H\"older regularity for a class of doubly non linear PDEs

TL;DR

This paper establishes local H"older continuity for solutions to a class of doubly nonlinear parabolic PDEs, expanding understanding of their regularity properties in mathematical analysis.

Contribution

It introduces a novel approach combining positivity expansion and DeGiorgi-type lemmas to prove regularity for a specific class of doubly nonlinear PDEs.

Findings

Proved local H"older continuity for solutions.

Developed new techniques involving positivity expansion and exponential shifts.

Enhanced understanding of the intrinsic geometry of the PDEs.

Abstract

We prove local H\"older continuity for non negative, locally bounded, local weak solutions to the class of doubly nonlinear parabolic equations $\partial_t (u_q) - \text{div} (|Du|^{p-2} Du) = 0$ for $p > 2$, $ 0 < q < p-1$. The proof relies on expansion of positivity results combined with the study of an alternative (related to DeGiorgi-type lemmas) and an exponential shift which allows us to deal with the intrinsic geometry associated to the problem.