On the singularities of solutions of the Dirichlet problem in the exterior of a~slender cone
Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 415-437
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The power singularities of solutions of the Dirichlet problem for strongly elliptic differential systems of order $2m$ outside a slender cone $k_\varepsilon$ are studied, where $\varepsilon$ is a small positive parameter which characterizes the angle vertex of the cone. In essence, the asymptotics as $\varepsilon\to0$ of the small eigenvalues $\lambda_j(\varepsilon)$ of the first boundary value problem on the unit sphere with a small hole are discussed for a differential operator depending polynomially on a complex parameter $\lambda$. As an application of the asymptotic formulas for $\lambda_j(\varepsilon)$, a theorem is obtained on the validity of a bound on the maximum modulus of a solution of the Dirichlet problem in a region with a slender conical notch.
Bibliography: 22 titles.
@article{SM_1985_50_2_a7,
author = {V. G. Maz'ya and S. A. Nazarov and B. A. Plamenevskii},
title = {On the singularities of solutions of the {Dirichlet} problem in the exterior of a~slender cone},
journal = {Sbornik. Mathematics},
pages = {415--437},
publisher = {mathdoc},
volume = {50},
number = {2},
year = {1985},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1985_50_2_a7/}
}
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V. G. Maz'ya; S. A. Nazarov; B. A. Plamenevskii. On the singularities of solutions of the Dirichlet problem in the exterior of a~slender cone. Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 415-437. http://geodesic.mathdoc.fr/item/SM_1985_50_2_a7/