Singularities of solutions of the Neumann problem at a conical point.

*(English. Russian original)*Zbl 0701.35021
Sib. Math. J. 30, No. 3, 387-396 (1989); translation from Sib. Mat. Zh. 30, No. 3(175), 52-63 (1989).

The paper treats the singularities of the first order derivatives of the solution of Neumann’s problem in a domain \(\Omega\) with a conic point, such that close to it \(\Omega\) can be regarded as the complement of a thin cone \(\Gamma_{\epsilon}\). The asymptotic behavior of the solution in a neighborhood of such a point is known. It depends on the eigenvalues of Neumann’s problem for the Beltrami operator in \(G_{\epsilon}=\Gamma_{\epsilon}\cap S^ 2.\)

In the present paper the authors give explicit formulas for the principal parts of the asymptotics as \(\epsilon\) tends to 0 of all eigenvalues belonging to (0,1) which is sufficient to establish the behavior of the first derivatives of the solution. The work is related to a previous paper of the same authors and Plamenevskij on the singularities of the solution of the Dirichlet problem in a similar domain.

In the present paper the authors give explicit formulas for the principal parts of the asymptotics as \(\epsilon\) tends to 0 of all eigenvalues belonging to (0,1) which is sufficient to establish the behavior of the first derivatives of the solution. The work is related to a previous paper of the same authors and Plamenevskij on the singularities of the solution of the Dirichlet problem in a similar domain.

Reviewer: J.Madjarova

##### MSC:

35B40 | Asymptotic behavior of solutions to PDEs |

35J25 | Boundary value problems for second-order elliptic equations |

35C20 | Asymptotic expansions of solutions to PDEs |

##### Keywords:

singularities; first order derivatives; Neumann’s problem; conic point; thin cone; eigenvalues; Beltrami operator
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\textit{V. G. Maz'ya} and \textit{S. A. Nazarov}, Sib. Math. J. 30, No. 3, 387--396 (1989; Zbl 0701.35021); translation from Sib. Mat. Zh. 30, No. 3(175), 52--63 (1989)

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##### References:

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