Eigenvalue problem for the Laplace operator in $s$-dimensional unit ball $\Omega\subset\R^{s+1}$ with displacements in derivatives II
A. V. Gerasimov1, B. V. Loginov2, N. N. Yuldashev3
| Annotation | In the class of continuous and continuously differentiable up to the second order functions the boundary eigenvalue problem for the Laplace operator in s-dimensional unit ball $\Omega$ with displacements in derivatives along the radii $0<r_0<1$ and 1 of the concentric spheres is considered, i.e. $u\in C^{2+\alpha}(\Omega)$ and $\frac{\partial u(r_0,\theta)}{\partial r}=\frac{\partial u(1,\theta)}{\partial r}$. In the previous work of the authors \cite{svmo1} were found eigenvalues and for $s=2$ eigen- and adjoint functions (Jordan chains) for the direct problem; and their length does not exceed three. In this work, calculated Jordan chains for the conjugate problem when $s=2$, the direct and conjugate problems when $s>2$, and it is proved that if $s>2$ they are terminated at the second elements. |
|---|---|
| Keywords | Laplace operator, unit ball in $\R^{s+1}$, eigenvalues, eigen- and adjoint functions, Jordan chains, direct and conjugate problems for $s=2$ and $s>2$. |
1Postgraduate student of Applied Mathematics, Differential Equations and Theoretical Mechanics Chair, Ogarev Mordovia State University, Saransk; gerasimov_artyom@mail.ru.
2Professor of Higher Mathematics Chair, Ulyanovsk State Technical University, Ulyanovsk; bvllbv@yandex.ru.
3Docent of Higher Mathematics Chair, Tashkent Institute of Textile and Light Industry, Tashkent; nurilla1956@mail.ru.
Citation: A. V. Gerasimov, B. V. Loginov, N. N. Yuldashev, "[Eigenvalue problem for the Laplace operator in $s$-dimensional unit ball $\Omega\subset\R^{s+1}$ with displacements in derivatives II]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,16:4 (2014) 7–22 (In Russian)
