On viscosity and weak solutions for non-homogeneous p-Laplace equations

TL;DR

This paper explores the connection between viscosity and weak solutions for non-homogeneous p-Laplace equations, extending previous results and establishing a two-way correspondence under certain conditions.

Contribution

It generalizes the relationship between viscosity and weak solutions to non-homogeneous p-Laplace equations with lower order terms, including a converse statement under additional regularity assumptions.

Findings

Locally bounded viscosity solutions are weak solutions.

Extension of previous homogeneous case results to non-homogeneous equations.

A converse relationship is established under extra regularity assumptions.

Abstract

In this manuscript we study the relation between viscosity and weak solutions for non-homogeneous p-Laplace equations with lower order term depending on $x$, $u$ and $\nabla u$. More precisely, we prove that any locally bounded viscosity solution constitutes a weak solution, extending previous results by Juutinen, Lindqvist and Manfredi on the homogeneous case, and Julin and Juutinen for a linear right hand side. Moreover, we provide a converse statement in the full case under extra assumptions on the regularity of the solutions.