CC BY
20
УДК 517.98, 517.55
Negative Sobolev Spaces in the Cauchy Problem for the Cauchy-Riemann Operator
Ivan V.Shestakov* Alexander A.Shlapunov^
Institute of Mathematics Siberian Federal University, av. Svobodny 79, Krasnoyarsk, 660041,
Russia
Received 10.11.2008, received in revised form 20.12.2008, accepted 29.01.2009
Let D be a bounded domain in Cn (n ^ 1) with a smooth boundary dD. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for the Cauchy-Riemann operator d in D. In particular, we describe traces of the corresponding Sobolev functions on dD and give an adequate formulation of the problem. Then we prove the uniqueness theorem for the problem, describe its necessary and sufficient solvability conditions and produce a formula for its exact solution.
Keywords: negative Sobolev spaces, ill-posed Cauchy problem
As it was understood in the 50-th of XX-th century, it is very natural to consider generalized formulations of boundary value problems and to solve the problems in spaces of generalized functions (see, for instance, [1], [2]). There are two principal reasons for this: the advantage of using very powerful mathematical apparatus of functional analysis and the needs of applications (in modern models it is practically impossible to point-wisely measure the data of boundary value problems; in the best case one can interprete them as functionals). In the present paper we want to consder the Cauchy problem for the Cauchy-Riemann operator d in spaces of distributions with restrictions on the growth near the boundaries of domains (the last condition is imposed in order to define traces of such distributions on domain's boundaries, see, for instance, [3], [4], [5]).
It is well-known that the Cauchy problem is ill-posed (see, for instance, [6], [7]). However it naturally appears in applications: in Hydrodynamics, in Tomography, in Theory of Electronic Signals. Beginning from the pioneer work [8], the problem was actively studied through the XX-th century (see [7] and [4] for a rather complete bibliography). Here we present the approach developed in [9] for the Cauchy problem for holomorphic functions (cf. [10]); but we consider the non-homogeneous Cauchy problem (cf. [11]). Of course, it is easy to see that these problems are equivalent for n =1. On the other hand, if n > 1 then the Cauchy-Riemann system is overdetermined, and the equivalence takes place only if we have information on the solvability of the d-equation in a domain where we look for a solution to the problem. Therefore, the problems are not equivalent in domains which have no convexity properties (see, for example, [12], [13]). We emphasize that in the present paper we impose no convexity conditions on the domain D.
* e-mail adress: [email protected] t e-mail adress: [email protected] © Siberian Federal University. All rights reserved
1. Functional Spaces
Let Rn be n-dimensional Eucledian space and Cn be n-dimensional complex space with points being n-vectors z = (zi,..., zn), where zj = Xj + %/—1xj+n, j = 1, ...,n, 1 being imaginary unit and x = (x1,..., x2n) G R2n. We tacitly assume n > 1, though n = 1 is formally possible.
Let d be the Cauchy-Riemann operator in Cn. It is known (see, for instance, [13] or [14]) that it induces the differential compatibility complex (Dolbeault complex):
0_► a(°'°) —U A(0'1} —U A(0'2) —U ... —U A(0-n) _► 0;
here A(q'r) is the set of all the complex exterior forms of bi-degree (q, r) and d be the (graduated) Cauchy-Riemann operator extended to the differential forms.
Let D be a bounded domain (i.e., an open connected set) in R2n, and let D be its closure. We always assume that the boundary dD of D is of class C
As usual we denote by CTO(D) the Frechet space of infinitely differentiable functions in D and by C»mp(D) the space of smooth functions with compact supports in D. Besides, let CTO(D) stand for the set of smooth functions in D with any derivative extending continuously to D and, for an open (in the toplology of dD) subset r C dD, let C^mp(D U r) be the set of all CTO(D)-functions with compact supports in D Ur. Everywhere below the set of differential forms of bi-degree (q, r) with the coefficients from a space 6(D) is denoted by 6(D, A(q'r)).
Recall that for every R G CTO(D, A(n'n-1)) there is such a function r G (dD) that R = rds on dD (ds being the volume form on dD induced from Cn). We will write R—ds for r.
Let us denote by * the Hodge operator for differentail forms (see, for instance, [14, §14]); it is convenient to set = *f for f G CA(q'r)). If f G CA(0'1)) then n(f) = (*f)—ds is called normal part of f on dD and then n(dv) = dnv is called the complex normal derivative of a function v G CTO(D) on dD.
We write L2(D) for the Hilbert space of all the measurable functions in D with a finite norm
¡' dz A dz
(u,v)l2(d) = / u(z)v(z)-
lD w v (^v/-l)r
Then the Hermitian form
(u, v)L2(D,A(q,r)) = u A *v J D
defines the Hilbert structure on L2(D, A(q'r)).
We also denote by Hs(D) the Sobolev space of distributions over D, whose weak derivatives up to the order s G N belong to L2(D). For a non-integer positive s G R+ we define Sobolev spaces Hs with the use of the standard interpolation procedure (see, for example, [15] or [4, §1.4.11]). It is known (see, for instance, [15]) that functions of Hs(D), s G N, have traces on dD of class Hs-1/2(dD) and the corresponding trace operator is continuous.
Sobolev spaces of negative smoothness may be defined in many different ways (see, for instance, [16]). The standard Sobolev space H-s(D), s G N, is the completion of CTO(D) with respect to the norm
|(u,v)L2(D)|
u
I H-s(D) = sup
vec- (D) llvll Hs(D)
However we prefer to use the ones allowing us to consider boundary value problems, to use integral representations and to get boundedness of standard potentials. That is why we follow [3]
(cf. [5], [17]). More precisely, let CTO'0(D) be the subspace of CTO(D) having zero values on dD. Apart from the standard one, two more types of negative norms may be defined for functions from CTO(D):
||u||_s = sup u u-, |u|-s = sup -rr-r-.
veo^(D) iivnh s(D) v£C»,«(D) iivnhs(d)
It is more correct to write || • ||-s,D and | • |-s,D but we prefer to drop the index D if this does not cause misunderstandings. Denote the completions of CTO(D) with respect to these norms by H(D, || • ||-s) and H(D, | • |-s) respectively. By definition, it is aslo natural to call these Banach spaces negative Sobolev spaces (cf. [3]). Obviously, H(D, || • ||-s) ^ H(D, | • |-s) ^ H-s(D), because ||u||_s > |u|-s > ||u||h-=(d).
For s € N the Banach space H(D, || • ||-s) is (topologically) isomorphic to the dual space (Hs(D))' for the Hilibert
space Hs(D) (see, for instance, [4, Theorem 1.4.28]). This allows us to define a Hilbert structure on the space H(D, || • ||-s). Indeed, this Banach space is a Hilbert one with the scalar product
(u, v)-s = 1 (||u + V | -g - ||u - V | -g + ||iu + v||-s - ||iu - v||_s) ,
related to the norm ||.||-s, because the mentioned above [4, Theorem 1.4.28]) implies the parallelogramm identity.
Given a bounded domain Q D D, it is easy to see that any u € H(D, || • ||-s) extends to an element U € H(Q, || • ||-s); for instance, one can set
(U,v)n = (u,v>d for all v € Hs(Q)
(here (.,.)d is the pairing between H and H' for a space H of distributions over D). It is natural to denote this extention by xdu; obviously, the support of the distribution xdu lies in D. The linear operator
xd : H(D, HU) ^ H(Q, H|_s),
defined in this way is continuous.
In [3] these spaces were used to study Dirichlet problem for the scalar elliptic partial differential operators in Rn (see also [17] for more general operators); we briefly expose the corresponding results and slightly modify the results [3] for Cn because we will need them now. To this end, define pairing (u, v) for u € H(D, || • ||-s), v € CTO(D) as follows. By the definition, one can find such a sequence {uv} in CTO(D) that ||uv - u||-s ^ 0 if v ^ to. Then
|(uv - uM,v)l2(d)| < ||uv - um||_s||v||h=(d) ^ 0 as v ^ to.
Set (u,v) = lim (uv, v)l2(d). It is clear that the limit does not depend on the choice of the sequence {uv}, for if ||uv||-s ^ 0, v ^ to, then
|(uv,v)L2(D)1 < ||uvy_s|v|Hs(D)
tends to zero, too. This implies that for u € H(D, || • ||-s) and v € CTO(D) we have the inequality: |(u, v)| ^ |u|-s|v|hs(d). Similarly, one defines pairing (u, v) for u € H(D, | • |-s) and v € CTO'0(D) and, obviously one has |(u, v)| ^ |u|-s||v||hs(d). Of course, the scalar product (., .)L2(D,a(o,i}) induces pairings (.,.) on H(D,
A^Ml • II_,)
x CTO(D,A(0,1)) and
H(D, A(0'1), | • |-s) x C)TO'0(D, A(0'1)).
2n 2
Now, given F and u°, consider Dirichlet problem for the Laplace operator A = in
j=o Xj
n.
Am = F in D,
u = u° on dD.
More exactly, let F e H(D, | • |_s_2), u0 e H_s_1/2(dD), s e Z+. One says that u e H(D, || • ||_s) is a strong solution to (1) if there is a sequence {uv} e CTO'°(D) such that
||uv - u|_s ^ 0, ||uv - U° || _s_1/2,dD ^ 0, ||5nUv - U°|_s_3/2iSD ^ 0, |AMv - F | _s_2 ^ 0,
v ^ to, where U° e H_s_3/2(dD) is arbitrary.
Given F e H(D, | • |_s_2), u° e H_s_1/2(dD), we say that a function u is a weak solution to (1) if it belongs to H(D, || • ||_s/) with a number s' e Z+ and, according to the Green formula in complex form,
(u, Av) = (F, v) - 2(u°,dnv)dD for all v e CTO'°(D). Clearly, any strong solution to (1) is a weak one.
Theorem 1. Let s e Z+. If F e H(D, | • |_s_2), u° e H _s_1/2(dD), then there is the unique weak solution u to the problem (1). In particular, the weak solution to (1) is the strong one and
||u||_s < c (|F|_s_2 + ||u°y_s_i/2ia^ ,
where the constant c does not depend on F, u° and u.
Proof. See, for instance, [17] (cf. [3] for the real case). □
Denote by P(D) : H s 1/2 (dD) ^ H(D, || • ||_s) the continuous operator, mapping u° and F = 0 to the unique solution to Dirichlet problem (1). Of course, on a sufficiently smooth u°, this is nothing but the Poisson integral of the Dirichlet problem. Similarly, denote by :
H(D, | • |_s_2) ^ H(D, || • ||_s) the continuous operator, mapping F e H(D, | • |_s_2) to the unique solution to Dirichlet problem (1) with the zero boundary data.
Now we want to solve the Cauchy problem for d in spaces H(D, || • ||_s). For an element u of Hs(D), H(D, || • ||_s) or H(D, | • |_s) we always understand du in the sense of distributions in D. Of course, d continuously maps Hs(D) to Hs_1(D), s e Z.
Lemma 1. The differential operator d induces a linear bounded operator
d : H(D, A(0'r), || • ||_s) ^ H(D, A(0'r+1), | • |_s_1).
Proof. It immediately follows from Stokes' formula. □
However there is no need for elements of H(D, || • ||_s) to have traces on dD and there is no need for d to map H(D, || • ||_s) to H(D, A(0'1), || • ||_s^1).
For this reason we introduce two more types of spaces (cf. [4, §9.2, 9.3]). Namely, denote the completion of (D) with respect to the graph norms
||u||_s,a = (||u||2_s + Pu||2_s_1)1/2 , ||u||_s,6 = (||u||2_s + ||u|_g_1/2,dD) 1/2
by Hg(D, || • |_s) and H6(D, || • ||_s) respectively. Obviously, H^(D, || • ||_s) and H6(D, || • ||_s) are Hilbert spaces with scalar products
(u,v)_s Q = (u, v)_s + (du, dv)_s_1, (u,v)_Sj6 = (u, v)_s + (u,v)_s_1/2i3D
respectively. Clearly, elements of these spaces are more regular in D than elements of H(D, || •H-s). Moreover, by the definition, the differential operator d induces a bounded linear operator
d-s : Hd(D, || • H-s) - H(D, A^, || • ||-s-i),
and the trace operator ts : Hs(D) — Hs-1/2(dD) induces a bounded linear trace operator
t-s : Hb(D, || • |-s) — H-s-1/2(dD).
Theorem 2. The linear spaces H^(D, || • ||-s) and Hb(D, || • ||-s) coincide and their norms are equivalent.
Proof. By the definitions of the spaces we need to check the equivalence of norms on CTO(D) only. Let d g = —*d*g be the formal adjoint for d. Then because of Stokes' formula we have:
(dv,g) = (v,d*g) + / v(*g) for all g e CTO(D, A(0'1)), v e CTO(D). JdD
Hence, for all v e CTO(D) we have:
|| V | -s + ||dv||-s-1 < (P^H2 + |ns+1|2 + 1)(|v|2-s + HvH2-s-1/2,ЭD),
where ns+1 is the continuous operator ns+1 : Hs+1(D, A(0,1)) — Hs+1/2(dD) induced by the normal operator n, and ds+1 : Hs+1(D, A(0,1)) — Hs(D) is the continuous operator induced by d*.
Back, fix a defining function p e of the domain D; without loss of a generality we may assume |dp| = 1 on dD. For a function g0 e CTO(dD), set
Go = E P(D) j=i
/ \
dp go gzj
E
k=i
dp
ÔZfc
1/2
/
Due to [14, lemma 3.5] and the properties of the Poisson integral P(D), we see that Go G CA(o'1})) with n(Go) = go on dD and
with a constant y = y(s), not depending on go and Go. Then, by Stokes' formula, we have:
I vgods(x) = / v(*Go) = (dv, Go) - (v, d*Go) for all v G Cœ(D). ./ôD JdD
Hence
llvh-s + l|vy-s_i/2 < (1 + Y2 + 72ydS+iy2)(yvy2_s + pvllVi). □
2. Weak Boundary Values of Sobolev functions
Consider now the weak extension of d on the scale H(D, || • ||-s). Namely, denote by Hw (D, || • ||-s) the set of functions u from H(D, || • ||-s) such that du e H(D, A(0,1), || • ||-s-1). As d is linear, then the set is linear too; we endow it with the graph norm
= (lMl-s + l|duH-s-l)1/2-
2
u
It is not difficult to see that the normed space Hw (D, || • ||_s) is complete. Clearly,
H(D, || • |_s) C H|(D, || • H_s), (2)
Besides, the differential operator d induces the linear bounded operator
rs : HW(D, H|_s) - H(D, A^, || • ||_a_i).
The unions U^=1H|(D, |H|_s) and U^=1H(D, |H|_s) we denote by Hd(D) and H(D) respectively. As before, let r be an open (in the topology of dD) connected subset of dD.
Definition 1. We say that a function u G Hg-(D) has weak boundary value t^ (u) = w0 G D'(r) with respect to the operator d on r if
(du,g)D - (u,d*g)D = (uo,n(g)) for all g G C£mp(D U r, A(0,1)).
Stokes' formula implies that every u G Hg(D, || • ||_s) has a weak boundary value on dD in the sense of Definition 1, coinciding with the trace t_s(u) G H_s_1/2(dD).
Theorem 3. For every function u G Hg(D) there is the weak boundary value tgD (u) in the sense of Definition 1, coinciding with limit boundary value of the harmonic function (u +G(D) (2d*(d u))) of finite order of growth near 3D.
Proof. Let u G Hg(D). Then there is such an s G N that u G H|"(D, || • ||_s), and then du G H(D, A(0,1), || • N_s_i).
First of all, we note that by Lemma 1 and Theorem 1 the operator G(D) d continuously maps the space H(D, A(0'1), || • ||_s_i) to H^(D,^ ||_s) .
Then any element w of the image G(D)d (H(D, A(0,1), || • ||_s_i)) has zero trace t_s(w), and hence it has zero weak boundary value on dD in the sense of Definition 1. Now it is clear that a function u G Hq(D) has weak boundary value tgD(u) in the sense of Definition 1 if and only if the function v = (u + G(D)(2d*(du))) does. Since A = —2d*d we see that, by the construction, v G Hq(D, || • ||_s) satisfies
Av = —2d*du + 2d*(du) =0 in D.
In particular, as v G H(D, || • ||_s), it has a finite order of growth near dD and has weak limit value t(v) = v0 G D'(dD) (see [5]). More precisely, set De = {x G D : p(x) < — e}. Then, for a sufficiently small e > 0, the sets D£ c D c D_£ are domains with smooth boundaries dD±£ of class and vectors ^ev(x) belong to dD±e for every x G dD (here v(x) is the external normal unit vector to the hypersurface dD at the point x). It is said that v = v0 in the sense of weak limit values on r if
< vo, w >=
lim w(y)v(y - ev(y))ds(y) for all w G C~ (r).
dD
Further, as it is explained above,
(xDv,w) = (v, w)D for all w G CTO(C"), (xD(dv), *G) = (dv,G)D for all G G C, A(0'X)).
By the construction, (d © d )dv = 0 in D and the components of dv are harmonic functions with a finite order of growth near dD. Ellipticity of the operators A and d © d [4, Theorem 9.4.7] implies that there is a positive sequence |ev}, converging to zero and such that
{xDv, w) = lim v(x)w(x)dx for all w e CTO(C"),
{xD(dv), *G) = lim i dv A *G for all G e CTO(C", A(01)).
By the Whitney Theorem, any smooth function on D can be extended to a smooth function on Cn. Therefore
(v,w)D = lim v(x)w(x)dx for all w e CTO(D),
(dv, G)D = lim / dv A (*G) for G e C™(D, A(01)). Hence, by Stokes' formula, for all g e CA(01))
we have:
(dv, g) d - (v, d*g)D = lim (dv A (*g) - v*d*g) =
lim / v*g = lim / v((*g)-ds£v )ds£v = (vo,n(g)),
which was to be proved. □
Corollary 1. For every function u e Hw(D, || • ||-s) there is weak limit value V^d(u) on dD in the sense of Definition 1, belonging to H-s-
with
PsD (u) || -s-1/2,dD < C |u|_Sia, (3)
where the constant C does not depend on u.
Proof. We have already proved the existence of weak boundary values in the sense of Definition
1 in the class of distributions for elements of the space Hw (D, || • ||-s). We need to prove that they belong to the corresponding Sobolev spaces on dD. Fix g0 e CTO(dD). Then, as we have seen proving Theorem 2,
|(uo, go)l = |(«o,n(Go))| = (du, Go) - (u, d*Gq)
<
< ldu!-s-1|G0|ff s+1(D,A(»,1)) + ||u||-s||d G0|Hs(D,A(0,1)) < C ||u||-s,d ^0 || H* + i/2(dD) with a constant C not depending on g0. Hence
|u0|H-s-i/2(dD) = SUp —- < G |u|-sd.
0eC~mp(dD) |9|Hs + 1/2(dD)
Thus, we have proved that Vqd(u) e H-s-1/2(dD) and the estimate (3) holds true. □
Corollary 2. The spaces Hd(D, || • |-s) and Hw (D, || • |-s) coincide.
Proof. Because of (2), it is sufficient to prove that
Hf(D, H|-s) C Hd(D, H|-s).
Fix a function u G Hw(D, || • ||-s). By Corollary 1, it has weak boundary value t^D(u) € Let us show that u is a weak solution to the Dirichlet problem (1) with the
data
F = —2d*(du) G H(D, | • | —s—2) and t^(u) G H-s-1/2(dD). Indeed, Corollary 1 implies that, for all v G CTO'0(D), we have:
(u, Av) = —2(u,d*(dv)) = — 2(du,dv) + 2(u0,dnv). (4)
On the other hand, as du G H(D, A(0,1), || • ||-s-1), then there is a sequence {fv} C CTO(D, A(0,1)) converging to du in this space. By Lemma 1, the sequence {d fv} C CTO(D) converges to d (du) in the space H(D, | • |-s-2). That is why
(du, dv) = lim (fv, dv) = lim (d fv, v) = (d (du),v). (5)
v—v—
Combining (4) and (5) we conclude that u is the weak solution to the Dirichlet problem.
Finally, Theorem 1 implies that u is the strong solution to the Dirichlet problem and hence it belongs to Hd(D, |H|-s). □
Corollary 3. Let u G H(D). If
du G H(D, || • ||-s-1), t-s(u) G H-s-1/2(dD)
then u G Hd(D, || • ||-s). Moreover, if du G Hs-1(D), t(u) G Hs-1/2(dD) then u G Hs(D).
As we have seen above, the space H^(D, || • ||-s) is a suitable class for stating the Cauchy problem for the Cauchy-Riemann operator. In order to do this we need to choose proper spaces for the boundary Cauchy data on a surface r C dD. As we are interesting in the case r = dD, we will use one more type of Sobolev spaces: Sobolev spaces on closed sets (see, for instance, [4, §1.1.3]). Namely, let H-s(r) stand for the factor space of H-s(dD) over the subspace of functions vanishing on a neighbourhood of r. Of course, it is not so easy to handle this space, but its every element extends from r up to an element of H-s(dD). Further characteristic of this space may be found in [4, Lemma 12.3.2]). We only note that if r has a C^-smooth boundary (on dD), then
H(r, || • |-s,r) ^ H-s(r) ^ H-s(r).
Corollary 4. For every function u G H-g-(D, || • ||-s) and every r C dD there is boundary value tr(u) in the sense of Definition 1, belonging to H-s-1/2(r).
As dD is compact, U^=1H(dD, || • ||-SjaD) = D'(dD). Set U^=1H-s(r) = D'(r).
Corollary 5. For every u G H^(D) and every r C dD there is boundary value tr(u) in the sense of Definition 1, belonging to D'(r).
3. The Martinelli-Bochner Formula
Let $ be the standard fundamental solution to the Laplace operator in R2n and U be the Martinelli-Bochner kernel (see, for instance, [14]):
$(*-y) = i 1 n =1' U(Z,z)=
(2-2n)|i-y|2n-2' n> 1' ^ (2nv/=1)nj=l > |Z - z|2n z
where a„ is the square of the unit sphere in Rn. We use the same notation $ also for the operator corresponding to the introduced fundamental solution kernel.
For z G dD, denote by Mv0(z) the Martinelli-Bochner transform of a density v0 G V(dD), i.e., the action's result of the distribution vo to the function n(U(-, z)) with respect to the variable Z G dD. As the kernel U is harmonic with respect to z = Z, the transform is harmonic everywhere in Cn outside the support supp vo of the density vo.
Further, for a form f G CTO(D, A(0,1)) denote by TDf the following volume potential:
(Tdf)(z) = ($5*xdf)(z) = i f (Z) AU(Z'Z).
J D
Lemma 2. For every bounded domain Q c Cn with dQ G and dD O Q = 0, the potential Td induces the bounded linear operator
Td,q : H(D, A(0'1), || • H-s-i) - Hd(Q, II • N —)-Moreover, for any form f G H(D, A(0,1), || • H-s-i), the function To^f is harmonic in Q \ D. Proofs. For all g G CTO(D, A(0'1)) and ^ ^ G Cwe have:
(Tdg, = ($d*XD= (xdg, d$xQ^)en' (5Td g,^)o = (d$d*xD g,Xn^)en = (Xd g,d = (Xd g,dTo^)en.
As pseudo-differential operators $xQ and Tq are continuous on the scale of Sobolev spaces Hs(Q), s G Z+ (see, for instance, [4, theorem 2.4.24] then
I|TDg||_s,s,n < C ||g||-s,D for all g G C~(D, A(0,1)), (6)
with a constant C > 0 does not depending on g G CTO(D, A(0,1)).
Now, if f G H(D, A(0'1), || • ||-s-1) then there is a sequence {fv} C CTO(D, A(0>1)) converging to f in H(D, A(0'1), || • ||-s-1). By the inequality (6), the sequence {TDfv} is fundamental in Hg(Q, || • ||-s); we its limit denote by TD,Qf. Clearly this limit does not depend on the choice of the sequence {fv} C CTO(D, A(0,1)), and the estimate (6) guarantees the boundedness of the linear operator TD,of defined in this way. Moreover, since every potential Tdfv is harmonic in Cn \ D then Stiltjes-Vitali Theorem implies that the sequence {Td fv} converges uniformly together with all the derivatives on compacts in Q \ D and its limit is harmonic in Q \ D. □
Lemma 3. For any domain Q c Cn such that dQ G and D C Q, the transform M defined above induces bounded linear operators
Md : H-s-1/2(dD) - Hd(D, || • h-s), Mq : H-s-1/2(dD) - H(Q, || • |-s). Besides, for every function v G H^(D) the Martinelli-Bochner formulae hold true:
Md (t(v)) + Td,d dv = v, Mn(t(v)) + Td,q5v = xd v. (7)
Proof. As we have seen before, for any v0 G H s 1/2(dD), the Poisson integral P(D)(v0) G || • ||_s), satisfies t(P(D)(v0)) = v0 (see Theorem 1). We set
Md = P(D) - dP(D) : H_s_1/2(dD) ^ H^(D, || • ||_s),
= xdP(D) - TD,QdP(D) : h_s_1/2(dD) ^ H(fi, || • ||_s).
By Lemma 2, Theorem 1 and the continuity of the operator xd, the operators Md, Mq, defined in this way, are continuous. Let us show that Md and Mq coincide with the transform M on CTO(dD). Indeed, if v0 G CTO(dD) then P(D)v0 G (D) and
Mv0 = M (t(P (D)v0)).
Then, by Martinelli-Bochner formula for smooth functions (see [14]), we have:
XdP (D)V0 = Mv0 + Td dP (D)V0,
which was to be proved.
As CTO(dD) is dense in H_s_1/2(dD) then M continuously extends from CTO(dD) to the space H_s_1/2(dD) up to the operators MD, Mq defined above. Moreover, it is easy to see that the functions MdV0, MqV0 coincide with the transform Mv0 in D and fi \ supp V0 respectively. Indeed, let ^ G Cg°(fi\suppv0). We approximate the distribution v0 G H_s_1/2(dD) by smooth functions v0v) with supports in a neighbourhood of supp V0 in such a way that supp v(v) Hsupp ^ = 0. Then, by easy computations, limv—TO(M(v0 — v(v)), ^)q = 0, and hence
(Mv0,^)q = lim (Mv0v),^)q = (MqV0,^)q
because Mq is continuous. Similarly, if supp ^ C D then
(Mv0, = lim (Mv(v), = (Mdv0, ¿)d.
v—
Let now v G Hg(D). Then v G H^(D, || • ||_s) with a number s and there is a sequence {vv} C (D) converging to v in H^(D, || • ||_s). Therefore the Martinelli-Bochner formula for smooth functions implies
M (t(vv)) + Td dvv = Xd vv. (8)
Passing to the limit with respect to v ^ to in the spaces H^(D, || • ||_s) and H(fi, || • ||_s) in (8) we obtain (7) because of Lemma 2 and the continuity of Md, Mq, proved above. □
Remark 1. Let f G H(D, A(0,1), || • ||_s_1). If fi, fi1 and fi2 are bounded domains in Cn containing D, having smooth boundaries and such that fip C fi, p=1,2, then functions Tp,Qf G H(fi, || • ||_s), TD Qi\Df G H(fi1 \ D, || • ||_s) and TD Q2\Df G H(fi2 \ D, || • ||_s) are harmonic in fi \ D, fi1 \ D and fi2 \ D respectively. As each of them is constructed as a limit of the same sequence of functions, they coincide in (fi1 H fi2) \ D. Actually, as fi, fi1, and fi2 are arbitrary, all these limits harmonically extend their to Cn \ D and all these extentions coincide, too. Since the operators Mq, Mq1 , and Mq2 are constructed with the use of the operators Td,q Tp q1 \p and Td q2\d respectively, this remark is valid for potentials of the type Mqv0 with v0 G H_s_1/2(dD). This allows us to consider functions Tpf and Mv0 harmonic in Cn \ D, having finite orders of growth near dD (outside D!) and such that Tpf = Td,q G H(fi, || • ||_s), Mv0 G H(fi, || • ||_s) for any domain fi D D.
4. The Cauchy problem
Set H(D, A(0'1)) = U^=1H(D, A(0'1), || • ||-s).
Problem 1. Given f G H(D, A(0'1)), u° G D'(r), find u G Hd(D) with
(u,3*>) = (f,^) — (u°,n(^)) for all $ G Cc~mp(D U r, A(0,1)). (9)
As we have seen before, a function u G H(D) is a solution to the Cauchy Problem 1 if and only if du = f in the sense of distributions in D and tr(u) = u° on r. Moreover, Corollary 3 means that, for sufficiently smooth data f and u°, Problem 1 becomes the classical Cauchy problem for the Cauchy-Riemann operator. Besides, we easily get the Uniqueness Theorem for Problem 1.
Theorem 4. Problem 1 has no more than one sulution.
Proof. Indeed, if u° =0, f = 0 then corollary 1 implies that a solution to 1 is a holomorphic Sobolev function in D having zero limit values on r. As it has a finite order of growth near r (see, for instance, [4, theorem 9.4.8]), we conclude that u = 0 in D because of [4, theorem 10.3.5]). □ As d =0, then df = 0 in D if the Cauchy problem is solvable. Besides, if n > 1, the Cauchy-Riemann operator d induces the tangential operator dT on dD (see, for instance, [14, §11]). This means that the Cauchy data u° and f have to be coherent. Namely, taking in (9) as ^ a differential form with £ G C~mp(D U r, A(0'2)), we see that
(u°,n(d*£)) = (f,d*£) for all £ G C£mp(D u r, A(0,2)), (10)
if Problem 1 is solvable. For f = 0 it means that u° is a CR-function on r.
We want to get a solvability criterion for Problem 1. With this aim, let us choose a domain D+ in such a way that the set fi = D U r U D+ would be a bounded domain with a smooth boundary; it is convenient to set D- = D. For a function v G C(D+ U D-) we denote by v± its restriction to D±.
For u° G H-s-1/2(r) we fix an element u° G H-s-1/2(dD) of its equivalence class. As we have explained in Remark 1, the distribution F = Mu° + Td f is harmonic outside D and belongs to H(fi, || • |-s).
Theorem 5. The Cauchy problem 1 is solvable if and only if condition (10) holds and there is a harmonic in fi function F of finite order of growth near dfi coinciding with F in D+.
Proof. Let Problem 1 be solvable and u be its solution. The necessity of condition (10) is already proved. Set
F = F — xd u. (11)
By the definition, F is harmonic in D+. Then, by Martinelli-Bochner formula (7), Lemma 3 and Remark 1, we have:
F = MQu° + TD,nf — Xd u = MQ(u° — t(u)).
As u° = t(u) on r then Mq(u° — t(u)) = M(u° — t(u)) is harmonic in fi \ r as parameter depending distribution. Hence F has the same property. It has a finite order of growth near dD because of the structure of the kernel U(Z, z) and the compactness of dD: the kernel U(Z, z) is harmonic outside the diagonal {Z = z} and it grows as |z — Z|1-2n near the diagonal; besides,
the compactness of dD implies that the distribution (Uo — t(u)) has a finite order of singularity on dD.
Back, let there be a harmonic in Q function F of finite order of growth near dQ coinciding with F in D+. Set
u = TDd f + MD Uo — F(12)
As f e H(D, A(0,x)) and u0 e V(r), Lemmata 2 and 3 imply that TD Df + MDU0 e Hd(D). Moreover, since D C Q then F is harmonic in D and has a finite order of growth near dD. Therefore t(F) e V'(dD) (see [4, Theorem 9.3.16]). Hence F- = P(D)(t(F)) and F- e Hd(D) because of Theorem 1. Thus, by the construction, the function u belongs to H^(D). According to Corollary 1, there is boundary value t(u) on r in the space of distributions which can be calculated by Definition 1.
Let take sequences {fv} C CTO(D, A(0,1)) and {u0v)} C CTO(dD) approximating functions f e H(D, A(0,1), || • ||-s-1) and u0 e V'(dD) respectively in these spaces. Then, because TDfv e C2(Q) (see [4, Theorem 2.4.24] ) and because of the Jump Theorem for Martinelli-Bochner integral (see, for instance, [14, Theorem 2.3 or Theorem 3.1]),
(Tdfv)- — (Tdfv)+ =0 on r, (Mu0v))- — (Mu0v))+ = u0v) on r.
Now, using Stokes' formula and Lemmata 1, 3, 2 and Remark 1, we conclude that for all g e
C~mp(D U r, A(0,1)) we have: _
(du,g)D — (u, 5 g)D =
= lim (d(Tdfv + M-U0v) — F),g)D — (Tdfv + (Mu0v))- — F, d*g)D =
V—
= lim (U0 + (Tdfk)+ + (Mu0k))+ — F), n(g)) = (u0, n(g)),
k—
i.e., u = U0 on r.
In order to finish the proof we need to convince ourselves that du = f in D. To this end consider the form P = xd(f — du) belonging to H(Q, A(0,1)). It is clear that C~mp(Q, A(0,2)) C C~mp(D U r, A(0,2)). Then, by (10) and Definition 1 we have for all p e C~mp(Q, A(0,2)):
(P,d*p)fi = (f,d*P)D — (du,d* P)d = (u0,n(*d*p)) — (u0,n(*d*p)) — (u,d*d*p )d = 0,
because d2 = 0. Hence dP = 0 in Q.
On the other side, by Definition 1, for all v e C^mp(Q) we have:
(P,5v)n = (f,dv)D — (du,dv)D = (f, 5v)d — (u, 2av)d — (u0, 5„v). (13)
Since F is harmonic in Q and coincides with F in D+, then (see Remark 1)
(F, 2aV)d = —(F, 2aV)d+ = —(TD,f + MqU0, 2aV)d+ . (14)
Besides, as $ is bilateral fundamental solution to the Laplace operator in Cn then 1ATD f = d xdf. Hence again taking a sequence {fv} C CTO(D, A(0,1)), approximating f e H(D, A(0,1), || • ||-s-1) in this space, and using Lemmata 1, 2 and Remark 1, we see that
(TD,f, 2av)d+ + (Td,d f, 1av)d = v—m ((Td fv, 1av)d+ + (Td fv, 2av)d) =
1 —* — -lim (tdfv, -Av)Q = lim (d xdfv, v)n = lim (fv, dv)D = (f, dv)D. (15)
v—2 v—v—
Combining (13), (14), (15), we conclude that
(P, 3v)n = (MUo, 2av)d+ + (MUo, 1Av)d - (uo, dnv). Finally, by the Stokes' formula, we have in the sense of weak limit values on r: (MUo, 2av)d+ + ((MUo)-, 1av)d = ((MUo)- - (MUo)+, 5„v)aD + + (5„((MUo)- - (MUo)+),v) = (uo,dnv),
because in the sense of weak limit values on r there are the jumps on r:
(MUo)- - (MUo)+ = uo, 5„((MUo)- - (MUo)+) = 0,
see [18] and [9] respectively.
—* /— —* \ Thus, d P = 0 in Q, and hence (d © d )P = 0 in Q. As every such a form has harmonic
coefficients in Q, the uniqueness theorem for harmonic functions yields P = 0 in Q. In particular,
f = du in D. □
Corollary 6. Let f e H(D, A(o'1), || • ||-s-i), uo e H-s-1/2(r). The Cauchy Problem 1 is solvable in Hg-(D, || • ||-s) if and only if (10) holds and there is harmonic in Q function F e H(Q, || • ||-s) coinciding with F in D+.
Proof. Indeed, if the Cauchy Problem 1 is solvable in H^(D, || • ||-s) then it is solvable and F = mq(uo - uo) (see the proof of Theorem 5). Hence, according to Lemma 3, function F belongs to H(Q, || • ||-s).
Back, if F e H(Q, || • ||-s) is harmonic and coincides with F in D+ then the Cauchy Problem 1 is solvable. Therefore, its unique solution u is given by (12) and F is given by (11). In particular, XDu = (F-F) e H(Q, ||-||_s). Take v e CTO(D). Then there is V e Cwith ||V||s,n = ||v|s,D and v = V in D. By the definition,
|(u,v)d| = |(xdu, V)n| < ||xdu|_s,n|v|SjD,
i.e., u e H(D, || • ||_s). Finally, as du = f e H(D, A(o'1), || • ||_s-1) we see that u e Hd(D, || • ||_s). □
At the end, we note that Corollary 6 allows us to use the bases with double orthogonality property in order to construct Carleman's formulae for Cauchy Problem 1 in the same way as in [10] or [11].
The authors were supported in part by grant 2427.2008.1 of Leading Scientific Schools, by RFBR, grant 08-01-90250, and by Siberian Federal University.
References
[1] S.L.Sobolev, Equations of Mathematical Physics, Fizmatgiz, Moscow, 1954 (in Russian).
[2] O.A.Ladyzhenskaya, Boundary Value Problems for Equations of Mathematical Physics, Nauka, Moscow, 1973 (in Russian).
[3] M.Schechter,, Negative norms and boundary problems, Ann. Math., 72(1960), no. 3, 581593.
[4] N.N.Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Berlin, Akademie Verlag, 1995.
[5] E.J.Straube, Harmonic and analytic functions admitting a distribution boundary value, Ann. Sc. Norm. Super. Pisa, Cl. Sci., 11(1984), no. 4, 559-591.
[6] J.Hadamard, Le probleme de Cauchy et les equations aux derivees partielles lineares hyperboliques, Paris, Gauthier-Villars, 1932.
[7] L.A.Aizenberg, Carleman's Formulas in Complex Analysis. First Applications, Kluwer, Dordreht-Boston-London, 1993.
[8] T.Carleman, Les Fonctions Quasianalytiques, Gauthier-Villars, Paris, 1926.
[9] L.A.Aizenberg, A.M.Kytmanov, On possibility of holomorphic extension to a domain of functions, given on a part of its boundary, Mat. USSR Sb., 72(1992), no. 2, 467-483.
10] A.A.Shlapunov, N.N.Tarkhanov, On the Cauchy problem for holomorphic functions of the Lebesque class L2 in a domain, Sib. Math. J, 33(1992), no. 5, 914-922.
11] D.P.Fedchenko, A.A.Shlapunov, On the Cauchy problem for multi-dimensional Cauchy-Riemann operator in Lebesque L2 in a domain, Mat. Sb., 199(2008), no. 11, 141-160 (in Russian).
12] L.Hormander, Notions of Convexity, Berlin, Birkhauser Verlag, 1994.
13] G.M.Khenkin, Integral Representation Method in Complex Analysis, Results of Science and Technics. Modern problems of Mathematics (fundamental directions), VINITI, Moscow, 7(1985), 23-124.
14] A.M.Kytmanov, The Bochner-Martinelli Integral and its Applications, Birkhauser Verlag, Basael-Boston-Berlin, 1995.
15] Yu.V.Egorov, M.A.Shubin, Linear Partial Differential equations. Foundations of Classical Theory, Results of Science and Technics, Modern problems of Mathtmatics, Fundamental directions, Moscow, VINITI, 30(1988).
16] R.A.Adams, Sobolev Spaces, Academic Press. Inc., Boston et al., 1978.
17] A.A.Shlapunov, N.N.Tarkhanov, Duality by reproducing kernels, International Jour. of Math. and Math. Sc., 2003(2003), no. 6, 327-395.
[18] E.M.Chirka, Analytic representation of CR-functions, Mat.Sb., 98(1975), 591-623.
