Advances in Differential Equations

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

Juha Kinnunen, Niko Marola, Michele Miranda Jr., and Fabio Paronetto

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In this paper we study problems related to parabolic partial differential equations in metric measure spaces equipped with a doubling measure and supporting a Poincaré 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.

Article information

Adv. Differential Equations Volume 17, Number 9/10 (2012), 801-832.

First available in Project Euclid: 17 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 30L99: None of the above, but in this section 31E05: Potential theory on metric spaces 35K05: Heat equation 5K99 49N60: Regularity of solutions


Kinnunen, Juha; Marola, Niko; Miranda Jr., Michele; Paronetto, Fabio. Harnack's inequality for parabolic De Giorgi classes in metric spaces. Adv. Differential Equations 17 (2012), no. 9/10, 801--832.

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