$\newcommand{\SO}[1]{\text{SO}(#1)}$
$\newcommand{\dist}{\operatorname{dist}}$


Let $\mathbb{D}^n$ be the closed $n$-dimensional unit ball, and let $f:\mathbb{D}^n \to \mathbb{R}^n$ be smooth.


Set 
$$X=\text{GL}^+\_n \cup \{ A \in M\_n \, | \text{ the singular values of } \, A \text{ are distinct }\}$$
Here $M\_n$ is the space of real $n \times n$ matrices.



> 
> Do there there exist $f\_n \in C^{\infty}(\mathbb{D}^n, \mathbb{R}^n)$ such that $f\_n \to f$ in $W^{1,2}(\mathbb{D}^n, \mathbb{R}^n)$ and $df\_n \in X$ everywhere on $ \text{int}(\mathbb{D}^n) $? 
> 
> 
> 


Can we at least perturb $f$ to make the points where the are recurring singular values isolated? We need to understand what happens to the zeroes of the [discriminant](https://en.wikipedia.org/wiki/Discriminant#Zero_discriminant) of the characteristic polynomial of $df^Tdf$ under perturbation.



Can we perturb a map $\mathbb{R}^n \to \mathbb{R}^n$ to have distinct singular values?
Here is a positive answer for $n=2.$ The argument doesn't seem to generalize easily to higher $n.$ The idea is to write $df\in X$ as $\star df\_1 + df\_2\neq 0$ and make use of Hodge decomposition.


The set $X$ consists of the matrices not of the form $(\begin{smallmatrix}a&b\\b&-a\end{smallmatrix}),$ in the $(dx,dy)$ basis (for matrices with $\sigma\_1=\sigma\_2,$ the middle part of an SVD is scalar, so the matrix has to be a scalar multiple of an orthogonal matrix). So we want to find an approximating sequence $g^{(n)}$ with


$$\star dg^{(n)}\_1 + dg^{(n)}\_2\neq 0$$
on the unit ball, with the usual Hodge star operator $\star(a\;dx + b\;dy)=-b\;dx+a\;dy.$.


For the Hodge decomposition I want to replace $\mathbb D^2$ by a more convenient space. I'll use $\mathbb R^2,$ but using a non-compact manifold is not essential an the argument could use a compactification $\mathbb R^2/\Lambda\mathbb Z^2.$


We can assume $f$ extends to a function $\bar f$ in $C^\infty\_c(\mathbb R^2,\mathbb R^2),$ for example using the Sobolev extension theorem plus a standard result on density of compactly-supported smooth functions in Sobolev spaces.
The combination $\zeta=\star d\bar f\_1+d\bar f\_2$ is a compactly-supported smooth $L^2$ vector field.


By a perturbation we can arrange that, on a neighborhood of $\mathbb D^2,$ $\zeta$ is non-zero except at isolated points Specifically...


* pick a bounded open neighborhood $U$ of $\mathbb D^2$
* pick a $\psi$ in $C^\infty\_c(\mathbb R^2,\mathbb R^2)$ that is strictly positive on $U$
* define $\phi:U\times \mathbb R^2\to\mathbb R^2$ by $\phi(x,M)=(M-\zeta(x))/\psi(x)$
* and consider a regular value $N\approx (0,0)$ for the restriction $\phi|\_{U\times\{(0,0)\}}$


The preimage $\phi^{-1}(\{N\})$ is the graph $\{(x,\zeta(x)+N\psi(x))\}.$
The preimage $\phi|\_{U\times\{(0,0)\}}^{-1}(\{N\})$ consists of isolated points $(x,(0,0))$ such that $\zeta(x)+N\psi(x)=(0,0).$ Projecting from the graph $\{(x,\zeta(x)+N\psi(x))\}\subset U\times \mathbb R^2$ to $U$ is a diffeo, so projecting a set of isolated points gives a set of isolated points. So the points $x\in\mathbb D^2$ with $\zeta(x)+N\psi(x)=(0,0)$ are isolated.


By [pushing these out of the unit ball](https://math.stackexchange.com/questions/3433870/can-we-approximate-a-vector-field-on-the-plane-with-non-vanishing-vector-fields/3439203#3439203) - we can approximate $\zeta$ in $L^2$ by a sequence of smooth $L^2$ vector fields $\gamma\_n$ such that $\gamma\_n\neq 0$ everywhere in the unit ball.


Each $\gamma\_n$ has an orthogonal Hodge decomposition which we can write as $\gamma\_n=\star dg^{(n)}\_1 +dg^{(n)}\_2$ where $g^{(n)}\_1,g^{(n)}\_2$ are determined up to additive constants.
The components $\star dg^{(n)}\_1$ and $dg^{(n)}\_2$ are ["longitudinal and transverse" fields](https://en.m.wikipedia.org/wiki/Helmholtz_decomposition#Longitudinal_and_transverse_fields) defined by pointwise projections in Fourier space, and since $\gamma\_n$ has bounded Sobolev norms $(\int(1+|\xi|^2)^k|\hat\gamma\_n(\xi)|^2d\xi)^{1/2}$ (where $\hat \cdot$ is Fourier transform), so do $\star dg^{(n)}\_1$ and $dg^{(n)}\_2.$ So they're smooth.


The functions $g^{(n)}\_i$ are unique if we add the requirement $\int\_{\mathbb D^2} g^{(n)}\_i=\int\_{\mathbb D^2} f\_i$ for $i=1,2.$ Because Hodge decomposition is an orthogonal decomposition, $\star dg^{(n)}\_1+dg^{(n)}\_2\to \star d\bar f\_1+d\bar f\_2$ in $L^2$ implies $dg^{(n)}\to d\bar f$ in $L^2.$ The Poincaré–Wirtinger inequality then gives $g^{(n)}|\_{\mathbb D^2}\to f$ in $W^{1,2}(\mathbb D^2,\mathbb R^2).$


