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

On the Banach manifold of simple domains in the Euclidean space and applications to free boundary problems