Non-classical boundary layers for fourth-order equations with singular limit solution. (English) Zbl 1021.35026

The author deals with the following fourth order linear model \[ \begin{cases} \varepsilon\Delta^2u^\varepsilon + u^\varepsilon=H\quad \text{in }\Omega,\\ u^\varepsilon=\phi_0,\quad \frac{\partial u^\varepsilon}{\partial\nu} = \phi_1\quad \text{on }\Gamma,\end{cases}\tag{1} \] where \(\phi_0\) and \(\phi_1\) are two given (smooth) functions, \(\Omega\) is a channel of \(\mathbb{R}^3\) and \(H\) is the Heaviside function in the direction orthogonal to the walls (\(z\)-direction). In the \(x\) and \(y\) directions the boundary condition is periodic for \(u^\varepsilon\). The author derives the first-order term in the asymptotic expansion of \(u^\varepsilon\) with respect to \(\varepsilon\).


35C20 Asymptotic expansions of solutions to PDEs
35J40 Boundary value problems for higher-order elliptic equations
35R05 PDEs with low regular coefficients and/or low regular data
35B25 Singular perturbations in context of PDEs
35A20 Analyticity in context of PDEs