  We prove that various notions of supersolutions to the porous medium equation
are equivalent under suitable conditions. More spesifically, we consider weak
supersolutions, very weak supersolutions, and $m$-superporous functions defined
via a comparison principle. The proofs are based on comparison principles and a
Schwarz type alternating method, which are also interesting in their own right.
Along the way, we show that Perron solutions with merely continuous boundary
values are continuous up to the parabolic boundary of a sufficiently smooth
space-time cylinder.
