# H\"older regularity of the 2D dual semigeostrophic equations via analysis of linearized Monge-Amp\`ere equations

## TL;DR

This paper establishes the H"older regularity of solutions to the 2D dual semigeostrophic equations and related maps, using new interior estimates for linearized Monge-Ampère equations, extending previous results.

## Contribution

It provides the first interior H"older estimates for linearized Monge-Ampère equations in 2D with inhomogeneous terms, leading to improved regularity results for dual semigeostrophic equations.

## Key findings

- Proved H"older regularity of the time derivative of solutions.
- Established H"older regularity of polar factorization maps.
- Extended regularity results to densities bounded away from zero and infinity.

## Abstract

We obtain the H\"older regularity of time derivative of solutions to the dual semigeostrophic equations in two dimensions when the initial potential density is bounded away from zero and infinity. Our main tool is an interior H\"older estimate in two dimensions for an inhomogeneous linearized Monge-Amp\`ere equation with right hand side being the divergence of a bounded vector field. As a further application of our H\"older estimate, we prove the H\"older regularity of the polar factorization for time-dependent maps in two dimensions with densities bounded away from zero and infinity. Our applications improve previous work by G. Loeper who considered the cases of densities sufficiently close to a positive constant.