A New Proof for the Equivalence of Weak and Viscosity Solutions for the<i>p</i>-Laplace Equation

Abstract

In this paper, we give a new proof for the fact that the distributional weak solutions and the viscosity solutions of the p-Laplace equation −div(|Du| p−2 Du) = 0 coincide. Our proof is more direct and transparent than the original proof of Juutinen et al. [8 Juutinen , P. , Lindqvist , P. , Manfredi , J.J. ( 2001 ). On the equivalence of viscosity solutions and weak solutions for a quasi-linear equation . SIAM J. Math. Anal. 33 : 699 – 717 .[Crossref], [Web of Science ®] , [Google Scholar]], which relied on the full uniqueness machinery of the theory of viscosity solutions. We establish a similar result also for the solutions of the non-homogeneous version of the p-Laplace equation.