My question is based on Change of Variables in Multiple Integrals II Peter D. Lax > It is not necessary to read the paper before answering this question.The author tried to prove change of variables formula using elementary method but turns out to be buried in complexity.
My question is essentially theorem 2(on this post)
To ask my question , I will first state and prove the obvious result in one-dimension:
$$\quad \epsilon>0, n>0,x \in R,[a b] \subset R,b>a,y:R->R$$
Theorem 1: Given any continuously differentiable function $y(x)$ and if $\frac{dy}{dx}>0 $ for $x \in [a, b]$: Then there exists a variable $\varphi_n$, and $\varphi_n(x)$ is continuously differentiable over $\mathbb{R}$ with the following properties:
i. $|\varphi_n(x)-y(x)| \le \epsilon$ for $x \in (a, b)$
ii. $\varphi_n(x)=x$ for sufficiently large $|x|$
iii. $\lim\limits_{n\mapsto 0}\varphi_n(x)=y$ and $\lim\limits_{n\mapsto 0}\frac{d\varphi_n}{dx}=\frac{dy}{dx}>0$ for $x \in (a, b)$
iv. The mapping $\varphi:x\to R$ is one-to-one, so it is invertible
Proof:
$r(x)=-{\frac {n}{x^2-a^2}}$
$h(x)=-{\frac {n}{b^2-x^2}}$
$\psi_n(x) = \begin{cases} e^{r(x)+h(x)}, & \text{if $a <x< b$ } \\ 0, & \text{otherwise} \end{cases}$
define $\varphi_n=x(1-e^{-nx^2})+\psi_n(x) y(x)+\int_{-\infty}^x2nt(\frac{-1}{(t^2-a^2)^2}+\frac{1}{(b^2-t^2)^2})\psi_n(t)y(t)dt$
it is easily seen $\frac{d\varphi_n}{dx}=(1-e^{-nx^2})+2nx^2e^{-nx^2}+\psi_n(x)\frac{dy}{dx}$
all the terms on the right hand side are greater than zero
Taking limits and using inverse function theorem in one dimension, the theorem follows
Theorem 2: $f:\mathbb{R}^N\to\mathbb{R}^N$, $x \in R^N$, $y=f(x)$, $y=[y_1(x), y_2(x),y_3(x),..y_N(x)]$, $x=[x_1,x_2,\dots,x_N]$, and the the determinant of the Jacobian matrix of $y$ denoted as $\det J$ . Under what conditions can we always find a conitnuously differentiable $\varphi_n: \mathbb{R}^N\to\mathbb{R}^N$ such that:
i. $|\varphi_n(x)-y(x)| \le \epsilon$ inside some bounded region A
ii. $\varphi_n(x)=x$ for sufficiently large norm $|x|$
iii. $\lim\limits_{n\mapsto 0}\varphi_n(x)=y$ and $\lim\limits_{n\mapsto 0}\det J_n =\det J$ inside some bounded region and $|\det J_n|>0$ for all $x$,where $\det J_n$ is determinant of Jacobian matrix of $\varphi_n(x)$ ?
This is of practical importance because if the above properties are satsified by $\varphi_n$, then according to Hadamard's global inverse function theorem then the mapping $\varphi_n :R^N->R^N$ is globally bijective.
This is not an answer (also too long for a comment I guess) but what I was only able to do:
$i \in N$
$a_i<b_i$
$n>0,\epsilon>0,\delta>0$
$r(x_i)=-{\frac {n}{x_i^2-a_i^2}}$
$h(x_i)=-{\frac {n}{b_i^2-x_i^2}}$
$\psi(x_i) = \begin{cases} e^{r(x_i)+h(x_i)}, & \text{if $a_i <x_i< b_i$ } \\ 0, & \text{otherwise} \end{cases}$
$A_i=${$x: a_i <x_i< b_i$}
$A=\bigcap_{i=1}^N A_i$
$z^2=\sum_{i=1}^N x_i^2$
$p_i(x)=\psi(x_1)\psi(x_2)\psi(x_3)...\psi(x_N)y_i(x)$
$p_i(x)=0$ for $x \notin A$
$g_i(x)=x_i(1-e^{-nz^2})$
$\phi_i(x)=p_i(x)+g_i(x)$
$\varphi_n(x)=[\phi_1(x),\phi_2(x),\phi_3(x),...\phi_N(x)]$
$p_n(x)=[p_1(x),p_2(x),p_3(x),...p_N(x)]$
$g_n(x)=[g_1(x),g_2(x),g_3(x),...g_N(x)]$
$J,J_{p_n},J_{g_n},J_{\varphi_n}$ denote the Jacobian Matrix of $y(x),p_{n}(x),g_{n}(x),\varphi_n(x)$ respectively
I was able to show $\lim\limits_{n\mapsto 0}\varphi_n(x)=y(x)$ for every $x \in A$ and $\lim\limits_{n\mapsto 0}|detJ_{\varphi_n}|=|detJ|$ for every $x \in A$
and for sufficiently large $|x|$ i.e large $|z|$, $\varphi_n(x)=x$
Using Matrix determinant lemma, I was able to show $detJ_{\varphi_n}=detJ_{g_n}>0$ for $x \notin A$
I was also able to show for sufficiently small $n$ ,
$\mu^*(A \bigcap B) \le \epsilon$
where $B=${x:$|detJ_{\varphi_n}|=0$}
How can I fix my $\varphi_n(x)$ so that it maintains all the stated properties and $|detJ_{\varphi_n}|>0$ for all $x$ ?