# Improved H\"older regularity for strongly elliptic PDEs

## TL;DR

This paper demonstrates enhanced H"older regularity for solutions to certain elliptic PDEs in the plane by linking them to nonlinear Beltrami equations and establishing new relations that improve classical regularity bounds.

## Contribution

It introduces novel relations between elliptic PDEs and Beltrami equations, leading to improved regularity results beyond classical estimates.

## Key findings

- Solutions to autonomous Beltrami equations have higher H"older regularity than classical bounds.
- New relations between nonlinear Beltrami and elliptic PDEs enable regularity improvements.
- Results are specific to planar strongly elliptic PDEs and their associated Beltrami equations.

## Abstract

We establish surprising improved Schauder regularity properties for solutions to the Leray-Lions divergence type equation in the plane. The results are achieved by studying the nonlinear Beltrami equation and making use of special new relations between these two equations. In particular, we show that solutions to an autonomous Beltrami equation enjoy a quantitative improved degree of H\"older regularity, higher than what is given by the classical exponent $1/K$.