The proof of Harnack inequality using DeGiorgi method has a great flexibility and it can be exteneded even to doubling metric measure spaces that support Poincare inequalities. 

The elliptic case has been treated in:

J. Kinnunen, N. Shanmugalingam, <A HREF="https://link.springer.com/article/10.1007%2Fs002290100193"><FONT FACE="Arial">Regularity of quasi-minimizers on metric spaces</FONT></A><FONT FACE="Arial">. *Manuscripta Math.* 105 (2001), 401–423.
(<A HREF="https://mathscinet.ams.org/mathscinet-getitem?mr=1856619"><FONT FACE="Arial">MathScNet review</FONT></A><FONT FACE="Arial">.)



The parabolic case has been treated in:

J. Kinnunen, N. Marola, M. Miranda, Jr., F. Paronetto,
<A HREF="https://arxiv.org/abs/1106.5356"><FONT FACE="Arial">Harnack's inequality for parabolic De Giorgi classes in metric spaces</FONT></A><FONT FACE="Arial">.
*Adv. Differential Equations* 17 (2012), 801–832. 
(<A HREF="https://mathscinet.ams.org/mathscinet/search/publdoc.html?pg1=INDI&s1=349676&sort=Newest&vfpref=html&r=25&mx-pid=2985675"><FONT FACE="Arial">MathScNet review</FONT></A><FONT FACE="Arial">.)







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