Asymptotic of the solution of a boundary value problem in a thin cylinder with nonsmooth

S. A. Nazarov

Izvestiya. Mathematics , Tome 42 (1994) no. 1, pp. 183-217

Résumé

The mixed boundary value problem is considered for a selfadjoint elliptic second order equation in a three-dimensional cylinder $Q_\varepsilon$ of small height $\varepsilon$, with Dirichlet conditions on the lateral surface and Neumann conditions on the bases. The cross-section $\Omega$ of the cylinder has a corner point at 0. The full asymptotic expansion of the solution in a series of powers of the small parameter $\varepsilon$ is derived. In contrast to the iterative processes for a smooth boundary $\partial\Omega$, here there arises an additional (corner) boundary layer in the neighborhood of 0. This layer is described by means of the solutions of the boundary value problem in the domain $t=K\times(-\frac12, \frac12)$, where $K$ is a plane angle. The solvability of the problem is investigated in some Hilbert spaces of functions with weighted norms, and asymptotic representations of the solutions at infinity are established. The construction of the asymptotics of the solution with respect to $\varepsilon$ is based on the method of redistribution of residuals between the right-hand sides of the limiting problems.