A complete asymptotics of the solution of the Dirichlet problem for a two-dimensional Laplace equation with rapidly oscillating boundary data (Q1375003)

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.