Calculus of Variations and Geometric Measure Theory

J. Kinnunen - N. Marola - M. Miranda Jr - F. Paronetto

Harnack's inequality for parabolic De Giorgi classes in metric spaces

created by miranda on 28 Jun 2011
modified by marola on 31 Jan 2013


Published Paper

Inserted: 28 jun 2011
Last Updated: 31 jan 2013

Journal: Adv. Differential Equations
Volume: 17
Number: 9-10
Pages: 801-832
Year: 2012


In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincar\'e inequality. We give a definition of parabolic De Giorgi classes and compare this notion with that of parabolic quasiminimizers. The main result, after proving the local boundedness, is a scale and location invariant Harnack inequality for functions belonging to parabolic De Giorgi classes. In particular, the results hold true for parabolic quasiminimizers.