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We prove a comparison principle for the porous medium equation in moregeneral open sets in $\mathbb{R}^{n+1}$ than space-time cylinders. We applythis result in two related contexts: we establish a connection between apotential theoretic notion of the obstacle problem and a notion based on avariational inequality. We also prove the basic properties of the PME capacity,in particular that there exists a capacitary extremal which gives the capacityfor compact sets.

A comparison principle for the porous medium equation and its consequences