A complete asymptotics of the solution of the Dirichlet problem for a two-dimensional Laplace equation with rapidly oscillating boundary data
type: Item
instance of: scholarly article
title: A complete asymptotics of the solution of the Dirichlet problem for a two-dimensional Laplace equation with rapidly oscillating boundary data
zbMATH Open document ID: 0887.35023
author: A. S. Demidov
published in: Doklady Mathematics
publication date: 1998-06-03T00:00:00Z
review text: We present two methods for constructing the asymptotics with respect to \(\varepsilon\to 0\) in the metric \(C^k(\overline\Omega)\) for \(k\in\mathbb{N}\) for the solution of the problem \[ \Delta u= 0\quad\text{in }\Omega\Subset\mathbb{R}^2,\quad u|_{s\in\partial \Omega}= f_\varepsilon(s),\tag{1} \] where the domain \(\Omega\) has the analytic boundary \(\Gamma=\partial\Omega\) that is diffeomorphic (for simplicity) to a circle, \(s\) is a point of \(\Gamma\) that is identified with the length of an arc along \(\Gamma\), \(f_\varepsilon(s)= f(s/\varepsilon)\), and \(f\in C^\infty(\mathbb{T}^1)\), \(\mathbb{T}^n\) is an \(n\)-dimensional torus.
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B20
zbMATH DE Number: 1099466
zbMATH Keywords: asymptotics
zbMATH Keywords: \(n\)-dimensional torus
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1099466