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

Abstract

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.