A sharp H\"older estimate for elliptic equations in two variables

TL;DR

This paper establishes a precise H"older continuity estimate for solutions to certain two-dimensional elliptic equations with measurable, symmetric coefficient matrices of unit determinant, extending prior results using a Wirtinger inequality.

Contribution

It provides a sharp H"older estimate for elliptic equations in two variables with measurable coefficients, advancing the understanding of regularity in this context.

Findings

Proves a sharp H"older estimate for solutions.

Extends previous work by Piccinini and Spagnolo.

Uses a Wirtinger type inequality in the proof.

Abstract

We prove a sharp H\"older estimate for solutions of linear two-dimensional, divergence form elliptic equations with measurable coefficients, such that the matrix of the coefficients is symmetric and has {\em unit determinant}. Our result extends some previous work by Piccinini and Spagnolo. The proof relies on a sharp Wirtinger type inequality.