Let $s(theta), b(theta)$ be two smooth **non-constant** real-valued functions on $mathbb{S}^1$, and assume that $s$ never vanishes.

Do there exist a map $h:(0,1) times mathbb{S}^1 to mathbb{S}^1$, such that for every $r in (0,1)$, $h(r,cdot)$ is a diffeomorphism of $mathbb{S}^1$,

$$

h_{theta}=s(theta)/s(h(r,theta)), tag{1}

$$

and

$$

h_{theta}^2+ 1/h_{theta}^2+big(rh_r+b circ h-bcdot h_{theta}big)^2/s^2 tag{2}

$$

is independent of $r,theta$?

Are there some conditions on $s,b$ that imply such a solution $h$ exists?

Using equation $(1)$, the expression in $(2)$ can be written as

$$

big(frac{s}{s circ h}big)^2+ big(frac{s circ h}{s}big)^2+ big(rh_r+bcirc h-b cdot frac{s}{s circ h}big)^2/s^2. tag{3}

$$

This PDE arises when trying to build ‘concentric’ area-preserving diffeomorphisms of a given $2D$ shape, having constant singular values.