This article is cited in 2 papers

Brief communications

The study of boundary value problems for a singular $B$-elliptic equation by the method of potentials

E. V. Chebatoreva

Chair of Mathematical Analysis, Tatar State University of Humanities and Education, Kazan, Russia

Abstract: In this paper we apply the method of potentials for studying the Dirichlet and Neumann boundary value problems for a $B$-elliptic equation in the form
$$ \Delta_{x''}u+B_{x_{p-1}}u+x_p^{-\alpha}\frac\partial{\partial x_p}\left({x_p^\alpha\frac{\partial u}{\partial x_p}}\right)=0, $$
where $\Delta_{x''}=\sum^{p-2}_{j=1}\frac{\partial^2}{\partial x_j^2}$, $B_{x_{p-1}}=\frac{\partial^2}{\partial x_{p-1}^2}+\frac k{x_{p-1}}\frac\partial{\partial x_{p-1}}$ is the Bessel operator, $0<\alpha<1$ and $k>0$ are constants, $p\ge3$. We prove the unique solvability of these problems.

Keywords: Bessel operator, $B$-elliptic equation, Dirichlet problem, Neumann problem, method of potentials.

UDC: 517.946

Received: 01.12.2009

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