On the singularities of solutions of the Dirichlet problem in the exterior of a slender cone

V. G. Maz'ya   ; S. A. Nazarov   ; B. A. Plamenevskii

Sbornik. Mathematics, Tome 50 (1985) no. 2, pp. 415-437

Résumé

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.

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