On the spectral properties of a non-coercive mixed problem associated with ∂-operator Текст научной статьи по специальности «Математика»

УДК 517.95, 517.5

On the Spectral Properties of a Non-coercive Mixed Problem Associated with d-operator

Alexander N. Polkovnikov* Alexander A. Shlapunov^

Institute of Mathematics and Computer Science, Siberian Federal University, Svobodny, 79, Krasnoyarsk, 660041,

Russia

Received 10.01.2013, received in revised form 10.01.2013, accepted 20.01.2013 We consider a non-coercive Sturm-Liouville boundary value problem in a bounded domain D of the complex space Cn for the perturbed Laplace operator. More precisely, the boundary conditions are of Robin type on dD while the first order term of the boundary operator is the complex normal derivative. We prove that the problem is Fredholm one in proper spaces for which an Embedding Theorem is obtained; the theorem gives a correlation with the Sobolev-Slobodetskii spaces. Then, applying the method of weak perturbations of compact self-adjoint operators, we show the completeness of the root functions related to the boundary value problem in the Lebesgue space. For the ball, we present the corresponding eigenvectors as the product of the Bessel functions and the spherical harmonics.

Keywords: Sturm-Liouville problem, non-coercive problems, the multidimensional Cauchy-Riemann operator, root functions.

Introduction

Non-coercive boundary value problems for elliptic differential operators attract attention of mathematicians since the middle of XX-th century (see, for instance, [1,2]). One of the typical problems of this type is the famous d-Neumann problem for the Dolbeault complex (see [3]). The investigation of the problem resulted in the discovery of the subellipticity phenomenon which greatly influenced to the development of the Theory of Partial Differential Equations (cf. [4]).

As it is known (under reasonable assumptions) the Spectral Theory gives both the solvability conditions and the formulae for the exact and the approximate solutions to boundary value problems via expansions over (generalized) eigenfunctions related to the corresponding linear operators (see, for instance, [5] and elsewhere). This is well understood for the coercive boundary value problems in smooth domains for both self-adjoint and non-selfadjoint cases (see [6-8]). For the Spectral Theory related to the elliptic problems in Lipschitz domains we refer to the survey [9] and its bibliography (see also [10,11] for the domains with the conic and edge singularities). Recently Agranovich [12] noted that the use of the negative Sobolev spaces gives an additional advantage proving the completeness of the root functions related to the coercive boundary value problems in non-smooth domains.

*[email protected] [email protected] © Siberian Federal University. All rights reserved

The aim of the present paper is to extend the results to the non-coercive boundary value problem for the weakly perturbed Laplace operator in the complex space Cn (= R2n). First, using the standard methods of the Functional Analysis (see [13,14] and elsewhere) we prove that the problem is a Fredholm one in the proper Sobolev type spaces. Then, applying the method of weak perturbations of compact self-adjoint operators (see [6]), we prove the completeness of the generalized eigenvectors related to the boundary value problem in the Lebesgue space. Examples of the eigenfunctions related to the problem in the ball are constructed.

1. The mixed problem

Let D be a bounded domain in the complex space Cn = R2n with a Lipschitz boundary, i.e., the surface dD is locally the graph of a Lipschitz function. In particular, the boundary dD possesses a tangent hyperplane almost everywhere.

Let the complex structure in Cn be given by Zj = Xj + a/—1xn+j with j = 1,..., n and d stand for the Cauchy-Riemann operator corresponding to this structure, i.e., the column of n complex

d 1 / d _ d \ - -

derivatives —- = - ( —--h \f—1—-). The formal adjoint d* of d with respect to the usual

ozj 2 V dxj dxn+j J

1 / d ,_ d \

Hermitian structure in the space L2(Cn) is the line of n operators —-{7:--\f—-I =:

2 V dxj oxn+jJ

d z z

— ——. Then an easy computation shows that d*d just amounts to the —1/4 multiple of the

dzj

2n / d \ 2

in R2n.

{ d \

Laplace operator A2n = y d—j J

j=i _ We consider complex-valued functions defined in the domain D and its closure D. We write Lq(D) for the Lebesgue space, i.e. the set of all measurable functions u in D, such that the integral of |u|q over D is finite. We also write Hs(D), s e N, for the corresponding Sobolev space of functions with all the weak derivatives up to order s belonging to L2 (D). For non-negative non-integer s we denote by Hs(D) the Sobolev-Slobodetskii space, see, for instance, [14].

Consider the second order linear partial differential operator A in the domain D associated with the Cauchy-Riemann operator:

Au = - A2„ + aj (z) — + ao (z)u,

the coefficients aj and a0 being of class Lœ(D). Consider also a first order boundary operator

B = bi(z)dv + bo(z)

- n d where dv = ^^ (vj(z) — \f—îvj+n(z^) —— is the complex normal derivative and v(z) =

- -1 dzj

j=i

(v1(z),... v2n(z)) is the unit normal vector to dD at the point z (cf. with the usual normal d 2r\ d

derivative — = V^ v,-(z) —— ). The coefficients b0(z) and b1(z) are assumed to be bounded dv ^ d—,

j=i j

measurable functions on dD satisfying |b0|2 + |b1|2 = 0. We allow the function b1(z) to vanish on an open connected subset S of dD with piecewise smooth boundary dS.

n

Consider the following boundary value problem with the Robin-type condition on the surface dD. Given a distribution f in D, find a distribution u in D which satisfies in a proper sense

Au = f in D, ...

Bu = 0 on dD. ( )

Note that in general the Shapiro-Lopatinskii condition is violated on the smooth part of dD \ S for the pair (A, B) because if S = 0, aj = 0 for all j = 0,..., n and b0 = 0 problem (1) is a version of the famous d-Neumann problem (cf. [3]).

Denote by HX(D, S) the subspace of H1 (D) consisting of those functions whose restriction to the boundary vanishes on S. This space is Hilbert under the induced norm. It is easily seen that smooth functions on D vanishing in a neighborhood of S are dense in H 1(D, S); then the space H1 (D, dD) is usually denoted Hq (D). Since on S the boundary operator reduces to B = b0 and b0(z) = 0 for z G S, the functions of H1 (D) satisfying Bu = 0 on dD belong to H1 (D, S).

As we want to study perturbations of self-adjoint operators we split both a0 and b0 into two parts a0 = a0,0 + Ja0, b0 = b0,0 + Jb0, where a0,0 is a non-negative bounded function in D and b0,0 a bounded function on dD satisfying b0,0/b1 > 0. Consider now the Hermitian form

(u,v)+ =4^ ( ^J + (aoiou,v)i2(D) +4(6o,o61 Vv)L2(dD\s)-

j=1 V °zj/L2(D)

on the space HS). It follows from the Uniqueness Theorem for holomorphic functions that the form defines a scalar product on HS) if one of the following conditions holds true:

1) the open set S c dD is not empty;

2) a0,0 > c0 in U with some constant c0 > 0 on an open non-empty set U C D;

3) b0,0 > ci in V with some constant c0 > 1 on an open non-empty set V C dD \ S.

Then we denote by H+ (D) the completion of H1 (D, S) with respect to the norm || • || + coherent with the scalar product (•, •)+ .

From now on we assume that the space H +(D) is continuously embedded into the Lebesgue space L2(D), i.e.,

||u||l2(d) < c for all u G H+(D), (2)

where c is a constant independent of u. It is true under rather weak assumptions (see Theorem 1 below). Now we need the continuous inclusion

i: H+(D) ^ L2(D), (3)

to specify the dual space of H +(D) via the pairing in L2(D). More precisely, let H-(D) be the completion of H 1(D,S) with respect to the negative norm (cf. [15])

II || |(v,u)L2(D)|

||u||i = suP -m-M-•

veH\D,s) ||v|| +

v=0

Lemma 1. The space L2(D) is continuously embedded into H-(D). If inclusion (3) is compact then the space L2(D) is compactly embedded into H-(D).

Proof. By definition and estimate (2) we get

I, || . ||u||L2(D) ||v||L2(D) ^ ,, ,,

||u|- < SUp -—- < c ||u|L2(D)

v£H1(D,S) ||v| +

v=0

for all u e L2(D), i.e., the space L2(D) is continuously embedded into H-(D) indeed.

Suppose (3) is compact. Then the Hilbert space adjoint i* : L2(D) ^ H + (D) is compact, too. As HX(D,S) is dense in H+(D) and the norm || • || + majorizes || • \\l2(d) we conclude that

1 (i(v), u)L2(D)1 |(v, i*(u))+| l|u||- = sup -rr-r—^ = sup -rr-r-= ||i (u)|| + (4)

veH1(D,S) ||v|| + v£H+(D)

v=0 v = 0

for all u e L2(D). Therefore, any weakly convergent sequence in L2(D) converges in H (D), which shows the second part of the lemma. □

Since CC^mp(D) is dense in L2(D) and the norm || • ||L2(D) majorizes the norm || • ||-, we conclude that CCOOmp(D) is dense in H-(D), too.

Lemma 2. The Banach space H-(D) is topologically isomorphic to the dual space H + (D)' and the isomorphism is defined by the sesquilinear pairing

(v,u) = lim (v, u„)l2(d) (5)

v—

for u e H-(D) and v e H+(D) where {uv} is any sequence in H 1(D, S) converging to u. Proof. See, for instance, [16, Theorem 1.4.28]. □

Note that H + (D) is reflexive, since it is a Hilbert space. Hence it follows that (H-(D))' = H+(D), i.e., the spaces H+ (D) and H-(D) are dual to each other with respect to (5).

From now on the Sobolev space H-s(D), s > 0, stands for the dual to Hs(D) via the pairing induced by the scalar product (•,-)L2(D) as in Lemma 2 above. Similarly, let H-s(D), s > 0, stands for the dual to H0s(D). Obviously H-s (D) c H-s(D). We also denote by hs(D) the space of all the harmonic functions in the domain D belonging to the Sobolev space Hs(D).

Theorem 1. Let dD be a Lipschitz surface. Then

1) the space H 1(D,S) is continuously embedded into H+(D) if 60,0 6-1 e (dD \ S);

2) the elements of H+(D) belong to H1:Oc(D U S, S); in particular, if S = dD then the space H +(D) is continuously embedded into Hq(D);

3) the space H +(D) is continuously embedded into L2(D) if

a0,0 ^ c0 in D with some constant c0 > 0; (6)

4) the space H +(D) is continuously embedded into h1/2-e(D) © Hq(D) with any e > 0 if

60,06-1 ^ ci in dD \ S with some constant c: > 0. (7)

Moreover, if dD e C2 then, under (7), the space H+(D) is continuously embedded into the space h1/2(D) © Hq(D). In particular, estimate (7) implies that i is compact.

Proof. If 60,06-1 e LTO(dD \ S) then, according to the Trace Theorem for the Sobolev spaces, we obtain

||u||+ < Wa/^qo\l~(d)\u\l2(d) + ^^V/60,06-1|L~(dD\S)|u|L2(dD) + ||u||H 1(D) < c ||u|H 1(D)

for all u e H1 (D, S) with some positive constant c independent on u. This proves 1).

The statement 2) follows from the fact that the Dirichlet problem for the Helmholtz operator (a0,0 — A2n) is coercive.

+

Now using the definition of the norm || • || + we see that

IMI+ > > a/^0 ||u||2(D)

i.e. estimate (2) holds true and the embedding i is continuous under estimate (6).

Further, let (7) holds. Then the norm || • ||+ is not weaker than the norm || • ||h on H 1(D,S) defined by

\ i/2

+ 4|u||2(dD\s)) ,u G H 1(D, S).

L2(D) 7

||u||h = (4

n n

du

j=i

dzj

Fix a number e > 0. Let us show that the norm || • ||h is not weaker than the norm || • ||Hi/2-e(D) on H 1(D, S). Indeed, let ^2n denote the two-sided fundamental solution of the convolution type for the Laplace operator A2n in R2n. Then the volume potential

$v(x) = ^2n(x - y)v(y)dx,v G L2(D), (8)

D

induces the bounded linear operator $ : L2(D) ^ H2(X) for any bounded domain X containing D. It is clear that any element u G H-s(D) extends up to an element U G H-s(R2n) via

(U,v)R2n = (u,v>D for all v G Hs(R2n);

here (•, •)D is the pairing on H x H' for a space H of distributions over D. It is natural to denote it by xDu. Thus, the defined in this way linear operator xD : H-s(D) ^ H-s(R2n), s G R+, is obviously bounded. The distribution xDu is supported in D, so we actually may reduce our consideration to a smooth closed manifold. This allows us to conclude that the volume potential (8) induces the bounded linear operator

$ oxD : He-1/2(D) ^ He+3/2(X), 0 < e < 1/2,

for any bounded domain X containing D (see, [17]). Hence, the operators

d —

— o $ o XD : He-1/2(D) ^ He+1/2(X) and dv o $ o xd : He-1/2(D) ^ He(<9D) azj

are bounded, too, if 0 < e < 1/2 (the last one is bounded because of the Trace Theorem for the Sobolev spaces). For e = 0 the arguments fail because the elements of the space H1/2(X) may have no traces on dD c X.

Now the integration by parts with u G H 1(D, S) and v G L2(D) yields

n (d$v du \

(v,u)L2(D) = (A2n$v,u)L2(D) = 4^ ( , d)Z- J + 4(dv$v,u)L2(dD\S). (9)

j=1\ dZj dzj / L2 (D)

Take a sequence {vk} c CTO(D) converging to v in the space He-1/2(D), 0 < e < 1/2. As the

d

space Hs(D) is reflexive for each s, using (9) and the continuity of the operators -7— o G o xd,

dv o G o xd above, we obtain for u G H 1(D, S):

,, l(v,u)| |(vfc,u)l2(d)| ||H1/2-£(D) = suP ]T-n- = suP -n—n- =

v£He-1/2(D) ||v||H-1/2(D) v£He-1/2(D) ||v||H-1/2(D)

v=0 v=0

IEn=1( afr o $ o XDv, jf-)l2(d) + (dv o $ o xdv, u)l2(sd\s)| 4 sup -^-- <

v£He-1/2(D) ||v||He-1/2(D)

v=0

u

< ME

— o $ o xd dzi

du

dzi

+ ||dV o $ o XD || ||uyL2(SD\sJ •

L2(D)

with a constant c > 0 being independent on u. Thus, there are constant CQ > 0, C2 > 0 such that

i|u||h 1/2-e(D) < Ci ||u||fc < C2 ||u|| + for all u G H Q(D,S).

This proves the continuous embedding H + (D) ^ H1/2-e(D) with any e > 0.

Further, let G and P stand for the Green function and the Poisson integral of the Dirichlet Problem for the Laplace operator A2n in D respectively. Then they induce the bounded operators (cf. [13,15])

Gi : H-1(D) ^ H0(D), Pi : H 1/2(D) ^ h1(D). As the operator A2n extends to the continuous linear operator A2n : H 1(D) ^ H-1(D) via

(A2nu,v> = 4(du, dv)l2(d), u G H 1(D),v G H(D),

we see that u = PQu + G1 A2nu for each u G H 1(D). Hence, for u,v G H 1(D,S), we obtain:

(u, v)h = (Pu,Pv)L2(dD\s) + (dGA2„u, (dGA2„v)L2(D). (10)

In particular,

I|u|| + > ||u|h = |Piu||2(dD\s) + ||5GiA2„u||!2(D) for all u G H 1(D,S).

On the other hand, the Garding inequality yields

||v||Hi(D) < |dv|L2(D) for all u G Hi^ (11)

Therefore, using (10) and (11) we conclude that any sequence {uk} c H 1(D,S) converging to u G H +(D) in the space H +(D) can be presented as

uv = Piufc + GiA2„ufc

where the sequence {G1A2nuk} converges in HQ(D) c H 1(D,S) to an element wQ. Now the already proved part of the theorem yields that {PQuk} converges to an element w2 in H 1/2-e(D). According to the Stiltjes—Vitali Theorem the element w2 is harmonic in D. Hence

u = wi + W2, A2„u = A2„w, u = Pu + Gi A2„u; (12)

here Pu is the Poisson integral of the trace u|dD G L2(dD) related to u G H+(D). This proves the continuous embedding H +(D) h1/2-e(D) © HQ (D).

Finally, if dD G C2 then we may use the regularity of the solutions to the Dirichlet Problem for the Laplace operator in D. More precisely, in this case we have the bounded linear operators

G2 : L2(D) ^ H2(D), dv o G2 : L2(D) ^ H 1/2(dD), P2 : H3/2(dD) ^ H2(D);

for a Lipschitz boundary these may be not true in general.

To finish the proof we will show that the Poisson integral P induces the bounded linear operator P1/2 : L2(dD) ^ H1/2 (D). With this aim, for u 0 G H 1/2 (dD) take a sequence {u0k} c H 1/2(dD) converging to u0 in H-1/2(dD). Then, integrating by parts we obtain:

IID ,, |(v,P1u0fc )L2 (D)1 |(A2nG2v,P1u0fc )L2(D)1 ^ ||Piu0fc^l2(d) = sup -—-= sup -—- <

v£L2(D) ||v||L2(D) v£L2(D) ||v||L2(D)

v=0 v=0

n

|(d v G2v,u0k )l2(8D)1 . ||d v G2v||H!/2(dD)||u0k IIH-!/2(dD) .

< sup -—-^-< sup -^^-^-<

v£L2(D) ||v||L2(D) v£L2(D) ||v||L2(D)

v=0 v=0

< |dvG2IIIK k nH 1/2(dD) •

Hence the sequence {Piu0k} converges in L2(D) and the Poisson integral P induces the bounded linear operator P0 : H-1/2(dD) ^ L2(D). Now we may use the interpolations arguments (see [14], [18]). Indeed, by the interpolation, the Poisson integral P induces the bounded linear operators

Pe : [H-1/2(dD), H 1/2(SD)]0 ^ [L2(D), H 1(D)]0, 0 < 0 < 1, where [H0,H1]g means the interpolation between the pair H0 and H1 of Hilbert spaces. But

[L2(D), H1 (D)]e = H0 (D), [H-1/2(dD), H1/2 (dD)]e = H1/2-0 (dD),

see, for instance, [14, Ch. I, Theorems 9.6 and 12.5]. Therefore, choosing 0 = 1/2 we conclude that the Poisson integral P induces the bounded linear operator P1/2 : L2(dD) ^ H1/2 (D). Hence (12) implies the continuous embedding H+(D) ^ h1/2(D) © Hq(D) if dD G C2. □

We emphasize that the space H + (D) is not continuously embedded into H 1(D) unless S = dD, because the Shapiro-Lopatinskii condition is violated on the smooth part of dD\S. Actually the embeddings described in Theorem 1 are sharp at least for the ball (see Examples 1 and 2 below).

Further, on integrating by parts we see that

(Au,v)L(D) = 4jC (Jj, Jj)^ +4 (6-1&0u'vWd\S) + (j=> Jj + «0u>v)L2(D)

for all u G H2(D) and v G H 1(D) satisfying the boundary condition of (1). Suppose that

|Jb0| < c1| b0,0| on dD \ S with a positive constant c1. (13)

Then, if

|Ja0| < c2| a0j0| on D with a positive constant c2. (14)

or (7) is fulfilled, we have

n du

(6-1 ¿&0u,v) L2(dD\S) + d" + ^a0 u,v) l2(D) < C ||u|| + ||v|| + (15)

for all u,v G H 1(D,S), where c is a positive constant independent of u and v. Therefore, in these cases for each fixed u G H +(D), the sesquilinear form

Q(u, v) = 4 (J|, lj)L2(D) + 4 (6-160u' V)L2(dD\S) + ( X> dZj + ^ v) L2(D)

determines a continuous linear functional f on H+(D) by f (v) := Q(u, v) for v G H+(D). By Lemma 2, there is a unique element in H-(D), which we denote by Lu, such that

f (v) = (v,Lu)

for all v G H+(D). We have thus defined a linear operator L : H+(D) ^ HFrom (15) it follows that L is bounded. The bounded linear operator L0 : H +(D) ^ H-(D) defined in the same way via the sesquilinear form (•, •)+, i.e.,

(v, u) + = (v, L0u) (16)

for all u, v G H+(D), corresponds to the case a,j = 0 for all j = 1,..., n, a0 = a0j0, and b0 = b0,0.

We are thus lead to a weak formulation of problem (1). Given f G H-(D), find u G H+(D), such that

Q(u,v) = (v,f> for all v G H+(D). (17)

Now one can handle problem (17) by standard techniques of functional analysis, see for instance [13, Ch. 3, §§ 4-6]) for the coercive case. As the properties of the Dirichlet problem are well known, we will be concentrated on the study of the mixed problem under condition (6) or condition (7) of Theorem 1 in the case S = dD.

Lemma 3. Assume that aj = 0 for all j = 1,...,n, £a0 = 0, and £60 = 0. If (6) or (7) hold then for each f G H-(D) there is a unique solution u G H + (D) to problem (17), i.e., the operator L0 : H + (D) ^ H-(D) is continuously invertible. Moreover, the norms of both L0 and its inverse L-1 are equal to 1.

Proof. Under the hypotheses of the lemma, (17) is just a weak formulation of problem (1) with A and B replaced by A0 = —A2n + a0 0, B0 = b1dv + b0 0, respectively. The corresponding bounded operator in Hilbert spaces just amounts to L0 : H+(D) ^ H-(D) defined by (16). Its norm equals 1, for, by Lemma 2, we get

||L0u||- = sup = sup = ||u||+ (18)

v£H+(D) ||v||+ v£H+(D) ||v|| +

v=0 v=0

whenever u G H + (D).

The existence and uniqueness of solutions to problem (17) follows immediately from the Riesz theorem on the general form of continuous linear functionals on Hilbert spaces. From (18) we conclude that L0 is actually an isometry of H-(D) onto H+(D), as desired. □

Corollary 1. Let estimates (7), (13) be fulfilled and the constant c in (13) satisfy 0 < c1 < 1. Then problem (17) is Fredholm.

Proof. If aj = 0 for all 1 < j < n and ¿a0 = 0 then, under the hypothesis of the corollary, estimate (15) holds with 0 < c < 1. In this case the operator L1 : H+(D) ^ H-(D) corresponding to problem (17) is easily seen to differ from L0 by a bounded operator ¿L1 : H+(D) ^ H-(D) whose norm does not exceed 0 < c < 1. As L0 is invertible according to Lemma 3 and the inverse operator L-1 has norm 1, a familiar argument shows that L1 is invertible, too.

n du

On the other hand, as ¿a0 and aj, 1 < j < n belong to (D), the term ¿a0 aj (z)-^-

j=1 dzj

induces the bounded linear operator £L2 : H +(D) ^ L2(D). Then Theorem 1 and Lemma 1 imply that the operator ¿L2 : H +(D) ^ H-(D) is compact. This means that the operator L2 : H+(D) ^ H-(D) corresponding to problem (17) differs from the invertible operator L1 by the compact operator ¿L2 : H+(D) ^ H-(D), i.e. L2 is a Fredholm operator. □

2. Spectral properties of the problem

As estimate (6) does not provide the compactness of the embedding i, we are to study the spectral properties of problem (17) under condition (7) of Theorem 1 in the case S = dD. With this aim we consider the sesquilinear form on H-(D) given by

(u, v)H-(D) := (L-1 u, v) for u,v e H-(D).

Since

(L-1u,v) = (L-1u, L0L-1v) = (L-1u,L-1v)+ for all u, v e H-(D), (19)

the last equality being due to (16), this form is Hermitian. Combining (18) and (19) yields

V/(u,u)- = ||u|- for all u e H-(D).

From now on we endow the space H-(D) with the scalar product (•, •)_.

We recall that a compact self-adjoint operator C is said to be of finite order if there is 0 < p < to, such that the series v |Av lp converges where {Av} is the system of eigenvalues of the operator C (its existence is provided by Hilbert-Schmidt Theorem, see, for instance, [5] and elsewhere).

Lemma 4. Suppose that (6) or (7) is fulfilled. Then the inverse L-1 of the operator given by (16) induces positive self-adjoint operators

i'iL-1 : H-(D) ^ H-(D), iL-11' : L2(D) ^ L2(D), L-1 i'i : H+(D) ^ H+(D)

which have the same systems of eigenvalues and eigenvectors; besides, the eigenvalues are positive. Moreover, if (7) holds true then they are compact operators of finite orders and there are orthonormal bases in H+(D), L2(D) and H-(D) consisting of the eigenvectors.

Proof. According to Theorem 1 the embedding i is continuous. As i', L-1 are bounded, all the operators (i'iL-1), (iL-11'), (L-1 i'i) are bounded, too. Then, by (19),

(i'iL-1u, v)- = (v,i'iL-1u)- = (L-1v, i'iL-1u) = (iL-1u, il-1v)l2(d), (20)

(u, i'iL-1v)- = (i'iL-1v,u)- = (iL-1u, il-1v)l2(d)

for all u,v e H-(D), i.e., the operator (i'iL-1) is self-adjoint. Using (16) we get

(iL-1 i'u, v)l2(d) = (i(L-1(i'u)), v)l2(d) = (L-1(i'u), i'v) = (L-1(i'u), L-1(i'v))+,

(u, iL-1 i'v)l2(d) = (iL-1 i'v, u)l2(d) = (L-1(i'u), L-1 (i'v))+

for all u, v e L2(D), i.e., the operator (iL-11') is self-adjoint. On applying (16) once again we obtain

(L-1 i'iu, v)+ = (L-1(i'iu), v)+ = (i'iu, v) = (iu, iv)l2(d), (21)

(u, L-1 i'iv)+ = (v,u)+ = (iu, iv)l2(d)

for all u, v e H +(D), which establishes the self-adjointness of (L-1 i'i).

Finally, as the operator L-1 is injective, so are the operators (i'iL-1), (iL-11') and (L-1 i'i). Hence, all their eigenvectors {uv} (if exist!) belong to the space H+(D), for L-1 uv lies in H+(D)

and all the eigenvalues are positive. From the injectivity of i and i' we also conclude that the systems of eigenvalues and eigenvectors of (I'lL-1), (iL-1i') and (L-1 i'i) coincide.

If (7) holds true then Theorem 1 implies that the embedding i is compact. Then all the operators (i'iL-1), (iL-11'), (L-1 i'i) are compact, too. Now we refer to [7] (see also Proposition 5.4.1 in [17]) that if there is S > 0 such that a compact operator C maps Hs(D) continuously to Hs+<5(D), then it has a finite order (actually, one may choose p = n/S + t for each t > 0). But, under estimate (7), Theorem 1 implies that the operator (iL-V) actually maps L2(D) to H 1/2-e(D) with any e > 0. Hence it has a finite order. As the operators (i'iL-1) and (L-1 i'i) have the same eigenvalues, their orders are finite, too.

The last part of the lemma follows from Hilbert-Schmidt Theorem. □

Our next goal is to apply Keldysh's Theorem (see [6] or [5, Ch. 5, § 8]) for studying the completeness of root functions of weak perturbations of the finite order compact self-adjoint operators.

Theorem 2. Let estimate (7) be fulfilled and S60 = 0. Then, for any invertible operator L : H +(D) ^ H- (D) related to problem (17) the system of root functions of the compact operator (i'iL-1) : H-(D) ^ H-(D) is complete in the spaces H-(D), L2(D) and H+(D).

Proof. By assumption there is a bounded inverse L-1 : H-(D) ^ H+(D). Since I — L0L-1 = (L — Lo) L-1, we conclude that

(i'iL-1) — (i'iL-1) = (i'iL-1) ((L — Lo) L-1) . (22)

n

As Sb0 = 0, the operator (L — L0) : H +(D) ^ H-(D) is induced by the term Sa0 + J2 aj (z) JZ-.

j=1 j

Then, as we have seen in the proof of Corollary 1, this operator is compact. Since L-1 is bounded, it follows that the operator (L — L0) L-1 : H-(D) ^ H-(D) is compact, too.

Hence, (i'iL-1) is an injective weak perturbation of the compact self-adjoint operator (i'iL-1) of finite order (see Lemma 4). Then Keldysh's Theorem [6] or [5, Ch. 5, § 8]) implies that the countable system {uv} of root functions related to the operator (i'iL-1) is complete in the Hilbert space H-(D).

Pick a root function uv of the operator (i'iL-1) corresponding to an eigenvalue Av. Note that Av = 0, for the operator L-1 is injective. By definition there is a natural number m, such that ((i'iL-1) — AvI)muv = 0. Using the binomial formula yields

m

uv = {j

j=i

(m) A-j (i'iL-1)j Uv.

Hence, uv G H+(D) because the range of the operator L-1 lies in the space H+(D).

We have thus proved that {uv} c H+(D). Our next concern will be to show that the linear span L({uv}) of the system {uv} is dense in H+(D) (cf. Proposition 6.1 of [9] and [12, p. 12]). For this purpose, pick u G H +(D). As L maps H +(D) continuously onto H-(D), we get Lu G H-(D). Hence, there is a sequence {fk} c L({uv}) converging to Lu in H-(D). On the other hand, the inverse L-1 maps H-(D) continuously to H+(D), and so the sequence

L 1fk = L Vt/fc

converges to u in H +(D).

If now uv0 G L({uv}) corresponds to an eigenvalue A0 of multiplicity m0 then the vector vV0 = (t'(L-1)uV0 satisfies

((i'iL-1) — A01 )m0 vvo = ((i'iL-1) — A0/)mo+1uvo + A0((i'iL-1) — A0/)m0 u^ = 0.

Thus, the operator (i'iL-1) maps L({uv}) to L({uv}) itself. Therefore, the sequence {t'iL-1/k} still belongs to L({uv}) and we can think of {L-Q/k} as sequence of linear combinations of root functions of i'iL-1 converging to u. These arguments show that the subsystem L-1 L({uv}) c L({uv}) is dense in H+(D).

Finally, since the space Cq° (D) of the functions with compact supports is included into H +(D) and C0°(D) is dense in the Lebesgue space L2(D), the space H+(D) is dense in L2(D) as well. This proves the completeness of the system of root functions in L2(D). □

If ¿60 = 0 then the corresponding perturbation may be non-compact (see Example 2 below). In this case one may use another methods to study the root functions (see, for instance, [9,11]). However these methods are beyond the scope of this paper.

3. Examples for the unit ball

Let S = 0 and = 0 for all 1 ^ j ^ n, = 1 and a0, b0 be constants. Then we obtain the mixed problem for the Helmholtz equation. In the generalized setting the corresponding spectral problem reads as

n ( du dv \

4^ (d^d^) +460(u,v)L2(SD) + («0 - A)(u,v)L2(D) =0 for all v G H+(D). (23)

j = l V j j/ L2(D)

In particular, applying the last identity with u = v we conclude that A > a0j0 if ¿a0 = ¿60 = 0.

We are going to study the Sturm-Liouville problem on the unit ball D = B in Cn. Actually, the matter is quite similar to the coercive mixed problem for the Laplace operator in the ball (see [19, Suppl. II, P. 1, §2]).

To this end, we pass to spherical coordinates x = r S(y) where y are coordinates on the unit sphere dD = S in Cn. The Laplace operator A in the spherical coordinates takes the form

A2n = r2((r|:)2 + (2n-2)(r|;) - as), (24)

where AS is the Laplace-Beltrami operator on the unit sphere. On the other hand, in the unit ball we have

— = r— d =Vz- — = 1 (r — + Bg dv v j dzo 2 V dr S

j=i j v

where the operator Bg depends on the coordinates on the sphere S only. If, for instance, n = 1

- 1 ( d d \ then, in polar coordinates, dv = - r ——+ \f—1 -7— .

2 \ dr ay/

To solve the homogeneous equation (—A2n + a)u = 0 we make use of the Fourier method of separation of variables. Writing u(r, y) = g(r)h(y) we get two separate equations for g and h, namely

- (r^) +(2 - 2n^r^)+ar2) g = cg

Agh = ch,

c being an arbitrary constant.

The second equation has non-zero solutions if and only if c is an eigenvalue of Ag. These are well known to be c = k(2n + k — 2), for k = 0,1,... (see for instance [19]). The corresponding eigenfunctions of AS are spherical harmonics hk(y) of degree k, i.e.,

AShk = k(2n + k - 2) . (25)

The number of the linearly independent spherical harmonics of the degree k is finite and equals

(2n + 2k - 2)(2n + k - 3)! ^ , ,

to J (k) = -—---. In the complex space Cn we may choose the harmonics

k!(2n - 2)!

in accordance with the complex structure. Namely, it possible to find an orthonormal basis {Hpj<j} in L2(S) of consisting on the polynomials of the form

H$(z,z)= £ cj z^

with complex coefficients cj (see, [20]). Let J (p, q) stands for the number of the polynomials of the bi-degree (p, q) in the basis; of course J(p, q) < J(p + q). Clearly,

dvHpjq=qHpjq, bshj = (q - p)Hp«q. (26)

Consider now the following Sturm-Liouville Problem for ordinary differential equation with respect to the variable 0 < r < 1 (see [19, Suppl. II, P. 1, §2])

( 1 ( 9\2 + (2 2 d\ , (p + q)(2n + p + q - 2) + \ ^ ( ) (27)

\- r^ lrdrJ +(2 - 2n)l ~r~d~r) +-r^-+av g(r) =A g(r), (27)

dr (1) + ^260 + (q - p)) g(1) = 0 and g(r) is bounded at the point r = 0. (28)

Actually, if a0, A G R then (27) is a version of the Bessel equation, and its (real-valued) solution g(r) is a Bessel function defined on (0, while the space of all the solutions is two-dimensional. For example, if A = a0 then g(r) = arp+q + pr2-p-q-n with arbitrary constants a and p is a general solution to (27). In the general case the space of solutions to (27) contains a one-dimensional subspace of functions bounded at the point r = 0, cf. [19].

For a triple (p, q,j), fix a non-trivial solution gpq^r) to (27), (28) corresponding to an eigenvalue Apq^. Then the function upq^ = gPjqi)(r)HPj^(^) satisfies

(-A2n + (ao - Aj)) uj^ = 0 on Cn, (29)

(bo + dv)uPjqi) =0 on dD. (30)

Indeed, by (24), (25), (27) and the discussion above we conclude that this equality holds in Cn \ {0}. We now use the fact that up'q^ is bounded at the origin to see that (29) holds. On the other hand, (30) follows from (26) immediately.

Theorem 3. Let ¿a0 = ¿b0 = 0 and a0 , 0 + b0 , 0 = 0. The system {upjqi)}; i G N, p, q G Z+, 1 < j ^ J(p, q), coincides with system of all the eigenvectors of the Sturm-Liouville problem (17) in the ball B. In particular, it is an orthogonal basis in H +(B), L2(B) and H-(B).

Proof. As a^ , 0 + , 0 = 0, Theorem 1 implies that H +(B) is continuously embedded to L2(B).

Now we note that the system {upq^} consists of eigenvectors of the Sturm-Liouville problem (17) in the ball B. Moreover, according to [21, Lemma 7.1], the system {upq^} is orthogonal with respect to the Hermitian forms (•, -)L2(S) (•,-)L2(B) and ^)[L2(B)]n. In particular, it is orthogonal in H +(B). The orthogonality of the system in H-(B) is fulfilled because (20) and Lemma 4 imply

(ujq°,u jf)- = (A^rViL^ujUjf)- = Ajfju jf)L2(B).

By the very construction, the system {Hp^}, p, q G Z+, 1 < j < J(p, q), is an orthonormal basis in L2(S). As it is known, if ¿a0 = 0 then A pj^ > a0j0 and the countable system {gpfq ^r)}^ of eigenfunctions is an orthogonal basis in the weighted space LR([0,1], r) of real valued functions with the scalar product (v^, v/r^)L2([0j1]) (see [19, Suppl. II, P. 1, §2]) for each fixed triple (p, q, j) with p, q G Z+, 1 < j < J(p, q). Easily, it is also is an orthogonal basis in the weighted space L2([0,1],r) (consisting of complex-valued functions). Hence, by the familiar arguments, the system = gPjqi)(r)HPj()(y>)}, p, q G Z+, is an orthogonal basis in L2(B) = L2(S x [0,1]),

see, for instance, [23, Ch. VII, §3.5, Theorem 1].

Now, as the system {upjq)} is an orthogonal basis in L2 (B) there are no other eigenvalues of the problem (17) besides the already mentioned Ap^. Hence there are no eigenvectors corresponding to a value A0 besides the linear combinations of the already constructed eigenfunctions related to this value.

As we already mentioned, the space L2(B) is dense in H-(B). Hence the system {up^} is complete in H-(B), too. Finally, let a function u G H+(D) is orthogonal to each vector upjgi) with respect to (•, •)+. Then, using Lemma 4 and (21) we conclude that (u, upjqi))L2(B) = (u,L-Vi,upjq))+ = Apjq^(u,upjqi))+ = 0 i.e. u is orthogonal to each vector wpj^ in L2(B). Therefore u = 0 in L2(B) and, consequently in the space H + (D). This exactly means that the system {upjqi)} is complete in H + (B). □

We note that, as opposed to the coercive case, in this way we can not provide that the multiplicities of the eigenvalues of problem (17) are finite (cf. Example 1 below).

Example 1. Let a0 = a0 0 = 1, b0 = b0 0 = 0. Then the space H + (D) is continuously embedded to L2(D) (see Theorem 1 above). It follows from (23) that the eigenvalues (if exists) are equal or more than 1; moreover the eigenvalue A =1 corresponds to the space O2 (D) of holomorphic functions from the Lebesgue space L2(D). The dimension of the eigenspace O2(D) (i.e. the multiplicity of the eigenvalue A = 1) is not finite and hence the embedding i is not compact. However, Theorem 3 allows us to construct an orthogonal basis in H + (B), L2(B) and H-(B) consisting of the eigenvectors of problem (17).

Let us see that the corresponding embedding in Theorem 1 is sharp for the ball B. Indeed, if

TO zk

D = B and n =1 then the series ue(z) = + 1)e/2, e > 0, converges in H+(D) and ||ue|| + =

TO 1 2 2s

KH^B) = n E0 (k + 1)i+e. According to [22, Lemma L^ ue Hs(B) > ^k=0 (fc+fci)i+e, 0 <

s ^ 1, i.e. for each s G (0,1) there is e > 0 such that ue G Hs(B). Therefore H+(B) can not be continuously embedded to Hs(B) for any s > 0. □

Actually, the embedding corresponding to (7) in Theorem 1 is sharp for the ball B, too.

Example 2. Let first a0j0 = 0 and 60j0 = b0 = 1. Then H +(B) is continuously embedded to H1/2 (B) and the corresponding operator L0 is of finite order (see Theorems 1 and Lemma 3). If £a0 G C then, according to Theorem 2, the system of the root functions related to problem (17) is complete in H +(B), L2(B) and H-(B). On the other hand, problem (27), (28) may be treated in a similar way as (17) with H0 = L2([0,1],r), i.e. the term ¿a0 induces a weak perturbation of the self-adjoint problem (27), (28) with a0 = a0j0 > 0. Hence, by Keldysh's Theorem the corresponding system {sp^} of its (complex-valued) root functions is complete in the weighted space L2([0,1],r). Thus we conclude that the system {upjqi)} of the root functions related to problem (17) is complete in the Lebesgue space L2(B) (and then in H-(B)) for every a0 G C.

If n = 1 then the series ue(z) = —Z (1+e)/2, e > 0, converges in H+(B) and ||ue|| +

k=0 ( + )

cxj

œ 1

112 ~ — 1

"£ I

fc = Q

¿2(s) = 2n £ +i)1+e, We ^s(B) ^ n k=Q (k+ryr^, 0 < s < 1, i.e. for each s G (1/2,1)

H

k=Q

there is e > 0 such that w£ G Hs(B). Therefore H + (B) can not be continuously embedded to Hs(B) for any s > 1/2.

Let now 0 = |5&o| < bo,o = 1. Then problem (17) is still a Fredholm one (see Corollary 1). Take n =1 and the sequence {zp}. It is bounded in H+(B) because ||zp|| + = ||zp||L2(S) = i/2n. As ||zp — zk|| + = 4n for every G Z+ we conclude that the sequence contains no fundamental subsequences. On the other hand, for the corresponding bounded operator ¿L0 we have

||£Lo(zp — )||_ =4 sup |(v,<№o(zp — ))L2(S)I > 4|^o|||zp — |L2(S) = 8|<№o|Vn

v£H1(D) ||v|| +

v=o

i.e. the sequence {¿Lozp} contains no fundamental subsequences, too. Hence the operator ¿Lo can not be compact. □

The work was supported by RFBR, grant 11-01-91SS0-NNIQ_a.

2

œ

References

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__гл гл u

спектральных свойствах одной некоэрцитивнои

«-» -л

смешанной задачи, ассоциированной д-оператором

Александр Н. Полковников Александр А. Шлапунов

Мы рассматриваем некоэрцитивную задачу Штурма-Лиувилля в некоторой ограниченной области D комплексного пространства Cn для возмущенного оператора Лапласа. Более точно, мы ставим на границе условия Робиновского типа, в которых член первого порядка пропорционален комплексной нормальной производной. Доказывается фредгольмовость задачи в подходящих пространствах, для которых получена теорема вложения, дающая соотношения со шкалой пространств Соболева-Слободецкого. Затем, используя метод слабого возмущения компактных самосопряженных операторов, мы доказываем полноту корневых функций, ассоциированных с краевой задачей в пространстве Лебега. Для шара соответствующие собственные векторы представлены как произведение функций Бесселя и сферических гармоник.

Ключевые слова: задача Штурма-Лиувилля, некоэрцитивные задачи, многомерный оператор Коши-Римана, корневые функции.