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On the Harnack inequality for parabolic minimizers in metric measure spaces. (English) Zbl 1293.30096

The main result of the paper under review is the verification of a Harnack inequality in the parabolic setting.
Let me first describe the situation in the Euclidean setting, where the result can be stated in the realm of partial differential equations. The authors show that (for \(1<p<\infty\) and \(0<\delta<1\)) a (suitably integrable) weak solution \(u\) of \[ \frac{\partial(|u|^{p-2}u)}{\partial t}-\nabla \cdot (|\nabla u|^{p-2}\nabla u)=0, \] satisfying \(u>0\), being locally bounded and locally bounded away from zero satisfies \[ \mathop{\roman{ess\;sup}}_{\delta Q^-} u\leq C\mathop{\roman{ess\;inf}}_{\delta Q^+} u, \] where \(C\) is independent of \(u\) (the authors give a list of quantities on which \(C\) depends). The sets \(\delta Q^-\) and \(\delta Q^+\) are certain positive and negative space-time cylinders, respectively.
However, let me underline that the focus of the authors lies on the setting of quite general metric measure spaces and that the above stated differential equation does not enter the proof of the Harnack inequality. Solving the partial differential equation is translated into solving a minimization problem, which has an interpretation in the framework of metric measure spaces as well. The Harnack inequality holds then for the minimizer of the minimization problem.

MSC:

30L99 Analysis on metric spaces
35K55 Nonlinear parabolic equations
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References:

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