Functions satisfy Neumann boundary condition

Question

I have a question about Neumann boundary condition.

Let $X=\{x=(x_1,\ldots,x_d)\mid x_d>0\}$. That is, $X \subset \mathbb{R}^d$ is the upper half space of $\mathbb{R}^d$. The topological boundary $\partial X$ of $X$ is expressed as $\partial X=\{x=(x_1,\ldots,x_d)\mid x_d=0\}$. Fix $p \in \partial X$ and $U$ be a open ball of $\mathbb{R^d}$ centered at $p$.

My question

Can we find $C^2$ functions $\{f_{\alpha}\}_{\alpha \in A}$ satisfying the following conditions?:

• For each $\alpha \in A$, $\text{supp}[f_{\alpha}]\subset U$,
• For each $x,y \in U$ with $x \neq y$, there exists $\alpha \in A$ such that $f_{\alpha}(x) \neq f_{\alpha}(y)$,
• For each $x \in U$, there exists $\alpha \in A$ such that $f_{\alpha}(x) \neq 0$,
• For each $\alpha \in A$, $\sum_{i=1}^{d}\frac{\partial f_{\alpha}}{\partial x_i}n_i=0$ on $\partial X$, where $n=(n_1,\ldots,n_d)$ is the outward normal to $\partial X$.

Answer 1

Yes if $A$ is allowed to be uncountable.

For each point $x\in U\setminus \partial X$ put into the set all bump functions centered at $x$ and supported in a ball of radius stricly smaller than $\mathrm{dist}(x, \partial X\cup \partial U)$. These functions vanish identically on a neighbourhood of $\partial X$, so no problem with the Neumann boundary condition. Moreover, these functions separate points on $U\setminus \partial X$.

The true difficulty, as you surely know, lies on $\partial X$. It would be easier if $U$ were an open rectangle, that is $$ U=\partial X \times (-1, 1),\quad \partial X=(-1,1) \times \ldots \times (-1, 1).$$ Choose $\phi\colon \mathbb R\to \mathbb R$ supported in $(-1, 1)$ and such that $\phi'(0)=0$. Then for each function $g$ supported on $\partial X$ you can construct a function supported on $U$ by multiplication: $$ f(x_1, \ldots, x_{d-1}, x_d)=g(x_1, \ldots, x_{d-1})\phi(x_d), $$ and the function $f$ thus constructed satisfies $$ \left.\frac{\partial f}{\partial x_d}\right|_{x_d=0}= g(x_1,\ldots ,x_{d-1})\phi'(0)=0.$$

So choosing a system of functions $g_\alpha$ on $\partial X$ that are smooth and separate points, this construction produces a set of functions supported on $U$ that satisfies the same property and also a Neumann condition on $\partial X$.