In this paper we study the Banach manifold made up of simple \(C^{m+\mu }\)-domains in the Euclidean space \(\mathbb{R}^{n}\). This manifold is merely a topological or a \(C^{0}\) 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.

中文翻译：

---

欧氏空间中简单域的Banach流形及其在自由边界问题中的应用

在本文中，我们研究了由欧几里得空间\（\ mathbb {R} ^ {n} \）中的简单\（C ^ {m + \ mu} \）-域组成的Banach流形。该流形只是一个拓扑或\（C ^ {0} \）Banach流形不具有可区分的结构。现在，一些研究人员已经认识到，在这个歧管中，某些点是可以区分的，因为在这种情况下仍然可以引入切向量和切空间的概念。但是，仔细研究表明，这些概念的定义并不像乍看起来那样简单。实际上，要在此流形上建立有用的演算理论，必须克服某些技术难题。在本文中，我们使用差分拓扑的标准语言对该流形进行了系统研究，并将其构建为准可微Banach流形。因此，可以在此Banach流形中考虑微分方程。