We prove a comparison principle for the porous medium equation in more general open sets in ℝn+1 than space-time cylinders. We apply this result in two related contexts: we establish a connection between a potential theoretic notion of the obstacle problem and a notion based on a variational inequality. We also prove the basic properties of the PME capacity, in particular that there exists a capacitary extremal which gives the capacity for compact sets.
- Elliptic comparison principle
- Obstacle problem
- Parabolic capacity
- Porous medium equation