Equivalence of viscosity and weak solutions for a p-parabolic equation

Abstract

We study the relationship of viscosity and weak solutions to the equation∂tu-Δpu=f(Du),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\smash {\\partial _{t}u-\\varDelta _{p}u=f(Du)}, \\end{aligned}$$\\end{document}where p>1 and fin C({mathbb {R}}^{N}) satisfies suitable assumptions. Our main result is that bounded viscosity supersolutions coincide with bounded lower semicontinuous weak supersolutions. Moreover, we prove the lower semicontinuity of weak supersolutions when pge 2.