Iii республикалық студенттік ғылыми-практикалық конференциясының баяндамалар жинағЫ
жүктеу/скачать 13,94 Mb.
|
- Бұл бет үшін навигация:
- SPECTRAL INEQUALITIES FOR THE NEWTON POTENTIALS Suragan D.
- Short proof.
- References T.Sh. Kal’menov and D. Suragan, To spectral problems for the volume potential, Doklady Mathematics , 80
- О ДВУМЕРНЫХ ВЕЙВЛЕТ – КОЭФФИЦИЕНТАХ ФУНКЦИИ ОГРАНИЧЕННОЙ ВАРИАЦИИ Тажибаева Ш.Д.
Литература Байрак Л. Г. Интегральная формула Коши для кватернионов. http://scolium.narod.ru Бранец В. Н., Шмыглевский И. П. Применение кватернионов в задачах ориентации твердого тела. – М.: Наука, 1973, 320 с. UDK 517.956 SPECTRAL INEQUALITIES FOR THE NEWTON POTENTIALS Suragan D. Al-Farabi Kazakh National University, Almaty Supervisor: Prof. Baltabek Kanguzhin First, consider the one-dimensional Laplace operator  in the Hilbert space  .
Consider the spectral problem for the one-dimensional Newton potential ()  . (1)
We have  .
Integrating by part, we obtain 
 .
Thus,
 ,  .
Therefore, the boundary conditions for the one-dimensional Newton potential are  ,  .
So the spectral problem for the one-dimensional Newton potential is equivalent to the following boundary value spectral problem ,  , (2)
with the one-dimensional Newton potential boundary conditions  ,  . (3)
Solving the boundary value spectral problem (2), (3) we find two series of eigenvalues  and  where  ,  .
We enumerate these eigenvalues in increasing order and denote by Theorem 1. If , we have  , . (4)
Short proof. Dirichlet boundary conditions:   , (5)
with  . (5) has eigenvalues
  ,
Neumann boundary conditions:   , (6)
with  .
Newton potential boundary conditions:  ,  , (7)
with  ,
 , , .
Fourier analysis shows that the eigenfunctions form a basis. Furthermore, all eigenvalues , and From a), b) and с) it is easy to check (4). Now consider the spectral problem on eigenvalues of the Newton potential in the bounded Lipschitz domain with the boundary  , (8)
is a fundamental solution of the Laplace equation i.e.,  in , is the delta-function,  is the distance between two points  and  in -dimensional Euclidean space ,  is the area of the unit sphere in .
In work [1] we explicitly computed the eigenvalues of the Newton potential (8) in the 2-disk and 3-disk. It is known that the Newton potential has positive discrete spectrum in the bounded domain and we denote eigenvalues of the Newton potential by  , , and enumerate their eigenvalues in increasing order (with multiplicity taken into account). By using the proof of Filonov’s Theorem [2], the max-mini principles and some new lemmas for the Newton potential now we shall compare the eigenvalues of the Newton potential with the Dirichlet eigenvalues and the Neumann eigenvalues in any bounded Lipschitz domain  .
The main result of this work is the following generalization of Theorem 1. Theorem 2. Let ,  is a bounded Lipschitz domain then
, , (9)
where and are eigenvalues of Neumann and Dirichlet Laplasian correspondingly. References T.Sh. Kal’menov and D. Suragan, To spectral problems for the volume potential, Doklady Mathematics, 80 (2009), 646-649. N. Filonov, On an inequality between Dirichlet and Neumann eigenvalues for the Laplace Operator, St. Petersburg Math. J., Vol. 16 (2005), No. 2, 413-416. Suragan D. Eigenvalues and eigenfunctions of the Newton potential, 7th International ISAAC Congress, London, 2009, pp. 56. УДК 517.51 О ДВУМЕРНЫХ ВЕЙВЛЕТ – КОЭФФИЦИЕНТАХ ФУНКЦИИ ОГРАНИЧЕННОЙ ВАРИАЦИИ Тажибаева Ш.Д. жүктеу/скачать 13,94 Mb. Достарыңызбен бөлісу:
|
. (6) has eigenvalues
. (7) has eigenvalues
and the whole spectrum is discrete.