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.
Reviewer: J.Madjarova


35B40 Asymptotic behavior of solutions to PDEs
35J25 Boundary value problems for second-order elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
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[1] V. A. Kondrat’ev and O. A. Oleinik, ?Boundary problems for partial differential equations in nonsmooth regions,? Usp. Mat. Nauk,38, 3-76 (1983).
[2] V. G. Maz’ya, S. A. Nazarov, and V. A. Plamenevskii, ?Singularities of solutions of the Dirichlet problem in the exterior of a thin cone,? Mat. Sb.,122, No. 4, 453-456 (1983).
[3] G. Polya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Kraus Reprints (1951).
[4] V. G. Maz’ya, ?Functions with a finite Dirichlet integral in domains with a peak on the boundary,? in: Linear Operators and the Theory of Functions [in Russian], Vol. 126, Zap. LOMI, Leningrad (1983), pp. 117-137.
[5] I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Non-Self-Adjoint Operators, Amer. Math. Soc. (1967).
[6] V. G. Maz’ya, S. A. Nazarov, and V. A. Plamenevskii, Asymptotic Expansions of Solutions of Elliptic Boundary Problems for Singular Perturbations of the Domain [in Russian], Izd. TGU, Tbilisi (1981).
[7] V. A. Kondrat’ev, ?Boundary problems for elliptic equations in regions with conic or corner points,? Tr. Mosk. Mat. Ova.,16, 219-292 (1967).
[8] M. I. Vishik and L. A. Lyusternik, ?Solution of several problems concerning peturbations in the case of matrices and self-adjoint and non-self-adjoint differential equations,? Usp. Mat. Nauk,15, No. 4, 37-95 (1960).
[9] M. M. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations, Noordhoff International Pub., Leyden (1974). · Zbl 0274.47033
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