Twisted Harnack inequality and approximation of variational problems with a convexity constraint by singular Abreu equations

TL;DR

This paper introduces a new approach to approximate solutions of variational problems with convexity constraints using singular Abreu equations, employing a novel Harnack inequality for singular Monge-Ampère type equations.

Contribution

It develops a new Harnack inequality for singular linearized Monge-Ampère equations and applies it to approximate convexity-constrained variational problems with singular Abreu equations.

Findings

Established a Harnack inequality for singular Monge-Ampère equations.

Proved uniform approximation of convexity-constrained minimizers by singular Abreu solutions.

Addressed singularities that cannot be removed by transformations.

Abstract

We show in all dimensions that minimizers of variational problems with a convexity constraint, which arise from the Rochet-Chon\'e model with a quadratic cost in the monopolist's problem in economics, can be approximated in the uniform norm by solutions of singular Abreu equations. The difficulty of our Abreu equations consists of having singularities that occur only in a proper subdomain and they cannot be completely removed by any transformations. To solve them, we rely on a new tool which we establish here: a Harnack inequality for singular linearized Monge-Amp\`ere type equations that satisfy certain twisted conditions.