On approximating minimizers of convex functionals with a convexity constraint by singular Abreu equations without uniform convexity
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.
Keywords
Cite
@article{arxiv.1910.01486,
title = {On approximating minimizers of convex functionals with a convexity constraint by singular Abreu equations without uniform convexity},
author = {Nam Q. Le},
journal= {arXiv preprint arXiv:1910.01486},
year = {2020}
}
Comments
v2: final version incorporating suggestions from the referee report; the derivations of (3.3) and (3.5) are included; to be published in Proc. Roy. Soc. Edinburgh Sect. A