Regularity of weak solutions and supersolutions to the porous medium equation

Abstract: We study the relations between different regularity assumptions in the definition of weak solutions and supersolutions to the porous medium equation. In particular, we establish the equivalence of the conditions $u^m \in L^2_{\rm loc}(0,T;H^1_{\rm loc}(\Omega))$ and $u^\frac{m+1}2 \in L^2_{\rm loc}(0,T;H^1_{\rm loc}(\Omega))$ in the definition of weak solutions. Our proof is based on approximation by solutions to obstacle problems.