A proof of the parabolic Schauder estimates using Trudinger's method and the mean value property of the heat equation

TL;DR

This paper demonstrates a simplified proof of the parabolic Schauder estimates by adapting Trudinger's mollification method and utilizing the mean value property of the heat equation, paralleling the elliptic case.

Contribution

It introduces a novel approach that employs the heat equation's mean value property to derive Schauder estimates for parabolic equations, simplifying the existing proof techniques.

Findings

Provides a straightforward derivation of interior Schauder estimates for parabolic equations.

Adapts Trudinger's mollification method to the parabolic setting.

Highlights the use of the heat equation's mean value property in regularity theory.

Abstract

One method available to prove the Schauder estimates is Neil Trudinger's method of mollification. In the case of second order elliptic equations, the method requires little more than mollification and the solid mean value inequality for subharmonic functions. Our goal in this article is show how the mean value property of subsolutions of the heat equation can be used in a similar fashion as the solid mean value inequality for subharmonic functions in Trudinger's original elliptic treatment, providing a relatively simple derivation of the interior Schauder estimate for second order parabolic equations.