C(D+) and has a zero in a neighbourhood of T and f(z) =

o[(R-rr>)

315

when r —> R where 72 > 2|ReA|. Then the homogenous equation (1) has an infinite number of linearly independent solutions of type Uk(z) = zk(R~r)+2X,

k = 0,1,2.

The non-homogenous equation (1) has always a solution, its general continuous solution contains one analytic function and is given by the following formula

+

+

(4)

"w - ><•> <* ^ ')"*(•> ; //'f^'Vlx?-^ D+

where <&(z) is an arbitrary analytic function. Theorema 1 and 2 are proved using the connection of integral equation (1) and the following elliptic first order system with singular and super-singular coefficients on the circle T dU

j_

+

^

XeiV

rrr s

d

f

U

W=rF M=di'

(^

(5)

where £ = | ( £ + i £ ) . At first we find the solution of equation (5). In the regular case an analogous solution of equation (5) is searched in the form U{z) = V(z)exp[-2XLJa(r)],

(6)

when a > 1, and in the form U(z) = V(z)(R-r)2X,

(7)

when a = 1 , where V(z) is new unknown function. Then we substitute the valued U(z) from the formula (6) when a > 1 and from the formula (7) when a = 1 in the equation (5). As the function exp[—2Xwa(r)} when a > 1 and (R — r)2X when a = 1 are solution of homogenous equation (1) we find the function V(z) correspondingly from the following equalities:

W=e2X„a{r)dl oz ^ = (R-r)~2X^, oz

whena>l

(8)

dz when a = 1. oz

(9)

316

If the equations (8) and (9) have solutions, which can be represented in the form [1]

1 d(

c-

)+

V(z) = {z) - -

•

J

z

-2Xd<; fdf(R / ^ = --p) j - 1 d( z

c-

D+

5

-,

when a =1,

(11)

where $(z) is an arbitrary analytic function. Let function f(z) have zero in the neighbourhood of T and its behaviour is determined by following asymptotic formula f(z) = o[exp(—2\u)a{r)){R — r) 7 1 ],

whereji>a

— l,

when a > 1,

r —> R and f(z) = o[(R — r) 7 1 ],

where 71 > 2,

when a = 1, r —> i?.

Then we find V(z) = * ( , )

+

f(z)eM2X^(r))

+

±ff

^t^™™,

(12)

D+

when a > 1 and

vM-«,H/W(«-o-»+*//v^
(13>

00

when a = 1, by integrating in the right sides of the equalities of the formulas (10), (11) in according to Poincare-Bertrann formula and conditions for the function f(z) (exp(2\uia{r)f(z))r=R ((R-r)-2Xf(z))r=R

= 0, = 0,

ReA > 0 if a > 1, ReA>0

ifa = l,

Then we substitute the value of V(z) from the equalities (12), (13) into the formulas (6) and (7) accordingly to a > 1 and a — 1. Hence we find the solution of integral equation (1) correspondingly for a > 1 and a = 1. Let in equation (1) ReA < 0. In this case, if exist the solution of the equation (1) from class C(D+), then it represented in the form (3) by $(z) = 0, when a > 1 and in the form (4) by $(z) = 0 , when a = 1. Besides, easily it is possible see, that U(Relv) = 0. Prom here follows, that in this case

317

U(z) in neighbourhood V has first order zero. But at a > 1 it is necessary that the solution equation (1) U(z) has zero order 5 > a - 1. If f(z) in neighbourhood of T has zero order 5\ > a - 1, then solution of the type (3) U(z) in neighbourhood of T has zero order 5 > a — 1. Then from theorem 1 and 2 is follows: Theorem 3 Suppose a > 1, ReA < 0. Besides, let f(z) G C ( 5 + ) and / ( z ) = 0[(i? - r)s], S > a - I, when r —> R, a > 1. Then the homogeneous equation (1) has no non-trivial solution. The non-homogeneous equation (1) has a unique solution, which is given by the formula (3) with $(z) = 0 a > 1 and by the formula (4) with $(z) = 0 a = 1, Besides U(z) = 0[(R - r)s], 5 > a — 1 by a > 1. 2

Explicit solution of the non-model complex integral equation for the disc

At present we consider the following integral equation:

D+

where A+(r) is given function in D+,A+(r) £ C{D+). To find a solution of the equation (14), we represent it in the following form A+(R)

f f

——J

J

u(z}

e^UjQdC

_

(R-Pnc-*)-fix)-

D+

HI 1 f

fe^(A+(R)-A+(p))U(C)dC

(R-pnc-z)

D

We assume, that function A+(r) fulfils the following inequality in a neighbourhood of circle T: \A+(R)-A+(r)\
7 > a - 1.

Moreover, assuming that,

n*) = s(*)-lj je^(A+(R)-A+(p))U(C)dC (R-p)»(C-z) D

(15)

318

is a known function, we resolve the model integral equation

vw-Z&ff A+(R)

f f

e^U{QdC (iJ-p)a(C-z)

F(z).

D+

Writing the solution of this equation by a > 1 and a = 1, after solving the corresponding integral equations with weakly singular kernels ? ' ? we shall obtain the folloving solution of equation (14) U(z) =

(16) f(z) + exp

2A+(R)cja(r)-W+(z) e^A+(p)exp{2A+(R)Lva(p) (R-p)*(£-z)

•w+i// =

+

W+(0}f(Od(

K+[$(z),f{z)],

if a > 1, and (17)

U(z) = 2A+

f(z) + {R- r)

^

*(z) + \f j D+

=

n J Jr. K+[$(z)J(z)],

exp [ - W%{z)] x A+W (R-p)^+(R)+1

exp[W+(0] f(Q
if a = 1, where

"CfM-i//

(A+(JZ)-A+(p))e^dC (i?-p)«

(c-*)- 1

L>+

The solution of the forms (16) and (17) exists if functions f(z), satisfy in neighbourhood of T the inequality (15) and f(z) = o[exp(-2ReA+(R)iJQ(r))(R Re(A+(R))

> 0

- r)™],

72 > a - 1

A(r) (18)

if a > 1, and f(z) = o[(R-r)'»],

if a = 1. Consequently, we have proved:

K>2\ReA+(R)\,

(19)

319 Theorem 4 Suppose a > 1, A+(r) e C(D+), ReA+(ft) > 0, / ( z ) 6 C(D ) and has a zero in a neighbourhood ofY, and the behaviour of A+(r) and f(z) is given correspondingly by the inequality (15) for a > 1 and equality (18) for r —> R. Then the homogeneous equation (1) has an infinite number of linearly independent solutions of type +

Uk(z) = zk exp[-2A+(R)ua(r)

- W+(z)],

k = 0,1,2,...

The non-homogenous equation (1) is always solvable, its general solution contains one analytic function and is determined by the formula (16). Theorem 5. Let a = 1, A+(r) e C(D+), ReA+(R) > 0, f(z) € C(D+) and has a zero in a neighbourhood ofT, and the behaviour of A+(r) and f(z) is given correspondingly by the inequality (15) for a = 1 and equality (19) for r —> R. Then the homogenous equation (1) has infinity a number of linearly independent solutions of type Uk(z) = zk exp[-W+(z)](R

- r)2A+(R\

k = 0,1,2,...

The non-homogenous equation (1) is always solvable, its general solution contains one analytic function and is determined by formula (17). Let in equation (14) a > 1, A+(r) £ C(D+), ReA+{R) < 0 and in a neighbourhood of circle T fulfile the unequality (15). Then from theorem 4 and 5 are follows: Theorem 6 Let a > 1, A+{r) G C(D+), ReA+{R) < 0, f{z) e C{D+). The behaviour of A+(r), f(z) is defined correspondingly by the inequality (15) and the following asumptotic formula f(z) = o[(R-r)5>},

S1>a-1.

Then the equation (14) has a unique solution, which is given by the formula U(z) = K+[0,f(z)},

when

a>\

U(z) = K+[0,f(z)},

when a = 1.

and by the formula

Now we consider the integral equation transposed to equation (14), that is the following equation

D+

320

Introducing the new unknown function ip(z) = (R-

r)-ae-ie(4+(r))-1V(.z))

we come to an integral equation of the type (14) with the right-hand side = g(z)(R rrae-ie(A+(r))-\ gi(z) It follows from the theorem 3 that the particular solution of the homogenous equation (20) are the following functions: Vk(z) = A+(r)ei@(R

- r)~azk exp[-2A+(R)wa(r)

- W+(z)},

k = 0,1, 2 , . . . (21)

when ReA+(R) > 0 and a > 1. General solution of non-homogenous equation (20) is the following function: (22)

V(z) = g(z) + eieA+(r)(R

- r)~a exp

-2A+(R)cja(r)-W+(z)

W + ljj exp[2^+(JZ)a,C (p) + W + ( C ) ] a — z

g(c)tfc

D+

M+ when ReA+(R) > 0, g(z) in a neighbourhood F has zero, and its behaviour is defined by the following asymptotic formula g(z) = o[exp(-2ReA+(R)uia(r)},

if

r -* R

(23)

The above investigation can be stated as follows: Theorem 7 Suppose in equation (20) a > 1, A+(r) G C(D+), g{z) G C{D), RQA+(R) > 0. The function A+(r) satisfies in a neighbourhood of V the inequality (15). The function g(z) has zero in a neighbourhood ofT and its behaviour is determined by the assymptotic formula (23). Then the homogenous equation (20) has an infinite number of linearly independent solutions of the type (21). The non-homogenous equation (20) is always solvable, its general solution contains one analytic function and is determined by the formula (22) where $i(z) an arbitrary analytic function. The solvability of the equation (20) for a — 1 is formulated in the following assertion. Theorem 8 Suppose in equation (20) a — 1, A+(r) G C(D+), g(z) £ C(D), ReA+(R) > 0. The function A+(r) satisfies in a neighbourhood ofT

321

the inequality (15). The function g{z) has zero in a neighbourhood ofY and its behaviour is determined by the assymptotic formula g{z) = o[(R - r ) 7 4 " 1 ] ,

74 > 2ReA+(R)

as r -> R.

Then the homogenous equation (20) has an infinite number of independent solutions of type: Vk{z) = zk(R - r)2A+W-lA+{r)ei@

exp(-W+(z)),

k = 0,1,2,...

The non-homogenous equation (20) is always solvable, its general solution contains one analytic function and is determined by the formula V(z) =

(24) g(z) + (R-

2

l+ R

1

+

r) ^ ( )- A (r)e

i0

exp ( -

W+(z)}

$(*) + ! / / eMwA(0)g(OdC (R-p)2A+(0(C-Z)

= M+\^i(z),g(z)], where $(z) is an arbitrary analytic function. Suppose a > 1, A+{r) e C(D+), g{z) G C(D+), ReA+{R) < 0. Function A+(r) satisfies in a neighbourhood of T the inequality (15). Then from theorems 7 and 8 follows: Theorem 9 Consider in equation (20). Suppose a > 1, A+(r) G C(D+), g(z) G C(D+), HeA+(R) < 0. Function A+(r) satisfies in a neighbourhood of r the inequality (15). Then the equation (20) has a unique solution, which is given by the formula V(z) = M+[0,g(z)],

when

a > 1

V{z) = M?[0,g{z)],

when

a = 1

and formula

3

Explicit solution of a complex nonmodel integral equation for unbounded domains

In D

we shall consider the integral equation _ 1 f

f D~

e^A-(p)U(QdC

M,

(25)

322

where a = constant > 0, A~(r), f(z) are given functions in D~, if = arg£. By the analogy with p.l. it is possible to prove the following assertions about solvability of the equation (25): Theorem 10 Consider in equation (25). Suppose a > 1, A~(r), f(z) £ C(D~), Re(^4 _ (i?)) < 0. The function A~(r) satisfies in a neighbourhood of r the following inequality \A-(r)-A-((R)\
73

> a - 1.

(26)

The function f(z) has a zero in a neighbourhood of T and its behaviour is determined by the following asymptotic formula f{z) = o[exp(2ReA-(R)wa{r))(r

- R)-*],

73

> a - 1,

A~(oo) — Ao = constant. Then any solution of the equation (25) of class C(D~) can be represented in the form U-(z) =

(27) f{z) + exp [2A-(R)wa(r) [ ]

=

1 f

+ W^{z)^ x

f exp[-2A-(R)uja(p)

W J

- WX(Q + i)f(Qd(

(p-RWC-z)

K-[*-(z),f(z)],

where w a (r) = [(a - l)(p - i ? ) " " 1 ] - 1 ,

wiM^-JJ (A-(p)-A-(R))dC (p-R)°(C-z)

'

D-

$(z) is an arbitrary analytic function in D~. Theorem 11 Consicer in equation (25). Suppose a = 1, A~(r), f(z) G C(D~), BB(A~(R)) < 0. The function A~(r) satisfies in a neighbourhood ofV the condition (26) by a = 1. The function f(z) has a zero in a neighbourhood of r and its behaviour is determined with the following asymptotic formula f(z) = o[(r - R)-*],

74 >

2\ReA-(R)\,

A~(oo) = Ao, /(oo) = /o is bounded. Then any solution of the equation (25) of class C(D~) can be represented in the form U~(z) = f(z) + * - ( z ) ( r - R)'2A~^

^//'^)"' D-

,a)expiw (

exp(WX(z))+

( iy /(cK

(28

t i)
323

where $ (z) is an arbitrary analytic function in D and its behaviour in a neighbourhood of z = oo determined by the following assumptotic formula $-(*) = 0[r~6], 6 > 2\ReA-{R)\, SeN. It follows from the representations (27) and (28) of the solution of equation (25) the following statement: Suppose in equation (25) a > 1, A~(r) £ C(D~), ReA~(R) > 0, and suppose the condition (26)is satisfied. Then we get from theorem 10 and 11: Theorem 12 Consider in equation (25). Suppose a > 1, A~(r), g(z) € C(D~), ReA~(R) > 0. Besides, A~(r) the condition (26) satisfies. The function f(z) satisfies in a neighbourhood ofT be the condition

f(z) = o (R-r)

5x > a - 1.

Suppose, further, that in a neighbourhood infinity the following condition is satisfied: f(z) — o(r~e),

e>0

a = 1, r—>oo;

/ is bounded for a > 1, /(oo) = /o. Then equation (25) has a unique solution in the class C(D+), which is given by the formula U(z) = K-\0,f{z)},

when

a > 1

U(z) = # r [ 0 , /(z)], when

a = 1.

and formula

In this case the behaviour of the solution of equation (25) in a neighbourhood of r and of the point at infinity correspondingly are given by the following asumptotic formulas: U(z)=o[(r-Rf% U(z) = o(l), U(z) = o{r~e),

52>a-l,

byr-+R,a>l;

by r —> oo, a > 1; e > 0, by r -> oo, a = 1.

Remark 1 Any solution of integral equation (25) of class C(D~) has zero in r and its behaviour is determined by the following asymptotic formula U{z)=o[(r-R)2\RelA~W\

for

a = l

and U(z) = o[exp{-2\Re{A-(R))\)},

for

a >1

324

Remark 2 Solution of the form (27) unlimited for \z\ —» oo and its behaviour is determined by the following asymptotic formula

0\r^Re^^\

U{{z) =

Solution of the form (27) is limited for \z\ —» oo and its value is determined by following equality [/-(oo) = / ( o o ) + $~(oo). 4

Explicit solution of a class of complex model integral equations for kernels with two singular lines

Let Df be the ring D\ = {z : R\ < \z\ < R2}, I \ the circle \z\ = Ri, T2 the circle \z\ = R2. In D~i we consider the integral equation, whose kernel has singularities on Ti and I V

where A is a piece-wise constant function: A = Ai, for R\ < r < R, A = X2 for R < r < R2, R! < R < R2. As in p.l., using the connection of the integral equation (29) with the generalised Cauchy-Riemann system dU_ \ei&U(z) dz + (r-R1)(R2-r)

Of ~ dl'

we prove that the function UQ{Z) — <&(z)k(r), where Rr> — r

2X

i

k(r) = ( )«2-*i for r — R\

R1

and (Ro

— r \ «2-«l

*>-(;=*)

Rr, — R 2(A1-A2)

t^ft)

"•-/-•*<'<«•.

$(z) is an arbitrary analytic function for ReAi < 0, and ReA2 > 0 gives the general solution of the homogeneous equation (29) of class C(D^).

325

By a first-hand calculation is proved that if the function f{z) has zeros in neighbourhoods of Ti and T% and its behaviour is determined by the following asymptotic formulas f{z)=o[{r-R{f% ^ > ^ ~ ^ -

1

'

for r-*Ru

(30)

f(z) = 0{(R2 - r)
- 1,

for r - R2,

(31)

S2 > ^ ~ ^

then the particular solution of equation (29) is given by the following equality

^ " W M + ^ Z /k<j>y Q c-z -d( D+

Then the general solution of integral equation (29) of class C(D^) is represented in the form U(z) = ${z)k{r) + M(f),

(32)

where Q(z) is an arbitrary analytic function. The solution of type (32) has zero in neighbourhoods of Ti and T2 and its behaviour is determined by the following asymptotic formulas U(z) = o[{r - R^1^1*^11},

if r-^Rx

U(z) = o[{R2-r)^2^lReX%

if

r-^R2.

Therefore we have the following result: Theorem 13 Let in equation (28) ReAi < 0, and Re\2 > 0, f(z) G C(D^) and f has zeros in neighbourhoods ofT\ and T2 and its behaviour is determined by the asymptotic formulas (30), (31). Then any solution of the integral equation (29) of class C(D^) can be represented in the form (32), where $(z) is an arbitrary analytic function in D^. In the case, when ReAi > 0, ReA2 < 0, we have the following assertion. Theorem 14 Let in equation (28) ReAi > 0, ReA2 < 0, f(z) £ C{D+), has zeros in neighbourhoods ofTi, T2 and its behaviour is determined by the formulas /(z)=o((ili-r)ei),

£i>0,

for r -> R±

f{z) = o((r-R2y*),

e2>0,

for r -> R2.

326

Then the integral equation (29) has a unique solution, which is given by the formula U(z) =

5

M(f).

Boundary value problems

The above obtained solutions of the integral equations (1), (14), (25), (28) containing an arbitrary analytic functions in the case a = 1 and a > 1, permits us to study a number of boundary value problems of Schwarz, RiemannHilbert and of linear conjugation type. For example the linear conjugation boundary value problem for integral equations (14), (25) if a = 1 and a > 1 sets as follows: Problem Ri Let a = 1. Find the solutions of integral equation (14) in D+ and of integral equation (25) in D~ of classes C(D+ UT) and C(D~ UT) such that the function V~(z) = U~(z)(r — R)2A ^ is bounded at infinity and satisfies the boundary condition [U(z)exp(W+(z))(R-r)-2A+W]^R G(t)[U(z)exp(-WX(z))(r

- R)2A~^];=R

= +g(t),

(Rl)

where G(t) and g(t), t GT, are given functions that satisfy Holder's condition and G{t) ^ 0. Problem Ri Let a > 1. Find the solutions of integral equation (14) in D+ and integral equation (25) in D~ of classes C(D+UT) andC(D~lir) such that the function V~(z) = U~(z) exp(—2A(R)uja(r)) is bounded at infinity, satisfies the boundary condition [U(z)exp(2A-(R)wa(r)

+ WA(z))}+=R

=

G(t)[U(z) exp(-2A-(R)wa(r)

- W+(z))];=R

+ g(t).

(R2)

where G(t) and g(t), t £T, are given functions that satisfy Holder's condition and G(t) ^ 0. We can analogously formulate various other problems, prescribing conditions of a problems of linear conjugation type for some values of a dependence and conditions of an interior and exterior Riemann-Hilbert problem, or of an interior and exterior Schwarz type problem. Using the general solutions of the integral equations (14) and (25) for a = 1 and a > 1 from the formulas (16), (17), (26), (27) the above problems can be

327

reduced to the successive solution of linear conjugation problems for analytic functions. Problems Ri, R2 are reduced to the following linear conjugation problems for analytic function: *+(*) = G(t)-(t)+9l(t)

(33)

where 9i(t) =

(34)

u\ + 9Vl I ( exp[-^(C) + iy]A-(p)/(QdC (p-R)1-2*-W(<:-Re**)

7T J J D~

exp[W+(C)+M.4+(p)/(CR {p-Ry+2A+(R){£-Re*®)

Ml

D+

if a = 1 (for problem Ri) and 5i (*) =

(35) 9{t) +

~ J

J

(p-RWC-R**)

_ 1 f f exp[2A+(R)cja((p) nj J

mdC

+ W+(Q +iip}A+(p)f(Qd( (R-p)«((-Ret®)

D+

if a > 1 (for problem R2). Let K = IndG(i). If K > 0, then the problem (32) is always solvable and its general solution contains K + 1 arbitrary complex constants and is given by the following formula 6 : *(z) = X(z)[PK{z) + *(z)], where *[Z)

2niJ r

X+(t)(t-zy

when z e D+ and x(z) — z~Ker(z\

X[Z

>

when z G D _ ,

(36)

328 K

G(t)]dt r(*) = - Wln[t^(t)(t-z) If K < 0, then problem (33) is solvable if and only if K — 1 solvability conditions tgi

J r

^dt

X+(t)

= 0,

k = l,2,...,K-l

(37)

are satisfied. In this case the solution of problem (33) is given by the formulas (36) if PK^(z)=0. Then we obtain the solutions of problems R^ and Ri replacing the values of $(z) which are found in this way in the representations (16), (27) for a > 1 and representations (17), (28) for a = 1. Therefore we conclude: Theorem 15 Let a > 1, K = IndG(i) and functions A±(r),f(z) from integral equations (14), (25) satisfy conditions of the theorem 3 and 7. If K > 0, then problem R2 is always solvable and its general solution contains K + 1 arbitrary complex constants. If K < 0, then problem R2 is solvable if and only if K — 1 solvability conditions (37) are satisfied (gi(t) - are given by equality (35) ) . In this case general solution of problem R2 are given by formulas (16), (26), where analytic function $>(z) are defined by the formulas (36), (35) (at K<0, PK{Z) = 0). Theorem 16 Let a = 1, n = IndG(£) and functions A±(r),f(z) from integral equations (14), (25) satisfy conditions of the theorem 4 and 8. If K > S, then problem R\ is always solvable and its general solution contains K — 5 arbitrary complex constants. If K < 5, then problem R2 is solvable if and only if S — K solvability conditions (37) are satisfied (g\ (t) - is given by equation (34) )• In this case the general solution of problem R\ is given by formulas (17), (27), where the analytic function $(z) is defined by the formulas (36), (34) (at K < 0, PK-i(z) = 0).

References 1. Vekua I. N., Generalized analytic functions, Fizmatgiz, Moscow, 1959, English transl., Pergamon Press, Oxford, and Addison-Wesley, Reading, Mass., 1962, 628 p.

329 2. Rajabov N., On the one methods solution of the Vecua I.N. integral equation. Dokl. An. Taj. SSR, vol. 4, N 5 (1961), pp. 3-7. 3. Rajabov N., Riemann-Hilbert boundary value problems for the second order linear elliptic systems with singular coefficients. Boundary value and initial value problems in complex analysis: studies in complex analysis and its applications to partial differential equation I, Longman, Scietific and Technical, New York 1991, pp. 13-28. 4. Rajabov N., To theory second order linear elliptic systems which that all boundary is singular lines. Mathematical Herald v. 40 N 3-4 (1988), p. 301-307 (Materials third international symposium-Complex analysis and appications Herceg-Novi, Yugoslavia, May 23-29, 1988). 5. Rajabov N., Integral representations and boundary value problems for some second order linear elliptic systems with regular and singular coefficients. Complex analysis and applications 87, Sofia 1989, pp. 431-441. 6. Gachov F. D., Boundary value problems, "Nauka", M. 1977, 640 p.

330

A N EXPLICIT SOLUTION TO A CLASS OF S E C O N D K I N D L I N E A R SYSTEMS OF COMPLEX I N T E G R A L EQUATIONS WITH SINGULAR A N D SUPER-SINGULAR KERNELS (ABSTRACT) N. RAJABOV Tajik State National University Dushanbe 734025, Tajikistan E-mail: [email protected] Let D+ be the disk \z\ < R,T the circle \z\ = R. In D+ we shall consider the following system of integral equations

R - l

D

±

where ajk (j, k = 1,2) - are given real constants,