Use of very weak approximate boundary layer solutions to spatially nonsmooth singularly perturbed problems (Q2236022)

The author studies singularly perturbed spatially nonsmooth semilinear second-order boundary value problems (BVPs) of the type \[ \varepsilon^2\frac{\mathrm{d}}{\mathrm{d}x}\left(a(x,\varepsilon)\frac{\mathrm{d}}{\mathrm{d}x}u(x)\right) = f(x,u(x),\varepsilon), \ x\in[0,1],\ 0 < \varepsilon \ll 1, \] \[ u(0) = u^0,\ u(1) = u^1. \] Using the abstract implicit function theorem, the author proves in Theorem 1.1 the existence and local uniqueness of solutions \(u = u_{\varepsilon}\) to the Dirichlet BVP above which are close to a given family of zeroth-order approximate boundary layer solutions of the type \[ \bar{u}_{\varepsilon}(x) =: \bar{u}(x) + \varphi_0(x/\varepsilon) + \varphi_1((1 - x)/\varepsilon), \] where \(\bar u\) is a solution of a degenerate problem \(f(x,\bar{u},0) = 0\) obtained by taking \(\varepsilon = 0\) in the differential equation.