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The mathematical problem

The model problem we propose is to find the solution of the equation
\varepsilon \Delta u - u_x = 0
\end{displaymath} (1)

for some parameter $\varepsilon >0$. The equation is defined on the exterior of the unit circle, i.e., on the region
\Omega = \left\{ (x,y) ~\vert~ x^2+y^2 > 1 \right\} \subset \mathbb{R}^2
\end{displaymath} (2)

satisfying the boundary conditions
$\displaystyle u(x,y;\varepsilon ) =1$ $\textstyle \mathrm {for}$ $\displaystyle x^2+y^2=1$ (3)
$\displaystyle u(x,y;\varepsilon ) =0$ $\textstyle \mathrm {for}$ $\displaystyle x^2+y^2\to\infty$ (4)