  Singular fourth-order Abreu equations have been used to approximate
minimizers of convex functionals subject to a convexity constraint in
dimensions higher than or equal to two. For Abreu type equations, they often
exhibit different solvability phenomena in dimension one and dimensions at
least two. We prove the analogues of these results for the variational problem
and singular Abreu equations in dimension one, and use the approximation scheme
to obtain a characterization of limiting minimizers to the one-dimensional
variational problem.
