General double-control problem for orthogonal curve foliations

Determine whether, for a rectangle R = I × J and sufficiently small perturbations f : I → I and g : J → J of the identity, there exist two mutually orthogonal foliations of R by green and red curves such that each green curve starting at (a,t) ends at (b,g(t)) and each red curve starting at (x,c) ends at (f(x),d), together with the boundary and regularity properties required in Theorem 1.1.

Background

The main theorem constructs orthogonal green and red foliations in a rectangle when the green curves preserve their starting height from the left edge to the right edge, while the red curves realize a prescribed small perturbation f of the identity between the bottom and top edges. Remark 1.4 asks whether the horizontal endpoint condition can be generalized so that the green curves realize a second prescribed increasing diffeomorphism g of the vertical interval.

The paper does not establish the general result. It states that the proof of the main theorem does not extend directly, although a special case is subsequently solved in Section 11 when f − Id and g − Id each have an interior zero of order k, including the case of perturbations supported in strict subintervals. Thus the unrestricted double-control problem remains unresolved.

References

We do not know whether the analogue of Theorem \ref{t1} is still true in this context. Our proof does not extend, but if we give ourselves some room around $R$ for the curves to organize themselves (instead of forcing them to be the straight lines), we can use Theorem~\ref{t1} to get the result.

— On a control problem for two orthogonal nets of curves in the plane  (2610.11569 - David et al., 8 Oct 2026) in Remark 1.4 ("A question about double control"), Section 1.2; revisited in Section 11