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JOURNALS // Zhurnal Srednevolzhskogo Matematicheskogo Obshchestva // Archive

Zhurnal SVMO, 2014, Volume 16, Number 4, Pages 7–22 (Mi svmo501)

Eigenvalue problem for the Laplace operator in $s$-dimensional unit ball $\Omega\subset\R^{s+1}$ with displacements in derivatives II

A. V. Gerasimova, B. V. Loginovb, N. N. Yuldashevc

a Ogarev Mordovia State University
b Ulyanovsk State Technical University
c Tashkent Institute of Textile and Light Industry

Abstract: 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 [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 $s>2$.

UDC: 517.9

Received: 29.12.2014



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