Lipschitz Regularity for a Homogeneous Doubly Nonlinear PDE
Lipschitz Regularity for a Homogeneous Doubly Nonlinear PDE
We study the doubly nonlinear PDE $ |\partial_t u|^{p-2}\,\partial_t u-\operatorname{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 …