Let $\Omega_1$ and $\Omega_2$ be (simply connected) domains on $\mathbb{R}^2$, with coordinates $(x,y)$ and $(X,Y)$ respectively. Given a (smooth) function $Z(X,Y)$ such that $Z\left(\partial \Omega_2 \right)=0$, consider the system of PDEs

\begin{align} \left(\frac{\partial X}{\partial x} \right)^2+\left(\frac{\partial Y}{\partial x} \right)^2+\left(\frac{\partial Z\left(X,Y \right)}{\partial x} \right)^2=1 \end{align} \begin{align} \left(\frac{\partial X}{\partial y} \right)^2+\left(\frac{\partial Y}{\partial y} \right)^2+\left(\frac{\partial Z\left(X,Y \right)}{\partial y} \right)^2=&1, \end{align}

subjected to

\begin{align} \left(X,Y\right)\left(\partial \Omega_1 \right)=\partial \Omega_2. \end{align}

Does a solution exist ?

This problem comes as a particular case of Chebyshev mappings, where a surface is covered by a net of lines whose tangents are unit vectors at all its points. In this case the boundary of the surface is constrained to lie on plane. Generically, the (local) existence of solutions is shown by obtaining an equivalent system that is explicitly hyperbolic (by cross-differentiation), so the problem is hyperbolic in nature. In this case the condition is not hyperbolic but elliptic, so the existence of solutions is not secured