A quasilinear elliptic equation with small parameter (Q1076228)

Let u(x,y,\(\epsilon)\) be \(2\pi\)-periodic in y solution of the singularly perturbed problem \[ \epsilon \Delta u+\partial u/\partial x+(\partial u/\partial y)^ 2=0;\quad u|_{x=0}=\phi_ 0(y),\quad u|_{x=1}=\phi_ 1(y), \] where \(\phi_ 0(y)\) and \(\phi_ 1(y)\) are \(2\pi\)-periodic and \(0\leq x\leq 1\). Under some conditions it is shown that the problem is regularly degenerating as \(\epsilon\) \(\to 0\) to the problem \[ \partial u/\partial x+(\partial u/\partial y)^ 2=0,\quad u|_{x=1}=\phi_ 1(y). \]