On the equivalence of viscosity solutions and weak solutions or a quasi-linear equation

We discuss and compare various notions of weak solution for the p-Laplace equation -div(\delu\(p-2)delu) = 0 and its parabolic counterpart u(t) - div(\delu\(p-2)delu) = 0. In addition to the usual Sobolev weak solutions based on integration by parts, we consider the p-superharmonic (or p-superparabolic) functions from nonlinear potential theory and the viscosity solutions based on generalized pointwise derivatives ( jets). Our main result states that in both the elliptic and the parabolic case, the viscosity supersolutions coincide with the potential-theoretic supersolutions.