# Lipschitz regularity for a homogeneous doubly nonlinear PDE

## TL;DR

This paper establishes Lipschitz and Hölder continuity properties for viscosity solutions of a specific doubly nonlinear PDE, providing insights into their regularity and long-term behavior.

## Contribution

It proves spatial Lipschitz and temporal Hölder regularity for solutions of a homogeneous doubly nonlinear PDE, advancing understanding of their regularity properties.

## Key findings

- Proved spatial Lipschitz continuity of solutions.
- Established Hölder continuity in time of order (p-1)/p.
- Provided pointwise control of large time behavior.

## Abstract

We study the doubly nonlinear PDE $$ |\partial_t u|^{p-2}\,\partial_t u-\textrm{div}(|\nabla u|^{p-2}\nabla u)=0. $$ This equation arises in the study of extremals of Poincar\'e inequalities in Sobolev spaces. We prove spatial Lipschitz continuity and H\"older continuity in time of order $(p-1)/p$ for viscosity solutions. As an application of our estimates, we obtain pointwise control of the large time behavior of solutions.