# Approximation of a Sobolev map with fixed singular values by smooth maps with the same singular values

Let $$0<\sigma_1<\sigma_2$$, and let $$D \subseteq \mathbb{R}^2$$ be the closed unit disk.

Let $$f \in W^{1,\infty}(D,\mathbb{R}^2)$$, and suppose that the singular values of $$df$$ are a.e. equal to $$\sigma_1,\sigma_2$$.

Do there exist $$f_n \in C^{\infty}(D^o,\mathbb{R}^2)$$ such that $$f_n \to f$$ in $$W^{1,2}$$ and the singular values of $$df_n$$ are everywhere equal to the $$\sigma_i$$?

Can we also get the $$f_n$$ to converge uniformly to $$f$$? ($$f$$ is continuous, since it is in $$W^{1,\infty}$$).

The standard mollification process won't work here-it is an averaging process, and averaging matrices with given singular values reduces the norm. (all these matrices lie on Euclidean sphere of radius $$\sqrt{\sigma_1^2+\sigma_2^2}$$, which is strictly convex).

Here is a concrete example:

Suppose that $$\frac{\sigma_2}{\sigma_1}=n$$ is a natural number.

The map $$f:re^{i \theta} \to \sigma_1re^{i(\sigma_2/\sigma_1) \theta}$$ (which in general is smooth on the disk only after removing a ray) has constant singular values $$\sigma_i$$.

Since we assumed $$\frac{\sigma_2}{\sigma_1}=n$$, $$f(z)=f(re^{i\theta})= re^{in\theta}=\frac{z^n}{|z|^{n-1}},$$ is smooth on the entire disk without the origin, and is in $$W^{1,\infty}(D,\mathbb{R}^2)$$.

Can we approximate $$f$$ in $$W^{1,2}$$ with smooth maps having the fixed singular values $$\sigma_i$$?