See my other answer for the caveat that I am writing about things I have learned from student presentations, so they may be flawed.

Suppose we have a block of raw material which has some shape $X \subset \mathbb{R}^3$. We want to mold it into a new shape $Y \subset \mathbb{R}^3$, but, if the material is stretched too much, it will break. How can we model this?

A possible way of molding $X$ into $Y$ is a smooth bijection $f : X \to Y$. At any point, this map has a Jacobian matrix $Df$. If $Df$ is orthogonal, then we are just rotating the material, so there is no problem at all. How can we numerically impose that $Df$ is nearly orthogonal?

We choose some parameter $\epsilon$, and insist that all singular values of $Df$ are in the range $(1-\epsilon, 1+\epsilon)$. Then we have a problem in PDE's: To map $f : X \to Y$ while imposing that the singular values of $Df$ stay in a bounded range.