A New Proof of the Equivalence of Weak and Viscosity Solutions to the Homogeneous $\texttt{p}(\cdot)$-Laplacian in $\mathbb{R}^n$

TL;DR

This paper introduces a new, direct proof demonstrating the equivalence between weak and viscosity solutions for the variable exponent p-Laplace equation in Euclidean space, extending previous fixed exponent results.

Contribution

It provides a novel proof using infimal convolutions to establish the equivalence in the variable exponent case, building on and extending prior methods.

Findings

Established equivalence between weak and viscosity solutions for p(·)-Laplace in R^n

Extended fixed exponent proof techniques to variable exponents

Utilized infimal convolutions for a direct proof approach

Abstract

We present a new proof for the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}(\cdot)$-Laplace equation in $\mathbb{R}^n$. The proof of the equivalence in the variable exponent case in Euclidean space was first given by Juutinen, Lukkari, and Parviainen (2010) and extended the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}$-Laplace equation in $\mathbb{R}^n$, given by Juutinen, Lindqvist, and Manfredi (2001). In both the fixed exponent case and the variable exponent case, the main argument is based on the maximum principle for semicontinuous functions, several approximations, and also applied the full uniqueness machinery of the theory of viscosity solutions. This paper extends the approach of Julin and Juutinen (2012) for the fixed exponent case, and so we employ infimal convolutions to present a direct, new proof for the equivalence of potential theoretic weak solutions and viscosity solutions to the $\texttt{p}(\cdot)$-Laplace equation in $\mathbb{R}^n$.