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

Does 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 $$ h_{\theta}^2+ 1/h_{\theta}^2+\big(rh_r+b \circ h-b\cdot h_{\theta}\big)^2/s^2 \tag{2} $$ 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 $$ \Big(\frac{s}{s \circ h}\Big)^2+ \Big(\frac{s \circ h}{s}\Big)^2+ \frac{\Big(rh_r+b\circ h-b \cdot \dfrac{s}{s \circ h}\Big)^2}{s^2}. \tag{3} $$


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

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