Lipschitz continuity and equivalence of positive viscosity and weak solutions to Trudinger's equation

TL;DR

This paper proves that positive viscosity supersolutions to Trudinger's equation are also weak solutions and establishes their Lipschitz continuity in space, filling a gap in the existing literature.

Contribution

It demonstrates the equivalence of viscosity and weak solutions for Trudinger's equation and provides the first valid proof of their Lipschitz continuity.

Findings

Viscosity supersolutions are also weak supersolutions.

Both viscosity and weak solutions are Lipschitz continuous in space.

First valid proof of Lipschitz regularity for solutions to Trudinger's equation.

Abstract

We show that stricty positive viscosity supersolutions to Trudinger's equation are local weak supersolutions when $1<p<\infty$. As an application, we show using the Ishii-Lions method that not only viscosity but also weak solutions are Lipschitz continuous in the space variable. To the best of our knowledge, there is no preceding valid Lipschitz proofs in the literature.