# Does this geometric PDE has a solution?

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.