  In this paper we study the Banach manifold made up of simple
$C^{m+\mu}$-domains in the Euclidean space $\mathbb{R}$. This manifold is
merely a topological or a $C^0$ Banach manifold. It does not possess a
differentiable structure. We introduce the concept of differentiable point in
this manifold and prove that it is still possible to introduce the concept of
tangent vector and tangent space at a differentiable point. Consequent, it is
possible to consider differential equations in this Banach space. We show how
to reduce some important free boundary problems into differential equations in
such a manifold and then use the abstract result that we established earlier to
study these free boundary problems.
