Interior Harnack inequality and H\"older estimates for linearized Monge-Amp\`ere equations in divergence form with drift

TL;DR

This paper establishes interior Harnack inequalities and Hölder estimates for solutions to linearized Monge-Ampère equations with drift, extending previous results to include divergence form equations with additional drift terms.

Contribution

It extends existing interior regularity results for Monge-Ampère equations to a broader class with divergence form and drift, applicable in meteorology and calculus of variations.

Findings

Proves interior Harnack inequality in 2D and higher dimensions under certain conditions.

Establishes Hölder continuity estimates for solutions.

Extends prior results to equations with drift terms.

Abstract

In this paper, we study interior estimates for solutions to linearized Monge-Amp\`ere equations in divergence form with drift terms and the right-hand side containing the divergence of a bounded vector field. Equations of this type appear in the study of semigeostrophic equations in meteorology and the solvability of singular Abreu equations in the calculus of variations with a convexity constraint. We prove an interior Harnack inequality and H\"older estimates for solutions to equations of this type in two dimensions, and under an integrability assumption on the Hessian matrix of the Monge-Amp\`ere potential in higher dimensions. Our results extend those of Le (Analysis of Monge-Amp\`ere equations, Graduate Studies in Mathematics, vol.240, American Mathematical Society, 2024) to equations with drift terms.