An Implicit Function Theorem and Applications to Nonsmooth Boundary Layers

Abstract

First we present an abstract result of implicit function theorem type. Then we apply this to singularly perturbed boundary value problems of the type

$$ \begin{array}{l} \varepsilon ^2\left( a(x)u'(x)\right) '+ b(x,u(x),\varepsilon )= 0, \quad x \in (0,1),\\ u(0)=u'(1)=0, \end{array} $$

which are spatially nonsmooth, i.e. such that $x \mapsto a(x)$ is allowed to be discontinuous and that $x \mapsto b(x,u,\varepsilon )$ is allowed to be non-differentiable. We show existence and local uniqueness of boundary layer solutions $u_\varepsilon$ close to zeroth order approximate boundary layer solutions $u^0_\varepsilon$, i.e. such that $\Vert u_\varepsilon -u^0_\varepsilon \Vert _\infty \rightarrow 0$ for $\varepsilon \rightarrow 0$. These results are straightforward generalizations of those which are known for spatially smooth problems. But the rate of convergence $\Vert u_\varepsilon -u^0_\varepsilon \Vert _\infty \sim \varepsilon$, which is known for spatially smooth problems, is not true anymore for spatially nonsmooth problems, in general.

This paper is dedicated to Bernold Fiedler’s 60th birthday.