On the H\"older regularity of signed solutions to a doubly nonlinear equation

We establish the interior and boundary H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is $$\partial_t\big(|u|^{p-2}u\big)-\Delta_p u=0,\quad p>1.$$ The proof relies on the property of expansion of positivity and the method of intrinsic scaling, all of which are realized by De Giorgi's iteration. Our approach, while emphasizing the distinct roles of sub(super)-solutions, is flexible enough to obtain the H\"older regularity of solutions to initial-boundary value problems of Dirichlet type or Neumann type in a cylindrical domain, up to the parabolic boundary. In addition, based on the expansion of positivity, we are able to give an alternative proof of Harnack's inequality for non-negative solutions. Moreover, as a consequence of the interior estimates, we also obtain a Liouville-type result.