Proof request: Extending vector valued function of unit outer normals to the whole space $\Bbb R^n$.

Question

I was wondering if anyone could provide a proof or a reference of the following proposition:

Let $C$ be an open subset of $\mathbb{R}^n$ with $C^2$ boundary so that the unit outer normal vector $v(x)$ will be a $C^1$ function i.e $v: \partial C \to \mathbb{S}^{n-1}$, then it can be extended to a $C^1$ function $N: \mathbb{R}^n \to \mathbb{R}^n $ such that $|N(x)| \leq 1$.

Unfortunalety I don't know much differential geometry so it is not obvious to me. For the context, this is used when proving that the characteristic function of a $C^2$ open bounded set $C$ is of bounded variation and its total variation equals $\mathcal{H}^{n-1}(\partial C)$.

Thanks in advanced

Answer 1

Given the regularity of the boundary of the open set involved, in my opinion the best way to solve directly the problem is to use Whitney's extension theorem. This has to be done by following the two steps here below

1. First step: compute the normal vector $v:\partial C\to \Bbb S^{n-1}$. Since the set $C$ has a $C^2$ boundary, this step does not really involve any particular difficulty, and is indeed classical.

2. Second step: apply Whitney's extension theorem to each of the vector components $v_s$, $s= 1,\ldots,n$ of the normal $v(x)$, $x\in\partial C$ and find a $C^1(\Bbb R^n)$ vector field $N(x)$ such that $$ N |_{x\in\partial C}= v $$ Indeed, in its sharpest and most recent version due to Charles Fefferman, this result (theorem C, p. 510) states that given $m, n \ge 1$, there exists $k$, depending only on $m$ and $n$, for which the following holds.
   Let $f : E \to\Bbb R$ be given, with $E$ an arbitrary subset of $\Bbb R^n$. Suppose that, for any $k$ distinct points $x_1,\ldots, x_k \in E$, there exist $(m−1)$^rst degree polynomials $P_1,\ldots, P_k$ on $\Bbb R^n$, satisfying

   1. $P_i(x_i) = f(x_i)$ for $i = 1,\ldots,k$;
   2. $|\partial^\beta P_i(x_i)|\le M$ for $i=1,\dots,k$ and $|\beta|\le m−1$; and
   3. $|\partial^\beta(P_i − P_j)(x_i)| \le M|x_i − x_j|^{m−|\beta|}$ for $i, j = 1,\ldots,k$ and $|\beta| \le m − 1$; with $M$ independent of $x_1,\ldots, x_k$.

   Then $f$ extends to a $C^{m−1,1}$ function on $\Bbb R^n$.

   Let's have a sketchy intuition how this extension result applies in our case: first note that since $v_s\in C^1(\partial C)$ for all $s=1,\ldots,n$ by construction, the polynomials $P_{s1},\ldots, P_{sk}$ are simply constants defined by the relationships $$ P_{si}(x)=P_{si}=v_s(x_i)\quad\forall\; i=1,\ldots,k \,\wedge\, s=1, \ldots, n $$ and since $v_s$ is the $s$-th component of a unit vector (the normal) we have $$ |v_s(x)|\le 1\quad \forall\; s=1, \ldots, n\quad \forall x\in \partial C $$ Therefore all the three conditions above are trivially satisfied and then there exist $n$ functions $N_1(x), \ldots,N_n(x)\in C^{0,1}(\Bbb R^n)$ such that the vector function $N(x) =(N_1(x), \ldots, N_n(x))\in C^{0,1}(\Bbb R^n,\Bbb R^n)$ such that $$ N|_{x\in\partial C} = v \iff \|N(x)\| = 1\text{ on }\partial C. $$

Notes

The way of constructing the extension $N$ of the normal $v$ to a $C^1$-manifold described above it is not elementary as it implies the use of powerful theorems: however, there is another, perhaps simpler approach due to Ennio De Giorgi. De Giorgi defines an approximate normal to the boundary of an (almost arbitrary) set by applying the Weierstrass transform to its characteristic function, calculate its gradient and normalize it respect to the vector norm of the gradient itself. This approach has several merits, to be detailed in the next edits

References

[1] Charles Fefferman, "A sharp form of Whitney’s extension theorem". (English) Annals of Mathematics, Second Series 161, No. 1, pp. 509-577 (2005), MR2150391, Zbl 1102.58005.