Twisted Harnack inequality and approximation of variational problems with a convexity constraint by singular Abreu equations
Analysis of PDEs
2023-10-03 v2
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.
Keywords
Cite
@article{arxiv.2212.05905,
title = {Twisted Harnack inequality and approximation of variational problems with a convexity constraint by singular Abreu equations},
author = {Nam Q. Le},
journal= {arXiv preprint arXiv:2212.05905},
year = {2023}
}
Comments
v2: final version published in Adv. Math