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

Jarkko Siltakoski

doi:10.1007/s00028-020-00666-y

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.

Highlights

A classical solution to a partial differential equation is a smooth function that satisfies the equation pointwise
We show that bounded viscosity supersolutions to (1.1) coincide with bounded lower semicontinuous weak supersolutions
Equations with gradient terms are studied for example in [2], where comparison principle is shown for the equation ∂t u − Δpu − |Du|β = 0 when p > 2 and β > p − 1

Summary

Introduction

A classical solution to a partial differential equation is a smooth function that satisfies the equation pointwise. One such class is the extensively studied distributional weak solutions defined by integration by parts Another is the celebrated viscosity solutions based on generalized pointwise derivatives. We prove estimates for the essential supremum of a subsolution using Moser’s iteration technique We use those estimates to deduce that a supersolution is lower semicontinuous at its Lebesgue points. The equivalence of viscosity and weak solutions for the p-Laplace equation and its parabolic version was first proven in [12]. Equations with gradient terms are studied for example in [2], where comparison principle is shown for the equation ∂t u − Δpu − |Du|β = 0 when p > 2 and β > p − 1.

Preliminaries

Weak solutions are viscosity solutions

Viscosity solutions are weak solutions

Lower semicontinuity of supersolutions