On the Equivalence of Viscosity Solutions and Weak Solutions for a Quasi-Linear Equation

We discuss and compare various notions of weak solution for the p-Laplace equation -\text{div}(|\nabla u|^{p-2}\nabla u)=0 and its parabolic counterpart u_t-\text{div}(|\nabla u|^{p-2}\nabla u)=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.

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