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

TL;DR

This paper introduces a new approximation scheme for convex functionals with convexity constraints, removing the need for uniform convexity assumptions, and applies it to economic and elasticity models.

Contribution

The paper develops a novel approximation method for convex minimizers that does not require uniform convexity, extending previous PDE-based approaches.

Findings

Successfully removes uniform convexity requirement.

Applicable to economic models like Rochet-Choné.

Relevant for elastic shell wrinkling analysis.

Abstract

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