Dirichlet problem in an angular domain with rapidly oscillating boundary: Modeling of the problem and asymptotics of the solution

S. A. Nazarov

Algebra i analiz, Tome 19 (2007) no. 2, pp. 183-225.

Résumé

Leading asymptotic terms are constructed and justified for the solution of the Dirichlet problem corresponding to the Poisson equation in an angular domain with rapidly oscillating boundary. In addition to an exponential boundary layer near the entire boundary, a power-law boundary layer arises, which is localized in the vicinity of the corner point. Modeling of the problem in a singularly perturbed domain is studied; this amounts to finding a boundary-value problem in a simpler domain whose solution approximates that of the initial problem with advanced precision, namely, yields a two-term asymptotic expression. The way of modeling depends on the opening $\alpha$ of the angle at the corner point; the cases where $\alpha\pi$, $\alpha\in(\pi,2\pi)$, and $\alpha=2\pi$ are treated differently, and some of them require the techniques of selfadjoint extensions of differential operators.

Détail

Keywords: Dirichlet problem, oscillating boundary, corner point, asymptotics, selfadjoint extension.

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