I would like to ask about the equivalence between these two definitions for a $C^1$ domain. In the book Vector Analysis Versus Vector Calculus, we have:

Definition 8.2.1: Let $\mathbb{H}^k=\{(t_1,\ldots,t_k)\in\mathbb{R}^k:\,t_k\geq 0\}$. Let $2\leq k\leq n$ and $M\subseteq\mathbb{R}^n$, $M\neq\emptyset$, be given. Then $M$ is said to be a regular $k$-surface with boundary of class $C^p$ if for every $x\in M$ there exists an injective mapping $$ \varphi: \mathbb{A}\subseteq\mathbb{H}^k\rightarrow M\subseteq\mathbb{R}^n $$ of class $C^p$ in the relatively open set $\mathbb{A}$ such that $x\in\varphi(\mathbb{A})$ and the following hold:

  1. For every relatively open subset $\mathbb{B}\subseteq \mathbb{A}$, $\varphi(\mathbb{B})$ is relatively open in $M$.

  2. For every $t\in\mathbb{A}$, the set $\{\partial_{t_1}\varphi (t),\ldots,\partial_{t_k}\varphi(t)\}$ is linearly independent.

In the book Partial Differential Equations by Evans, we have:

Definition in appendix C: Let $U\subseteq \mathbb{R}^n$ be open and bounded. We say that $\partial U$ is $C^1$ if for each point $x^0\in\partial U$ there exist $r>0$ and a $C^1$ function $\gamma:\,\mathbb{R}^{n-1}\rightarrow\mathbb{R}$ such that - upon relabeling and reorenting the coordinate axes if necessary - we have $$U\cap B(x^0,r)=\{x\in B(x^0,r):\,x_n>\gamma(x_1,\ldots,x_{n-1})\}.$$

My questions:

  1. Are both definitions equivalent? (when $k=n$, $p=1$ and $M$ bounded in the first definition).

  2. The second definition can be extended to a Lipschitz domain, usually used in PDE. Is there a version of definition 1 for "Lipschitz surfaces with boundary"?

  3. In the definition by Evans, should we add the condition $\partial U\cap B(x^0,r)=\text{Graph}(\gamma|_\mathcal{N})$, for some open set $\mathcal{N}\subseteq\mathbb{R}^{n-1}$ ?

  • 1
    $\begingroup$ I didn't read your question carefully enough at first. The definitions are not equivalent, because the first definition does not require $M$ to be bounded. $\endgroup$ – Ben McKay Jan 14 '18 at 11:52
  • $\begingroup$ @BenMcKay Ok, that's correct. I'll add $M$ to be bounded in question 1. $\endgroup$ – user39756 Jan 14 '18 at 11:53
  • $\begingroup$ The basic idea is $\phi(t_1,\dots,t_{n-1})=(t_1,\dots,t_{n-1},\gamma(t_1,\dots,t_{n-1})$. $\endgroup$ – Ben McKay Jan 14 '18 at 11:59
  • $\begingroup$ @BenMcKay So you suggest that both definitions are equivalent, and that the proof requires the implicit function theorem, right? $\endgroup$ – user39756 Jan 14 '18 at 12:01
  • $\begingroup$ Yes, the implicit function theorem. $\endgroup$ – Ben McKay Jan 14 '18 at 12:05

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