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

We consider singularly perturbed Dirichlet problems which are, in the simplest nontrivial case, of the type epsilon(2)u '' (x) = f(x, u(x)) for x is an element of [0, 1], u(0) = u(0), u(1) = u(1). For small epsilon > 0we prove existence and local uniqueness of solutions u = u(epsilon), which are close to functions of the type (u) over bar (x)(x) = (u) over bar (x) + phi(1) ((1-x)m) with f(x, (u) over bar (x)) = 0 for x is an element of [0, 1] and with phi(0)'' (xi) = f(0, (u) over bar (0) + phi(0)(xi)) for xi is an element of [0, infinity), (u) over bar (0) + phi(0)(infinity) = u(0), theta(0)(infinity) = 0, phi(1)'' (xi) = f(1, u(1) + phi(1)(xi)) for xi is an element of [0, infinity), u(1) + phi(1)(0) = u(1), phi(1)(infinity) = 0, and we show that parallel to u(epsilon)- (u) over bar (epsilon)parallel to(infinity) -> 0 for epsilon -> 0. We do not suppose monotonicity of the correctors theta(0)and theta(1). And, mainly, we do not suppose any regularity besides continuity of the functions f(., u) and (u) over bar. Hence, the functions (u) over bar (epsilon) approximately satisfy the boundary value problem for epsilon approximate to 0 in a very weak sense only, and one cannot expect that parallel to u(epsilon) - (u) over bar (epsilon)parallel to(infinity) = O(epsilon) for epsilon -> 0, in general. For the proofs we use an abstract result of implicit function theorem type which was designed for applications to spatially nonsmooth singularly perturbed boundary value problems with nonsmooth approximate solutions. (C) 2021 Elsevier Inc. All rights reserved.