A Green's function approach to linearized Monge-Amp\`ere equations in divergence form and application to singular Abreu type equations

TL;DR

This paper develops Green's function-based regularity estimates for linearized Monge-Ampère equations in divergence form and applies these results to solve singular Abreu type equations with convexity constraints.

Contribution

It introduces a Green's function approach to establish regularity estimates and proves solvability of singular Abreu equations in all dimensions.

Findings

Established local and global regularity estimates for the equations.

Proved solvability of second boundary value problem for singular Abreu equations.

Extended analysis to all dimensions under certain convexity conditions.

Abstract

In this paper, we establish local and global regularity estimates for linearized Monge-Amp\`ere equations in divergence form via critical Lorentz space estimates for the Green's function of the linearized Monge-Amp\`ere operator and its gradient. These estimates hold under suitable conditions on the data and the convex Monge-Amp\`ere potential is assumed to have Hessian determinant bounded between two positive constants. As an application, we obtain the solvability in all dimensions of the second boundary value problem for a class of singular fourth-order Abreu type equations that arise from the approximation analysis of variational problems subject to convexity constraints.