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

Abstract

We consider singularly perturbed Dirichlet problems which are, in the simplest nontrivial case, of the typeε2u″(x)=f(x,u(x)) for x∈[0,1],u(0)=u0,u(1)=u1. For small ε>0 we prove existence and local uniqueness of solutions u=uε, which are close to functions of the typeu¯ε(x)=u¯(x)+φ0(x/ε)+φ1((1−x)/ε) with f(x,u¯(x))=0 for x∈[0,1] and withφ0″(ξ)=f(0,u¯(0)+φ0(ξ)) for ξ∈[0,∞),u¯(0)+φ0(0)=u0,φ0(∞)=0,φ1″(ξ)=f(1,u¯(1)+φ1(ξ)) for ξ∈[0,∞),u¯(1)+φ1(0)=u1,φ1(∞)=0, and we show that ‖uε−u¯ε‖∞→0 for ε→0. We do not suppose monotonicity of the correctors φ0 and φ1. And, mainly, we do not suppose any regularity besides continuity of the functions f(⋅,u) and u¯. Hence, the functions u¯ε approximately satisfy the boundary value problem for ε≈0 in a very weak sense only, and one cannot expect that ‖uε−u¯ε‖∞=O(ε) for ε→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.