Bounded C^k domains and boundary charts

Assume Countable Choice for the Sobolev interfaces used by consumers of this definition. Let $k\ge 1$ and $n\ge 2$. A bounded $C^k$ domain in ${\mathbb{R}}^n$ is a nonempty bounded open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ with the following local graph property. For every boundary point $x\in \mathrm{\partial }\mathrm{\Omega }$ there are an open neighbourhood $W\subseteq {\mathbb{R}}^n$ of $x$, a rigid motion $R(p)=Qp+b$ with orthogonal $Q$ and $b\in {\mathbb{R}}^n$, an open ball $B\subseteq {\mathbb{R}}^{n-1}$, and a function $h\in C^k(B;\mathbb{R})$ such that, after shrinking $W$ so that $R(W)\subseteq B\times \mathbb{R}$, $R(\mathrm{\Omega }\cap W)=R(W)\cap \{(y,s)\in B\times \mathbb{R}:s<h(y)\}\mathrm{.}$ In the coordinates $z=R(p)$ the domain therefore lies locally strictly below the graph $s=h(y)$, and the boundary is that graph. The graph convention, including the requirement that the domain occupy exactly the one-sided subgraph, is the one of Bounded C1 domains and their outward normals; the regularity $h\in C^k$ is the only strengthening here, and connectedness is not required.

Write $R(p)=(y,s)$. The flattening chart is $\mathrm{\Phi }:W⟶\mathrm{\Phi }(W)\subseteq B\times \mathbb{R},\mathrm{\Phi }(p)=(y,s-h(y)),$ with inverse ${\mathrm{\Phi }}^{-1}(y,t)=R^{-1}(y,t+h(y))$ on $\mathrm{\Phi }(W)$. Thus $\mathrm{\Phi }(\mathrm{\Omega }\cap W)=\mathrm{\Phi }(W)\cap \{t<0\}$. Both maps are of class $C^k$; the coordinate shear $(y,s)\mapsto (y,s-h(y))$ has Jacobian determinant one. If $B^{\mathrm{\prime }}\subset \overline{B^{\mathrm{\prime }}}\subset B$ is a compactly contained concentric ball, then every derivative $D^{\beta }h$ with $\mathrm{\mid }\beta \mathrm{\mid }\le k$ is continuous on $\overline{B^{\mathrm{\prime }}}$, hence bounded there, and consequently the derivatives through order $k$ of $\mathrm{\Phi }$ and of ${\mathrm{\Phi }}^{-1}$ are bounded on the corresponding compact patch. This is the precise sense in which a boundary chart is said to have bounded derivatives through order $k$ on the compact patches used; it does not assert any bound uniform in the chart, the point, or a boundary atlas.

In dimension $n=1$, the bounded sets satisfying this local one-sided condition are finite disjoint unions of bounded open intervals. Each endpoint has an interval neighbourhood on which the domain occupies one side; the coordinate is the identity or its reflection, and the graph function and derivative bounds are vacuous. General bounded open subsets of $\mathbb{R}$ need not have this property. A bounded $C^1$ domain in the sense of Bounded C1 domains and their outward normals is a bounded $C^1$ domain in this sense when $n\ge 2$.

This definition asserts no extension theorem, no trace operator, and no uniformity of chart constants: it fixes only the regularity of the boundary and the exact one-sided graph convention that later local statements use.

Laugesen, Definition 3.10, printed pp. 58–59, fixes the $C^m$ boundary-graph convention and the local one-sided subgraph placement. Oh, Proposition 11.13 and Remark 11.14, printed pp. 157–159, use $C^k$ boundary charts and record that the extension construction depends on the order $k$ through the chart regularity. The shear determinant and the compact derivative bounds recorded here are immediate from the definition of $C^k$ and are used by C^k boundary flattening preserves local W^{k,p}.