Local H\"older continuity for some doubly nonlinear parabolic equations in measure spaces

TL;DR

This paper proves local H"older continuity for nonnegative weak solutions of specific doubly nonlinear parabolic equations in measure spaces, addressing singularities and degeneracies using intrinsic scaling methods.

Contribution

It introduces a novel approach to establish regularity results for doubly nonlinear equations in measure spaces with singular and degenerate features.

Findings

Proved local H"older continuity of solutions

Applicable to equations with singular time derivatives and degenerate principal parts

Utilized intrinsic scaling in measure spaces with Poincaré inequality

Abstract

We establish the local H\"older continuity for the nonnegative weak solutions of certain doubly nonlinear parabolic equations possessing a singularity in the time derivative part and a degeneracy in the principal part. The proof involves the method of intrinsic scaling and the setting is a measure space equipped with a doubling non-trivial Borel measure supporting a Poincar\'e inequality.