Singular Abreu equations and linearized Monge-Amp\`ere equations with drifts

TL;DR

This paper proves the solvability and regularity of higher-dimensional singular Abreu equations by transforming them into linearized Monge-Ampère equations with drifts, extending previous results beyond two dimensions.

Contribution

It introduces a method to solve higher-dimensional singular Abreu equations via transformation into linearized Monge-Ampère equations with drifts, establishing global Hölder estimates.

Findings

Established global Hölder estimates for linearized Monge-Ampère equations with drifts.

Proved solvability of second boundary value problems for singular Abreu equations in higher dimensions.

Extended solvability results to cases with general right-hand sides.

Abstract

We study the solvability of singular Abreu equations which arise in the approximation of convex functionals subject to a convexity constraint. Previous works established the solvability of their second boundary value problems either in two dimensions, or in higher dimensions under either a smallness condition or a radial symmetry condition. Here, we solve the higher dimensional case by transforming singular Abreu equations into linearized Monge-Amp\`ere equations with drifts. We establish global H\"older estimates for the linearized Monge-Amp\`ere equation with drifts under suitable hypotheses, and then use them to the regularity and solvability of the second boundary value problem for singular Abreu equations in higher dimensions. Many cases with general right-hand side will also be discussed.