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

We show that stricty positive viscosity supersolutions to Trudinger’s equation are local weak supersolutions when $1<p<\mathrm{\infty }$. 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.