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

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 $\mathbb{R}^n$.