Lipschitz regularity for a homogeneous doubly nonlinear PDE

Abstract

We study the doubly nonlinear PDE $$|{\partial }_{t}u{|}^{p-2}{\partial }_{t}u-\text{div}(|\nabla u{|}^{p-2}\nabla u)=0.$$ This equation arises in the study of extremals of Poincaré inequalities in Sobolev spaces. We prove spatial Lipschitz continuity and Hölder 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.