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Harnack estimates for weak supersolutions to nonlinear degenerate parabolic equations

Tuomo Kuusi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this work we prove both local and global Harnack estimates for weak supersolutions to second order nonlinear degenerate parabolic partial differential equations in divergence form. We reduce the proof to an analysis of so-called hot and cold alternatives, and use the expansion of positivity together with a parabolic type of covering argument. Our proof uses only the properties of weak supersolutions. In particular, no comparison to weak solutions is needed.

Heat kernel on manifolds with ends

Alexander Grigor’yan, Laurent Saloff-Coste (2009)

Annales de l’institut Fourier

We prove two-sided estimates of heat kernels on non-parabolic Riemannian manifolds with ends, assuming that the heat kernel on each end separately satisfies the Li-Yau estimate.

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