О некоторых обратных спектральных задачах для произвольного возмущения бигармонического оператора с сингулярными коэффициентами

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ON SOME INVERSE SPECTRAL PROBLEMS FOR AN ARBITRARY PERTURBATION OF THE BI-HARMONIC OPERATOR
WITH SINGULAR COEFFICIENTS
V.S. Serov, Dr. Sc. (Physics and Mathematics), Professor (University of Oulu, P.O. Box 3000, Fin-90 014, Oulu, Finland, vserov@pc. oulu. fi)
Received 17. 04. 2014
The classical inverse boundary spectral problem for any perturbation of the bi-harmonic operator with singular coefficients from some Sobolev spaces is considered. The problem is formulated as follows: Do the Dirichlet eigenvalues and some derivatives of the corresponding normalized eigenfunctions at the boundary of smooth bounded domain uniquely determine the coefficients of this operator?
We proved the first step of this problem, i.e. we proved that the Dirichlet eigenvalues and the derivatives up to the second order of the normalized eigenfunctions at the boundary uniquely determine the so-called Dirichlet-to-Neumann map that corresponds to the Friedrichs self-adjoint extension of any perturbation of the bi-harmonic operator with singular coefficients. The main role in this proof is played by the existence of the Green'-s function and its estimates up to the boundary of the domain. These facts will allow us to prove the classical Borg-Levinson theorem for the operators of fourth order of such type.
Keywords: Green'-s function, Friedrichs extension, Dirichlet-to-Neumann map.
УДК 517. 95
О НЕКОТОРЫХ ОБРАТНЫХ СПЕКТРАЛЬНЫХ ЗАДАЧАХ ДЛЯ ПРОИЗВОЛЬНОГО ВОЗМУЩЕНИЯ БИГАРМОНИЧЕСКОГО ОПЕРАТОРА С СИНГУЛЯРНЫМИ КОЭФФИЦИЕНТАМИ Серов В. С., д.ф. -м.н., профессор (Университет прикладных наук Оулу, P.O. Box 3000, Fin-90 014, г. Оулу, Финляндия)
Дата подачи статьи: 17. 04. 2014
Аннотация. Рассматривается классическая обратная граничная спектральная задача для произвольного возмущения бигармонического оператора с сингулярными коэффициентами из некоторых пространств Соболева. Задача формулируется следующим образом: определяют ли однозначно собственные значения задачи Дирихле и некоторые производные соответствующих нормированных собственных функций на границе гладкой ограниченной области коэффициенты этого оператора?
Доказано (как первый шаг в решении этой проблемы), что собственные значения задачи Дирихле и производные до второго порядка нормированных собственных функций на границе однозначно определяют так называемое пре-
образование от Дирихле к Нейману, которое соответствует самосопряженному расширению по Фридрихсу произвольного возмущения би-гармонического оператора с сингулярными коэффициентами. Главную роль в этом доказательстве играет существование функции Грина и ее оценки вплоть до границы области. Эти факты позволят нам доказать классическую теорему Борга-Левинсона для операторов четвертого порядка такого типа.
Ключевые слова: функции Грина, расширение по Фридрихсу, преобразование от Дирихле к Нейману.
The subject of this work concerns to the classical inverse spectral problem. This inverse problem can be formulated as follows: do the Dirichlet eigenvalues and the derivatives (which order?) of the normalized eigenfunctions at the boundary determine uniquely the coefficients of the corresponding differential operator? For operators of order two this type of theorem is called Borg-Levinson theorem. In the case of the Schrodinger operators the knowledge of the Dirichlet eigenvalues and the normal derivatives of the normalized eigenfunctions at the boundary uniquely determine unknown potential. Borg-Levinson theorem for the Schrodinger operators was proved for the first time by Nachman, Sylvester and Uhlmann [1] for the potentials from the space C& quot- (Q). Their proof remains however valid if one
assumes that potential just from L& quot-(Q). This problem was reduced finally to the fact that the Dirichley-to-Neumann map uniquely determines such potentials. The same result was obtained independently by Novikov [2]. For singular potentials from the space Lp (Q) — n/2& lt-p<-x>-, this theorem was proved by Paivarinta and Serov [3]. For inverse boundary spectral problems on Riemannian manifolds some related results were proved by Kachalov, Kurylev and Lassas [4] (see also [5]).
For the magnetic Schrodinger operator with singular coefficients Borg-Levinson theorem was proved for the first time by Serov [6]. The proof of this result for the magnetic Schrodinger operators uses the same technique as for the Schrodinger operators with singular potentials. It must be mentioned here that the magnetic Schrodinger operator cannot be considered as a & quot-small"- perturbation of the Schrodinger (or Laplace) operator.
For the operator of order 4 which is the first order perturbation of the bi-harmonic operator with Navier boundary conditions on a smooth bounded domain in Rn, n& gt-3, it is proved by Krupchyk, Lassas and Uhlmann [7] that the Dirichlet-to-Neumann map uniquely determine this first order perturbation.
The main goal of present work is to show that the knowledge of the discrete Dirichlet spectrum and some special derivatives up to the second order of the normalized eigenfunctions at the boundary uniquely determine the Dirichlet-to-Neumann map that corresponds to an arbitrary perturbation of the bi-harmonic operator with singular coefficient from some Sobolev spaces on the smooth bounded domain in Rn, n& gt-2. The next step might be to prove that the knowledge of the Dirichlet-to-Neumann map uniquely determines the coefficients of the operator H4 which is an arbitrary perturbation of the bi-harmonic operator.
The solution of this problem will be given in future publications.
Below we use the following notations. The space W/(Q), t& gt-0,1<-p<-x, denotes the Lp — based Sobolev space in the domain Q and the space B'-pfi (dQ), t g R, 1 & lt- p, 9 & lt- & lt-«, denotes the Besov space on the boundary of the domain Q. The Besov space B'-pfi (SQ) for negative t is defined as the adjoint space
to the corresponding Besov space with positive index of smoothness — t. Throughout this article we use the trace type theorem for the functions from the Sobolev spaces at the boundary (see, for example, [8]). More precisely, for any function f from the Sobolev space W'- (Q) with t & gt- 1 and p & gt- 1 there is the trace on the
boundary of Q from the Besov space Bppp (dQ). Green'-s function
Let Q be a bounded domain with smooth boundary in Rn, n& gt-2. We consider in this domain the following operator of order four that we will call the & quot-magnetic"- operator of 4th order with variable coefficients
H4 = H4 (x, d) := A2 + /V (V (A (x) ¦ V)) + /V • (A (X)A) --V • (F (x) V) — i V • (G (x)) — iG (x) • V + V (x), (1) where V denotes the gradient in Rn, A denotes the Laplacian in Rn and where the coefficients A (x), F (x), G (x) and V (x) are assumed to be real-valued. We assume also that
A (x) e (W (Q))& quot-, F (x) eW (Q, G*) e (W (Q)& quot- (2)
and V (x) e Lp (Q),
P & gt--, n & gt- 2, 2
(3)
where it is assumed (WLG) that the value of p is the same for all these spaces. It is well-known that under the conditions (2) and (3) for the coefficients the following Garding'-s inequality holds (see, for instance, [9]):
(H4u, u) l2 & gt- cJmL — c2||4
iL2 (n),
(4)
where 0& lt-ci<-1, c2& gt-0. This inequality allows us to define symmetric operator (1) by the method of quadratic forms. Then H4 has a self-adjoint Friedrichs extension denoted by (H4)F with domain
D ((H 4) f) = {f (x) e W2 (Q): H f (x) e L2 (Q)},
where W22 (Q) denotes the closure of the space C- (Q) by the norm of Sobolev space W22 (Q). It is
i
t-
possible to prove actually that under the conditions (2) and (3)
D ((H 4) f) = W2 (Q) n W24 (Q). (5)
The spectrum of this extension is purely discrete, of finite multiplicity and has an accumulation point only at the +»: X1 & lt-X2<-… & lt-Xk<- …
The corresponding orthonormal eigenfunctions (x)}^ form orthonormal basis in L2(Q). The Garding'-s inequality (4) allows us also to conclude that there is a positive constant | such that the operator (H4)F+^o1 is positive.
Therefore for any X & gt- |0 the operator (H4)F+XI is positive and its inverse
((H4)F + XI)-1: L2(Q) ^ L2(Q) (6)
is compact. It is an integral operator with kernel denoted by G (x, y, X). If we use for this integral operator the symbol G (X) then we have
((H4)f + XI) G (X) = I, G (1)((H4)f +11) = I,
G (x, y, X) = G (x, y, X). (7)
Definition 1. The kernel G (x, y, X) of the integral
operator G (X) is called the Green'-s function of the
operator (H4)F + XI. Our first result is:
Theorem 1. Suppose that A (x), F (x), G (x) and V (x) satisfy the conditions (2) and (3). Then for any X& gt-|0 the Green'-s function of the operator (H4)F+XI exists and satisfies the following estimates:
|G (x, y, X)|& lt- C | x — y |4-n e
4-n -Sx-y V
n & gt- 5, (8)
|G (x, y, X)& lt- C n = 4,
i Л
1+ log (x — y X4)
-5|x-y|X4
C
and 1 G (x, у, X) & lt-~- e
-S|x-ylX
4-n
X ~
n & lt- 3,
'-(9)
(10)
where x, & gt->-eQ and constants C& gt-0 and 8& gt-0 do not depend on x, & gt->-eQ and X.
Proof. It is well-known that the fundamental solution of the operator A2+XI, X& gt-0, has exactly these estimates (8)-(10). The conditions (2) and (3) for the coefficients that are in front of the derivatives of order one and higher of the operator H4 allow us to conclude that they belong to the Kato space Kn-1(Rn) (if we extend them by zero outside of the domain Q), i.e.
K-i (R& quot-) = {f (x): sup J | x-y |1-n| f (y) |dy & lt- «& gt-}.
Rn Rn
Using this fact we can easily prove the existence of the fundamental solution for the operator H4+XI. Moreover, the same estimates (8)-(10) will be fulfilled for this fundamental solution too. In order to obtain the same estimates (8)-(10) for the Green'-s function G (x, y, X) we refer to the paper [10] (with
some changes that are connected to the operator of order 4). Theorem 1 is proved.
We have three immediate corollaries of Theorem 1 (see again [10]).
Corollary 1. Assume that A (x), F (x), G (x) and n
V (x) are as above and a& gt- -, n& gt-2. Then for any function f (x)eL2(Q) the following inequality holds
|| ((H4)f +Wr fc (Q) & lt- Cx8 If where X& gt-^ with as in Theorem 1.
Iii2 (Q) —
Corollary 2. Assume that a & gt- -, n & gt- 2. Then there is a constant C& gt-0 depending only on Q, such that the estimate T 1 yk (& lt- CX4 holds
S (Xt +xr
uniformly in xeQ and X& gt-|0.
n
Corollary 3. Assume that a & gt- -, n& gt-2. Then the
following series «1
^TT+T^ (11)
k=1 (Xk)
converges.
Remark 1. It can be mentioned here that the estimates (8)-(10) of the Green'-s function of the & quot-magnetic"- operator (1) are obtained in Theorem 1 for quite weak conditions of the coefficients of H4. As far as we know they never appeared in the literature.
Dirichlet-to-Neumann map and eigenfunctions
Lemma 1. Under the conditions (2) and (3) for the coefficients of H4 we have that for any two functions u and ^ from Wp4 (Q) the following equality holds
(H4u, 4& gt-) 2 — (U, H42 = (A. u, 2 +
v 4 T L2(a) 4T l2(a) 1 T/l2(sq)
+(A2u, dv2 — (u, A,^) 2 — (dvu, A24) 2 ,
V 2 & gt- VTZ l2 (SQ) 1T/L2(3Q) v V '- 2T/ Z2(3Q)'-
where A1 and A2 are defined as
A1u (x)=dv (Au)(x)+idv (A -Vu)(x)+iv- A A u (x)--Fdvu (x)-iv- G u (x), xedQ (12)
and A2u (x)=-Au (x)-i A -Vu (x), xedQ, (13)
respectively. Here v is unit outward normal vector at the boundary of the domain Q.
Proof. The proof of this lemma is straightforward and is based only on the divergence theorem (& quot-integration by parts& quot-). And the main purpose of this lemma is to define the map A=(A1, A2). It must be mentioned also here that the conditions (2) and (3) for the coefficients of the operator H4 and the conditions for the functions u and ^ from this lemma guarantee the existence of the traces of the corresponding terms in A1u (x), A1^(x), A2u (x) and A2^(x) on the boundary dQ from some Besov spaces. This fact justifies the
n
e
correct application of the integration by parts and proves this lemma.
Let X& gt-|0 with | as in Chapter 1. Consider the following Dirichlet problem:
((H4)F+X/)w (x)=0- xeQ, u (x)=f0(x), dvw (x)=/J (x),
xedQ, (14)
where the boundary functions f0(x) and fj (x) satisfy
the following conditions:
3−1 2 1 «
f (x) e Bp/ (SQ), f (x) e Bp/ (SQ), p & gt- -, (15)
where B'-pp (SQ), t e R, denotes Besov space on the
boundary and p is the same as in (2) and (3).
Using the technique of the multipliers from Sobolev spaces (see, for example, [11]) it can be proved that there exists a unique solution of the Dirichlet boundary value problem (14)-(15) from the spaces
u (x)? W» (Q) n WP (Q).
(16)
Thus, we may define the Dirichlet-to-Neumann map Ax.
Definition 2. The Dirichlet-to-Neumann map Ax for Dirichlet boundary problem (14)-(15) is defined as the following two-dimensional vector
AJ/, № x): =
(dv (Au)(x) + idv (A ¦ Vu)(x) + iv ¦ AAu (x) --Fdvf (x) — iv-Gf0 (x), — Am (x) — iA-Vu (x)), (17)
where v is outward normal vector at the boundary dQ.
Conditions (2), (3), (15) and (16) imply that the Dirichlet-to-Neumann map (17) acts as (for fixed X)
3−1 2−1 -1 -1
A: Bp/ (SQ) x Bp/ (SQ) ^ BJ (SQ) x BJ (SQ) (18)
with p as in (15).
The following theorem can be considered as one of the main results of this work.
Theorem 2. Assume that A1, A2, Fl, F2, Gl, G2 and V, V2 satisfy the conditions (2), (3) andf0, f satisfy the condition (15). In addition we assume that
A1 (x) = A2 (x), F1 (x) = F2 (x), Gj (x) = G2 (x) on the
boundary dQ. Then, for any 0 & lt- 8 & lt- 1 — -
P
, 4 {J0, f} {J0, Jin Bs (яп)
X^+x& quot- HBnn (an)
lim KXi) {f, f} -ЛХ2) {f, f}|
= 0,
(19)
where Aj denotes the corresponding Dirichlet-to-Neumann map for A, F., G, V. + X, j = 1,2.
Proof. Let a (x): =Mi (x)-w2(x), where uj (x), j=1,2, solves the problem (14) with Aj, Fj, G, V,., respectively. We denote the corresponding operators (1) by H (4j& gt-. Then a (x) solves the boundary value problem (Hf + XI) ra (x) = (H42) — Hf) u2 (x), x eQ, ra (x)=0, dvra (x)=0, xedQ.
This problem can be rewritten in the domain Q as
(A2 + X/)ra + (iV (V (AjVra)) + /V (4 Ara) — V (F1Vra) —
-/V (G1ra) + V1ra) = iV (V (A2 — A1) Vu2) + (20)
+iV ((A2 — A1) Au2) — V ((F2 — F) Vu2) —
-iV ((G2 — G1) u2) + (V2 -V1)u2
and with Dirichlet boundary conditions
ra (x)=0, dvra (x)=0, xedQ. (21)
Denote by G0 (X) the integral operator with the kernel which is the Green'-s function of the operator A2+XI with Dirichlet boundary conditions (21). Applying G0 (X) to the left and to the right hand-sides in (20) we obtain the following integral equation
(I + K) ra (x)=F (x), (22)
where the integral operator K and the function F are given by
K := G (X)(iV (V (4V)) + iV (A, A) —
-V (fV) — /V (G)+v)
and
(23)
F (x) := G0 (X)(iV (V (A2 — A1) Vu2) +
+iV ((A2 — A) Au2) — V ((F2 — F) Vu2) — (24)
-iV ((G2 -G)u2) + (V2 -V,)u2)(x). We consider this equation (22) in the space of functions from the Sobolev space Wp4 (Q) which are vanishing with their first normal derivatives at the boundary dQ.
Due to the assumptions (2) and (3) for the
coefficients Aj, Fj, Gj and Vj, j=1,2, and embedding (16) we may conclude that F belongs to this space and K is compact there. Since the operator H4(1) + XI is positive for X& gt-|0 then the boundary value problem
(Hf +XI)ra (x) = 0, x eQ, ra (x)=0, dvo (x)=0, xedQ
has only trivial solution ra=0. The same is true for the homogeneous equation corresponding to (22). By the Fredholm'-s alternative the operator I+K has a bounded inverse in the indicated Sobolev space and therefore the solution ra of the equation (22) satisfies the following inequality
(25)
INI * CF lwP (0) & gt-
where constant C is independent on X& gt-|0.
n
Since p satisfies the inequality p & gt- - then the following embeddings hold W3p (Q) c WP (Q) c r (Q).
This fact and the conditions (2) and (3) allow us to obtain from (24) and (25) that
M W
& lt- C\u
21Wp (Q) ¦
(26)
We apply now the result from [12] and obtain that
l|(A2 (n) ^ CMIHLp (O) ¦ (27)
4
By combining the inequalities (26) and (27) we get the following inequality
c
Hp& lt-Q) * YI|M2|W& gt-)¦ (28)
The interpolation of (26) and (28) leads us to the inequality
C
INW (q) — INwl (ay-X 4
(29)
where 0& lt-s<-4. Using the definition (17) and the equalities for the terms A1 (x) = A1 (x), F (x) = = F2 (x), Gj (x) = G2 (x) on the boundary dD we have
11^ ^{fo, /}-^{f, /}[S (8Q) & lt-||dv (Affl)|| bS +
PP PP
+|S v (AjVro)| +1A, Vro|| я
1 DO /Л
+ vAjAro
+ IMI ^
Using trace theorem and (29) we can estimate the latter terms as follows:
ЛП/о, /}-Л? {/, /} bS & quot-V
C ,
& lt- C m s+3+~ -_
— Ч HI Wp p (a) ~ I=b_
(30)

4 4 p
U2Iwj (Q) •
Since
1
S & lt- 1 — taking into account the
P
boundedness of the norm of u2 in X we may conclude from (30) that Theorem 2 is completely proved.
We are in the position now to estimate the normalized eigenfunctions of the magnetic Schrodinger operator.
Lemma 2. Under the assumptions (2) and (3) for the coefficients A, F, G and V the orthonormal eigenfunctions 9k (x) satisfy the estimate
Ik
s n -±
(q) ^ C& amp-k + *& gt-)4 8,
(31)
where 0& lt-s<-4, p & gt- and | is as in Theorem 1.
Proof. Let Xk be an eigenvalue and 9k (x) corresponding orthonormal eigenfunction. Then Corollary 2 from Theorem 1 allows us to obtain quite
n
easily that ||9jr (Q} & lt- C (Xk)8 and
n
IklU (Q) ^ C (^ +*& gt-)8, (32)
where 1& lt-p<-o>- and constant C& gt-0 depends only on n, p and Vol (D).
Rewriting the equation for the eigenfunctions
9k (x) in the form (A2+|V+6(x, d))9k (x)=(Xk+|o)9k (x), xeQ, 9k (x)=0, 9v9k (x)=0, xedQ, where Q (x, d) is the rest of the operator H4, and applying the inequality
n
(25), we obtain for any p & gt- - that
& lt- C (X k +^0)|K|U (Q) & lt- C (X k +^0 Z8. (33)
Now by interpolation of (32) and (33) we obtain (31). Thus, Lemma 2 is proved.
The next lemma shows us the representation for the kernel of the operator Ax.
n
Lemma 3. For I =
2
+1 (here [a] denotes the
entire part of a) and f0 and f as in (15) we have for the vectorJ-j (Ax {/, /}(x)) := (A®, A®) the
following representations
Ag = J g (11) (x, y, X) f (y)dc (y) +
+ J g (12) (x, у, Щ (y)d CT (y),
3Q
Л$ = J gi (21) (x, y, Vfo (y)dv (y) +
3Q
+ J g (22) (x, y, X) f (y)d CT (y),
(34)
where the kernels gf, i, j = 1,2, are defined by
¦ (a, (A-v9t (x)) + v-Ah. jfk (x)) (a, (A ¦ v^kw)+v ¦ aa^c/)
gr (x, y, X) = (-1)& quot- 1! Z-
(h + h
¦-& gt- (x, y, X) = (-1)4 !X
^ ! «(idv (A ¦ V& lt-pt (x)) + iv ¦ AA% (x)) (AVt (y) — U- V& lt-f>-t (y))
(Xt +ХГ
j — & quot- (Лф"(x)+(x))(& gt-av (A. v<-pt (y)) + iv-(y))
¦& gt-(x, y, X) = (-1)'-+1г! ^
t.i (Xk + X)
_ ^ (Афк (x) + iA ¦ Уфк (x)) (Афк (y) — ?4 ¦ Уфк (y)) (35)
'-& gt-(x, y, X) _ (-1)'-l!^, +
4.1 (Xk + X)
and where for the right-hand sides of (35) are converging in Lp (dQxdQ).
Proof. Solution u of the problem (14)-(15) definitely depends on X and we will denote it from now on as u (x, X) with Integration by parts for the problem (14) with the boundary conditions f0 and fi from (15) leads to
u (x, X) = j (av (AyG (x, y, Xj) — idv (A-VrG (x, y, X) — ivy ¦ AA yG (x, y, X)) /» (y)do (y) —
y y /
r/ - (36)
— J (ayG (x, y, X) — iA ¦ VyG (x, y, X)) f,(y)do (y), V 7
ao
where G (x, y, X) is the Green'-s function of (H4)F+X/ defined in (6)-(10) and vy denotes the outward normal vector in y. In our case the Green'-s function is given by
G (x, y, X) = X
Ф k (х)Фк (У)
k=1 X k +X
(37)
Since u solves the problem (14)-(15) then using J. von Neumann spectral theorem it can be easily proved by induction that
dj u (x, X) = (-1)11!((H4)f +XIu (x, X), (38) I = 1,2,…
The operator ((H4)F + XI)1 is well-defined by the spectral theorem and it is the integral operator with kernel denoted by Gj (x, y, X)
su
+
5
5
SQ
& lt-
4−1
k=1

Gi (x у, '-Х) = Z
Ф* (*)ф* (у)
k=1 (X k + X)
This fact allows us to represent ((H4)F+X/)-1w (x, X) as follows
((H4)F + XI) '- u (x, X) = G,(x, y, X) u (y, X) dy =
= S
Фк (x)uk (X)
(39)
(X * +X)'-
where uk (X) is given by uk (X) = J9k (y)u (y, X) dy.
Q
Integration by parts in the last equality gives us
ut (A) = J (sv (A^M) — idv (A ¦ V^W) -iv ¦ AA^Cy)) f (y)da (y) +
Ak + A 5Q
+ I (-iA-V^kCv)) f (y)do (y). (40)
Ak + A go
Combining (39) and (40) we obtain the following equality ((H4)F+XI)-1u (x, X)=
. V (X) (sv (Aikw) — idv (A ¦ vikty)) — iv ¦ AA^Jy)) / (y)
-Js-
(4 + яу+
-da (у) +
— %(x)(y)-iA-Vifk (y)) fi (y) (41)
+ J ^ f, i v+i da (y)
SQt-l (K + K)
which coincides for 1=0 with (36).
Since u solves the boundary value problem (15)-(16) using (17) we can obtain
d
Ш (л& gt-!/"-'-fikx,)=
= d.

ГА (du
V Id
^ d1 m d X'-
+ id.
A-V
f d1 и ^
VdX yy
+ iv- AA
^ d1 и d X1
— iA-V
^ d '- m d X'-
(42)
Thus, the equalities (38), (41) and (42) give us that formally we have for the vector ^(f0, /}) the following relations =
, 1. (a,(A-v& lt-ft (x)) + v ¦ ААщ (x))(a,(A-v& lt-ft (y)) + v ¦ ААщ (y))f (y) = (-1)1 ! fYA---п---
J /1,1 V+1
?Qk=1 (Лк + K)
| «(idv (A-V9k (x)) + iv-ЛДфк (x))(ДфксУ)-?4-V^kCy))f (y)
+(-1)'-l! iZ& quot--,. ---dc (y)
da (y) +
(Як +A)l+
and Л® =
x (Дф4(x) + ii-Vct (x))(idv (A-V9t (y)) + IV-AA9t (y))/(y)
=(-1) 1! JE& quot--^----da (y) +
SQk=1 (kk + k)
l x (A9t (x) + ii-V9t (x))(A9t (y) — iA-V9t (y)) fi (y) + (-1) 1 ! J S--TTi±y& gt-
3Qk=i (kk + k)
Here we have used the following equalities for the eigenfunctions 9k (x) at the boundary (see (15) and (31)): 9v9k=0, 9v (A9k)=A (9v9k), xe9Q.
The latter equalities show that this lemma will be proved if we show the convergence of all series (35) in Lp (dQx9Q). To this end, the inequality (31) from Lemma 2, the conditions (2) and (3) for the coefficients and Sobolev imbedding theorem allow us to conclude that
dv (ALP (8n) ^C\9ktn2+?(n) & lt-C (Xk +ц0)
2±P
1 n
2 4 p 8
IN
* CI ФАГ 1 (a) * C k)
2 4 p 8
l =
kttlP (3Q)
By using these estimates and taking now n 2
+1, we have for k, m=l, 2 and j=0,1
J (x, y, X) fj (y)d ct (y)
& lt-
Lp (3Q)
& lt- с E-
Wp P (Q)
f
[ n ]+2\J i\lf (3Q)
(Xt +X)
^ cf, Y
глУрр («в) k=i [n
* + h& gt-)
n — n
Г 2 p 4
where t = 3 — - for j=0 and t = 2 — - for j=1. Thus,
P P
due to Corollary 3 of Theorem 1 (see estimate (11))
the latter series converges since p & gt- 2, n & gt- 2, and
therefore Lemma 3 is completely proved.
From spectral data to Dirichlet-to-Neumann map
Now we are in the position to formulate and to prove the second main result of this work. By symbol
Xk (A, F, G, V) we denote the discrete Dirichlet spectrum of the operator H4 defined in (1) with the coefficients A, F, G, V and by (x- A, F, G, V) the corresponding eigenfunctions accounting their multiplicities.
Theorem 3. Assume that Aj (x) e (Wp (Q))& quot-,
Fj (x) e (Q), GJ (X) e (Wj (Q))& quot-, VJ (X) e LL (Q), n
p & gt--, n & gt- 2, for /=1, 2. Assume in addition that 2
A (x) = A2 (x) and dv A1 (x) = dv A1 (x) at the boundary 3Q. Assume also that for each k=1, 2, …
Xk (A, Fi, G, Vi) = Xk (A, F2, G2, F2) (43)
and V^k (x- A, Fi, G = V^k (x- Aj, F2, G2, F2), A9k (x- A, F, G,) = A9k (X- A& gt-, F2, G2, V2), (44) dv (V9k (x- A, F, & lt-5, V,)) = dv (V9k (x- A, F2, G2, V2)),
x e 9Q.
Then for all X& gt-^0
, /i& gt- = A2). /o, /,} (45)
3-i 2−1
for any fo e (SQ) and f e (SQ).
Q
«
SQ
2
k «1
ii 2±
& lt-

Proof. The conditions (43) and (44), Lemma 3 (see formulas (34), (35)) imply that for all X& gt-|0 and
for l =
n 2
+1 we haveJ-j (ЛЧ/о, ЛКx) —
3- 1
-Af {f, fiXx)) = 0 for any /0 e Bppp (5Q) and
f e Bppp (3D.). This equality can be read as
i _
A®{f, fl}(x)-Af {f, fixx) = ?^Lm{f, f} (46)
m= 0
where two-dimensional vector-valued operators Lm
are bounded from Bppp (3D) x Bppp (3Q) to
Lp (3Q)xLp (3Q). But Theorem 2 (see (19)) shows us that the polynomial in the right-hand side of (46) is zero. Hence, the equality (45) holds. It means that Theorem 3 is proved.
Remark 2. This theorem means that the inverse boundary spectral problem for the operator H4 is reduced to the problem of the reconstruction of the unknown coefficients of this operator by the knowledge of Dirichlet-to-Neumann map.
Acknowledgments. This work was supported by the Academy of Finland (Application No. 250 215, Finnish Programme for Centres of Excellence in Research 20 122 017).
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