On a singular perturbation problem (Q1355815)

The author considers the following Dirichlet problem: \[ \varepsilon^2{\partial^2\over\partial x^2} u_\varepsilon-{\partial^2\over\partial y^2} u_\varepsilon+ u_\varepsilon= f_\varepsilon\quad\text{in} \quad \Omega,\quad u_\varepsilon= 0\quad\text{on} \quad \Gamma=\partial\Omega, \] where \(\omega=]0,\pi[\times]0,\pi[\) and \(f_\varepsilon\) is in the space \(L^2(\Omega)\). It is shown that as \(f_\varepsilon\) converges to \(f\) in \(L^2(\Omega)\), the solution of the Dirichlet problem converges to the solution of \[ -{\partial^2\over\partial y^2} u+ u=f\quad\text{in} \quad\Omega,\quad u(x,0)= u(x,\pi)= 0\quad\text{for} \quad 0<x<\pi. \] A further asymptotic result is also proved.