On the H\"older regularity of signed solutions to a doubly nonlinear equation. Part II

TL;DR

This paper proves local H"older continuity for solutions to a class of doubly nonlinear parabolic equations, including boundary regularity, using two different proof methods that simplify and extend previous results.

Contribution

It provides two novel proofs for the H"older regularity of solutions to doubly nonlinear equations, improving understanding of their boundary behavior.

Findings

Established local H"older continuity for sign-changing solutions.

Provided boundary regularity results for Dirichlet and Neumann problems.

Simplified proof approach using positivity expansion.

Abstract

We demonstrate two proofs for the local H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-\Delta_p u=0,\quad p>2,\quad 0<q<p-1. \] The first proof takes advantage of the expansion of positivity for the degenerate, parabolic $p$-Laplacian, thus simplifying the argument; whereas the other proof relies solely on the energy estimates for the doubly nonlinear parabolic equations. After proper adaptions of the interior arguments, we also obtain the boundary regularity for initial-boundary value problems of Dirichlet type and Neumann type.