zbMATH — the first resource for mathematics

Singularities of the gradient of the solution of the Neumann problem at the vertex of a cone. (English. Russian original) Zbl 0639.35018
Math. Notes 42, No. 1-2, 555-563 (1987); translation from Mat. Zametki 42, No. 1, 79-93 (1987).
Let us consider the Neumann problem for Poisson’s equation in a domain which coincides with a cone in the neighbourhood of the origin. There is studied the asymptotic behaviour of the solution to this problem near the origin.
Reviewer: P.Drábek

35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B40 Asymptotic behavior of solutions to PDEs
35J67 Boundary values of solutions to elliptic equations and elliptic systems
Full Text: DOI
[1] V. A. Kondrat’ev, ?Boundary-value problem for elliptic equations in domains with conic or corner points,? Tr. Mosk. Mat. Obsh.,16, 209-292 (1967).
[2] V. A. Kondrat’ev and O. A. Oleinik, ?Boundary-value problems for partial differential equations in nonsmooth domains,? Usp. Mat. Nauk,38, No. 2, 3-76 (1983).
[3] V. A. Kondrat’ev, I. Kopachek, M. D. Lekevishvili, and O. A. Oleinik, ?Nonimprovable estimates in Holder spaces and exact Saint-Venant principle for the solutions of the biharmonic equation,? Tr. MIAN SSSR,166, 91-106 (1984).
[4] V. G. Maz’ya, S. A. Nazarov, and B. A. Plamenevskii, ?On the singularities of the solutions of the Dirichlet problem in the exterior of a thin cone,? Mat. Sb.,122, No. 4, 435-456 (1983).
[5] M. G. Dzhavadov, ?Asymptotics of the solution of the boundary-value problem for second-order elliptic equations in thin domains,? Differents. Uravn.,5, No. 10, 1901-1909 (1968). · Zbl 0165.12302
[6] S. A. Nazarov, ?Structure of the solutions of elliptic boundary-value problems in thin domains,? Vestn. Mosk. Gos. Univ.,7, 65-68 (1982). · Zbl 0509.35008
[7] I. E. Zino and É. A. Tropp, Asymptotic Methods in Heat Conduction and Thermal Elasticity [in Russian], Leningrad State Univ. (1978).
[8] S. A. Nazarov, Introduction to Asymptotic Methods of Elasticity Theory [in Russian], Leningrad State Univ. (1983).
[9] V. L. Verdichevskii, ?High-frequency long-wave oscillations of membranes,? Dokl. Akad. Nauk SSSR,236, 1319-1322 (1977).
[10] V. V. Kucherenko and V. A. Popov, ?High-frequency oscillations of membranes,? Dokl. Akad. Nauk SSSR,244, No. 4, 819-823 (1979). · Zbl 0496.73058
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.