Bounded C1 domains and their outward normals

Throughout this surface-integration page assume ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for the earlier Lebesgue and polar measure machinery. Let $n\ge 2$. A bounded $C^1$ domain is a nonempty bounded open set $\mathrm{\Omega }\subset {\mathbb{R}}^n$ whose boundary is locally, after a rigid change of coordinates, the graph $z=h(y)$ of a $C^1$ function, with $\mathrm{\Omega }$ locally exactly the subgraph $z<h(y)$. Connectedness is not required. The outward normal in these coordinates is $\nu =(-Dh,1)\mathrm{/}\sqrt{1+\mathrm{\mid }Dh{\mathrm{\mid }}^2}$, transported by the orthogonal coordinate map. Its overlap agreement is justified with surface charts below.

The convention $F\in C^1(\bar{\mathrm{\Omega }})$ means F is continuously differentiable in $\mathrm{\Omega }$, and F and its first derivatives extend continuously to its closure. For $C^2$ require the same for derivatives through order two. No ambient extension across the boundary is required. Derivatives use The total (Fréchet) derivative $Df(a)$ as the linear first-order approximation with $o(\mathrm{\parallel }h{\mathrm{\parallel }}_2)$ remainder; products are the real Euclidean inner products of Real and complex inner product spaces, with the inner product linear in the first argument. Write $\mathrm{div}F={\sum }_i{\mathrm{\partial }}_i{F}_i$, $Du=({\mathrm{\partial }}_{i}u{)}_i$ and $\mathrm{\Delta }u={\sum }_i{\mathrm{\partial }}_i^{2}u$.

Hunter, §1.10 Definitions 1.34–1.35 and §1.10.3, printed pp. 13–16 (PDF pp. 19–22). Interior-up-to-boundary regularity is the local convention.