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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.