Singular Abreu equations and minimizers of convex functionals with a convexity constraint

Abstract: We study the solvability of second boundary value problems of fourth order equations of Abreu type arising from approximation of convex functionals whose Lagrangians depend on the gradient variable, subject to a convexity constraint. These functionals arise in different scientific disciplines such as Newton's problem of minimal resistance in physics and monopolist's problem in economics. The right hand sides of our Abreu type equations are quasilinear expressions of second order; they are highly singular and a priori just measures. However, our analysis in particular shows that minimizers of the 2D Rochet-Chon\'e model perturbed by a strictly convex lower order term, under a convexity constraint, can be approximated in the uniform norm by solutions of the second boundary value problems of singular Abreu equations.