On singular strictly convex solutions to the Monge-Amp\`ere equation

TL;DR

This paper proves the existence of strictly convex solutions to the Monge-Ampère equation with singular Hessian parts, extending previous work and resolving an open question in the field.

Contribution

It demonstrates the existence of convex solutions with singular Hessian measures for the Monge-Ampère equation, advancing understanding of solution regularity.

Findings

Existence of convex solutions with singular Hessian parts.

Construction of solutions with prescribed Monge-Ampère measure.

Extends previous theoretical results and answers open questions.

Abstract

We show the existence of a strictly convex function $u: B_1 \to \mathbb{R}$ with associated Monge-Amp\`ere measure represented by a function $f$ with $0 < f < 1$ a.e. whose Hessian has a singular part. This extends the work [13] and answers an open question of [14,Sec. 6.2(1)].