# Global H\"older estimates for 2D linearized Monge-Amp\`ere equations with right-hand side in divergence form

## TL;DR

This paper proves global H"older continuity estimates for solutions to 2D linearized Monge-Amp ext`ere equations with divergence-form right-hand side, relevant in meteorology and convex functional approximation.

## Contribution

It introduces affine invariant, degenerate H"older estimates for these equations under natural boundedness assumptions, extending classical second order elliptic estimates.

## Key findings

- Estimates hold under natural domain and boundary conditions.
- Results are affine invariant and applicable to degenerate cases.
- Applicable in meteorology and convex functional approximation.

## Abstract

We establish global H\"older estimates for solutions to inhomogeneous linearized Monge-Amp\`ere equations in two dimensions with the right hand side being the divergence of a bounded vector field. These equations arise in the semi-geostrophic equations in meteorology and in the approximation of convex functionals subject to a convexity constraint using fourth order Abreu type equations. Our estimates hold under natural assumptions on the domain, boundary data and Monge-Amp\`ere measure being bounded away from zero and infinity. They are an affine invariant and degenerate version of global H\"older estimates by Murthy-Stampacchia and Trudinger for second order elliptic equations in divergence form.