On the Harnack inequality for parabolic minimizers in metric measure spaces

Abstract. In this note we consider problems related to parabolic partial dif-ferential equations in geodesic metric measure spaces, that are equipped with a doubling measure and a Poincare ́ inequality. We prove a location and scale invariant Harnack inequality for a minimizer of a variational problem related to a doubly non-linear parabolic equation involving the p-Laplacian. Moreover, we prove the sufficiency of the Grigor’yan–Saloff-Coste theorem for general p> 1 in geodesic metric spaces. The approach used is strictly variational, and hence we are able to carry out the argument in the metric setting. 1