In this paper, we study minimizers of the Choné--Rochet variational problem in dimension two. We first establish global C^1 regularity on arbitrary bounded convex domains, and then prove global C^{1,1} regularity on bounded strictly convex domains or, more generally, whenever the zero set of has positive measure. Next, we construct smooth bounded convex domains with a flat boundary segment for which no prescribed modulus of continuity controls the gradient; this shows that, without additional geometric assumptions, global C^1 regularity is optimal. Finally, we prove that the tamed free boundary (that is, the interface between the strictly convex and non-strictly convex regions of the solution) is locally a C^1 embedded curve, significantly strengthening previously known regularity results.
