Let \( m \in \mathbb{N}_0 \) and \( \alpha \in (0,1] \). We say that a bounded domain \(\Omega \subseteq \mathbb{R}^d\) has a \(C^{m,\alpha}\)-boundary if, for any boundary point \(x_0\in \partial \Omega\), there exists a neighborhood \(U\) of \(x_0\) such that, after translating and rotating the coordinate system, the set \(U\cap \partial \Omega\) can be represented as the graph of a function \(h:U' \to \mathbb{R}\), where \(U' \subseteq \mathbb{R}^{d-1}\) and \(h \in C^{m,\alpha}(\overline{U'})\). More precisely, each \(x\in U\) can be written uniquely as

\[ (x', x_d) = (y', y_d + h(y')) \]

for some \(y' \in U'\). Moreover,\(y_d > 0\) for \(x \in U \cap \Omega\), \(y_d < 0\) for \(x \in U \cap \Omega^c\), and \(y_d = 0\) for \(x \in U \cap \partial \Omega\).

If \(\Omega\) has a \(C^{m,\alpha}\)-boundary, we write \(\partial \Omega\in C^{m,\alpha}\).

For a domain \(S\subseteq \Omega\), we say \(S\) has a \(C^{m,\alpha}\)-boundary with respect to \(\Omega\), if the above condition holds for every \(x_0\in \partial S \cap \Omega\), where neighborhoods are taken with respect to the subspace topology \(\Omega\).

Remarks
  • Suppose \(\Omega\) is bounded and has a \(C^{m,\alpha}\)-boundary. Since \(\Omega\) is compact, there is a finite cover \((U_i)\) of \(\partial \Omega\) and a partition of unity \((\varphi_i)\) with \[ \sum \varphi_i \equiv 1 \text{ on } \bigcup U_i. \] We say that \((U_i, \varphi_i)\) is a \(C^{m,\alpha}\)-localization of \(\Omega\).
  • It is \(\partial \Omega \in C^{m,\alpha}\) if and only if \(\partial \Omega\) is a [\(C^{m,\alpha}\) manifold](smooth manifold.md) [1, Lemma 6.6].

See also Link to heading

References Link to heading

  1. M. Dobrowolski, Angewandte Funktionalanalysis: Funktionalanalysis, Sobolev-Räume und elliptische Differentialgleichungen. Berlin, Heidelberg: Springer Berlin Heidelberg, 2010. doi:10.1007/978-3-642-15269-6