On the Cauchy problem for operators with injective symbols in Sobolev spaces Текст научной статьи по специальности «Математика»

УДК 517.955

On the Cauchy Problem for Operators with Injective Symbols in Sobolev Spaces

Ivan V.Shestakov* Alexander A.Shlapunov^

Institute of Mathematics, Siberian Federal University, av. Svobodny 79, Krasnoyarsk, 660041,

Russia

Received 11.10.2007, received in the revised form 20.11.2007, accepted 05.12.2007 Let D be a bounded domain in Rn (n > 2) with a smooth boundary dD. We describe necessary and sufficient solvability conditions (in Sobolev spaces in D) of the ill-posed non-homogeneous Cauchy problem for a partial differential operator A with injective symbol and of order m > 1. Moreover, using bases with the double orthogonality property we construct Carleman's formulae for (vector-) functions from the Sobolev space Hs(D), s > m, by their Cauchy data on г and the values of Au in D where г is an open (in the topology of dD) connected part of the boundary.

Key words: ill-posed Cauchy problem, Carleman's formula, bases with double orthogonality.

It is well-known that the Cauchy problem for an elliptic system A is ill-posed (see, for instance,

[1]). However it naturally appears in applications: in hydrodynamics (as the Cauchy problem for holomorphic functions), in geophysics (as the Cauchy problem for the Laplace operator), in elasticity theory (as the Cauchy problem for the Lame system) etc., see, for instance, the book

[2] and its bibliography. The problem was actively studied through the XX-th century (see, for instance, [3], [4], [5], [6], [7], [8], [9], [10] and many others); it stimulated the development of the theory of conditionally stable problems.

In this paper we present the approach developed in [9] for the homogeneous Cauchy problem for overdetermined elliptic partial differential operators. However we consider the non-homogeneous Cauchy problem. Of course, it is easy to see that these problems are equivalent (at least, locally) for systems with the invertible principal symbol. But, if the system is overdetermined, the equivalence takes place only if we have information on the solvability of the equation Au = f in a domain where we look for a solution of the problem. Therefore, even for operators with constant coefficients, the problems are not equivalent in domains which have no convexity properties with respect to the operator A (see, for example, [11]). Moreover, if the coefficients of the operator A are C-smooth (and not real analytic) then there are no general results even on the local solvability of the equation Au = f (see, for instance, [12, §0.0.2, §1.3.13]).

We emphasize that in the present paper we impose no convexity conditions on the domain D.

1. The Problem

Let X be a CTO-manifold of dimension n with a smooth boundary dX. We tacitly assume that it is enclosed into a smooth closed manifold XX of the same dimension.

For any smooth C-vector bundles E and F over X, we write Diffm(X; E ^ F) for the space of all the linear partial differential operators of order < m between sections of E and F. Then, for an

о о

open set O С X (here X is the interior of X) over which the bundles and the manifold are trivial, the sections over O may be interpreted as (vector-) functions and A G Diffm(X; E ^ F) is given

* e-mail: Shestakov-V@yandex.ru t e-mail: shlapuno@lan.krasu.ru © Siberian Federal University. All rights reserved

as (l x k)-matrix of scalar differential operators, i.e. we have A = aa(x) , x G O, where

| a| <m

aa(x) are (l x k)-matrices of CTO(O)-functions, k = rank(E), l = rank(F).

Denote E* the conjugate bundle of E. Any Hermitian metric (., .)x on E gives rise to a sesquilin-ear bundle isomorphism (the Hodge operator) : E ^ E* by the equality (*ev,u)x = (u, v)x for all sections u and v of E; here (., .)x is the natural pairing in the fibers of E* and E.

Pick a volume form dx on X, thus identifying the dual and conjugate bundles. For A G Diffm(X; E ^ F), denote by A' G Diffm(X; F* ^ E*) the transposed operator and by A* G Diffm(X; F ^ E) the formal adjoint operator. We obviously have A* = *-1A'*F, cf. [2, 4.1.4] and elsewhere.

Write <r (A) for the principal homogeneous symbol of the order m of the operator A, a (A) living on the (real) cotangent bundle T*X of X. From now on we assume that a(A) is injective away from the zero section of T*X. Then we will say that A is elliptic if rank(E) = rank(F) and overdetermined elliptic otherwise. Hence it follows that the Laplacian A*A is an elliptic differential operator of the order 2m on X.

o

We always assume that A satisfies the so-called uniqueness condition in the small on X.

o

(i) if u is a distribution in a domain D C X with Au = 0 in the sense of distributions and u = 0 on an open subset O of D then u = 0 in D.

It holds true if, for instance, all the objects under consideration are real analytic.

For an open set O C X, we write L2(O, E) for the Hilbert space of all the measurable sections of E over O with a finite norm (u,u)L2(O,E) = /0(u,u)xdx. We also denote Hs(O,E) the Sobolev space of the distribution sections of E over O, whose weak derivatives up to the order s G N belong

o

to L2(O, E). Let D be a bounded domain in X, and r be a C^-smooth open (in the topology of dD) connected part of dD. As usual, let Hfoc(D U r, E) be the set of sections in D belonging to Hs(a, E) for every measurable set a in D with a C D U r. For u G Hfoc(O, E), we always

o

understand Au in the sense of distributions in O. Given any open set O in X we let Sol^(O) stand for the space of all the weak solutions to the equation Au = 0 in O.

Further, for non-integer positive s we define Sobolev spaces Hs(O, E) with the use of the proper interpolation procedure (see, for example, [2, §1.4.11]). In the local situation we can use other (equivalent) approach. For instance, if X C Rn and the bundles E and F are trivial, we may we denote H1/2(O, E) the closure of CTO(O, E) functions with respect to the norm (see [13]):

Ih1/2(o,e)

II ||2 + f f Kx) - u(y)|2dxdy

IIMIIl2(0,e) +J J |x - y|2n+1 .

O O

Then, for s G N, let Hs 1/2 (O,E) be the space of functions from Hs 1(O, E) such that weak derivatives of the order (s — 1) belong to H1/2 (O,E).

It is well-known that if dD is sufficiently smooth then the functions from the Sobolev space Hs(D), s G N, have traces on the boundary in the Sobolev space Hs-1/2 (dD) and the corresponding trace operator tr : Hs(D) ^ Hs-1/2(dD) is bounded and surjective (see, for instance, [13]). In particular, this means that for every u G Hfoc(D U r), s G N, there is a trace trr(u) on r belonging

to H- 1/2(r).

Fix a Dirichlet system Bj, j = 0,1,..., m — 1, of the order (m — 1) on the boundary of D. More precisely, each Bj is a differential operator of the type E ^ Fj and order mj < m — 1, mj = m; for j = i, in a neighbourhood U of dD. Moreover, the symbols a(Bj), if restricted to the conormal bundle of dD, have ranks equal to the dimensions of Fj. From now on we assume that mj = j and

set t(u) = e"=o1ßju G e"=01Hs-j-1/2(öD,Fj) for u G Hs(D,E), s > m.

Problem 1. Let N 3 s > m. Given boundary data em-o1uj G ©m-,1Hfocj 1/2(r,F^-) n L2(r, Fj) and f G HfoCm(D U r, F) n L2(D, F), find a section u G Hfoc(D U r, E) such that

Au = f in D, (1)

u

t(u) = em=—0 1uj on r. (2)

As usual, we say that the problem is homogeneous if f = 0 in D and non-homogeneous otherwise. It is well known that problem 1 has no more than one solution under the Uniqueness condition (i) (see, for instance, [9, theorem 2.8]). We reduce this problem to the problem of the extension as a solution to an elliptic system from a small domain to a bigger one. In this way we generalize [9, theorems 5.2 and 10.3] related to the homogeneous Cauchy problem. We also construct formulae for the approximate and exact solutions of the problem.

2. Necessary Solvability Conditions

As far as we consider the overdetermined systems, it is natural to assume that operator A is included into an elliptic differential complex

0 4 CTO(E) 4 CTO(F) 4 CTO(G).

This means that Ai o A = 0 and the corresponding symbolic complex is exact away from the zero section of T*X. It is possible, for instance, if the operator A is sufficiently regular (see, for instance, [12, Definition 1.3.7]). For example, every operator with constant coefficients is sufficiently regular. Also the operators with real analytic coefficients and injective symbol may be included into an elliptic complex under mild assumptions (see [14]). Of course, if A is elliptic then A1 = 0.

Now due to the properties of the complex, A1f = 0 in D if the Cauchy problem is solvable. Besides, for l > k the operator A induces tangential operator AT on dD (see, for instance, [12, §3.1.5]). This means that the Cauchy data emT)1Uj and f should be coherent.

More exactly, it is well-known that under our assumptions on the domain D there exists a real valued CTO-smooth function p with |Vp| =0 on dD and such that D = {x G X : p(x) < 0}. Without loss of a generality we can always choose the function p in such a way that |Vp| = 1 on a neighborhood of dD.

Fix a Green operator Ga attached to A, i.e. an operator Ga(., .) G Diffm-1(X; (F*, E) 4 A"-1) such that

dGA(g, v) = ((g, Av)y — (A'g, v)y) dy for all g G CTO(X, F*), v G CTO(X, E);

here Ap is the bundle of the exterior differential forms of the degree 0 < p < n over X.

The Green operator always exists (see [12, Proposition 2.4.4]) and (as dD is not characteristic for A in our sutuation) it may be written in the following form:

m— 1

GA(g, v) = (Cjg, Bjv)yds(y) + dp A Gv(g, v) (3)

j=o

in a neighbourghood U of dD, where p is a defyning function of D, Gv(g, v) G Diffm—1(U; (F*, E) 4 An—2|U) and {Cj}m=01 is a Dirichlet system of the order (m — 1) on dD, with operators Cj G Diffm—j —1(U; F|4 F*) (see [15, Lemma 8.3.2]); here ds is the volume form on dD induced from X.

Now let CCOmp(D U r, E) stand for the set of CTO(D, E)-functions with compact support in D U r. Then for the solvability of problem 1 it is necessary that

,, m— 1 ,,

J ^ (CjA1A Uj)yds(y) = J (A1^, f)ydy for all p G C^omp(D U r, G*). (4)

r j=0 d

In fact, dp = 0 on dD. Hence, if problem 1 is solvable and u is its solution then, by Stokes' formula, we have for each section p G C^mp(D U r, G*):

J mr(CjA1Puj, )yds(y) = J Ga(A1P,u) = J(A1P,Au)ydy = J(A1p,f )ydy

r j=0 dd d d

where G is a domain in D with a smooth boundary such that supp v C G.

3. Solvability Criterion

From now on we assume that the Laplacian A* A satisfies the Uniqueness condition (i). Then it

o

has a two-sided (i.e. left and right) pseudo-differential fundamental solution, say, on X (see, for instance, [2, §4.4.2]). In particular, L = $A* is a left pseudo-differential fundamental solution for A.

Let Mrv be the Green integral with a density v = ®m-11Vj G em-)1L2(r, Fj):

/m—1

^ (Cj(y)L(x, y)), Vj)y ds(y), x G r (5)

r j=o

(here L(x,y) is the Schwartz kernel of L (see, for instance, [12, 1.5.4]). It is known that if dD is smooth enough (e.g. dD G CTO) then the Green integral induces a bounded linear operator

MdD : em—01Hs—j—1/2(dD, Fj) ^ Hs(D,E), s G Z+, s > m

(see, for instance, [16, 2.3.2.5]). In particular, we easily see that in our case Mr(©m=01uj) G Hfoc(DU r, E).

Further, for a section f G L2(D, F) we denote by TDf the following volume potential:

Tdf = LXdf

where xD is the characteristic function of the domain D. If dD is smooth enough (e.g. dD G CTO) then the potential Td induces a bounded linear operator

TD : Hp(D,F) ^ Hp+m(D,E), p G Zt

(see, for example, [16, 1.2.3.5]). Moreover, for p = 0 we can extend f by zero onto X obtaining thus a form f G L2(X) and therefore the potential Td induces actually a continuous linear operator

o

Td : L2(D,F) ^ Hmc(X, E). (6)

o

In particular, in our case we easily see that TDf G Hfoc(D U r, E) n Hmc(X, E).

Further, if dD is smooth then for every section u G Hm(D, E) we have the Green formula:

MdD (®m=o1Bj u) + TdAu = XD u (7)

(see [2, lemma 10.2.3])).

It is clear that the integrals Mrv and TDf satisfy A*A(Mrv) = 0 and A*A(TDf) = 0 everywhere outside D as parameter dependent integrals. Hence the section

F = Mr(em=o1uj)+ Td f

o _

belongs to So1a*a(X \ D). The Green formula (7) shows that the potential F contains a lot of information on solvability conditions of problem 1.

Now we would like to obtain necessary and sufficient conditions for the solvability of the Cauchy

o

problem 1 with the use of function F. For this purpose we choose a set D+ C X in such a way

o

that ^ = D U r U D+ is a bounded domain with piece-wise smooth boundary dD+ in X.

Denote by F± the restrictions of F onto D± (here D— = D). By the definition, F + belongs to So1a*a(D+). Besides, defining v in formula (5) by zero on the boundary of a large enough domain D D, we see that, if dD is smooth enough (e.g. dD G Cthen the Green integral induces a bounded linear operator

M+D : ®m=01Hs—j—1/2(5D, Fj) ^ Hs(il \ D, E), s G N

(see, for instance, [16]). In particular, we easily see that in our situation M]t(©m—11uj) G Hfoc(D+ U r, E). Thus, F± G Hfoc(D± U r, E).

Let A* © A1 be the standard differential operator of type F ^ (E, G) mapping g to the pair (A*g,A1g).

Theorem 1. Let both A* A and A* © Ai satisfy the Uniqueness condition (i). Then the Cauchy problem 1 is solvable if and only if condition (4) holds true, and there is F 6 So1a*a(Q) coinciding with F + on D+.

Proof. Let problem 1 be solvable and u be its solution. The necessity of condition (4) is already proved. Set

F = F - xdu.

By the definition, the function F satisfies A*AF = 0 in D+ and belongs to Hfoc(D± U r, E).

Take a domain G C D with a smooth boundary such that G n dD C r. Then according to the Green formula (7) we have in D+ U G:

F = Mr(ejm=01Uj) + Tdf - xgu = = Mr(e™=01 Bju) + TgAu + TD\Gf - MaG(©m=o1Bju) - TGAu = = -MdG\r(©Jm=o1Bj u) + TD\Gf.

This identity implies that F extends from D+ to D+ U G U (r n dG) as a solution to the operator A*A since the integrals MdG\r(©™r01Bju) and TD\Gf are solutions to this operator everywhere outside the integration sets as parameter depending integrals.

Finally, since for every point x 6 D there is a domain G 3 x with the described properties, we see that in factF belongs to So1a*a(Q) and coincides with F+ on D+.

Back, let there be a section F 6 So1a*a(Q) coinciding with F + on D+. Set

u = F - -F-. (8)

By the construction, the section u belongs to H®oc(D U r, E). Moreover, since the section F is CTO-smooth in Q and the potential TDf belong to HmOc(Q) (see (6)), we see that t+(F+) = tr(F-) and t+(T+f) = tr(TDf); here t+ : Hfoc(D+ U r, E) ^ e!mL"01Hfo~j-1/2 (r, Fj) is the corresponding trace operator. Hence the jump theorem for the Green integral (see [9, lemma 2.7]) gives:

tr(u) = tr(M- jo1 uj) - t+(M+ ©mTo1 uj) + tr(T-f) - t+(T+f) = joS.

In order to finish the proof we need to check that Au = f in D. For this purpose we consider the section g = f - Au belonging to Hfo-m(D U r, F). Condition (4), in particular, means that f satisfies A1f = 0 in D, and therefore the section g has the same property.

Moreover, g satisfies A* g = 0 in D. Indeed, as $ is a two-sided fundamental solution of the Laplacian A* A, we have

A* (x Df - ATDf) = A* (x Df - A$A*x Df) =0 in xX, (9)

A*g = A*f - A* AMr (©m=o1uj) - A*ATDf = 0 in D.

Thus we have proved that (A* © A1)g = 0 in D.

Now let VE 6 Diff1 (X; E 4 E® (T*X)c) and VG 6 Diff1(X; G 4 G® (T*X)c) be connections in the bundles E and G respectively compatible with the corresponding Hermitian metrics (see [17, Ch. III, Proposition 1.11]). Let m1 be the order of A1. Set,

m1—m — 1

VE (VEVE) 2 , if (m1 - m) is positive and odd; Qe = ^ (VEVE)(mi-m)/2, if (m1 - m) is positive and even; I, if mi < m,

m—m i — 1

V(VGVg) 2 , if (m - mi) is positive and odd; Qg = ^ (VGVG)(m-mi)/2, if (m - m1) is positive and even; I, if m < mi.

Denote m = max(m, m.1). Clearly, Qe G Diff,—m(X; E ^ Be) and Qq G Diff,—mi (X; G ^ Bq) have injective symbols; here Be and Bq are the corresponding vector bundles. Then, the ellipticity of the complex means that

P = QeA* © QqA1

belongs to Diffm(X; F ^ (BE,Bq)) and has the injective symbol (cf. [12, §2.1.4]).

Since P(f — ATDf) = Pg = 0 in D, we conclude that both g and (f — ATDf) are smooth in D. As g G L2oc(D U r, F) n SolP(D), it has a finite order of growth near r (see [9, theorems 2.6 and 4.4]).

Set De = {x G D : p(x) < —e}. Then for all the sufficiently small e > 0 the sets De CC D CC D—e are domains with smooth boundaries dD±e and vectors ^ev(x) belong to dD±e for every point x G dD (here v(x) is the outward unit normal vector to dD at the point x). Now using Stokes' formula, we easily obtain

J Gai (£, g) = y (Aip, (Au — f ))y dy =

3De De

= — y^f )y dy + J GA(A,1^,u) for all p G Ccc^p(D U r,G*) (10)

De 3De

Then, using (3) and condition (4), we get for all p G C^ (D U r, G*):

Д1+ I - J (Aiß, f )y dy + J GA(Aiß, u) I = V De ôDe /

/n m 1

(AiP, f )y dy + £ (CjAiP,Uj)yds(y) = 0. (11)

d r j=0

Combining (10) and (11), we obtain:

£lim0 y GAl (P,g)=0 for all £ G (D U r,G*). (12)

Similarly, using Stokes' formula and [12, Proposition 2.4.5], we get for all h G C^mp(D U r, E* ):

y GA* (h,g) = -y<(A*)'h,/)y dy + J ((A*)'h,ATD/)y dy+

ôDe De De

/m—1

^ (Cj *F(AMr(©m=o1uj) — AF), Bj *-1 h)ydse(y). (13)

SDe j=o

Let h G Co(Q, E*) such that h = h in D. Then, according to (9), we have

— y ((A*)'h, f)y dy + J((A*)'h, ATdf)y dy = — J ((A*)'h, ATdf)y dy. (14)

D D n\D

Moreover, since TDf G So1a*a(D+), and F G CTO(^,E), Stokes' formula implies:

/mi л

^ (Gj AF,Bj h)ydse(y)= / ((A*)'h, ATdf) dy+

j — 0 ^ öDe j—0 Q\D

/m—i_

£ (Cj (AMr(©m—0iUj),Bj 1 h)yds—e(y). (15)

_n

£

Hence, using (13), (14), and (15), we obtain:

~ if m—1_

£limoy Ga*(h,g) = £limQ U £ (Cj *F(AMr(©m=o1Uj),Bj 1 h>ydse(y)-

ÖDe We j=0

/m—1__\

£ (Cj *F(AMr(©m=o1Uj),Bj 1 h>yds—e(y) I =0

dD- j=0 /

for all h e C°Omp(D U r, E*), because of the lemma on the weak jump of Green integrals (see [9, Lemma 2.7]). Thus,

£lim/ Ga*(h,g) = 0 for all h e C°Omp(D U r,E*). (16)

Choose a Dirichlet system {Bj j!"=—1 of the order (m — 1) in a neighbourhood of dD and denote by {Cj jj=—/ a dual Dirichlet system for it, i.e. such that the Green operator GP is presented in the form

m—1

Gp (¿,V0= E (C^,Bj ^>yds(y)e + dp a G„ (g,f), ^ e C°° (F), ^ e C° ((BE ,BG))

j=o

in a neighbourhood of dD (see [15, Lemma 8.3.2] and the discussion in § above).

Using [12, Proposition 2.4.5], (12), (16), and the fact that (A* © A1 )g = 0 in D, we see:

* m—1

/ Yj (Cj ^ Bj g>y dSe(y) = / GP g) =

öDe j=0 öDe

= e1i>I+0 (/ Gqgä1 (^G,g)+ / GqeA* ,g)l =

WE öDe /

= e1imo ^ J Gqg (^g, A1g) + Gai (Q'^g, g) + Gqe (¿e, A*g) + Ga* (QE^e, g) | =

= £1nno ^ J Gai (Q'g^g, g) + Ga* (QE, g)| = 0,

for all $ e C°0mp(DUr, (BG, BE)); here $ = (¿e, ¿g), ^e e C°(DUr, BE), ^g e C°(DUr, BG). Hence,

1imo| Gp(¿,g) = 0 for all $ e C^D U r, (BE,BG)). (17)

e

ÖD,

As {<5j j™"1 is a Dirichlet system on dD, for every — 6 CIT(r,Fj*) there is ^ 6 G~mp(D U r, (BE, BG)) with Cj^ = 0 for i = j, Cj^ = —j on dD and therefore the famous theorem by Banach and Steinhaus yields that is equivalent to the following:

lim / (—j, Bjg(y - ev(y)))y ds(y) = for every —j 6 CD(r, F*) and for each 0 < j < m - 1,

e >+0 J dD

i.e. ©™=01JBjg = 0 on r in the sense of the weak boundary values (see [9, Definition 2.2]).

Now the uniqueness theorem [9, theorem 2.8] for the Cauchy problem for systems with injective symbols implies that the section g = f - Au equals to zero in D identically because the Uniqueness

o

condition (i) for the operator A* © A1 holds true in X. □

For f = 0 and the operators with real analytic coefficients, theorem 1 was obtained in [9, theorem 10.3].

Remark 1. Theorem 1 easily implies conditions of local solvability of the Cauchy problem. Indeed, fix a point xo G r. Let V be a (one-sided) neighbourhood of xo in D and r = dV n r. Set F = Mr(©m=o1uj) + Tvf. As

F = F + Mr\f (©m=o1uj) + Td\v f

we see that Ft extends as a solution to the Laplacian A*A in n = V U r U D+ if and only if the potential .F+ does. Hence, under condition (4), the solution of the Cauchy problem exists in the neighbourhod V where the extention of the potential F + does.

Also we would like to note that theorem 1 gives not only the solvability conditions to problem 1 but the solution itself, of course, if it exists (see (8)). It is clear that we can use the theory of functional series (Taylor series, Laurent series, etc.) in order to get information about extendability of the potential F + (cf. [8], [2]). However in this paper we will use the theory of Fourier series with respect to the bases with the double orthogonality property (cf. [18], [2] or elsewhere). Moreover, using formula (8) we can construct approximate solutions of problem 1 (see below).

4. Bases with Double Orthogonality in the Cauchy Problem and Carleman's Formula

It is often important in applications to look for a solution of problem 1 in the class Hs (D, E). For this purpose in the present section we assume that uj G Hs -j - 1/2(r, Fj), f G Hs - m(D, F). Then Whitney's theorem implies that for each 0 < j < m — 1 there is a section Vj G Hs -j - 1/2(dD, Fj) coinciding with uj on r. We can always choose such a section Vj vanishing outside a given neighborhood of r. Now fix such functions ©m=o1Vj. Set j=0

E = MdD (©m=o1Vj)+ Td f.

The boundedness theorems for potential operators in Sobolev spaces (see [16, 1.2.3.5 and 2.3.2.5]) imply that fF± G Hs(D±, E).

Corollary 1. Let both A*A and A* © A1 satisfy the Uniqueness condition (i) and let be piece-wise smooth. In addition, let uj

G Hs-j-1/2(r,Fj), f G Hs-m(D). Then the Cauchy problem 1 is solvable in Hs(D, E) if and only if condition (4) is fulfilled and there is a function F G Hs(Q, E) n SolA*A(n) coinciding with F + in D+.

Proof. Let problem 1 be solvable in Hs(D, E). Then theorem 1 implies that condition (4) holds and there is a function F G so1a*a(^) coinciding with F+ in D+. Clearly,

F = F + MdD\r(©m=o1Vj). (18)

Since the potential MdD\r(©m=o1Vj) belongs to so1a*a(^) we conclude that the function F+ extends to a solution from so1a*a(^) if and only if the function .F+ does. Therefore, the function

F = F + MdD\r(©m=o1Vj) = F + MdD\r (©m=o1vj) — XD u = F — xd u (19)

belongs to SolA*A(^) and coincides with F + in D+. Moreover, as F G Hfoc(Q, E) n Hs(D±, E) we easily see that F G Hs(Q, E).

Back, formula (18) and theorem 1 imply that, under the hypothesis of the corollary, problem 1 is solvable. In order to finish the proof we will show that its solution u, given by (8), is, in fact, the solution of problem 1 in Hs(Q, E). However, using (8), (18) and (19) we immediately obtain that

u = fF- — . (20)

Since F G Hs(D, E) and F G Hs(0,E) we see that u G Hs(D,E). □

Now recall the notion of bases with the double orthogonality property in spaces of solutions of elliptic systems (cf. [18], [2] or [9]).For this purpose we denote by hs(Q) the space So1a*a(Q) h H s(Q,E ).

Lemma 1. If w C Q is a domain with a piece-wise smooth boundary and Q \ w has no compact (connected) components then there exists an orthonormal basis in hs(Q) such that is an orthogonal basis in hs(w).

Proof. These {bv}£=1 are eigen-functions of compact self-adjoint operator R(Q, w)*R(Q, w), where R(Q, w) : hs(Q) ^ hs(w) is the natural inclusion operator (see [2] or [9, theorem 3.1]). □ Now we can use the basis {bv} in order to simplify corollary 1. For this purpose fix domains

w C D+ and Q as in lemma 1 and denote by cv(È+) = ( .. ' ("'E), v G N, the Fourier coefficients of E+ with respect to the orthogonal basis {bv|w} in hs(w).

Corollary 2. Let both A* A and A* ® Ai satisfy the Uniqueness condition (i). In addition, let U G Hs-j-1/2(r,f1j), f G Hs-m(D). The Cauchy problem 1 is solvable in Hs(D,E) if and only

œ ^

if condition (4) is fulfilled and the series |cv(^+)|2 converges.

V=1

Proof. Indeed, if problem 1 is solvable in Hs(D, E) then, according to corollary 1 condition (4) is fulfilled, and there exists a function F G hs(Q) coinciding with E+ in w. By lemma 1 we conclude that

œ

F"(x) = ^ kv(F>v(x), x G Q, (21)

v=1

where kv (F") = (F, bv)hs(Q'E), v G N, are the Fourier coefficients of F with respect to the orthonor-

œ „

mal basis {bv} in hs(Q). Now Bessel's inequality implies that the series ^ |kv(F)|2 converges.

V=1

Finally, the necessity of the corollary holds true because

c ,p +) = (R(Q,w)^,fl(Q,w)6y)g.(„'E) = (F,R(Q,w)*R(Q,w)6v)g.(n,E) = k (-)

Back, if the hypothesis of the corollary holds true then we invoke the Riesz-Fisher theorem. According to it, in the space hs(Q) there is a section

œ

F(x) = ^ cv(F +)bv(x), x G Q. (22)

v=1

By the construction, it coincides with E+ in w. Therefore, using theorem 1, we conclude that problem 1 is solvable in Hs(D,E). □

The examples of bases with the double orthogonality property be found in [9], [2], [18]. Let us obtain Carleman's formula for the solution of problem 1. For this purpose we introduce the following Carleman's kernels:

N

CN(y, x) = L(y, x) - ^ Cv(L(y, -))&v(x), N G N, x G Q, y G w, x = y.

v=1

Corollary 3. Let both A*A and A* ® A1 satisfy the Uniqueness condition (i). Then, for every section v G Hs(D, E), s G N, the following Carleman's formula holds true:

lim

N^oo

v - v(N)

= 0, (23)

, m — 1

/110- 1 n

Y (CCn(.,x)),vj)yds(y) + y (Cn(.,*), Au)ydy

dD j=0 D

and Vj € Hs-j-1/2(dD, Fj) are (arbitrary) sections coinciding with Bjv on r for each 0 < j < m — 1.

Proof. Indeed for the Cauchy data f = Av and ©m-01Mj = (Bjv)|r the Cauchy problem 1 is solvable in Hs(D,E). Hence corollary 1 implies that a solution of this problem u is given by formula (20). Then the Uniqueness theorem for the problem (see, for instance [9, theorem 2.8]) gives u = v in D.

As w fl D = 0 we may use Fubini theorem and obtain for all v € N:

kv(F+) = (- f m-;(Cj(y)cv(L(y,.)), Vj)yds(y) + f (cv(L(y,.)),f )dy ) . (24)

V ¿D j=0 D /

Moreover (see the proof of corollary 2) we know that the function FF is given by formula (21) with the coefficients (24), the series converges in Hs(Q, E) to F and hence in Hs(Q, D) to F1- — u, i.e. we have:

lim

N^oc

v - MdD(©^To1 Vj) - TdAv-

N

I-

V=1

m—1 » \

J E (Cj(y)cv(L(y, •)),Vj)yds(y) + y (cv(L(y, •)),/>„dy I 6„ 3D j=0 D /

= 0.

h s(D,E)

This exactly gives identity (23) after regrouping the summands. □

Remark 2. Formula (20) means that v = MdD(®m=01Vj) + TdAv — F. As F and each function bv are solutions of the elliptic system A*A in Q, the Stiltjes-Vitali theorem implies that the series (22) converges in C~(Q, E) Therefore, if Av € Hp(D, F), s < p + m, then TD Av € Hp+m(D, E), MdD(©m=01vj) € CI0c(D,E) and we additionally have: 1) Avn converges to Av in HpOc(D U r, F); 2) vn converges to v in Hp+m(D,E).

It is worth emphasizing that in fact we obtain the same type of Carleman kernel as for f = 0 (cf. [9, theorem 12.6]). In particular, if A is a Dirac type operator and D is a part of a unit ball in Rn cut off by smooth hypersurface r ^ 0 we easily construct both exact and approximate solutions of the Cauchy problem 1 by using the decomposition for harmonic functions with respect to spherical harmonics (see [9, §13]). For the Cauchy-Riemann operator on the complex plane this formula for exact solution is the well-known formula by Goluzin and Krylov (see, for instance, [6, Theorem 1.1]).

The first author was suppoted by Krasnoyarsk regional scientific fund, grant 17G102; the second author was supported in part by RFBR, grant 05-01-00517.

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