an estimate for the completeness of products of solutions of pde
TRANSCRIPT
Applied Mathematics and Computation 147 (2004) 799–804
www.elsevier.com/locate/amc
An estimate for the completenessof products of solutions of PDE
Hakan S�ims�ekAtat€uurk €UUniversitesi, Fen-Edebiyat Fak€uultesi Matematik B€ool€uum€uu 25240 Erzurum, Turkey
Abstract
In this study we prove that on estimate for the completeness of products of solutions
of PDE.
� 2002 Elsevier Inc. All rights reserved.
1. Introduction
The property of completeness of products for harmonic functions have
important role in theory of PDE. The property of completeness of products forharmonic functions in Rn nP 2, was observed by Calderon [1]. Apparently, it is
not vaild in R1 (Linear combinations of linear functions are not dense).
Suppose that X be a bounded domain in Rn nP 2. Consider the function,
u1; u2 that harmonic near X. We will show that span fu1u2g, is dense in L1ðXÞ,for nP 3. Assume that it is not so. Then there is a non-zero measure l sup-
ported in X such thatRu1ðyÞu2ðyÞdlðyÞ ¼ 0 for all such u1; u2. The functions
u1; u2. The functions u1ðyÞ ¼ jx� yj2�n ¼ u2ðyÞ are harmonic near X when xoutside X, so the Riesz potential
Rx� yja dlðyÞ ¼ 0, (a ¼ �2) when x is outside
X. It was proven using asymptotic behaviour of this potential at infinity by
Riesz [3] or using entension to higher dimensions by Lavrentiev [2] or using
Fourier transform by Isakov [5] that when a ¼ 2k, aþ n 6¼ 2k þ 2, the exterior
Riesz potential determines a measure l in a unique way, so we have l ¼ 0,
which is contradiction [4].
E-mail address: [email protected] (H. S�ims�ek).
0096-3003/$ - see front matter � 2002 Elsevier Inc. All rights reserved.
doi:10.1016/S0096-3003(02)00813-5
800 H. S�ims�ek / Appl. Math. Comput. 147 (2004) 799–804
Now we demonstrate Calderon�s approach which turned out to be fruitfull
for some equations variable coefficients. Our main goal is to suggest two es-timate for the on PDE. We give some auxilary results.
2. Completeness of products of solutions of PDE
In this section we will give some auxiliary Lemma�s and theorems.
Lemma 2.1. Let nð0Þ 2 R2 n f0g, nP 2. Then are fð1Þ, fð2Þ 2 Cn such that
fð1Þfð1Þ ¼ 0 ¼ fð2Þfð2Þ; fð1Þ þ fð2Þ ¼ nð0Þ: ð2:1Þ
If P 3, then for any R > 0 then are such nð1Þ; nð1Þ with the additional property
Rþ jnð0Þj2
6 knðjÞj; jTnðjÞj6 jnð0Þj2
���� þ R: ð2:2Þ
Proof. The proof of this lemma can be found in [5]. �
Now we can demonstrate Calderon�s proof of completeness of products of
harmonic functions.
Since linear combinations of exponential functions expðinð0Þx; nð0Þ 2Rn n f0g are dense (in L2ðXÞ), it sufficies to find two harmonic functions whose
product is this exponential functions.
We let u1ðxÞ ¼ expðifð1ÞxÞ; u2 ¼ expðifð2ÞxÞ where the nðjÞ satisfy conditions(1), which guarantee that u1; u2 are harmonic. Such fðiÞ exist by Lemma 2.1.
By multiplying the exponential functions and using condition (2.1) again, we
obtain expðinð0ÞxÞ. This approach was transferred by Sylevester and Ulhlmann
[6] on to solutions of the multidimensionl Sch€oodinger equaiton (�Du� þ cju�j ¼0 in X, cj ¼ a�1=2
j Da�1=2j ).
They suggested using almost exponential solutions to this equation
ujðAÞ ¼ expðifðjÞxÞð1þ wjðxÞÞ ð2:3Þ
with the property that wj goes to zero (in L2ðXÞ) when R (from Lemma 2.1)
goes to infinity. Isakov extend this method to the more general equations,
ðPjð�ioÞ þ cjÞuj ¼ 0: ð2:4Þ
Lemma 2.2. Let P be a linear partial operator of order m in Rn with constantcoefficients. Then there is a bounded linear operator E from L2ðXÞ into itself suchthat
Pjð�ioÞEf ¼ f ; for all f 2 L2ðXÞ ð2:5Þ
H. S�ims�ek / Appl. Math. Comput. 147 (2004) 799–804 801
and
kQEf k2ðXÞ6 c supQðnÞP ðnÞ
� �kf k2ðXÞ ð2:6Þ
where c depends only m, n, diamX and sup over n 2 Rn.Here
PðnÞ ¼Xjaj6m
joanP ðnÞj2
!1=2
:
For the complete proof can be found in [7].
Theorem 2.1. Let R0 be an open non-void subset of Rn. Suppose that for anynð0Þ 2 R0 and for any number R then there are solutions fðjÞ to the algebraicequations
fð1Þ þ fð2Þ ¼ nð0Þ; PjðfðjÞÞ ¼ 0 with jfðjÞj > R ð2:7Þ
and there is a positive number c such that for there nðjÞ we have
1
jfðjÞj 6 cP jðnþ fðjÞÞ for all n 2 Rn: ð2:8Þ
If f 2 L1ðXÞ and
ZXfu1u2 ¼ 0 ð2:9Þfor all L2––solutions uj then the Eq. (2.4) near X, then f ¼ 0. In other words, thistheorem says that linear combinations of products of solutions of these equationsare dense in L1ðXÞ.
Theorem 2.2. Suppose that conditions (2.7), (2.8) are satisfied. Then for any
nð0Þ 2 R0 there are solutions to Eq. (2.4) near X of the form (2.3) wherekwjk2 6 c=jfðjÞj and depends only on kcjk1ðXÞ and on diam X.
The proof of the theorem can be found in [5].
Corollary 2.1. Let nP 3. If (2.9) is valid for all solutions of uj to the equations�Duj þ cjuj ¼ 0, j ¼ 1; 2, near X, then f ¼ 0.
Proof. To prove this result we check the conditions of Theorem 2.1. Let
nð0Þ 2 Rn. Due to rational invariancy we may assume
802 H. S�ims�ek / Appl. Math. Comput. 147 (2004) 799–804
nð0Þ ¼ n1ð0Þ2
; in1ð0Þ2
4
0@ þ R2
!1=2
;R; 0; . . . ; 0
1A
the vectorsnð1Þ ¼ n1ð0Þ2
; in1ð0Þ2
4
0@ þ R2
!1=2
;R; 0; . . . ; 0
1Anð2Þ ¼ n1ð0Þ
2;
0@ � in1ð0Þ2
4
þ R2
!1=2
;� R; 0; . . . ; 0
1A
are solutions to the equation nf ¼ 0 with the absolute values grater than R, socondition (2.7) is satisfied. To check condition (2.8) we observe thatP2ðnþ fÞP j2n1 þ 2f1j2 þ � � � þ j2fn þ 2nnj2 þ 12
P 4ðjsn1j2 þ � � � þ jsnnj2ÞP jfj2
provided that nf ¼ 0. This completes the proof [5]. �
In the fundemental paper [6] the authors used the more standard funda-
mental solution E (The Fadeev Green�s functions) and the obtained estimates
(2.6) in the Sobolev spaces Hmd ðRnÞ, which are constructed from the weighted
L28ðRnÞ––spaces with the norm kf ðxÞð1þ jxj2Þd=2k2ðRnÞ.
3. Main results
In this section we will prove two estimate for solutions of the equationP ¼ �D� 2fr, ff ¼ 0.
Lemma 3.1. Let l ¼ �D� 2ifr and E be its regular fundamental solution. Thenthe estimate kEwkð1ÞðXÞ6 ckwkð0Þ is valid for all functions w 2 L2ðXÞ that arezero outside X with c depending only on diam X and the dimension n of the space.
Proof. By use of Lemma 2.2, we have
kQEwk2ðXÞ6 c1 supn
QðnÞP ðnÞ
� �kwk2ðXÞ ð3:1Þ
where c1 depends only on diam X and dimension n of the space and sup is over
n 2 Rn also
H. S�ims�ek / Appl. Math. Comput. 147 (2004) 799–804 803
PðnÞ ¼Xjaj6m
jDxnP ðnÞj
2
!1=2
:
First of all we can write
ePP ðnÞ ¼ n21 þ � � � þ n2n þ 2f1n1 þ � � � þ 2fnnnonkP ¼ 2n1 þ � � � þ 2nn þ 2f1 þ � � � þ 2fn þ 2ðfk þ nkÞ;
where
onjofk ¼2; j ¼ k0; j 6¼ k
�:
Thus we obtain,
ePP 2ðnÞ ¼ ðjfþ nj4 þ 4jfþ nj2 þ 4nÞP 4Xkjfk þ nkj2:
If we take fk ¼ ak þ ibk then
ePP 2ðnÞP 4Xkjfk þ nkj2 þ jbkj2 P 4
Xk
ðnkÞ2 þ 2ðaknkÞ þ ðakÞ2 þ ðbkÞ2
P 4Xk
ðnkÞ2 þ 2ðaknkÞ þ 2jakj2:
Since ff ¼ 0 then Rja2k j ¼ Rjbkj2we can write
ePP 2ðnÞP 4Xk
ðnkÞ2 þ 2ðaknkÞ þ 2jakj2
P 4Xk
1
2ððnkÞ2 þ 2ðakÞðnkÞ þ 2ðakÞ2Þ þ
1
2ðnkÞ2
P1
2ðnk þ 2akÞ2 þ
1
2n2k P 2ðnkÞ2:
Namely
ePP 2ðnÞP 2ðnkÞ2: ð3:2ÞBy the definition we write
kEwk2ð1ÞðXÞ ¼ZX
Xjaj6 1
jDanEwj
2
!¼
ZXjEwj2
�þZXjDnEwj2
�: ð3:3Þ
By taking Q ¼ 1 in (1) we have bound for the integralRX
Pjaj6 1 jEwj
2;
kEwk22ðXÞ ¼ZXjEwj2 6 c sup
1
~llðnÞ kwk2ðXÞ:
804 H. S�ims�ek / Appl. Math. Comput. 147 (2004) 799–804
There ePP ðnÞ uniformly bound for all n by (3.2). Thus we obtain
kEwk22ðXÞ6 c1kwk2ðXÞ: ð3:4Þ
Also similarly by taking Q ¼ ink in (3.1) we have the bound for the integralRX jDnwj2;
kQEwkð2ÞðXÞ ¼ZXjDnEwj2 6 c sup
n
eQQðnÞePP ðnÞ kwk2ðXÞ:
We know that all rations bounded by (2), therefore we writekQEwkð2ÞðXÞ6 c2ðkwk2ðXÞÞ: ð3:5Þ
Then by plugging (3.4), (3.5) into (3.3) one can have
kEwkð1ÞðXÞ6 c1kwk2ðXÞ þ c2kwk2ðXÞ6 c3kwk2ðXÞ
or the same as
kEwkð1ÞðXÞ6 ckwkð0ÞðXÞ:
There c depends only on c and dimension n. �
References
[1] A.P. Calderon, On an inverse Boundary Value Problem, Seminar on Numerical Analysis and its
Application to Continium Physics, Rio de Janerio, 1980, pp. 65–73.
[2] M.M. Lavrentiev, Some Improperly Posed Problems of Mathematical Physics, Springer, Berlin,
1967.
[3] M. Riesz, Integrales de Riemann-Lioville et Potentials, Acta Szeyed 9 (1938) 1–42.
[4] V. Isakov, Inverse problems for partial differential equations, Applied Mathematical Sciences,
vol. 127, Springer, Berlin, 1998.
[5] V. Isakov, Inverse Source Problems, Mathematical Surveys And Monographs no. 34, American
Math. Society Providence Rhode Island, 1990.
[6] J. Sylvester, G. Ulhmann, Global uniqueness theorem for an inverse boundary problem, Ann.
Math. 125 (1987) 153–169.
[7] L. N€oormender, The Linear Analysis of Linear Partials Operators, Springer Verlag, New York,
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