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We revisit the problem of approximating minimizers of certain convexfunctionals subject to a convexity constraint by solutions of fourth orderequations of Abreu type. This approximation problem was studied in previousworks of Carlier-Radice (Approximation of variational problems with a convexityconstraint by PDEs of Abreu type. Calc. Var. Partial Differential Equations. 58(2019), no. 5, Art. 170) and the author (Singular Abreu equations andminimizers of convex functionals with a convexity constraint,arXiv:1811.02355v3, Comm. Pure Appl. Math., to appear), under the uniformconvexity of both the Lagrangian and constraint barrier. By introducing a newapproximating scheme, we completely remove the uniform convexity of both theLagrangian and constraint barrier. Our analysis is applicable to variationalproblems motivated by the original 2D Rochet-Chon\'e model in the monopolist'sproblem in Economics, and variational problems arising in the analysis ofwrinkling patterns in floating elastic shells in Elasticity.

On approximating minimizers of convex functionals with a convexity constraint by singular Abreu equations without uniform convexity