Iii республикалық студенттік ғылыми-практикалық конференциясының баяндамалар жинағЫ

1. 
   Байрак Л. Г. Интегральная формула Коши для кватернионов. http://scolium.narod.ru
   
2. 
   Бранец В. Н., Шмыглевский И. П. Применение кватернионов в задачах ориентации твердого тела. – М.: Наука, 1973, 320 с.
   

UDK 517.956

Al-Farabi Kazakh National University, Almaty

We have

Integrating by part, we obtain

Thus,

with the one-dimensional Newton potential boundary conditions

Solving the boundary value spectral problem (2), (3) we find two series of eigenvalues

We enumerate these eigenvalues in increasing order and denote by , .

1. 
   Dirichlet boundary conditions:
   

2. 
   Neumann boundary conditions:
   

with . (6) has eigenvalues

3. 
   Newton potential boundary conditions:
   

with , . (7) has eigenvalues

Fourier analysis shows that the eigenfunctions form a basis. Furthermore, all eigenvalues , and are positive and ,, and the whole spectrum is discrete.

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

Where

In work [1] we explicitly computed the eigenvalues of the Newton potential (8) in the 2-disk and 3-disk.

where and are eigenvalues of Neumann and Dirichlet Laplasian correspondingly.

1. 
   T.Sh. Kal’menov and D. Suragan, To spectral problems for the volume potential, Doklady Mathematics, 80 (2009), 646-649.
   
2. 
   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.
   
3. 
   Suragan D. Eigenvalues and eigenfunctions of the Newton potential, 7^th International ISAAC Congress, London, 2009, pp. 56.
   

УДК 517.51