Local H\"older continuity for doubly nonlinear parabolic equations

TL;DR

This paper proves that weak solutions to specific degenerate doubly nonlinear parabolic equations are locally H"older continuous in measure spaces with minimal assumptions, using a combination of Harnack inequalities and intrinsic scaling methods.

Contribution

It provides a new proof of H"older continuity for solutions under weak measure and geometric assumptions, extending previous results to more general measure spaces.

Findings

Weak solutions are locally H"older continuous.

The proof applies to measure spaces with doubling measure and Poincaré inequality.

Uses a combination of Harnack inequality and intrinsic scaling methods.

Abstract

We give a proof of the H\"older continuity of weak solutions of certain degenerate doubly nonlinear parabolic equations in measure spaces. We only assume the measure to be a doubling non-trivial Borel measure which supports a Poincar\'e inequality. The proof discriminates between large scales, for which a Harnack inequality is used, and small scales, that require intrinsic scaling methods.