LAUSR

In this paper we study the Banach manifold made up of simple C m + μ $C^{m+\mu }$ -domains in the Euclidean space R n $\mathbb{R}^{n}$ . This manifold is merely a topological or a C 0 $C^{0}$ Banach manifold not possessing a differentiable structure. It has now been recognized by some researchers that in this manifold some points are differentiable in the sense that it is still possible to introduce the concepts of tangent vectors and the tangent space at such a point. However, a careful study shows that definitions of these concepts are not as simple as it might look at first sight. In fact, to establish a useful calculus theory on this manifold certain technical difficulties must be overcome. In this paper we use standard language of differential topology to make a systematic investigation to this manifold and build it into a quasi-differentiable Banach manifold. Consequent, it is possible to consider differential equations in this Banach manifold. As an application we also discuss how to reduce some important free boundary problems into differential equations in such a manifold or some of its vector bundles.